The Structure and Stability of
Vortices in Astrophysical Discs
A.D.Railton
Department of Applied Maths and Theoretical Physics
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
Pembroke College April 2015

To Mum, Dad, Ella, Jess and Toby–
I couldn’t have done it without you.

Declaration
I hereby declare that, except where specific reference is made to the work of others, the
contents of this dissertation are original and have not been submitted in whole or in part
for consideration for any other degree or qualification in this, or any other University. This
dissertation is the result of my own work and includes nothing which is the outcome of work
done in collaboration, with the exception of the algebra in Sections 7.5.1 and 7.5.2 and the
order of magnitude calculations in Section 8.6.1, which were done by JCBP.
Parts of Chapters 5 to 8 make up the paper Railton and Papaloizou (2014).
A.D.Railton
April 2015

Acknowlegements
In producing this thesis, I used the LATEX template created and maintained by Krishna Kumar
(Kumar, 2014). References were managed using the open source reference manager JabRef
(JabRef Development Team, 2014). Graphs were made using gnuplot (Williams et al., 2010),
MATLAB (MATLAB, 2010), IDL (Liu et al., 2013) and VisIt (Childs et al., 2012) and some
diagrams with Inkscape and Gimp.
The book ‘How to Write a Better Thesis’ (Evans et al., 2011) did in fact help me write a
better thesis, and I’m grateful to Ginny Haskell for giving it to me. I have also been greatly
influenced by Mike McIntyre’s writing on Lucidity and Science (McIntyre, 1997) and have
attempted to follow his advice. The Pomodoro technique1 took months off my submission
date, while Dr Inger Mewburn’s blog ‘The Thesis Whisperer’2 made me realise that other
people struggle too.
In no particular order, I would like to thank: Adrian Barker for his help getting PLUTO set
up, the Maths department IT helpdesk for all the stupid questions I asked them over three
and a half years, Lizzie Polgreen and Dr Nilanjana Datta for putting up with my crises, my
family for their unending support and Dad for making a door so that I didn’t freeze to death
while writing up. Pembroke College has been a source of support and comfort throughout my
time at Cambridge and will always feel like home.
Prof. J. C.B. Papaloizou was a better supervisor then I deserved or could have hoped for
and I am infinitely grateful to him for his patience and kindness. Furthermore, I would like to
thank the Science and Technology Facilities Council for funding this PhD studentship.
Finally, I am indebted to the various, uncountable people who persuaded me not to quit.
1http://pomodorotechnique.com/
2http://thesiswhisperer.com/

Abstract
This thesis finds that vortex instabilities are not necessarily a barrier to their potential as sites
for planetesimal formation.
It is challenging to build planetesimals from dust within the lifetime of a protoplanetary
disc and before such bodies spiral into the central star. Collecting matter in vortices is a
promising mechanism for planetesimal growth, but little is known about their stability under
these conditions. We therefore aim to produce a more complete understanding of the stability
of these objects.
Previous work primarily focusses on 2D vortices with elliptical streamlines, which we gener-
alise. We investigate how non–constant vorticity and density power law profiles affect stability
by applying linear perturbations to equilibrium solutions. We find that non–elliptical stream-
lines are associated with a shearing flow inside the vortex.
A ‘saddle point instability’ is seen for elliptical–streamline vortices with small aspect ratios
and we also find that this is true in general. However, only higher aspect ratio vortices act
as dust traps. For constant–density vortices with a concentrated vorticity source we find
parametric instability bands at these aspect ratios. Models with a density excess show many
narrow bands, though with less strongly growing modes than the constant–density solutions.
This implies that dust particles attracted to a vortex core may well encounter parametric
instabilities, but this does not necessarily prevent dust–trapping.
We also study the stability and lifetime of vortex models with a 2D flow in three dimensions.
Producing nearly–incompressible 3D models of columnar vortices, we find that weaker vortices
persist for longer times in both stratified and unstratified shearing boxes, and stratification is
destabilising. The long survival time for weak, elongated vortices makes it easier for processes
to create and maintain the vortex. This means that vortices with a large enough aspect
ratio have a good chance of surviving and trapping dust for sufficient time in order to build
planetesimals.

Contents
Contents xi
List of Figures xv
List of Tables xix
Nomenclature xxi
1 Introduction 1
2 Protoplanetary discs as an environment for planet formation 3
2.1 Historical context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Circumstellar discs and star formation . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Observational properties of discs . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Disc structure and evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 The role of dust in protoplanetary discs 25
3.1 Dust in PP discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Beyond planetesimal formation: creating terrestrial and gas giant planets . . . 30
3.3 Planetesimal formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 The Safronov–Goldreich–Ward mechanism . . . . . . . . . . . . . . . . . . . . . 38
3.5 Beyond SGW: Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Research design 55
5 Calculating equilibrium solutions 57
5.1 Governing equations for well–coupled dust . . . . . . . . . . . . . . . . . . . . . 57
5.2 Functions specifying the vorticity and density profiles . . . . . . . . . . . . . . 65
5.3 An analytic solution: the Kida vortex . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 The polytropic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xii Contents
5.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Calculating equilibrium solutions: numerical approach and results 87
6.1 Setting up the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Calculating equilibrium vortex solutions using vortex.f90 . . . . . . . . . . 89
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 Stability Analysis 109
7.1 Perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.2 Time–independent wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.3 Horizontal stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.4 Time–dependent wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.5 Vertical stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8 Numerical treatment of stability analysis 141
8.1 First numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.2 Second numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.3 Point vortex stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.4 Polytropic model stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.6 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9 A study of the stability of vortex models with a 2D flow to 3D perturbations175
9.1 The PLUTO code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.2 2D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9.3 3D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10 Conclusions and further work 209
10.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.2 Directions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Appendix A Elliptic coordinates 213
Appendix B Floquet theory as solution of an initial value problem 215
Appendix C Growth rates in the shearing sheet for k ∝ t 221
Appendix D PLUTO runs 225
Contents xiii
Bibliography 231

List of Figures
2.1 PP disc around HL Tau showing gaps formed by protoplanets . . . . . . . . . . 7
2.2 Fraction of YSO with accretion discs vs. stellar age . . . . . . . . . . . . . . . . 10
2.3 Variation of Mdisc with M∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 The flared PP disc around HH 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Overview of PP disc evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Spiral structure and dust vortex in the PP disc around SAO 206462 . . . . . . 27
3.2 A large dusty vortex in the system SR21 . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Dust drift velocity as a function of stopping time τs . . . . . . . . . . . . . . . 37
5.1 Geometry of the Keplerian disc . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 The shearing sheet model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 The symmetry of the shearing sheet . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4 Vorticity profile of the Kida vortex . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 The elliptic coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 Kida vortex pressure profiles as a function of the aspect ratio χ . . . . . . . . . 81
5.7 The behaviour of the Kida pressure distribution for different χ . . . . . . . . . 82
6.1 Program design for vortex.f90. . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2 Vortex profiles with ωm = 0.09, α = {0, 1, 2, 4} and constant density. . . . . . . 95
6.3 Streamlines of the vortex {α, β, ρm} = {4, 0, 0} with ωm = 0.0183. . . . . . . . . 96
6.4 The variation of χ with ymax for different strength, constant density vortices
with α = 1, 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.5 Vortices with a Kida Bernoulli vorticity profile, ωm = 0.09 and nonzero density 101
6.6 Vortices with a non–Kida Bernoulli vorticity source, ωm = 0.09 and nonzero
density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.7 Vortex profiles for the class {0.25, 1, 1, 0.5} with varying ωm . . . . . . . . . . . 105
6.8 Variation in central density enhancements against χ for various α values. . . . 107
6.9 The variation of ρ with χ for the vortex class {0, 1, 0.1} . . . . . . . . . . . . . 108
7.1 Definition of arclength σ and total arclength Σ . . . . . . . . . . . . . . . . . . 118
xvi List of Figures
7.2 Geometrical meaning of horizontal and vertical stability . . . . . . . . . . . . . 119
7.3 Geometric realisation of θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.4 Epicyclic frequency for the Kida vortex in the horizontal limit . . . . . . . . . . 125
7.5 Saddle point direction for saddle point instability . . . . . . . . . . . . . . . . . 126
8.1 Calculating a contour using arclength.f90 . . . . . . . . . . . . . . . . . . . 144
8.2 Stability plots from the first approach for vortices with no density . . . . . . . 148
8.3 Stability plot for the Kida vortex . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.4 Stability plots from the first approach, for vortices with a central density en-
hancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.5 Stability plot for {0.25, 0, 0} using the first numerical method . . . . . . . . . . 151
8.6 Problems with the first numerical approach . . . . . . . . . . . . . . . . . . . . 152
8.7 Stability plots from the second approach for vortices with uniform density . . . 156
8.8 Maximum growth rate vs. χ for vortex solutions with uniform density . . . . . 157
8.9 Comparison of streamline choice for {0.25, 0, 0} . . . . . . . . . . . . . . . . . . 158
8.10 Effect of streamline choice for class {0, 1, 2} . . . . . . . . . . . . . . . . . . . . 159
8.11 Stability plots for vortices with a central density enhancement and Kida vorticity
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.12 Maximum growth rate as a function of χ for vortices with a central density
enhancement and Kida vorticity profile . . . . . . . . . . . . . . . . . . . . . . . 161
8.13 Stability of non–Kida vorticity profiles with a central density enhancement . . 162
8.14 Maximum growth rate plots for vortices with non–Kida vorticity profiles . . . . 163
8.15 Stability plot for the point vortex model . . . . . . . . . . . . . . . . . . . . . . 164
8.16 Stability plots for polytrope solutions . . . . . . . . . . . . . . . . . . . . . . . . 167
8.17 Plot of maximum growth rate against χ for the polytropic vortex solutions . . 168
8.18 Growth rate of the parametric instability of the polytropic model for kz = 0 . . 168
9.1 Testing the PLUTO shearing box implementation . . . . . . . . . . . . . . . . . 177
9.2 Time evolution of vorticity for Run 1h . . . . . . . . . . . . . . . . . . . . . . . 180
9.4 Time evolution of vorticity for Run 2h . . . . . . . . . . . . . . . . . . . . . . . 183
9.6 The χ–profiles for the α = 1 equilibrium solutions from Chapter 6, compared to
those produced by PLUTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.7 Run 1i, L = (4, 8), N = (512, 1024) snapshot showing smaller vortex structures 185
9.8 Effects of box size and resolution of time evolution on an imposed χ = 5 Kida
vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9.9 Effects of box size and resolution of time evolution on an imposed χ = 8 Kida
vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.10 ωmin for 2D boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.11 Initial columnar vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
List of Figures xvii
9.12 Initial unstratified box tests to test the effect of a Gaussian density perturbation.191
9.13 Effect of different stratification regimes . . . . . . . . . . . . . . . . . . . . . . . 194
9.14 Investigating use of full and half boxes in stratified runs . . . . . . . . . . . . . 195
9.15 Onset of instability for the unstratified χ = 3.5 case . . . . . . . . . . . . . . . 197
9.16 Onset of instability for the stratified cases ρ′ ∼ 0.1, χ = 8 and ρ′ ∼ 0.1, χ = 8 . 198
9.17 Determining the onset of instability using ⟨v2y⟩ . . . . . . . . . . . . . . . . . . 200
9.18 Density in stratified boxes for different ρ′ . . . . . . . . . . . . . . . . . . . . . 201
9.19 Investigating different cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.20 Time evolution of vorticity for cs = 1, χ = 8 run . . . . . . . . . . . . . . . . . 202
9.21 ωmin for the stratified box with ρ′ ∼ 0.1 . . . . . . . . . . . . . . . . . . . . . . 203
9.22 ωmin for the stratified box with ρ′ ∼ 0.5 . . . . . . . . . . . . . . . . . . . . . . 204
9.23 Results for unstratified models . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.24 Instability time τunstable as a function of χ . . . . . . . . . . . . . . . . . . . . . 206

List of Tables
2.1 Summary of observational PP disc properties . . . . . . . . . . . . . . . . . . . 23
3.1 Vortical dust trapping of different particle types . . . . . . . . . . . . . . . . . . 44
3.2 Summary of vortex models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Important timescales for PP discs . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9.1 Initial 2D runs, testing box size and resolution . . . . . . . . . . . . . . . . . . 178
D.1 Details of PLUTO runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Nomenclature
Roman Symbols
vd dust velocity
vg, v gas velocity
Fdrag Epstein drag force acting on a single dust particle of mass m•, see equation (3.3.4),
page 33
A(ψ) Bernoulli source term in equation (5.2.4) such that A(ψ) = dF1dψ
B(ψ) density source term in equation (5.2.4) such that B(ψ) = d log ρdψ
M Mach number, typically M = vϕ/cs in a disc
Z variable in the Hill equation describing vertical stability for the internal shear free case,
see equation (7.5.23), page 137
Mdisc disc mass
M⊙ mass of the Sun, M⊙ = 1.989× 1030kg
v velocity of combined fluid, see equation (5.1.5), page 60
SA phase function in Lagrangian WKBJ ansatz, see equation (7.1.13), page 112
P˜ period round a vortex streamline, see equation (5.3.32), page 83
P˜kida = 2πS (χ−1), period round a streamline of a Kida vortex solution,, see equation (5.3.35),
page 84
fν viscous force per unit volume, see equation (5.1.1), page 58
k wavenumber
u mean velocity of the dust with respect to the gas, see equation (5.1.6), page 60
vd velocity of dust component in two–fluid equations, page 59
xxii Nomenclature
vK Keplerian velocity, vK =
√
GM∗/rϕˆ
A scaling constant in the Bernoulli source term A(ψ), see equation (5.2.5), page 66
B scaling constant in the density source term B(ψ), see equation (5.2.5), page 66
b density scaling factor in vortex polytropic model, see equation (5.4.3), page 84
cs local sound speed
F (ψ) arbitrary function that appears in the Poisson equation form of the momentum equa-
tion (5.1.29), see equation (5.1.24), page 64
G gravitational constant
h scaling constant for elliptic coordinate system, see equation (5.3.0), page 73
j specific angular momentum, j = r2Ω (l and h are also commonly used)
k⊥ magnitude of the horizontal wavenumber, k⊥ =
√
k2x + k2y, see equation (7.2.19), page 117
M∗ mass of central star
m• dust particle mass, page 33
M⊕ mass of the Earth, M⊕ = 5.97× 1024kg
MJ Jeans mass, see equation (2.2.0), page 5
n polytropic index for density, see equation (5.4.3), page 84
N1 The number of iterations in the first loop of vortex.f90, page 91
N2 The number of iterations in the second, density-enhancement loop of vortex.f90,
page 91
P gas pressure
Q ≡ csΩ/πGΣ, Toomre Q parameter, page 39
Q′ the Eulerian perturbation of a quantity Q, page 110
qHill buoyancy frequency in Hill equation, page 138
r0 origin of shearing coordinates, distance to the central star., page 61
S = −r dΩdr , the shear of a background shearing flow u = (0,−Sx, 0). In a Keplerian disc
S = 32Ω. q and A are frequently used in the literature.
Nomenclature xxiii
t0 time at which y passes through its maximum value in the Fourier series expansion of
particle trajectories x and y, see equation (7.4.3), page 128
Ts Stokes number, Ts = Ωτs
W rescaled wavenumber used in vertical stability analysis, W = µ|k|2, page 135
Greek Symbols
α power law index in the Bernoulli source term A(ψ), see equation (5.2.5), page 66
αν Shakura-Sunyaev α, a dimensionless parameter measuring the efficiency of angular mo-
mentum transport due to turbulence, see equation (2.4.26), page 21
ν¯ scaled effective kinematic viscosity ν¯ = ρgρg+ρd ν, page 69
β power law index in the density source term B(ψ), see equation (5.2.5), page 66
βP pressure scaling factor in vortex polytropic model, see equation (5.4.4), page 85
ω vorticity of a flow with velocity v, ω = ∇× v, page 64
σ stress tensor in the Navier-Stokes equations, see equation (2.4.3), page 17
ξ the Lagrangian displacement, page 110
χ aspect ratio of a vortex
∆Q the Lagrangian perturbation of a quantity Q, page 110
δ dimensionless diffusion parameter inside vortex, parametrises turbulence in vortices,D = δcsH
(e.g Lyra and Lin, 2013)
η angular elliptic coordinate, see equation (5.3.0), page 73
γ growth rate
λ large parameter in Lagrangian WKBJ ansatz, see equation (7.1.13), page 112
λJ Jeans length, see equation (2.2.0), page 5
µ Scalar quantity in vertical stability analysis v′ = (µky,−µkx), page 134
µj characteristic, or Floquet, exponents. See also Appendix B, page 122
ν effective kinematic viscosity, see equation (2.4.6), page 17
νt root mean square turbulent velocity, see equation (5.2.6), page 67
xxiv Nomenclature
Ω angular velocity, Ω = vϕ/r
ωm a measure of the total imposed vorticity in a numerical vortex solution, see equa-
tion (6.1.3), page 88
ΩGMC typical angular velocity in a GMC
Ωcore angular velocity of a GMC core
ωkida the vorticity of a Kida vortex, S + c = −ωkida, page 72
ρ density of combined fluid, ρ = ρg + ρd, see equation (5.1.5), page 60
Φ the ‘combined potential’ of the gravitational and centrifugal potentials, Φ = Φgr −
1
2Ω2r2, see equation (5.1.2), page 59
Φgr gravitational potential due to central star, Φgr = GM∗|r|
ψσ Value of the streamfunction at (0, ymax), page 142
ψ the Stokes’ streamfunction, see equation (5.1.21), page 64
ψ(ex) the exterior streamfunction for the Kida solution, page 73
ψ(in) the interior streamfunction for the Kida solution, page 73
ψ0 the streamfunction of the background flow. In the shearing sheet, ψ0 = Sx2/2, see
equation (5.1.21), page 64
ψb value of the streamfunction ψ evaluated on the vortex boundary at (0, 1), see equa-
tion (5.2.5), page 66
Γd = Σd/Σg, dust-to-gas ratio by mass
ρ0 constant background density, page 66
ρd dust density
ρg gas density
ρm a measure of the total imposed mass in a numerical vortex solution, see equation (6.1.3),
page 88
ρ• dust particle density, page 33
ρmax central density enhancement in an equilibrium vortex solution, page 99
σ coordinate determining location on a streamline, related to some fraction of the ar-
clength, page 117
Nomenclature xxv
Σ total circumference of a streamline (see σ the arclength parameter), page 121
τν viscous timescale, see Table 3.3
τcoag coagulation or dust sticking timescale, see Table 3.3
τcollide dust collision timescale, see Table 3.3
τdecay orbital decay timescale, see Table 3.3
τdyn dynamical or orbital timescale, see Table 3.3
τff freefall time, ∼ (Gρ)−1/2, page 5
τsc sound crossing time, ∼ L/cs for characteristic lengthscale L
τth thermal timescale, see Table 3.3
τcorr turbulent correlation time, see equation (5.2.6), page 67
τs dust stopping time or frictional timescale, see Table 3.3
τdisc disc lifetime, see Table 3.3
Θ scaled ratio of horizontal and vertical wavenumbers, see equation (7.2.19), page 117
θ angle between the perturbation wavevector k and zˆ, see equation (7.2.23), page 118
τsettle settling or sedimentation timescale, see Table 3.3
τunstable time at which instability occurs for the 3D vortex columns in Chapter 9
ϖ frequency while travelling round a streamline, P˜ = 2π/ϖ, page 128
ϱj characteristic multipliers in Floquet theory, ϱj = eµjΣ. See also Appendix B, page 122
ξ radius elliptic coordinate, see equation (5.3.0), page 73
ξb the bounding streamline with ψ = ψb is given by ξ = ξb, page 73
Other Symbols
{α, β, ρm} the three numbers that define a vortex class
Acronyms / Abbreviations
a radius of a dust particle, see Section 3.1.1
ALMA Atacama Large Millimeter Array
xxvi Nomenclature
AU astronomical unit, the average distance between the Earth and the Sun,
1AU ≈ 1.496× 1011m
GI gravitational instability
GMC giant molecular cloud
GNG Goodman–Narayan–Goldreich model vortex solution, Goodman et al. (1987)
IR infrared
ISM interstellar medium
MHD magnetohydrodynamics
MMSN minimum mass solar nebula, page 10
MRI magnetorotational instability
Myr megayear, a million years, 1Myr = 106yr
pc parsec, the distance at which one astronomical unit subtends an angle of one arcsecond,
1 pc ≈ 3.086× 1016m ≈ 2.063× 105AU
Re Reynolds number, a dimensionless quantity that describes the ratio of inertial forces to
viscous forces, Re ≡ UL/ν
RWI Rossby wave instability
SBI Subcritical baroclinic instability
SED spectral energy distribution, the distribution of flux as a function of frequency or wave-
length
SGW Safronov-Goldreich-Ward mechanism of planetary formation, see Chapter 3
UV ultraviolet
YSO young stellar objects
Chapter 1
Introduction
Humanity has always wanted to know where we came from and how we came to exist. In an
age where exoplanets are discovered almost daily and the likelihood of life elsewhere in the
universe increases, the question of how they and our own planet came to exist is ever more
salient.
The formation of planets from a dusty, gaseous disc around a young star has been inves-
tigated since the inception of the ‘nebular hypothesis’ in the 1700s. The formation of small
boulders from dust in the disc and the formation of planets from large planetesimals are largely
well understood processes, but getting between these two states is still an outstanding prob-
lem. The most promising route to enable us to bypass what is known as the ‘metre gap’ is by
collecting matter in pressure maxima until sufficient concentration is reached and gravitational
instability can take over.
Local pressure maxima at the centre of some vortices attract dust but the general stability
of these structures with and without density enhancements is not really known. After all it
is necessary for vortices to persist long enough in the presence of dust to build up sufficient
concentrations of matter.
We therefore aim to produce a more complete understanding of the stability of vortices in
protoplanetary discs.
We limit this study to the stability of equilibrium vortices to elliptical stabilities, beyond
the analytical Kida case, in fluids which are incompressible, inviscid, isothermal and two-
dimensional and with dust modelled as perfectly coupled to the gas. We later extend this to
hydrodynamical simulations of columnar vortices in stratified shearing boxes.
The structure of the PhD thesis ‘The Structure and Stability of Vortices in Astrophysical
Discs’ is as follows:
The first chapter examines the literature on protoplanetary (PP) discs as an environment
for planet formation. It examines development and evolution of the PP disc and how it is closely
related to star formation. Chapter 3 looks in more detail at the properties of dust in the disc
and how it interacts with the gas. We consider the limitations of the Safronov–Goldreich–Ward
2 Introduction
mechanism of planet formation and how vortices and other pressure maxima in the disc can
help overcome these difficulties. It then examines the instabilities that can occur in the disc
to form these vortices (and perhaps destroy them) and finally explores existing vortex models
and approaches to their analysis. Chapter 4 briefly outlines our research design.
In order to perform a comprehensive study of the stability of equilibrium vortices in PP
discs, it is necessary to create a large variety of stable vortex configurations to then perturb
about. Chapter 5 therefore details the equations of motion in an idealised shearing sheet
model of the PP disc and a well-known analytical vortex model often used as the starting
point of stability investigations. We show how to extend this approach to find models that
have more physically realistic configurations. Chapter 6 shows the numerical procedures in
actually producing these configurations and details the results of these.
In Chapter 7 we take these new equilibrium vortex models and formulate the stability
analysis of these solutions to perturbations localised on streamlines while in Chapter 8 we
show how we implement the numerics of this and present the results. Chapter 9 looks towards
more realistic, 3D models in the disc and uses a compressible hydrodynamic code to do so.
The overall approach is evaluated and discussed, with conclusions drawn in Chapter 10.
Chapter 2
Protoplanetary discs as an
environment for planet formation
The study of disc evolution has important consequences for theories of both star and planet
formation. Circumstellar discs, discs around stars, perform two crucial roles in both their
evolution. Firstly, they perform a dynamical role in a star’s formation by providing a disc
from which the star accretes most of its mass. Secondly, in the latter part of its lifetime, the
disc plays an important chemical role in processing various gas species and in the growth of
dust grains into planetesimals and beyond; in this way, the circumstellar disc is characterized
as a protoplanetary (PP) disc1.
In this chapter we discuss the role PP discs play as an environment for planet formation.
In Section 2.1 we will give an overview of the historical development of the ‘solar nebula’
disc model for planetary system formation before briefly outlining the role discs play in star
formation in Section 2.2. We will review observations of PP discs and the properties we can
infer from these in Section 2.3, including their lifetime, mass, size and structure and how
these place some quite stringent constraints on the planet formation process. In light of these
constraints we will review the structure and evolution of gas discs in Section 2.4, including a
general evolutionary pathway in Section 2.4.1, before looking in detail at the equations which
govern them.
2.1 Historical context
Circumstellar discs as the site of planet formation is an idea that is centuries old. The Swedish
scientist and philosopher Emanuel Swedenborg is credited with first proposing the ‘nebular
hypothesis’ as a model for the formation of the Solar System, suggesting that planets formed out
of a nebular ‘crust’ that surrounded the Sun then broke apart (Swedenborg, 1734). Observation
1The phrases circumstellar disc and protoplanetary (PP) disc are used interchangeably in this work, although
not all circumstellar discs may have the potential to evolve into a planetary system.
4 PP discs as an environment for planet formation
that the solar system planets were roughly coplanar and orbit the sun in the same direction
lead to the belief that the planets had a common origin in a rotating disc. Twenty years later,
Kant (1755) developed this model with the idea of a nebulous cloud in slow rotation, pulled
apart by gravitational forces and flattened to a spinning disc from which stars and then planets
form. Laplace (1796), independent of Kant, proposed a similar model but with the planets
forming before the Sun, condensing from rings of gas thrown off by the disc.
The Kant-Laplace theory dominated for the 19th Century, but a major difficulty was that
the model failed to account for the angular momentum distribution observed in the Solar
System; the planets contain 99% of the total angular momentum despite containing less than
1% of the mass (Woolfson, 1993). Furthermore, Maxwell showed that if all the matter in the
known planets was distributed around the Sun in a disc, shearing forces (the “different rotation
between the inner and outer parts of a ring2”) would have prevented material condensing
to individual planets without rings of hundreds of times more mass than the planets they
produced.
Little progress was made on the problem until the solar nebula disc model of Safronov
(1969), the first full, quantitative explanation of the formation of the Sun and Solar system
which still forms the basis for modern theory. Details of this model are given in Sections 3.2
and 3.4.
2.2 Circumstellar discs and star formation
Circumstellar discs are an essential component of star formation, and indeed the creation of
such discs is intimately tied up with the process for forming the central star from a giant
molecular cloud (GMC).
GMCs are a type of interstellar cloud which are of a size and density such that the gas
within them exists in a molecular form. This is in contrast to the majority of the interstellar
medium that contains predominantly atomic gas, as the interiors of GMCs are shielded from
stellar radiation by dust which allows the gas to be molecular. When part of a GMC reaches
a critical size, mass or density it begins to collapse under its own gravity, creating protostars3.
These highly variable young stars, also known as T Tauri stars after their prototype, were
discovered near molecular clouds by Joy (1945). Soon after discovering that T Tauri stars emit
more IR radiation than bare photospheres should, Lynden-Bell and Pringle (1974) suggested an
explanation via an accreting circumstellar disc. In this landmark work, the first full disc model
calculations were done and they proposed the mechanism of angular momentum transport
through the disc as a way of removing it from circumstellar material to allow it to accrete onto
2See the letters of Maxwell in the collection Harman (2002), pages 438–479.
3This process is not very well understood as both turbulence and a magnetic field B support the GMCs
against gravity. For example, ambipolar diffusion of charged particles in the GMC’s plasma could reduce the
effect of B and cause support failure.
2.2 Circumstellar discs and star formation 5
the star. The basic principles of disc evolution were later reviewed in Pringle (1981), which
remains one of the best introductions to the topic.
Where do these discs come from? GMCs are not homogeneous structures, with density
and velocity fields exhibiting structure across wide range of lengthscales that is characteristic
of turbulence (Larson, 1981; McKee and Ostriker, 2007). Any collapsing region will therefore
possess some intrinsic angular momentum. With no rotation, gas in a core could, in principle,
freely collapse towards the centre forming a relatively compact protostar. However, small
amounts of angular momentum prevents such radial collapse, only allowing gas to sink down
to a minimum radius from the centre.
Most of the infalling matter will therefore not fall directly onto the protostar but form
a disc around it (e.g Terebey et al., 1984). Random gas motions average out in favour of
the direction of the cloud fragment’s net angular momentum. The stabilisation of these orbits
forms a disc with thickness much smaller than its radius on a timescale ∼ 104−105yr (Shu et al.,
1993). These discs survive as quasi-equilibrium structures since material in the disc has specific
angular momentum increasing with radius and thus is Rayleigh-stable (see Section 2.4.5).
A significant proportion of the mass of the system is initially in orbit around the protostar
(Shu et al., 1987) and therefore some mechanism is required to shed angular momentum so
matter can fall onto the star. Much of the final stellar mass is accreted through this disc and
so discs are essential to a star’s genesis.
Consideration of the Jeans length, λJ , of a GMC gives us a handle on the expected size of
a circumstellar disc. λJ is the maximum size of a cloud core that is stable against gravitational
collapse, with thermal pressure balancing gravity. It is found by equating the sound crossing
time τsc ∼ λJ/cs with the freefall timescale τff ∼ (Gρ)−1/2, where ρ is the gas density.
λJ =
2πcs√
4πGρ, (2.2.1)
which implies a Jeans mass of MJ = ρλ3J . For MJ ≈ M⊙, where M⊙ is the mass of the Sun,
cs =
√
kT/µmp (with k the Boltzmann constant, mp the proton mass, µ ≈ 2.3 the mean
molecular weight and T ≈ 10K) and a number density n ∼ 105cm−3 (Shu et al., 1987) we find
λJ ≈ 0.1pc, the observed size of typical molecular cloud cores (McKee and Ostriker, 2007).
With knowledge of typical angular velocities in a GMC, ΩGMC (e.g. Goodman et al., 1993),
we can use this λJ to estimate the specific angular momentum of the gas per unit mass, j.
Material in the disc will initially circulate the protostar on eccentric orbits but will lose energy
though shocks and dissipation until they travel on the minimum energy, circular orbit for a
given j. Then, equating the angular momentum of the infalling fluid element to that of a
Keplerian orbit at a distance R from a central star of mass M∗ we can infer a disc radius Rdisc
when M∗ = 1M⊙:
Rdisc =
j2
GM∗
≈ (ΩGMCλ
2
J)2
GM∗
≈ 102 − 104AU. (2.2.2)
6 PP discs as an environment for planet formation
This is much larger than the radius of the protostar so some redistribution of angular momen-
tum is required for mass to accrete further into the centre.
Discs inherit initial mass, size and chemical composition from their star formation environ-
ment, with a large variation in core accretion rates suggesting that a large diversity of initial
disc masses and sizes is expected (Goodman et al., 1993). Furthermore, the evolution of a disc
can be shaped by environmental effects such as stellar radiation (from the central star or other
stars in the locality), gas accretion, stellar flybys or binary companions (Lodato, 2008).
2.3 Observational properties of discs
The most common way of detecting PP discs is via their infrared (IR) emission from hot
gas in the inner regions r ≲ 0.1AU (Hartigan et al., 1995; Najita et al., 2007), warm dust
at temperatures around 100K (Kenyon and Hartmann, 1987; Chiang and Goldreich, 1997) at
r ≲ 1AU (Kenyon and Hartmann, 1995; Haisch et al., 2001; Hartmann et al., 2005) and cold
(T ≃ 10K) dust in the outer disc r ≳ 50AU (Beckwith et al., 1990).
Discs began to be directly observed in the mid-1980s, first with dusty debris discs4 (Aumann
et al., 1984; Smith and Terrile, 1984) and then gas-rich discs in the survey of Sargent and
Beckwith (1987). Despite the difficulty of their observation due to their small size, low mass
and temperature, subsequent advances in optics and telescope technology have found discs
to be prevalent around young stars. There are currently 170 resolved discs known (130 discs
around T Tauri stars and 40 debris discs), of which there is an online catalogue5.
Observations of protoplanetary discs only came about with the invention of infrared (IR)
detectors, with the first statistical survey of PP disc occurrence in star forming regions by
Cohen et al. (1989) and Strom et al. (1989). It was then found by Weintraub et al. (1989)
that many of these discs contained large dust grains, followed by the calculations of Beckwith
et al. (1990) showing there was enough mass in these grains to form systems like our own
Solar System. In the nineties, improvements in the sensitivity of optical and IR telescopes
and enhancements of spatial resolution resulted in the first direct image of a PP disc (and the
confirmation of a flat disc structure) by O’dell and Wen (1994). We can now even observe
asymmetric and radial structure in the disc, with spiral structures in the disc around SAO
206462 (see Section 3.1.2, Figure 3.1a, Muto et al., 2012) and the possible presence of gaps
opened up by protoplanets6 in HL Tau (Figure 2.1, Vlahakis et al., 2014).
2.3.1 Classification of young stellar objects
To obtain large samples of young stars (and hence PP discs) one must observe rich star forming
regions such as Ophiuchus, Taurus or Orion, between 120 − 410 pc from the Sun. Resolving
4See Section 2.4.
5http://circumstellardisks.org/
6For the precise definition of ‘protoplanet’ and ‘planetismal’ etc., see Section 3.1.1.
2.3 Observational properties of discs 7
Figure 2.1 PP disc around HL Tau showing gaps possibly produced by orbiting protoplanets.
HL Tau is a Class I object with a mass of around 0.33M⊙ and a substantial disc mass of
0.13M⊙ (Kwon et al., 2011), which we expect to be gravitationally unstable. It is one of the
brightest known discs (Beckwith et al., 1990), which extends to around 100 AU (e.g Lay et al.,
1997). Greaves et al. (2008) claimed to find a protoplanetary candidate of around 8MJupiter at
75 AU, though Nero and Bjorkman (2009) suggested it could be a compact feature due to disc
fragmentation. The bright rings in the disc are separated by gaps likely to have been formed
by the interaction with orbiting protoplanets, which is especially interesting as the system is
around 105 years old, implying very rapid planet formation. (ESO/ALMA).
8 PP discs as an environment for planet formation
even large discs at these distances is difficult so statistical measures of disc occurrence and
lifetime are often derived using the IR spectral energy distribution (SED) of Young Stellar
Objects (YSOs; Lada and Wilking, 1984, Adams et al., 1987 and Beckwith, 1999).
YSOs are also classified by the strength of their emission lines, which are due to gas
accreting onto the stellar surface. Observation of accretion is important as it is the easiest way
of inferring the presence of gas in PP discs, albeit only in the innermost regions of the disc
r ≲ 0.1AU (Najita et al., 2007). The classification is as follows:
• Class 0: The least evolved objects observed during first stages of cloud collapse (André
et al., 1993), where the protostar is deeply embedded in the optically thick cloud of dust
and gas. The disc forms very early on in this phase and grows extremely rapidly on a
timescale of 104 − 105 yr (Shu et al., 1993; Yorke et al., 1993; Hueso and Guillot, 2005).
Most of the mass is accreted onto the object in this embedded stage (Hartmann et al.,
1993).
• Class I: The first objects around which discs are detected, though the cold, optically
thick envelope of dust and gas still dominates emission.
These objects are around an order of magnitude less luminous than expected for a steady
release of gravitational energy as the envelope falls onto the star. They also undergo short
outbursts of activity due to periods of high accretion (Herbig, 1977). Both of these point
to young discs in this embedded phase being (possibly) gravitationally unstable. Outflows
and jets are also detected which can further hinder the measurement of disc properties.
This phase lasts ∼ 0.5Myr (Evans et al., 2009).
• Class II: Star formation is essentially over for these objects and the envelope is com-
pletely dispersed (André and Montmerle, 1994). Emission in this case is largely from
a disc, either because it is actively accreting and thus luminous or because it is being
heated by stellar radiation. These are known as ‘classical T Tauri’ stars. The majority
of planet formation occurs in the discs around Class I and II objects.
• Class III: The circumstellar material around these objects has largely disappeared so
there is little accretion and thus little emission. However, some do display the signatures
of an optically thin remnant debris disc. These are known as ‘weak line T Tauri’ stars.
Note that a ‘transitional disc’ is an object in an evolutionary stage between the gas-rich discs
around Class II YSOs and the gas-poor ‘debris discs’ around Class III objects (Kley and Nelson,
2012 and Espaillat et al., 2014).
2.3.2 Disc lifetimes
The lifetime of a PP disc is a fundamental parameter, both as a timescale for the physical
processes driving disc evolution and as the time available for planet formation.
2.3 Observational properties of discs 9
We can infer the typical disc lifetime τdisc from studying the fraction of stars in stellar
clusters in star forming regions (assumed to have a relatively uniform age) with various disc
signatures (Haisch et al., 2001; Sicilia-Aguilar et al., 2006). Disc properties that are measurable
as a function of age are IR excess in the SED (Carpenter et al., 2006; Mamajek et al., 2004)
(with observations of cold dust in the outer disc correlating reassuringly well with observations
of inner dust discs (Andrews and Williams, 2005)), the steady decline of gas accretion rate M˙
onto the star (Hartmann et al., 1998; Muzerolle et al., 2000; Calvet et al., 2000), measurements
of dust mass (Carpenter et al., 2005; Wyatt et al., 2003) and size distribution (Kessler-Silacci
et al., 2005; van Boekel et al., 2005). A review of these observations is given in Hillenbrand
(2005).
There is remarkable similarity in average disc properties from region to region (Williams
and Cieza, 2011). There is also good agreement between the ages of T Tauri discs and the
inferred formation time of Solar System meteorites (Podosek and Cassen, 1994). However, we
do find disc lifetimes are about twice as short around higher mass stars due to higher accretion
rates and a stronger radiation environment (Hillenbrand, 2008).
The median disc lifetime is 2− 3Myr, but with a wide dispersion. Desipte this, there is a
fairly stringent upper limit of ∼ 10Myr on the lifetime of PP discs, with Strom et al. (1989)
finding that ≲ 10% of YSOs with ages in excess of 10Myr have discs, confirmed by more recent
Spitzer surveys (Williams and Cieza, 2011). A plot of the fraction of YSOs with accretion
discs vs. stellar age can be seen in Figure 2.2 which shows a steep drop-off in this fraction for
protostars older than 5Myr.
The small amount of objects with an outer but no inner disc (due to photoevaporation,
discussed in Section 2.3.10) show that the dissipation timescale of the entire disc once accretion
stops is ≲ 0.5Myr (Skrutskie et al., 1990; Wolk and Walter, 1996; Cieza et al., 2007). This
implies that the transition between Class II and Class III YSOs is very rapid and across the
entire radial extent of the disc. A discussion of this process and the fate of PP discs is given
in Section 2.3.10.
2.3.3 Disc mass
Discs are found around stars of very different masses, from brown dwarfs (M∗ < 0.08M⊙) to
massive stars (M∗ > 8M⊙). In general, the most detailed information is available for discs
round young objects of about 1M⊙. The mass of the disc, Mdisc, is inferred from the disc’s
dust mass.
The dust-to-gas ratio Σd/Σg (which we will hereafter call Γd) in PP discs is initially assumed
to be the canonical interstellar value of 1% (Draine, 2003), though this is expected to increase
as the disc evolves. This means the dynamics of the disc are almost completely dominated
by the gas while the dust component of the disc dominates the opacity and thus the emission
properties of the disc. In practice this means that around 99% of the disc is functionally
10 PP discs as an environment for planet formation
Figure 2.2 The fraction of YSO in stellar clusters with accretion discs vs. stellar age, from
Hillenbrand (2005). Individual clusters are treated as containing stars of the same age. It
shows that the fraction of stars with discs in young stellar clusters decreases with age and after
5Myr very few stars have discs left; those that do are mostly low-mass T Tauri stars.
invisible. It also dominates the ionisation state of the disc and thus the coupling to the
magnetic field. Note also that there are no strong constraints on Γd, which need not match
the interstellar medium (ISM) (Youdin, 2010, Section 3.1.2).
The surveys of André and Montmerle (1994); Andrews and Williams (2005, 2007); Eisner
and Carpenter (2006) have found disc-to-star mass ratios of
Mdisc/M∗ ∼ 10−3 − 101, (2.3.1)
with the medianMdisc/M∗ = 0.009 and an average disc massMdisc ≈ 0.02M⊙, which is roughly
the minimummass solar nebula (MMSN)7 (Kusaka et al., 1970; Weidenschilling, 1977; Hayashi,
1981). Figure 2.3 shows a plot of the variation of Mdisc with the mass of the central star.
These masses are likely to be underestimates as there will be hidden mass in large, difficult-
to-detect grains (Draine, 2006). There are also not enough massive discs observed in nearby
7The minimum mass solar nebula is a PP disc that contains the minimum amount of solids to build the
planets of the Solar system, by no means an agreed quantity (e.g. Crida, 2009).
2.3 Observational properties of discs 11
Figure 2.3 Variation of Mdisc with M∗, from Williams and Cieza (2011). Almost all the discs
with M∗ = 0.04− 10M⊙ have Mdisc/M∗ = 10−3 − 10−1.
star-forming regions to account for the amount of giant exoplanets (Greaves and Rice, 2010)
which also implies a mass underestimate. Furthermore, we expect Mdisc to decrease with time
due to both accretion onto the central star and mass loss through jets and winds (see Figure 2.4
and Section 2.3.9).
2.3.4 Disc size
Constraining the disc size requires the ability to resolve it, made even more difficult as the
outer regions of discs are cool and produce weak emission. Observations in Taurus by Andrews
and Williams (2007) found disc sizes of a few hundreds up to 1000AU, agreeing with the back-
of-the-envelope calculation using the Jeans length in Section 2.2. We expect discs around more
massive stars to be bigger (Andrews et al., 2009, 2010) and for the disc to spread viscously as
the system evolves (the necessity of matter accreting onto the central star means some matter
must carry the excess angular momentum outwards, see Section 2.4.6, Pringle, 1981).
12 PP discs as an environment for planet formation
Figure 2.4 The flared PP disc around the young star HH 30 in Taurus, an edge-on disc which
appears as a flattened cloud of dust split into two halves by a dark region, with H ≈ 6.3AU
at 50AU (Watson and Stapelfeldt, 2004). The two red jets could be produced by the star’s
magnetic field (Stapelfeldt et al., 1999) channelling gas from the disc along the rotation axis
(Hubble, NASA).
2.3.5 Disc geometry and scale height
The disc scale height H depends on the balance between thermal pressure in the disc and
the vertical component of gravity due to the central star (derivation in Section 2.4.4). Discs
flare gently, with an inner edge at some distance from the star (Kenyon and Hartmann, 1987;
Beckwith, 1999) so the scale height depends on radius from the star, H = H(r). Direct
evidence for this comes from Hubble images of disc silhouettes in both Taurus and Orion (e.g.
Burrows et al., 1996; Smith et al., 2005). For example, Figure 2.4 shows a Hubble image of
the flared disc around HH 30 which has H ≈ 6.3AU at 50AU (Watson and Stapelfeldt, 2004).
By comparison, the rather more flattened disc around HK Tau B has H ≈ 3.8AU at 50AU
(Stapelfeldt et al., 1998). Most observations are from dust at large vertical heights, leading to
a H/r ≃ 0.1. H/r = 0.05 is a more appropriate value for the bulk of the disc mass distribution.
In Section 2.4.2 we go into more detail about the thin disc approximation that leads on
from this.
2.3.6 Velocities in the disc
We saw in Section 2.2 that we expect gas and dust velocities in PP discs to be approximately
Keplerian, confirmed by the observations of Hughes et al. (2011). This study, with observations
2.3 Observational properties of discs 13
of various discs at high spectral resolution, found subsonic turbulence, near–perfect Keplerian
rotation profiles and disc self–gravity not appearing to be significant (Section 3.4.1).
However, there is some evidence of deviation from Keplerian velocities, with Grady et al.
(1999), Fukagawa et al. (2004) and Corder et al. (2005) finding evidence of spiral structures
and Lin et al. (2006); Piétu et al. (2005) showing streaming motions along spiral arms. This
was later confirmed by images from the Subaru telescope showing spiral structure in the disc
around SAO 206462 (Figure 3.1a, Section 3.1.2).
2.3.7 Temperature, cooling and heating
The relatively large size of discs means that temperatures vary considerably within them, with
inner regions reaching a few 103K and the outer disc at a few 10K. The temperature profile
is important as it sets a typical scale for the disc thickness H. It also drives opacity and
ionization and therefore disc emission signatures.
The disc cools mainly via thermal emission from dust grains (Dullemond et al., 2007); the
large surface area of the disc means this is particularly efficient and thus we can model PP discs
as isothermal in z. Simultaneously, discs are heated via absorption of direct stellar radiation
and the viscous dissipation of gravitational energy in the disc due to accretion. For most discs,
radiation dominates the heating, except in the inner-most regions of the disc and in strongly
accreting periods such as outbursts.
The snow line is the inner edge of the region where the temperature falls below the con-
densation temperature of water (Lecar et al., 2006). It moves inwards as the disc accretion
rate drops with time and it lies within the planet forming region 0.1 ≲ r ≲ 5 − 50AU for at
least some fraction of the protoplanetary disc phase (Martin and Livio, 2012). The heavier
silicates and metallic materials are better suited to condense at higher temperatures and thus
inside the snow line we see formation of the inner, terrestrial planets made almost entirely of
rock and metal. Outside the snow line, hydrogen compounds condense into relatively sticky
ices which can rapidly form large, icy cores around which gas giants can form.
2.3.8 Opacity and ionisation
In PP discs, dust is the dominant opacity source and thus it dominates observational char-
acteristics. Only in the innermost regions of the disc where T ≃ 1500K and dust particles
are destroyed do other sources dominate the opacity. At any point in the disc the total dust
opacity will be a function of disc temperature, chemical composition and the size distribution
of the particles (Beckwith et al., 1990).
Temperatures in the majority of the PP disc are not high enough to fully ionise the gas
so at first approximation we can assume that PP discs are neutral. However, even negligible
ionisation fractions suffice to couple magnetic fields dynamically to the gas (Wardle, 2007),
which may be critical for particular angular momentum transport mechanisms (Section 2.4.6).
14 PP discs as an environment for planet formation
2.3.9 Accretion rates
Accretion rates are estimated from the strength of emission lines emitted as gas hits the star.
This is therefore the accretion rate M˙ as the gas hits the star and not necessarily the accretion
rate at larger radii; M˙ could differ considerably with the presence of matter sinks in the disc
such as a wind or a massive planet which could act as a dam for accretion flow by opening up
a gap in the disc.
During the initial collapse of part of a GMC, the accretion rate is initially very high
M˙ ∼ 10−6 − 10−5M⊙/yr (Hartmann et al., 1993), with most of the mass accreted onto the
YSO while in this embedded (Class 0) phase. This quickly drops to M˙ ∼ 10−9 − 10−7M⊙/yr
for Solar type T Tauri stars once the infall phase is over (Nakamoto and Nakagawa, 1994;
Hueso and Guillot, 2005).
2.3.10 Fate of the gas disc
The depletion of the gaseous PP disc due to accretion onto the central star is predicted to
be a gradual process taking place over a few viscous timescales, τν (Section 2.4.6). However,
the relative scarcity of systems transitioning from Class II to Class III YSOs suggest that
dispersal phase of PP discs is fast, ∼ 105 years (Simon and Prato, 1995; Wolk and Walter,
1996). This implies that there must be other physical processes contributing to loss of gas
from the disc. The most likely cause is photoevaporation, where UV or X-ray radiation, from
either the central star or an ambient radiation field, heats the disc surface to the point where
it becomes hot enough to escape the gravitational potential as a thermally driven wind (Clarke
et al., 2001; Alexander et al., 2006a,b). This occurs for the duration of the disc’s lifetime but
only becomes the dominant source of mass loss when the viscous accretion rate falls below the
photoevaporation rate (Dullemond et al., 2007).
2.4 Disc structure and evolution
We have established that most knowledge about PP discs has been derived from spatially
unresolved IR SEDs in nearby star forming regions. Interpreting these to extract the disc
properties we have seen requires the use of theoretical models.
2.4.1 Overview of disc evolution
At this point it is instructive to give an overview of the evolution of a disc in the most general
sense, from its formation shortly after the infall of a GMC to a debris disc and possible planetary
system around a main sequence, adult star. A summary of this can be seen in Figure 2.5.
We have seen that discs form almost immediately after the freefall collapse of a GMC as
a natural consequence of its specific angular momentum. They are unstable at early times,
2.4 Disc structure and evolution 15
(a) Massive flared disc around a Class I YSO
(b) Settled disc around a Class II YSO
(c) Photoevaporating or transitional disc
(d) Debris disc around a Class III YSO
Figure 2.5 Evolution of a general PP disc, based on a diagram in Williams and Cieza (2011).
For details see Section 2.4.1.
16 PP discs as an environment for planet formation
accreting in bursts and with mass outflows along jets. After ∼ 104 − 105yr the envelope is
accreted and a quasi-stable, flared disc exists, losing mass through accretion onto the star and
photoevaporation of the outer disc (Figure 2.5a). During this and the initial phase, matter
will spread outwards to r ≳ 100AU (Nakamoto and Nakagawa, 1994; Hueso and Guillot, 2005)
due to viscous spreading, as for accretion to occur some matter must carry angular momentum
outwards (Lynden-Bell and Pringle, 1974). This spreading only stops when photoevaporation
truncates the disc (Scally and Clarke, 2001).
During this protoplanetary phase, primordial dust present in the disc agglomerates into
larger bodies, reducing the dust scale height Hd and the flared disc becomes flatter (Fig-
ure 2.5b). Solids from dust to planets settle into the midplane of the disc and can migrate
inwards under gas drag (see Section 3.3).
The gradual process of accretion-driven gas depletion in the disc continues until the accre-
tion rate drops below the photoevaporation rate, whereupon the disc is rapidly eroded from
inside to out and we enter the transitional disc phase. With the outer disc unable to resupply
the inner disc with material an inner hole is formed in the disc and accretion halts. The rest
of the gas disc then dissipates. This transitional phase is very rapid, occurring on a timescale
of a few 105yr (Figure 2.5c).
Once the gas disc photoevaporates the a ≲ 1µm sized dust grains are blown off by radiation
pressure. Slightly larger ones spiral inwards due to Poynting-Robertson effect and eventually
evaporate. This leaves a gas-poor debris disc containing large grains, planetesimals and planets
(Figure 2.5d). These are not always detectable and the lifetime of such a disc is poorly
constrained (Wyatt, 2008).
2.4.2 Thin disc approximation
Modelling our PP disc requires some assumptions and approximations to be made, the first of
which is the thin disc approximation. In Section 2.3.5 we found the disc scale height H to be
H/r ≲ 0.1, which allows us to treat the disc to first approximation as infinitesimally thin, or
at the very least that vertical thickness of disc much less than the orbital radius. We can then
introduce the small quantity H/r ≪ 1 into our equations of motion. Under this approximation,
most equations can then simply be integrated in the vertical direction.
This approximation follows from the fact that a disc has a large surface area and thus can
cool via radiative losses efficiently. Efficient cooling implies relatively low disc pressure and
temperatures which are unable to support the gas against gravity except in a geometrically
thin disc (Pringle, 1981).
2.4.3 Gas dynamics of viscous discs
We have see that gas evolution in PP discs is driven by viscous accretion and photoevaporation
while grain growth, settling and radial drift (intimately connected processes, see Section 3.3),
2.4 Disc structure and evolution 17
drive dust evolution. However, for the moment we will ignore the effect of dust altogether and
consider the dynamics of disc gas.
The evolution of the gas disc can be described by the continuity and Navier-Stokes equa-
tions:
∂ρ
∂t
+∇ · (ρv) = 0 (2.4.1)
∂v
∂t
+ v · ∇v = −1
ρ
(∇P −∇ · σ)−∇Φgr, (2.4.2)
where ρ is the density, v denotes the gas velocity and P the pressure distribution. The
gravitational potential is denoted by Φgr which, in most cases, is dominated by the gravitational
attraction from the central star (see Section 3.4.1 for when this doesn’t apply). Therefore
Φgr = −GM∗
R
(2.4.3)
where G is the gravitational constant, M∗ the mass of the central star and R the spherical
radius (to distinguish it from the cylindrical radius r). Finally, the stress tensor σ describes
other physical effects such as viscous forces, magnetic fields, self-gravity etc.
The geometry of the disc means we adopt a cylindrical polar coordinate system (r, ϕ, z)
centred on the star. It is also assumed that as a first approximation the disc is axisymmetric
so no fluid quantities depend on the azimuthal angle ϕ. We define the surface density Σ(r, t)
as the density per unit surface area, the vertically and azimuthally averaged density under the
thin disc approximation, equal to
Σ(r, t) = 12π
∫∫
ρdzdϕ. (2.4.4)
Integrating equation (2.4.1) in the vertical direction (e.g. Pringle, 1981) gives
∂Σ
∂t
+ 1
r
∂
∂r
(rΣvr) = 0, (2.4.5)
where vr is the radial velocity. In its simplest form, the stress tensor σ can be assumed to be
the classical shear viscosity, whereby its only nonzero component is
σrϕ = ρνr
dΩ
dr
, (2.4.6)
where ν is the kinematic viscosity, Ω = vϕ/r is the angular velocity and rdΩ/dr is the rate of
strain of the flow. This is a fundamental property of the flow, so much so that we define the
18 PP discs as an environment for planet formation
shear rate, S8:
S = −rdΩ
dr
(2.4.7)
so that σrϕ = −ρνS. For particles in Keplerian motion, the base state we expect our gas to be
in from Sections 2.2 and 2.3.6, we note that
Ω =
√
GM∗
r3
⇒ S = 32Ω. (2.4.8)
A Keplerian flow is therefore also a shearing flow, with fluid closer to the star moving faster
than material in the outer disc.
From the three components of equation (2.4.2) we can now derive various features of PP
discs, namely the centrifugal balance from the radial component, hydrostatic equilibrium in
the vertical direction and angular momentum conservation from the azimuthal component.
2.4.4 Vertical structure
We found in Section 2.3.3 that it is safe to make the assumption that Mdisc ≪ M∗ and thus
we begin by neglecting the gravitational potential of the disc. This is marginal for some of the
most massive discs with Mdisc ≃ 0.1M⊙ and fails completely at very early epochs, but for now
assume Mdisc ≪ 0.1M⊙ as this is a perfectly reasonable assumption during the PP disc phase.
We will cover disc self-gravity in Section 3.4.1.
Therefore, considering the vertical component of equation (2.4.2), we neglect the left hand
side altogether, as confining the disc to the equatorial plane in the thin disc approximation
means that vz ≪ 1. Therefore in the z-direction, equation (2.4.2) reduces to
1
ρ
∂P
∂z
= −∂Φgr
∂z
= −GM∗z
R3/2
= − GM∗z
(r2 + z2)3/2
. (2.4.9)
In the case of an isothermal disc (a reasonable assumption in the thin disc approximation
H/r ≪ 1)
P = c2sρ (2.4.10)
where c2s is the isothermal sound speed. In the thin disc approximation, z ≪ r and r ∼ R and
so
gz =
GM∗z
(r2 + z2)3/2
≃ Ω2z (2.4.11)
where Ω is the Keplerian angular velocity (from now on this will always be the case unless
otherwise stated). We define vK = Ωr to be the Keplerian (azimuthal) velocity. The differential
8Throughout this work we will retain both S and Ω, despite the simple relation between them, as they
parametrise different effects; shear and rotation.
2.4 Disc structure and evolution 19
equation (2.4.9) is then easily solved to find
ρ = ρ0 exp
(
−Ω
2z2
2c2s
)
= ρ0 exp
(
− z
2
2H2
)
(2.4.12)
where ρ0 = 1√2π
Σ
H is the midplane (z = 0) density and we’ve found the disc scale height
H ≡ csΩ . (2.4.13)
This gives the neat equivalence
H
r
= cs
vK
= 1M , (2.4.14)
where
M = vϕ
cs
≈ vK
cs
, (2.4.15)
is the Mach number. The expected derivation of vϕ from vK will be derived in the next
Section, 2.4.5. Thus, requiring the disc to be thin is the same as requiring the disc rotation to
be supersonic, which is consistent as a thin disc allows efficient cooling.
2.4.5 Radial balance
The radial component of equation (2.4.2) gives the radial momentum equation:
∂vr
∂t
+ vr
∂vr
∂r
− v
2
ϕ
r
= −1
ρ
∂P
∂r
− ∂Φgr
∂r
. (2.4.16)
Using the thin disc approximation H/r ≪ 1 we can deduce from equation (2.4.14) that
cs (≡ ΩH)≪ vϕ (= Ωr), while the condition that accretion must occur over a long timescale
(as seen in Section 2.3.9) implies that vr ≪ cs. Therefore, an important ordering of velocities
due to the thin disc approximation arises:
vr ≪ cs ≪ vϕ. (2.4.17)
Under the assumption that the gas is isothermal, the supersonic nature of vϕ means that we
can neglect the first two terms in equation (2.4.16) and
v2ϕ
r
∼ c
2
s
r
+ GM∗
r2
=
v2ϕ
M2r +
GM∗
r2
(2.4.18)
where we have also made the approximation dΩ/dR ≈ dΩ/dr (as in Section 2.4.4). Thus:
vϕ =
√
GM∗
r
[
1−O
( 1
M2
)]
= vK(1− η), (2.4.19)
20 PP discs as an environment for planet formation
where vK is the Keplerian velocity and η ∼ (H/r)2 is small (η is a dimensionless measure of
pressure support). This small deviation from Keplerian flow has profound effects for the radial
drift of dust, discussed later in Section 3.3.4.
Therefore, to first approximation the flow is Keplerian and radial equilibrium is simply
centrifugal balance. The pressure near disc midplane normally decreases outwards since discs
are generally hotter and denser closer to the YSO, so 1ρ
dP
dr is generally negative and vϕ is
slightly less than the Keplerian velocity.
We also note that the specific angular momentum of gas in a Keplerian disc is
j = rvϕ =
√
GM∗r (2.4.20)
is increasing function of radius. Thus for gas to move inwards and be accreted by the star
it needs to lose angular momentum. Understanding the mechanisms that result in angular
momentum loss is a central problem in accretion disc physics (Papaloizou and Lin, 1995).
Finally, Keplerian discs are also Rayleigh stable. Rayleigh (1917) found that a rotating flow
in a non-magnetised, non-self gravitating disc is stable to infinitesimal axisymmetric hydro-
dynamic perturbations if and only if the specific angular momentum j increases with radius9:
dj
dr
= d
dr
(r2Ω) > 0. (2.4.21)
A Keplerian flow has j ∝ √r so it is predicted to be hydrodynamically stable.
2.4.6 The surface density evolution equation and viscous disc models
Finally, we consider the azimuthal component of equation (2.4.2) to derive an evolution equa-
tion for the surface density Σ(r, t). For the sake of brevity, we skip the bulk of the derivation
(see e.g. Ogilvie (2005) for details) and, after some algebra:
∂Σ
∂t
+ 1
r
∂
∂r
{(
dj
dr
)−1 [ ∂
∂r
(
νΣ r3dΩ
dr
)
+ r(T − Sj)
]}
= S (2.4.22)
where T (r, t) parametrise external torques (e.g. tides or magnetic effects on the inner disc)
and S(r, t) are mass sources such as jets or infall from the envelope at early times. Again, the
specific angular momentum of the flow is j = r2Ω. We define an internal torque G(r, t) due to
the viscosity of the fluid, parametrised via the vertically averaged kinematic viscosity ν (von
Weizsäcker, 1948):
G = 12π
∫∫
rσrϕ dϕdz = νΣ r3
dΩ
dr
. (2.4.23)
This is assumed to be positive as ν ≥ 0 so angular momentum transport is outwards. For
Keplerian orbits and in the absence of external torques or mass sources/sinks T = S = 0, so
9For a derivation see e.g. Pringle and King (2007).
2.4 Disc structure and evolution 21
equation (2.4.22) becomes
∂Σ
∂t
= 3
r
∂
∂r
[
r1/2
∂
∂r
(
νΣr1/2
)]
. (2.4.24)
This is a key equation in accretion disc evolution. First presented by von Weizsäcker (1944),
there was a general solution found by Lüst (1952)10 with the first full treatment in the landmark
work of Lynden-Bell and Pringle (1974).
For a constant ν there exists an analytic solution to this diffusion equation which shows
that a ring of matter will spread viscously, as is observed in PP discs (Section 2.4.1); the bulk
of the mass flows inwards while a small proportion of the mass carries angular momentum
outwards (Pringle, 1981).
From this equation we also get a viscous timescale
τν ∼ r
2
ν
(2.4.25)
which for PP discs around Solar-type stars is around 1Myr. Thus the evolution of Σ(r, t) is
on a timescale equivalent to the disc lifetime τdisc so, to a good first approximation, our disc
is steady.
The study of steady accretion discs began with the work of Lynden-Bell (1969), applied to
galactic discs. The so-called α-disc model was devised by Shakura and Sunyaev (1973) to create
realistic disc models while avoiding solving the viscosity problem in detail (i.e. determining the
mechanisms driving it). This was done by introducing the α–prescription for the disc viscosity,
writing the vertically averaged turbulent viscosity throughout the entire disc as
ν = ανcsH. (2.4.26)
αν is a dimensionless parameter measuring the efficiency of angular momentum transport due
to turbulence, which we give the subscript ν to prevent confusion with an α used later in this
work. In reality, αν may vary with temperature, density and composition of disc gas and is
not well constrained (Papaloizou and Lin, 1995).
Angular momentum transport is thought to be mainly driven by magnetohydrodynamic
(MHD) turbulence as a result of the magnetorotational instability (Velikhov, 1959; Chan-
drasekhar, 1960; Balbus and Hawley, 1991; Balbus and Hawley, 1998). This instability arises
as coupling a magnetic field to the disc supplies additional degrees of freedom. The presence
of a poloidal magnetic field modifies the Rayleigh criterion (Section 2.4.5) such that instability
requires dΩ2/dr < 0, which a Keplerian flow satisfies. Typically, αν ≃ 10−2 for sufficiently
ionized discs atmospheres, while αν ≲ 10−4 in the MRI dead zone in the midplane (Gammie,
10The solution was dedicated to Heisenberg in honour of his 50th birthday. Posterity does not tell us if
Heisenberg would’ve preferred socks.
22 PP discs as an environment for planet formation
1996).
Other contributions to αν could be from hydromagnetic winds (Blandford and Payne, 1982),
external tidal forces from binary companions (Papaloizou and Pringle, 1977) or embedded
protoplanets (Lin and Papaloizou, 1993), disc self-gravity (Lynden-Bell and Kalnajs, 1972;
Toomre, 1964, 1981), nonlinear or transient instabilities in purely hydrodynamic flows (Lin
and Papaloizou, 1980), vortices and their interactions (Adams and Watkins, 1995; Godon and
Livio, 1999b) or the density waves produced by large scale vortices (Johnson and Gammie,
2005). We return to the subject of vortices in Section 3.5.
Detailed models of viscous accretion (Pringle, 1974; Lynden-Bell and Pringle, 1974; Hart-
mann et al., 1998) are consistent with observational constraints for disc masses, sizes and the
decrease of accretion rates over time (Hartmann et al., 1998; Hueso and Guillot, 2005). They
are however just first order approximations in more complex evolution.
The need for accurate models of Σ(r, t) is crucial for the study of PP discs as average radial,
vertical and velocity profiles can only be determined from resolved observations and models of
the surface density.
2.4.7 Important timescales governing disc evolution
From the surface density evolution equation (2.4.24) we found the viscous timescale τν ∼ r2/ν.
There are however other important timescales determining the evolution of PP discs. Orbital
motion occurs on a timescale Ω−1 = H/cs (equation (2.4.13)) so is the same as the sound
crossing time – the time to achieve (vertical) hydrostatic equilibrium. We therefore define the
fundamental orbital or dynamical timescale
τdyn ∼ Ω−1 (2.4.27)
which obeys the empirical relation (Heng and Kenyon, 2010)
τdyn ≈ 0.2 yr
(
M∗
M⊙
)−1/2 ( r
AU
)3/2
. (2.4.28)
It is convenient to compare the sizes of other disc timescales to this quantity. For instance,
the viscous timescale τν from Section 2.4.6 (the timescale on which matter diffuses through
the disc under the effect of viscous torques) is related to τdyn by
τν ∼ r
2
ν
∼ r
2
ανcsH
∼ r
2
ανΩH2
∼ 1
αν
(
H
r
)−2
τdyn. (2.4.29)
Thus for a thin disc with subsonic turbulence, αν < 1, we have τdyn ≪ τν . Similarly, we can
find a local thermal timescale on which the disc cools (Pringle, 1981), given by the ratio of the
2.5 Summary and conclusions 23
Property Size
mass 10−3 − 10−1M∗
size 102 − 104AU
thickness H/r ≪ 1, typically H/r ≃ 0.05
temperature 10K (outer disc) – 1500K (inner regions)
accretion rate
M˙ (M⊙/yr)
∼ 10−6 − 10−5 (embedded phase),
Hartmann et al. (1993)
∼ 10−11 (Brown dwarfs),
Alexander and Armitage (2006)
∼ 10−9 − 10−7 (T-Tauri, M∗ ∼M⊙),
Nakamoto and Nakagawa (1994)
∼ 10−5 (massive stars, M∗ > 8M⊙),
Alexander and Armitage (2006)
dust fraction, Γd 0.01− 0.02Mdisc
Table 2.1 Summary of observational PP disc properties given in this chapter.
thermal energy of the gas, Σc2s, to the rate of viscous heating, νΣΩ2:
τcool = τth ∼ c
2
sΣ
νΣΩ2 ∼
cs
ανHΩ2
∼ α−1τdyn. (2.4.30)
We therefore find that (Lynden-Bell and Pringle, 1974; Pringle, 1981)
τdyn < τth ≪ τν , (2.4.31)
so centrifugal balance in the radial direction (orbital timescale) and hydrostatic balance in
the vertical direction (dynamical timescale) are very rapidly achieved, followed by the disc
temperature evolving on a longer timescale while the evolution of Σ(r, t) is very slow indeed.
Also note that all three timescales are functions of disc radius, scaling like r3/2 (H/r ≃ const).
Thus the evolution of the inner disc is generally much more rapid than that in the outer disc.
2.5 Summary and conclusions
A summary of important physical properties deduced from the observations of PP discs can
be found in Table 2.1, while one summarising the timescales found in PP disc structure and
evolution can be found in Table 3.3.
24 PP discs as an environment for planet formation
We have found that discs are a natural consequence of star formation and, consequently,
are expected to occur around most protostars. They should therefore not be thought of as
particularly special nor uncommon objects. In the protoplanetary phase, they are geometrically
thin structures which, to first approximation, we can treat as isothermal and 2D; the ratio
H/r ≪ 1 is a useful small quantity for scaling equations of motion.
These PP discs form rapidly (104 − 105 yr), persist for around 1Myr and are efficiently
cleared by photoevaporation ∼ 105 yr. In astronomical terms, 1Myr is not a long time11,
imposing a relatively small window for planet formation.
Gas in the disc orbits the star with approximately Keplerian velocity so we can consider
PP discs to also be a Keplerian disc with an intrinsic shearing flow. We will find in the next
chapter that the slightly sub–Keplerian orbital speed achieved by the disc gas has profound
effects when we consider the role of dust.
Circumstellar discs do not just contain gas: they initially have a dust fraction Γd ≃ 0.01
inherited from the ISM. Dust dominates the opacity and therefore most of the observational
properties of discs but due to the low dust–to–gas ratio Γd, the dynamics of the disc are driven
mainly by the gas component. At a first approximation we can therefore consider the disc to
be a single fluid, an approach which produces self–consistent disc models.
Up to this point we have largely neglected the role of dust grains in PP disc evolution. Since
this dust is the very matter we build planetesimals and planets out of, we will now consider it
in more detail.
11The lifetime of a 1M⊙ star is around 11,000Myr.
Chapter 3
The role of dust in protoplanetary
discs
In the previous chapter we reviewed the observational properties of PP discs and the models
frequently used to describe their structure and evolution. We now examine the role of dust in
these discs; on the evolution of the disc itself, the mechanism for building ∼ 104km planets
from ∼ 0.1µm dust and the severe barriers to overcome for this to occur, namely the ‘metre
gap’. Finally, we will investigate how local pressure maxima in vortices offer a promising route
around these problems.
In Section 3.1 we will review the role of dust in protoplanetary disc evolution and the
evidence for its existence. After briefly discussing where planetesimal formation fits into a
broader planet formation (both terrestrial and gas giant), in Section 3.2 we will investigate
possible mechanisms of planetesimal formation in Section 3.3. In order to build planetesimals
we need some mechanism of grain growth (Section 3.3.1) and to understand how it interacts
with the surrounding gas (Section 3.3.2). The key effects of this aerodynamic drag and how
this creates a ‘metre gap’ are discussed in Sections 3.3.3 and 3.3.4.
We discuss the Safronov–Goldreich–Ward (SGW) mechanism as a planetesimal and planet
forming mechanism that can bypass the metre gap in Section 3.4, its shortfalls (Section 3.4.2)
and possible routes around these problems (Section 3.4.3). The most promising of these
is the role of vortices, producing both density enhancements which can locally support the
SGW mechanism (Section 3.5) and trapping dust to help build planetesimals within the strin-
gent timescales imposed (Section 3.5.1). Finally, we investigate how to produce these struc-
tures (Section 3.5.2), how long they can persist (Section 3.5.3), the various instabilities which
threaten them (Section 3.5.4) and how to model them (Section 3.5.5). There is a summary of
the timescales governing PP discs in in Table 3.3.
26 The role of dust in PP discs
3.1 Dust in PP discs
3.1.1 A note on definitions
In the literature, dust can mean particles anything from a ≃ 0.1µm to a ≃ 1m in size. Different
definitions of rocks, boulders, pebbles etc. will only serve to confuse so we will just refer their
radius, a, instead.
Throughout, we will adopt the definition of a planetesimal1 that was agreed at the workshop
organised by Dullemond and Klahr (2006):
“A planetesimal is a solid object arising during the accumulation of planets
whose internal strength is dominated by self-gravity and whose orbital dynamics
is not significantly affected by gas drag. This corresponds to objects larger than
approximately 1 km in the solar nebula.”
The definition of a protoplanet is less well constrained, requiring that the body not only be
bound by self-gravitation but also change the path of the rocks a few radii away from them.
In reality this means a ≃ 100− 1000km. They are also known as protoplanetary embryos. For
completeness, the IAU definition of a planet is
“[...] a celestial body that: (a) is in orbit around the Sun, (b) has sufficient mass
for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic
equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its
orbit.”
The planet forming region is anywhere solid and/or gas can accumulate. This excludes
regions very close to the star (≲ 0.1AU) where it is too hot for solids to exist or for gravitational
instability (GI) to occur. The outer limit is more difficult to constrain (the literature gives
anything from 5-50AU) but since the outer disc contains less processed dust (Section 3.1.2)
with longer collisions times, it is of less interest for planet formation.
3.1.2 Evidence of dust in PP discs
Discs have dust features seen in emission spectra from their optically thin outer layers heated
by stellar radiation (Calvet et al., 1991; Chiang and Goldreich, 1997). As we have seen, the
absorption and emission profiles of PP discs are overwhelmingly dominated by this trace dust
component (Semenov et al., 2003).
The composition of dust grains in Class 0 and I YSOs is similar to the a ≃ 0.1µm dust
in the ISM, with both the models of Kruegel and Siebenmorgen (1994) and observations of
Beckwith and Sargent (1991) confirming this. In discs around more evolved objects we find
1The word planetesimal was first used by Chamberlin (1900) for solid bodies that had subsequently accumu-
lated into planets.
3.1 Dust in PP discs 27
(a) Spiral structure and dust cavity (NAOJ/Subaru).
(b) Dust in a large vortex (Pérez et al. 2014,
ESO/ALMA).
(c) Possible positions of the vortex, planet and
spiral waves.
Figure 3.1 The spiral structure observed in the disc around SAO 206462 by Muto et al.
(2012) was the first clear case of spiral arms around an individual star. The disc extends to
140AU, with evidence of a dust depleted cavity at r ≲ 46AU. Pérez et al. (2014) later observed
signatures of an asymmetric dust concentration (Figure 3.1b) which suggests the spiral gravity
waves are produced by the vortex, which is in turn induced by a planet.
28 The role of dust in PP discs
Figure 3.2 A large dusty vortex in the system SR21 (Pérez et al. 2014, ESO/ALMA).
large masses of dust particles a ≃ 0.1 − 1cm (Prosser et al., 1994) and signatures of dust
processing and coagulation (Kessler-Silacci et al., 2007, review Natta et al., 2007). The best
evidence for grain growth to a ≃ 1cm sizes is from the detection of 3.5cm dust emission in the
face-on disc around TW Hya (Wilner et al., 2005). Frustratingly, the radiative inefficiency of
a ≳ 1cm objects makes it difficult to observe when planetesimals actually form.
From observations of silhouette discs there is some evidence for the vertical stratification
of grains, with the smallest a ≃ 0.1µm closest to the disc’s surface. There is also some radial
dependence, with grains in the outer disc less processed then those closer to the star (van
Boekel et al., 2004).
Reassuringly, the typical size of dust particles also appears to increase on a timescale ≃ τdisc
(D’Alessio et al., 2001; Shuping et al., 2003; McCabe et al., 2003). However, there is however
a large amount of variation in dust properties among different sources, with no correlation
between grain size, a, and stellar properties such as M∗, luminosity or age, nor with the
properties of the disc such as its mass Mdisc or accretion rate M˙ .
Furthermore, small dust a ≃ 0.1µm − 1cm is seen to persist for periods ≃ τdisc. This
survival of a large mass of small grains to large times is a significant challenge to planet
formation theory as it suggests that the process of planetesimal formation is less efficient than
predicted by models, or, if forming planetesimals from small dust is as fast as predicted (see
Section 3.3.1), it must be at the end of a long quiescent phase. Another option is that the
3.1 Dust in PP discs 29
grain growth process is in fact slow (with inefficiency due to fragmentation and bouncing), but
as we shall see in Section 3.3.4 this is a problem for some intermediate grain sizes due to rapid
radial drift.
Finally, as predicted (e.g. Wolf and Klahr, 2002), telescopes have recently begun to resolve
asymmetric dust features in discs (Andrews et al., 2011; Birnstiel et al., 2013; Casassus et al.,
2013; Fukagawa et al., 2013; Isella et al., 2013; van der Marel et al., 2013; Pérez et al.,
2014; Espaillat et al., 2014), possible signatures of dust trapping in large-scale vortices (see
Section 3.5). Observations are largely limited to transitional discs which are not shrouded
in optically thick material (Section 2.4.1) but there is no reason to believe that large scale
asymmetries are limited to these late-epoch discs.
Two examples from the systems SAO206462 and SR21 can be seen in Figures 3.1 and 3.2,
with the former also exhibiting spiral density waves and a cleared inner disc. This suggests
a combination of a planet and induced vortex. We expect a vortex with approximately sonic
velocities to produce density waves in this manner (Nelson et al., 2000), suggesting the con-
figuration displayed in Figure 3.1c. Figure 3.2 is another recent example of an observed large
vortex structure.
3.1.3 Importance of dust in PP discs
The evolution of dust in PP discs is governed by a plethora of physical processes such as growth
via coagulation, erosion through sublimation or collisional fragmentation, radiation pressure,
photoevaporation and photoionisation, interaction with magnetic fields and interactions with
the gas disc. It is this last effect that is the most dominant and the process in which we are
most interested.
As we saw in the last chapter, grains dominate disc emissions, therefore we need to under-
stand the dust in order to probe the thermal and geometric structure of the disc and understand
our observations. The dominant source of ionisation is the ionisation of heavy elements by high
energy stellar photons or cosmic rays (Glassgold et al., 2000; Fromang et al., 2002). These
heavy elements are locked up in dust grains and grain growth is an essential step in retaining
these heavy elements (Weidenschilling and Cuzzi, 1993). Grains also act to shield the disc
midplane from this radiation which could possibly lead to a central ‘dead zone’ where the MRI
cannot operate; this is important for angular momentum transport processes (Section 2.4.6).
There may however be weak hydrodynamic turbulence in the dead zone from vertical-shear
instability (Nelson et al., 2013). Also note that the layered disc model of Gammie (1996) has
the quiescent midplane dead zone bounded by MRI-active surface layers, so care needs to be
taken establishing where our models are applicable.
As we will see, solids also play a dynamical as well as chemical role in the disc (Section 3.3.2
and hereafter), modifying properties of turbulence, introducing two-fluid instabilities (Sec-
tion 3.4.2) and seeding gravitational instability in areas of particle overdensity (Section 3.4.1).
30 The role of dust in PP discs
Finally, dust is the source of matter from which planets are formed. Over thirteen orders of
magnitude separate a ≃ 0.1µm dust from terrestrial planets and many different areas of physics
play a role throughout this process. The very existence of the Solar system and the increasing
amount and diversity of exoplanet systems found2 (Weidenschilling and Cuzzi, 1993; Beckwith
et al., 2000) show that the planet formation process appears to be a robust one, albeit not
fully understood.
3.2 Beyond planetesimal formation: creating terrestrial and
gas giant planets
Modern theories of planet formation are based on the work of Safronov (1969) and can be di-
vided into three main stages: planetesimal formation (of which this thesis is chiefly concerned),
terrestrial planet formation and giant planet formation, with different physical processes dom-
inating each one. Returning to planetesimal formation in due course, we shall look ahead to
the last two stages.
Once planetesimals have been formed, they are (by definition) massive enough to be more-
or-less decoupled from the gas disc. The dominant physical process governing them is therefore
gravity. Terrestrial planet formation is a well–posed problem because of this, but still a chal-
lenging one due to the large number of bodies and long timescales involved. Starting from
large population of planetesimals, of uncertain dimensions, there are two distinct phases.
The first is runaway growth where gravitationally focussed pairwise collisions between plan-
etesimals results in progressively larger objects; unlike with collisions between tiny dust grains
(Section 3.3.1), the gravity of the bodies in these collisions is strong enough to assume most of
the mass will end up agglomerating into a larger single object. This is followed by a period of
oligarchic growth where a few large protoplanets consume planetesimals in their own indepen-
dent ‘feeding zones’ to make approximately earth mass cores M ≃M⊕ within τdisc (Chambers
and Wetherill, 1998; Goldreich et al., 2004).
In order to accrete a large envelope of gas, giant planets have to be formed before the gas
disc disperses in a time ≲ τdisc. Giant planet formation has two qualitatively different, but not
necessarily mutually exclusive, theories:
• Core accretion theory (Cameron, 1973; Pollack et al., 1996), whereby a core of ice and
rock acquires a giant envelope of gas, with the icy core formed in an identical process to
terrestrial planet formation. The timescale, and indeed viability of this model, depends
on how quickly the core is assembled and how rapidly the gas in the envelope can cool
and accrete onto the core (Papaloizou and Terquem, 2006).
2An up-to-date list of NASA’s Kepler exoplanet discoveries can be found at
http://kepler.nasa.gov/Mission/discoveries/.
3.3 Planetesimal formation 31
• Disc instability theory (Kuiper, 1951; Cameron, 1978; Boss, 1997) has giant planets
rapidly forming via the gravitational fragmentation of an unstable PP disc, similar to
Safronov–Goldreich–Ward theory of planetesimal formation (Section 3.4). It requires the
disc to cool rapidly on a timescale ≃ τdyn.
3.3 Planetesimal formation
We now go backwards in the planet formation process to look at the key processes governing
the creation of planetesimals from primordial dust. The mechanism for doing so is necessarily
robust, given how commonplace planets are, but finding this mechanism has proved to be
elusive.
3.3.1 Grain growth
Protoplanetary discs are dust-processing factories, building planets from tiny a ≃ 0.1µm grains
of icy, dusty particles into planetesimals, protoplanets and finally planets.
The hierarchical growth of dust grains occurs when pairwise collisions result in two particles
sticking and not bouncing nor fragmenting. The likelihood of a collision involves considering
detailed velocity distributions (e.g. Garaud et al., 2013; Hubbard, 2012, 2013), but once 1mm-
sized particles have formed and begun to settle out of the gas there is an order of magnitude
approximation for a collisional timescale (Heng and Tremaine, 2010):
τcollide ∼ aρ•ΣdΩ (3.3.1)
where Σd ≃ 0.01Σ is the surface density of solids and ρd is the density of the dust particles.
This corresponds to τcollide ≃ 0.005−0.5yr for a ≃ 1mm−10cm, while τcollide ≃ 5yr for a ≃ 1m
(Heng and Kenyon, 2010). We therefore expect collisions to be frequent on astrophysical
timescales.
Once a collision has taken place, it must then be determined whether it resulted in net
growth (sticking), no growth at all (bouncing) or, worst of all, negative growth (fragmentation).
The sticking efficiency in any given pairwise collision depends on the shape of the colliding
bodies, the porosity of the grains (how compacted they are, Okuzumi et al., 2012) and the
velocity of the collision (Windmark et al., 2012b) and is a field in its own right.
For small dust, growth via Brownian motion is reasonably well understood from both
laboratory and numerical experiments (Blum, 2004; Henning et al., 2006), with typical relative
velocities low enough for sticking. In general, sticking efficiencies tend to be high (10− 100%)
at small grain sizes a ≲ 1cm, and as relative velocities increase, the aggregates can compact
and runaway growth is possible. However, for a ≳ 1cm sticking becomes less efficient and
growth slows.
32 The role of dust in PP discs
Moreover, short range van der Waals interactions only explain the sticking of individual
grains at speeds < 1ms−1 (Dominik and Tielens, 1997), with grain collisions destructive at
speeds in excess of 10ms−1 (Dominik et al., 2007). There is also a bouncing barrier between
these two limits (Güttler et al., 2010; Zsom et al., 2010, Windmark et al., 2012a). For a ≳ 1m,
relative velocities are in excess of 50ms−1, above the destruction threshold (Wurm et al., 2005).
Thus the relative velocities of the particles regulate both collision rates and sticking prop-
erties. They are driven by the differential coupling to the disc gas with respect to particle
diameter a (Weidenschilling and Cuzzi, 1993; Beckwith et al., 2000, Section 3.3.2). Collision
speeds increase monotonically with a until decoupling occurs. Grain growth therefore cannot
be fully understood without an understanding of the interaction with the gas velocity field,
especially turbulence.
For example, Weidenschilling (1984) found that including turbulence initially increases
coagulation rates, with aggregates quickly reaching a ≃ 0.1− 1cm, before subsequent erosion,
fragmentation and collisional bouncing prevents growth to larger sizes. Turbulent stirring also
hinders dust settling to the midplane (Section 3.3.3). Furthermore, Nelson and Gressel (2010)
found that MRI turbulence could seed destructive collisions between dust grains.
Coagulation models of grain growth in PP discs find the sticking process to be extremely
efficient with rapid growth from a ≃ 0.1µm to a ≃ 1mm on a coagulation timescale (τcoag ≃
103 − 104yr) within a few AU of the star (Blum and Wurm, 2008; Zsom et al., 2010). This is
in agreement with grain properties observed in early disc evolution (e.g. Kessler-Silacci et al.,
2007).
One problem with these coagulation-only models is they predict similarly rapid growth be-
yond a ≃ 1cm which is inconsistent with observations. Furthermore, the sticking process alone
is so efficient that all particles with a ≲ 100µm are cleared from the disc in ≃ 104yr, contrary to
the presence of µm-sized dust observed in older discs (Dullemond and Dominik, 2005). Small
grains must therefore be replenished by some mechanism. The continual collisional fragmen-
tation of agglomerates (Dullemond and Dominik, 2008) or more inefficient sticking is a way to
maintain the observed population of small dust to large times (Brauer et al., 2008; Birnstiel
et al., 2011). However, the work of Garaud et al. (2013) finds that it is also possible to pro-
duce populations of both small and large particles by including both motion due to dust-gas
interaction (midplane settling and radial drift) and stochastic motion (from Brownian motion
and turbulence) without needing to overly rely on fragmentation.
The scenario whereby small particles stick to a few ‘runaway’ bodies does not work either.
The collision of small grains (tightly coupled to the sub-Keplerian gas) with large solids (largely
decoupled so orbiting at approximately vK) will face the full brunt of a headwind at speeds
comparable to a sandblaster3 which causes rapid radial drift as well as no net growth. This is
covered in more detail in Section 3.3.4.
3Eugene Chiang is to thank for this great analogy, Youdin (2010).
3.3 Planetesimal formation 33
In conclusion:
“The direct formation of kilometer-sized planetesimals cannot (yet?) be under-
stood via sticking collisions.” – Blum and Wurm (2008)
3.3.2 Aerodynamic drag
We’ve found that considering grain growth independent of gas dynamics in the disc gives
predictions inconsistent with observations. We must therefore try to understand the coupling
between the gas and dust, necessary not only for grain growth but the vertical distribution
and radial motion of dust and planetesimals. It is also not a one-sided arrangement; large
concentrations of dust will also have an impact on the dynamics of the gas.
We will begin by considering a fundamental quantity of the gas; its mean free path γmfp.
This characteristic lengthscale is the average distance travelled by a gas particle before en-
countering another. For the MMSN, Cuzzi et al. (1993) finds the expression
γmfp =
(
r
1AU
)11/4
cm. (3.3.2)
A good ballpark figure is γmfp ≃ 1m for typical disc conditions in the planet forming region.
This is important as the drag force exerted by the gas on a particle contained within it
depends on the size of the particle a compared to γmfp (Whipple, 1972). If a≪ γmfp then drag
is the result of random collisions of gas molecules with dust particles, known as the Epstein
regime (Epstein, 1924). At the other end of the scale, if a≫ γmfp then particles feel the gas as
a fluid and experiences drag through the wake it creates. This is known as the Stokes regime,
and drag is dependent on the Reynolds number, Re4.
Consider spherical grains of radius a, particle density ρ•, particle mass m• = 4πρ•a3/3 and
a macroscopic, fluid density ρd, moving with velocity vd. Their velocity relative to that of the
gas vg is therefore
u = vd − vg. (3.3.3)
In the Epstein regime, the drag force on a particle is given by
Fdrag = −m•
τs
u (3.3.4)
where the dust friction/stopping time is
τs =
ρda
ρgcs
. (3.3.5)
Gas density is denoted by ρg and cs is again the sound speed. Though a precise value of τs
4See e.g. Batchelor (2000), page 233.
34 The role of dust in PP discs
depends on the size of the dust, typical conditions at the midplane, at 1 AU for 1µm dust gives
τs ≈ 10 s≪ τdyn (Papaloizou and Terquem, 2006), well coupled to the gas.
We define the Stokes number Ts, a dimensionless stopping time
Ts = Ωτs =
τs
τdyn
. (3.3.6)
Then, rewriting equation (3.3.5) using equation (2.4.13) for the disc scale height:
a =
(
ρg
ρd
)
TsH ≃ (100Ts) cm ⇒ Ts ≃
(
a
100cm
)
, (3.3.7)
for typical values for the MMSN at 5AU (Section 2.3.3, 2.3.5, Papaloizou and Terquem, 2006).
In the Epstein regime where Ts ≪ 1, the evolution of the dust particle distribution can be
modelled as a pressureless fluid (Garaud and Lin, 2004).
Once a ≥ 9γmfp/4, we move to the Stokes regime and the drag force takes the form
Fdrag = −CD2 πa
2ρguu. (3.3.8)
The drag coefficient CD depends on how aerodynamic the particle is. For spherical particles,
as we are considering here, CD only depends on the Reynolds number, with
CD ≃

24Re−1 Re < 1
24Re−0.6 1 < Re < 800
0.44 Re > 800
(3.3.9)
(Weidenschilling, 1977). For large spherical bodies, acceleration via gas drag scales with a−1
so it eventually becomes negligible once bodies of planetesimal size have formed. We find that
the Stokes regime applies in the inner disc and Epstein in the outer regions, with the transition
region for specific particle size, a at
rc (a) =
(4
9
a
1cm
)4/11
AU (3.3.10)
(Cuzzi et al., 1993; Chavanis, 2000). In this transition region it is possible to use an interpo-
lation between the two regimes (see Woitke and Helling, 2003 and Paardekooper, 2007). We
find that the primordial a ≃ 0.1µm dust is effectively always in the Epstein regime while for
particles a ≃ 10 − 100cm, the critical radius is 1.7AU < rc < 3.9AU. This is of particular
importance when we consider the problem of radial drift in Section 3.3.4.
3.3 Planetesimal formation 35
3.3.3 Midplane settling
While growing, dust particles start to sediment due to the vertical component of the stellar
gravitational field, creating an efficient feedback mechanism for accelerating collisional growth
(e.g. Dullemond and Dominik, 2005; Schräpler and Henning, 2004).
As we’ve seen, small a ≃ 0.1µm grains have a large surface area to mass ratio and are
therefore strongly coupled to the gas. They collide and coalesce, forming grains with a smaller
surface area to mass ratio which experience greater gas drag and settle to the midplane.
This is interrupted by turbulence causing some degree of vertical stirring and mixing of grains
(Dullemond and Dominik, 2005).
Out of the plane of the disc, the gas is supported against gravity by a pressure gradient
but this does not act on dust particles (Weidenschilling, 1977). Therefore, neglecting any
turbulence in the disc for now, and considering small, tightly coupled dust particles we can
assume that any relative vertical velocity in the fluid is the dust velocity, uz ≃ vz,d since in
the vertical direction the gas is approximately in equilibrium (Section 2.4.4).
Balancing the z-component of gravity due to the central star with the gas drag we find the
dust particle’s terminal settling velocity to be vsettle = τsΩ2z (equation (2.4.11)), implying a
settling timescale of
τsettle =
z
vsettle
≃ 1Ω2
ρgcs
ρda
= 1ΩTs
, (3.3.11)
in terms of the Stoke’s number Ts. A more careful analysis (Youdin, 2010) gives
τsettle ≃ 1 + 2T
2
s
ΩTs
. (3.3.12)
If z ∼ H, we find a good first approximation at 1AU of
τsettle ≃

105yr a ≃ 1µm
104yr a ≃ 0.1− 1cm
103yr a ≃ 0.1− 1m,
(3.3.13)
uniformly short compared to the disc lifetime, τdisc (Chiang and Goldreich, 1997). Note that
this is strictly the minimum time to settle since vertical mixing by turbulence can hold particles
aloft.
Therefore, in absence the of turbulence we expect micron-sized dust to rapidly sediment
out of the upper layers of the disc. Since τsettle ∝ ρg, settling will be faster at high z where gas
is less dense (Section 2.4.4), and as τsettle ∝ (ρda)−1, any grain growth will hasten the settling
process.
In reality, small scale turbulence strongly reduces sedimentation (Weidenschilling, 1980;
Cuzzi et al., 1993; Dubrulle et al., 1995; Johansen and Klahr, 2005; Carballido et al., 2005;
36 The role of dust in PP discs
Turner et al., 2006 and Cuzzi et al., 2008) or can prevent settling occurring at all; Weiden-
schilling (1984) and Supulver and Lin (2000) found that particles a ≃ 0.1−1cm were too small
to settle in the presence of turbulence. Porous or fractal particles, which we would expect to
find as a result of coagulation, can also slow settling (Blum and Wurm, 2008; Zsom et al.,
2010).
3.3.4 Radial drift
The fact that dust particles do not experience the same pressure forces as the gas has an even
more profound effect on the radial dynamics of solids (Weidenschilling, 1977). As we found
in Section 2.4.5, gas in the disc orbits the star at a slightly sub-Keplerian velocity due to the
radial pressure gradient, with
vϕ,g = vK(1− η), (3.3.14)
where
η ≡ − 1
rΩ2ρg
∂P
∂r
≈
(
H
r
)2
∼ 10−3 (3.3.15)
(Takeuchi and Lin, 2002, 2005). In the planet forming region, the Ts ≪ 1, Epstein, limit
corresponds to a ≲ 1µm, dust well coupled to the gas. Particles in this regime migrate slowly
inwards at the same speed as the gas.
In the opposite Stokes regime limit Ts ≫ 1 (corresponding to a ≳ 1km), the bodies have
so much inertia that they barely experience the headwind and thus only migrate inwards very
slowly. Between these two limits we have Ts = 1, where a ≃ γmfp ≃ 1m.
Following the analysis of Adachi et al. (1976) and Weidenschilling (1977) and the notation
of Garaud and Lin (2004) and Youdin (2010) we find the particle drift speed
vr,d = −2TsηvK1 + T 2s
, (3.3.16)
so a particle with Stokes number Ts experiences a headwind of
− ηvK1 + T 2s
(3.3.17)
This implies a headwind −ηvK/2 and maximum inward drift speed of −ηvK for a Ts = 1
particle. This appears small, but the Keplerian velocity at 1AU from a M∗ = M⊙ star is
30kms−1. We therefore expect a ≃ 1m particles and 1AU moving at vK to experience a
headwind ∼ 100ms−1. It’s the sandblaster analogy again.
Figure 3.3 shows the inward radial drift velocity vr,d as a function of Ts and highlights this
problematic Ts ≃ 1 region. Note that Ts corresponds to different sized particles at different
distances from the star: smaller solids drift faster in the outer disc, while particles a ≃ 1m
drift the fastest at 1AU.
3.3 Planetesimal formation 37
Figure 3.3 Dust drift velocity as a function of stopping time Ts = τsΩ, for two different
Shakura-Sunyaev α, from the review paper of Alexander (2008). This clearly shows the radial
drift problem for Ts ≃ 1 particles, which will spiral into the central star in < 100yr.
We therefore find a drift timescale of
τdrift =
r
vr,d
= r(1 + T
2
s )
2TsηvK
= 1 + T
2
s
2ηTsΩ
. (3.3.18)
While the size of the fastest migrating particles varies with r, the peak drift speed does not
(Youdin, 2010):
τdrift,max ≡ rmax(vdrift) ≃ 85(r/AU)yr. (3.3.19)
In summary:
τdrift ∼
< 100τdyn if a ≃ 1m (Ts ≃ 1)> τdisc if a ≲ 1µm or a ≳ 1km (Ts ≪ 1 or Ts ≫ 1), (3.3.20)
with the shortest drift timescale at 1AU ≲ 100yr. These particles are not safe even if the mass
of dust at the midplane is sufficient to dominate the gas dynamics (so vϕ,g ≃ vK). In this case,
collective drag on the surface of the subdisc layer then dominates and τdrift ∼ 103yr for a ≃ 1m
at 1AU (Goldreich and Ward, 1973), not much better.
Therefore, growth through the a ≃ 0.01 − 1m scale must be exceptionally rapid else solid
38 The role of dust in PP discs
material will drift towards the star and be destroyed. To bypass this barrier would need
dust settling and growth on a timescale τcoag, τsettle < τdrift (Youdin, 2010): quite a serious
constraint.
3.4 The Safronov–Goldreich–Ward mechanism
As we have found, there are two major stumbling blocks in the process of creating planetesimals
from micron-sized dust. The first is that due to large relative velocities, fragmentation makes
the creation of bodies a ≳ 1m almost impossible under normal disc conditions. The second is
that the headwind experienced by these same bodies around 1AU quickly scrubs off angular
momentum and will rapidly deposit them into the central star. The combination of these two
problems is known as the metre gap5 and have been a major hurdle in planet formation theory
for some years.
So far we’ve made the two assumptions that dust particles are unimportant for the evo-
lution of the gas disc and that the only important reactions between particles are pairwise
collisions. However, these assumptions are only valid if the dust particles are both small and
distributed uniformly throughout the disc. The Safronov–Goldreich–Ward (hereafter SGW)
mechanism, proposed independently by Safronov (1969) and Goldreich and Ward (1973), has
the gravitational instability (GI) of a dust-rich subdisc in the midplane as a mechanism for
rapidly producing planetesimals to bypass the metre gap. It was also initially thought it could
bypass the complicated and problematic process of dust sticking altogether:
“...the fate of planetary accretion no longer appears to hinge on the stickiness
of the surface of dust particles” – Goldreich and Ward (1973)
though as we shall see, both processes are necessary. The mechanism can be outlined thus:
1. There is initially a well-mixed disc of gas and dust with a high Toomre parameter Q (see
Section 3.4.1). Effects of self-gravity have no role in disc evolution.
2. The dust settles to midplane, with some collisional growth assumed to have occurred to
overcome the effect of turbulent stirring (Section 3.3.3). Inward radial drift contributes
to midplane particle density in inner disc (Youdin and Shu, 2002).
3. High dust surface density Σd and/or low velocity dispersion results in the dust-rich
subdisc having Q < 1 and becoming gravitationally unstable. This leads to the formation
of bound clumps of particles which quickly agglomerate to planetesimals, decouple from
the gas disc and therefore do not suffer rapid radial drift.
5We remark that this is quite inaccurately named since the precise size of particles that form this barrier
depends on local disc properties. However,we will continue to use it regardless since it does apply to roughly
meter-sized bodies in the planet forming region of the disc. It’s also quite a snappy name.
3.4 The Safronov–Goldreich–Ward mechanism 39
Initially considered to be the silver bullet of planetesimal formation, the SGW process soon
ran into difficulties.
3.4.1 Disc self-gravity and the Toomre Q parameter
As discs become more massive one can no longer neglect the effect of self-gravity when deter-
mining the gravitational potential Φ. This effect was first studied by Toomre (1964) in the
case of a stellar disc6 who found that as a disc becomes more massive, there is a tendency for
its own self-gravity to result in the formation of overdense clumps.
A disc with surface density Σ is unstable to the effect of self-gravity if the timescale for
gravitational collapse is shorter than the timescales on which sounds waves can cross a clump
or shear destroys it, since both pressure forces and shear tend to resist clump formation. For
axisymmetric perturbations, this can be realised as
csΩ
πGΣ < 1, (3.4.1)
where Q = csΩ/πGΣ is the Toomre Q parameter7. Generalising to include nonaxisymmetric
modes produces additional numerical factors which mean that instability sets in more easily
(i.e. at a higher Q), with the development of spiral arms rather than axisymmetric rings.
Therefore instability occurs when Q ≲ Qcrit, where Qcrit ∼ O(1) still.
Since H = cs/Ω (Section 2.4.4) and estimating the disc mass as Mdisc ∼ πr2Σ this insta-
bility condition can be written as
Mdisc
M∗
≳ H
r
. (3.4.2)
It is necessary to work out the enhancement we need in the gas disc density, while maintaining
the same disc thickness, H to get GI; this density enhancement is due to increasing the surface
density of solids in the gas. We find we require a dust-to-gas ratio of around
Γd ≃ 20− 100 (3.4.3)
for GI, though the precise value of necessary enhancement remains uncertain (Youdin and Shu,
2002; Gómez and Ostriker, 2005; Johansen et al., 2009; Shi and Chiang, 2013).
Observations of Class II YSOs have found in general that their discs – protoplanetary discs
– have Q≫ 1 (Andrews et al., 2010; Isella et al., 2009), though this is not true of early epoch
discs (Section 2.3.9, Hartmann et al., 1993).
6See Binney and Tremaine (2011) and Bertin (2000) for treatment and derivation of the fluid case and
Appendix B of Chavanis (2000) for a turbulent, rotating disc.
7For a derivation see e.g. Armitage (2010).
40 The role of dust in PP discs
3.4.2 Self-excited turbulence in the dust subdisc
As we found in Sections 3.3.3 and 3.3.4, in the absence of turbulence, particles would settle
to the midplane to arbitrarily high densities in times < τdrift, making the onset of GI trivial
(Youdin, 2010). However, fully turbulent regions of the disc prevent settling to these densities
(Youdin and Shu, 2002).
Furthermore, Weidenschilling (1980) showed that dust at these high densities would dom-
inate the flow in the dust subdisc, carrying the gas at nearly Keplerian velocities. This would
result in a vertical velocity shear between the midplane layer and the slower moving gas above
and below it. This shear drives the Kelvin-Helmholtz instability between the layers (KHI,
Kelvin, 1871; Helmholtz, 1868).
This self-excited, dust-driven turbulence results in a mixing of the dust and gas, preventing
settling for reasonable values of Γd (Cuzzi et al., 1993; Sekiya, 1998; Sekiya and Ishitsu, 2000).
However, if Γd is enhanced above the canonical, ISM value of ∼ 1%, GI could occur before the
shearing instability disrupts the layer (Garaud and Lin, 2004), though it is unclear why this
would occur.
In the absence of rotation, the hydrodynamic stability of a stratified shearing flow against
KHI is usually assessed using the Richardson number (Chandrasekhar, 1961; Miles, 1961;
Howard, 1961):
Ri ≡ gz ∂ log ρ
∂z
(
∂vϕ
∂z
)−2
= N2
(
∂vϕ
∂z
)−2
, (3.4.4)
where gz is the vertical gravitational acceleration, ρ = ρg + ρd and N2 is the Brunt-Väisälä
frequency8. For a purely Cartesian flow with no rotational forces, instability occurs when
Ri < 0.25. Naively applying this criterion to a PP disc, turbulence will disrupt the subdisc
long before Q ≃ 1 (Supulver and Lin, 2000; Weidenschilling, 1980, 2003).
However, the fluid in PP discs is not just affected by this vertical shear due to the subdisc
– rotational forces and radial shear act on it too. It is therefore not clear that Ri < 0.25 is
sufficient for instability. For example, Gómez and Ostriker (2005) and Johansen et al. (2006a)
included the Coriolis force (but not radial shear) in models of the dust subdisc and found that
the onset of KHI occurs at a value of Ri more than an order of magnitude greater than the
traditional value of 0.25. It therefore occurs for a thicker dust subdisc, meaning GI is even
more difficult to achieve, although large Ri dust layers could be destabilised by sufficient cooling
(Garaud and Lin, 2004). Inclusion of the radial shear (as well as Coriolis and centrifugal forces)
by Chiang (2008) muddied the waters further, finding that the Richardson number alone does
not determine stability, with the critical Ri dependent on Γd. The prospects of SGW robustly
bypassing the metre gap are not looking good.
8For a derivation see Li et al. (2003) or Drazin and Reid (2004).
3.4 The Safronov–Goldreich–Ward mechanism 41
3.4.3 SGW despite turbulence
A couple of global mechanisms could contribute to Γd enhancement; gas loss by photoevapora-
tion (Throop and Bally, 2005) and, perversely, radial drift; the r-dependence of the radial drift
velocity could result in the global redistribution and concentration of small dust as it drifts
inwards and piles up at smaller r (Youdin and Shu, 2002).
However, we conclude that globally in a PP disc it is very difficult to achieve the Γd in the
subdisc such that Q < 1. However, it is possible that locally this may not be the case. Firstly,
KHI can only stir finite amounts of solids (Sekiya, 1998; Youdin and Shu, 2002). Secondly, while
turbulence diffuses particles on average, it can also create intermittent particle clumps and the
enhanced self-gravity of such an overdense clump could lead to GI. The two-fluid streaming
instability (SI) caused by radial and azimuthal drift (Goodman and Pindor, 2000; Youdin and
Goodman, 2005; Johansen and Youdin, 2007; Youdin and Johansen, 2007) is found to be a
strong particle clumping mechanism, with particle overdensities generating perturbations to
the drag force which sustain the instability.
Recall from Section 3.3.1 that τcollide ∝ Σ−1d so the likelihood of coagulation will increase if
particle density in enhanced. Growth (and thus potential particle overdensities) can therefore
be made more efficient if dust can be trapped in locally overdense regions of the disc. Consider
the radial equation of motion of the gas from Section 2.4.5:
v2r,g
r
= GM∗
r
+ 1
ρg
∂P
∂r
(3.4.5)
The presence of the pressure gradient term implies that if gas has some local structure (e.g.
spiral density waves, vortices), it will orbit at slightly enhanced velocities at the inner edge of
pressure enhancement. This means that dust coupled to the gas in this region will experience
a reduced headwind. Therefore, dust will migrate outwards towards local pressure maxima
(Whipple, 1972). The real beauty of this process is that the very particles that suffer the most
severe radial drift are also the most susceptible to this behaviour.
Confirming this phenomena, Rice et al. (2004, 2006b) find that spiral density waves in
self-gravitating discs can lead to strong density enhancements of a ≃ 1m bodies in spiral arms.
Similar results are found Durisen et al. (2005), with accelerated particle growth in dense rings.
However, these both require the disc to be globally gravitationally unstable with Q < 1 – as
we’ve seen, this is unlikely in the PP disc phase. Pressure maxima near the snow line (Kretke
and Lin, 2007) or at the edge of the dead zone (Dzyurkevich et al., 2010) could also be suitable
global locations, as could the zonal flows produced by MRI turbulence (Johansen et al., 2009)
on a more intermediate scale.
The presence of a planet in a disc could produce similar behaviour, generating spiral
density waves that creates a pressure maximum next to the gap opened up by that planet
(Paardekooper and Mellema, 2004, 2006). These particular local pressure maxima prove to be
42 The role of dust in PP discs
a ‘size sorting’ mechanism in discs, as the Ts ≃ 1 grains tend to clump near the local maximum
while well coupled, smaller dust moves with the gas and larger bodies orbit independently (Rice
et al., 2006a; Garaud, 2007; Alexander and Armitage, 2007). However, this obviously requires
a planet to already exist – even if a single planet can seed many others, this initial body needs
to come from somewhere.
3.5 Beyond SGW: Vortices
Finally, another form of localised pressure maxima are disc vortices. We expect them to
be common in PP discs (Dowling and Spiegel, 1990; Abramowicz et al., 1992; Adams and
Watkins, 1995), with the observations of coherent vortices in rotating, turbulent fluids existing
for many decades (e.g. Hopfinger et al., 1982). We will discuss how vortices can be produced
in circumstellar discs in Section 3.5.2.
Based on Descartes (1644) and the writings of Kant, von Weizsäcker (1944) and Alfven and
Arrhenius (1976) were the first modern works to suggest the role of vortices in the formation of
the Solar System, with the PP disc organised into a highly organised (and implausible) system
of vortices where matter could interact and condense. Their role in planet formation theory
was then largely ignored until Adams and Watkins (1995), Barge and Sommeria (1995) and
Tanga et al. (1996) all independently proposed that the trapping of solids in vortices in PP
discs enhances particle growth to make the SGW mechanism locally viable (see also Fromang
et al., 2006; Bodo et al., 2007; Mamatsashvili and Rice, 2009). It has been found to be possible
to create up to a ≃ 1km bodies in pressure maxima by just coagulation and no GI (Brauer
et al., 2008), but on timescales that could be too slow (≃ 1000yr, Lyra et al., 2009). Therefore,
the growth of larger particles and collapse via GI should be thought of as complementary
processes.
The approximately 2D nature of PP discs is of crucial importance here; in 3D, eddies are
quickly damped by an energy cascade towards small scales while 2D turbulence persists without
energy dissipation – instead they form larger and larger vortices until a steady, solitary vortex
is formed9.
Also, Barge and Sommeria (1995) found that small vortices with radial extent R ≪ H
with typical vorticity Ω have velocity v ∼ ΩR≪ cs = ΩH (Chavanis, 2000) and are therefore
approximately incompressible, avoiding strong shocks and large density wave losses. They also
appear to persist for large times after a period of vortex merging to form single, coherent
structures. This growth desists when R ≃ H (so M ≃ 1), beyond which energy losses by
sound waves become problematic (Barge and Sommeria, 1995).
As we saw in Section 3.1.2 and Figures 3.1 and 3.2, there is also an observational justi-
fication for studying these objects since they can both trap dust and produce spiral density
9e.g. Jupiter’s persistent red spot (Ingersoll, 1990).
3.5 Beyond SGW: Vortices 43
waves, both of which can now be observed thanks to the resolution afforded to large arrays of
telescopes such as ALMA.
3.5.1 Dust trapping
In a shearing flow like that in a PP disc, cyclonic (vorticity, ω > 0) vortices are rapidly
elongated and destroyed by the shear, while anticyclones (ω < 0) persist for long times until
they are ultimately destroyed by viscosity or instability. Furthermore, these two different
structures have different effects on any dust particles suspended in the fluid.
In a cyclone, both centrifugal and Coriolis forces are positive and push particles outwards,
while the Coriolis force in anticyclonic vortices pushes dust inwards. If the vortex rotates
rapidly, the centrifugal force dominates and particles are expelled while the opposite is true if
it is slowly rotating (i.e. if it is a weak vortex).
This can also be seen in terms of the pressure distribution inside the vortex; particles
travel up a pressure gradient towards local maxima but are driven away from vortex cores
with a central minimum. Particles do not feel the pressure gradient directly (as discussed in
Section 3.3.2) - they circulate inside the vortex in epicycles with the Keplerian orbital frequency.
However, the rotation frequency of a pressure–supported vortex is smaller than the Keplerian
epicyclic frequency so these particles experience an headwind and spiral towards the centre of
the vortex. This is entirely analogous to radial drift in the disc (Section 3.3.4).
Furthermore, the numerics of Bracco et al. (1999) and Godon and Livio (1999a,b, 2000)
support this notion of particle capture, which find very efficient capture and concentration
inside weak anticyclones.
The effect on potential planetesimal formation is three-fold. Firstly, the concentration of
density may enhance collision rates by more than an order of magnitude (Bracco et al., 1999;
Godon and Livio, 2000; Johansen et al., 2004; Klahr and Bodenheimer, 2006; Heng and Kenyon,
2010), enhancing grain growth (recall τcollide ∝ Σ−1d , Section 3.3.1). Secondly, the larger Γd
could trigger gravitational instabilities, as is seen in Lyra et al. (2009). Thirdly, the local
pressure perturbation of the self-gravitating structure will lead to density enhancement that
will migrate significantly slower or not at all (Johansen et al., 2006b). Therefore the growing
particles could remain trapped in vortices until they are large enough to decouple when the
characteristic stopping time Ts ≫ 1. At this point the radial drift is negligible and we avoid
the catastrophic inspiral of Ts ≃ 1 planetesimals into the central star.
Considering these forces acting on particles in a vortex, Chavanis (2000) established the
details of the dynamics of dust trapping. He found a capture timescale τcapt, the characteristic
time for a particle of a certain size to become trapped in the closed streamlines of a vortex. In
the ‘light’ particle limit τs → 0, where dust is perfectly coupled to the gas, τcapt ∝ (ΩTs)−1.
These particles remain in epicycles close to the vortex boundary, or can diffuse away from
vortices due to turbulent fluctuations. In addition, the capture time for primordial, well-
44 The role of dust in PP discs
Particle type Characteristic Dynamics
Light Ts ≪ 1, τs ≪ τdyn
Grains are strongly coupled to gas so follow its
streamlines. Stops at epicycles close to the vor-
tex edge.
Intermediate Ts ≃ 1, τs ≃ τdyn Optimal – grains can reach deeper epicyclesclose to the vortex core.
Heavy Ts ≫ 1, τs ≫ τdyn
Can take a long time to connect with vortex
epicycle and can even escape as their motion
nearly unaffected by the vortex.
Table 3.1 Vortical dust trapping of different types of particles (Chavanis, 2000).
coupled 1µm dust is > τdisc. ‘Light’ dust is therefore not strongly concentrated. From this we
can deduce that some degree of particle sticking and growth is necessary before vortex capture.
In the opposite ‘heavy’ particle limit τs →∞, the associated timescale is τcapt ∝ τs. These
heavy particles can even pass through a vortex without being captured by it as their momentum
is large enough they are barely effected by the pressure perturbation. The optimal stopping
time for capture is, unsurprisingly, Ts ≃ 1 (Chavanis, 2000; Godon and Livio, 2000; Youdin and
Goodman, 2005), the same particles that settle the fastest and are most susceptible to radial
drift (Section 3.3.4). Chavanis (2000) found that the capture time is minimised for vortices
with aspect ratio χ ≃ 4, where
τmincapt =
8
3Ω = O(τdyn). (3.5.1)
Table 3.1 summarises dynamics of dust trapping for different Stokes number Ts.
Due to the two different aerodynamic regimes in the disc (Section 3.3.2), there are two
regions in a PP disc where τcapt is minimised (Chavanis, 2000). Also recall that the size of
particle for which Ts ≃ 1 depends on location in the disc. The optimum size for trapping is
1–50cm in the planet forming region around 1AU. Building on this, Heng and Kenyon (2010)
found that the annulus where vortex capture is most favourable decreases in width and strength
with time. This known as ‘vortex ageing’, where more evolved discs prefer to capture smaller
dust.
Considering the turbulent diffusion of particles, Chavanis (2000) finds that these particles
also form the maximum dust concentrations and it is theoretically possible to get sufficient
density enhancements Γd ≃ 100 for particles with this Ts, i.e. particles in the 1–50cm range.
This is without the need for an initially large Γd, which gives it an advantage over streaming
instabilities which require Γd ≃ O(1) near midplane to seed GI.
Finally, also note that vortices do not have the vertical shear that causes the KHI in the
3.5 Beyond SGW: Vortices 45
subdisc (Klahr and Bodenheimer, 2006) because of the presence of a pressure extremum. Thus
sedimentation in vortices is more efficient (Lyra et al., 2009).
Therefore, at certain disc radii, for sufficiently large dust, we conclude that vortices can
‘rehabilitate’ the SGW mechanism.
3.5.2 Producing vortices in protoplanetary discs
We have established that vortices are potentially very useful for particle concentration. How-
ever, this is academic if there are no mechanisms for producing these structures in PP discs.
Taking the curl of the momentum equation (2.4.2) and defining vortensity ω/ρ, we have a
2D vortensity evolution equation where, in absence of magnetic fields and negligible viscosity:
D
Dt
(
ω
ρ
)
=


*0(ω
ρ
)
· ∇v + 1
ρ2
∇ρ×∇P, (3.5.2)
where ρ is surface density. If the source term on the right hand side vanishes, as it does for
a barotropic flow P = P (ρ), any vortensity is just advected with the fluid flow. Thus total
vorticity is advectively conserved and is it difficult to both create and destroy10.
Using the ideal gas equation P = ρRT , with R the ideal gas constant, we find that the
source term is ∝ ∇ρ × ∇T . We found in the previous chapter that, in general, there is a
uniform, radial, temperature distribution in the disc due to stellar irradiation (Sections 2.3.7
and 2.4.2) so we expect PP discs to be approximately locally barotropic (Adams and Watkins,
1995). Sources of vorticity will therefore arise from any asymmetries in density ρ.
Turbulence in the PP disc has long been proposed as a means of angular momentum
transport (Pringle, 1981). In stratified PP discs there are special locations where flow is
both turbulent and quasi–2D. Under these conditions, the fluid organises itself into coherent,
long-lived vortices (i.e. lasting many τdyn) without the need for special initial conditions
(Carnevale et al., 1991; Weiss and McWilliams, 1993; Tabeling, 2002). This is because the
‘vortex stretching’ term is absent in 2D, allowing an inverse cascade of energy. Indeed, random
initial velocity fields applied to a Keplerian shearing flow were found to produce large scale,
long-lived anticyclones by Bracco et al. (1999); Godon and Livio (1999a,b, 2000).
Furthermore, sheared, rotating fluids support Rossby waves (Rossby, 1945; Dickinson,
1978). The Rossby Wave Instability is an edge mode instability, similar to KHI, which converts
excess shear into vorticity, resulting in large scale vortices. It has a long history in astrophysi-
cal contexts (Lovelace and Hohlfeld, 1978; Toomre, 1981; Papaloizou and Pringle, 1984, 1985;
Hawley, 1987) and first applied to thin accretion discs by Lovelace et al. (1999) and Li et al.
(2000).
This led Varnière and Tagger (2006) to suggest that the Rossby wave instability (hereafter
RWI) could be at work in the transition between MRI-active and MRI dead zones (Gammie,
10This is Kelvin’s circulation theorem, see Landau and Lifshitz (2013).
46 The role of dust in PP discs
1996) to form dust-trapping disc vortices. This is supported by the work of Inaba and Barge
(2006); Méheut et al. (2012b) who produced enhanced particle growth through the accumula-
tion of dust in RWI generated, self-sustained vortices.
With a sharp enough transition region, RWI leads to the formation of multiple vortices,
vortex merging and finally a single, large structure (Inaba and Barge, 2006; Li et al., 2001)
which is columnar and extends throughout the vertical extent of the disc (Richard et al., 2013).
The 3D simulations of barotropic, stratified discs by Méheut et al. (2010, 2012a) again found
strong and persistent RWI-generated vortices with interesting meridional circulation patterns
that reinforce the need to model these vortical structures in 3D (Méheut et al., 2010; Méheut
et al., 2012b). Despite this, instability in three dimensions was investigated by Umurhan
(2010), Méheut et al. (2012c), Lin (2012) and Lin (2014), finding that RWI is a nominally 2D
instability, with negligible difference between growth rates in 2D and 3D calculations.
As touched on in Section 3.4.3, the pressure maxima at planetary gap edges are seen to
excite vortices (de Val-Borro et al., 2006; Rice et al., 2006a), later explained in terms of the
RWI (de Val-Borro et al., 2007; Li et al., 2009; Lyra et al., 2009; Lin and Papaloizou, 2010;
Lin and Papaloizou, 2011). This is the proposed explanation for the lopsided dust distribution
seen in the transitional disc around Oph IRS 48 (van der Marel et al., 2013). Zhu et al. (2014)
again found that vortices of this nature behaved in an intrinsically 2D manner and were able
to concentrate a wide range of grain sizes 0.02 < Ts < 20. These gap edges could also act to
filtrate and trap dust in the outer disc (Zhu et al., 2012; Pinilla et al., 2012), further increasing
Σd and Γd.
The Zombie Vortex Instability of Marcus et al. (2013), whereby vortices in stratified shear
flows self-replicate, could be another source of vortex production. However, the specificity of
conditions to produce it could limit its viability.
3.5.3 Vortex lifetime
Vortices in 2D PP discs have been found to be stable, persistent structures (Godon and Livio,
1999b; Umurhan and Regev, 2004; Johnson and Gammie, 2005), with Davis et al. (2000);
Inaba and Barge (2006); Lyra et al. (2009) finding that in the absence of viscosity, 2D vortices
can persist indefinitely.
However, in 3D, vortices are generally subject to hydrodynamic instabilities that may
destroy them on a timescale < τν . Close to the disc midplane the flow exhibits more 3D
structure and vortices appear to be destroyed rather quickly (Shen et al., 2006). However,
Barranco and Marcus (2005) found that they have a better survival time z > 2H away from
the midplane, where ρ drops rapidly with z and the fluid behaves more 2D. This also suggests
the possibility that vertical stratification has a stabilising effect. However, off-midplane vortices
are not observed in PP disc simulations (Chiang, 2008; Johansen et al., 2009) so producing
them may be problematic.
3.5 Beyond SGW: Vortices 47
Investigating columnar PP disc vortices, Lithwick (2009) found that vortices live indefinitely
when their vertical extent is less than their length ∆z ≲ ∆y. In light of this, the Barranco
and Marcus (2005) result can then be understood because the local scale height H is reduced
above midplane. It was also found that weak vortices with aspect ratios χ≫ 1 can survive for
long times because they are stabilised by rotation and behave as Taylor–Proudman columns
(Proudman, 1916; Taylor, 1917). The Taylor–Proudman theorem states that when a solid
body (in this situation a vortex at or near the midplane) is moved within a fluid that is being
rotated with angular velocity Ω (with Ω large in comparison to the movement of the body) the
resulting fluid velocity will be uniform along any line parallel to the axis of rotation. When
considered in an isothermal, 3D box, 2D vortices would therefore act like column vortices over
the height of the box.
3.5.4 Vortex instabilities
The periodic nature of circulation round a vortex means that these structures are subject to
the elliptical instability. This instability is a form of parametric resonance observed when the
background flow follows closed streamlines, with the instability, crucially, localised on these
streamlines. Laboratory and numerical experiments both show it is a robust mechanism,
leading to flow becoming highly complicated and full of small scale disorder. It is invariably
3D, with linear growth rates that scale with the shear, S (Kerswell, 2002).
The form most relevant to astrophysical context was first presented in the numerical work of
Pierrehumbert (1986) on an unbounded, strained vortex. This was followed by the analytical
work of Bayly (1986) and Craik and Criminale (1986), who performed Floquet analysis on
disturbances taking the form of Kelvin waves (Kelvin, 1880),
{u(x, t), p(x, t)} = {uˆ(t), pˆ(t)} exp[ik(t) · x]. (3.5.3)
The periodic nature of the coefficients appearing in the differential equations governing {uˆ(t), pˆ(t)}
means this is a Floquet problem (Floquet, 1883), related to the Hill and Mathieu equations
(Hill, 1886; Mathieu, 1868).
The effect of background rotation, relevant to our rotating disc, was investigated by Craik
(1989), who found that anticyclonic, elliptical flows can be stable for some rotation rates. Fur-
thermore, Miyazaki and Fukumoto (1992), found that growth rates of the elliptical instability
were reduced by stable exponential stratification in the zˆ direction, albeit not working in a
rotating frame.
The elliptical instability however need not completely destroy our PP disc vortices. After
vortices develop bursts of 3D turbulence in their cores, the subcritical baroclinic instability
(SBI, Lesur and Papaloizou, 2010) offers a mechanism for amplifying the vortices, allowing
them to survive as weaker eddies. In this way, the SBI can produce large scale vortices up to
48 The role of dust in PP discs
Model Stream-
lines
Vorticity Ωv Validity Notes
Kida Elliptical −3Ω(χ
2 + 1)
2χ(χ− 1)
3Ω
2(χ− 1) χ > 1
Valid with Keplerian
background shear
GNG Elliptical −
√
3Ω(χ2 + 1)
χ
√
χ2 − 1 Ω
√
3
χ2 − 1 χ > 2
Zero pressure gradient
along streamlines
Point Circular κδ(r)
κ
2πr χ = 1
Irrotational, not valid
with shearing back-
ground
Table 3.2 Summary of vortex models
a size of order H in the radial direction.
Since dust is driven into centre of vortices by drag forces, diffusion is needed to maintain a
steady state over the lifetime of the disc (Klahr and Henning, 1997; Chavanis, 2000). In this
context, the ratio of the relative strengths of drag and diffusion becomes important (Cuzzi
et al., 1993; Dubrulle et al., 1995; Lyra and Lin, 2013). This is expressed as Ts/δ, where δ
is the turbulence in the vortex11, where decoupling of dust and gas becomes important when
Ts/δ > 1. Note also that this ratio increases with both t and r, and with decreasing mass
accretion rate M˙ (Jacquet et al., 2012), so we expect the optimally coupled Ts ≃ 1 dust to
be ejected from vortices at late epochs and far away from the central star. In addition, the
presence of turbulence in vortex cores will act to diffuse out concentrations of dust, potentially
leading to insufficient particle concentrations required to seed GI.
3.5.5 Existing vortex models and stability calculations
Most analyses of individual vortices in a shearing background begin with either a Kida (1981)
or Goodman–Narayan–Goldreich (GNG, Goodman et al., 1987) vortex model.
Based on the rotating elliptical vortex solution of Kirchhoff (1876), and related to the
Moore–Saffman vortex (see Moore and Saffman, 1971, Neu, 1984 and Saffman, 1995), the
Kida solution finds an analytical solution to an elliptical, constant-density patch of uniform
vorticity in a background Keplerian shearing flow, S = 3Ω/2. Its streamlines are concentric
ellipses of constant aspect ratio χ and its power as a model solution stems from these well-
defined elliptical streamlines which allow for a degree of analytical stability analysis. It is also
useful as an initial condition, with Lin and Papaloizou (2011) finding that RWI vortices excited
at planetary gap edges resemble vortices formed by perturbing the disc with the Kida solution.
11i.e. not αν , the Shakura–Sunyaev α from Section 2.4.6.
3.5 Beyond SGW: Vortices 49
This model is described in detail in Section 5.3.
A similar model is the polytropic GNG solution (with their ε = χ−1) which exactly solves
the compressible Euler equations with shear
S =
√
3(χ− 1)
χ+ 1 Ω. (3.5.4)
This is a also an attractive model as it has a more straightforward pressure distribution than
the Kida solution; its pressure gradient is zero along streamlines. (For the Kida solution this
is only the case when χ = 7.)
We discuss these two models in more detail in Sections 5.3.2 and 5.4. Their properties are
summarised in Table 3.2, along with that for the classical point vortex solution (Batchelor,
2000, pages 93–96).
Lyra and Lin (2013) extended the Kida solution to include vortex-trapped dust in a tran-
sitional disc, solving for the dust distribution in steady state between gas drag (driving dust
inwards in certain cases, Section 3.5.1) and diffusion, which acts to expel it. With the fluid
constrained to elliptical streamlines of constant χ, like the Kida and GNG vortices, they find
a Gaussian for the dust density ρd with standard deviation Hv, the ‘dust vortex scale height’.
This scale height is a function of gas scale height H, χ and Ts/δ.
They also found that it was safe to assume that dust is approximately axisymmetric along
streamlines, with only small non-axisymmetric corrections. They derived a density enhance-
ment of ρd,max = Γdρ0(Ts/δ + 1)3/2 and thus a constraint on the strength of turbulence, δ, in
the vortex core. A similar relationship to Ts is found in a much larger scale vortex in Birnstiel
et al. (2013).
With regard to analysing the linear stability of vortex models, at this point it is useful to
look in detail at two previous approaches to this problem; the work of Lesur and Papaloizou
(2009) (Section 3.5.5.1) and Chang and Oishi (2010) (Section 3.5.5.2).
3.5.5.1 The approach of Lesur and Papaloizou (2009)
With the existence, numerous methods of production and important dust-trapping property
of vortices in PP discs long established, there is a need for a more in-depth understanding of
the stability of these structures.
Lesur and Papaloizou (2009) approached this problem by producing simple, steady 2D
vortices based on Kida vortices. A Floquet analysis of the linearised equations governing 3D
perturbations in the vortex core was then performed. This was then solved numerically, with
analytical expressions existing for the horizontal and vertical perturbation limits. They found
a small, linearly stable regime for Kida vortices with aspect ratio 4 < χ ≲ 5.9.
This was then investigated in 3D for both stratified and unstratified models. Vertical
stratification was found to suppress but not eliminate the elliptical instability for weak (large
50 The role of dust in PP discs
χ) vortices (see their Figure 6). They also found that results obtained from incompressible
models were unaffected by introduction of moderate compressibility. In general, elliptical (and
in general, parametric) instabilities were found to be always present, but often involve small
radial wavelength compared to H and small growth rates compared to τdyn so were hard to
pinpoint numerically.
3.5.5.2 The approach of Chang and Oishi (2010)
This paper aimed to build upon the work of Lesur and Papaloizou (2009) by including the
influence of a density gradient; the main shortcoming of the Kida (1981) solution in this
context is that it applies to a constant density fluid.
Chang and Oishi (2010) perturb around both the Kida and GNG solution with an imposed
density profile given by
∂ log ρ
∂ log b = c, ⇒ ρ = ρ0b
c, (3.5.5)
where b > 0 is the semi-minor axis of the elliptical vortex patch and ρ0, c are constants. They
find an instability, which they call the ‘Heavy Core Instability’ (HCI), for cases with c < 0, so
vortex density gradients are destabilising for vortices with sufficiently heavy cores. Instability
occurs if Γd ≳ 1.2. They therefore expect that the HCI instability will disrupt these vortices
long before seeding the GI (which requires Γd ≃ 20 − 100, Section 3.4.1). However, it is
important to note that these solutions do not correctly match the background flow as they just
assumed Kida streamlines with an arbitrary density superposed.
They base their perturbation wavenumber on the Kida wavenumber of Lesur and Pa-
paloizou (2009)
k = k0
(
cos(ωt+ ϕk,0),−χ−1 sin(ωt+ ϕk,0)
)
, (3.5.6)
(so kz = 0, considering only perturbations in the plane of the vortex), acting on streamlines
x = b (cos(ωt+ ϕ0),−χ sin(ωt+ ϕ0)) , (3.5.7)
However, this form of perturbation wavenumber in equation (3.5.6) applies to the Kida case
only and not the generic equilibrium. A cause for concern is their final stability equations
contain explicit dependence on the phase of k, i.e.
dρ˜
dt
= −a(b, t) sin (ϕk,0 − ϕ0) ∂ρ
∂b
, (3.5.8)
where ρ˜ ∝ ρ′, the Eulerian perturbation of ρ. This 2D instability has also, as yet, not been
observed in any numerical simulations. They outline a potential pseudospectral numerical
method for searching for this instability in Oishi and Chang (2013), but as of yet no results
are forthcoming.
3.6 Summary and conclusions 51
3.6 Summary and conclusions
A summary of important timescales for disc and dust evolution is given in Table 3.3.
In this chapter, we have found that dust dominates the observations of discs so there is
plenty of evidence for its existence, general structure and evolution. There are also an increasing
amount of observations of dust asymmetries in discs such as rings (Figure 2.1) and vortices
(Figures 3.1, 3.2). This observational evidence reinforces the need to study these structures and
suggests that dust–laden vortices could be stable, or quasi–stable, under certain conditions.
Furthermore, grain growth is relatively a well understood and robust mechanism up to
about a ≃ 1cm in size, where it is limited by destructive collisions and bouncing. Fragmentation
is required to sustain the population of small dust grains observed at late times but also means
that growth beyond a ≃ 1m is next to impossible by sticking alone. At the other extreme,
growth from planetesimals to terrestrial and gas giant planets is a well understood process,
though still not a straightforward one.
Aerodynamic drag couples dust particles to the gas in the disc, while the action of the
star’s gravity causes dust to settle vertically in the disc, a process disrupted by turbulence.
Disparities between the mean angular velocity of the gas and dust due to pressure gradients
causes an inward radial drift of grains and this inward motion can be very fast: < 100yr for
Ts ≃ 1 particles. This, combined with the wholly destructive collisions between a ≃ 1m bodies,
is a barrier to planetesimal formation known as the ‘metre gap’. There is therefore a real need
for a robust mechanism for rapid growth from metre–sized objects to planetesimals.
The Safronov–Goldreich–Ward mechanism proposes that the gravitational instability of
a self–gravitating disc can collapse directly from dusty gas to planetesimals. However, it
appears unlikely that, globally at least, the dust–to–gas ratio Γd required to seed gravitational
collapse can be achieved before self–generated KHI disrupts the layer. This implies a more
local approach is necessary.
Dust–trapping, anticyclonic vortices can bridge the gap between sticking processes and
planetesimal collisions in the planet formation process. For certain grain sizes, capture by
vortices is fast (≃ τdyn for Ts ≃ 1, a ≃ 1−50cm dust around 1AU) and the consequence of this
is twofold; particles coagulate faster in the higher Σd while the enhanced Γd could make the
SGW mechanism locally viable. Meanwhile, the initial sticking of particles to sizes a ≳ 1cm
is an indispensable step in this process as these grains are preferentially trapped within the
necessary timescales. Furthermore, there are numerous ways of producing vortices in PP
discs (MRI, RWI, gap edges) so we can assume they are commonplace objects. Despite this,
vortices are subject to potentially destructive instabilities, so their lifetime with and without
the presence of dust is unclear. Therefore, further study on the stability of these structures in
Keplerian shearing flows is required.
Finally, existing vortex solutions with background shear are based around elliptical stream-
lines and can, to a certain extent, include density enhancements. These solutions are neces-
52 The role of dust in PP discs
sarily periodic and are therefore subject to elliptical, parametric instabilities. These can be
investigated using Floquet analysis of the perturbed equations, as in the work of Lesur and
Papaloizou (2009) and Chang and Oishi (2010). We will attempt to build upon this work to
form a more complete understanding of the stability of more general vortices.
3.6
Sum
m
ary
and
conclusions
53
Table 3.3 Important timescales for PP discs.
Timescale Symbol Scaling Notes
dynamical, orbital,
shearing
τdyn Ω−1
Time to reach centrifugal equilibrium; ∼ r3/2 for Keplerian discs.
Since Ω−1 = H/cs, this is also the vertical sound crossing time,
or the time to reach hydrostatic balance in the vertical direction.
Also the growth time for MRI and gravitational instability.
thermal τth
τdyn
α
Time for disc to modify thermal structure; balance of τcool from
cooling processes and τheat from energy release from accretion.
viscous τν
r2
ν
∼ α−1
(
H
r
)−2
τdyn
Timescale on which matter diffuses through the disc under the
effect of viscous torques. Sets scale for evolution of surface density.
Around 1Myr for Solar-type stars.
disc formation – ∼ 104 − 105yr e.g. Shu et al. (1993)
disc lifetime τdisc few ×106yr There is large scatter in this value (e.g. Hillenbrand, 2005).
disc dispersal – 105yr
Lifetime of transitional discs; time take for photoevaporation to
clear a disc of gas.
coagulation timescale τcoag τcoag ≳ 103yr Timescale for growth from a ≃ 0.1µm−1mm by sticking processes.
collisional timescale τcollide ≃ 0.005− 5yr Typical time between dust-dust collisions.
54
T
he
role
ofdust
in
PP
discs
frictional,
dust stopping time
τs
ρda
ρgcs
Typical time for a particle with initial velocity different to the gas
to stop in a frame moving with the gas velocity. τs ≃ 3s for 1µm
particle at 1AU in disc midplane.
dimensionless
dust stopping time
Ts τsΩ Particles with Ts ≃ 1 are most susceptible to radial drift and
vortex capture.
sedimentation,
dust settling
τsettle
1 + 2T 2s
TsΩ
Time for dust to settle to midplane due to aerodynamic drag.
τsettle ∼ 105yr (a ≃ 1µm)− ∼ 103yr (a ≃ 0.1 − 1m) at 1AU from
solar mass star.
radial drift,
orbital decay
τdrift
1 + T 2s
2ηTsΩ
, ∼ τsettle/η Typical time for particle to spiral into the central star, τdrift ≲
100τdyn (a ≃ 1m), > τdisc (a ≲ 1µm, a ≳ 1km).
dust capture timescale τcapt τmincapt ∼ τdyn Typical time for a coherent vortex to capture dust particles.
Chapter 4
Research design
We have established it is challenging to produce the necessary planetesimals in protoplane-
tary discs, from which to build planets. It is difficult because of both timescale constraints
(radial drift, a relatively short τdisc) and the actual physical mechanisms of growth themselves
(coagulation/fragmentation and collapse via gravitational instability).
Vortices appear to be a promising solution, but despite their prevalence both in global
numerical simulations and now observationally, little is known about their stability in this
context. The analysis of Kida vortices in Lesur and Papaloizou (2009) is a solid base from
which to generalise.
Therefore, the questions we would like to tackle in this thesis are:
(i) What happens when vortex streamlines are not elliptical, as they are in the Kida case?
(ii) How do non–constant profiles of vorticity and density affect stability?
(iii) Can the ‘heavy–core’ instability of Chang and Oishi (2010) be reproduced?
(iv) Does the stability of these more general models have any bearing in 3D?
With problems like these there are broadly two approaches; a ‘top down’ or ‘bottom up’
approach. The former involves large, global models including as much physics as possible. In
this case, that could be a fluid with an N -body model for the dust grains, perhaps including
coagulation, fragmentation, self-gravity, and radiative effects. Good examples of this approach
in this context is the work of Lyra et al. (2009), with N -body dust, or Zhu and Stone (2014),
which includes MHD effects.
On the other hand, one can isolate different features of a problem and try to understand
the underlying processes governing them and how these depend on the problem’s parameters.
In this way, one hopes to get a global view of the problem and to be able to come up with
some generic features. This is difficult with the ‘kitchen sink’ approach, where there are a large
number of parameters and a complexity limiting consideration to only a few models on a case
by case basis. This work therefore predominantly falls into the second camp.
56 Research design
Our approach to trying to answer the above questions is to firstly formulate a set of local,
equilibrium vortices in a shearing background, including more generalised vorticity and den-
sity profiles. We establish our equations of motion and general models in Chapter 5 while in
Chapter 6 we demonstrate how we implement these numerically to find steady vortex solu-
tions. Secondly, we will analyse the stability of these models to perturbations with a general
wavenumber k, with the stability analysis given in Chapter 7 and its numerical implementation
in Chapter 8. In Chapter 9 we will then investigate the relative lifetimes of different columnar
vortices in 3D using the hydrodynamical code PLUTO and try to gain some insight into how
the instabilities manifest themselves in 3D. Finally, we draw conclusions to what this means
in the context of planetesimal formation in PP discs in Chapter 10.
Chapter 5
Calculating equilibrium solutions
As highlighted in the previous chapter, a comprehensive stability analysis of a large variety of
vortex configurations is necessary to investigate the instabilities that occur within them, since
these could be a threat to their survival or dust-attracting ability. In order to do this, we must
first develop a system for calculating equilibrium vortices which we can then investigate the
stability of.
In this chapter we give the basic equations governing the fluid model we use in Section 5.1.1,
its limitations and constraints. We also show how we parametrise different vortex configura-
tions based on their vorticity and density profiles (Section 5.2). In Section 5.3 we then present
the well-known analytical Kida solution which we use to verify our method and form the
starting point of our stability analysis. Finally, in Section 5.4 we discuss a polytropic model
that can be used to consider solutions in a non-Keplerian background flow and in a Keplerian
background flow for the special value of the vortex aspect ratio χ = 7.
5.1 Governing equations for well–coupled dust
We begin by considering a two–fluid model of the dust and gas circulating in a protoplanetary
accretion disc. In the Epstein regime it is useful to treat dust particles collectively and describe
them using a two–fluid dust and gas flow where the two fluids have different flow velocities
and consequently exchange momentum through drag forces (Section 3.3.2, Cuzzi et al., 1993;
Garaud and Lin, 2004).
5.1.1 Basic equations for the two–fluid model
In order to do this we must make two key assumptions. Firstly, we will assume particles are
of a single size, a. This is so Epstein drag is constant across the fluid - else we will need a
multi-fluid model and any analytic progress is impossible. Secondly, we will assume dust is
collisionless, i.e. that the grains do not directly interact and consequently their size remains
58 Calculating equilibrium solutions
constant. As we saw in Section 3.3 these assumptions are reasonable if
1. The frequency or sticking probability of collisions is sufficiently small such that τcoag ≫
τsettle, τdrift, i.e. growth occurs on timescales longer than orbital changes due to interac-
tion with the gas. This is reasonable if we consider particles with a ≲ 1cm (Section 3.3,
Table 3.3).
2. We work in a region of disc are not exposed to intense external radiation and the particles
are charge neutral. We can then neglect interaction with the magnetic field and global
disc radiation, i.e. there is no MRI. This is satisfied in opaque planet forming regions
0.1 ≲ r ≲ 5AU near the midplane (Section 2.3.8). We can therefore also neglect Lorentz
forces.
3. The disc is sufficiently cold to be well below sublimation temperature. We can then
neglect molecular phase changes and grains losing or gaining mass by evaporation or
condensation. This is satisfied away from the snow line, which is < 5AU in at least some
protoplanetary phases (Section 2.3.7).
The basic equations for the gas component are those of continuity and momentum conservation.
We consider flow in a frame rotating with angular velocity Ω = Ωzˆ, with zˆ being the unit
vector in the fixed direction of rotation (here called the vertical direction - see Figure 5.1) and
Ω corresponding to the magnitude the angular velocity of Keplerian rotation at some radius.
These take the form:
∂ρg
∂t
+∇ · (ρgvg) = 0 (5.1.1a)
∂vg
∂t
+ (vg · ∇)vg + 2Ω× vg +Ω× (Ω× r) = − 1
ρg
∇P −∇Φgr (5.1.1b)
+fν
ρg
− ρd
ρg
Fdrag
m•
Here, P is the gas pressure, ρg is the gas density, ρd the dust density, Φgr is the gravitational
potential due to the central mass of the star, M∗, and r being the position vector measured
from the star. The gas velocity is vg and fν could be taken to be an anomalous viscous force
per unit volume. This could be associated with stochastic forcing due to weak turbulence.
The drag force per unit mass acting on the gas is −ρdFdrag/ρg, with the drag force acting
on a single dust particle of mass m• being Fdrag (see Section 3.3.2).
Writing
Ω× (Ω× r) = (Ω · r)Ω− Ω2r = −Ω2r = −∇
(1
2Ω
2r2
)
(5.1.2)
5.1 Governing equations for well–coupled dust 59
Figure 5.1 The geometry of our Keplerian disc and central star.
we incorporate the centrifugal term with the gravitational potential Φgr so Φ = Φgr − 12Ω2r2.
The momentum equation for the gas component is therefore:
∂vg
∂t
+ (vg · ∇)vg + 2Ω× vg = − 1
ρg
∇P −∇Φ+ fν
ρg
− ρd
ρg
Fdrag
m•
. (5.1.3)
The equations governing the evolution of the dust are that of a pressureless fluid:
∂ρd
∂t
+∇ · (ρdvd) = 0 (5.1.4a)
∂vd
∂t
+ (vd · ∇)vd + 2Ω× vd = ∇Φ+ Fdrag
m•
(5.1.4b)
with vd the velocity of the dust component. Note that the drag forces acting on the dust and
gas are related such that the total momentum of the two components is conserved locally.
We saw in Section 3.3.2 that in the Epstein regime, the drag force Fdrag acting on a single
dust particle is given by equation (3.3.4):
Fdrag =
m•
τs
(vg − vd) .
This drag force is proportional to vg − vd, the relative velocity between the gas and dust
components and the inverse of the stopping time τs = ρda/ρgcs (equation (3.3.5)).
60 Calculating equilibrium solutions
5.1.2 Governing equations for the dust and gas in the τs → 0 limit
In the limit τs → 0, the two fluid description reduces to that of a single combined fluid with
density ρ and velocity v. These are defined through
ρ = ρg + ρd and ρv = ρgvg + ρdvd. (5.1.5)
The mean velocity of the dust with respect to the gas is
u = vd − vg. (5.1.6)
We can then find
vg = v − ρdu
ρ
, (5.1.7a)
vd = v +
ρgu
ρ
. (5.1.7b)
Adding the continuity equations for the gas and dust (5.1.1a and 5.1.4a) gives the continuity
equation for the combined fluid in the form
∂ρ
∂t
+∇ · (ρv) = 0. (5.1.8)
Taking a linear combination of equations (5.1.3) and (5.1.4b) that eliminates the drag force
gives the momentum equation for the combined fluid. Assuming v−vd = O(τs) and neglecting
terms of order τ2s and higher, this takes the form
ρ
(
∂v
∂t
+ (v · ∇)v + 2Ω× v
)
= −∇P − ρ∇Φ+ fν . (5.1.9)
We now use equation (5.1.4b) to find u. Consistent with neglecting contributions of order τ2s ,
we may set vd = v in the left hand side of equation (5.1.4b) and, recalling from equation (3.3.4)
that Fdrag ∝ u, we readily obtain (correct to first order in τs)
u = τs
ρ
(∇P − fν) (5.1.10)
The first term on the right hand side of (5.1.10) gives rise to the drift caused by a pressure
gradient which tends to lead to particles concentrating at pressure maxima. The second term
that arises from the prescribed anomalous viscosity could be taken to originate from the pres-
ence of a low level of turbulence. Using equation (5.1.7b) to eliminate vd in equation (5.1.4a)
we obtain
∂ρd
∂t
+∇ · (ρdv) +∇ ·
(
ρdρgu
ρ
)
= 0 (5.1.11)
5.1 Governing equations for well–coupled dust 61
Equations (5.1.9), (5.1.10) and (5.1.11) thus provide a description of the system, that is correct
to first order in τs, using only the mean flow variables and the dust density.
We consider vortices with small length scale, such that with reference to the sound speed in
the gas, relative velocities are highly subsonic. Then we expect the gas to move incompressibly
(Section 3.5). When the dust is tightly coupled to the gas as τs → 0 this will also move
incompressibly.
In this limit, with u is set to zero and we see that each of ρ, ρg, and ρd satisfy the same
continuity equation. For a fluid moving incompressibly we have
∇ · v = 0 (5.1.12)
and hence
∂ρ
∂t
+ v · ∇ρ = 0. (5.1.13)
The equations governing an incompressible gas and dust in the limit τs → 0 are seen to be
(5.1.9), (5.1.12) and (5.1.13). These are seen to be identical to those for a single incompressible
fluid with a variable density that is conserved on fluid elements.
5.1.3 Steady state solutions in the inviscid limit
In a steady state for which viscous forces may be neglected, the equation of motion (5.1.9)
reduces to
v · ∇v + 2Ω× v = −∇P
ρ
−∇Φ. (5.1.14)
We also make two further simplifications when considering a PP disc. Firstly, we will use
the thin disc approximation from Section 2.4.2, so to first approximation we consider 2D,
isothermal state. Secondly, we impose a low disc mass Mdisc compared to the mass of the star
M∗, such that
Mdisc
M∗
<
H
r
≪ 1 (5.1.15)
so that it can be considered to be non-self-gravitating (Q > 1, Sections 2.3.3 and 3.4.1).
5.1.4 The shearing sheet approximation
In order to consider local steady state solutions within a 2D Keplerian disc in detail, we adopt
a local shearing box with origin centred on a point of interest and rotating with its Keplerian
angular velocity Ω(r0) = Ω0, where r0 is the distance to the central star, as shown in Figure 5.2.
Its symmetry can be seen in Figure 5.3. This is a commonly used local model of a differentially
rotating disc, first developed by Goldreich and Lynden-Bell (1965) for the study of galactic
discs. A discussion of how it is implemented numerically is given by Hawley et al. (1995) and
its limitations in Regev and Umurhan (2008). Considering an arbitrary reference point with
62 Calculating equilibrium solutions
Figure 5.2 The shearing sheet model with x = r − r0 and y = r0 [ϕ− (ϕ0 +Ω0t)].
Figure 5.3 The shearing sheet is invariant to rotations by π around the z-axis; the sheet
doesn’t know the value or sign of r0, only Ω0 and S0.
cylindrical coordinate (r0, ϕ0 +Ω0t, 0), we use this as the origin of a local Cartesian coordinate
system:
x = r − r0 (5.1.16a)
y = r0 [ϕ− (ϕ0 +Ω0t)] (5.1.16b)
z = z (5.1.16c)
Orbital motion is represented locally as uniform rotation (Ω0) plus a linear shear flow −S0x ey,
where, as established in Section 2.4.3, the shear rate is given by equation (2.4.7):
S(r) = −rdΩ
dr
,
S0 = S(r0) and in the Keplerian case S = 3/2Ω. The length scale associated with each
dimension of the box can be taken to be the vertical scale height, which, in the thin disc
approximation, is assumed to be H ≪ r0.
5.1 Governing equations for well–coupled dust 63
5.1.5 The effective potential
We want to find the effective potential Φ (i.e. the combined centrifugal and gravitational Φgr
potentials) in this rotating frame as a function of our new variables. Applying our disc mass
assumption from Section 5.1.3, the point mass gravitational potential in cylindrical coordinates
around a central star of mass M∗ is
Φgr(r, z) = − GM∗√
r2 + z2
, (5.1.17)
which is both axisymmetric and symmetric about the z = 0 plane. Additionally, the angular
velocity for a body moving in Keplerian motion at radius r is Ω = (GM∗/r3)1/2. Using
r = r0 + x we expand our effective potential
Φ = Φgr(r, z)− 12Ω
2
0r
2.
to second order in x and z:
Φ = Φgr(r0, 0) + x
∂Φgr
∂r
∣∣∣∣
(r0,0)
+ 12x
2 ∂
2Φgr
∂r2
∣∣∣∣
(r0,0)
+ 12z
2 ∂
2Φgr
∂z2
∣∣∣∣
(r0,0)
− 12Ω
2
0(r20 + 2r0x+ x2) + · · · (5.1.18)
where ∂Φgr/∂z|(r0,0) = 0 due to the symmetry of Φgr in the z–direction. The two terms
Φgr(r0, 0) and 12Ω20r20 are constant so can be ignored and we note that
1
r0
∂Φgr
∂r
∣∣∣∣
(r0,0)
= 1
r0
GM∗r0
r30
= GM∗
r30
= Ω20, (5.1.19)
so the x coefficients cancel. Hence our effective potential is
Φ = 12x
2
[
∂
∂r
(rΩ2)− Ω2
] ∣∣∣∣
(r0,0)
+ 12z
2
[
GM∗
r3
] ∣∣∣∣
(r0,0)
= 12x
2 [−2Ω0S0] + 12z
2
[
Ω20
]
= −Ω0S0x2 − 12Ω
2
0z
2
= −12Ω
2
0
(
3x2 − z2
)
, (5.1.20)
where we used equation (2.4.8) to eliminate S0. From now on we will drop the subscript ‘0’
from Ω and S since inside the shearing box these will only serve to confuse.
In the case of a 2D disc the fluid state variables are independent of z and the z–dependence
of the effective potential Φ is ignored. Although our final solutions will therefore not depend on
z, they may apply to horizontal planes of an isothermal disc for which hydrostatic equilibrium
64 Calculating equilibrium solutions
holds in the vertical direction (Lesur and Papaloizou, 2009). As we saw in Section 2.4.4, given
the form for the effective potential Φ in equation (5.1.20) in the isothermal, P = c2sρ case,
ρ ∝ exp
(
−Ω
2z2
2c2s
)
= exp
(
− z
2
2H2
)
. (5.1.21)
The factor exp
[−z2/(2H2)] may be applied to the two dimensional solutions for ρ and P
obtained from Equations (5.1.12)-(5.1.14). Then, when we restore the z–dependence to Φ,
hydrostatic equilibrium will hold in the z–direction.
The description of local solutions in vertical hydrostatic equilibrium has been found nu-
merically to be applicable to vortices generated by the RWI (Section 3.5.2, e.g. Lin, 2012).
Note too that in the limit of zero stopping time τs → 0 considered above, dust is frozen into
the fluid so midplane settling (Section 3.3.3) does not occur.
5.1.6 The Stokes’ streamfunction
We look for solutions of equations (5.1.12)–(5.1.14) in the case of a 2D disc where the fluid state
variables are independent of z. In order to satisfy the incompressibility condition ∇ · v = 0 we
introduce a Stokes streamfunction ψ = ψ(x, y) such that
v = ∇× (ψzˆ) =
(
∂ψ
∂y
,−∂ψ
∂x
, 0
)
. (5.1.22)
The vorticity of the flow is therefore
ω = ∇× v = −∇2ψzˆ. (5.1.23)
For the undisturbed background Keplerian flow v0 = (0,−Sx, 0), we have the streamfunction
of the background, ψ0:
ψ0 =
1
2Sx
2. (5.1.24)
5.1.7 Working form of the momentum equation
For a steady state, equation (5.1.13) becomes
v · ∇ρ = 0. (5.1.25)
For a two–dimensional flow this implies that density is constant along streamlines and thus is
a function of ψ alone; ρ = ρ(ψ).
This cannot be determined if τs = 0 as it is then an invariant that must be input externally.
However, when τs ̸= 0 but small and slow diffusive processes are included, this is no longer the
case and it may be considered to be the result of evolutionary processes taking place on a long
5.2 Functions specifying the vorticity and density profiles 65
time scale. This is discussed in detail in Section 5.2.2.
Using the identity
v · ∇v = ω × v +∇
(1
2 |v|
2
)
, (5.1.26)
the steady momentum equation (5.1.14) can be written as
(2Ω+ ω)× v = − P
ρ2
∇ρ−∇
(
P
ρ
+Φ+ 12 |v|
2
)
. (5.1.27)
Expressing quantities in terms of ψ, equation (5.1.27) becomes(
−∇2ψ + 2Ω + P
ρ2
dρ
dψ
)
∇ψ = −∇
(
P
ρ
+Φ+ 12 |∇ψ|
2
)
≡ −∇F, (5.1.28)
where we have now dropped the overbars for clarity. Since we are ignoring the z–dependence
of Φ in equation (5.1.20) the combined potential in the shearing sheet is
Φ = −32Ω
2x2. (5.1.29)
As both sides of equation (5.1.28) are proportional to ∇ψ, it follows that F = F (ψ). Similar
to the density, in the absence of diffusive processes the arbitrary function F (ψ) has to be input
externally (see Section 5.2.2). In the absence of a density gradient, the derivative of F (ψ)
represents a conserved vorticity.
Equation (5.1.28) can then be written as a second order partial differential equation for the
streamfunction ψ thus:
∇2ψ = dF
dψ
+ 2Ω + P
ρ2
dρ
dψ
. (5.1.30)
Once F (ψ) is specified, the pressure is expressed in terms of the streamfunction through the
relation
P
ρ
= −Φ− 12 |∇ψ|
2 + F (ψ). (5.1.31)
Solutions of equation (5.1.30) corresponding to local vortices with central dust concentrations
may be sought once the arbitrary functions ρ(ψ) and F (ψ) and appropriate boundary condi-
tions are specified. In this context, note that after making an appropriate adjustment to F ,
equation (5.1.28) is invariant to adding an arbitrary constant to P .
5.2 Functions specifying the vorticity and density profiles
We begin by separating the solution corresponding to an undisturbed Keplerian background
flow for which ψ = ψ0. We write the streamfunction ψ as the sum of a contribution from the
66 Calculating equilibrium solutions
background flow and a deviation ψ1 that will correspond to a superposed vortex.
ψ = ψ0 + ψ1 =
3
4Ωx
2 + ψ1. (5.2.1)
Consider the function F (ψ) for the background flow:
F0 =
P
ρ
∣∣∣∣
0
+Φ+ 12 |∇ψ|
2
= P
ρ
∣∣∣∣
0
− 32Ω
2x2 + 12 ·
9
4Ω
2x2
= P
ρ
∣∣∣∣
0
− Ωψ2 . (5.2.2)
We now choose an arbitrary additive constant for P such that the pressure is zero for the pure
background flow (Section 5.1.7). Thus we set
F = −Ωψ2 + F1, (5.2.3)
where F1 vanishes for the background flow. Equation (5.1.30) then yields
∇2ψ1 = dF1
dψ
+ P
ρ2
dρ
dψ
= A(ψ) + P
ρ
B(ψ), (5.2.4)
where
B(ψ) = d log ρ
dψ
. (5.2.5)
5.2.1 The arbitrary source terms
We now denote A(ψ) as the Bernoulli source term and B(ψ) as the density source in the Poisson
equation for the background flow, equation (5.2.4). These are both functions of the overall
streamfunction ψ. As we are in the τs = 0 limit, these functions are invariants that have to be
specified (Section 5.2.2).
Physically these profiles are expected to be determined by prior evolution governed by
effective viscosity, particle diffusion and friction, and hence they are constrained when they
are included. However, as weak turbulence is likely to have an important role (Lyra and Lin,
2013), finding forms of A(ψ) and B(ψ) from evolutionary calculations of gas/dust systems that
can be used to characterise steady state solutions with adequate resolution is not practical,
particularly if local stability is to be considered.
We therefore consider specifications of A(ψ) and B(ψ) that enable a large class of steady
state solutions with varying vorticity and density profiles to be considered. The Bernoulli and
density sources are superposed on horizontal planes, on which there is a uniform background
5.2 Functions specifying the vorticity and density profiles 67
Keplerian flow with density ρ0. They are both non-zero only on streamlines that circulate
interior to some bounding streamline ψb.
When we solve these equations numerically, we will specify the point where this streamline
crosses the y–axis. The arbitrary unit of length is chosen so that this point is at (0, 1), while
the unit of time is chosen so that Ω = 1 and we set ρ0 = 1. From now on, the ignorable
z–coordinate is suppressed.
We adopt power law functions for our sources of the form
A(ψ) = A|ψ − ψb|α (5.2.6a)
ρ(ψ)− ρ0 = B|ψ − ψb|β. (5.2.6b)
which will provide us with solutions covering a wide range of aspect ratios with a variety of
vorticity and density profiles by varying A, B, α and β.
The functions A(ψ) and ρ(ψ) − ρ0 are set to be zero on streamlines exterior to those
with ψ = ψb. The constants A and B are chosen to scale the total vorticity and relative
mass excesses associated with the Bernoulli and mass sources respectively, while α and β are
constant indices. Note that A > 0 as this gives rise to an anticyclonic vortex and B > 0 to
give a mass-loaded vortex. We specify ρ(ψ) instead of B(ψ) as this enables a more meaningful
scaling later (Section 6.1.2).
When α = β = B = 0 we obtain the analytical Kida solution (Section 3.5.5, Kida, 1981;
Lesur and Papaloizou, 2009) which is a useful test case.
5.2.2 Constraints on the functions F (ψ) and ρ(ψ)
When τs → 0, the functions F (ψ) and ρ(ψ) have to be input externally. However, when
frictional processes are included, these functions are expected to be determined by the evolu-
tionary history of the system; as we are studying steady state solutions we do not study this
in this work.
However, we can show that there are two constraints on every streamline which must be
satisfied and which in principle allow F (ψ) and ρ(ψ) to be determined if the stopping time τs
(equation (3.3.5)) and the viscous force fν (equation (5.1.1b)) are known.
5.2.2.1 Constraints on every streamline arising from dust diffusion
We begin with equation (5.1.11) for the dust density in the form
∂ρd
∂t
+∇ · (ρdv) +∇ ·
(
ρdρgu
ρ
)
= 0.
68 Calculating equilibrium solutions
The dust-to-mass ratio, Γd, defined in Section 2.3.3 is related to ρg, ρd and ρ thus:
ρd = Γd ρ (5.2.7a)
ρg = (1− Γd)ρ. (5.2.7b)
Using equation (5.1.8) to eliminate the gradients of ρ and equation (5.1.10) to eliminate the
relative velocity u we obtain the following equation for Γd:
ρ
(
∂Γd
∂t
+ v · ∇Γd
)
+∇ · [Γd(1− Γd)τs(∇P − fν)] = 0. (5.2.8)
When turbulence is present there will be a stochastic component to u which is expected to
lead to dust diffusion. Assuming this is due to the action of fν , we model its effect over long
times by introducing a diffusion term for Γd. Thus we replace equation (5.2.8) with
ρ
(
∂Γd
∂t
+ v · ∇Γd
)
+∇ · [Γd(1− Γd)τs∇P − ρD∇Γd] = 0. (5.2.9)
where the diffusion coefficient is given in terms of the root mean square of the turbulent velocity
νt and the turbulent correlation time τcorr
D = ⟨ν2t ⟩τcorr. (5.2.10)
Equation (5.2.9) can be interpreted as an advection-diffusion equation for Γd. Note that
when there is dependence on z we could introduce an anisotropic diffusion coefficient to allow
diffusion in the vertical direction to independently balance the tendency for the dust to settle
due to the vertical pressure gradient (e.g. Dubrulle et al., 1995; Fromang et al., 2006, and
references therein). This would remove derivatives with respect to z from the steady state
form of equation (5.2.9).
Using the mass conservation equation for the average density ρ (equation (5.1.8)) and
repeated use of equation (5.2.7b), the steady state form of (5.2.9) is:
∇ ·
[
ρdv +
ρg
ρ
(
1− ρg
ρ
)
τs∇P + ρD∇
(
ρg
ρ
)]
= 0. (5.2.11)
We integrate it over a cylindrical volume for which the boundary of the cylinder in the (x, y)
plane is a streamline. An arbitrary interval may be taken in the z–direction as there is no
effective variation in z.
As the vertical surface is composed of streamlines, there is no contribution from the term
involving the combined fluid velocity v = ∇× ψzˆ. With the z integration performed along a
5.2 Functions specifying the vorticity and density profiles 69
line of unit length, the resulting surface integral becomes the following contour integral:
0 =
∫
V
∇ ·
[
ρdv +
ρg
ρ
(
1− ρg
ρ
)
τs∇P + ρD∇
(
ρg
ρ
)]
dV
=
∮
S
[
ρg
ρ
(
1− ρg
ρ
)
τs∇P + ρD∇
(
ρg
ρ
)]
· ∇ψ|∇ψ| ds
=
∮
S
[(
1− ρg
ρ
)
τs∇P −D∇ρ
]
· ∇ψ|∇ψ| ds. (5.2.12)
To reach the last line we have used the fact that ρg does not vary in horizontal planes and
ρ = ρ(ψ) so is constant round the line integral. Therefore, ρg/ρ may be removed as a factor.
Here ds is the line element on a streamline and we have used the fact that to lowest order in
τs and D, the integral around any streamline can be calculated using solutions for τs = 0.
Equation (5.2.12) can be viewed as expressing the balance between the rate of accumulation
of dust driven towards a pressure maximum by the pressure gradient and outward diffusion.
It gives a relation between D and other equilibrium quantities.
In addition, by considering the signs of the two terms comprising equation (5.2.12) we can
see that this condition is consistent with there being a density maximum at the centre of the
vortex if there is also a pressure maximum there and an inconsistency if there is a central
pressure minimum. When the pressure has a saddle point at the centre of the vortex then a
more detailed calculation is needed.
Also note that if D is assumed to be a function of ψ alone, it may be taken outside the
integral and calculated directly. Alternatively, if D is specified in terms of other quantities,
the constraint could be regarded as specifying the function dρ(ψ)/dψ. This can be seen by
writing equation (5.2.12) in the form
dρ
dψ
∮
S
D|∇ψ| ds =
∮
S
(
1− ρg
ρ
)
τs∇P · ∇ψ|∇ψ| ds. (5.2.13)
However, as both the form of ψ and the right hand side of equation (5.2.13) depend on the
choice of dρ(ψ)/dψ, there is an implicit dependence of these quantities on dρ(ψ)/dψ.
5.2.2.2 A constraint on every streamline arising from viscous diffusion
We return to the two–fluid momentum equation (5.1.9):
ρ
(
∂v
∂t
+ (v · ∇)v + 2Ω× v
)
= −∇P − ρ∇Φ+ fν .
We shall adopt the variable-density incompressible limit. It is possible to add correction
terms ∝ τs to this, which would lead to additional contributions to the discussion below. For
simplicity these will be neglected so that discussion applies to vortices with dust frozen into the
fluid. Furthermore, for convenience we shall drop the overbars in equation (5.1.9). Assuming
70 Calculating equilibrium solutions
the standard form (Batchelor, 2000), the i component of the viscous force per unit mass fν is
taken to be
fν,i =
∂
∂xj
[
ρgν
(
∂vi
∂xj
+ ∂vj
∂xi
)]
, (5.2.14)
where ν is the effective kinematic viscosity that we assume is independent of z (or can be
vertically averaged due to the thin disc approximation, Section 2.4.2). Defining the scaled
viscosity ν¯:
fν,i =
∂
∂xj
[
ρν¯
(
∂vi
∂xj
+ ∂vj
∂xi
)]
, (5.2.15)
where, recalling equation (5.1.5), ρ = ρg + ρd:
ν¯ = ρg
ρg + ρd
ν. (5.2.16)
Taking the scalar product of equation (5.1.9) with v and assuming a steady state we obtain
∇ ·
[
v
(
ρΦ+ 12ρ|v|
2 + P
)]
= v · fν . (5.2.17)
Now integrate this over a cylindrical volume, V , with a horizontal cross-sectional area A that
is bounded by a fixed streamline labelled by ψ. As we consider any steady states for which any
z–dependence of the density can be factored out (see Section 5.1.5) we assume independence
of z so, again, the height of this cylinder is arbitrary.
Since the velocity is parallel to the bounding surfaces (by definition of ψ), the use of the
divergence theorem shows that the left hand side of equation (5.2.17) gives no contribution to
the integral. Thus we obtain
0 =
∫
V
v · fν dA
=
∫
S
ρν¯vi
(
∂vi
∂xj
+ ∂vj
∂xi
)
ej |dS|
− 12
∫
V
ρν¯
(
∂vi
∂xj
+ ∂vj
∂xi
)(
∂vi
∂xj
+ ∂vj
∂xi
)
dA, (5.2.18)
where ej is the unit vector in the j direction. Taking the height of the cylinder to be of unit
length, as before, we may write dV → dA and |dS| → |ds| where ds is the line element parallel
to the bounding streamline.
The first term on the right hand side of equation (5.2.18) represents the work done by
viscous stresses on the bounding streamline. The second term represents the interior rate of
energy dissipation.
We can removed vi from the constraint (5.2.18) and instead express it in terms of the
5.2 Functions specifying the vorticity and density profiles 71
streamfunction since
v =
(
∂ψ
∂y
,−∂ψ
∂x
)
. (5.1.22)
In doing so, we allow the scaled kinematic viscosity ν¯ to vary with position; such a dependence
could occur through a dependence on ρ, in which case we could consider ν as a function of
ψ. The use of the divergence theorem, integration by parts and moving around some terms
results in: ∮
C
ρν¯
(
∇2ψ
)
|∇ψ||ds| =
∫
A
{
ρν¯
[
(∇2ψ)2 + 2∇ψ · ∇(∇2ψ)
]
+ ∇(ρν¯) · ∇
(
|∇ψ|2
) }
dA (5.2.19)
This condition is satisfied for the flow associated with an analytic Kida vortex (see Section 5.3)
when ν¯ is constant, as long as A is interior to the vortex. However, as the vorticity is dis-
continuous at the vortex boundary in this case, it cannot be satisfied more generally if it is
assumed that ρ and ν¯ are smoothly varying because the left hand side is discontinuous as the
boundary is passed through.
For general vortices, the forms of ν¯ and ψ can be seen to be connected by equation (5.2.19).
5.2.2.3 The case without dust
In this case the Poisson equation (5.1.30) is simply
∇2ψ = −ω(ψ) = 2Ω + dF
dψ
. (5.2.20)
Integrating the left hand side of equation (5.2.19) by parts again and using
∂ψ
∂xi
dω
dψ
= − ∂ψ
∂xi
d
dψ
(
∇2ψ
)
⇒ dω
dψ
∇ψ = −∇
(
∇2ψ
)
(5.2.21)
we find that the constraint due to viscous diffusion, equation (5.2.19), simplifies to∫
A
ρν¯ |∇ψ |2 dω
dψ
dA =
∫
A
∇(ρν¯) ·
[
∇
(
|∇ψ|2
)
−∇ψ(∇2ψ)
]
dA (5.2.22)
This can be used to specify dω/dψ in terms of ψ. Recalling that the integration is over an area
A enclosed by the streamline labelled by ψ,
dA = |ds|dψ|∇ψ| , (5.2.23)
72 Calculating equilibrium solutions
Figure 5.4 The vorticity profile inside the Kida vortex, also known as the ‘tophat’ vorticity
profile. This corresponds to an anticyclonic vortex in the inertial frame, hence the decrease in
vorticity.
so after differentiating equation (5.2.22) with respect to ψ we find
dω
dψ
∮
C
ρν¯ |∇ψ| dA =
∮
C
∇(ρν¯) ·
[
∇
(
|∇ψ|2
)
−∇ψ(∇2ψ)
] |ds|
|∇ψ| . (5.2.24)
This is now of the form of a constraint on a closed streamline. This indicates that variations
in vorticity require variations in ν¯ and that a smoothly varying ν¯ is associated with a smooth
vorticity profile. If a steady state vortex exists in a viscous fluid produced by e.g. RWI, the
vorticity profile is constrained by the form of the viscosity which also has to be consistent with
the RWI.
5.3 An analytic solution: the Kida vortex
There exists an exact solution of the incompressible Euler equations (5.1.14), (5.1.25) con-
sisting of an elliptical vortex patch of constant vorticity ωkida that matches to a Keplerian
shearing flow at large distances. The streamlines are concentric elliptical epicycles (Kida,
1981, Section 3.5.5). The contents of this section are not new work, but are useful to have in
one place.
5.3 An analytic solution: the Kida vortex 73
5.3.1 Calculating the streamfunction ψ
As established in Section 5.1.6, the Stokes’ streamfunction of a background Keplerian shearing
flow v0 = (0,−Sx, 0) is ψ0 = 12Sx2 (equation (5.1.24)).
Following the approach of Section 5.2 we split our streamfunction into the sum of its
background and superposed vortex parts:
ψ = ψ0 + ψ1 =
1
2Sx
2 + ψ1. (5.3.1)
This is a constant density solution, so B(ψ) = 0 and the Poisson equation (5.2.4) reduces to
∇2ψ1 = dF1
dψ
= A(ψ) =
c inside vortex0 outside vortex, (5.3.2)
where c is a constant obeying S + c = −ωkida and ωkida is the total vorticity inside the vortex
patch. The vorticity profile takes the form of a ‘top hat’ shape, as in Figure 5.4. We let ψ(in)
and ψ(ex) be the overall potentials interior and exterior to the vortex respectively:
∇2ψ(in) = S + c (5.3.3a)
∇2ψ(ex) = S. (5.3.3b)
Inside the vortex we look for a solution with elliptical streamlines of constant aspect ratio, so
we try a solution of the form
ψ(in) = A1x2 +A2y2 ⇒ 2(A1 +A2) = S + c (5.3.4)
In order to make progress it is now convenient to switch to an elliptical coordinate system1. In
this system the x– and y–coordinates are parameterised by a radius-like unit ξ and angular-like
unit η thus:
x = h sinh ξ sin η (5.3.5a)
y = h cosh ξ cos η. (5.3.5b)
The scaling quantity h is the focus of some specified ellipse. How these two parameters map to
the Cartesian system can be seen in Figure 5.5. Note that this coordinate system is a system
of confocal ellipses of different aspect ratios, χ, whereas the Kida solution contains streamlines
of the same χ throughout.
We get around this problem by scaling to the bounding streamline. We define the bounding
streamlines ψ = ψb of the vortex to pass through (x, y) = (0, 1) with ξ = ξb. The semi-minor
1For more details see Appendix A
74 Calculating equilibrium solutions
Figure 5.5 The elliptic coordinate system. The constant η curves are in green, the constant
ξ curves are the concentric ellipses. The boundary of the Kida vortex is given by ξ = ξb.
5.3 An analytic solution: the Kida vortex 75
and semi-major axes a and b are therefore
a = h sinh ξb
b = h cosh ξb = 1 (5.3.6)
and the aspect ratio χ is
χ = b
a
= 1tanh ξb
. (5.3.7)
with χ > 1 everywhere. The boundary between interior and exterior solutions is at
ξb = tanh−1
( 1
χ
)
= 12 log
(
χ+ 1
χ− 1
)
, χ > 1. (5.3.8)
Using standard hyperbolic identities,
(h sinh ξb, h cosh ξb) =
(
h√
χ2 − 1 ,
hχ√
χ2 − 1
)
(5.3.9)
and we define
h˜ = h√
χ2 − 1 . (5.3.10)
Our adapted elliptical coordinate system inside the vortex, for h˜ ∈ (0, χ−1] is therefore
x = h˜ sin η (5.3.11a)
y = h˜χ cos η. (5.3.11b)
Outside the vortex, ξ > ξb, we use the standard elliptical coordinates given by equation (5.3.5).
We will now consider the external solution then match ψ(in) and ψ(ex) on the boundary
ξ = ξb to determine our unknowns A1 and A2 given by equation (5.3.4). Outside the vortex,
ψ1 obeys Laplace’s equation and since[
∂2
∂x2
+ ∂
2
∂y2
]
ψ
(ex)
1 =
1
h2
(
sinh2 ξ + sin2 η
) [ ∂2
∂ξ2
+ ∂
2
∂η2
]
ψ
(ex)
1
= χ
2 − 1
h˜2
(
sinh ξ + sin2 η
) [ ∂2
∂ξ2
+ ∂
2
∂η2
]
ψ
(ex)
1 = 0
(see Appendix A) we are left with [
∂2
∂ξ2
+ ∂
2
∂η2
]
ψ
(ex)
1 = 0.
The boundary conditions at infinity are given by requiring the flow tends to the background
76 Calculating equilibrium solutions
shear for large x and y, so
∂ψ
(ex)
1
∂x
→ 0, ∂ψ
(ex)
1
∂y
→ 0 as x, y →∞ (5.3.12)
in the new coordinate system (again, see Appendix A) become
∂ψ
(ex)
1
∂η
→ 0, ∂ψ
(ex)
1
∂ξ
→ 0 as ξ →∞. (5.3.13)
Thus the solution of our Lagrangian for ψ(ex)1 plus the background term and an arbitrary
constant d gives our overall exterior potential
ψ(ex) = ψ0 + ψ(ex)1 =
1
2Sx
2 + α0ξ +
∑
m>0
αme
−mξ cos (mη) + d. (5.3.14)
We now need to match the interior and exterior solutions on the boundary of the ellipse,
insisting that ψ and the tangential velocity ∂ψ/∂ξ are continuous there. On the boundary
ξ = ξb and
A1x
2 +A2y2 = A1h2 sinh2 ξb sin2 η +A2h2 cosh2 ξb cos2 η = E.
For an elliptical streamline on the boundary, we need constant E:
A1h
2 sinh2 ξb = A2h2 cosh2 ξb
⇒ tanh ξb =
√
A2
A1
= 1
χ
. (5.3.15)
The exterior solution should also be constant on ξ = ξb since we want ξb to be a streamline.
Note that this requires αm = 0 for m ≥ 3 thus
ψ(ex)|ξ=ξb =
1
2Sh
2 sinh2 ξb sin2 η + α0ξb + α2e−2ξb cos 2η + d
= 14Sh
2 sinh2 ξb(1− cos 2η) + α0ξb + α2e−2ξb cos 2η + d
⇒ α2 = 14Sh
2e2ξb sinh2 ξb. (5.3.16)
We also require the continuity of ∂ψ/∂ξ on ξ = ξb:
∂ψ(in)
∂ξ
∣∣∣∣
ξ=ξb
= 2h2 cosh ξb sinh ξb
[
A1 sin2 η +A2 cos2 η
]
= h2 cosh ξb sinh ξb [(A1 +A2) + (A2 −A1) cos 2η]
∂ψ(ex)
∂ξ
∣∣∣∣
ξ=ξb
= 12Sh
2 cosh ξb sinh ξb (1− cos 2η) + α0 − 12Sh
2 sinh2 ξb cos 2η
5.3 An analytic solution: the Kida vortex 77
Comparing the coefficients of cos 2η we find
2
S
(A1 −A2) = 1 + tanh ξb. (5.3.17)
Therefore, with equations (5.3.4) and (5.3.15), we find:
c = S(χ+ 1)
χ(χ− 1) (5.3.18)
ωkida =
S(χ2 + 1)
χ(1− χ) . (5.3.19)
This shows that (as expected), the aspect ratio, χ, of a Kida vortex is a function of the
background shear S and the vorticity of the patch ωkida = −(S+c). The opposite signs of ωkida
and S shows that only anticyclonic vortices of this constant-vorticity type occur in Keplerian
flows. There are a range of pressure profiles associated with different χ, as can be seen in
Figure 5.6 which are calculated explicitly in Section 5.3.2. As we have found in Sections 3.4.3
and 5.2.2.1, this is particularly important when considering dusty gases as particles tend to
drift towards pressure maxima.
We can also find the values of α0, A1, A2 and d:
α0 =
Sh2
2(χ− 1)2 (5.3.20a)
A1 =
Sχ2
2χ(χ− 1) (5.3.20b)
A2 =
S
2χ(χ− 1) (5.3.20c)
d = Sh
2(1− 2ξb)
4(χ− 1)2 . (5.3.20d)
and therefore finally we arrive at our complete solution for ψ:
ψ(in) = S2χ(χ− 1)
(
χ2x2 + y2
)
(5.3.21a)
ψ(ex) = Sh
2
4 sinh
2 ξ(1− cos 2η) + Sh
2
4(χ2 − 1)e
2(ξb−ξ) cos 2η
+ Sh
2ξ
2(χ− 1)2 +
Sh2(1− 2ξb)
4(χ− 1)2 . (5.3.21b)
78 Calculating equilibrium solutions
5.3.2 The pressure distribution
5.3.2.1 The interior solution
To find the pressure distribution inside the vortex we apply equation (5.1.31) to the stream-
function. Thus for ψ(in) we find (dropping the superscripts)
∇ψ = (2A1x, 2A2y) ⇒ 12 |∇ψ|
2 = 2(A21x2 +A22y2),
so
P
ρ0
= 23S
2x2 − 12 |∇ψ|
2 + (c− S3 )ψ +
P0
ρ0
.
The constant P0/ρ0 term can be taken to be zero without any loss of generality. Substituting
for ψ and c using equation (5.3.4):
P
ρ0
=
[2
3S
2 + 2A1A2 − 43SA1
]
x2 +
[
2A1A2 − 43SA2
]
y2, (5.3.22)
so P/ρ0 is a quadratic form and we expect the pressure contours to be either elliptical or
hyperbolic. We make the substitutions
αp=
2
3S
2 + 2A1A2 − 43SA1=
S2(7− 4χ)
6(χ− 1)2 (5.3.23a)
βp= 2A1A2 − 43SA2 =
S2(4− χ)
6χ(χ− 1)2 (5.3.23b)
such that
P
ρ0
= αp x2 + βp y2. (5.3.24)
5.3.2.2 Pressure gradients inside the vortex
It is useful to consider gradients of P in the h˜– and η–directions. Using the adapted elliptical
coordinates described in equation (5.3.5) we find that
P = ρ0h˜
2S2
6(χ− 1)2
[
(7− 4χ) sin2 η + χ(4− χ) cos2 η
]
(5.3.25a)
∂P
∂h˜
∣∣∣∣
η
= ρ0h˜S
2
3(χ− 1)2
[
(χ− 1)(χ− 7) sin2 η + χ(4− χ)
]
(5.3.25b)
∂P
∂η
∣∣∣∣
h˜
= ρ0h˜
2S2
3(χ− 1)2 (χ− 7) sin η cos η (5.3.25c)
5.3 An analytic solution: the Kida vortex 79
Considering equation (5.3.25b), let a = sin2 η and
f(χ, a) = (χ− 1)(χ− 7)a+ χ(4− χ), (5.3.26)
where f(χ, 0) = χ(4− χ) and f(χ, 1) = 7− 4χ. For a ̸= 1,
f(χ, a) =
[
9
(
a− 12
)2
+ 74
]
− (1− a)
[
χ+ 2a(2a− 1)1− a
]2
(5.3.27)
so for a ∈ [0, 1) this is a parabola with a maximum point. We are interested in the behaviour
of f(χ, a) as a = sin2 η varies as this will indicate whether the core of a Kida vortex contains
a minimum, maximum or saddle point in its pressure distribution. The positive root (since
χ > 1) of f(χ, a) = 0 occurs at
χ∗ =
−2(2a− 1) +
√
9
(
a− 12
)2
+ 74
1− a . (5.3.28)
χ∗ decreases monotonically from χ∗ = 4 in the range a ∈ [0, 1), with χ∗ → 7/4 as a → 1.
Therefore, as can be seen in Figure 5.7, f(χ, a) = f(χ, sin2 η) will be negative ∀ η if χ > 4 and
positive ∀ η if χ < 7/4.
Since h˜ increases outwards from the vortex centre, we find that we have:
pressure minima 1 < χ < 7/4
saddle 7/4 < χ < 4
pressure maxima χ > 4.
The first case is of the least astrophysical interest as dust is drawn up pressure gradients to
pressure maxima (Sections 3.4.3 and 3.5) so these vortices will not attract dust to their core
(e.g Chavanis, 2000). The χ→ 1 limit is also that of the infinitely strong vortex, so we expect
that the creation of such vortices is also difficult in practice. Correspondingly, the weaker
vortices with χ > 4 are of the most interest in a planetesimal formation context, while the
transition cases 7/4 < χ < 4 prove to exhibit interesting stability behaviour (see Section 3.4.3).
The pressure distributions for a variety of Kida vortices of different aspect ratios can be
found in Figure 5.6, where the minima, maxima and saddle regions can be seen.
Considering equation (5.3.25c) for the pressure gradient around a streamline, we find it is
zero for an entire streamline when χ = 7. This special case has particular importance when we
look at the parametric model in Section 5.4. It also means that the frozen–in dust distribution
is axisymmetric around streamlines inside the vortex for Kida vortices when χ = 7 (Lyra and
Lin, 2013).
80 Calculating equilibrium solutions
-2
-1
0
1
2
-2 -1 0 1 2
y
x
0
1
2
3
4
5
6
7
P
(a) χ = 3/2
-2
-1
0
1
2
-2 -1 0 1 2
y
x
0
0.5
1
1.5
2
2.5
P
(b) χ = 13/8
-2
-1
0
1
2
-2 -1 0 1 2
y
x
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P
(c) χ = 7/4
-2
-1
0
1
2
-2 -1 0 1 2
y
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
P
(d) χ = 2
-2
-1
0
1
2
-2 -1 0 1 2
y
x
0
0.05
0.1
0.15
0.2
0.25
0.3
P
(e) χ = 3
-2
-1
0
1
2
-2 -1 0 1 2
y
x
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
P
(f) χ = 4
5.3 An analytic solution: the Kida vortex 81
-2
-1
0
1
2
-2 -1 0 1 2
y
x
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
P
(g) χ = 5
-2
-1
0
1
2
-2 -1 0 1 2
y
x
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
P
(h) χ = 8
Figure 5.6 The pressure distributions of Kida vortices with a range of aspect ratios. As the
aspect ratio increases (or, the amount of additional vorticity inside the vortex patch decreases),
a pressure maximum appears inside the vortex.
In Figure 5.6a we see a pressure minimum at the centre of the vortex, which by Figure 5.6c
has moved to a transitive state where the pressure has no x–dependence in the vortex core.
In Figure 5.6d we have a saddle point, with hyperbolic pressure contours inside the vortex,
while we see y–independence for P in Figure 5.6f, the second transitory case. For χ > 4 there
is a pressure maximum, as can be seen in Figures 5.6g and 5.6h. This is important as dust is
drawn towards local pressure maxima.
82 Calculating equilibrium solutions
-15
-10
-5
0
5
10
1 74 4
f
(χ
,a
) χ
a = 0
a = 1/4
a = 1/2
a = 3/4
a = 1
Figure 5.7 The behaviour of f(χ, a) = (χ− 1)(χ− 7)a+ χ(4− χ), where a = sin2 η. f(χ, a)
governs the Kida pressure distribution in the ‘radial’ direction h˜. When χ > 4, f < 0 and there
is a pressure maximum, while for 1 < χ < 7/4, f > 0 and the core contains a dust–rejecting
pressure minima. There is a saddle point between these two regions.
5.3.2.3 The exterior solution
For completeness, here the pressure distribution obeys
P
ρ0
= 23S
2x2 − 12 |∇ψ|
2 − 13Sψ (5.3.29)
with ψ = ψ(ex). In order to work out the 12 |∇ψ|2 term we need
|∇ψ|2 =
(
∂ψ
∂x
)2
+
(
∂ψ
∂y
)2
= 1
h2(sinh2 ξ + sin2 η)
[(
∂ψ
∂η
)2
+
(
∂ψ
∂ξ
)2]
.
5.3 An analytic solution: the Kida vortex 83
(see Appendix A). Thus, for ψ(ex)
(
∂ψ
∂ξ
)2
= S
2h4
4
[
sinh 2ξ sin2 η (5.3.30a)
− e
2(ξb−ξ)
χ2 − 1 cos 2η +
1
(χ− 1)2
]2
= S
2h4
4 D
2
ξ
(
∂ψ
∂η
)2
= S
2h4
4 sin
2 2η
[
sinh2 ξ − e
2(ξb−ξ)
χ2 − 1
]2
= S
2h4
4 D
2
η (5.3.30b)
and
P
ρ0
= 23S
2h2 sinh2 ξ sin2 η − S
2h2
8(sinh2 ξ + sin2 η)
[
D2ξ +D2η
]
− 13Sψ
(ex) (5.3.31)
5.3.3 Period round a streamline
It is useful to investigate the internal shear within the vortices by calculating the period
P˜ =
∮
dσ
|∇ψ| (5.3.32)
around streamlines, where σ is our arclength (see Section 6.3 for results and Section 8.1.1.3 for
how it is calculated from the gridded data). Kida vortices have no internal shear so we expect a
constant period throughout the vortex patch. Parametrising our streamline using the adapted
elliptical coordinates of Section 5.3.1, equation (5.3.5), we have (x, y) = (h˜ sin η, h˜χ cos η) and
therefore our arclength element is
dσ =
√
(dx)2 + (dy)2
= h˜
√
cos2 η + χ2 sin2 η dη (5.3.33)
while equation (5.3.21a) implies that
|∇ψ| = S
χ(χ− 1)
√
χ4x2 + y2 (5.3.34)
so substituting for the elliptical coordinates again, equation (5.3.32) yields
P˜kida =
∮
dσ
|∇ψ|
=
∫ 2π
0
h˜
√
cos2 η + χ2 sin2 η
Sh˜
(χ−1)
√
χ2 sin2 η + cos2 η
dη
=
∫ 2π
0
χ− 1
S
dη
= 2π
S
(χ− 1), (5.3.35)
84 Calculating equilibrium solutions
which is indeed independent of streamline choice inside the vortex.
5.4 The polytropic model
It is possible to consider different values of S (i.e. a non–Keplerian background flow) while
retaining the potential Φ appropriate for a Keplerian disc. This requires the pressure gradient
to be non-zero in the background flow and consequently enables us to consider situations where
the vortex is centred on a background where there is pressure extremum. This is of interest
as dust is expected to accumulate at the centre of a ring where there is a pressure maximum
(Whipple, 1972). The Rossby wave instability can also result in vortices forming at such
locations (see e.g. Méheut et al., 2010, 2012b).
For example, if we set
S =
√
3(χ− 1)
χ+ 1 Ω (5.4.1)
then we have (to within a constant in the vortex core) that
P
ρ
= − Sψ
χ(χ− 1) + F (ψ). (5.4.2)
This makes P/ρ a function of ψ alone; it will be a linear function of ψ provided that F (ψ) is.
However, when such a solution is matched to an exterior function (see Lesur and Papaloizou,
2009) the background flow will correspond to one with vy = −Sx. From equation 5.4.1, this
corresponds to the Keplerian case strictly only when χ = 7. For other values of S, consideration
of the exterior Kida solution implies that there is an implied background pressure maximum
at the co-orbital radius for χ > 7 and a background pressure minimum (which will not attract
dust) at χ < 7.
This turns out to be useful for constructing a vortex model with non-uniform density. This
will have a stream function of the form in equation 5.3.21a inside the vortex core with S given
by equation (5.4.1). The density will take the form
ρ =
[
1− b(ψ − ψb)
ψb
]n
(5.4.3)
inside the vortex, where ψb is the stream function on the core boundary and the background
density is taken to be unity. The quantities b and n are constants determining the profile and
magnitude of the density excess above the background. At the vortex centre ρ = (1+b)n while
at the boundary ρ = 1, the background value.
Following the standard form taken by the pressure in polytropic models P ∝ ρ1+1/n (Chan-
5.5 Summary and conclusions 85
drasekhar, 1939), the pressure is assumed to take the form
P = βPψbρ
1+1/n
b(n+ 1) , (5.4.4)
where βP is a constant determined such that the equilibrium conditions apply. Substituting
for ∇2ψ and ρ in equation (5.1.30) yields:
S(χ2 + 1)
χ(χ− 1) =
dF (ψ)
dψ
+ 2Ω− nβP
n+ 1 (5.4.5)
while substituting for P , ρ and ψ in equation (5.4.2) gives
βPψb
b(n+ 1)
[
1− b(ψ − ψb)
ψb
]
= − Sψ
ψ(ψ − 1) + F (ψ). (5.4.6)
Taking the derivative of equation (5.4.6) with respect to ψ and eliminating dF/dψ from equa-
tion (5.4.5) results in an expression for βP :
βP = Ω
(
2− χ
√
3
χ2 − 1
)
. (5.4.7)
The parameter n > 0 can be specified arbitrarily, while b can then be chose to scale the density
excess above the background in the centre of the vortex provided χ > 2 (βP disappears when
χ = 2.) Note that this is related to the GNG solution of Goodman et al. (1987) with their
variable ε = χ−1.
5.5 Summary and conclusions
In this chapter, starting from the two–fluid equations we formulate the equations of motion
for a dusty gas in a 2D, Keplerian, shearing flow. In the incompressible limit we consider the
case where the friction/ stopping time τs → 0, so the dust and gas are tightly coupled. We
can then formulate a Poisson equation for the Stokes’ streamfunction for a vortex patch in the
local shearing sheet approximation.
The source term of this Poisson equation can be split into a vorticity and density contri-
bution, for which we choose to adopt power law forms of ψ. This will give us the freedom to
investigate the stability of a large variety of vortex solutions, with and without mass.
We also detail the analytical Kida solution, which will provide a good test case for sub-
sequent numerical work. The polytropic solution of Goodman et al. (1987), although not
generally true for a Keplerian shearing flow, is useful in future stability analysis.
In the next chapter we will discuss finding numerical solutions to these equations and the
results of this.

Chapter 6
Calculating equilibrium solutions:
numerical approach and results
In Chapter 5 we described our mathematical model for our set of equilibrium solutions. We
now need a method of calculating these. In this chapter we present how we solved for these
solutions numerically and tested their validity and convergence.
In Section 6.1 we show how we set up the problem, presenting the various parametrisations
and scalings used. Section 6.2 examines the numerical routines used to iteratively solve the
Poisson equation with discussion of the problems found, while Section 6.3 demonstrates a selec-
tion of equilibrium solutions, with discussion of the importance of their pressure distributions
and internal shear.
6.1 Setting up the problem
Since analytical solutions to the Poisson equation (5.2.4) only occur for the constant-density,
constant-vorticity Kida case we must integrate this equation numerically to produce different
vortex solutions. We will organise them into different aspect ratio solutions within different
vortex classes.
6.1.1 Parametrising the vortex class
We need a way of prescribing different combinations of different A(ψ) and B(ψ). We do this
through four numbers:
α : the vorticity power law index, equation (5.2.6a)
β : the density power law index, equation (5.2.6b)
ρm : a measure of the total mass added
ωm : a measure of the total vorticity added
88 Calculating equilibrium solutions: numerical approach and results
We will refer to the first three numbers {α, β, ρm} as a vortex class. We produce different
aspect ratio, χ, solutions inside this vortex class by varying ωm.
Another secondary quantity is ρmax, the central density enhancement, given by
ρmax = max
ψ
[ρ(ψ)− ρ0] . (6.1.1)
Its relationship to χ is shown in Section 6.3.3.
6.1.2 Scalings
As mentioned in Section 5.2.1, we chose our arbitrary units of length, time and mass such that
we have:
• a length scale such that bounding streamline passes though (x, y) = (0, 1)
• a time scale such that Ω = 1
• a mass scale such that ρ0 = 1.
We need to scale both the Bernoulli and density sources in the momentum equation (5.2.4)
and we do this via the two quantities ωm and ρm. We define ωm as
πωm =
∫
A(ψ) dS, (6.1.2)
where the integral is taken over the total area inside the vortex core, i.e. inside the bounding
streamline that passes through (0, 1). This quantity is a measure of the total vorticity inside
the vortex. In order to rescale the Bernoulli source (equation (5.2.6a)), we recalculated A after
every iteration to ensure that πωm remained fixed. This procedure was iterated until there
was no discernible difference in successive streamfunction and pressure contours; the global
streamline pattern will then have converged. For vortices with B = 0 this generally required
about 50 iterations.
Comparison with the analytic solution for the Kida vortex case (Section 5.3) enabled our
numerical procedure to be checked.
In a similar way we scale the mass added to the vortex by imposing a fixed value of ρm,
where
ρm =
∫
[ρ(ψ)− ρ0] dS, (6.1.3)
where again integration is over the vortex core. Note that scaling additional mass in this way
(as to be consistent with the treatment of total vorticity) results in different strength vortices
having different central mass enhancements, ρmax; for the same ρm, more elongated structures
will have a greater ρmax (Section 6.3.3).
6.2 Calculating equilibrium vortex solutions using vortex.f90 89
6.2 Calculating equilibrium vortex solutions using vortex.f90
The program vortex.f90 is designed to take the four descriptors {α, β, ρm, ωm} and a grid
size [n,m] and return the equilibrium vortex solution. The general procedure for this can
be seen in the flow chart in Figure 6.1. Due to the extra computation it takes to calculate
the pressure distribution at each iteration, initially an equilibrium solution is found with zero
density perturbation ∀β,B (i.e. the first loop in the flow chart). Only once this has been
found is density added to the solution, if B ̸= 0. In order that the addition of this mass does
not perturb the new solution too far away from the previous one at each iteration, mass is
added gradually to enable convergence (see Section 6.2.4 for details).
We perform our calculations on a 512× 2048 grid that covers the vortex core. The x– and
y–coordinates are both in the range [−1, 1]. This grid was chosen to ensure enough resolution to
undertake a local stability analysis (see Chapters 7 and 8). Very similar results were obtained
when the grid resolution was reduced by a factor of two. We have different resolutions in the
x– and y–directions because of the geometry of our vortex; it always has a semi-major axis of
length 1 along the y–axis.
An unavoidable result of this is that for elongated vortices there are significantly fewer grid
points resolving the core on the x–axis than on the y–axis. Consequently, in the latter stability
analysis we usually use a streamline that passes through (0, 0.85), where the streamline is still
contains a large number of grid points.
6.2.1 Calculating the Poisson integral
We initially solve a reduced version of the Poisson equation with B = 0 that is applicable to
the uniform density case, namely
∇2ψ1 = A(ψ). (6.2.1)
We start by defining the Bernoulli source term A to be a nonzero constant inside the unit
circle passing through (0, 1) and zero outside this. We then solve equation (6.2.1) using the
2D Green’s Function, obtaining
ψ1(r) =
1
2π
∫∫
log |r− r′| A(ψ) d2r′. (6.2.2)
Calculation of the Green’s function 12π log |r − r′| is an O(n2m2) operation. Therefore, to cut
down computational time, the Green’s function is only calculated for x ∈ [0, 1], y ∈ [0, 1]. The
symmetry of the solution is then exploited to find ψ for the other three quadrants.
We add on the background shear ψ0 = 12Sx2 to ψ1 to get the total streamfunction ψ. Using
the value of ψ at (0, 1) we find the coordinates of the bounding streamline. Inside this boundary
we then rescale the Bernoulli source as indicated in Section 6.1.2 and equations (6.1.2) and
(5.2.6a). and apply the Green’s function to obtain an updated solution for ψ1. This procedure is
90 Calculating equilibrium solutions: numerical approach and results
restart? define initial source
read in initial
distributions
calculate
bounding
streamline
scale vorticity source calculateA(ψ) using ψ
do Poisson integral, update ψ1
add background to find ψ,
calculate bounding streamline
value
of i?
calculate
P/ρ using
eqn (6.2.3)
calculate ρ distribution using eqn (5.2.6b)
scale density
calculate B(ψ) and total source
do Poisson integral, update ψ1
add background to find ψ,
calculate bounding streamline
value of
j?
calculate P/ρ, output results
no
yes
i ∈ [1, N1)
i = N1
j = N2
j ∈ [1, N2)
Figure 6.1 Program design for vortex.f90. The two main loops are controlled by the
variables N1 and N2, which typically took the values 40 and 20 respectively. The boxes
coloured pink are procedures expanded in detail in the text. Restarted runs already have the
correct equilibrium vorticity solution found so the first loop is bypassed when the restart option
is set.
6.2 Calculating equilibrium vortex solutions using vortex.f90 91
repeated for N1 iterations; usually a N1 = 40 was sufficient to obtain a density-free equilibrium
vortex solution.
6.2.2 Calculating the pressure distribution
We now have the total streamfunction for the flow ψ = ψ0 + ψ1 for a constant density equi-
librium. To find solutions with increased density in the central parts of the vortex core the
pressure distribution has to be calculated at each iteration. This is found from equation (5.1.31)
which yields
P
ρ
= 23S
2x2 − 12 |∇ψ|
2 − 13S (ψ − ψb) +
∫ ψ
ψb
A(ψ′) dψ′, (6.2.3)
where ψb is the value of the streamfunction on the boundary of the vortex. The terms ∂ψ/∂x
and ∂ψ/∂y are calculated using first order finite differences. We use the trapezium rule to
approximate this integral for each ψ on grid points inside the vortex
∫ ψ
ψb
A(ψ′) dψ′ ≈ h
[
A(ψb) +A(ψ)
2 +
n−1∑
i=1
A(ψi)
]
(6.2.4)
where
h = ψ − ψb
n
(6.2.5a)
ψi = ψb + ih. (6.2.5b)
Throughout our calculations, n = 100.
6.2.3 Calculating B(ψ) and the total source term
With a form for ψ we use equation (5.2.6b) to find ρ(ψ). After calculating |ψ − ψb|β we scale
the mass added by this contribution using equation (6.1.3). We set ρ = 1 and ∇ρ = 0 outside
the vortex and prescribe dρ/dψ inside the vortex exactly. We can then assemble the density
source B(ψ) = d log ρ/dψ and hence the total source term
A(ψ) + P
ρ
B(ψ). (6.2.6)
We can now integrate this using the 2D Green’s function of the form
ψ1(r) =
1
2π
∫∫
log |r− r′|
[
A(ψ) + P
ρ
B(ψ)
]
d2r′ (6.2.7)
where the P/ρ distribution is calculated from the previous iteration via equation (6.2.3). This
loop is repeated N2 times for each ρm (which is itself increased incrementally to aid convergence
– see Section 6.2.4). Typically N2 = 20.
92 Calculating equilibrium solutions: numerical approach and results
6.2.4 Convergence
Convergence for models with constant density was straightforward, although resolution issues
arise for a fixed grid when α becomes too large due to the peaking of the vorticity profile in
the centre of the vortex. To avoid this we used α ≤ 4.
For models with enhanced density inside the vortex core, convergence was more difficult
and required a starting model close to the final one. We began by imposing only small increases
to either, or both of β and ρm from the values appropriate to an existing equilibrium solution.
These changes moved them in the direction of our target parameters. This procedure was
especially necessary for the cores of vortices with non-Kida, α ̸= 0, vorticity profiles. Having
done this, a new form of ψ1 was obtained as before by iterating the Green’s function solution
N2 ≥ 10 times. At this point additional small increments of the order of 1% were made to ρm
and the process repeated until the target values were attained. The Green’s function solution
could then be iterated to convergence.
Convergence was tested for in a systematic way by observing the percentage change in the
value of the source term A(ψ) at the origin. Once the change in A(ψ) was beneath a fixed
value (10−4) between iterations, convergence was deemed to have occurred1. For solutions that
didn’t converge (e.g. because too much mass was added in one go), one saw large and often
periodic variations in this value. There was a ceiling imposed on N2 to terminate such cases.
6.3 Results
We begin by reproducing the streamlines for Kida vortices. Since analytical expressions exist
for the streamfunction and pressure distributions (Section 5.3) our numerical procedure can
be validated.
The aspect ratio, χ, of a vortex is a good way to parametrise its relative strength. Weak
vortices will be sheared out more by the background flow and will therefore have a large aspect
ratio, while strong vortices have a small aspect ratio. We are also interested in cases for which
the resulting pressure distribution has a maximum at the centre of the vortex as this location
will then attract dust (Section 3.5, Whipple, 1972). Recall that while {α, β, ρm} are fixed, ωm
is varied to produce vortices with different χ.
The aspect ratio for each vortex stated in the figures is measured by evaluating the ratio of
the largest y– and x–coordinates for the streamline passing through (0, 0.85). The boundary
itself was not used as in the stability calculations in Chapters 7 and 8, calculating certain
derivatives on the boundary proved problematic (due to e.g. discontinuities in the vorticity
profile). Calculating derivatives on a streamline a little away from the boundary solved these
problems, so the χ calculations in this chapter are consistent with this streamline choice.
1A Kida vortex with χ = 5 would have A(ψ) = 0.45 inside the vortex patch. α > 0 solutions would have
larger values.
6.3 Results 93
We also investigate the internal shear within the vortices by calculating the period around
streamlines (see Section 5.3.3 and equation (5.3.32)). Kida vortices have no internal shear so
a constant period throughout the vortex patch, P˜kida = 2π(χ− 1)/S. The results for the Kida
vortex are in good agreement with analytic expectation (see Figure 6.2a). We plot P˜ against
the value of the positive y–coordinate ymax where each streamline crosses the y–axis.
Note that the periods we plot were obtained by locating coordinate extrema from the results
of numerical integrations of fluid particles moving around streamlines stored on a relatively
coarse temporal grid. This results in some low level jitter at a relative level ∼ 10−3. This is also
a measure of the departure from the constant value of the period P˜kida obtained numerically
in the Kida vortex case.
For each equilibrium solution displayed we will show the vorticity/density distribution, the
ψ contours, the pressure distribution and P˜ against ymax.
6.3.1 Solutions with constant density
In Figure 6.2 we illustrate results for constant-density vortices with the same value of ωm. We
increase the power law index α of the Bernoulli source term (A(ψ), equation (5.2.6a)) over the
range 0 ≤ α ≤ 4, with α = 0 corresponding to the Kida vortex. This results in an increased
concentration of vorticity in the centre of the vortex.
Vortices with larger α but equivalent ωm have characteristics of stronger Kida vortices as
they are sheared less by the surrounding flow and accordingly have smaller χ. This process
also decreases the pressure maximum (or removes it altogether), making these vortices less
susceptible to dust collection.
Apart from the case for α = 4 in Figure 6.2d, there is a pressure maximum at the centre
of the vortex patch, with a saddle point in the latter case. We find that the transition from
central saddle point to maximum occurs at smaller aspect ratios as α increases and the vorticity
distribution becomes more centrally concentrated.
In addition, streamlines for vortices with centrally concentrated vorticity sources tend to
become pinched towards the y–axis near the boundary, compared to the Kida case. This leads
to a relative increase in circulation period. Another result of this is that the χ of streamlines
near the centre of the vortex are smaller than that of boundary streamlines. This needs to
be considered when plotting stability curves against aspect ratio in subsequent chapters. This
effect is demonstrated in Figures 6.3 and 6.4.
The introduction of a nonzero α also introduces a shear inside the vortex. A fluid parcel
on a streamline in a Kida vortex core will move round with the same period as all other fluid
parcels and thus there is no internal shear in the vortex. However, the introduction of a non-
uniform vorticity profile destroys this configuration and we see the development of a nonlinear
relationship between the period around a streamline and its position in the vortex core. This
effect becomes more noticeable as α increases.
94 Calculating equilibrium solutions: numerical approach and results
0
0
.1
0
.2
0
.3
0
.4
0
.5
-1
-0
.5
0
0
.5
1
−∇
2
ψ
1
y
-1
-0
.50
0
.51
-1
-0
.5
0
0
.5
1
y
x
-0
.1
00
.1
0
.2
0
.3
0
.4
0
.5
0
.6
0
.7
0
.8
ψ
-1
-0
.50
0
.51
-1
-0
.5
0
0
.5
1
y
x
00
.0
1
0
.0
2
0
.0
3
0
.0
4
0
.0
5
0
.0
6
0
.0
7
P
1
6
.7
1
6
.8
1
6
.9
0
0
.2
0
.4
0
.6
0
.8
1
P˜
y
P˜
k
i
d
a
=
1
6
.7
5
M
o
d
e
l
P˜
k
i
d
a
(a
)
{α
,β
,ρ
m
,ω
m
}=
{0
,0
,0
,0
.0
9}
,t
he
K
id
a
vo
rt
ex
w
he
re
χ
=
5
0
0
.2
0
.4
0
.6
0
.81
-1
-0
.5
0
0
.5
1
−∇
2
ψ
1
y
-1
-0
.50
0
.51
-1
-0
.5
0
0
.5
1
y
x
-0
.1
00
.1
0
.2
0
.3
0
.4
0
.5
0
.6
0
.7
0
.8
ψ
-1
-0
.50
0
.51
-1
-0
.5
0
0
.5
1
y
x
00
.0
1
0
.0
2
0
.0
3
0
.0
4
0
.0
5
0
.0
6
0
.0
7
P
9
1
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
0
0
.2
0
.4
0
.6
0
.8
1
P˜
y
P˜
k
i
d
a
=
1
2
.5
5
M
o
d
e
l
P˜
k
i
d
a
(b
)
{1
.0
,0
,0
,0
.0
9}
,w
ith
χ
=
4.
0.
0
0
.2
0
.4
0
.6
0
.81
1
.2
1
.4
1
.6
-1
-0
.5
0
0
.5
1
−∇
2
ψ
1
y
-1
-0
.50
0
.51
-1
-0
.5
0
0
.5
1
y
x
-0
.1
00
.1
0
.2
0
.3
0
.4
0
.5
0
.6
0
.7
0
.8
ψ
-1
-0
.50
0
.51
-1
-0
.5
0
0
.5
1
y
x
00
.0
1
0
.0
2
0
.0
3
0
.0
4
0
.0
5
0
.0
6
0
.0
7
0
.0
8
P
6789
1
0
1
1
1
2
1
3
1
4
1
5
1
6
0
0
.2
0
.4
0
.6
0
.8
1
P˜
y
P˜
k
i
d
a
=
1
0
.8
8
M
o
d
e
l
P˜
k
i
d
a
(c
)
{2
.0
,0
,0
,0
.0
9}
,w
ith
χ
=
3.
6.
F
ig
ur
e
6.
2
(C
on
tin
ue
d
on
ne
xt
pa
ge
.)
6.3 Results 95
0
0
.51
1
.52
2
.53
3
.54
4
.5
-1
-0
.5
0
0
.5
1
−∇
2
ψ
1
y
-1
-0
.50
0
.51
-1
-0
.5
0
0
.5
1
y
x
-0
.1
00
.1
0
.2
0
.3
0
.4
0
.5
0
.6
0
.7
0
.8
ψ
-1
-0
.50
0
.51
-1
-0
.5
0
0
.5
1
y
x
00
.0
1
0
.0
2
0
.0
3
0
.0
4
0
.0
5
0
.0
6
0
.0
7
0
.0
8
0
.0
9
P
2468
1
0
1
2
1
4
1
6
1
8
2
0
2
2
0
0
.2
0
.4
0
.6
0
.8
1
P˜
y
P˜
k
i
d
a
=
9
.2
4
F
it
P˜
k
i
d
a
(d
)
{4
.0
,0
,0
,0
,0
.0
9}
,r
es
ul
tin
g
χ
=
3.
2.
F
ig
ur
e
6.
2
Vo
rt
ic
es
w
ith
ω
m
=
0.
09
an
d
de
ns
ity
eq
ua
l
to
th
e
ba
ck
gr
ou
nd
va
lu
e.
Fr
om
le
ft
to
rig
ht
we
ha
ve
:
(i)
th
e
vo
rt
ic
ity
di
st
rib
ut
io
n,
(ii
)
th
e
ψ
di
st
rib
ut
io
n,
(ii
i)
th
e
pr
es
su
re
di
st
rib
ut
io
n
an
d
(iv
)
a
pl
ot
of
th
e
pe
rio
d
P˜
ar
ou
nd
a
st
re
am
lin
e
ag
ai
ns
t
th
e
va
lu
e
of
th
e
po
sit
iv
e
y
–c
oo
rd
in
at
e
w
he
re
it
in
te
rs
ec
ts
th
e
y
–a
xi
s.
In
cr
ea
sin
g
α
fro
m
0
to
4
re
su
lts
in
a
vo
rt
ex
th
at
is
st
ro
ng
er
(i.
e.
sh
ea
re
d
le
ss
by
th
e
ba
ck
gr
ou
nd
flo
w
)a
nd
ac
co
rd
in
gl
y
ha
sa
sm
al
le
ra
sp
ec
tr
at
io
.N
ot
e
th
at
al
lt
he
se
vo
rt
ic
es
,e
xc
ep
tt
he
on
e
ill
us
tr
at
ed
in
th
e
bo
tt
om
pa
ne
ls,
ha
ve
a
pr
es
su
re
m
ax
im
um
at
th
e
vo
rt
ex
ce
nt
re
.
T
he
pr
es
su
re
di
st
rib
ut
io
n
in
th
e
la
tt
er
vo
rt
ex
ha
s
a
sa
dd
le
po
in
t.
T
he
re
is
al
so
sig
ni
fic
an
t
sh
ea
r,
as
in
di
ca
te
d
by
th
e
va
ria
tio
n
in
th
e
pe
rio
d
P˜
to
ci
rc
ul
at
e
ar
ou
nd
a
st
re
am
lin
e.
96 Calculating equilibrium solutions: numerical approach and results
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
y
x
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ψ
Figure 6.3 The streamlines of the vortex {4, 0, 0} with ωm = 0.0183 (χkida = 10) where one
can see significant variation of streamline aspect ratio with position in the vortex.
A consequence of this is that disturbances that overlap adjacent streamlines are expected
to shear out. Therefore localised disturbances might be expected to have different behaviour
to that exhibited in a Kida vortex and thus the local stability may be different.
6.3 Results 97
1
2
3
4
5
6
7
8
9
0 0.2 0.4 0.6 0.8 1
χ
ymax
(0.018, 10)
(0.026, 8.6)
(0.032, 7.8)
(0.050, 6.4)
(0.090, 5.0)
(0.333, 3.0)
(1.125, 2.0)
(4.592, 1.4)
(a) The variation of χ with ymax for different strength vortices with α = 1.
1
2
3
4
5
6
7
0 0.2 0.4 0.6 0.8 1
χ
ymax
(0.018, 10)
(0.026, 8.6)
(0.032, 7.8)
(0.050, 6.4)
(0.090, 5.0)
(0.333, 3.0)
(1.125, 2.0)
(4.592, 1.4)
(b) The variation of χ with ymax for different strength vortices with α = 4.
Figure 6.4 The pair of (ωm, χkida) describes each vortex, where ωm is given by equation (6.1.2)
and χkida is the Kida aspect ratio you would achieve for a vortex patch with ωm added.
Here we can see the significant variation in the aspect ratio for streamlines within the same
vortex when α = 4 caused by the ‘pinching’ of streamlines close the boundary. Recall that in
the Kida case these lines will be horizontal as the streamlines are ellipses. Figure 6.3 shows
the streamlines of the vortex with ωm = 0.018333 where one can clearly see this ‘pinching‘.
98 Calculating equilibrium solutions: numerical approach and results
6.3.2 Solutions with a central density enhancement
In our models the dust is completely coupled to the gas (i.e. the τs → 0 limit in a two fluid
model), so density increases above the background value model a concentration of dust.
In Figure 6.5 we give results for vortices for which the Bernoulli vorticity source A(ψ) is
identical to that of the Kida case. A density enhancement is imposed through the application
of a scaled density profile extending over the whole vortex. Results are given for models for
which the central density ranges from between four and 16 times the background value ρ0 = 1.
Results for vortices with variable Bernoulli sources of the same form as those illustrated in
Figure 6.2 together with a density enhancement are given in Figure 6.6.
In all these cases the solutions were made by taking the mass–free versions of the vortex
class as our initial solution (e.g. if we wanted to make vortices with {0, 1, 1} we would start
from {0, 0, 0}), then changing β to its intended value and adding small increments to ρm, as
detailed in Section 6.2.4.
Figures 6.5 and 6.6 show that increasing ρm results in a higher pressure maximum at the
centre of the resulting vortex. However, and somewhat surprisingly, varying this parameter has
very little impact on either the form of the streamlines or the shear profile within the vortex.
In fact, increasing ρm for a vortex with the Kida, α = 0, Bernoulli source reduces the variation
of P˜ inside the vortex. The reason for this is that while the pressure profile varies by a large
amount, P/ρ varies by very little and this is the quantity that appears in equation (5.2.4).
Figure 6.7 also demonstrates that the transitional behaviour seen in the Kida case in
Figure 5.6 also appears with different Bernoulli and density profiles. In Figure 6.7a we see
a strong pressure maximum but as we increase ωm and produce stronger vortices we see the
aspect ratio decrease and the pressure distribution going through a transitional phase with a
saddle point (Figure 6.7c) until in Figure 6.7d have a vortex exhibiting a pressure minima at
its centre.
We encountered significant difficulties in finding solutions with significant central density
enhancements (ρmax > 10). Typically the largest we could consistently get to converge had
ρmax ≃ 4, as can be seen in Figure 6.8. Similarly, it was difficult to get solutions with β > 1
to converge successfully.
6.3.3 Relationship between the central density enhancement, ρmax, and as-
pect ratio, χ
A consequence of the way we have chosen to scale the density added to these vortices (i.e. using
equation (6.1.3)) is that the central density enhancement has a dependence on the vortex aspect
ratio χ. Figures 6.8a–6.8c show how the central density enhancement ρmax varies with vortex
class and χ. Figure 6.9 shows the variation of ρ viewed as a slice through the y–axis against
aspect ratio χ for class {0, 1, 0.1}. The act of adding a fixed amount of mass to each vortex
6.4 Summary and conclusions 99
across a class has resulted in significant variation in the density enhancement at the centre
of the vortex. In the limit χ → ∞, the vortex becomes a segment of an infinitesimally thin
circular ring. In this case, finite mass implies infinite central density which is consistent with
the previous observation.
6.4 Summary and conclusions
In this chapter, we have parametrised each vortex solution using four numbers, with α and
ωm controlling the vorticity profile and β and ρm the density profile. The variable α controls
the steepness of the vorticity profile, with α = 0 corresponding to the flat, Kida case, while
ωm parametrises the amount of vorticity in the patch and therefore its strength. Accordingly,
increasing vortex strength ωm is related to a decreasing χ.
Similarly, the parameter β regulates the the steepness of the density profile while additional
mass in the vortex is scaled by ρm. In practice, we are limited to β = 1 for convergence to
occur.
We solve the Poisson equation (5.2.4) iteratively for the streamfunction ψ using the 2D
Green’s function. Initially this is done with no density enhancement, regardless of the final
desired set of parameters {α, β, ρm, ωm}. Then, for solutions with a central density enhance-
ment, mass is added gradually in order that the solution does not deviate too much between
iterations and convergence can occur. We systematically tested convergence by observing the
change in A(ψ) at the origin, as well as changes to the streamline shape.
Despite this, there was some difficulty in getting very large ρmax solutions to converge.
This limitation on density enhancements is a numerical issue and we later use the analytic
polytropic solution to argue that there is no reason to expect such a limitation in reality
(Section 8.4). Convergence is worse when there is a density enhancement as the right hand side
of equation (5.2.4) involves higher derivatives of ψ through the pressure terms; such derivatives
are prone to amplifying errors. Trying to improve the scheme would probably involve moving
to an iterative implicit scheme which would be difficult to implement at the required resolution
and with no guarantee of success. We leave this approach for future investigation.
We also note that the recent paper of Raettig et al. (2015)2 finds that dust strongly coupled
to the fluid (i.e. Ts ≪ 1, which is under consideration here since we are working in the
τs = 0 limit) is not strongly concentrated in 2D vortices. Instead of the very strong localised
concentrations of particles which occur in the Ts ≃ 1 case, small grains are instead spread
over the entire vortex. Thus being limited to a power law index of β = 1 may not be too
unreasonable. The Ts ≃ 1 case certainly needs more study, with a different approach to the
one presented here.
Finally, we find that constant density solutions, for the same total vorticity inside the
2Uploaded to ArXiv while writing up.
100 Calculating equilibrium solutions: numerical approach and results
0
0.
050.
1
0.
150.
2
0.
250.
3
0.
35
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.100.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
01
0.
02
0.
03
0.
04
0.
05
0.
06
0.
07
P
16
.5
16
.6
16
.7
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
16
.4
9
M
od
el
P˜
k
i
d
a
(a
)
{α
,β
,ρ
m
}=
{0
,1
.0
,0
.1
},
w
ith
χ
=
4.
93
.
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
91
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.100.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
01
0.
02
0.
03
0.
04
0.
05
0.
06
0.
07
0.
08
P
16
.1
16
.2
16
.3
16
.4
16
.5
16
.6
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
16
.1
9
M
od
el
P˜
k
i
d
a
(b
)
{0
,1
.0
,0
.3
},
w
ith
χ
=
4.
84
.
F
ig
ur
e
6.
5
(C
on
tin
ue
d
on
ne
xt
pa
ge
.)
6.4 Summary and conclusions 101
0
0.
2
0.
4
0.
6
0.
81
1.
2
1.
4
1.
6
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.100.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
01
0.
02
0.
03
0.
04
0.
05
0.
06
0.
07
0.
08
P
15
.9
16
.0
16
.1
16
.2
16
.3
16
.4
16
.5
16
.6
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
16
.0
1
M
od
el
P˜
k
i
d
a
(c
)
{0
,1
.0
,0
.5
},
w
ith
χ
=
4.
80
.
F
ig
ur
e
6.
5
Vo
rt
ic
es
w
ith
a
K
id
a
vo
rt
ic
ity
pr
ofi
le
fo
r
th
e
Be
rn
ou
lli
so
ur
ce
te
rm
,w
ith
ω
m
=
0.
09
an
d
no
nz
er
o
de
ns
ity
en
ha
nc
em
en
t
pa
ra
m
et
er
ρ
m
.
T
he
ce
nt
ra
ld
en
sit
ie
s
in
th
es
e
vo
rt
ic
es
ra
ng
e
be
tw
ee
n
0.
3
an
d
6
tim
es
th
e
ba
ck
gr
ou
nd
va
lu
e.
N
ot
e
su
bs
ta
nt
ia
lc
ha
ng
es
in
th
e
pr
es
su
re
di
st
rib
ut
io
n
bu
t
re
la
tiv
el
y
lit
tle
va
ria
tio
n
in
th
e
st
re
am
lin
es
.
T
he
as
pe
ct
ra
tio
,χ
,c
al
cu
la
te
d
at
(0
,0
.8
5)
.
102 Calculating equilibrium solutions: numerical approach and results
0
0.
050.
1
0.
150.
2
0.
250.
3
0.
35
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.100.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
01
0.
02
0.
03
0.
04
0.
05
0.
06
0.
07
P
12
.0
12
.5
13
.0
13
.5
14
.0
14
.5
15
.0
15
.5
16
.0
16
.5
17
.0
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
13
.8
8
M
od
el
P˜
k
i
d
a
(a
)
{α
,β
,ρ
m
}=
{0
.5
,1
,0
.1
,0
.0
9}
,r
es
ul
tin
g
χ
=
4.
31
0
0.
2
0.
4
0.
6
0.
81
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.100.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
01
0.
02
0.
03
0.
04
0.
05
0.
06
0.
07
0.
08
P
12
.0
12
.5
13
.0
13
.5
14
.0
14
.5
15
.0
15
.5
16
.0
16
.5
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
13
.5
5
M
od
el
P˜
k
i
d
a
(b
)
{α
,β
,ρ
m
}=
{0
.5
,1
,0
.3
,0
.0
9}
,r
es
ul
tin
g
χ
=
4.
23
F
ig
ur
e
6.
6
Vo
rt
ic
es
w
ith
a
no
n–
K
id
a
Be
rn
ou
lli
vo
rt
ic
ity
so
ur
ce
,w
ith
ω
m
=
0.
09
an
d
no
nz
er
o
de
ns
ity
en
ha
nc
em
en
t
pa
ra
m
et
er
ρ
m
.
T
he
vo
rt
ic
ity
pr
ofi
le
s
in
th
es
e
vo
rt
ic
es
ar
e
no
n–
un
ifo
rm
,r
es
ul
tin
g
in
sig
ni
fic
an
tv
ar
ia
tio
n
of
th
e
pe
rio
d
fo
r
ci
rc
ul
at
in
g
ar
ou
nd
in
te
rn
al
st
re
am
lin
es
an
d
he
nc
e
sig
ni
fic
an
t
in
te
rn
al
sh
ea
r.
6.4 Summary and conclusions 103
vortex patch, behave like stronger vortices when α is increased and the vorticity profile is more
strongly peaked. There is also significant deviation from the constant period of circulation
observed for Kida vortices, with α ̸= 0 producing internal shear between vortex streamlines.
However, adding mass does not have the same effect and can even reduce internal shear. We
also observe transitional behaviour in the pressure distribution inside vortices as in the Kida
case. This has generic consequences for the ‘saddle point instability’ detailed in the next
chapter.
We now move to test the linear stability of these solutions.
104 Calculating equilibrium solutions: numerical approach and results
0
0.
51
1.
52
2.
5
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.100.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
00
5
0.
01
0.
01
5
0.
02
0.
02
5
0.
03
0.
03
5
0.
04
0.
04
5
0.
05
P
19
.5
20
.0
20
.5
21
.0
21
.5
22
.0
22
.5
23
.0
23
.5
24
.0
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
21
.1
6
F
it
P˜
k
i
d
a
(a
)
{α
,β
,ρ
m
,ω
m
}=
{0
.2
5,
1,
0.
5,
0.
05
},
re
su
lti
ng
χ
=
6.
05
0
0.
2
0.
4
0.
6
0.
81
1.
2
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.100.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
02
0.
04
0.
06
0.
08
0.
1
0.
12
0.
14
P
10
.2
10
.4
10
.6
10
.8
11
.0
11
.2
11
.4
11
.6
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
10
.3
8
F
it
P˜
k
i
d
a
(b
)
{0
.2
5,
1,
0.
5,
0.
16
},
re
su
lti
ng
χ
=
3.
48
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.200.
2
0.
4
0.
6
0.
8
1
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
05
0.
1
0.
15
0.
2
0.
25
0.
3
P
6.
6
6.
7
6.
8
6.
9
7.
0
7.
1
7.
2
7.
3
7.
4
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
6.
65
F
it
P˜
k
i
d
a
(c
)
{0
.2
5,
1,
0.
5,
0.
3}
,r
es
ul
tin
g
χ
=
2.
59
F
ig
ur
e
6.
7
(C
on
tin
ue
d
on
ne
xt
pa
ge
.)
6.4 Summary and conclusions 105
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
-1
-0
.5
0
0.
5
1
ρ(ψ)−ρ
0
y
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
-0
.6
-0
.4
-0
.200.
2
0.
4
0.
6
0.
8
1
ψ
-1
-0
.500.
51 -
1
-0
.5
0
0.
5
1
y
x
00.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
P
3.
20
3.
25
3.
30
3.
35
3.
40
3.
45
3.
50
3.
55
3.
60
3.
65
3.
70
0
0.
2
0.
4
0.
6
0.
8
1
P˜
y
P˜
k
i
d
a
=
3.
23
F
it
P˜
k
i
d
a
(d
)
{0
.2
5,
1,
0.
5,
1.
13
},
re
su
lti
ng
χ
=
1.
77
F
ig
ur
e
6.
7
D
iff
er
en
ta
sp
ec
tr
at
io
vo
rt
ic
es
,w
ith
no
n–
un
ifo
rm
bu
ts
m
oo
th
vo
rt
ic
ity
pr
ofi
le
s,
pr
od
uc
ed
by
va
ry
in
g
ω
m
.
T
he
pa
ra
m
et
er
s
de
te
rm
in
in
g
th
e
de
ns
ity
an
d
vo
rt
ic
ity
pr
ofi
le
s
ar
e
fix
ed
,a
pa
rt
fro
m
th
e
sc
al
in
g
of
th
e
Be
rn
ou
lli
so
ur
ce
,w
hi
ch
is
co
nt
ro
lle
d
by
ω
m
.
Q
ui
te
di
ffe
re
nt
be
ha
vi
ou
r
to
th
at
se
en
fo
r
th
e
K
id
a
ca
se
as
ill
us
tr
at
ed
in
Fi
gu
re
5.
6
is
se
en
.
106 Calculating equilibrium solutions: numerical approach and results
0
2
4
6
8
10
12
14
16
2 4 6 8 10 12 14
ρ
m
a
x
χ
{0.0, 1.0, 0.1}
{0.0, 2.0, 0.1}
{0.0, 1.0, 0.3}
{0.0, 1.0, 0.5}
{0.0, 1.0, 2.0}
(a) The variation of central density enhancement ρmax with χ for vortex classes with α = 0. The relationship
between ρmax and χ in all cases is linear.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
2 4 6 8 10 12 14
ρ
m
a
x
χ
{0.25, 1.0, 0.1}
{0.25, 2.0, 0.1}
{0.25, 1.0, 0.3}
{0.25, 1.0, 0.5}
(b) The variation in central density enhancement ρmax with χ for vortex classes with α = 0.25. Note that
equivalent vortices with α = 0 (comparing with Figure 6.8a) have lower ρmax then when α = 0.25 as α ̸= 0
solutions concentrate mass more.
Figure 6.8 (Continued on next page.)
6.4 Summary and conclusions 107
0
0.5
1
1.5
2
2.5
3
2 4 6 8 10 12 14
ρ
m
a
x
χ
{0.5, 1.0, 0.1}
{0.5, 2.0, 0.1}
{0.5, 1.0, 0.3}
(c) The variation in central density enhancement ρmax with χ for vortex classes with α = 0.5. Again, ρmax
is greater for these vortices with α = 0.5 than vortices with α < 0.5 for the same aspect ratio χ.
Figure 6.8 Variation in central density enhancement ρmax against χ for various α values.
Again, the three numbers in curly brackets related to {α, β, ρm}.
108 Calculating equilibrium solutions: numerical approach and results
-1
-0.5
0
0.5
1
2 4 6 8 10 12
y
χ
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
ρ
Figure 6.9 The variation of ρ with χ along the y-axis for the vortex class {0, 1, 0.1}. This
plot is made from combining vertical slices through the density distribution along the y-axis
for the entire range of aspect ratios. There is significant variation in the height of the central
density maximum from the most circular vortices χ ≈ 1 where ρmax ≈ 1 (which are have a
larger area therefore less density piled into the middle) to the elongated vortex where χ = 14
and ρmax ≈ 1.8.
Chapter 7
Stability Analysis
In the previous two chapters we set up our model and calculated a large set of equilibrium
solutions. In this chapter we formulate the stability analysis of these solutions to perturbations
localised on streamlines.
In Section 7.1 we consider making both Eulerian and Lagrangian perturbations to our
equilibrium solutions in the most general form. When perturbations can be associated with
a time–independent wavenumber they can lead to exponentially growing modes, as is found
for the known instabilities of the Kida vortex, which we generalise in Section 7.2. Various
parametrisations are made in Section 7.2.2.1 and 7.2.2.3 which leads to a working form of the
(Eulerian) stability equations which we can integrate in Section 7.2.2.4. The periodic nature
of the equilibrium solutions means that Floquet theory can be used to determine the growth
rates of instability, and this is detailed in Section 7.2.3.
We investigate the horizontal (kz → ∞) limit of the stability calculations in Section 7.3,
followed by a discussion of the ‘saddle point’ and parametric instabilities that we expect to
occur in Sections 7.3.2 and 7.3.3, respectively.
The generic case when the period of circulation around a streamline in a vortex is not
constant is explored in Section 7.4 with a wavenumber with magnitude that ultimately increases
linearly with time and its application discussed.
Finally we consider the vertical stability (i.e. perturbations that have kz = 0) of solutions
in Section 7.5 for vortices with and without internal shear. The polytropic model of Section 5.4
is used to investigate the latter case.
7.1 Perturbation analysis
We now have a large variety of equilibrium solutions and we are interested in their linear
stability. The introduction of shear in the vortex core via the Bernoulli source A(ψ) (see
equation (5.2.6a)) and a density profile via B(ψ) (see equation (5.2.6b)) provides a lot of new
solutions to investigate the stability profiles of. The stability of the Kida vortex is explored in
110 Stability Analysis
detail in Lesur and Papaloizou (2009), so this solution will again be used as a test case of our
method.
It is useful to consider both the Lagrangian and Eulerian formulations of the linear stability
problem as they are found to be convenient for different purposes. Throughout this work we
use Q′ to denote the Eulerian perturbation of Q, the perturbation of Q at a fixed point in space.
∆Q meanwhile denotes the Lagrangian perturbation of Q, the perturbation of a quantity as
experienced by a fluid element, taking into account its displacement by a distance ξ. These
perturbations are related by
∆Q = Q′ + ξ · ∇Q, (7.1.1)
to first order in ξ (Lynden-Bell and Ostriker, 1967).
7.1.1 Eulerian formalism
The time–dependent momentum equation in the rotating frame (see equation (5.1.9)) in the
case where viscous forces may be neglected is
Dv
Dt
+ 2Ω× v = 1
ρ
∇P −∇Φ, (7.1.2)
where we have dropped the overbars for clarity and Φ is the combined potential (Section 5.1.5,
equation (5.1.29)).
We begin by making the Eulerian perturbations:
v 7−→ v0 + v′
P 7−→ P0 + P ′ (7.1.3)
ρ 7−→ ρ0 + ρ′
where {v0, P0, ρ0} are the equilibrium quantities from solving our steady equation of mo-
tion (5.2.4). Perturbing equation (7.1.2) and linearising yields the linearised momentum equa-
tion in component form:
Dv′x
Dt
+ v′ · ∇v0,x − 2Ωv′y = −
1
ρ0
∂P ′
∂x
+ ρ
′
ρ20
∂P0
∂x
(7.1.4a)
Dv′y
Dt
+ v′ · ∇v0,y + 2Ωv′x = −
1
ρ0
∂P ′
∂y
+ ρ
′
ρ20
∂P0
∂y
(7.1.4b)
Dv′z
Dt
= − 1
ρ0
∂P ′
∂z
(7.1.4c)
where in these linearised equations the convective (or Lagrangian) derivative D/Dt = ∂/∂t+
7.1 Perturbation analysis 111
v.∇ is replaced by the expression
D
Dt
≡ ∂
∂t
+ v0 · ∇. (7.1.5)
Using this expression for D/Dt in the linearised equations has second order corrections to the
original equations.
Due to the local nature of these perturbations, we use Cowling’s approximation (Cowling,
1941) to neglect the variation of the gravitational potential. Note that in the z-direction we
have hydrostatic balance as v0 = (v0,x, v0,y, 0) = ∇ × (ψ0zˆ). We also have the continuity
equation from the two-fluid averaged version of equation (5.1.8)
∂ρ
∂t
+∇ · (ρv) = 0 (7.1.6)
so taking the Eulerian perturbation and linearising:
Dρ′
Dt
+ ρ′∇ · v0 + v′ · ∇ρ0 + ρ∇ · v′ = 0
⇒ Dρ
′
Dt
+ v′ · ∇ρ0 = 0 (7.1.7)
where we use the fact that v0 is the curl or a vector v0 = ∇× (ψ0zˆ) and ∇ · v′ = 0 since we
are insisting on incompressibility everywhere.
These equations (7.1.4, 7.1.7) expressed in terms of Eulerian variations were found to
be simpler to use when analysing vertical stability (see Section 7.5). They also are solved
numerically when considering vortex stability in Chapter 8 for the same reason.
7.1.2 Lagrangian formalism
While the Eulerian formulation is convenient in some contexts the Lagrangian formalism is
useful for some aspects, such as the analytic discussion of the saddle point instability (Sec-
tion 7.3.2).
Firstly, the Lagrangian derivative (equation (7.1.5)) commutes with the Lagrangian per-
turbation so that
D
Dt
(∆Q) = ∆
(
DQ
Dt
)
, (7.1.8)
as shown by Lynden-Bell and Ostriker (1967). The displacement ξ is the Lagrangian variation
of the fluid element position vector such that
∆r = ξ (7.1.9)
112 Stability Analysis
and
∆v = Dξ
Dt
. (7.1.10)
Taking the Lagrangian perturbation of equation (7.1.2) we obtain
D2ξ
Dt2
+ 2Ω× Dξ
Dt
= ∆F (7.1.11a)
with F = −∇P/ρ−∇Φ and
∆F = −∇P
′
ρ
+ ρ
′
ρ2
∇P − ξ ·
(∇P
ρ
+∇Φ
)
, (7.1.11b)
where we have again used Cowling’s approximation and equation (7.1.1). Considering the
continuity equation (7.1.6) and incompressibility
Dρ
Dt
= 0,
which implies that ρ is advected with the fluid and hence
∆ρ = ρ′ + ξ · ∇ρ0 = 0. (7.1.12)
Equations (7.1.11, 7.1.12), together with the incompressibility condition ∇ · ξ = 0 lead to
a system of equations for the horizontal components of ξ that is fourth order in time (see
Section 7.2.1).
7.1.3 Local analysis
We consider perturbations that are localised on streamlines. To do this we assume that any
perturbation quantity is of the form
∆Q = ∆Q0 exp(iλSA). (7.1.13)
Here we are adopting the WKBJ1 ansatz with amplitude factor ∆Q0 (the WKBJ envelope),
phase function SA and constant λ taken to be a large parameter. Discussion of this approach
in a variety of contexts can be found in Lifschitz and Hameiri (1991), Sipp and Jacquin (2000)
and Papaloizou (2005). The effective wavenumber is
k = λ∇SA (7.1.14)
which has a large magnitude. The amplitude factor ∆Q0 is assumed to be a function of time
and slowly varying relative to SA while we have rapid variation of the complex phase λSA.
1See the work of Wentzel (1926), Kramers (1926), Brillouin (1926) and Jeffreys (1925).
7.1 Perturbation analysis 113
The standard assumption of local analysis is that only the variation of this phase needs to
be considered when taking space derivatives. The only exception is when variation is along
streamlines since we require that v0 · ∇SA = 0 and so we need to consider the v0 · ∇(∆Q0)
contribution. Due to this rapid variation of the complex phase λSA we perform a WKBJ
analysis where the state variables ∆Q0 are expanded in inverse powers of λ, substituting
equation (7.1.13) into equation (7.1.11), recalling that ξ = ∆r:[
1
λ2
D2(∆r0)
Dt2
+ i
λ
(
2DSA
Dt
D(∆r0)
Dt
+ D
2SA
Dt2
∆r0
)
−
(
DSA
Dt
)2
∆r0
]
+2Ω×
[ 1
λ2
D(∆r0)
Dt
+ i
λ
DSA
Dt
∆r0
]
= 1
λ2
∆F0 (7.1.15)
where we have divided through by λ2 and cancelled the common factor exp(iλSA). In the term
∆F , consider ∇P ′:
P ′ = ∆P − ξ · ∇P
∇P ′ = iλ∇SAP ′ +∇ (∆P0 −∆r0 · ∇P ) exp(iλSA). (7.1.16)
Thus the lowest order term on the RHS of equation (7.1.15) is λ−2∇P ′ which is O(λ−1); the
rest of the terms are O(λ−2). Therefore, to lowest order, equation (7.1.15) gives
DSA
Dt
= 0. (7.1.17)
When working to the next order only the variation of the phase SA needs to be considered
when taking spatial derivatives, apart from when considering expressions involving D/Dt as
this annihilates SA. In this case, the contribution v ·∇(∆Q0) needs to be retained. Therefore,
equation (7.1.11), together with equation (7.1.12) yield
D2ξ
Dt2
+ 2Ω× Dξ
Dt
= − ikP
′
ρ
− ξ · ∇ρ
ρ2
∇P + ξ · ∇ (v · ∇v + 2Ω× v) , (7.1.18)
while the incompressibility condition ∇ · ξ = 0 gives
k · ξ = 0. (7.1.19)
We can also find an equation for the time evolution of the wavenumber k using equations (7.1.14)
and (7.1.19):
114 Stability Analysis
Dki
Dt
= λ
[
∂
∂t
∂SA
∂xi
+ v0,j
∂2SA
∂xixj
]
= λ
[
∂
∂xi
(
−v0,j ∂SA
∂xj
)
+ v0,j
∂2SA
∂xixj
]
= λ
[
−∂v0,j
∂xi
∂SA
∂xj
− v0,j ∂
2SA
∂xixj
+ v0,j
∂2SA
∂xixj
]
= −λ∂v0,j
∂xi
∂SA
∂xj
⇒ Dki
Dt
= −kj∇v0,j (7.1.20)
We can eliminate P ′ using the vertical component of equation (7.1.18) (see Section 7.2.1 and
equation (7.2.4)). Then equations (7.1.18)-(7.1.20) give a complete system for the evolution of
ξ and k as an initial value problem. Evolution consists of advection of data along streamlines
so it is possible to consider disturbances localised on individual streamlines (Papaloizou, 2005).
In general one could start with an arbitrary initial SA and then k would depend on time.
7.2 Time–independent wavenumber
An initial route into this problem is to look for solutions for which SA is independent of time
and a function of quantities conserved on unperturbed streamlines so v0 · ∇SA = 0. Then in
an Eulerian viewpoint, k = λ∇SA is fixed for all time and we only have to solve for ξ.
Our vortex solutions are two-dimensional with an initial state independent of z (see Sec-
tion 5.1.5) so we take
SA = g(ψ) + kzz
λ
. (7.2.1)
Here ψ is our unperturbed streamfunction, g an arbitrary function and kz the constant vertical
wavenumber. With this form of SA, our wavenumber is
k = λ dg
dψ
∇ψ + kzzˆ. (7.2.2)
For now we’ll assume that kz ̸= 0 which is appropriate to the physically realistic case where
perturbations are localised in z. The kz = 0 case will be dealt with separately in Section 7.5.
This form of k is not the most general solution of equation (7.1.17), except fortuitously in
the case where the velocity is linear in the coordinates (as in the Kida case, see Section 5.3).
For other solutions we would expect the magnitude of the wavenumber to ultimately increase
linearly with time. However, in that situation we expect that although there may be temporary
amplification, the system may not ultimately show growth of linear perturbations exponentially
7.2 Time–independent wavenumber 115
with time; this situation is well known in the context of the shearing box (Goldreich and
Lynden-Bell, 1965 and Appendix C). This is looked at in more detail in Section 7.4.
7.2.1 Lagrangian form
When k takes the form dictated by equation (7.2.2) the perturbed momentum equation 7.1.18:
D2ξi
Dt2
+ 2ϵi3jΩ
Dξj
Dt
+ ikiP
′
ρ
= Hi =
[
−ξ · ∇ρ
ρ2
∇P + ξ · ∇ (v · ∇v + 2Ω× v)
]
i
. (7.2.3)
Considering the z-component of this expression:
D2ξz
Dt2
= − ikzP
′
ρ
, (7.2.4)
and, noting that kz is constant, we rearrange the incompressibility condition (equation (7.1.19))
to find
ξz = −kxξx + kyξy
kz
= −kjξj
kz
. (7.2.5)
In this last expression, and throughout this section, summation is over j = 1, 2 only. Then
iP ′
ρ
= − 1
kz
D2ξz
Dt2
= 1
kz
D2
Dt2
(
kjξj
kz
)
= 1
k2z
[
kj
D2ξj
Dt2
+ 2Dkj
Dt
Dξj
Dt
+ ξj
D2kj
Dt2
]
(7.2.6)
Substituting this expression into equation (7.2.3) we arrive at
(
δij +
kikj
k2z
)
D2ξi
Dt2
+ 2
(
ϵi3jΩ+
ki
k2z
Dkj
Dt
)
Dξj
Dt
+ kiξj
k2z
D2kj
Dt2
= Hi (7.2.7)
with
Dki
Dt
= −kj ∂v0,j
∂xi
(7.2.8a)
D2ki
Dt2
= kl
∂v0,l
∂xj
∂v0,j
∂xi
− kjv0,l ∂
2v0,j
∂xj∂xl
(7.2.8b)
using equation (7.1.20). Neglecting the vertical stratification in this calculation can be justified
if it is assumed that k2z/(k2x + k2y) is large. Else, the modes can be assumed to be localised in
the vicinity of the midplane where the vertical stratification is least.
Though the presence of the arbitrary function g(ψ) may seem a cause for concern, since
the derivatives in equation (7.2.7) correspond to advection round a streamline and dg/dψ
is constant on streamlines it acts as a multiplicative constant scaling the magnitude of the
116 Stability Analysis
wavenumber:
k2x + k2y = λ2
(
dg
dψ
)2
|∇ψ|2 = Λ2|∇ψ|2, (7.2.9)
where Λ is another large constant. We return to the use of these equations in Section 7.3.
7.2.2 Eulerian form
We begin by remarking that with the Lagrangian perturbations given in the form of equa-
tion (7.1.13)
∆Q = ∆Q˜ exp(iλSA).
the corresponding Eulerian perturbation is
Q′ = ∆Q− ξ · ∇Q = ∆Q−∆r · ∇Q
= Q˜′ exp(iλSA). (7.2.10)
Our WKBJ approximation still holds as any gradients of ∆Q˜−∆r˜ · ∇Q will be an order in λ
lower than the gradient of exp(iλSA).
We again aim to eliminate v′z and P ′ to produce a matrix equation in our remaining
perturbed quantities x′ = (v′x, v′y, ρ′)T that we can integrate. Applying the incompressibility
condition ∇ · v′ = 0 when SA takes the form described by equation (7.2.2) gives:
iλ
dg
dψ
(
v′x
∂ψ
∂x
+ v′y
∂ψ
∂y
)
+ ikzv′z = 0. (7.2.11)
Rearranging to eliminate v′z:
v′z = −
λ
kz
dg
dψ
v′ · ∇ψ. (7.2.12)
We substitute this into equation (7.1.4c) to get an expression for P ′
Dv′z
Dt
= D
Dt
(
− λ
kz
dg
dψ
v′ · ∇ψ
)
= − ikzP
′
ρ0
(7.2.13)
⇒ P ′ = − iρ0λ
k2z
dg
dψ
D
Dt
(
v′ · ∇ψ) . (7.2.14)
Then eliminating P ′ from equations (7.1.4a,b) using the highest order term in equation (7.1.16)
so ∇P ′ = iλ∇P ′SA and
Dv′i
Dt
+ Mijv′j −
iλ
k2z
dg
dψ
D
Dt
(
v′ · ∇ψ) [iλ ∂ψ
∂xi
dg
dψ
]
= ρ
′
ρ20
∂P0
∂xi
,
Dv′i
Dt
+ Mijv′j +
λ2
k2z
(
dg
dψ
)2 D
Dt
(
v′ · ∇ψ) ∂ψ
∂xi
= ρ
′
ρ20
∂P0
∂xi
, (7.2.15)
7.2 Time–independent wavenumber 117
where the 2× 2 matrix Mij is given by
M =

∂v0,x
∂x
∂v0,x
∂y
− 2Ω
∂v0,y
∂x
+ 2Ω ∂v0,y
∂y
 (7.2.16)
and summation is over i = 1, 2, j = 1, 2 only. We can expand the DDt (v′ · ∇ψ) term to collect
terms of the form Dv′i/Dt, so:
Dv′i
Dt
+ Mijv′j +
λ2
k2z
(
dg
dψ
)2 ∂ψ
∂xi
[
Dv′j
Dt
∂ψ
∂xj
+ v′jv0,k
∂2ψ
∂xj∂xk
]
= ρ
′
ρ20
∂P0
∂xi
⇒
[
δij +
λ2
k2z
(
dg
dψ
)2 ∂ψ
∂xi
∂ψ
∂xj
]
Dv′j
Dt
=
−
[
Mij + λ
2
k2z
(
dg
dψ
)2 ∂ψ
∂xi
∂2ψ
∂xj∂xk
v0,k
]
v′j +
1
ρ20
∂P0
∂xi
ρ′. (7.2.17)
The density perturbation ρ′ cannot be simplified beyond equation (7.1.7):
Dρ′
Dt
= −∂ρ0
∂xi
v′i, (7.2.18)
recalling that D/Dt = ∂/∂t+ v0 · ∇.
7.2.2.1 Parametrisation via Θ and θ
At this point it is now convenient to introduce a variable Θ, a scaled ratio of horizontal and
vertical wavenumbers:
Θ ∝ k
2
x + k2y
k2z
= k
2
⊥|σ=0
k2z
. (7.2.19)
Without loss of generality, we have set our horizontal wavenumber to be evaluated at zero
arclength σ = 0 which corresponds to the point on the positive y axis where the chosen
streamline crosses, as in Figure 7.1. Also, note that k⊥ =
√
k2x + k2y is the relevant quantity
over kx and ky since any perturbations along streamlines result in no deformation to the
streamline and hence will produce no instability.
Since k = λ∇SA (equation 7.1.14) and SA depends only on ψ in the time–independent
case, there is only one parameter to be chosen here; choosing k⊥ makes the most sense as it is
independent of coordinate system. As
k⊥ = ∇⊥(λg) = λ dg
dψ
∇⊥ψ (7.2.20)
we find that
k2⊥|σ=0 = λ2
(
dg
dψ
)2
v20|σ=0. (7.2.21)
118 Stability Analysis
Figure 7.1 The coordinate σ determining the location on a streamline and total arclength Σ
for an arbitrary streamline.
We therefore define Θ by
Θ =
[
1
v20
k2⊥
k2z
]
σ=0
= λ
2
k2z
(
dg
dψ
)2
. (7.2.22)
The angle between k = (k⊥, kz) and the vertical zˆ pointing out of the plane of the disc/vortex
is given by
tan θ = k⊥
kz
∣∣∣∣
σ=0
=
√
Θ · v0|σ=0 (7.2.23)
and is one of the variables in the final stability plots (see e.g. Section 8.1.2). Its geometric
meaning can be seen in Figure 7.3. Note that k⊥ is, by definition, perpendicular to streamlines,
or k⊥ · v0 = 0.
7.2.2.2 Matrix form of the stability equations
Using equation (7.2.22), the stability equation (7.2.17) can now be written(
δij +Θ
∂ψ
∂xi
∂ψ
∂xj
)
Dv′j
Dt
= −
(
Mij +Θ ∂ψ
∂xi
∂2ψ
∂xj∂xk
v0,k
)
v′j +
1
ρ20
∂P0
∂xi
ρ′. (7.2.24)
7.2 Time–independent wavenumber 119
Figure 7.2 The orientation of the wavevector k in relation to the plane of the disc; zˆ is out
of the plane of the disc and vortex. In (a) we have the horizontal stability limit kz → ∞ ,
whereas in (b) we show the vertical stability limit kz = 0.
Note that this problem only depends on the direction of the wavevector k and not its magnitude
so is scale–independent. We will write equation (7.2.24) in matrix form as
Aij
Dv′j
Dt
= Bijv′j + Ciρ′ (7.2.25a)
where
Aij = δij +Θ ∂ψ
∂xi
∂ψ
∂xj
(7.2.25b)
Bij = −Mij −Θ ∂ψ
∂xi
∂2ψ
∂xj∂xk
v0,k (7.2.25c)
Ci = 1
ρ20
∂P0
∂xi
(7.2.25d)
This will eventually be solved, alongside equation (7.2.18) as an initial value problem, detailed
in the following sections. A quick calculation shows that
det(A) = Θ|∇ψ|2 + 1 ≥ 1 (7.2.26)
so A is invertible.
120 Stability Analysis
Figure 7.3 The geometric realisation of the parameter θ in terms of angle between the unit
vector zˆ (out of the plane of the disc and vortex) and wavevector k.
7.2.2.3 Parametrisation via arclength σ
In general, time t is not the most convenient choice of parameter for conducting our stability
analysis; instead we chose arclength σ.
|v0| = Dσ
Dt
⇒ D
Dt
= |v0| D
Dσ
Including the matrix form of our continuity equation:
Dρ′
Dt
= Div′i (7.2.27a)
Di = −∂ρ0
∂xi
. (7.2.27b)
the final form of our equation,
Dx′
Dσ
= 1|v0(σ, ψ)|Π(σ, ψ)x
′ (7.2.28a)
where
Π =
(
A−1B A−1C
D 0
)
(7.2.28b)
is a 3×3 matrix, A, B are 2× 2 matrices, C, D are 2D vectors and x′ = (v′x, v′y, ρ′)T .
7.2 Time–independent wavenumber 121
7.2.2.4 The working form of equation (7.2.28)
Expanding v0 as v0 = ∇× (ψzˆ) = (∂ψ/∂y,−∂ψ/∂x) we find that
A−1 =
(
a1 a2
a3 a4
)
,
a1 =
1
1 + Θ|∇ψ|2
[
1 + Θ
(
∂ψ
∂y
)2]
a2 = a3 = − Θ1 +Θ|∇ψ|2
∂ψ
∂x
∂ψ
∂y
a4 =
1
1 + Θ|∇ψ|2
[
1 + Θ
(
∂ψ
∂x
)2]
(7.2.29)
and
B = −
(
M+Θ ∂ψ
∂xi
∂2ψ
∂xj∂xk
v0,k
)
= −

∂2ψ
∂x∂y
+Θ∂ψ
∂x
B1
∂2ψ
∂y2
− 2Ω + Θ∂ψ
∂x
B2
2Ω− ∂
2ψ
∂x2
+Θ∂ψ
∂y
B1
∂2ψ
∂x∂y
+Θ∂ψ
∂y
B2
 , (7.2.30)
where
B1 =
∂2ψ
∂x2
∂ψ
∂y
− ∂
2ψ
∂x∂y
∂ψ
∂x
B2 =
∂2ψ
∂x∂y
∂ψ
∂y
− ∂
2ψ
∂y2
∂ψ
∂x
.
The vectors C = 1
ρ20
∂P0
∂xi
and D = −∂ρ0
∂xi
have no simplified versions.
7.2.3 Use of Floquet theory
Recall equation (7.2.28):
Dx′
Dσ
= 1|v0(σ, ψ)|Π(σ, ψ)x
′
where x′ = (v′x, v′y, ρ′)T and σ is the arclength as measure round a vortical streamline from
the point where the streamline crosses the positive y axis. We let Σ be the total circumference
of a chosen streamline. Around any particular streamline all the matrix elements Πij will be
Σ–periodic, so Π(σ) = Π(σ + Σ). We can analyse this sort of periodic system using Floquet
theory (see e.g. Whittaker and Watson, 1996, and Appendix B).
We have three perturbed quantities, v′x, v′y and ρ′ so consider n = 3 vectors x′k, all solutions
122 Stability Analysis
to equation (7.2.28). Let X ′ki be the ith element of x′k so
X ′ =
(
[x′1], [x′2], [x′3]
)
⇒ DX
′
ki
Dσ
= 1|v0|ΠijX
′
kj .
Our approach is to make X ′(σ = 0) the principal fundamental matrix2 so
X ′(σ = 0) = I (7.2.31)
and integrate equation (7.2.28) to find X ′(σ = Σ). This is equivalent to integrating from
an initial state of x′1 = (v′x, v′y, ρ′)T = (1, 0, 0)T then x′2 = (0, 1, 0)T and x′3 = (0, 0, 1)T so
any initial perturbation can be reconstructed. The eigenvalues of the matrix X ′(σ = Σ) are
the three characteristic multipliers ϱj for the matrix Π. The corresponding characteristic (or
Floquet) exponents are µj ∈ C satisfying
ϱj = eµjΣ.
Floquet theory shows that the entire solution is stable if all the characteristic multipliers satisfy
|ϱj | ≤ 1. For a given vortex solution, once we have calculated the three eigenvalues ϱj ∈ C we
convert to the exponential complex form with modulus |ϱj | and argument θj :
ϱj = |ϱj |eiθj = esjΣeiθj = eγj P˜ eiθj . (7.2.32)
We define the growth factor per orbit, s and temporal growth rate, γ by taking the maximum
|ϱj |, with
s = 1Σ maxj {log |ϱj |} (7.2.33a)
γ = 1
P˜
max
j
{log |ϱj |} (7.2.33b)
More details of how this is implemented numerically are detailed in Chapter 8.
According to this theory, if some internal mode with a natural oscillation frequency depen-
dent on k is described by either the Eulerian perturbation equation (7.2.28) or the Lagrangian
form in equation (7.2.7), unstable bands of exponential growth are expected as k is varied to
allow resonances of this frequency with the frequency of motion around the streamline.
2For relevant Floquet theory definitions and theorems please see Appendix B.
7.3 Horizontal stability 123
7.3 Horizontal stability
We will begin by specialising to the case when kz ≫
√
k2x + k2y, what is known as the horizontal
instability case. In this limit, motion occurs in uncoupled horizontal planes which will enable
us to investigate the instability described in Section 7.2.3.
It can be seen from the definition of Θ in equation (7.2.22) that in the kz → ∞ limit,
Θ → 0. Therefore, the Eulerian form of the stability equations (7.2.24) become a system of
equations with constant coefficients for the components of x′.
In the kz → ∞ limit we find that the Lagrangian form of the stability equations equa-
tion (7.2.7) reduces to
δij
D2ξi
Dt2
+ 2ϵi3jΩ
Dξj
Dt
= Hi (7.3.1)
or
D2ξ
Dt2
+ 2Ω× Dξ
Dt
= −ξ · ∇ρ
ρ2
∇P + ξ · ∇ (v · ∇v + 2Ω× v) . (7.3.2)
Note that the ∇·ξ = 0 condition yields ξz → 0 so ξ = (ξx, ξy). Since the steady state equation
of motion (equation (5.1.14)) is
v · ∇v + 2Ω× v = −∇P
ρ
−∇Φ
we find that the ξ · ∇ρ contributions disappear:
D2ξ
Dt2
+ 2Ω× Dξ
Dt
= −ξ · ∇ρ
ρ2
∇P − ξ · ∇
(∇P
ρ
+∇Φ
)
= −ξ · ∇ρ
ρ2
∇P −
(
ξ
ρ
· ∇
)
∇P + ξ · ∇ρ
ρ2
∇P − (ξ · ∇)∇Φ
= −
(
ξ
ρ
· ∇
)
∇P − (ξ · ∇)∇Φ. (7.3.3)
When ρ is constant and P and Φ are quadratic in x and y, equation (7.3.3) becomes an
equation with constant coefficients. As this is always the case arbitrarily close to the centre of
any regular vortex where there is a stagnation point there are some generic consequences (see
Section 7.3.2).
7.3.1 Kida horizontal limit
Applying the system of equations given in Section 7.2.2.4 in this horizontal case gives some
insight into the saddle point instability which appears in the next Section 7.3.2. Remember
that in this horizontal limit, Θ = 0 and in the absence of a density gradient the matrix D = 0.
Therefore ρ′ is a constant which, without loss of generality, we can set to be zero and ignore.
(This also saves us having to find the matrix C.)
Firstly, recall equation (5.3.21a) for the streamfunction of the Kida vortex with aspect ratio
124 Stability Analysis
χ and background shear S:
ψ = S2χ(χ− 1)
(
χ2x2 + y2
)
With a little algebra we therefore find the forms of the matrices A and B, using 2Ω = 4S/3 to
find the final form of B:
AKida = I (7.3.4)
BKida =
S
χ− 1
 0
(2χ+ 1)(2χ− 3)
3χ
4− χ
3 0
 (7.3.5)
so
Dv′
Dt
= BKidav′. (7.3.6)
We found in Section 5.3.3 that the arclength element dσ in a Kida vortex is given by
dσ = b
√
cos2 η + χ2 sin2 η dη (7.3.7)
where η is the one of the elliptical coordinates and b is the semi-minor axes of a chosen
streamline. We remark that
|v0| = |∇ψ| = Sb
χ− 1
√
cos2 η + χ2 sin2 η = S
χ− 1
Dσ
Dη
(7.3.8)
so performing the change of variables from t to η (similar that that detailed in Section 7.2.2.3)
we have
D
Dt
= |v0| D
Dσ
= S
χ− 1
D
Dη
and finally
Dv′x
Dη
= (2χ+ 1)(2χ− 3)3χ v
′
y (7.3.9a)
Dv′y
Dη
= 4− χ3 v
′
x. (7.3.9b)
this system describes horizontal epicyclic oscillations with frequency
κ2kida = −
(2χ+ 1)(2χ− 3)(4− χ)
9χ . (7.3.10)
As shown in Figure 7.4, the epicyclic frequency κ2kida is only negative in the range 3/2 < χ < 4
corresponding to exponential growth in v′ and thus instability. In this range the pressure
distribution is in a transitional saddle point regime between a minima χ < 3/2 and maxima
for χ > 4 (see Figure 5.6). There is a generalisation of this instability seen in more generic
7.3 Horizontal stability 125
-5
0
5
10
15
20
25
30
35
2 4 6 8 10
χ
κ
2
Kida
Figure 7.4 Epicyclic frequency for the Kida vortex in the horizontal kz → ∞ limit. κkida
is only real outside the range 3/2 ≤ χ ≤ 4, corresponding to the location of the saddle point
instability.
vortices.
7.3.2 The saddle point instability
In the horizontal limit we solve equation (7.3.3) by setting
ξ = ξ0 exp iγht, (7.3.11)
where ξ0 is a constant vector, and finding an algebraic expression for γh. The right hand side
of equation (7.3.3) is
−
(
1
ρ
∂2P
∂xi∂xj
+ ∂
2Φ
∂xi∂xj
)
ξj = Eijξj (7.3.12)
where we note that the matrix E is symmetric. Therefore equation (7.3.3) becomes
−γ2hξi + 2iΩγhϵi3jξj = Eij . (7.3.13)
The two components of equation (7.3.13) are then
(γ2h + E11)ξx + (2iΩγh + E12)ξy = 0 (7.3.14a)
(−2iΩγh + E21)ξx + (γ2h + E11)ξy = 0 (7.3.14b)
126 Stability Analysis
Figure 7.5 Geometry of saddle point for which the saddle point instability occurs, with a
maximum in the x-direction. This is the pressure distribution for a Kida vortex with χ = 2.
which implies
(γ2h + E11)(γ2h + E22) = −(2iΩγh + E12)(2iΩγh − E21)
γ4h + γ2h(E11 + E22) + E11E22 = 4Ω2γ2h + E12E21 − 2iΩγh(E12 − E21). (7.3.15)
We exploit the symmetry of E in equation (7.3.15) to cancel the imaginary term to leave us
with
⇒ γ4h +
[
Tr(E)− 4Ω2
]
γ2h + det(E) = 0, (7.3.16)
where ‘Tr’ and ‘det’ denote the trace and determinant of E respectively.
We have instability if κ has at least one complex root. A sufficient condition for this to
occur is det(E) < 0. In the limit approaching the vortex centre |r| → 0, the elements of E are
Eij = −1
ρ
∂2
∂xi∂xj
(
P − Sρx2
)
, (7.3.17)
recalling the form of the effective potential given in equation (5.1.20). The det(E) < 0 condition
is then equivalent to P − Sρx2 having a saddle point at the centre. This will occur when the
P has a saddle point that appears as a maximum along the x–coordinate and an minimum
along the y–coordinate line, as shown in Figure 7.5. These occur in the analytic solution for
Kida vortices with aspect ratio 3/2 < χ < 4, as detailed in Section 7.3.1. Saddle points of this
type are generically associated with instability in all cases, independent of density or vorticity
profile (Section 8.2.2).
7.4 Time–dependent wavenumber 127
7.3.3 The parametric instability
Equation (7.3.3) applies in the limit kz → ∞ and is an equation with constant coefficients
when ρ is constant and both P and Φ are quadratic in x and y. This is always the case for
any streamline in a Kida vortex core. However, in more general cases, P , ρ and hence E will
be represented by a power series in x2 and y2. In this case, E will be periodic with period
P˜ /2, one half of that associated with circulating round a streamline (see equation (5.3.32)).
Then equation (7.3.3) will have coefficients that are periodic in time and parametric instability
becomes possible (Section 3.5.4).
Following the discussion in Papaloizou (2005), Appendix B, close to the vortex centre the
time–dependence can be treated as a perturbation and the parametric instability can be derived
analytically, albeit in terms of unknown coefficients.
The parametric instability is first expected to occur when the epicyclic oscillation period
is equal to the period P˜ . Higher order bands are expected to be generated when the ratio of
epicyclic oscillation period to P˜ is 1/2, 1/3, . . . . For a vortex with a core like the Kida solution,
these resonances occur when χ = 4.65, 5.89 and 7.32 respectively (Lesur and Papaloizou, 2009).
7.4 Time–dependent wavenumber
We now consider a more general k that is a function of time t.
7.4.1 The general from of SA
The general form of SA is obtained from the solution of equation (7.1.17):
DSA
Dt
= 0.
For the case where the background flow is independent of z we can write
SA = S⊥(x, y, t) + kzz
λ
. (7.4.1a)
where S⊥(x, y) satisfies
∂S⊥
∂t
+ ∂ψ
∂y
∂S⊥
∂x
− ∂ψ
∂x
∂S⊥
∂y
= 0. (7.4.1b)
The general solution of equation (7.4.1b) requires S⊥ to be a function of only quantities that
are invariants of the particle trajectories. These trajectories are found by solving
dx
dt
= ∂ψ
∂y
(7.4.2a)
dy
dt
= −∂ψ
∂x
. (7.4.2b)
128 Stability Analysis
The solutions for x and y define streamlines that are periodic in time and on which ψ is
constant, as per the definition of the streamfunction. The period is P˜ = 2π/ϖ, as in Section 6.3,
where in general ϖ will be a function of ψ. We can therefore express x and y as Fourier series:
x =
∞∑
n=−∞
xn(ψ) exp(inϕ) (7.4.3a)
y =
∞∑
n=−∞
yn(ψ) exp(inϕ) (7.4.3b)
with
ϕ = ϖ(t− t0). (7.4.3c)
Without loss of generality we will take t0 to be the time at which y passes through its maximum
value. Equation (7.4.1b) states that S⊥ is constant on an orbit defining a streamline. The
general solution to this is that S⊥ is an arbitrary function of ψ and t0, S⊥ = S⊥(ψ, t0). As
the orbits are by definition periodic this function should too be periodic in t0 with period
P˜ = 2π/ϖ. Therefore S⊥ can also be written as a Fourier series:
S⊥ =
∞∑
n=−∞
Sn(ψ) exp(−inϖt0) =
∞∑
n=−∞
Sn(ψ) exp(in[ϕ−ϖt]). (7.4.4)
We can now find the time–dependent wavenumber k = λ∇SA where
∇S⊥ = ∂S⊥
∂ψ
∇ψ + ∂S⊥
∂ϕ
∇ϕ = ∂S⊥
∂ψ
∇ψ + ∂S⊥
∂ϕ
ϖ∇t. (7.4.5)
We removed the explicit t dependence using equation (7.4.2):
∇t =
(
∂t
∂x
,
∂t
∂y
)
=
(
ϖ
∂y
∂ψ
,−ϖ∂x
∂ψ
)
, (7.4.6)
arriving at
kx = λ
(
∂ψ
∂x
∂S⊥
∂ψ
+ϖ ∂y
∂ψ
∂S⊥
∂ϕ
)
(7.4.7a)
ky = λ
(
∂ψ
∂y
∂S⊥
∂ψ
−ϖ∂x
∂ψ
∂S⊥
∂ϕ
)
(7.4.7b)
Quantities are either expressed as functions of r = (x, y) or with (ϕ, ψ) as independent
variables. Transforming between these two representations is more straightforward than would
initially appear as
v = Dr
Dt
= dϕ
dt
∂r
∂ϕ
= ϖ∂r
∂ϕ
(7.4.8)
7.4 Time–dependent wavenumber 129
and the Jacobian is
∂(ϕ, ψ)
∂(x, y) = det

∂ϕ
∂x
∂ϕ
∂y
∂ψ
∂x
∂ψ
∂y
 = det

ϖ
∂y
∂ψ
−ϖ∂x
∂ψ
∂ψ
∂x
∂ψ
∂y
 = 2ϖ. (7.4.9)
In addition, if only the n = 0 term is considered in equation (7.4.4) then S⊥ = S⊥(ψ) ⇒
∂S⊥/∂ϕ = 0 and we recover the time–independent wavenumber equation (7.2.1) from Sec-
tion 7.2:
SA = g(ψ) + kzz
λ
.
7.4.2 Wavenumber increasing with time
We rewrite equations (7.4.7) for the components of k so the time dependence is more explicit.
If terms with n ̸= 0 occur and dϖ/dψ = 0 (the condition for no internal shear inside the vortex,
see Section 5.3.3) then the wavenumber is expected to depend on t as well as the coordinates.
Considering the form of S⊥ in equation (7.4.4) we see that
∂S⊥
∂ψ
= ∂S⊥
∂ψ
∣∣∣∣
ϖ
+ dϖ
dψ
∂S⊥
∂ϖ
= ∂S⊥
∂ψ
∣∣∣∣
ϖ
− tdϖ
dψ
∂S⊥
∂ϕ
, (7.4.10)
where ∂S⊥/∂ψ|ϖ results in the production of ∂Sn/∂ψ in the Fourier series expansion of S⊥.
Therefore, substituting for ∂S⊥/∂ψ in equations (7.4.7a,b) we find
kx = λ
[
∂ψ
∂x
∂S⊥
∂ψ
∣∣∣∣
ϖ
−
(
dϖ
dψ
∂ψ
∂x
t−ϖ ∂y
∂ψ
)
∂S⊥
∂ϕ
]
(7.4.11a)
ky = λ
[
∂ψ
∂y
∂S⊥
∂ψ
∣∣∣∣
ϖ
−
(
dϖ
dψ
∂ψ
∂y
t+ϖ∂x
∂ψ
)
∂S⊥
∂ϕ
]
(7.4.11b)
In the limit t→∞ we have
k2x + k2y ∼ λ2
(
dϖ
dψ
)2 (∂S⊥
∂ϕ
)2
|v|2t2 (7.4.12)
which is a product of |v|2t2 and a factor that is constant on a streamline, indicating that the
magnitude of the wavenumber increases to arbitrary large values at all points on it. We expect
the system to be stable to these perturbations at large t, behaving like the shearing waves
detailed in Goldreich and Lynden-Bell (1965). In Appendix C we demonstrate this is the case
for a vortex with infinite aspect ratio (ymax → ∞ with xmax finite). It is stable at large t to
shearing waves, with the ky ̸= 0 modes in particular displaying transient growth.
In short, although some transient growth will occur, ultimately the ‘shearing–up’ of the
130 Stability Analysis
waves will lead to stability.
7.4.3 A note on the Kida wavenumber form
The form of the wavenumber given by equation (7.2.2) in Section 7.2 differs from that adopted
in the analysis of the Kida solution in Lesur and Papaloizou (2009), which uses a special form
of k unique to the Kida streamlines.
Following from equations (7.1.17), (7.1.20) and
v0 =
S
χ− 1
(
y
χ
,−χx
)
,
we find that
Dkx
Dt
= −kx


7
0
∂vx
∂x
− ky ∂vy
∂x
= Sχky
χ− 1 (7.4.13a)
Dky
Dt
= −kx∂vx
∂y
− ky


7
0
∂vy
∂y
= − Skx
χ(χ− 1) (7.4.13b)
⇒ D
2kx
Dt2
= − S
2
(χ− 1)2 kx. (7.4.13c)
Without loss of generality we chose
kx = k0χ sin ζ cosϕ(t) (7.4.14)
where
ϕ(t) = S
χ− 1(t− t¯) (7.4.15)
and ζ, t¯ are integration constants. Substituting equation (7.4.14) into equation (7.4.13a) we
find the form of ky. With kz = k0 cos ζ we recover an equivalent form of the k used in Lesur
and Papaloizou (2009):
k(t) = k0
(
χ sin ζ cosϕ(t),− sin ζ sinϕ(t), cos ζ) (7.4.16)
It is important to note that equation (7.4.16) only works when the velocity components are
linear functions of x and y. In spite of this restriction, Chang and Oishi (2010) used it in the
problem of stability of a vortex core with a density gradient where this condition would not
be expected to be self-consistently satisfied. Thus an assessment of the situation that occurs
when this is not the case should be carried out.
From the definitions of Θ (equation (7.2.22)) and θ (equation (7.2.23)) we have that the
horizontal kz →∞ limit corresponds to Θ, θ, ζ = 0 while the vertical limit kz = 0 has Θ→∞
and θ, ζ = π/2. Thus ζ and θ are equivalent and from now on we will use θ instead of the
7.4 Time–dependent wavenumber 131
redundant ζ:
k(t) = k0
(
χ sin θ cosϕ(t),− sin θ sinϕ(t), cos θ). (7.4.17)
For a Kida vortex in a Keplerian background recall that the period to circulate round stream-
lines (equation (5.3.35)) is
P˜kida =
2π
S
(χ− 1)
which is independent of ψ and thus there is no internal shear. Thus ϖkida is a constant;
ϖkida =
2π
P˜kida
= S
χ− 1 ⇒
∂
∂ψ
ϖkida = 0, (7.4.18)
which is why the terms ∝ t in equation (7.4.11) are absent.
For completeness’ sake, we would like to find the form of SA(ψ, t) to which this k relates.
Recall that the parametric representation of an ellipse given in Section 5.3 has
(x, y) = (b sin η, bχ cos η)
For reasons that will become clear shortly, transform the elliptical coordinate η thus:
η = π2 + ϕ(t), (7.4.19)
so
(x, y) =
(
b cosϕ(t),−bχ sinϕ(t)). (7.4.20)
Given that b is constant on the ellipse around which the fluid parcel moves, b = b(ψ). Elimi-
nating ϕ we find that
x2
b2
+ y
2
b2χ2
= 1
⇒ b2χ2 = χ2x2 + y2 = 2χ(χ− 1)
S
ψ = 2χ
ϖkida
ψ
⇒ b =
√
2
χϖkida
ψ1/2, (7.4.21)
where we used the expression for the Kida streamfunction, equation (5.3.21a) and the definition
of ϖkida, equation (7.4.18). Then we have
x =
√
2
χϖkida
ψ1/2 cos[ϕ(t)] (7.4.22a)
y = −χ
√
2
χϖkida
ψ1/2 sin[ϕ(t)], (7.4.22b)
which can be consistency checked against equation (7.4.2). Comparing with the general form
132 Stability Analysis
of the particle trajectories given in equation (7.4.3) we see that in this form only terms with
n = ±1 are present. Taking the ansatz
S⊥ = S⊥,0ψ1/2f(ϕ) (7.4.23)
for some currently unknown f(ϕ), applying equation (7.4.11) we find
k⊥ =
1
2λS⊥,0ϖkida
√
2
χϖkida
(
χ
[
f cosϕ− f ′ sinϕ] ,− [f sinϕ+ f ′ cosϕ] ). (7.4.24)
Comparing this to the form of equation (7.4.17), if we set f(ϕ) = 1 we can match coefficients:
k0 sin θ =
1
2S⊥,0λ
√
2ϖkida
χ
(7.4.25)
k0 cos θ = kz (7.4.26)
We therefore reproduce the wavenumber of Lesur and Papaloizou (2009) (their equation (16),
with elliptical coordinates defined with semi-major axis in the x-direction).
Although this time–dependent wavenumber was derived from terms with n = ±1, and the
time–independent form is derived from adopting n = 0, you end up with the same equations
governing the stability of the Kida vortex regardless (these are given in Section 7.4.3.1). This
is because the Kida stability equations are invariant to a shift in the origin of time (it doesn’t
depend where on the streamline you start) and are thus independent of t¯. We may specify
(kx, ky) ∝ ∇ψ and exactly the same equations are recovered.
7.4.3.1 General stability equations for the Kida vortex
Following the approach of Section 7.3.1 ,the linearised equations for the Kida vortex with
aspect ratio χ are:
Dv′
Dη
= ΠKida(η, ψ)v′ (7.4.27)
where ΠKida = A−1KidaBKida and
ΘKida =
S2
(χ− 1)2Θ (7.4.28a)
A−1Kida =
1
1 + ΘKida(χ2 sin2 η + cos2 η)
(
1 + ΘKida cos2 η −ΘKidaχ sin η cos η
−ΘKidaχ sin η cos η 1 + ΘKidaχ2 sin2 η
)
, (7.4.28b)
7.5 Vertical stability 133
and
BKida =

−ΘKida sin η cos η (2χ+ 1)(2χ− 3)3χ +ΘKidaχ sin
2 η
4− χ
3 −ΘKidaχ cos
2 η ΘKida sin η cos η
 . (7.4.28c)
Note that these equations are independent of semi–minor axis b, i.e. independent of streamline
choice. This is because P˜Kida is the same on all streamlines.
7.4.4 Generic use of the time–independent wavenumber
In the generic case, ϖ is not the same on different streamlines. However, as discussed in
Section 7.4.2, if a wavenumber increases linearly with time only temporary exponential growth
is expected, with perturbations ultimately subject to at most weaker-than-exponential growth
with time. This requires nonlinear analysis to determine the outcome. This can shown to be
the case for the systems considered here. Furthermore, the time–dependent wavenumber in
the t→∞ limit is equivalent to the vertical stability limit kz = 0 (see the next section, 7.5).
Accordingly, we can extend the linear stability analysis for the Kida vortex (Lesur and
Papaloizou, 2009) to more general cases by adopting the time–independent wavenumber n = 0
and using the form for SA given by equation (7.2.1).
The form given by equation (7.4.17) however should only be reserved for cases where ϖ is
constant.
There is also an analogy with the situation that occurs in differentially rotating discs
for which the fluid elements orbit on circles. Time–independent wavenumber modes here
correspond to axisymmetric (m = 0) modes in the disc while modes with wavenumbers that
increase with time correspond to non-axisymmetric modes (m ̸= 0).
7.5 Vertical stability
We return to consider vertical stability (see Figure 7.2) for which kz = 0: the vertical velocity
perturbation is zero and we only consider perturbations in the plane of the vortex. From the
discussion in Section 7.4.2 and equation (7.4.12) we expect that
lim
t→∞
k2z
k2x + k2y
= 0
for choices of wavenumber that increase linearly with time. Our Θ parametrisation used in
Section 7.2.2 has Θ→∞ as t→∞.
We will start from the perturbed momentum and continuity equations (7.1.4, 7.1.7), in the
134 Stability Analysis
Eulerian formalism. The incompressibility condition ∇ · v′ = 0 gives k · v′ = 0 so
kxv
′
x + kyv′y = 0 (7.5.1a)
and we can set
v′ = (µky,−µkx) (7.5.1b)
for some scalar µ. The linearised form of the perturbed continuity equation is then
Dρ′
Dt
= −v′ · ∇ρ = −v′ · ∇ψ dρ
dψ
= −µ (ky,−kx) ·
(
∂ψ
∂x
,
∂ψ
∂y
)
dρ
dψ
= µ
(
kx
∂ψ
∂x
− ky ∂ψ
∂y
)
dρ
dψ
= µk · v0 dρ
dψ
(7.5.2)
Starting from equations (7.1.4a,b) we have
Dv′i
Dt
+ v′j
∂vi
∂xj
− 2ϵi3jΩv′j = −
1
ρ0
∂P ′
∂xi
+ ρ
′
ρ20
∂P0
∂xi
(7.5.3)
and will use
vi = ϵij3
∂ψ
∂xj
(7.5.4a)
v′i = µϵij3kj (7.5.4b)
∂P ′
∂xi
= ikiP ′ (7.5.4c)
with this last expression from equation (7.1.16). Equation (7.5.3) then becomes
ϵij3
D(µkj)
Dt
+ µϵjk3kk
∂
∂xj
(
ϵil3
∂ψ
∂xl
)
− 2ϵi3jΩµϵjk3kk = − 1
ρ0
(ikiP ′) +
ρ′
ρ20
∂P0
∂xi
. (7.5.5)
Using both
ϵjk3ϵil3 = δijδlk − δikδjl
ϵi3jϵjk3 = δik,
equation (7.5.5) becomes
ϵij3
D(µkj)
Dt
+ µ
[
kj
∂2ψ
∂xi∂xj
− ki∂
2ψ
∂x2j
]
− 2µΩki = − ikiP
′
ρ0
+ ρ
′
ρ20
∂P0
∂xi
. (7.5.6)
7.5 Vertical stability 135
We then take the dot product of equation (7.5.6) with v′ to eliminate P ′:
ϵij3
D(µkj)
Dt
(ϵik3kk) + µkj
∂2ψ
∂xi∂xj
(ϵik3kk) =
ρ′
ρ20
∂P0
∂xi
(ϵij3kj)
⇒ kjD(µkj)
Dt
− µkjkk ∂
∂xj
(
ϵki3
∂ψ
∂xi
)
= ρ
′
ρ20
ϵ3ij
∂P0
∂xi
kj
⇒ kjD(µkj)
Dt
− µkjkk ∂vk
∂xj
= ρ
′
ρ20
(∇P × k) · zˆ
⇒ kjD(µkj)
Dt
+ µkj
Dkj
Dt
= ρ
′
ρ20
(∇P × k) · zˆ,
using the expression for the time evolution of k (equation (7.1.20)) in the last step. Finally,
then
⇒ D
Dt
(µ|k|2) = ρ
′
ρ20
(∇P × k) · zˆ (7.5.7)
where we’ve cancelled the superfluous µ term, since we’re assuming µ ̸= 0, and zˆ is the unit
vector in the z-direction.
The two equations (7.5.7,7.5.2) provide a pair of first order differential equations to inte-
grate around streamlines. In the general case, with k taking the form in equation (7.4.11),
the wavenumber k is not a periodic function of time so these equations do not lead to a Flo-
quet problem. It is also worth noting that although k grows linearly with time, its form in
equation (7.4.11) implies that k · v0 remains bounded as the terms proportional to t cancel.
7.5.1 Vertical stability for the general vortex with internal shear
Considering the variable
s = t−1, ⇒ dt
ds
= −s−2 (7.5.8)
and let W = µ|k|2. Rescaling both equations (7.5.7 and 7.5.2) we find
Dρ′
Ds
= −W k · v0|k|2
dρ
dψ
s−2 (7.5.9)
DW
Ds
= − ρ
′
ρ20
(
∇P × kˆ
)
· zˆ|k|s−2 (7.5.10)
We are interested in the asymptotic behaviour of this equation. Vortex solutions with internal
shear have dϖ/dψ ̸= 0 so, provided ∂S⊥/∂ϕ ̸= 0 we have k ∼ t ∼ s−1. Since k · v0 is bounded
(Section 7.5), along with the background values ∇P , ρ0 and dρ/dψ we define the coefficients
I = k · v0 dρ
dψ
(7.5.11a)
J = (∇P × kˆ) · zˆ
ρ20
(7.5.11b)
136 Stability Analysis
and equations (7.5.9 and 7.5.10) have the asymptotic forms
Dρ′
Ds
∼ −I|k|−2s−2W ∼ −IW (7.5.12)
DW
Ds
∼ −J |k|s−2ρ′ ∼ −J s−3ρ′. (7.5.13)
Then
D2ρ′
Ds2
∼ −IDW
Ds
∼
[
IJ s−3
]
ρ′, (7.5.14)
For small s (i.e. large time t), the bracketed coefficient on the right hand side will be large.
Applying WKBJ analysis to equation (7.5.14) we find
ρ′(s) ∼ s3/4 exp
[
±2IJ s−1/2
]
. (7.5.15)
Therefore, for some 2|IJ | = K > 0 we find
ρ′(t) ∼ t−3/4 exp
[
Kt1/2
]
. (7.5.16)
The growth rate of ρ′ is ∼ t−1/2 → 0 as t → ∞ so there can be no exponentially growing
solutions that apply at large times. Weaker growth could occur, however.
7.5.2 Vertical stability for vortices with no internal shear
We now consider the vertical stability of the polytropic model (Section 5.4), where vortex has
the Kida streamlines given by equation (5.3.21a) and non-constant ρ = ρ(ψ). Note that we
also have P = P (ψ) and no internal shear, dϖ(ψ)/dψ = 0. We therefore adopt the Kida form
of the wavenumber given by equation (7.4.17) with ζ = θ = 0, corresponding to kz = 0 without
encountering the problem of wavenumber increasing linearly with time:
k(t) = k0
(
χ cosϕ(t),− sinϕ(t), 0). (7.5.17)
The coordinates on the streamline can be specified as indicated by equation (7.4.22b) so
x =
√
2(χ− 1)
Sχ
ψ1/2 sinϕ
y =
√
2χ(χ− 1)
S
ψ1/2 cosϕ.
7.5 Vertical stability 137
With the Kida velocity field given by
v0 = vkida =
S
χ− 1
(
y
χ
,−χx
)
=
√
2S
χ(χ− 1) ψ
1/2( cosϕ(t),−χ sinϕ(t)), (7.5.18)
and defining
Γ =
√
2Sχ
χ− 1 (7.5.19)
we find that k · v0 = Γk0ψ1/2 and equation (7.5.2) becomes
Dρ′
Dt
= µΓψ1/2k0
dρ
dψ
. (7.5.20)
Meanwhile, tackling equation (7.5.7) we have
|k|2 = k20
(
sin2 ϕ+ χ2 cos2 ϕ
)
∇P × k = −(k ×∇ψ)dP
dψ
= −Γk0ψ1/2dP
dψ
zˆ
so the equation for the time evolution of µ|k|2 is
D
Dt
(
µ|k|2
)
= D
Dt
[
µk20
(
sin2 ϕ+ χ2 cos2 ϕ
)]
= − ρ
′
ρ2
Γk0ψ1/2
dP
dψ
. (7.5.21)
Then, scaling equation (7.5.20):
(
sin2 ϕ+ χ2 cos2 ϕ
) Dρ′
Dt
= µk0
(
sin2 ϕ+ χ2 cos2 ϕ
)
Γψ1/2 dρ
dψ
⇒ D
Dt
[(
sin2 ϕ+ χ2 cos2 ϕ
) Dρ′
Dt
]
= −Γ
2ψ
ρ2
dP
dψ
dρ
dψ
ρ′ (7.5.22)
using the time-independence of ψ and v · ∇ψ = 0. Note that the pre-factor of ρ′ on the right
hand side of this equation is constant on streamlines. If we now set
Z =
(
sin2 ϕ+ χ2 cos2 ϕ
) Dρ′
Dt
(7.5.23)
then (recalling Dρ/Dt = 0) we find that Z satisfies a form of the Hill’s equation (Hill, 1886;
Whittaker and Watson, 1996) that can be written in the form
D2Z
Dt2
= − Γ
2ψ(
sin2 ϕ+ χ2 cos2 ϕ
)
ρ2
dP
dψ
dρ
dψ
Z = − qHillsin2 ϕ+ χ2 cos2 ϕ Z, (7.5.24)
138 Stability Analysis
where the quantity qHill is
qHill =
Γ2ψ
ρ2
dP
dψ
dρ
dψ
= 2Sχψ(χ− 1)ρ2
dP
dψ
dρ
dψ
. (7.5.25)
This equation (7.5.24) can be interpreted as describing the evolution of a gravity wave with a
time–dependent wavenumber, with qHill the square of the radial buoyancy or Brunt-Väisälä
frequency. The possibility of parametric instability is expected when this frequency is large
enough to be comparable to the vortex frequency ϖ. Solutions to this are discussed in Sec-
tion 8.4.
7.6 Summary and conclusions
In this chapter we have generalised the stability analysis applied to core of Kida vortex in Lesur
and Papaloizou (2009) so that it can be extended to the more general vortex models calculated
in Chapter 6. We consider both Lagrangian and Eulerian approaches to perturbation analysis.
The former more useful for drawing analytical conclusions (such as behaviour in the horizontal
stability limit in Section 7.3 and the vertical limit in Section 7.5), while the latter is more
useful for future numerical calculations.
The special Kida case is a useful one as the stability problem becomes separable for a specific
choice of time–dependent wavenumber (Section 7.4.3). Although this does not apply in general,
it is a useful test of the numerical work performed in the next chapter and as a starting point of
our perturbation analysis. Furthermore, we show the parity between the time–dependent and
–independent wavenumber form for this special, shear–free vortex solution. Our treatment of a
more general SA is therefore consistent with Lesur and Papaloizou (2009), despite apparently
very different forms for the wavenumber k.
In the general vortex case, where we may have a non–constant vorticity profile, density
profile, or both, it is still possible to look for modes localised on streamlines. From an Eulerian
viewpoint, these modes can have either a time–independent wavenumber (Section 7.2) or a
time–dependent one (Section 7.4).
However, when the circulation period is not constant on streamlines (i.e. the vortex con-
tains internal shear), the time–dependent wavenumber ultimately increases linearly with time.
These cannot be associated with conventional, exponentially growing linear modes, a situa-
tion familiar in shearing box calculations (Goldreich and Lynden-Bell, 1965). However, time–
independent wavenumber modes can be exponentially growing; for only these can the Kida
analysis be extended to more general vortices, contrary to the analysis of Chang and Oishi
(2010). We therefore justify extending the use of time–independent wavenumber to cases other
than the Kida vortex.
In the horizontal limit (θ = 0, perturbations out of the vortex plane), we find a general
7.6 Summary and conclusions 139
‘saddle point instability’, where a streamline’s epicyclic frequency is negative. This occurs
when the pressure distribution has a saddle point and this instability occurs independent of
density or vorticity profile. In the vertical limit (θ = π/2, perturbations in the vortex plane),
we find that vortices with Kida streamlines (where a time–dependent wavenumber applies)
will be stable. However, this is not true in general and so we expect to find some exponential
growth in this region. We need to move to numerical methods to establish behaviour inbetween
these two limits.

Chapter 8
Numerical treatment of stability
analysis
In this chapter we demonstrate how we implemented the numerics required for the stability
analysis described in the previous chapter. Two different approaches were used for determining
the growth rate of instability for the equilibrium vortex solutions detailed in Chapters 5 and
6.
The first method is detailed in Sections 8.1, based on extracting the various derivatives
needed in equation (7.2.28) directly from the numerical vortex solutions in Chapter 6. We
show the results of this approach in Sections 8.1.3. This method only gives good results
for small θ ≲ 30◦ (high kz) and no central density enhancement (Section 8.1.4). In order
to overcome these shortcomings, we formulate a second approach (Section 8.2), built around
calculating the matrix coefficients from an analytical fit of ψ. The results of this are presented
in Section 8.2.2 and we discuss how the two numerical approaches support each other.
The point vortex model and its role as a limiting case are discussed in Section 8.3, while
in the penultimate section we investigate the stability of the polytropic model. Finally, we
present the results of our stability analysis and discuss why the instabilities of the type found
here may not prevent a significant dust accumulation in vortices with a large aspect ratio.
8.1 First numerical approach
Our first method for finding growth rates in the (χ, θ) plane (where θ is the angle between the
vertical zˆ and the wavevector k, see Section 7.2.2.1 and Figure 7.3) for a vortex with class
{α, β, ρm} involves first picking a streamline inside the vortex and finding the coordinates of
that contour. We then calculate the matrix elements in the perturbation equation (7.2.28)
along that streamline. These two procedures are done in the program arclength.f90 ,
detailed in Section 8.1.1.
142 Numerical treatment of stability analysis
Then, for each value in a set of θ, we integrate each Floquet equation round a streamline
and calculate the eigenvalues of the resulting 3 × 3 matrix to find the growth rates. This is
done using a program called int_gr.f90 and is detailed in Section 8.1.2. Finally, we plot
the resulting stability curve in the (χ, θ) plane.
8.1.1 arclength.f90 : Finding quantities along streamlines
In Chapter 6 we describe our method for producing equilibrium vortex solutions from the four
parameters {α, β, ρm, ωm}. From these variables we produce a grid of values for the stream-
function ψ, the pressure distribution P , the vorticity source A(ψ) and the density distribution
ρ(ψ). From these fields we need to find the various derivatives found in the matrix perturbation
equation (7.2.28), namely
∂ψ
∂xi
,
∂2ψ
∂xi∂xj
,
∂P
∂xi
and ∂ρ
∂xi
{i, j = 1, 2}.
8.1.1.1 Calculation of the streamline contour
Given a position ymax ∈ (0, 1) on the y–axis we then calculate the streamline that passes
through this point. The typical value of ymax chosen was ymax = 0.85 for two reasons: it is far
enough away from the bounding streamline that passes through (0, 1), avoiding any potential
boundary issues and it is far away enough from the origin that the resulting streamline passes
through enough grid points to be of an appropriate resolution. This reduces the amount of
interpolation inside our integration program. The effect of changing ymax is investigated in
Section 8.2.2.
From this position (0, ymax), we find the value of the streamfunction at this point, ψσ. We
now follow this streamline around one revolution of the vortex, finding the coordinates of this
curve and then evaluate the quantities ∂ψ/∂xi, ∂P/∂xi etc. on this contour. We calculate the
fields of all the required derivatives over the original grid using the following finite difference
8.1 First numerical approach 143
schemes:
fx(x, y) =
1
12hx
[
− f(x+ 2hx, y) + 8f(x+ hx, y) (8.1.1a)
−8f(x− hx, y) + f(x+ 2hx, y)
]
+O(h4x),
fxx(x, y) =
1
h2x
[
f(x+ hx, y)− 2f(x, y) + f(x− hx, y)
]
+O(h2x), (8.1.1b)
fyy(x, y) =
1
h2y
[
f(x, y + hy)− 2f(x, y) + f(x, y − hy)
]
+O(h2y), (8.1.1c)
fxy(x, y) =
1
4hxhy
[
f(x+ hx, y + hy)− f(x+ hx, y − hy) (8.1.1d)
−f(x− hx, y + hy) + f(x− hx, y − hy)
]
+O(hxhy),
where hx is the grid spacing in the x–direction, hy is the grid spacing in the y–direction.
However, the equilibrium vortex solutions created in Chapter 6 are expressed over a rect-
angular grid (typically of 512× 2048 points) and not a grid that follows the contours (like an
elliptical grid in the Kida case). We therefore need to calculate the coordinates of the chosen
streamline.
For |y| ∈ [ymax/2, ymax], the curve of the streamline contour is tightest (especially for large
χ vortices) so we find the contour here by fixing x then interpolating between grid points in the
y–direction. As in Figure 8.1, if we have some x = X where we want to find the y–coordinate
of the streamline ψ = ψσ we begin by iterating in the y–direction to find the two grid points
either side of the streamline. We then use a simple linear interpolation to find the y–coordinate
Y where the streamline crosses x = X:
Y = yn +
yn+1 − yn
ψ(X, yn+1)− ψ(X, yn) × (ψσ − ψ(X, yn)) , (8.1.2)
where ψ(X, yn) is the value of the streamfunction at grid point (X, yn). Evaluating quantity Q
(which could be ∂ψ/∂xi, ∂P/∂xi etc.) at the point (X,Y ) also requires linear interpolation:
Q(X,Y ) = Q(X, yn) +
Q(X, yn+1)−Q(X, yn)
yn+1 − yn × (Y − yn) , (8.1.3)
where again Q(X, yn) is the value of Q at grid point (X, yn). In the range |y| < ymax/2,
we do an entirely similar calculation, fixing y = Y instead of x due to the steep gradient of
the contour in this region. This helps achieve a relatively even spacing of points around the
streamline, important for avoiding large jumps in our arclength variable which could lead to
inaccurate interpolation.
Once we have the coordinates of a complete contour, we calculate the arclength σ around
144 Numerical treatment of stability analysis
Figure 8.1 Calculating a point on the contour ψ = ψσ while fixing x = X or y = Y .
8.1 First numerical approach 145
the vortex streamline using the simple distance norm:
σn+1 = σn +
√
(xn+1 − xn)2 + (yn+1 − yn)2. (8.1.4)
This was deemed accurate enough due to the grid size; for the Kida vortex, a streamline passing
through (0, 0.85) had approximately 400 points for χ = 10 and approximately 3500 points for
χ ≃ 1.
Finally, we have arrays of all our derivative quantities Q as functions of σn. Therefore, we
can proceed to integrate the matrix stability equation (7.2.28) for different θ (Section 8.1.2).
8.1.1.2 Calculation of aspect ratio
The aspect ratio for the chosen streamline is calculated using the formula
χ = ymax
xmax
, (8.1.5)
where xmax is the maximum extent of the streamline along the positive x–axis. Note that χ
can vary throughout the vortex (Section 6.3.1). Unless otherwise stated, the value χ for a
given vortex will be the value at ymax = 0.85.
8.1.1.3 Calculation of the period
Recall from Section 5.3.3 a vortex’s internal shear can be visualised using the period around
streamlines, calculated using equation (5.3.32):
P˜ =
∮
dσ
|∇ψ| .
For a set of ymax ∈ (0, 1), for each ymax we calculate 1/|∇ψ| as a function of the arclength
using the approach outlined above. By definition of the streamfunction this shouldn’t be zero
anywhere so singularities won’t be a problem. With f(σ) = 1/|∇ψ| and N points {σn : n =
1, N} around our streamline contour, we calculate the period:
P˜ =
∮
dσ
|∇ψ| =
N−1∑
n=1
σn+1∫
σn
f(σ) dσ ≈
N−1∑
n=1
(σn+1 − σn)
[
f(σn) + f(σn+1)
2
]
. (8.1.6)
This enables us to produce plots of P˜ vs. ymax for our equilibrium solutions, as in Chapter 6.
The method was verified using the analytical Kida solution, which has period constant with χ
(equation 5.3.35):
P˜kida =
2π
S
(χ− 1).
146 Numerical treatment of stability analysis
8.1.2 int_gr.f90 : Integrating solutions along streamlines to find growth
rates
This program takes the various derivatives found along streamlines by arclength.f90 ,
performs a numerical integration of equation (7.2.28) for a range of θ (via the related variable
Θ, see equation (7.2.23)), and find the resulting growth rate of instability. Following the
Floquet approach detailed in Section 7.2.3, we approach this as an initial value problem (IVP)
in arclength σ, solving for a matrix of three perturbation vectors x′i = (v′x, v′y, ρ′)T such that
X ′ =
(
[x′1], [x′2], [x′3]
)
.
The initial condition is set to be
X ′(σ = 0) = I,
so that x′1 = (1, 0, 0)T etc., spanning the entire space of perturbations. Then, equation (7.2.28)
is integrated using the Bulirsch–Stoer algorithm (Stoer and Bulirsch, 2002) once around the
vortex to find X ′(σ = Σ). This is done using various routines from ‘Numerical Recipes’ (Press
et al., 1993), namely ODEINT, MMID, RAN1 and BSSTEP. Linear interpolation is again used to
supply the value of any quantity for any arclength value in the range [0,Σ], as required by the
integration routine. Care has to be taken to ‘wrap around’ the coefficients outside 0 ≤ σ ≤ Σ ,
since they are all periodic functions, with period Σ.
We perform the above integration for each of the three initial values of x′, for a set of values
of Θ = tan2 θ corresponding to equally spaced θ ∈ {0, 85◦}. Once the matrix X ′(σ = Σ) has
been calculated for each θ, we then find its eigenvalues using the Numerical Recipes routines
ELMHES and HQR. The first of these routines converts X ′(σ = Σ) into an upper Hessenberg
matrix, while the second finds the eigenvalues of this matrix. It returns the three characteristic
Floquet multipliers, ϱj , in the form
ϱj = ℜ(ϱj) + ℑ(ϱj)i,
which, according to equation (7.2.33b), we can extract our final growth rate γ:
γ = 1
P˜
max
j
{log |ϱj |}. (8.1.7)
Repeating this for all the ωm in our vortex class {α, β, ρm}, we plot γ as a heat plot against
the axes χ (calculated as in Section 8.1.1.2) and θ.
8.1.3 Results from first approach
Stability plots for vortices with variable vorticity profile and constant density were, for the most
part, of acceptable quality. The results of these can be seen in Figure 8.2. They are plotted
8.1 First numerical approach 147
for the regions 1 < χ ≲ 10, 0◦ ≤ θ < 70◦ and with growth rates in the range 10−4Ω < γ < 1Ω.
We show α in the range 0 ≤ α ≤ 4 (see Figure 6.2 for the equilibrium distributions of these).
As discussed, the chosen streamline passes through (0, 0.85), unless otherwise stated.
Figure 8.2a shows the Kida case which we can compare to Figure 8.3, produced using
analytic derivatives for the matrix coefficients. The main band emerging from 3/2 < χ < 4 (the
so called ‘saddle point’ instability) is reproduced, as is the band emerging from θ = 0◦, χ ≈ 5.9.
However, there is a spurious, phantom band above this main one. This phantom band persists
throughout the stability curves for vortices with constant density. Furthermore, when we
extend the plots to the full range of θ ∈ [0◦, 90◦], above θ ≈ 70◦ the factor Θ = tan2 θ
appearing in the matrix coefficients becomes very large. This magnifies interpolation errors,
especially for vortices with χ > 5. The result of this is a broad, spurious band for large θ, as
shown in Figure 8.5. For these reasons we restrict θ < 30◦ for this first method. This range is
sufficient however for a meaningful comparison with the second method detailed later.
Increasing α leads to more bands appearing for χ ≳ 4, but for the {α = 2, 4} cases it is
difficult to know what are real resonant bands and what are spurious. Noise from small aspect
ratio cases, χ ≲ 2, starts to creep in due to convergence issues in the original equilibrium
solutions.
There are real issues when a central density enhancement is introduced; typical output is
shown in Figure 8.4. The main saddle point instability band is picked up but everything else
was obscured by noise that no amount of filtering could remove. Therefore a new approach
needed to be found.
8.1.4 Summary of the difficulties with this approach
In Figures 8.5 and 8.6 we show the various problems that occur using this method. We find
the following problems in the regions marked in Figure 8.6:
• Region (1): Affected by convergence issues with the original equilibrium solutions. In
order to produce solutions in the region of χ = 1, increasingly large ωm need to be added
to existing solutions. This makes both convergence and producing many solutions in this
region difficult.
• Region (2): Interpolation in arclength.f90 and int_gr.f90 results in errors mag-
nified for large θ/Θ.
• Region (3): The fixed grid over which the equilibrium solutions are generated creates
resolution difficulties for large χ vortices which contain less points. We also have the
appearance of ‘phantom’ resonance band(s) at θ ≈ 40◦.
Furthermore, increasing the resolution of these plot in the χ–direction is very costly as it
requires calculating further equilibrium solutions; even starting from nearby solutions this
148 Numerical treatment of stability analysis
 
 
2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
60
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
χ
θ
log10 γ
(a) Plot of log10 γ for vortex class {α, β, ρm} = {0, 0, 0}
 
 
2 3 4 5 6 7 8 9
0
10
20
30
40
50
60
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
χ
θ
log10 γ
(b) Plot of log10 γ for vortex class {0.25, 0, 0}
 
 
2 3 4 5 6 7 8
0
10
20
30
40
50
60
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
χ
θ
log10 γ
(c) Plot of log10 γ for vortex class {0.5, 0, 0}
 
 
2 3 4 5 6 7
0
10
20
30
40
50
60
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
χ
θ
log10 γ
(d) Plot of log10 γ for vortex class {1, 0, 0}
 
 
2 3 4 5 6 7 8
0
10
20
30
40
50
60
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
χ
θ
log10 γ
(e) Plot of log10 γ for vortex class {2, 0, 0}
2 3 4 5 6 7 8
0
10
20
30
40
50
60
 
 
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
χ
θ
log10 γ
(f) Plot of log10 γ for vortex class {4, 0, 0}
Figure 8.2 Results of the first approach, finding stability plots in the (χ, θ) plane for vortex
solutions with no density. The region due the saddle point instability appears on the left
hand side as a wide band 3.2 < χ < 4 which shrinks in width and travels to smaller χ before
disappearing.
8.1 First numerical approach 149
0
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
1
.
5
4
6
1
0
1
0
0
θ
χ
-
4
-
3
.
5
-
3
-
2
.
5
-
2
-
1
.
5
-
1
-
0
.
5
0
log10 γ
F
ig
ur
e
8.
3
St
ab
ili
ty
pl
ot
fo
r
th
e
K
id
a
vo
rt
ex
,p
ro
du
ce
d
us
in
g
an
al
yt
ic
co
effi
ci
en
ts
.
T
hi
s
is
in
ag
re
em
en
t
w
ith
th
e
pl
ot
in
Le
su
r
an
d
Pa
pa
lo
iz
ou
(2
00
9)
(t
he
ir
Fi
gu
re
3)
.
N
ot
e
th
e
lo
gs
ca
le
al
on
g
th
e
x
–a
xi
s;
sin
ce
us
in
g
an
al
yt
ic
ex
pr
es
sio
ns
fo
r
th
e
m
at
rix
co
effi
ci
en
ts
we
ar
e
no
t
co
nfi
ne
d
by
th
e
nu
m
be
r
of
hi
gh
qu
al
ity
eq
ui
lib
riu
m
so
lu
tio
ns
we
ca
n
cr
ea
te
so
ca
n
ex
te
nd
th
e
χ
–a
xi
s
ar
bi
tr
ar
ily
fa
r.
150 Numerical treatment of stability analysis
 
 
2 3 4 5 6 7 8 9
0
10
20
30
40
50
60
70
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
χ
θ
log10 γ
(a) Stability plot for vortex class {0, 1, 0.5}
 
 
2 3 4 5 6 7 8 9
0
10
20
30
40
50
60
70
−2.5
−2
−1.5
−1
−0.5
χ
θ
log10 γ
(b) Stability plot for vortex class {0.25, 1, 0.5}
Figure 8.4 Results of the first approach, finding stability plots in the (χ, θ) plane for vortex
solutions with a central density enhancement. Despite picking up the saddle point instability
1.5 ≲ χ ≲ 4, all other bands are obscured by noise.
takes considerable time. This is a real problem when searching for the fine resonant bands
which occur in the region χ > 4.
We therefore develop a second method. We needed a method less reliant on interpolation
of the original equilibrium data and which allowed for higher resolution along the χ–axis at
less cost.
8.2 Second numerical approach
Most of the problems with the first approach occur because of interpolation in both contouring
and the calculation of the derivatives in arclength.f90, to the detriment of the stability cal-
culations. Therefore, instead of interpolating to find quantities along streamlines, we produced
polynomial fits of the streamfunction ψ.
8.2.1 Implementation
The fits of ψ from the grid of values produced for each equilibrium solution are calculated using
the polyfitweighted2.m , MATLAB code of Rogers (2007). This finds a least–squares fit
of the 2D data ψ(x, y) with an nth order polynomial and weighting w(x, y). The original data
was weighted by either 0 or 1 depending on whether a point was outside or inside the vortex
boundary, respectively. Due to the symmetry of the solutions we fit polynomials in x2 and y2,
8.2 Second numerical approach 151
Figure 8.5 Stability plot for {0.25, 0, 0} using the first method, for full range of θ. The
topmost region, circled in blue, is a spurious horizontal band, caused by the large θ/Θ factor
that appears in the matrix coefficients, magnifying resolution and interpolation errors. The
lower highlighted region is a phantom resonance band. Note that there is nevertheless good
agreement with the Kida results shown in Figure 8.3 for θ ≲ 30◦.
trying quadratic, cubic and quartic fits. For example, the form in the quadratic case would be:
ψ(x2, y2) = a00 + a10x2 + a01y2 + a20x4 + a11x2y2 + a02y4.
In order to enforce ψ = 0 at the centre of the vortex, without loss of generality we set a00 = 0.
We successfully found quadratic and cubic fits, but the extra degrees of freedom associated
with the quartic fits meant they usually failed to converge. We used the cubic fit, checking
their validity against the quadratic fits, the results of the previous approach for θ < 30◦ and the
growth rates from the analytic Kida solution. The coefficients a10, a01 etc. were well behaved
and varied in a predictable way so could be straightforwardly interpolated as functions of χ
and ymax. This allowed for any amount of interpolation between χ, eliminating the problem
of calculating models ab initio at high resolution along the χ–axis.
With these fits found, the location on the streamline as the integration proceeds is specified
152 Numerical treatment of stability analysis
Figure 8.6 An infographic showing the problem areas in the (χ, θ) stability diagram while
using the first numerical approach. Full explanation is given in Section 8.1.4.
by solving the equations
Dx
Dt
= ∂ψ
∂y
(8.2.1)
Dy
Dt
= −∂ψ
∂x
, (8.2.2)
with the pressure field calculated using equation (6.2.3):
P
ρ
= 23S
2x2 − 12 |∇ψ|
2 − 13S (ψ − ψb) +
∫ ψ
ψb
A(ψ′) dψ′,
and the density distribution using equation (5.4.3):
ρ =
[
1− b(ψ − ψb)
ψb
]n
.
At the vortex centre ψ = 0 so ρmax = (1 + b)n. With the central ρmax taken from the gridded
density data and n = 1 we can reconstruct ρ(ψ). When n = 1, the different expressions for ρ
given in equations (5.2.6b) and (5.4.3) are equivalent, with n ≡ β.
The various derivatives are then calculated by differentiating the analytic expressions, there-
fore avoiding the interpolation errors introduced in the first method. Integration is done using
8.2 Second numerical approach 153
an analogous, Bulirsch–Stoer method as in the first approach and we also use the same θ to
parametrise k. After integration, the growth rate of any instability present is obtained by
solving the additional equation
Dγ2
Dt
= log |v′|. (8.2.3)
For a system with growth rate γ, we expect that ultimately γ2 → γt2/2 so we determine γ
by making a parabolic fit to γ2 at large times. We found that integrations running for 1000
circulations around streamlines could detect growth rates down to γ ∼ 10−4Ω.
In order to resolve fine parametric bands, regions of instability often require high resolution
in the (χ, θ) phase space; typically for a model {α, β, ρm} we require a 300 × 300 grid. This
method does allow for a significant increase in resolution in the χ–direction over the first
approach (where we were limited to around 50 vortex models).
8.2.2 Results from second approach
We present the results for the constant–density cases in Figure 8.7. The Kida solution case in
Figure 8.7a agrees with that of Figure 8.3 both in terms of structure and magnitude of growth
rate. Similarly we can see the same structures (provided θ ≲ 30◦) between the nonzero α cases
shown in Figure 8.2 and Figures 8.7b–8.7e.
In Figure 8.7 we show results for streamlines with ymax = 0.85 and α = {0, 0.25, 0.5,
1.0, 2.0, 4.0}. The associated maximum growth rate plot can be seen in Figure 8.8. As α
increases, the instability band originating from χ ≈ 4.65 widens and moves to smaller values of
χ, while the small-χ region associated with the saddle point instability shrinks and eventually
disappears, as can be clearly seen in the maximum growth rate plot in Figure 8.8b. This is
due to vortices with these steep vorticity profiles no longer containing the pressure distribution
saddle points upon which this instability depends. Several additional instability bands appear
at larger values of χ.
In Figure 8.7f, the deviation of the α = 4 case from the smaller α cases preceding it, the
results from the first approach (Figure 8.2f) and the limiting point vortex case (Figure 8.15)
is most probably due to the extreme streamline shape near the boundary of these vortices.
As was observed in Section 6.3.1, a vorticity profile this steep results in severely ‘pinched’
streamlines near the y–axis. Therefore, the fits of ψ in x2 and y2 will be less accurate near the
boundary of such vortices (we would, however, expect them to still be good near the vortex
core).
In order to study the effects of introducing a variable vorticity profile in the core, in
Figure 8.9 we illustrate the stability of vortices with α = 0.25. The stability of motion on
streamlines with ymax = {0.5, 0.85, 0.95} is shown in Figures 8.9a, 8.9b and 8.9c respectively.
As indicated in Section 7.3.3, moving outwards from the vortex centre, we expect parametric
instability to occur for χ ≈ 4.65. This is visible for the ymax = 0.5 streamline, where a narrow
154 Numerical treatment of stability analysis
2 4 6 8 10
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(a) Stability plot for vortex class {α, β, ρm} = {0, 0, 0}
2 4 6 8
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(b) Stability plot for vortex class {0.25, 0, 0}
Figure 8.7 (Continued over the next two pages.)
8.2 Second numerical approach 155
2 4 6 8
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(c) Stability plot for vortex class {0.5, 0, 0}
2 3 4 5 6 7
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(d) Stability plot for vortex class {1, 0, 0}
Figure 8.7 (Continued on next page.)
156 Numerical treatment of stability analysis
2 4 6 8
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(e) Stability plot for vortex class {2, 0, 0}
2 3 4 5 6
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(f) Stability plot for vortex class {4, 0, 0}: see note in main text.
Figure 8.7 Results of the second approach finding stability plots in the (χ, θ) plane for vortex
solutions with uniform density. Plots of the maximum growth rates for these solutions can be
seen in Figure 8.8. These plots were produced using IDL (Liu et al., 2013).
8.2 Second numerical approach 157
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 2 3 4 5 6 7 8 9 10
m
ax θ
(l
og
1
0
γ
)
χ
Kida
{0.25, 0.0, 0.0}
{0.5, 0.0, 0.0}
{1.0, 0.0, 0.0}
{2.0, 0.0, 0.0}
Point vortex
(a) Maximum growth rate over the whole χ range
-4
-3
-2
-1
0
1
1 1.5 2 2.5 3 3.5 4 4.5
m
ax θ
(l
og
1
0
γ
)
χ
Kida
{0.25, 0.0, 0.0}
{0.5, 0.0, 0.0}
{1.0, 0.0, 0.0}
(b) Zooming in on the saddle point instability
Figure 8.8 Plot of maximum growth rate against χ for vortex solutions with uniform density,
including comparison with the point vortex case discussed in Section 8.3. The solutions tend
to match the point vortex at large χ as we would expect.
Figure 8.8a shows this over the whole range of χ while 8.8b restricts it to the maximum growth
rate of the strong saddle point instability, showing its migration to smaller χ while remaining
the same strength. Note that this instability does not exist for the α = 2 solution nor the
point vortex, the limit of large α.
158 Numerical treatment of stability analysis
2 4 6 8
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(a) ymax = 0.5
2 4 6 8
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(b) ymax = 0.85
2 4 6 8
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(c) ymax = 0.95
Figure 8.9 The stability plot for vortex class {0.25, 0, 0} for different ymax, i.e. different
streamlines in the same vortex solution.
8.2 Second numerical approach 159
2 4 6 8 10 12
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(a) ymax = 0.5
2 4 6 8 10 12
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(b) ymax = 0.67
2 4 6 8 10 12 14
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(c) ymax = 0.85
2 4 6 8 10 12 14
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(d) ymax = 0.95
Figure 8.10 The stability plot for vortex class {0, 1, 2}, with different ymax.
vertical instability band appears at this location. As ymax is increased (i.e. moving away from
the vortex centre), this band is seen to broaden. Simultaneously, the instability region at large
χ extends towards the χ–axis, intersecting this axis at smaller values of χ. In addition, the
region associated with the saddle point instability (Section 7.3.2) shifts towards smaller values
of χ.
The stability of vortices with α = 0, β = 1 and varying ρm are illustrated in Figure 8.11,
while results for vortices with α = {0.25, 0.5} and a central density enhancement are demon-
strated in Figure 8.13. For these calculations, recall that χ was varied by changing ωm while
ρm was chosen such that a fixed mass per unit length was added to the vortex for all χ. This
160 Numerical treatment of stability analysis
2 4 6 8 10
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(a) Stability plot for vortex class {0, 1, 0.5}
2 4 6 8 10 12 14
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(b) Stability plot for vortex class {0, 1, 2}
Figure 8.11 Resulting stability plots when a central density enhancement is added to a Kida
vorticity profile. The maximum growth rate plots for these solutions can be seen in Figure 8.12.
The small ‘islands’ of instability are a result of both insufficient resolution to pick up the whole
band and IDL’s plotting algorithm.
8.2 Second numerical approach 161
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
2 4 6 8 10 12
m
ax θ
(l
og
1
0
γ
)
χ
Kida
{0.0, 1.0, 0.5}
{0.0, 1.0, 2.0}
Figure 8.12 Maximum growth rate as a function of χ for vortices with a Kida vorticity profile
and a central density enhancement. The dashed lines show these models, neither of which have
the pronounced stable gap that the Kida solution does. There is some noise for these solutions
due to the difficulty of resolving the fine resonance bands.
results in the central density ρmax increasing with χ (see Section 6.3.3), with central densities
ranging from 3 to 12 times the background level at large values of χ.
In Figure 8.11, we can see the effect of introducing a small density excess on the stability
of a Kida vortex. For the case with ρm = 2, Figure 8.10 shows the results for streamlines with
ymax = {0.5, 0.67, 0.85, 0.95}. In the case of ymax = 0.5, where departures from the Kida–like
quadratic streamfunction are smaller, a parametric instability band is seen to emanate from
the expected location χ ≈ 4.65. This appears to connect with the band originating from
large χ and θ in the Kida case (Figure 8.7a), leaving a region that is very weakly unstable (or
possibly stable) between them. Bands with these features seem to a be common feature in these
calculations, requiring time–consuming calculations at high resolution to locate them. (This
renders the first method we adopted inapplicable.) In this case, even with the much improved
second numerical approach, we found it impractical to resolve instabilities with growth rates
γ < 10−4Ω.
When the streamline at ymax = 0.85 is considered, the band originating from χ ≈ 4.85 has
broadened to produce and unstable region with growth rate γ ∼ 0.05Ω for 4 < χ < 5. Two
additional narrow bands appear at larger χ with characteristic growth rates γ ∼ 0.01Ω. The
instability band that appears between 2 < χ < 4 is similar to that seen in the models with
large α and no density excess.
162 Numerical treatment of stability analysis
2 4 6 8 10
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(a) Stability plot for vortex class {0.25, 1, 0.5}
2 4 6 8
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
(b) Stability plot for vortex class {0.5, 1, 0.3}
Figure 8.13 The stability of non–Kida vorticity profiles with a central density enhancement.
In both cases, ρmax ≈ 2.5 when χ = 8. The associated plots for the maximum growth rate can
be found in Figure 8.14.
8.3 Point vortex stability 163
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 2 3 4 5 6 7 8 9 10
m
a
x
θ
(l
o
g
1
0
γ
)
χ
{0.25, 0.0, 0.0}
{0.25, 1.0, 0.5}
(a) Maximum growth rate for the case {0.25, 1, 0.5}.
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 2 3 4 5 6 7 8 9 10
m
a
x
θ
(l
o
g
1
0
γ
)
χ
{0.5, 0.0, 0.0}
{0.5, 1.0, 0.3}
(b) Maximum growth rate for the case {0.5, 1, 0.3}.
Figure 8.14 Maximum growth rate plots for vortices with non–Kida vorticity profiles, taken
for the streamline with ymax = 0.85. The band on the left hand side in both cases is due to the
saddle point instability while the middle band is due to the parametric instability which begins
at χ ≈ 4.65 at the centre of the vortex. See Figures 6.8b and 6.8c for the relation between
ρmax and χ in these cases.
Results for ymax = 0.95 are a continuous extension from smaller values. Note that the
deviations from elliptical streamlines are small for small ymax and increase with ymax. We
therefore expect greater deviation from Kida with increasing ymax. Furthermore, this supports
the use of the polynomial fits described in Section 8.2 – fits must work for sufficiently small
ymax where a quadratic polynomial is accurate. Also, different order fits must agree sufficiently
near the vortex core and we checked that this was the case.
When the ensemble of streamlines is considered for this model, it is difficult to find any χ
for which there is stability for all θ, as can be seen in the plots of maxθ(γ) vs. χ in Figure 8.12.
The results for models with α = 0.25 and α = 0.5 are shown in Figure 8.13, with the
associated maximum growth rate plots in Figure 8.14. The behaviour seen is quantitatively
similar to the previous cases with density excesses, but with growth rates at large χ being
γ ∼ 0.05Ω, similar to the concentrated Bernoulli source (large α) cases with constant density.
8.3 Point vortex stability
We also examined a point vortex model. This adopts the streamfunction
ψ = K log |r|+ 34Ω
2x2 (8.3.1)
which corresponds to a Bernoulli source localised at the vortex centre/box origin (see e.g.
Batchelor, 2000). We consider it here as it can be thought of as the limiting case of large α. The
aspect ratio of the streamline at ymax = 0.85 is fixed by an appropriate choice of the constant
164 Numerical treatment of stability analysis
3 4 5 6
20
40
60
80
  -4.000
  -3.333
  -2.667
  -2.000
  -1.333
  -0.667
   0.000
χ
θ
log
10
γ
Figure 8.15 Stability plot for the point vortex model with streamfunction ψ = K log |r| +
3
4Ω2x2, the limiting case for large α. Note its similarities to Figure 8.7e. The maximum growth
rate for each χ is compared to the other constant density solutions in Figure 8.8a.
K. The plot is produced by varying K for a given ymax; requiring that ψ(xmax, 0) = ψ(0, ymax)
in equation (8.3.1), and given χ = ymax/xmax we find that
K = 34
Ω2y2max
χ2 logχ. (8.3.2)
Thus K = K(χ) provides a streamline with a specified χ that passes through a given ymax.
The stability properties are obtained using the same method as for the other models, with
the resulting stability plot is given in Figure 8.15. Growth rates for χ ≳ 5 are γ ∼ 0.05Ω.
The point vortex is the limiting case of a constant density vortex with infinite α. The
relationship is apparent comparing the large–χ solutions for the constant density, α = 2 case
shown in Figure 8.7e with the point vortex results in Figure 8.15. When α is large and the
vorticity source strongly peaked, a streamline close to the boundary experiences a potential
very similar to the point vortex, resulting in a similar shape and stability characteristics. We
note that the failure of the polynomial fits in x2 and y2 to describe ψ near the boundary in
the α = 4 case (Figure 8.7f) prevent it from exhibiting these same features; they are however
8.4 Polytropic model stability 165
seen in for θ ≲ 30◦ in the results of the first approach (Figure 8.2f).
Although these models (and those detailed in Figure 8.7) do not have any density excess,
they are relevant to streamlines outside a high–density core so are of generic significance for
vortices accumulating dust in their cores.
8.4 Polytropic model stability
In Figure 8.16 we show the stability of the polytropic models detailed in Section 5.4. We use
n = 1 (so there is a direct relationship to the earlier power law solutions – see Section 8.2.1)
and central densities of 3 and 6 times the background value of ρ. The results are for streamlines
with ymax = 0.95 as these show the most detail. As can be seen in Figure 8.17, a larger central
density enhancement does not result in any substantial increase in the maximum growth rate,
although there is the loss of a stable ‘gap’ as ρmax is increased.
Recall that for these models, vortices associated with values of χ ̸= 7 have to be considered
to be immersed in a non–Keplerian background flow. The results are qualitatively similar to
models with a central density excess in a Keplerian background flow, with more instability
bands appearing as ρmax is increased. The characteristic growth rates at χ ≈ 7 are γ ∼ 0.01Ω.
Similarly, the results of the constant density polytropic model shown in Figure 8.16a are
qualitatively similar to that of the Kida vortex shown in Figure 8.7a, with the strong saddle
point instability region shifted to smaller values of χ.
We also consider the vertical stability of the polytropic model, as discussed in Section 7.5.2.
Note that this model has no internal shear and so in this limit, assuming exponentially growing
modes, we can proceed analytically. Solving equation (7.5.24) leads to a standard Floquet
problem. qHill is constant on streamlines and can be interpreted as the square of a radial
buoyancy/Brunt-Väisälä frequency and is what the period round the streamline resonates
with. Note that qHill = qHill(b, n), where ρmax = (1 + b)n.
We consider the case when χ = 7 as this is associated with the Keplerian background case
(see equation (5.4.1)). The growth rate is plotted as a function of qHill/Ω2 in Figure 8.18. The
maximum growth rate of γ ∼ 0.12Ω occurs for qHill/Ω2 = 1.5, which is around an order of
magnitude greater than the growth rates seen in Figure 8.16 at this value of χ. Thus modes
with kz = 0 (as in Chang and Oishi, 2010) may dominate in this case. However, as noted in
Section 7.2.1, the neglect of vertical stratification is only justified if k2z/k2⊥ is large so this is
likely to only be valid close to the midplane where the vertical stratification is least.
8.5 Summary
In this chapter we took the analytical work of Chapter 7 and attempted to find the exponential
growth rate, γ, for the various vortex solutions found in Chapter 6.
166 Numerical treatment of stability analysis
2 4 6 8 10
20
40
60
80
  -4.000
  -3.372
  -2.744
  -2.116
  -1.488
  -0.861
  -0.233
χ
θ
log
10
γ
(a) Stability plot for polytrope solution no central density enhancement (ρmax = 1).
4 6 8 10
20
40
60
80
  -4.000
  -3.444
  -2.889
  -2.333
  -1.777
  -1.222
  -0.666
χ
θ
log
10
γ
(b) Stability plot for polytrope solution with n = 1 and ρmax = 3.
Figure 8.16 (Continued on next page.)
8.5 Summary 167
4 6 8 10
20
40
60
80
  -4.000
  -3.420
  -2.840
  -2.260
  -1.681
  -1.101
  -0.521
χ
θ
log
10
γ
(c) Stability plot for polytrope solution with n = 1 and ρmax = 6.
Figure 8.16 Polytrope solutions for ρmax = {0, 3, 6}. A plot of the maximum growth rates
for each aspect ratio are shown in Figure 8.17.
168 Numerical treatment of stability analysis
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 2 3 4 5 6 7 8 9 10
m
ax θ
(l
og
1
0
γ
)
χ
Kida
Polytrope, ρmax = 1
Polytrope, ρmax = 3
Polytrope, ρmax = 6
Figure 8.17 Plot of maximum growth rate against χ for the polytropic vortex solutions, with
the Kida case illustrated for comparison. Note that the polytropic solutions do not exist below
χ = 2 for ρmax > 1 (see Section 5.4 for details). There is some low–level noise due to the
difficulty of resolving the fine resonance bands for large χ.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 2 4 6 8 10
γ
qHILL/Ω
2
Figure 8.18 Growth rate of the parametric instability of the polytropic model for χ = 7 and
kz = 0.
8.5 Summary 169
Our first approach involved taking the gridded data for ψ, P and ρ, producing fields of
derivatives ∂ψ/∂xi etc. using a finite difference regime, finding a streamline contour via linear
interpolation then calculating the values of these derivatives along this streamline as functions
of arclength σ. With these we then performed a Bulirsch–Stoer integration once round the
vortex streamline for each desired value of θ. Finally, we extracted the associated instability
growth rate by exploiting Floquet theory.
As shown in Figure 8.6, there were significant problems with this method. The need to
calculate new equilibrium vortex models to increase the resolution in the χ–direction made this
approach computationally expensive, especially in the small–χ region. Errors introduced by
interpolation at various points in arclength.f90 and int_gr.f90 also caused significant
problems at large θ and for vortex solutions with a central density enhancement. Results
for θ ≲ 30◦ were however reliable and allowed comparison with the second method and the
analytical Kida case.
The second approach revolved around avoiding the repeated use of interpolation. This was
done by producing quadratic and cubic fits in {x2, y2} for each vortex solution’s streamfunction,
ψ. These coefficients could then be expressed as functions of χ, increasing the resolution along
this axis. The various quantities needed in the matrix coefficients of equation (7.2.28) were
then directly calculated around the chosen streamline contour and integration was again done
using Bulirsch–Stoer. The viability of this method was tested against the result shown in Lesur
and Papaloizou (2009) for the Kida vortex (Figure 8.3), the constant density results of the first
approach and what we knew analytically in the horizontal, kz →∞ limit from Chapter 7.
Using this second approach, we have generalised the Kida vortices’ strong ‘saddle point’
instability (Section 7.3.2) to the centre of general vortices. Apart from having a strong insta-
bility, vortices of this type are of less interest as the associated saddle point in the pressure
distribution means they will not attract dust (Section 3.5).
We have also shown how parametric instability bands should appear, moving outwards
from the vortex centre, and consistent with the numerical results varying ymax in Section 8.2.2.
Additionally, vortices with a concentrated vorticity source and no density excess have strong
instability bands at all aspect ratios with growth rates γ ∼ 0.05Ω. This is also seen in the
limiting case of the point vortex (Section 8.3). Meanwhile, models with a density excess can
show many narrow instability bands, though those with flatter (small α) vorticity profiles show
less strongly growing modes with γ ∼ 0.01Ω. In general, we find that including a non-zero
ρmax does not lead to an increase in maxθ γ, but it does reduce the size of any completely
stable gap or eliminates it completely (see Figure 8.14).
We also investigated the stability of the dust–laden polytropic model (Sections 5.4 and 8.4)
with χ = 7 to modes localised on streamlines, adopting a time–dependent wavenumber for the
vertical stability case with kz = 0. This latter form is the case considered by Chang and Oishi
(2010). Note that this analysis of the polytropic model is possible as it is a special case with no
170 Numerical treatment of stability analysis
internal shear that matches onto the background Keplerian shearing flow only when χ = 7. We
found that when kz = 0 (Figure 8.18), modes could occur with growth rates γ ∼ 0.1Ω, around
an order of magnitude greater than when a general wavevector is considered (Figure 8.16).
However, these results would be affected by shear inside the vortex if an attempt is made to
generalise them to other vortices. This model has no internal shear and so is special; if shear
is included, this may shear out disturbances before they can grow.
Even with the weak internal shear ∼ 0.01Ω shown by vortices in Figure 6.5, with putative
growth rates γ ∼ 0.1Ω, we might expect temporary growth for 10 growth times, or a temporary
amplification factor ∼ 104. A nonlinear analysis would be required to resolve the outcome in
this case. Also, as noted in Section 7.2.1, these polytropic kz = 0 modes are also likely to be
affected by vertical stratification unless we are considering streamlines very close to the disc
midplane. We will investigate this further in the next chapter.
8.6 Discussion and conclusions
We have seen that dust particles attracted from the outer disc to a vortex core with high aspect
ratio, χ, may well encounter parametric instabilities with characteristic growth rates of a few
10−2Ω up to 0.1Ω. This is the case even outside any high density core and so it is important
to assess potential consequences for dust accumulation in vortices.
8.6.1 Inhibiting parametric instability
The instabilities we have found are both parametric and local so they can be inhibited by
either dissipative effects or effects that disrupt the periodicity of the circulated motion. The
magnitude of the dissipative effects is uncertain (for example it is dependent on location in the
PP disc) so we will just consider the second effect.
Firstly, the accumulation of dust in vortices may occur rapidly (e.g. Lyra et al., 2009;
Méheut et al., 2012b), such that dust particle motion departs significantly from being periodic.
This may also be the case for gas motion (Méheut et al., 2012b). For these reasons, parametric
instability may not have been seen in simulations of dust trapping up to now.
Secondly, if we suppose this parametric instability is present, it is likely to lead to some
low–level turbulence. This is indicated by work of both Lesur and Papaloizou (2010) and Lyra
and Klahr (2011) who find that such instabilities do not have a strongly disruptive effect on
large aspect ratio vortices produced by the subcritical baroclinic instability (SBI). Therefore,
although parametric instability may act to cause a vortex to ultimately decay, it may be
successfully maintained if there is some mechanism to generate it such as the SBI or RWI.
We have just considered steady vortex solutions not supported or generated by any particular
mechanism.
For Kida vortices, a strong, exponentially growing and potentially rapid vortex–destroying
8.6 Discussion and conclusions 171
instability only occurs due to the saddle point instability for 3/2 < χ < 4. These vortices do
not attract dust due to their unfavourable pressure gradient. However, for larger aspect ratios,
we might expect there to be a balance between inward flow due to the mean pressure gradient
and turbulent diffusion (e.g. Lyra and Lin, 2013). We will do a rough, back-of-the-envelope
calculation for this. The inflow rate for small particles driven by the pressure gradient is
|v| ∼ |∇P | τs
ρ
, (8.6.1)
(e.g. Papaloizou and Terquem, 2006). Supposing a vortex has lengthscale Lv in the minor axis
direction (so χ ∼ L−1v ) we estimate that
P ∼ ρΩ2L2v (8.6.2)
|v| ∼ Ω2Lvτs. (8.6.3)
As the unstable modes are local, the wavelength should be |k| ≪ Lv. For the purposes of this
crude approximation we will take π|k|−1 = Lv/10. An estimate of the associated diffusion
coefficient based on dimensional scaling is
D = γ|k|2 (8.6.4)
where γ is our parametric growth rate. Balancing pressure–driven inflow against diffusion we
obtain
|∇ρ|
ρ
= 1
Lρ
∼ |v|
D
∼ (10πΩ)
2τs
γLv
, (8.6.5)
where Lρ is the density lengthscale. Hence Lρ ∼ fLv where
Lp
Lv
= f ∼ γ/Ω100π2Ωτs . (8.6.6)
Significant dust concentrations become possible once f < 1. For γ = 0.1Ω this becomes equiv-
alent to Ts = Ωτs ≳ 10−4 with characteristic inflow time (Ω2τs)−1. Recall from Section 3.5.1
that the most favourably trapped dust has Ts ≃ 1.
Although this estimate is highly uncertain, it does indicate that the existence of parametric
instabilities do not necessarily prevent the possibility of dust accumulation in vortices.
8.6.2 Including dust back-reaction
Recall that in this study we consider a fluid of perfectly coupled dust and gas, τs = 0 (Sec-
tion 5.1.2). This becomes less valid when Γd becomes large, or when large particles not in the
Epstein regime are trapped. Fu et al. (2014)1 finds that dust back-reaction becomes important
1Published in parallel to Railton and Papaloizou (2014).
172 Numerical treatment of stability analysis
for the dynamics of the flow when Γd ≳ 1, i.e. when our ρmax ≳ 1. This is because one cannot
neglect momentum transfer when there are large particle overdensities. They show that there
is a dynamic instability when both dust feedback is included and Γd exceeds unity, leading to
shorter vortex lifetimes for vortices with higher Γd and larger particle sizes (i.e. larger Ts ∝ τs).
We do not see evidence of this as we have worked in the τs = 0 regime and could not produce
any very high Γd equilibrium solutions.
A subsequent study by Raettig et al. (2015)2 on the feedback of dust on vortical flow
finds that particle trapping occurs in 2D, shearing sheet solutions for all Ts and initial Γd.
Furthermore, it is possible to produce a density enhancement from an initial Γd = 10−2 to
Γd = 100 with Ts = 1 particles, enough to trigger the streaming instability and possibly GI.
Initially elliptical streamlines are bent to more complex motions when dust back-reaction is
include (i.e. when τs > 0) and there are larger resulting Γd when τs > 0. However, very small,
well-coupled dust Ts ≃ 0.01 is harder to capture and the deviation from Kida streamlines is
smaller, preventing particles from accumulating too densely (Lyra and Lin, 2013).
Furthermore, they find that dust concentrations are very localised in the optimal Ts = 1
case (less so for Ts ≪ 1 but this dust is harder to capture anyway, Section 3.5.1), meaning
that any vortex instability would be over a few streamlines instead of the entire vortex. We,
with our tightly (perfectly!) coupled dust, are considering Ts ≪ 1 particles and have a density
contrast spread more evenly across the vortex patch. Therefore we have more streamlines
affected by instability, as is shown when we vary ymax in Figures 8.9 and 8.10.
Finally, one further complication is that when Γd in a vortex becomes high, some consid-
eration should be given to dust coagulation/fragmentation (Testi et al., 2014).
8.6.3 Conclusions
In conclusion, we find that all vortices have some instability somewhere. There are not stable
gaps (like in the Kida case for 4 < χ ≲ 4.85) when all streamlines ymax ∈ [0, 1] are considered3.
Dust particles attracted from the outer disc to a vortex core with high aspect ratio, χ, may
well encounter parametric instabilities with characteristic growth rates 0.01–0.1Ω. We also
find that, broadly speaking, the larger the aspect ratio, the more stable the vortex.
Small dust grains trapped by vortices end up in a more or less smoothed profile over the
vortex core (Lyra and Lin, 2013; Raettig et al., 2015). Therefore, most or all of the internal
streamlines will have some parametric instability at some k. However, this may not be sufficient
to disrupt the vortex, as argued in Section 8.6.1.
If we consider larger grains (i.e. Ts = 1, more preferentially captured by vortices), concen-
trations of mass are more localised on a few streamlines, either piled into the centre or in a
2Published after write up was started.
3Although not considered in this work, this includes streamlines outside the core. See Lesur and Papaloizou
(2009) for more details.
8.6 Discussion and conclusions 173
ring-like structure. This implies that only a few streamlines will be subject to potentially dis-
ruptive parametric instabilities, or we are in a high β regime where the majority of additional
mass is piled into the centre. Recall that including mass decreases the instances of stable ‘gaps’
in χ, though since instability occurs in bands, there will still be some k that are stable. We
would need further non–linear analysis, with dust considered as separate fluid (ideally with a
range of sizes and including coagulation/fragmentation/bouncing effects) to determine if these
concentrations are sufficient to disrupt the entire vortex. We leave this to further work.
We also conclude that the situation is not as bad as presented in Chang and Oishi (2010).
Their stability analysis started from solutions that never correctly matched the background
flow as they assumed a Kida solution with arbitrary density superposed – as we showed in
Section 5.4 this is only possible in some cases when χ = 7. Furthermore, they assume a form
of the wavenumber that can only be applied to the Kida case only, not a generic equilibrium, as
shown in Section 7.4.3. If we assumed kz = 0 and use the Kida wavenumber; since it increases
with time thus we do not get instabilities growing exponentially with time (Section 7.4.2). Only
by considering kz ̸= 0 and a time–independent wavenumber do we get exponentially growing
modes.
We note that in a PP disc, the generation of vortices will compete with weak instability.
Lin (2014) finds that the RWI produces columnar vortices from an initial radial density bump
in a time O(10Ω−1). As these results are linear we need to make a nonlinear study to determine
the outcome of the competition of these two processes – will the the destruction time be longer
than a few 10s of orbits? We investigate this in the next chapter.
It is worth remembering that we do not need every vortex capable of trapping dust to
persist for long times in order to be useful for planetesimal formation. Planetesimals captured
and grown in one vortex (which it may or may not destroy) could easily be captured by another
vortex, up to the point where they become large enough to decouple from the flow. Vortices
are not difficult to make in PP discs so we’d expect that any vortex would exist alongside (or
even help generate) neighbouring structures.
Furthermore, we have also ignored the fact that vortices are not just generated and then
left to their own devices; they can be sustained by the very mechanisms that made them, such
as the RWI. There is a subtle interplay between generation, dust trapping and the speed of
these two processes, the structure of trapped dust, whether vortices are sustained or not and
the instabilities they are subject to. The flip side of the statement that ‘all vortices have some
instability somewhere’ is that there are also going to regions and circumstances where vortices
are stable, or stable enough to quickly grow some dust grains before it is disrupted. A dust
profile does not always cause a vortex to be completely unstable and in this context that may
well be good enough that vortices are still useful and promising sites for planetesimal formation

Chapter 9
A study of the stability of vortex
models with a 2D flow to 3D
perturbations
The work up to this point has been concerned with the linear stability of steady vortices in 2D.
In this chapter we aim to move beyond equilibrium vortex solutions, using the PLUTO code
(detailed in Section 9.1) to investigate possible nonlinear behaviour in 2D (Section 9.2), the
form of 3D vortices with and without the presence of vertical gravity (Section 9.3), the lifetime
of such vortices under different conditions and for a range of aspect ratios and finally how
instability, if it occurs, manifests itself.
9.1 The PLUTO code
In this section we used the compressible hydrodynamical code PLUTO (Mignone et al., 2007).
We chose to use this code as it is widely used, well–documented and would allow us to inves-
tigate the stability of vortices in two– and three–dimensions.
PLUTO is a finite-volume/finite-difference, shock capturing code which implements both
Newtonian hydrodynamics and ideal MHD. Computations are done using double-precision
arithmetic and we use the static grid version of the code. It can allow for initial data to be
read in from previous calculations and has a shearing box module already built in (Hawley
et al., 1995; Balbus, 2003; Regev and Umurhan, 2008). Additional physics, such as special
relativity and non-ideal effects like viscosity, resistivity and thermal conduction and cooling
can also be included. It also has a series of built in test cases for verifying it is working correctly
(Section 9.1.3).
Throughout this chapter, we will be working in a shearing box of dimensionL = (Lx, Ly, Lz)
with the number of grid points given by N = (Nx, Ny, Nz). We also use a scaled time where
176 Stability of vortices to 3D perturbations
Ω = 1 so time is in terms of the dynamical timescale τdyn = Ω−1.
9.1.1 Grid
We use a uniform grid with spacing ∆x = Lx/Nx (similar in the y– and z–directions), corre-
sponding to grid points
x[i] =−Lx2 + ∆x(i− 1), i = {1, Nx} (9.1.1a)
y[j] =−Ly2 + ∆y(j − 1), j = {1, Ny} (9.1.1b)
z[k] = ∆z(k − 1), k = {1, Nz} (unless otherwise stated) (9.1.1c)
where e.g. x[i] defines the left hand side/bottom of the grid cell. Variables are defined at the
cell centre.
9.1.2 Boundary conditions
A variety of boundary conditions are applied to the boxes in this chapter, namely periodic,
shearing, outflow and reflective conditions.
Periodic boundary conditions in y–direction (also used in the z–direction in unstratified
boxes) take the form:
q(x, y, z, t) = q(x, y + Ly, z, t) (9.1.2)
The shearing boundary conditions always apply along the x–boundaries, with adjacent boxes
sliding past each other with relative velocity u = |SLx|:q(x, y, z, t) = q(x± Lx, y ∓ ut, z, t)vy(x, y, z, t) = vy(x± Lx, y ∓ ut, z, t)± u, (9.1.3)
where q represents all other quantities aside from vy.
In our 3D stratified simulations, we also sometimes use an outflow boundary condition
(zero gradient across the boundary) at the top of the box z = Lz:
∂q
∂z
= 0, ∂v
∂z
= 0, (9.1.4)
while at the midplane z = 0 we use a reflective boundary condition:
v = (vx, vy, vz)→ (vx, vy,−vz), q → q. (9.1.5)
Again, q represents all other quantities not explicitly stated.
9.2 2D models 177
(a) Steady case for v′ = 0 and magnetic field B = 0. (b) Unsteady case for v′y ∼ 10−3 Gaussian noise and
constant B = B0zˆ.
Figure 9.1 Testing the PLUTO shearing box implementation. As expected we produce a steady
shearing box when v′ = 0 is imposed (for zero or constant B) and get MRI turbulence when a
constant magnetic field is imposed and v′ ̸= 0. In these cases, L = (0.25, 1, 0.25) and cs = 1.14.
9.1.3 Initial tests
As verification that we were using the code correctly we performed a few initial tests. PLUTO provides
a variety of test cases with the code. The results of a couple of these can be seen in Figure 9.1.
The first (Figure 9.1a) is a 3D empty box run containing just the background shearing flow
with all magnetic terms set to zero and no random perturbations added1. This is to check that
the code maintains background shear flow correctly; as you can see this is the case as we only
see the shearing background flow for large times.
In Figure 9.1b we reproduce the MRI in a shearing box by imposing a constant, vertical
magnetic field B = B0zˆ and Gaussian noise in the y–direction of magnitude
∣∣∣v′y∣∣∣ ∼ 10−3
(Section 9.3.1 for how perturbations are implemented). B0, in units of v0
√
4πρ0 = SLx
√
4πρ0,
is given by B0 = cs
√
2(γβ)−1, where cs = 1.14, the adiabatic index γ = 1 and β = 10000.
Using the various functions available in the VisIt software2 (Childs et al., 2012, also used
to produce the two figures in Figure 9.1 and many others in this chapter) we also checked that
vorticity was conserved.
178 Stability of vortices to 3D perturbations
Run Lx Ly Nx Ny N/L χ cs
1a 2 4 128 128 – 5 5
1b 2 4 128 256 64 " "
1c 4 4 256 256 64 " "
1d 4 8 256 512 64 " "
1e 4 12 256 768 64 " "
1f 8 4 512 256 64 " "
1g 2 4 256 512 128 " "
1h 4 4 512 512 128 " "
1i 4 8 512 1024 128 " "
1j 4 12 512 1536 128 " "
1k 2 4 512 1024 256 " "
1l 4 4 1024 1024 256 " "
2{a− l} as Run 1 8 5
Table 9.1 Initial 2D runs, testing box size and resolution.
9.2 2D models
Once we had convinced ourselves that the shearing box module was working correctly, we
tested the affect of different box dimensions and resolutions on an imposed Kida vortex in a
2D shearing flow. This is in the xy–plane, so there is no stratification or z–dependence.
We considered four different box dimensions (Lx, Ly) and four resolutions Nx ×Ny, sum-
marised in Table 9.1. In all cases the imposed vortex lies within |x| ≤ χ−1, |y| ≤ 1.
The vortices were created by inputting the Kida velocity field
v = (vx, vy) =
( 3Ωy
2χ(χ− 1) ,−
3Ωχx
2(χ− 1)
)
(9.2.1)
inside an ellipse with boundary passing through (0, 1) = (0.5Lx, 0.75Ly). As in previous the
analysis, we work in a regime where Ω = 1. The z–component of vorticity is then calculated
by the code using a simple difference scheme:
vortz[i][j][k]=0.5*(vy[i+1][j][k]-vy[i-1][j][k])/dx[i]
-0.5*(vx[i][j+1][k]-vx[i][j-1][k])/dy[j]
(9.2.2)
where vortz[i][j][k] represents the vorticity ω = ∂xvy − ∂yvx at the centre of the grid
cell defined by x[i], y[j], z[k] (Section 9.1.1). The x− and y−velocities vx, vy are defined
similarly by vx[i][j][k] and vy[i][j][k]. Note that in this 2D case, k=0 is constant.
This initial vorticity profile has a discontinuity at the boundary, where numerical truncation
1The only source of perturbation is therefore from numerical truncation and rounding errors.
2Documentation can be found at http://visitusers.org/index.php?title=Main_Page
9.2 2D models 179
errors are significant. This can be seen in Figure 9.2a which shows the numerical representation
of the vorticity at the first timestep, t = 0.
We use an initial columnar Kida vortex with χ = 5 inside the vortex core, since Lesur and
Papaloizou (2009) found a small linearly stable region around this aspect ratio. This allows
for a straightforward comparison of resolution, box dimension and other model conditions.
We ignore the perturbation of the background shearing flow outside the core because the
analytical solution is large scale and does not satisfy the boundary conditions, leading to
numerical difficulties. Instead we let the solution relax to a steady solution.
We note that we are looking at an incompressible regime with a compressible code; this is
achieved by specifying a large cs in an isothermal equation of state P = c2sρ. We can estimate
how close to the incompressible regime we are by comparing cs with 12SLx (the shearing velocity
at the initial vortex boundary is −S/χey). For incompressibility to be reasonable, we expect
this to be small compared to cs. This is certainly true in these 2D cases where when Lx = 4
and cs = 5, cs is more than three times as large. We discuss this in more detail for the 3D
cases in Section 9.3.6.
9.2.1 General evolution of imposed χ = 5 Kida vortex
Here we evolve a simple imposed vortex patch in a 2D (Lx, Ly) = (4, 4) shearing box with
uniform initial density. Snapshots of the vorticity profile for a Nx × Ny = 512 × 512 run are
shown in Figure 9.2 after a range of simulation times to illustrate the general evolution we see.
Frame 9.2a is the initial state, a simple anticyclonic vortex patch. The high vorticity
around the vortex boundary is caused by the finite difference method used to calculate the
vorticity. The dynamical relaxation of the patch causes it to shear out in frame 9.2b, splitting
into three distinct anticyclonic vortices in frame 9.2c due to phase where the central vorticity
is temporarily increased in magnitude. The symmetry of the shearing box means we get
rotational symmetry with respect to rotation through 180◦ around the origin. These extra
vortices are both smaller and weaker in higher resolution runs.
The two outer vortices are dragged to the top of the box, loop back round (as we have
periodic boundary conditions at the x–boundaries) before getting absorbed by the main vortex
in frame 9.2d. The patch quickly tends to an equilibrium state as seen in frame 9.2e; an
anticyclonic patch lying on top of the rest of the background shearing flow (ω = −1.5), which
does not change much between t ≃ 100− 400Ω−1.
In Figure 9.3a we see the vorticity profile (compared to the initial solution) at t = 88Ω−1
along both the x– and y–axes. There has been some smoothing from the sharp-profiled ‘top hat’
of the Kida solution. A time series for the vorticity profile along the y-direction in Figure 9.3b,
showing the effect of diffusion with time. As expected the strength of the vortex decreases
with t under the effect of diffusion. This increases the aspect ratio since weaker vortices are
more susceptible to the background shear. This process is slower at higher resolution, again,
180
Stability
ofvortices
to
3D
perturbations
(a) Run 211d, t = 0 (b) Run 211d, t = 8 (c) Run 211d, t = 40
(d) Run 211d, t = 88 (e) Run 211d, t = 200
Figure 9.2 The time evolution of vorticity for Run 1h, an imposed Kida vortex with χ = 5, cs = 5, L = (4, 4) and N = (512, 512).
See Section 9.2.1 in the main text for details.
9.2 2D models 181
as expected.
Finally, Figure 9.3c shows the χ–profile through the vortex for the just-settled equilibrium
(i.e. once the initial transient behaviour has died down), calculated from the maximum x and
y extent of the closed contours. There has already been some numerical diffusion by this point
so only the innermost streamlines have χ close to the initial value χ = 5. Recall from Chapter 6
that vortices with this χ profile contain internal shear.
In Figure 9.6 we reproduce the χ–profiles of the α = 1, constant–density equilibrium solu-
tions of Chapter 6, with the profiles of Figures 9.3c and 9.5c superposed. We find qualitatively
similar profiles compared to these equilibrium solutions, demonstrating that the choice of
Bernoulli source term A(ψ) as a power law, given in equation (5.2.6a), is reasonable. There is
a quantitative disparity between initial profile (i.e. ωm) and final χ–profile due to the discon-
tinuity in the boundary in the PLUTO solutions removing vorticity from the system.
Also note from Figures 9.2 and 9.3a that just outside the core we have vorticity ω > −1.5,
as seen in Lesur and Papaloizou (2010).
9.2.2 General evolution of imposed χ = 8 Kida vortex
This is similar to Run 1, except a weaker, χ = 8 vortex is imposed. More structure is seen
inside the vortex core, and it persists for longer than the χ = 5 case (Figure 9.4). However, this
structure is only transient and a stable, smoothly-profiled core is still produced. In Figure 9.5
we again see the vorticity profile for the stable vortex (9.5a) and its time series (9.5b), which
is qualitatively similar to the stronger χ = 5 case. There is a somewhat smoother χ–profile
(9.5c).
9.2.3 Effect of box dimension
Stretching the shearing box in the y–direction (while keeping all other parameters the same)
causes pairs of vortices to travel further as they interact and loop past each other, as can be
seen in Figure 9.7. The only real difference between these and boxes with shorter Ly is that
the relaxation time to a steady state is longer.
Figures 9.8a and 9.9a are plots of the minimum vorticity in the box, ωmin3, found in the
box, related to the strength of the resulting anticyclonic vortex. Box dimension has a nonzero
but not hugely significant effect on the value of ωmin. The vorticity dip around t = 10Ω−1 is
due to the shearing out of the initial imposed vortex patch which piles up the majority of the
vorticity in a narrow, central structure (for example, see Figure 9.2b). Typically the vortices
relax to a steady state around t ≃ 40Ω−1 and then persist until t = 200Ω−1 and beyond.
3For most cases this is at the centre of the box. Exceptions are when the final vortex straddles the periodic
y–boundary and when strong, initially low χ vortices spilt into two or more vortices.
182 Stability of vortices to 3D perturbations
-2
-1.9
-1.8
-1.7
-1.6
-1.5
-1.4
-1.3
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
ω
Coordinate
Along xaxis
Along yaxis
yaxis (initial)
(a) Vorticity profile along x and y–axes
-2
-1.9
-1.8
-1.7
-1.6
-1.5
-1.4
-1.3
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
ω
ycoordinate
t = 0
t = 68
t = 80
t = 100
t = 120
t = 140
t = 160
t = 180
t = 200
(b) Times series for vorticity profile along y–axis
4.8
5
5.2
5.4
5.6
5.8
6
6.2
6.4
0 0.2 0.4 0.6 0.8 1 1.2
χ
ycoordinate
(c) χ as a function of y–coordinate for vortex when it first be-
comes stable
Figure 9.3 Vorticity and χ–profiles for the χ = 5 vortex in a L = (4, 4), 2562 box. We find
uniformly lower χ for higher resolution cases due to decreased numerical diffusion.
9.2 2D models 183
(a) Run 2a128, t = 68 (b) Run 2a128, t = 80
(c) Run 2a128, t = 152 (d) Run 2a128, t = 200
Figure 9.4 The time evolution of vorticity for Run 2h (see Table 9.1), an imposed Kida vortex
with χ = 8, showing the more complex structure that forms in the core. See Section 9.2.2 in
the main text for details.
184 Stability of vortices to 3D perturbations
-1.8
-1.75
-1.7
-1.65
-1.6
-1.55
-1.5
-1.45
-1.4
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
ω
Coordinate
Along xaxis
Along yaxis
yaxis (initial)
(a) Vorticity profile along x and y–axes
-1.8
-1.75
-1.7
-1.65
-1.6
-1.55
-1.5
-1.45
-1.4
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
ω
ycoordinate
t = 0
t = 88
t = 100
t = 120
t = 140
t = 160
t = 180
t = 200
(b) Times series for vorticity profile along y–axis
7
7.5
8
8.5
9
9.5
10
10.5
11
0 0.2 0.4 0.6 0.8 1 1.2
χ
ycoordinate
(c) χ as a function of y–coordinate for initial steady vortex (t =
88Ω−1)
Figure 9.5 Vorticity and χ profiles for the χ = 8 vortex in a (Lx, Ly) = (4, 4), 2562 box.
9.2 2D models 185
1
2
3
4
5
6
7
8
9
10
11
0 0.2 0.4 0.6 0.8 1
χ
ymax
(0.018, 10)
(0.026, 8.6)
(0.032, 7.8)
(0.050, 6.4)
(0.090, 5.0)
(0.333, 3.0)
(1.125, 2.0)
(4.592, 1.4)
Run 1
Run 2
Figure 9.6 Reproduction of Figure 6.4a with the χ–profiles of Figures 9.3c and 9.5c super-
posed. The lines show the various χ–profiles for equilibrium solutions with α = 1, while the
numbers in brackets are (ωm, χk), where χk is the expected Kida aspect ratio for that ωm.
(a) t = 76Ω−1, additional vortices which cross the
box and increase the relaxation time
(b) t = 200Ω−1, final, relaxed vortex
Figure 9.7 Run 1i, L = (4, 8), N = (512, 1024), showing smaller, extra vortices travelling
away from the central disturbance (9.7a) then the final structure (9.7b). Similar behaviour is
seen in the larger boxes L = (4, 12) but are not seen at all at the lowest resolution N/L = 64.
186 Stability of vortices to 3D perturbations
-2.4
-2.3
-2.2
-2.1
-2
-1.9
-1.8
0 50 100 150 200
ω
m
in
Time (Ω−1)
L, Nx ×Ny
(2, 4), 128× 256
(4, 4), 256× 256
(4, 8), 256× 512
(4, 12), 256× 768
(8, 4), 512× 256
(a) Box size tests, at fixed resolution
-2.4
-2.3
-2.2
-2.1
-2
-1.9
-1.8
-1.7
-1.6
-1.5
0 50 100 150 200
ω
m
in
Time (Ω−1)
L, Nx ×Ny
(2, 4), 128× 256
(2, 4), 256× 512
(2, 4), 512× 1024
(2, 4), 128× 128
(4, 4), 256× 256
(4, 4), 512× 512
(4, 4), 1024× 1024
(b) Resolution tests. The highest resolution runs for both box sizes are top of each other. The
lowest resolution case is an outlier (the uppermost curve).
Figure 9.8 Effects of box size and resolution of time evolution on an imposed χ = 5 Kida
vortex. Figure 9.8a shows less variation of ωmin with Ly than Lx and an increase in relax-
ation time with an increase of Ly. Figure 9.8b shows broadly similar behaviour for different
resolutions, with the exception of the lowest resolution case.
9.2 2D models 187
-2
-1.95
-1.9
-1.85
-1.8
-1.75
-1.7
-1.65
0 50 100 150 200
ω
m
in
Time (Ω−1)
L, Nx ×Ny
(2, 4), 128× 256
(4, 4), 256× 256
(4, 8), 256× 512
(4, 12), 256× 768
(8, 4), 512× 256
(a) Box size tests, at fixed resolution.
-2.1
-2
-1.9
-1.8
-1.7
-1.6
-1.5
0 50 100 150 200
ω
m
in
Time (Ω−1)
L, Nx ×Ny
(2, 4), 128× 256
(2, 4), 256× 512
(2, 4), 512× 1024
(2, 4), 128× 128
(4, 4), 256× 256
(4, 4), 512× 512
(4, 4), 1024× 1024
(b) Resolution tests
Figure 9.9 Effects of box size and resolution of time evolution on an imposed χ = 8 Kida
vortex. The conclusions are the same as for the χ = 5 case in Figure 9.8.
188 Stability of vortices to 3D perturbations
9.2.4 Effect of resolution
Resolution has a more profound effect on vortex evolution than choice of box dimension, as
can be seen in both Figures 9.8b and 9.9b. We looked at three different resolutions, N/L =
{64, 128, 256} for most of the box dimensions, as can be see in Table 9.1.
The total vorticity between different box dimensions and resolutions is the same (to 2 s.f.)
so this is not due to any substantial effects at the vortex core boundary, as initially suspected.
Instead, adding more grid points reveals more structure that takes vorticity away from the
more more crudely resolved extra vortices in the lowest resolution runs. AS can be seen in
Figures 9.8b and 9.9b, in both χ = 5, 8 cases, moving to the highest N/L = 256 resolution
caused very little deviation from the N/L = 128 case.
We conclude that for the purposes of creating a stable solution to use to investigate the
stability in 3D, a 2×4 box of 128×256 grid points is sufficient, with a subset of the calculations
being run at higher resolution to double check. Increasing the box dimension serves to only
increase the relaxation time, while increasing the resolution resolves finer structures which have
to be absorbed by the main disturbance before a stable configuration settles down, with little
effect on the final structure.
9.2.5 Stability of solutions
For a series of L = (2, 4) boxes (with N = (128, 256) for χ < 10 and N = (192, 256) for
χ ≥ 10) we evolved vortices of different strengths, starting with an initial Kida configurations
(see Figure 9.2a).
There was Gaussian noise added with v ∼ 10−3 to see how perturbations behave in 2D,
following the same procedure we will use in 3D (for how this is implemented, see Section 9.3.1).
In all cases stable structures were formed which persisted for t = 400Ω−1, as can be seen in
Figure 9.10a. We compare ωmin with the initial vorticity minimum (i.e. the Kida vorticity) in
Figure 9.10b, which shows slightly weaker vortices produced than expected from the original
Kida vorticity. This is partially due to the effect of numerical diffusion (which also causes the
slow reduction of this ratio with time) and also the boundary effects seen in Figure 9.2a which
act to remove some initial vorticity from the system.
We expect these solutions to be stable; this corresponds to the vertical stability limit
detailed in Section 7.5 and shown in the various plots of Chapter 8 for large θ.
9.3 3D models
Here we expand the calculations to include the z–direction. Moving to three dimensions re-
quires some decisions about the form the imposed vortex will take, how to implement strati-
fication (Section 9.3.1) and the boundary conditions at the two z–boundaries (Section 9.3.2).
9.3 3D models 189
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
0 50 100 150 200 250 300 350 400
ω
m
in
Time (Ω−1)
χ = 2
χ = 2.5
χ = 3
χ = 3.5
χ = 4
χ = 5
χ = 6
χ = 7
χ = 8
χ = 9
χ = 10
χ = 12
χ = 15
χ = 20
(a)
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
0 50 100 150 200 250 300 350 400
ω
m
in
/(
ω
m
in
) i
n
it
Time (Ω−1)
χ = 2
χ = 2.5
χ = 3
χ = 3.5
χ = 4
χ = 5
χ = 6
χ = 7
χ = 8
χ = 9
χ = 10
χ = 12
χ = 15
χ = 20
(b)
Figure 9.10 Results for the 2D L = (2, 4), N = (128, 256) box with v′ ∼ 10−3, showing ωmin
against time. For all cases the final configuration is stable with a steady decrease in ωmin due
to numerical diffusion.
190 Stability of vortices to 3D perturbations
Figure 9.11 A vorticity contour plot showing the initial imposed vorticity for a χ = 3.5 Kida
vortex with v′ ∼ 0.1.
9.3.1 Initial conditions
Random noise is added to the shearing boxes by generating additive white Gaussian noise with
zero mean and standard deviation of 1, some AGWN(). This white noise is generated using
the Box-Muller transform (Box and Muller, 1958; Pike, 1965) which transforms two uniformly
distributed random numbers (U1, U2) ∈ (0, 1] to (Z0, Z1), independent random variables with
a standard normal distribution
Z0 = R cosΘ =
√−2 logU1 cos (2πU2) (9.3.1a)
Z1 = R sinΘ =
√−2 logU1 sin (2πU2) . (9.3.1b)
The function AGWN() then returns AGWN()= Z0. The uniform distributions U1, U2 are gen-
erated using the ratio of rand() and RAND_MAX from the standard C library:
Ui =
rand()
RAND_MAX
, (9.3.2)
where rand() returns a random number between 0 and RAND_MAX. If 0 was returned by
rand(), the ratio was calculated again until a nonzero result was returned.
We then add multiples of this noise to both the velocity and density distributions at t = 0.
If v0, ρ0 are the velocity and density distributions of the initial imposed vortex structure then
9.3 3D models 191
-2.6
-2.4
-2.2
-2
-1.8
-1.6
-1.4
-1.2
0 50 100 150 200
ω
m
in
Time (Ω−1)
χ = 3.5, ρ′ = 0
χ = 5, ρ′ = 0
χ = 8, ρ′ = 0
χ = 3.5, ρ′ ∼ 0.1
χ = 5, ρ′ ∼ 0.1
χ = 8, ρ′ ∼ 0.1
Figure 9.12 Initial unstratified box tests, testing effect of 3D and adding a Gaussian density
perturbation. This shows that the 3D calculations remain 2D if no perturbation (or a very
small one) is introduced. Also, instability sets in after a time that increases with aspect ratio
χ. Done for a box with L = (4, 4, 4), 1283, cs = 5.
v → v0+ v′ and ρ→ ρ0 + ρ′, where we use the notation
v′i ∼ 10−3 ⇒ v′i = 10−3 × AGWN(), i = {1, 2, 3} (9.3.3a)
|ρ′| ∼ 0.1 ⇒ ρ′ = ∣∣ 0.1× AGWN()∣∣. (9.3.3b)
In the density case, the perturbation added is always positive to ensure that ρ < 0 never occurs.
The velocity perturbation remains at |v′| ∼ 10−3 throughout as any larger perturbation serves
to quickly overwhelm the higher χ (weaker) imposed vortices and prevent them from relaxing
to an initial steady state. Changing the magnitude of ρ′ served to perturb the box to greater
or lesser extent without preventing the weaker vortices from relaxing to an initial steady state.
The initial vortex solution takes the form of the 2D case and is thus independent of z, i.e.
it is a columnar vortex patch. Attempts to initialise the solution with a vortex not extending
over the full height of the box led to it quickly stretching to the full box height. We therefore
decided to initialise our vortex over the full height of the box: a typical initial vortex can be
seen in Figure 9.11. These columnar vortices relax to pseudo–stable (Section 9.3.5) structures
in an analogous way to the 2D case in Sections 9.2.1 and 9.2.2.
Moving to 3D we ran some initial unstratified tests over a L = (4, 4, 4), 1283 box, with and
192 Stability of vortices to 3D perturbations
without the additional density perturbation ρ′, which served to perturb the system. The results
of this are shown in Figure 9.12, demonstrating that under no or negligible perturbations, our
3D calculations effectively remain 2D.
These initial test runs proved to be under-resolved in the x–direction (Lx = 4, Nx = 128)
so we reduced the x extent to Lx = 2 with no effect on the onset of any instability. For high
aspect ratio vortices χ > 10, Nx was increased from 128 to 192 to ensure that the solution
relaxed to a coherent vortex and not a ‘strip’ of vorticity along the y–axis.
9.3.2 Boundary conditions
For unstratified, gz = 0 cases we impose periodic boundary conditions at the z–boundaries
(Section 9.3.2). This is the same approach as Lesur and Papaloizou (2009). In stratified cases
we use half-boxes modelling z ≥ 0 with a reflective boundary at the midplane z = 0 and
an outflow condition at z = Lz (see Section 9.3.4). This is equivalent to imposing reflection
symmetry with respect to the midplane.
In both cases, there are shearing boundary conditions at the x–boundaries and periodic
boundary conditions at the y–boundaries, unchanged from the 2D case.
9.3.3 Stratification
In the stratified cases, vertical gravity is imposed over the bottom 80% of the box where the
midplane z = 0 was the lower z–boundary4:
gz =
−Ω
2z 0 ≤ z ≤ 0.8Lz
0 z > 0.8Lz.
(9.3.4)
This introduces a small low–density buffer zone for 0.8Lz < z < Lz which is found to make
the upper boundary consistent with periodic boundary conditions in z and to maintain its
stability. Stone et al. (1996) found this approach to be effective in vertically stratified shearing
box simulations. Also, this is in line with the approach of Lesur and Papaloizou (2009).
We also undertook some additional tests changing how gz was implemented. Moving the
boundary to z = 0.9Lz had a negligible effect, as did smoothing gz in the region 0.8Lz < 0.9Lz:
gz =

−Ω2z 0 ≤ z ≤ 0.8Lz
−Ω2z
(0.9Lz − z
0.1Lz
)
0.8 ≤ z ≤ 0.9Lz
0 z ≥ 0.9Lz.
(9.3.5)
For the stratified cases, the vortex column in the region where gz = 0 behaves differently on the
onset of instability. The column remains largely vertical here, while below the gz transition it
4The middle 80% was stratified in the few full-box simulations of Section 9.3.4.
9.3 3D models 193
shears along the y–axis as in Figure 9.16e. However, as can be seen in Figure 9.13, the height
of the buffer zone does not play an important role in the stability. There is little difference in
the instability onset time τunstable (Section 9.3.5) between the three forms of gz used. Similarly,
only imposing a perturbation over the bottom half of the box z < 0.5Lz increased the time
until the vortex broke up (unsurprising since the box was less perturbed) but had no effect on
how the instability manifested itself.
We note that since H ≡ cs/Ω (see Section 2.4.4 and equation (2.4.12) for ρ(z)), and since
we use Ω = 1 throughout, effectively H ≡ cs.
9.3.4 Use of halfboxes for stratified models
We expect that the symmetry around the midplane z = 0 means we need to only consider
the positive z domain. In order to confirm this we compared the use of a full stratified box
(−3 ≤ z ≤ 3) with half boxes of two different dimensions, 0 ≤ z ≤ 3 and 0 ≤ z ≤ 6, with equal
resolutions across each. The full box had period boundary conditions in z while the half boxes
had an outflow condition at z = Lz and a reflective condition at the midplane.
The results of this can be seen in Figure 9.14, where we see little quantitative difference
between them. For this reason we will henceforth use the half box with 0 ≤ z ≤ 3 for our
stratified runs. We also tested the required resolution in the z–direction and found thatNz = 32
was sufficient for Lz = 3 for our stratified half-boxes (Section 9.3.4).
Figure 9.18 shows the density stratification at late times (after onset of instability) for two
different ρ′, with otherwise identical initial conditions. As we expect for isothermal simulations,
we get a resulting hydrostatic equilibrium with ρ(z) ∝ exp (−z2/2H2) (where we have scaled
each with the midplane value from the simulation). This profile is independent of the strength
of the imposed vortex. We have a stably stratified system with N2 = 0, where N2 is the
buoyancy or Brunt-Väisälä frequency
N2 ≡ −gz
[
∂ log ρ
∂z
− 1
γ
∂ logP
∂z
]
. (9.3.6)
Since a strict isothermal equation of state is used everywhere, the adiabatic index γ = 1.
However, there are vertical restoring forces towards the midplane with frequency ∼ Ω which
may play a role in the stratified cases.
9.3.5 Pseudo–stable solutions and the onset of instability
We call the columnar vortices formed after the initial Kida vortex column relaxes pseudo–stable
and they invariably have a period of stability in both stratified and unstratified boxes before
being destroyed by instability.
Looking at the time series for ωmin (as in Figure 9.12) as well as eyeballing the 3D evolution
of the vorticity distribution (Figures 9.15 and 9.16) it is clear how instability manifests itself.
194 Stability of vortices to 3D perturbations
-3
-2.8
-2.6
-2.4
-2.2
-2
-1.8
0 50 100 150 200
ω
m
in
Time (Ω−1)
t = 123
t = 127
Sharp transition at z = 0.8Lz
Sharp transition at z = 0.9Lz
Smooth transition at z = 0.8Lz
(a) Effect of different stratification on an imposed χ = 5 vortex. The models with sharp transi-
tions at z = 0.8, 0.9Lz both went unstable at τunstable = 123Ω−1.
-2.6
-2.4
-2.2
-2
-1.8
-1.6
0 50 100 150 200
ω
m
in
Time (Ω−1)
t = 113
t = 109
t = 115
Sharp transition at z = 0.8Lz
Sharp transition at z = 0.9Lz
Smooth transition at z = 0.8Lz
(b) Effect of different stratification on an imposed χ = 8 vortex. Like the χ = 5 case there was
very good agreement between models.
Figure 9.13 Effect of different stratification regimes on the onset of instability.
9.3 3D models 195
-2.4
-2.2
-2
-1.8
-1.6
-1.4
0 50 100 150 200
ω
m
in
Time (Ω−1)
t = 156
t = 153
t = 162
Full box
Half box 1
Half box 2
Figure 9.14 Investigating the use of full and half boxes in stratified runs. The full box has
−3 ≤ z ≤ 3, while the first half box has 0 ≤ z ≤ 3 and the second has 0 ≤ z ≤ 6 with
equal resolutions. There was little qualitative or quantitative difference between any of these
simulations, as is demonstrated by the similar τunstable found using the procedure detailed in
Section 9.3.5.
In unstratified boxes, where there is no preferred value of z, the unstable mode has a
particular kz as can be seen in Figure 9.15, as expected from the work in Chapter 8.
In stratified cases the onset of instability is profoundly different. The vortex column begins
to shear in the yz–plane before splitting into smaller structures and becoming incoherent as in
Figure 9.16.
We sought a consistent diagnostic for the onset of instability in the stratified case. Compar-
ing the time series of ωmin with 3D visualisations of the vorticity showed the stable columnar
vortices going unstable (unsurprisingly) with a net decrease and an increasing amount of un-
steadiness in ωmin5. This behaviour can be seen in Figure 9.17, where ωmin (purple line at the
top of the plot) of a 3D steady vortex slowly increases with time (under the action of numerical
diffusion) before dropping when instability occurs. However, it was sometimes difficult to self
consistently pinpoint an instability time using this measure, as can be seen from comparing
Figures 9.12 and 9.14.
Therefore, considering the volume averages ⟨v2x⟩, ⟨v2y⟩, ⟨v2z⟩, together with ωmin and the 3D
contours of ωz, we found that instability kicked in close to the point where there was often a
5The behaviour of ωmax proved less useful as this mainly reflected the background shear.
196 Stability of vortices to 3D perturbations
local minimum in ⟨v2y⟩. Some care needed to be taken to pick out the correct minimum but
overall this was a robust method for pinpointing the onset of instability for the stratified cases.
We define this time to be τunstable.
Note that this local minima in ⟨v2y⟩ also corresponds to the beginning of an increase in
⟨v2z⟩; this is expected since the pseudo–stable solutions largely move in parallel xy–planes
(Section 5.1.5) so any large scale vertical motions are associated with instability.
The different form of the onset of instability in the unstratified cases meant that this
method did not work; instead we pinpointed the time when the vortex column first broke up
in 3D vorticity contour plots.
9.3
3D
m
odels
197
(a) t = 112Ω−1, steady columnar vortex. (b) t = 132Ω−1. (c) t = 160Ω−1.
(d) t = 172Ω−1 (e) t = 196Ω−1.
Figure 9.15 Onset of instability for the unstratified |v′| ∼ 10−3, χ = 3.5 case. Instability occurs in a qualitatively different way to
the stratified cases displayed in Figure 9.16, with no preferred value of z in this unstratified case. The z–dependence indicates an
unstable mode with a particular kz as expected from the linear stability theory (Chapter 7).
The variable ⟨v2y⟩ was not a useful diagnostic in unstratified cases so determining the instability timescale was a case of observing the
3D contours over time. Note that in these plots we have imposed a ceiling on the value of ωz displayed so the columnar structure can
be seen.
198
Stability
ofvortices
to
3D
perturbations
(a) ρ′ ∼ 0.1, χ = 8, t = 80Ω−1, steady colum-
nar vortex.
(b) t = 112Ω−1, vortex is sheared in the yz-
plane, close to the onset of instability.
(c) t = 200Ω−1, the columnar vortex is dis-
rupted though some structure still remains.
(d) ρ′ ∼ 0.5, χ = 12, t = 40Ω−1, steady
columnar vortex.
(e) t = 120Ω−1, vortex is sheared in the yz
plane, close to the onset of instability.
(f) t = 200Ω−1, the columnar vortex is dis-
rupted
Figure 9.16 The onset of instability for the ρ′ ∼ 0.1, χ = 8 stratified case (a)–(c) (compare with Figure 9.17) and the ρ′ ∼ 0.5, χ = 12
case (d)–(f).
9.3 3D models 199
9.3.6 Effect of cs
Increasing cs has a stabilising effect in stratified boxes, as demonstrated in Figure 9.19. Since
disc scale height H ≡ cs/Ω, increasing cs is equivalent to increasing H and thus the ratio of
the y–extent of these vortices to H causes stabilisation (Lithwick, 2009). Since we are, for the
most part, close to the incompressible limit, we do not expect much dependence on cs.
Density waves are excited at a distance from the centre of the vortex where the background
shearing flow | − Sxey| = cs and propagate out of the box (Lesur and Papaloizou, 2010).
Therefore, as we reduce cs ≲ 1 this sonic boundary moves inside the box.
In models with cs = 1 we do find density waves crossing the box. Furthermore, we do
not find a stable columnar vortex structure found for the entire range of z. Instead, a much
shorter column is formed over the top 30% of the box, with turbulence around the midplane.
Columnar structures, as seen in the more incompressible cases with cs > 2.5, do not last long,
but a few coherent vortices do persist in this top region for long times t ≳ 200Ω−1. This is
shown in Figure 9.20.
9.3.7 Effect of initial vortex strength on vortex lifetime
Here we investigate how the strength of the imposed vortex (parametrised in terms of the
Kida aspect ratio χ of the imposed flow) affects the lifetime of the resulting vortex. This is
done for density perturbations ρ′ ∼ {0.1, 0.5}. As discussed in Section 9.3.3, we use the box
L = {2, 4, 3} with z ≥ 0 and
gz =
−Ω
2z 0 ≤ z ≤ 2.4
0 z > 2.4.
(9.3.7)
We plot the resulting ωmin as a function of t for different imposed vorticity patches for both
ρ′ ∼ 0.1 and ρ′ ∼ 0.5 models in Figures 9.21 and 9.22 respectively. Using the procedure
described in Section 9.3.5 we found the time τunstable at which each of these solutions first
became unstable. The results of τunstable as a function of χ are plotted in Figure 9.24 for
various stratified and unstratified cases.
There are few cases where resolution is increased fromN = (128, 256, 32) toN = (192, 384, 32)
and the resulting τunstable are also shown in Figure 9.24. For χ = {5, 8, 10}, τunstable increases
with resolution, while for χ = {15, 20} it decreases relative to the lower resolution models.
This latter effect is most likely due to the increased resolution picking up weaker instabilities.
However, note that we still observe the same trend of τunstable increasing with χ.
200 Stability of vortices to 3D perturbations
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
S
c
a
l
e
d
v
a
r
i
a
b
l
e
s
Time (Ω−1)
t = 105
ωmin
〈v2x〉
〈v2y〉
〈v2z〉
(a) Evolution of χ = 4 vortex, instability occurs at t ≃ 105.
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
S
c
a
l
e
d
v
a
r
i
a
b
l
e
s
Timestep (Ω−1)
t = 113
ωmin
〈v2x〉
〈v2y〉
〈v2z〉
(b) Evolution of χ = 8 vortex, instability occurs at t ≃ 113.
Figure 9.17 Determining the onset of instability using ⟨v2y⟩ for the ρ′ ∼ 0.1 stratified case.
The variables ωmin, ⟨v2x⟩, ⟨v2y⟩ and ⟨v2z⟩ have been scaled to the same scale and smoothed using
the Bézier smoothing algorithm in gnuplot (Williams et al., 2010). For these, and all cases
with χ ≳ 3, the system has settled to a stable columnar vortex by t ≃ 30Ω−1. The onset time
of instability is denoted τunstable.
9.3 3D models 201
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0 0.5 1 1.5 2 2.5 3
ρ
(z
)
z
ρ′ ∼ 0.1
ρ′ ∼ 0.5
1.128 exp
(
−z2/2H2
)
1.581 exp
(
−z2/2H2
)
Figure 9.18 Density in stratified boxes for different ρ′ in a L = (2, 4, 3) box with gz given
by equation (9.3.4), demonstrating hydrostatic equilibrium. The density becomes constant for
z > 0.8Lz, inside the buffer zone where vertical gravity is zero.
-2.6
-2.4
-2.2
-2
-1.8
-1.6
0 50 100 150 200
ω
m
in
Time (Ω−1)
t = 149
t = 123
t = 136
cs = 2.5
cs = 5.0
cs = 10
Figure 9.19 Investigating the different effects of cs on χ = 5 vortices. Increasing cs improves
lifetime of columnar vortex. The coloured bars show the stable regions for each model. When
cs = 2.5, there are a pair of vortices till t ≃ 100 when it finally becomes a single stable
structure. These runs were performed in the standard L = (2, 4, 3)N = (128, 256, 32) box.
202 Stability of vortices to 3D perturbations
(a) t = 60Ω−1, showing the approximately steady
vortex (split over the y–axis) that exists in the top
≈ 30% of the stratified box.
(b) t = 150Ω−1, showing the subsequent vortices
that persist in the box for t > 200Ω−1
Figure 9.20 The time evolution of vorticity for a compressible stratified model with cs = 1
and χ = 8. This shows a slice of the xy–plane at z = Lz for clarity.
9.3
3D
m
odels
203
-4
-3.5
-3
-2.5
-2
-1.5
0 50 100 150 200
ω
m
i
n
Time (Ω−1)
χ = 2
χ = 2.5
χ = 3
χ = 3.5
χ = 4
χ = 5
χ = 6
χ = 7
χ = 8
χ = 10
χ = 15
χ = 20
Figure 9.21 Results for the stratified box with L = (2, 4, 3)N = (128, 256, 32), cs = 5 and ρ′ ∼ 0.1, showing ωmin against time.
Note that the vortices 2 ≤ χ ≤ 3 do not reach a stable configuration before breaking up, while the larger aspect ratios χ < 15 survive
until t ≃ 100Ω−1 before going unstable. The χ = 15, 20 cases survive beyond t = 200Ω−1.
204
Stability
ofvortices
to
3D
perturbations
-4
-3.5
-3
-2.5
-2
-1.5
0 50 100 150 200
ω
m
i
n
Time (Ω−1)
Figure 9.22 Results for the stratified box with L = (2, 4, 3)N = (128, 256, 32), cs = 5 and ρ′ ∼ 0.5, showing ωmin against time.
The key in Figure 9.21 also applies to this plot. Again, vortices with 2 ≤ χ ≤ 3 do not reach a stable configuration before breaking
up, though under this stronger perturbation all vortices with 3 < χ ≤ 20 go unstable t < 150Ω−1; an earlier time compared to when
ρ′ ∼ 0.1 (Figure 9.24).
9.3
3D
m
odels
205
-3
-2.8
-2.6
-2.4
-2.2
-2
-1.8
-1.6
0 50 100 150 200
ω
m
i
n
Time (Ω−1)
χ = 3
χ = 3.5
χ = 4
χ = 5
χ = 6
χ = 7
χ = 8
χ = 10
χ = 12
χ = 15
Figure 9.23 Results for unstratified models in a box with L = (2, 4, 3)N = (128, 256, 32), cs = 5 and ρ′ ∼ 0.1. We find that
χ = {2, 2.5} never form stable solutions, χ = {3, 3.5} do and go unstable τunstable < 200, χ = 4 goes unstable τunstable ≃ 228 and all
weaker vortices remain stable for at least 300Ω−1. This is broadly consistent with the 2D models in Section 9.2.5.
206 Stability of vortices to 3D perturbations
50
100
150
200
250
300
2 4 6 8 10 12 14 16 18 20
τ u
n
st
a
b
le
,(
Ω
−1
)
χ
stratified, ρ′ ∼ 0.1
stratified, high res, ρ′ ∼ 0.1
stratified, ρ′ ∼ 0.1 (short)
stratified, ρ′ ∼ 0.5
unstratified, ρ′ ∼ 0.1
Figure 9.24 Instability time as a function of χ for stratified boxes which shows that weaker
(higher χ) vortices persist for longer times. In the larger perturbation ρ′ ∼ 0.5 case there is a
similar trend to the ρ′ ∼ 0.1 case, but instability sets in more quickly.
The χ = 20 results could be a result of under resolution, or too large a perturbation with
respect to ρ′, v′.
The dashed line is for a box of half the normal height for the ρ′ ∼ 0.1 case.
The few unstable unstratified cases are shown in green; like the stratified cases the very strong
vortices χ ≲ 2.5 never settled to coherent structures, while χ > 5 were stable for t > 300 so
are not plotted.
9.4 Summary and conclusions 207
9.4 Summary and conclusions
From the preliminary work on 2D models in the shearing box, we find that the use of power
law prescriptions for the Bernoulli source in Chapters 5 and 6 is justified as vortices naturally
relax to these forms. Numerical diffusion decreases the strength of these vortices to below
expected for the imposed vorticity profile, an effect that persists in 3D. The 2D models are
also stable, as we expect from the vertical stability limit discussed in Chapter 7 and 8.
Moving to three dimensions, we confirm that weaker vortices persist for longer times (Fig-
ure 9.24). We expect that this is due to the narrower (and weaker) instability bands we see at
larger χ in Chapter 8. Also, larger χ vortices are also not subject to the saddle point instability
(Section 7.3.2), which occurs χ ≲ 4.
As can be seen in Figures 9.23 and 9.24, stratification is destabilising. This is because
vertical restoring forces have frequency ∼ Ω which matches to the vortex turnover frequency
to destabilise the system. Reassuringly, the choice of buffer zone does not play an important
role in the stability and vortices subject to larger perturbations break up sooner (Figures 9.21,
9.22 and 9.24).
Increasing resolution picks up weaker instabilities, leading to a small reduction in vortex
lifetime (Figure 9.24). However, this does not prevent the RWI generating vortices (Méheut
et al., 2012a) since these structures still have a long survival time, which makes it easier for
competing processes to create and maintain the vortex. Lin (2014) finds that the RWI produces
columnar vortices from an initial radial density bump in a few 10s of orbits, i.e. O(10Ω−1).
Thus vortices with large enough aspect ratios – so that destruction time significantly exceeds
O(10Ω−1) – have a good chance of surviving.
For cs > 1 we find that increasing the sound speed has a small stabilising effect in stratified
boxes. This because it effectively moves us closer towards a strictly incompressible regime. For
small cs ≲ 1 we find that the sonic boundary moves inside the shearing box and we find density
waves crossing the box. There are no steady structures in this regime but a few coherent and
long-lived vortices do persist in the top 30% of the box.
As we are for the most part nearly incompressible, we do not expect (nor see) much density
change in the radial direction6. This is because subsonic velocities do not disturb the hydro-
static equilibrium and thus the equilibrium density profile is approximately maintained. To
get density enhancements we would have to consider the dust separately, which we leave to
further work. Despite this, the fact that compressibility has been included while our models
are not far from incompressibility makes this work more physically realistic then considering
a strictly incompressible regime, as in Lesur and Papaloizou (2009, 2010).
We have created 3D analogues of the 2D, α ̸= 0 vortices devised in Chapters 5 and 6 in
stratified boxes. The columnar and overwhelmingly planar nature of the flow reinforces our
6Since the model can extend over many vertical scale heights, we do however see that ρ has a z–dependence,
as shown in Figure 9.18.
208 Stability of vortices to 3D perturbations
use of 2D models for this work’s stability analysis. Despite the instability bands observed in
Chapter 8, such as those seen in Figures 8.7 and 8.8, the weaker vortices (χ ≳ 5) can have
long lifetimes, persisting for more than 120 orbits, after taking around 30 orbits to create
a stable structure. This implies that although growth rates of around 10−1.5 ≈ 0.03Ω are
sufficient to eventually disrupt the vortices, there is still a decent-sized window where they
are approximately stable. We have also only considered vortices without any mechanisms
sustaining them. This bodes well for the potential of these structures to capture dust and
provide a place for it to coalesce and grow.
The fact that these 3D models do not have a significant density structure reminiscent of
a profile of trapped dust does not detract from this as similarly sized instability growth rates
were found in both vortices with and without a central density enhancement. Also, as discussed
in Section 8.6, any trapped grains are likely to be localised on a few streamlines, limiting the
effect on the whole vortex.
Chapter 10
Conclusions and further work
This study aimed to produce a more complete understanding of the stability of vortices in
protoplanetary discs, chiefly whether parametric vortex instabilities could hinder these struc-
tures being sites for dust trapping and eventual planetesimal formation. Vortices are thought
to be the most promising route around the planet formation ‘metre gap’ and thus any threat
to the lifetime of these structures has ramifications for the formation theory of planetesimals
and thus planets.
10.1 Main results
10.1.1 Stable vortex structures
In Chapters 5 and 6 we developed a framework for producing equilibrium vortex solutions in
the shearing sheet for a range of parameters. Starting from the Kida solution with uniformly
elliptical streamlines, we make models of vortices with streamlines that were not elliptical,
corresponding to a flow with some internal shear. We find that adding mass does not alter
the shear profile nor the shape of streamlines to any great degree, which is surprising. It was
also difficult to construct equilibrium solutions with very large ρ–enhancements (greater than
about four times the background), due to our constraint that dust is perfectly coupled to the
gas and is a smooth function of ψ. We would expect less well–coupled dust grains to form
more localised structures and not be smoothed over the whole vortex (e.g. Raettig et al., 2015).
However, in the small-grain limit we consider this lack of large ρ–enhancement to be physically
realistic.
The use of power law prescriptions for the Bernoulli vorticity source is reinforced by the 2D
models in Chapter 9, which produced smooth power law profiles once an initial Kida solution
relaxed. The work in this chapter also supported our use of 2D models for the stability analysis,
as the 3D flow still largely behaved in a planar, 2D manner.
These models could potentially be of use as the starting points for other simulations,
210 Conclusions and further work
removing the restriction to just Kida vortices.
10.1.2 Generic vortex stability – vortices are probably ‘stable enough’
In this thesis we sought to answer the following questions posed in Chapter 4:
(i) What happens when vortex streamlines are not elliptical, as they are in the Kida case?
Non–uniformly elliptical or non–elliptical streamlines introduced an internal shear. We
find that a nonzero α produces a new instability band around χ ≈ 4.85 which increases
in width as the internal shear increases. There we also find a general band of instability
attributed to the saddle point of the pressure distribution, the width of which shrank as
streamlines became less uniformly elliptical. The magnitude of instability remains similar
to Kida, with growth rates ≃ 0.1Ω for χ ≲ 5 and ≃ 0.05Ω for χ ≳ 5.
(ii) How do non–constant profiles of vorticity and density affect stability? Results in the loss
of a completely stable band (as in the Kida case 4 ≤ χ ≲ 4.85) but there are no substantial
changes to the magnitude of the maximum growth rate for any χ. Parametric instability
bands move outwards from vortex centre, and weaker vortices have lower growth rates.
This is confirmed by 3D work in stratified and unstratified cases.
When there is no central density enhancement we get strong instability bands at all aspect
ratios. The limit of this is the point vortex case which reinforces our findings. When
there is a density enhancement, we get many narrow, weaker instability bands, again at
all χ. Adding additional mass using the polytropic model (which only matches correctly
to the background at χ = 7 but is still a useful testing ground) confirms that there is no
dramatic jump in growth rate when mass is added, with maximum growth rates for each
χ ≳ 5 around 0.05Ω.
(iii) Can the ‘heavy–core’ instability of Chang and Oishi (2010) be reproduced? We found no
evidence of this instability and caution against applying the Kida wavenumber of Lesur
and Papaloizou (2009) to general vortex solutions. Our wavenumber analysis in Chapter 7
shows that the use of this time–dependent wavenumber doesn’t work in the generic case
as they cannot be associated with exponentially growing linear modes except in both the
special Kida case and the polytropic solution with χ = 7. The only possibility for them
are Kida-like cases with χ = 7 for which growth rates ≃ 0.05Ω were obtained.
Furthermore, the two nonlinear studies of Fu et al. (2014); Raettig et al. (2015) do not
support their suggestion that vortices with significant dust concentrations cannot be
sustained.
(iv) Does the stability of these more general models have any bearing in 3D? Yes, it reinforced
the trend that weaker vortices with χ ≳ 5 were subject to weaker instability and therefore
10.2 Directions for future work 211
persisted for longer in three dimensions in stratified boxes, surviving for at least 100 orbits.
When we produced wholly 2D models in PLUTO we find them to be stable, as we expect
from the vertical stability limit discussed in Chapter 7 and 8. In 3D stratified boxes, we
find that χ ≳ 5 persist long enough that competing processes have time to create and
maintain the vortex (Lin, 2014).
Dust attracted to vortices may well encounter parametric instabilities but parametric in-
stabilities could be inhibited by dissipative effects or effects that disrupt periodicity of motion
(e.g. fast accumulation of matter). There is still a need to consider the interplay of vortex
generation and sustenance and the localisation of dust within vortex alongside instabilities. In
balance, vortices are probably still fine as local planetesimal formation sites.
10.2 Directions for future work
10.2.1 Back reaction of dust
In this thesis we considered dust that was perfectly coupled to the fluid. We found it difficult to
produce equilibrium solutions that had high central dust enhancements, which as Lyra and Lin
(2013) and Raettig et al. (2015) show, has some grounding in reality. However, the particles
most susceptible to both radial drift and vortex dust trapping are Ts = 1 particles. A promising
line of enquiry is to consider the stability of vortices with captured Ts = 1 particles, which
could be modelled either as a second pressureless fluid or via an N–body regime. Furthermore,
having dust partially decoupled from the flow would not limit us to smooth profiles of dust
on streamlines, although finding equilibrium solutions of dust not spread uniformly inside a
vortex may well be infeasible and thus a different approach would be necessary. We would
also then perhaps see additional instabilities from dust–gas coupling, namely the streaming
instability (Youdin and Goodman, 2005).
10.2.2 Relationship between instability and other processes
We have so far only considered parametric instability in isolation, though in reality there will
be a balance between competing processes both generating and sustaining vortices. It would
also be interesting to investigate vortices close to the end of their ‘useful’ life, when a high
Γd results in local gravitational instability. We would also expect to see some sort of cycle of
particle concentration and vortex destruction, with recently grown particles being kicked out
of, or destroying, their original vortices then being captured by neighbouring ones. Considering
vortices not as isolated structures but as part of some sort of network would also be interesting.
212 Conclusions and further work
10.2.3 More 3D realism
Finally, it would be instructive to consider the effect of different vertical structure in our naive
3D models, such as the presence of a thin dusty subdisc (perhaps formed of N -body dust)
inside a thicker gas structure. How and where columnar vortices could survive in this regime
is important to this problem.
There is also the perennial astrophysical question of ‘What about magnetic fields?’ to
address. Lyra and Klahr (2011) find that only at low ionisation can vortices be produced
and sustained by the baroclinic instability. If a magnetic field is included, these vortices only
survive until the MRI develops in the box, whereupon the strain of the turbulence destroys
it. However, Fromang and Nelson (2005); Lyra and Mac Low (2012) suggest that vortices
may occur when magnetic fields are present and so stability studies where magnetic fields are
included are still of interest.
Appendix A
Elliptic coordinates
In 2D we take
r = h sinh ξ sin ηi+ h cosh ξ cos ηj (A.0.1)
so
eξ =
∂r
∂ξ
= h cosh ξ sin ηi+ h sinh ξ cos ηj (A.0.2a)
eη =
∂r
∂η
= h sinh ξ cos ηih cosh ξ sin ηj (A.0.2b)
⇒ |eξ| = |eη| = h
√
sinh2 ξ + sin2 η (A.0.2c)
|eξ| = hξ and |eη| = hη are the two scale factors of this coordinate system (hz = 1 is the third),
from which all the vector calculus quantities can be calculated using the general curvilinear
coordinate formula (see e.g. Riley et al. (2006)).
For working out quantities like d/dx and d/dy there is a useful trick. With
x = h sinh ξ sin η (A.0.3)
y = h cosh ξ cos η (A.0.4)
we can formulate the following
h2 = x
2
sinh2 ξ
+ y
2
cosh2 ξ
(A.0.5a)
h2 = x
2
sin2 η −
y2
cos2 η . (A.0.5b)
214 Elliptic coordinates
Then we apply ∂
∂x
∣∣∣∣
y
and ∂
∂y
∣∣∣∣
x
to both equations (A.0.5a) and (A.0.5b), rearrange and find
∂ξ
∂x
∣∣∣∣
y
= cosh ξ sin η
hΓ (A.0.6a)
∂ξ
∂y
∣∣∣∣
x
= sinh ξ cos η
hΓ (A.0.6b)
∂η
∂x
∣∣∣∣
y
= sinh ξ cos η
hΓ (A.0.6c)
∂η
∂y
∣∣∣∣
x
= −cosh ξ sin η
hΓ (A.0.6d)
where
Γ = sinh2 ξ + sin2 η. (A.0.7)
Thus the chain rule gives us
∂
∂x
= cosh ξ sin η
hΓ
∂
∂ξ
+ sinh ξ cos η
hΓ
∂
∂η
(A.0.8a)
∂
∂y
= sinh ξ cos η
hΓ
∂
∂ξ
− cosh ξ sin η
hΓ
∂
∂η
. (A.0.8b)
We can then happily calculate quantities such as u =
(
∂ψ
∂y
,−∂ψ
∂x
)
.
Appendix B
Floquet theory as solution of an
initial value problem
The stability analysis we do on our equilibrium vortex solutions requires solving a series of
linear differential equations with periodic coefficients. This calls for Floquet Theory (Floquet,
1883).
We have a problem of the form
x′ = A(t)x (B.0.1)
where A(t) is a n× n periodic matrix with period T so
A(t+ T ) = A(t) ∀t. (B.0.2)
Then x need not be periodic but must be of the form eµtp(t) where p(t) is also T -periodic so
p(t+ T ) = p(t) ∀t. (B.0.3)
Let x1(t), . . .xn(t) be n solutions of (B.0.1). Furthermore, let
X(t) =
([
x1(t)
]
. . . [xn(t)]
)
(B.0.4)
such that X(t) is an n× n solution of X′ = A(t)X.
Definition 1 (Fundamental Matrix). If our x1(t), . . .xn(t) are linearly independent then X(t)
is non-singular and is called a fundamental matrix.
Definition 2 (Principle Fundamental Matrix). If X(t0) = I, where t0 is the initial time, then
X(t) is called the principle fundamental matrix.
Lemma 1. If X(t) is a fundamental matrix then so is Y(t) = X(t)B, where B a non-singular
216 Floquet theory as solution of an initial value problem
constant matrix.
Proof. X(t) and B are non-singular so their inverses exist.
Y−1(t) = B−1X−1(t)
so Y is non-singular too. Then
Y′ = X′B = AXB = AY
Lemma 2. Let the Wronskian W (t) of X(t) be the determinant of X(t). Then
W (t) =W (t0) exp
[∫ t
t0
tr (A (s)) ds
]
. (B.0.5)
Proof. Let t0 be some time. Expanding X(t) as a Taylor series:
X(t) = X(t0) + (t− t0)X′(t0) +O
(
(t− t0)2
)
= X(t0) + (t− t0)A(t0)X(t0) +O
(
(t− t0)2
)
= [I+ (t− t0)A(t0)]X(t0) +O
(
(t− t0)2
)
.
Then
det (X(t)) = det [I+ (t− t0)A(t0)] det (X(t0))
⇒W (t) = det [I+ (t− t0)A(t0)]W (t0)
Since det(I+ ϵC) = 1 + ϵtr(C) +O(ϵ2),
W (t) =W (t0) [1 + (t− t0)tr (A(t0))]
=W (t0) + (t− t0)W ′(t0) +O
(
(t− t0)2
)
from taking the Taylor series. We have made no assumption about t0 so
W ′(t0) =W (t0)tr (A(t0))
⇒W ′(t) =W (t)tr (A(t0))
⇒W (t) =W (t0) exp
[∫ t
t0
tr (A (s)) ds
]
Theorem 1. If X(t) is a fundamental matrix then:
217
1. X(t+ T ) is fundamental
2. ∃ a non-singular and constant matrix B s.t. X(t+ T ) = X(t)B
3. det(B) = exp
[∫ T
0 tr (A (s)) ds
]
.
Proof. 1. Let Y(t) = X(t+ T ). Then
Y′(t) = X′(t+ T ) = A(t+ T )X(t+ T )
= A(t)X(t+ T ) = A(t)Y(t)
⇒ X(t+ T ) is fundamental.
2. Let B(t) = X−1(t)Y(t). Then
Y(t) =
[
X(t)X−1(t)
]
Y(t) = X(t)B(t).
Let B0 = B(t0). By Lemma 1, Y0(t) = X(t)B0 is a fundamental matrix. Since both
Y0(t) and Y(t) are solutions of equation (B.0.1), by the uniqueness of the solution
Y0(t) = Y(t) ∀t.
Thus B0 = B(t0) and B is time-independent.
3. Lemma 2 gives
W (t) =W (t0) exp
[∫ t
t0
tr (A (s)) ds
]
⇒W (t+ T ) =W (t0) exp
[∫ t
t0
tr (A (s)) ds +
∫ t+T
t
tr (A (s)) ds
]
=W (t) exp
[∫ t+T
t
tr (A (s)) ds
]
=W (t) exp
[∫ T
0
tr (A (s)) ds
]
, (B.0.6)
where the last line used the periodicity of A. Also
X(t+ T ) = X(t)B
det (X(t+ T )) = det (X(t)) detB
⇒ W (t+ T ) =W (t) detB
⇒ detB = exp
[∫ T
0
tr (A (s)) ds
]
(B.0.7)
218 Floquet theory as solution of an initial value problem
Definition 3 (Characteristic Multipliers). The eigenvalues of B, ϱ1, . . . , ϱn are the charac-
teristic multipliers for equation (B.0.1).
Definition 4 (Characteristic Exponents). The characteristic (or Floquet) exponents are µ1, . . . , µn ∈
C satisfying
ϱ1 = eµ1T , . . . , ϱn = eµnT (B.0.8)
The characteristic multipliers ϱ1, . . . , ϱn of B = X(T ) with X(0) = I satisfy
detB = ϱ1ϱ2 . . . ϱn = exp
[∫ T
0
tr (A (s)) ds
]
(B.0.9)
The characteristic exponents are not unique since if ϱj = eµjT then
ϱj = exp
[(
µj +
2πin
T
)
T
]
, n ∈ Z (B.0.10)
Lemma 3. The characteristic multipliers ϱj are an intrinsic property of equation (B.0.1) and
do not depend on the choice of fundamental matrix.
Proof. Suppose Xˆ(t) is another fundamental matrix. Then
Xˆ(t+ T ) = Xˆ(t)Bˆ.
By Lemma 1, since X(t) and Xˆ(t) are fundamental matrices, ∃ a constant non-singular matrix
C such that
Xˆ(t) = X(t)C
⇒ Xˆ(t+ T ) = X(t+ T )C
⇒ Xˆ(t)Bˆ = X(t)BC
⇒ X(t)CBˆ = X(t)BC
⇒ CBˆ = BC
⇒ B = CBˆC−1 (B.0.11)
Thus the eigenvalues of B and Bˆ are the same.
Theorem 2. Let ϱ be a characteristic multiplier and let µ be the corresponding characteristic
exponent so that ϱ = eµT . Then ∃ a solution x(t) of equation (B.0.1) such that
1. x(t+ T ) = ϱx(t)
2. ∃ a T -periodic solution p(t) such that x(t) = eµtp(t).
219
Proof. 1. Let b be an eigenvector of B corresponding to eigenvalue ϱ:
Bb = ϱb.
Let x(t) = X(t)b. Then x′ = Ax.
x(t+ T ) = X(t+ T )b
= X(t)Bb
= ϱX(t)b
= ϱx(t)
2. Let p(t) = x(t)e−µt. We need to show that p(t) is T -periodic.
p(t+ T ) = x(t+ T )e−µ(t+T )
= ϱx(t)e−µte−µT
= ϱ
eµT
x(t)e−µt
= x(t)e−µt
= p(t)
Note that:
xj(t+ T ) = ϱjxj(t)
xj(t+NT ) = ϱNj xj(t)
Baring that in mind, each characteristic multiplier falls into one of the following categories:
1. If |ϱ| < 1 then ℜ(µ) < 0 and so x(t) t→∞−−−→ 0
2. If |ϱ| = 1 then ℜ(µ) = 0 and so we have a psuedo-periodic solution. If ϱ = ±1 then the
solution is periodic with period T .
3. |ϱ| > 1 then ℜ(µ) > 0 and so x(t) t→∞−−−→∞
The entire solution is stable if all characteristic multipliers satisfy |ϱj | ≤ 1.

Appendix C
Growth rates in the shearing sheet
for k ∝ t
This derivation is based on the lecture notes1 of Prof. Gordon Ogilvie, who gave the clearest
and most concise explanation I could find.
Consider a 3D homogeneous, inviscid2, incompressible fluid, unbounded or periodic in x, y, z
and rotating in a frame with Ω = Ωzˆ, in the shearing box of Goldreich and Lynden-Bell (1965).
The Navier-Stokes equations governing this are
∂u
∂t
+ u · ∇u+ 2Ω× u = −1
ρ
∇P −∇Φ (C.0.1a)
∇ · u = 0 (C.0.1b)
Neglecting vertical gravity we have
Φ = −ΩSx2, (C.0.2)
where S is the shear rate. The basic state of the shearing box is given by u = −Sxyˆ and
P = const. Let v be the velocity relative to the shearing sheet, P ′ be the pressure relative to
the background and Ψ = P ′/ρ, where ρ is constant. Then(
∂
∂t
− Sx ∂
∂y
+ v · ∇
)
vx −2Ωvy = −∂Ψ
∂x
(C.0.3a)(
∂
∂t
− Sx ∂
∂y
+ v · ∇
)
vy+(2Ω− S)vx = −∂Ψ
∂y
(C.0.3b)(
∂
∂t
− Sx ∂
∂y
+ v · ∇
)
vz = −∂Ψ
∂z
(C.0.3c)
1http://www.damtp.cam.ac.uk/user/gio10/dad.html
2The contribution due to ν∇2u can be included and dealt with via an integrating factor.
222 Growth rates in the shearing sheet for k ∝ t
Consider plane waves with time-dependent wavenumber k(t):
v(x, t) = v˜(t) exp[ik(t) · x] (C.0.4)
Ψ(x, t) = Ψ˜(t) exp[ik(t) · x] (C.0.5)
Then (
∂
∂t
− Sx ∂
∂y
)
v =
{
dv˜
dt
+ i
(
dk
dt
· x− Sxky
)
v˜
}
exp[ik(t) · x]. (C.0.6)
Choosing dk/dt = Skyxˆ means that this reduces to(
∂
∂t
− Sx ∂
∂y
)
v = dv˜
dt
exp[ik(t) · x] (C.0.7)
and v · ∇v = since incompressibility gives ik · v˜ = 0. Equation (C.0.3) then becomes
dv˜x
dt
−2Ωv˜y = −ikxΨ˜ (C.0.8a)
dv˜y
dt
+(2Ω− S)v˜x = −ikyΨ˜ (C.0.8b)
dv˜z
dt
= −ikzΨ˜ (C.0.8c)
with
ik · v˜ = 0 (C.0.8d)
and
k = (kx,0 + Skyt, ky, kz) , (C.0.8e)
where kx,0, ky, kz are constants. Eliminating Ψ˜, v˜y and v˜z results in the second order ODE:
d2
dt2
(
k2v˜x
)
+ κ2k2z v˜x = 0, (C.0.9)
with k2 = (k2x,0+k2y +k2z)+S2k2yt2 and the epicyclic frequency κ2 = 2Ω(2Ω−S). In Keplerian
discs, κ2 = Ω2 > 0, which is what we are concerned with in this work.
If ky = 0, we have axisymmetric disturbances, k is constant and the equation has constant
coefficients. The system will therefore oscillate with period κkz/k.
If ky ̸= 0 we have sheared, non-axisymmetric waves and the solutions to equation (C.0.9)
are Legendre functions. As t→∞, this equation acts like
d2
dt2
(
t2v˜x
)
+ a2v˜x = 0, a2 =
κ2k2z
S2k2y
. (C.0.10)
223
This is a Cauchy-Euler equation so we look for solutions of the form v˜x = tr. This gives
(r + 1)(r + 2) + a2 = 0, (C.0.11)
which has roots
r = −3±
√
1− 4a2
2 = −
3
2 ± σ (C.0.12)
Since we are interested in growth rates at large t, we are only concerned about the positive
root, |v˜x| ∝ tσ−3/2. Equation (C.0.8d) implies that
v˜y, v˜z ∼ tσ−1/2 (C.0.13)
while equation (C.0.8c) shows Ψ˜ ∼ tσ−3/2. Therefore, at late times, |v˜| ∼ tσ−1/2. The following
cases arise:
(i) If 1− 4a2 < 0, or
κ2 >
S2k2y
4k2z
(C.0.14)
then σ is imaginary and |v˜| ∼ t−1/2 → 0 as t→∞.
(ii) If 0 < 1− 4a2 < 14 , or
0 < κ2 <
S2k2y
4k2z
(C.0.15)
then 0 < σ2 < 14 and |v˜| ∼ tσ−1/2 → 0 as t→∞.
(iii) If 1− 4a2 > 14 , κ2 < 0 and |v˜| ∼ tσ−1/2 →∞ as t→∞.
We find that flows that are Rayleigh-stable (Rayleigh, 1917) with κ2 > 0, such as Keplerian
rotation, are also stable at large t to shearing waves.

Appendix D
PLUTO runs
Table D.1 The variables governing different PLUTO runs. Typically, t = 200Ω−1.
Run Lx Ly Lz Nx Ny Nz χ cs ρ′ v′ Notes
211a 4 4 – 256 256 – 5 5 – – Testing box size
211b 4 8 – 256 512 – " " – –
211c 4 12 – 256 768 – " " – –
211d 4 4 – 512 512 – " " – – Testing resolution
211e 4 8 – 512 1024 – " " – –
211f 4 12 – 512 1536 – " " – –
211g 8 4 – 512 256 – " " – –
211h 4 4 – 1024 1024 – " " – –
211i 2 4 – 128 256 – " " – –
211j 2 4 – 256 512 – " " – –
211k 2 4 – 512 1024 – " " – –
211l 2 4 – 128 128 – " " – –
2D boxes in line with 3D versions
228a 2 4 – 128 256 – 2 5 – 10−3
228b " " – " " – 2.5 " – "
228c " " – " " – 3 " – "
228d " " – " " – 3.5 " – "
228e " " – " " – 4 " – "
228f " " – " " – 5 " – "
228g " " – " " – 6 " – "
228h " " – " " – 7 " – "
228i " " – " " – 8 " – "
228j " " – " " – 9 " – "
228k " " – 192 256 – 10 " – "
228l " " – " " – 12 " – "
Continued on next page
226 PLUTO runs
Table D.1 – Continued from previous page
Run Lx Ly Lz Nx Ny Nz χ cs ρ′ v′ Notes
228m " " – " " – 15 " – "
228n " " – " " – 20 " – "
Unstratified 3D boxes – testing perturbations
213a 4 4 4 128 128 128 5 5 – – Testing 3D box
213b " " " " " " 8 " – –
213c " " " " " " 3.5 " – –
213d " " " " " " 5 5 – 10−3 Added v′
213e " " " " " " 8 " – 10−3
213f " " " " " " 3.5 " – 10−3
Stratified boxes, full
215a1 2 4 6 128 128 64 5 5 0.1 10−3 −3 ≤ z ≤ 3
215b1 2 4 6 128 128 64 8 5 0.1 10−3 −3 ≤ z ≤ 3
215b1 2 4 6 128 128 64 3.5 5 0.1 10−3 −3 ≤ z ≤ 3
Stratified half–boxes, z ≥ 0
216a1 2 4 3 128 128 32 5 5 0.1 10−3
216a2 2 4 3 128 128 64 " " " "
216a3 2 4 6 128 128 64 " " " "
216a4 2 4 3 128 256 32 " " " "
217a 2 4 3 128 256 32 5 2.5 0.1 10−3 Changing cs
217b " " " " " " 5 25 0.1 10−3
217d " " " " " " 5 10 0.1 10−3
218a 2 4 3 128 256 32 5 5 0.5 10−3 Changing ρ′, v′
218b " " " " " " " " 0.1 5× 10−3
218c " " " " " " " " 0.1 10−2
219a 2 4 3 128 256 32 2 5 0.1 10−3 See effect of χ on
vortex lifetime
219b " " " " " " 2.5 " " "
219c " " " " " " 3 " " "
219d " " " " " " 3.5 " " "
219e " " " " " " 4 " " "
219f " " " " " " 6 " " "
219g " " " " " " 7 " " "
219h " " " " " " 8 " " "
219i " " " " " " 10 " " "
219j " " " " " " 15 " " "
219k " " " " " " 12 " " "
Continued on next page
227
Table D.1 – Continued from previous page
Run Lx Ly Lz Nx Ny Nz χ cs ρ′ v′ Notes
219m " " " " " " 9 " " "
219j3 " " " 192 " " 15 " " "
219l3 " " " " " " 20 " " "
220{a− l3} 2 4 3 128 256 32 as 219 5 0.5 10−3 See effect of χ on
vortex lifetime, big-
ger ρ′
Stratified short half-boxes
223a 2 4 1.5 128 256 32 2 5 0.1 10−3 See if we can repro-
duce short box sta-
bility
223b " " " " " " 3 " " "
223c " " " " " " 4 " " "
223d " " " " " " 5 " " "
223e " " " " " " 8 " " "
223f 1 4 1.5 " " " 10 " " "
223g " " " " " " 15 " " "
223f3 2 4 1.5 192 " " 10 " " " t = 300
223g3 " " " " " " 15 " " " t = 300
High resolution boxes
226a 2 4 3 256 512 64 5 5 0.1 10−3 Stratified long
226b " " " 256 512 64 5 5 0.5 10−3 "
226c " " " 256 512 64 5 5 1.0 10−3 "
226g 2 4 3 256 512 64 5 5 – 10−3 Unstratified long
Low cs, looking for density waves
229a 2 4 3 128 256 32 5 0.2 0.1 10−3 crashed
229b " " " " " " " 0.3 " 10−3 ≈ sonic
229c " " " " " " " 0.5 " 10−3
229d " " " " " " " 1.0 " 10−3
229e " " " " " " " 0.2 0.5 10−3 crashed
229f " " " " " " " 0.3 " 10−3
229g " " " " " " " 0.5 " 10−3
229h " " " " " " " 1.0 " 10−3
Checking stratified interface, perturbations only for z ≤ 0.5Lz, compare to run 219
232a 2 4 3 128 256 32 5 5 0.1 10−3 Interface at 0.8Lz
232b " " " " " " 8 " " " "
232c " " " " " " 5 " " " Interface at 0.9Lz
Continued on next page
228 PLUTO runs
Table D.1 – Continued from previous page
Run Lx Ly Lz Nx Ny Nz χ cs ρ′ v′ Notes
232d " " " " " " 8 " " " "
232e " " " " " " 5 " " " Interface at 0.8Lz,
smoothed transi-
tion
232f " " " " " " 8 " " " "
Unstratified boxes
233a 2 4 3 128 256 32 2 5 0.1 10−3
233b " " " " " " 2.5 " " "
233c " " " " " " 3 " " "
233d " " " " " " 3.5 " " "
233e " " " " " " 4 " " "
233f " " " " " " 5 " " "
233g " " " " " " 6 " " "
233h " " " " " " 7 " " "
233i " " " " " " 8 " " "
233j3 " " " 192 " " 10 " " "
233k3 " " " " " " 12 " " "
233l3 " " " " " " 15 " " "
Checking stratified interface, perturbations for z ≤ Lz, compare to runs 219, 232
234a " " " " " " 5 " " " Interface at 0.9Lz
234b " " " " " " 8 " " " "
234c " " " " " " 5 " " " Interface at 0.8Lz,
smoothed transi-
tion
234d " " " " " " 8 " " " "
Investigating low cs, stratified cases, compare to runs 219, 220
235a 2 4 3 128 256 32 5 1.0 0.1 10−3 = run229d
235b " " " " " " 8 " " "
235c " " " 192 " " 10 " " "
235d " " " " " " 15 " " "
235e " " " " " " 20 " " "
Investigating a high resolution case – is there a higher χcrit, above which all columns are stable?
236a 2 4 3 192 384 32 5 5 0.1 10−3
236b " " " " " " 8 " " "
236c " " " " " " 10 " " "
236d " " " " " " 15 " " "
236e " " " " " " 20 " " "
229

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