Instabilities and Transport in
Magnetized Plasmas
Mark Saul Rosin
Darwin College
A Dissertation submitted for the degree of
Doctor of Philosophy
University of Cambridge 2010
For my mother, father and sister.
“Astrophysics is too important to be left in the hands of astrophysicists.”
– Hannes Alfve´n

Acknowledgments
First and foremost I would like to thank my supervisor Alex Schekochihin. With-
out his guidance, enthusiasm, boundless energy and support I would still be living
in the dark ages. I would also like to thank all members of my research group at
Cambridge (past and present). In particular Franc¸ois Rincon from whom I learnt
not only a lot of science, but a lot about how to be a good researcher. Similarly
I would like to thank Jon Mestel and Gordon Ogilvie, both of whom were very
patient with me. I would also like to thank James Binney, Tobias Heinemann,
Matt Kunz, Geoffroy Lesur, Henrik Latter, Felix Para, Mike Proctor and Rich
Wood for useful discussions over the years.
Thanks to Toby Wood, Chris Taylor, Ba´rbara Ferreira and Adam Nahum for
being supportive and inspiring peers. Special mention must also go to Rodney
Mellor and John Hannay who inspired my love of mathematics and physics.
I extend thanks to the UK’s Science and Technology Facilities Council for three
generous years of funding. Further thanks to the Cambridge Philosophical Soci-
ety and Leverhulme Network for providing assistance towards the end of my Ph.D.
None of this would have been possible without the constant support of my family
and friends. There are too many to mention you all individually. You know who
you are. However, particular thanks must go to my Mother, sister, Helen and
grandparents who have helped me get to where I am today. Similarly, thanks to
all the Goodfellows, Becky and Bertie. Finally, Kate. Thank you for everything.
You stood by me. For what it’s worth, I wouldn’t have got this far without you.
v
vi
Declaration
This dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration except where specifically indicated in
the text. It is based on research done at the DAMTP, Cambridge from October
2006 to July 2010. No part of this thesis has been or is being submitted for any
qualification other than the degree of Doctor of Philosophy at Cambridge. The
main collaborative parts of my thesis are the result of work with my supervisor
Dr. Alex Schekochihin. However, Jon Mestel, Franc¸ois Rincon and Steve Cowley
have also contributed to the publications that came out of this work:
• M. S. Rosin, A. A. Schekochihin, F. Rincon, & S. C. Cowley,
“A nonlinear theory of the parallel firehose and gyrothermal instabilities in
a weakly collisional plasma”, MNRAS, (Submitted, 2010).
[e-print:arXiv.org/pdf/1002.4017].
• M. S. Rosin, A. J. Mestel, & A. A. Schekochihin, “The global MRI with
Braginskii viscosity”, MNRAS, (In Preparation, 2010).
Chapter 2 is largely based on the latter of these and chapter 3 on the former. I
have also contributed to the following publications as part of my Ph.D:
• A. A. Schekochihin, S. C. Cowley, F. Rincon, & M. S. Rosin, “Magnetofluid
dynamics of magnetized cosmic plasma: firehose and gyrothermal instabili-
ties”, MNRAS. 405, 291 (2010) [e-print arXiv:0912.1359]
• A. A. Schekochihin, S. C. Cowley, R. M. Kulsrud, M. S. Rosin, & T. Heine-
mann, “Nonlinear growth of firehose and mirror fluctuations in astrophys-
ical plasmas”, Phys. Rev. Lett., 100, 081301, (2008).
[e-print:arxiv.org/pdf/0709.3828].
Work done in the former appears in several places in chapters 1 and 3, and work
done in the latter appears in chapter 3. They are cited respectively as Schekochi-
hin et al. (2010) and Schekochihin et al. (2008).
- M. Rosin, 18th October, 2010


Abstract
In a magnetized plasma, naturally occurring pressure anisotropies facilitate in-
stabilities that are expected to modify the transport properties of the system. In
this thesis we examine two such instabilities and, where appropriate, their effects
on transport.
First we consider the collisional (fluid) magnetized magnetorotational instabil-
ity (MRI) in the presence of the Braginskii viscosity. We conduct a global lin-
ear analysis of the instability in a galactic rotation profile for three magnetic
field configurations: purely azimuthal, purely vertical and slightly pitched. Our
analysis, numerical and asymptotic, shows that the first two represent singular
configurations where the Braginskii viscosity’s primary role is dissipative and the
maximum growth rate is proportional to the Reynolds number when this is small.
For a weak pitched field, the Braginskii viscosity is destabilising and when its ef-
fects dominate over the Lorentz force, the growth rate of the MRI can be up to
2
√
2 times faster than the inviscid limit. If the field is strong, an over-stability
develops and both the real and imaginary parts of the frequency increase with
the coefficient of the viscosity.
Second, in the context of the ICM of galaxy clusters, we consider the pressure-
anisotropy-driven firehose instability. The linear instability is fast (∼ ion cy-
clotron period) and small-scale (ion Larmor radius ρi) and so fluid theory is
inapplicable. We determine its nonlinear evolution in an ab initio kinetic calcu-
lation (for parallel gradients only). We use a particular physical asymptotic or-
dering to derive a closed nonlinear equation for the firehose turbulence, which we
solve. We find secular (∝ t) growth of magnetic fluctuations and a k−3‖ spectrum,
starting at scales & ρi. When a parallel ion heat flux is present, the parallel
firehose instability mutates into the new gyrothermal instability. Its nonlinear
evolution also involves secular magnetic energy growth, but its spectrum is even-
tually dominated by modes with a maximal scale ∼ ρilT/λmfp, (lT is the parallel
temperature gradient scale). Throughout we discuss implications for modelling,
transport and other areas of magnetized plasma physics.
ix

Contents
I Introduction 1
1 Introduction 3
1.1 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Basic plasma theory . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Moment equations . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 The pressure tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Determining the pressure anisotropy . . . . . . . . . . . . . . . . 19
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
II Magnetized MRI 29
2 Magnetized MRI 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Galactic discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Global stability analysis . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Azimuthal field, θ = 0 . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Vertical field, θ = pi/2 . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6 Pitched field, θ  1 . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
xi
III Firehose instability 87
3 Firehose instability 89
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2 Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.3 Kinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.4 Firehose turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.5 Gyrothermal turbulence . . . . . . . . . . . . . . . . . . . . . . . 123
3.6 Comparison with previous work and a collisionless plasma . . . . 134
3.7 Astrophysical implications . . . . . . . . . . . . . . . . . . . . . . 137
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
IV Conclusion 143
4 Conclusions 145
4.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
V Appendices 151
A Gyrophase dependent pressure 153
B Kinetic theory: Electrons 157
C Kinetic theory: Ions 161
D Commonly used symbols 181
Bibliography 184
xii


Part I
Introduction
1

Chapter 1
Introduction
“If reading Chapman and Cowling is like chewing glass, then reading
Braginskii is like chewing fibre-glass.”
P.J.Dellar, Oxford, 2010
1.1 Setting the stage
Neutral fluid theory relies on the asymptotic smallness of the inter-particle mean
free path λmfp to formally close the moment hierarchy of the underlying kinetic
equation (Chapman & Cowling, 1970)1. In a plasma another potentially asymp-
totically small parameter exists that allows the kinetic equation to be simplified
considerably. This parameter is the Larmor radius ρs.
In a plasma with a sufficiently strong magnetic field, charged particles execute
Larmor orbits about the field lines with a radius ρs and (cyclotron) frequency
Ωs. When the mean free path of the plasma is greater than the Larmor radius
λmfp  ρs and the collision frequencies less than the cyclotron frequency νs  Ωs,
the plasma is said to be magnetized. The smallness of ρs restricts motion across
magnetic field lines (but not along them) and the largeness of Ωs renders the
position of the charged particle about its Larmor orbit irrelevant to lowest order
1A profile of Sydney Chapman published in the Observer, July 2, 1957 and reprinted in
Brush (1976) quotes him as saying his book was very heavy going, “like chewing glass”. The
comments at the start of this chapter are in reference to this.
3
4 1. Introduction
Figure 1.1: In a magnetized plasma, charged particles are constrained to travel
along, but not across strong magnetic field lines. The ensuing flows and tem-
perature profiles are anisotropic with respect to the direction of the magnetic
field, and lead to highly filamentary structures. Left Panel: X-ray emission im-
age of solar coronal loops taken by the Transition Region and Coronal Explorer
(TRACE) (Reale & Peres, 2000; Reale et al., 2000; TRACE, 2000). Right Panel:
Dα emission images (visible light) from Edge Localised Modes (ELMS) on the
Mega Ampere Spherical Tokomak (MAST) (Kirk et al., 2005; Dudson et al.,
2008).
in an asymptotic expansion of the plasma distribution function. As a result the
number of independent variables necessary to fully describe a kinetic plasma can
be reduced by one (Kulsrud, 1962, 1983). Furthermore, the component of the
particle velocity perpendicular to the magnetic field is controlled by a constant
of motion, the first adiabatic invariant. The physics of such plasma differs con-
siderably from a neutral fluid (and indeed a collisionally dominated plasma, e.g.
one described by magnetohydrodynamics (MHD)) in three important ways.
First, on macroscopic scales (greater than λmfp in a collisional plasma), the trans-
port of momentum, heat and current become highly anisotropic with respect to
the direction of the local magnetic field. Cross-field diffusion is limited by a
random walk of step size ρs, whilst the along-field transport is often either un-
constrained or limited by λmfp  ρs. The effects of this can be clearly seen in
the elongated filamentary structures of the Sun’s magnetic coronal loops and the
guide fields of a terrestrial tokomak, Figure 1.1.
1.1 Setting the stage 5
Second, the manner in which free energy gradients, angular velocity and temper-
ature are relaxed can change. Well known instabilities like the magnetorotational
instability (MRI) are modified considerably (Balbus & Hawley, 1998; Quataert
et al., 2002; Sharma et al., 2003; Islam & Balbus, 2005), and new instabilities
appear: the magnetothermal instability (MTI), the heat flux buoyancy instabil-
ity (HBI) and the magnetoviscous instability (MVI) (Balbus, 2000, 2001, 2004;
Quataert, 2008).
Third, changing magnetic fields and heat fluxes arising from, say, an instability
or macroscopic transport, create pressure anisotropies with respect to the local
direction of the magnetic field, (see e.g. Chew et al., 1956; Kulsrud, 1983). In
turn, these drive fast growing instabilities on microscopic scales (between ρs and
λmfp) whose effect on the instabilities and macroscopic transport processes from
which the anisotropy arose is unknown (Schekochihin et al., 2005).
Understanding the rich interplay between these three realisations of magnetized
plasma phenomena constitutes an important and, as of yet, unsolved issue in
astrophysics; specifically in galaxy clusters, galaxies and accretion discs. Without
delving into the details too deeply (we will do so later) let us consider a sample
of questions whose answers fundamentally depend on a complete understanding
of the magnetized plasma physics referenced above:
• In galaxy clusters, to what extent does anisotropic transport help/hinder
proposed solutions to the cooling flow problem (Fabian, 1994; Peterson &
Fabian, 2006)? What effect do magnetized instabilities that feed off the
temperature gradient have? What is the resultant effective viscosity and
conductivity relative to the Spitzer values? Does this depend on the mag-
netic field configuration and strength (Balbus & Reynolds, 2008; Parrish &
Quataert, 2008; Bogdanovic´ et al., 2009; Parrish et al., 2010; Kunz et al.,
2010)?
• What is the nature of the fluid turbulence arising from magnetized insta-
bilities in the galaxy and accretion discs (Schekochihin & Cowley, 2006)?
What is its effective α-parameter (Shakura & Sunyaev, 1973)? Is this the
same in a collisionless or weakly collisional plasma (Sharma et al., 2006,
6 1. Introduction
2007)? Do global effects have a role to play (Rosin et al., 2010a)? Again,
to what extent does the strength and configuration of the magnetic field
matter (Quataert et al., 2002; Sharma et al., 2003; Balbus, 2004)?
• How does small-scale dynamo action proceed in a magnetized plasma (Shukurov
et al., 2006; Brandenburg & Nordlund, 2009)? Does the presence of mi-
croscale instabilities categorically change, or merely modify, the rate of field
amplification (Malyshkin & Kulsrud, 2002; Schekochihin & Cowley, 2006)?
What is the structure of the field and how does it affect the macroscopic
dynamics?
• How precisely do microscale instabilities affect all of the above (Schekochi-
hin & Cowley, 2006; Lyutikov, 2007; Schekochihin & Cowley, 2007; Kunz
et al., 2010)? Can they be parameterised or is a full microphysical theory
necessary? In either case, in what way will they appear in the equations gov-
erning the macroscopic plasma evolution (Schekochihin et al., 2008; Rosin
et al., 2010b)?
The existing literature (some of which is cited above) addresses some of these
questions, but an all-encompassing description is far from complete and even
existing work is open to reassessment2. The overall picture is a deeply inter-
connected one and the types of questions outlined above are probably pertinent,
to some degree, to most magnetized astrophysical processes. In Figure 1.2 we
present a grossly simplified, but nevertheless useful, framework for considering
these processes.
It is the purpose of the research presented here to answer two sub-questions
from the list above. In chapter 2 we consider the global nature of a magnetized
shear instability, locally identifiable as the MRI in the presence of Braginskii
viscosity, and its dependence on the magnetic field configuration, plasma beta,
rotation rate and ion collision frequency (Balbus, 2004; Islam & Balbus, 2005).
2For example, Sharma et al. (2006, 2007) have investigated the turbulent state of a collision-
less magnetized accretion disc. In doing so, they assumed micro-instabilities relax the pressure
anisotropy towards the marginal stability boundary via an effective collision process. In chapter
3 we show that this method of maintaining marginal stability is not the only possibility and,
instead, the pressure anisotropy can be regulated by the growth of microscale magnetic fields.
1.2 Basic plasma theory 7
In particular, we focus on its role in the historic Interstellar Medium (ISM) of
spiral galaxies. In chapter 3 we consider the nonlinear evolution of the pressure-
anisotropy-driven firehose instability and the newly discovered heat flux driven
gyrothermal instability (GTI) and their effect on transport (Schekochihin et al.,
2010). In this case we focus on the Intracluster Medium (ICM) of galaxy clusters.
Throughout, we attempt to draw together the implications of our results for those
issues we do not directly address. Because the two problems considered in this
thesis are quite different, we attempt to keep the content of this introduction as
broad as possible.
We start in section 1.2 by introducing the fundamental equations of plasma
physics before, in section 1.3, reducing them to a hierarchy of moment equations.
Because of the complicated nature of the pressure tensor we do not immediately
close the hierarchy and instead, in section 1.4, we examine its general form as-
suming only that the plasma is highly magnetized. However, in section 1.5 we
make use of the asymptotic smallness of ρs (largeness of Ωs) to derive, and pro-
vide an physical interpretation of, the modified CGL equations which we then use
to close the moment hierarchy (neglecting heat fluxes). These equations, along
with the lower order moments provide the conceptual starting point from which
the rest of our work proceeds. In section 1.6 we summarise and lay out our plan
for the remainder of this dissertation.
1.2 Basic plasma theory
1.2.1 Fundamental equations
At the most basic super-quantum level a plasma is simply a statistically large
collection of ionized particles whose macroscopic evolution is fundamentally gov-
erned by gravity (which we shall omit until chapter 2) and the electromagnetic
force at both short and long ranges.
The plasma is described by the evolution of the number density distribution
function fs(t, r,w) where the subscript s = i, e indicates the species type: ions
8 1. Introduction
Macroscopic Gradients: Temperature, Velocity
Magnetized
instabilities
HD & MHD
instabilities
Macroscopic turbulence
Pressure
Anisotropies
Fluctuation
Dynamo
Microscale
Instabilities
Figure 1.2: Schematic representation of the salient astrophysical magnetized
plasma processes in, for example, a galaxy cluster, galaxy or accretion disc. Free
energy gradients sustained by active galactic nuclei (AGN), supernovae or the
differential rotation of a disc or galaxy can be unstable to both MHD, HD (hydro-
dynamic) and magnetized instabilities. In the nonlinear stages these instabilities
leads to turbulence. The turbulent scale flows induce rapidly changing magnetic
fields and therefore fluctuation dynamos and pressure anisotropies. The pressure
anisotropy leads to micro-instabilities that changes both the pressure anisotropy
itself and, possibly, the fluctuation dynamo. One would expect these processes to
then feed back onto the macroscale turbulence which, in turn, affects the macro-
scopic gradients by changing the macroscopic transport properties of the system,
e.g. an enhanced viscosity or reduced conductivity.
1.2 Basic plasma theory 9
or electrons. The equation governing the evolution of fs is the kinetic equation,
Dfs
Dt
≡
∂fs
∂t
+w · ∇fs +
Zse
ms
(
E +
w ×B
c
)
·
∂fs
∂w
= C[fs], (1.1)
where D/Dt is the full convective derivative in time t, position r and particle
velocity w phase-space and C[fs] represents the combined effect of all collisions
on species s, including like-like collisions between particles of the same species.
The remaining undefined terms are, Zs the charge (Zi is unspecified, Ze = −1)
in units of e the unit of elementary charge, E the electric field, B the magnetic
field and c the speed of light. The term in parentheses is the total force acting
on the particles, which in this case is simply the Lorentz force.
It is convenient for what follows to make a change of variables and calculate
the distribution function in terms of peculiar (also known as random) velocities
v = w − u(t, r) where u is the mean “fluid” (not particle) velocity. Under this
change (t, r,w)→ (t, r,v), the kinetic equation (1.1) becomes
dfs
dt
+ v ·∇fs +
(
a+
Zse
ms
v ×B
c
− v · ∇us
)
·
∂fs
∂v
= C[fs], (1.2)
where d/dt = ∂/∂t+us ·∇ is the fluid convective derivative and a = (Zse/m)(E+
us ×B/c)− dus/dt represents the fluid Lorentz force and acceleration terms.
To self-consistently determine the electromagnetic fields in this equation, we must
also consider Maxwell’s equations
∂B
∂t
= −c∇×E, (1.3)
∑
s
Zseus = j =
c
4pi
∇×B, (1.4)
∑
s
Zsnse ' 0, (1.5)
∇ ·B = 0, (1.6)
where ns is the number density and j the current.
These differ from the standard set in that the plasma is quasi-neutral (equation
(1.5)) and the displacement current (∂E/∂t) /4pi has been neglected from Am-
10 1. Introduction
pere’s Law in equation (1.4). The first modification is valid if the typical length
scales of the problem are greater than λDs = ωD,s
√
Ts/ms where ωD is the plasma
frequency. In this case the ion and electron densities are approximately equal.
The second modifications holds for relativistically slow flows, as the ratio of the
displacement current to c∇ × B scales as (us/c)2. Both conditions are readily
met in the plasmas we consider here.
Equations (1.2)-(1.6) provide a complete description of the plasma. However, in
all but the simplest of cases, such a treatment is entirely impractical. Not only
are the equations highly nonlinear, but fs is seven dimensional (and it describes
multiple species). Instead, the general approach is to simplify this set of equations
by taking moments and invoking some appropriate small parameter to close or
truncate the ensuing hierarchy. As we now see, the resultant “fluid” equations
are considerably simpler to solve.
1.3 Moment equations
1.3.1 Moments
The zeroth, first and second moments of the distribution function (in peculiar
variables) are defined as follows
ns =
∫
d3vfs, (1.7)
0 =
∫
d3vvfs, (1.8)
Ps = ms
∫
d3vvvfs, (1.9)
where Ps is the pressure tensor for species s. By our change of variables w → v
we have removed the mean flow dependence from fs and it is now explicit in
equation (1.2). From the kinetic point of view however,
nsus =
∫
d3wwfs. (1.10)
As an aside, it is traditional in fluid dynamics to divide Ps into two parts. The first
1.3 Moment equations 11
is its trace, the isotropic pressure ps ≡ Trace(Ps)/3 = ms/3
∫
d3vTrace(vv)fs =
(1/2)msnsv2th,s where vth,s =
√
2Ts/ms is the particle thermal speed, Ts its tem-
perature and from which it follows ps = nsTs. The second part is the traceless
deviatoric, or viscous, stress tensor D = P− pI. Where appropriate, primarily in
chapter 2, we will make use of this notation.
Returning to the moments of fs, their evolution is governed by the corresponding
moment of the kinetic equation (in which us is now explicit). The zeroth and
first moments of equation (1.2) are
dns
dt
= −ns∇ · us, (1.11)
nsms
dus
dt
= −∇ · Ps + Zse ns
(
E +
us ×B
c
)
+Rs, (1.12)
where Rs = ms
∫
d3vvC[fs] is the mean inter-species collisional transfer of mo-
mentum (friction force) between particles of species s and all other species.
As we shall see later in this chapter, determining Ps is non-trivial in a magne-
tized plasma, but possible by virtue of the largeness of Ωs. In a frame oriented
along the magnetic field, its diagonal elements are governed by not just one, but
rather two equations, one for the parallel and one for the perpendicular pressure
(equivalently, one equation for the scalar pressure p and one for the pressure
anisotropy). Therefore we momentarily postpone taking the second moment of
the kinetic equation and instead examine the first moment. In doing so, it is
convenient to re-write the ion and the electron versions as two equations: the in-
duction equation which does not explicitly depend on Pi and Pe and the “total”
momentum equation, which depends on both. (Because ui appears on the left
hand side, we sometimes refer to it as the “ion momentum” equation.)
1.3.2 Electrons & ions: Induction and momentum equa-
tions
Considering the electron version of equation (1.12) first, we can eliminate the
flow ue = ui− j/ene in favour of the ion velocity and the current using equation
12 1. Introduction
(1.4). Doing so and rearranging yields
ene
(
E +
ui ×B
c
)
=
j ×B
c
−∇ ·Pe−Re− neme
d
dte
(
ui −
j
ene
)
, (1.13)
where d/dte = ∂/∂t+ue · ∇ = ∂/∂t+ (ui − j/ene) · ∇ is the electron convective
derivative. Equation (1.13) is known as the generalised Ohm’s Law (Spitzer,
1962). Depending on the regime, some, if not all, of the terms on the right hand
side can be neglected as small.
Let us assume for the purposes of this thesis that:
Ti ∼ Te, ui ∼ ue, mi/me  1. (1.14)
The first assumption is conditional on the ions and electrons having been in
collisional contact long enough to equilibrate; the second, that the friction force
Re is sufficient to keep electrons at a similar velocity to ions; and the third is self
evident since the proton mass is approximately 1800 times that of the electron
mass. For the time being we also assume that kρi M where k ∼ ∇ is one over
the typical gradient scale length andM = ui/vth,i is the Mach number (although
in chapter 3 we will consider a case where this does not hold, but proves to be
irrelevant).
Under this set of assumptions, the dominant force balance for electrons is given
by the ideal Ohm’s Law,
E +
ui ×B
c
= 0. (1.15)
Now combining it with Faraday’s Law, equation (1.3), yields an evolution equa-
tion for the magnetic field
∂B
∂t
= ∇× (ui ×B) , (1.16)
which is the familiar ideal induction equation. The many non-ideal versions of this
are simply the non-gradient terms from equation (1.13) that survive the ordering
elimination. For example, Re is responsible for resistivity whilst j ×B/c is the
Hall term (which we ordered out by assuming kρi M).
1.3 Moment equations 13
The induction equation readily couples to the ion version of the momentum equa-
tion (1.12) once we write the Lorentz force in terms of the current. To do so we
must either add the electron momentum equation to it (thereby eliminating the
electric field) or simply replace the ion Lorentz force by the appropriate Ohm’s
Law. Both processes are equivalent but the latter requires us to go to an order
at which the Lorentz force does not vanish (as it does in the ideal case, equation
(1.15)).
Adopting the former approach, we sum the ion and electron versions of equation
(1.12). Neglecting electron contributions to the momentum since miui  meue
and noting that inter-species friction forces Rs are equal and opposite (by mo-
mentum conservation), as are the particle electric forces on each species ZsnsE,
we find
nimi
dui
dt
= −∇ · (Pi + Pe) +
B × (∇×B)
4pi
, (1.17)
where we have used Ampere’s Law equation (1.4) to eliminate j in favour of B.
As they stand, equations (1.11), (1.16) and (1.17) are not a closed set because
we do not yet have an expression for Ps. Depending on the particular problem
there are many ways to find it.
In the highly collisional (neutral) regime Ps is described by the theory of Chap-
man and Enskog (Chapman & Cowling, 1970) and in the highly collisional, mag-
netized regime by its generalisation due to Braginskii (1965). Braginskii’s work
lays the foundations for much of collisional magnetized transport theory. How-
ever, because it is limited to M∼ 1, contributions from heat fluxes are omitted
from P. Later works rectified this by allowing for M 1 and thereby included
heat fluxes (see Mikhailovskii & Tsypin, 1971, 1984; Catto & Simakov, 2004).
Outside the collisional regime other descriptions have also been developed. In
the long-wavelength collisionless regime, double adiabatic theory, or CGL theory,
and its modifications to include Landau damping are applicable (Chew et al.,
1956; Snyder et al., 1997). Anisotropic gradient-scale-length theories also exist in
the form of gyrokinetics (Howes et al., 2006) as well as the Kinetic MHD simpli-
fication to the kinetic equation (Kulsrud, 1962, 1983). Many other theories exist
and in chapter 3 we develop one specifically tailored to the problem of pressure
14 1. Introduction
B
+ρi
Figure 1.3: In a magnetized plasma, a positively charged ion executes Larmor
orbits of radius ρi and frequency Ωi about the magnetic field B. Transport
becomes highly anisotropic as particles are constrained to move along, but not
across, field lines and the pressure tensor becomes anisotropic with respect to the
direction of the local field (Dellar, 2010)
anisotropy driven micro-instabilities. However, for now we do not wish to con-
centrate on the particulars of any one theory. Instead we restrict ourselves to
simply considering the general structure of the pressure tensor.
1.4 The pressure tensor
1.4.1 Asymptotic expansion
In a highly magnetized plasma the lowest order contribution to the distribu-
tion function (expanded as a series in the large ratio of the gyrofrequency Ωs =
ZseB/msc = ρsvth,s to the remaining frequencies in the system) is gyrotropic, i.e.
independent of the local phase angle along the Larmor orbit. This means that
the zeroth order contribution of fs to Ps is gyrotropic too and one must go to
higher orders in the expansion to recover any gyrophase dependency. Doing so,
one finds the finite Larmor radius (FLR) terms associated with the gyrophase
dependency of the distribution function at first order. It is relatively easy to
show the form of P under these rather general conditions and we do so now.
We formally assume that the frequencies associated with equation (1.2) can be
1.4 The pressure tensor 15
ordered to be less than Ωs:
kvth,s ∼ k us ∼ γ ∼ γe ∼ νs ∼ δ Ωs, δ  1 (1.18)
where γ ∼ d/dt is the typical response frequency, γe is the typical electric field
time scale and νs = λmfpvth,s is the typical collision frequency with which C[fs]
scales.
Now using δ as an expansion parameter we write fs as a series in increasing
powers,
fs = f
(0)
s + δf
(1)
s + δ
2f (2)s + . . . , (1.19)
where all terms f (n)s (n an integer) are of the same order. Substituting equation
(1.18) and equation (1.19) into equation (1.2) and grouping terms of the same
order in δ leads to a set of coupled differential equations. To lowest order we find
Ωs
∂f (0)s
∂ϑ
= 0, (1.20)
where ϑ is the gyrophase coordinate about a Larmor orbit, defined in Cartesian
velocity space by v = v‖b + v⊥ = v‖b + v⊥(cosϑ eˆx + sinϑ eˆy) so that (v ×
b) · ∂/∂v = −∂/∂ϑ. Here v‖ and v⊥ are the particle’s velocities parallel and
perpendicular to the direction of the magnetic field, b. As promised, f (0)s is
gyrotropic. Now proceeding to next order we find
Ωs
∂f (1)s
∂ϑ
=
df (0)s
dt
+ v ·∇f (0)s + (as − v · ∇us) ·
∂f (0)s
∂v
− C[f (0)s ], (1.21)
where only the gyrotropic f (0)s appears on the right hand side. It follows that
any gyrophase dependency on the right hand side is in the coefficients of f (0)s and
the derivative operators acting on it. This, and the fact that we do not need to
explicitly solve equations (1.20) and (1.21) for f (0)s and f
(1)
s is key to obtaining
the form of the Ps. Instead we can simply take velocity moments of the various
orders of the distribution function.
16 1. Introduction
1.4.2 Structure of the pressure tensor
Under an algebraic decomposition, the tensor vv can be written in terms of
gyrophase-independent (first and second terms) and gyrophase dependent (final
term) parts:
vv =
v2⊥
2
(I− bb) + v2‖bb+
∂T
∂ϑ
, (1.22)
where
T =
(
v‖b+
v⊥
4
)
(v⊥ × b) + (v⊥ × b)
(
v‖b+
v⊥
4
)
. (1.23)
Using this decomposition we can take the second moments of f (0)s and f
(1)
s to
determine P = P(0)s + P
(1)
s + . . .. The lowest order contribution to the pressure
tensor is therefore given by
P(0)s = ms
∫
d3vvvf (0)s = p
(0)
⊥,sI−
(
p(0)⊥,s − p
(0)
‖,s
)
bb, (1.24)
where
p(0)⊥,s =
∫
d3v
v2⊥
2
f (0)s , (1.25)
p(0)‖,s =
∫
d3vv2‖f
(0)
s , (1.26)
are the perpendicular and parallel pressures to zeroth order (similar expressions
exist for the first order perpendicular and parallel pressures), and
∫
d3v ∂T/∂ϑ f (0)s
vanishes upon integrating by parts and recalling f (0)s is gyrotropic. In a frame
orientated along the direction of the magnetic field b, equation (1.24) is diagonal.
Proceeding to first order in δ we find,
P(1)s = ms
∫
d3vvvf (1)s = p
(1)
⊥,sI−
(
p(1)⊥,s − p
(1)
‖,s
)
bb+ G, (1.27)
1.4 The pressure tensor 17
where, as detailed in Appendix A,
Gs =
∫
d3vf (1)s
∂T
∂ϑ
, (1.28)
=
1
4Ωs
[b×Hs · (I + 3bb)− (I + 3bb) ·Hs × b] +
1
Ωs
[bhs × b) + (hs × b)b] ,
and the tensor Hs and the vector hs are given by,
Hs =
(
p(0)⊥,s∇us +∇q
(0)
⊥,s
)
+
(
p(0)⊥,s∇us +∇q
(0)
⊥,s
)T
, (1.29)
hs =
(
p(0)⊥,s − p
(0)
‖,s
)(db
dt
+ b · ∇us
)
+
(
3q(0)⊥,s − q
(0)
‖,s
)
b · ∇b, (1.30)
and the superscript T denotes the transpose.
Here we have introduced the zeroth order parallel fluxes of the perpendicular and
parallel heat
q(0)⊥,s = bms
∫
d3vv‖
v2⊥
2
f (0)s , (1.31)
q(0)‖,s = bms
∫
d3vv3‖f
(0)
s . (1.32)
Combining equations (1.24) and (1.27), the total pressure tensor is given by
Ps = P
(0)
s + P
(1)
s
= p⊥,sI−
(
p⊥,s − p‖,s
)
bb+ Gs, (1.33)
where p⊥,s = p
(0)
⊥,s + p
(1)
⊥,s and p‖,s = p
(0)
‖,s + p
(1)
‖,s. Alternatively, splitting Ps into an
isotropic pressure ps and deviatoric stress Ds = [. . .],
Ps =psI + Ds (1.34)
=ps I +
[
1
3
(p⊥,s − p‖,s)(I− 3bb) + Gs
]
(1.35)
where ps = (p‖,s + 2p⊥,s)/3 and, because it is traceless, Gs contributes only to D.
It is also important to emphasize that G is a factor δ smaller than the gyrotropic
component of the pressure and so its significance is felt primarily at small scales
where k is large.
18 1. Introduction
Both equations (1.33) and (1.35) are accurate up to and including first order in
δ. However, as it stands neither helps close the kinetic hierarchy because we do
not yet have expressions for the pressures or heat fluxes. (In fact we shall not be
overly concerned with determining the heat fluxes in this thesis and, except in
the second half of chapter 3, we neglect them). Therefore, in the absence of such
expressions, one might ask what was to be gained by casting the pressure tensor
in this form. The answer to this question is twofold as we now explain.
1.4.3 Implications of the pressure tensor
Let us assume p⊥,s − p‖,s 6= 0, an assumption we will later show to be true
under a wide range of circumstances. Firstly, it is immediately obvious from Ds
that Ps, and therefore the momentum equation, is anisotropic, as per our initial
discussion. We therefore expect the dynamics of a magnetized plasma to differ
from those of an unmagnetized plasma.
In chapter 2 we consider exactly such a case using a collisional closure. The
anisotropy modifies the global linear behaviour of the collisional magnetized MRI
leading to an enhanced growth rate for some configurations of the magnetic field,
and viscous damping for others. Although no numerical simulations are yet pub-
lished in this regime analogous studies have been made in the local, collisionless
case by Sharma et al. (2006, 2007). They found, compared to an unmagnetized
plasma, that there is an additional anisotropic pressure stress which is well in
excess of the Reynolds stress, and comparable to the Maxwell stress. They go on
to suggest that this may moderately enhance the angular momentum transport.
Such differences, as well as many others, between a magnetized and unmagnetized
plasma count as sufficient motivation to determine under what general conditions
Ps is anisotropic.
The second reason for determining the general form of Ps is as follows, and
in chapter 3 we explore this in greater detail (including nonlinearly). Pressure
anisotropy driven instabilities, for example the firehose, can be shown to exist in-
dependently of an expression for the anisotropy, provided that (p⊥−p‖)/p < −2/β
where β = 8pinT/B2 is the plasma beta3 (Schekochihin et al., 2005). Similarly,
3The mirror instability may also be shown to exist when p⊥−p‖ > 1/β. Correctly capturing
1.5 Determining the pressure anisotropy 19
when 2q⊥,i − q‖,i 6= 0 heat fluxes are a destabilising influence and the newly
discovered gyrothermal instability can emerge (Schekochihin et al., 2010).
For both instabilities, the term Gs plays a crucial role, and so any less sophis-
ticated description of Ps that excludes it, for example double adiabatic theory
or lowest order Braginskii theory, is inadequate. Although Gs is negligible in
the long wavelength limit, on small scales it is crucial. For the firehose this is
because, in its absence, the dispersion relation is ill-posed and predicts an ultra-
violet catastrophe (infinite growth at infinitesimal wavelengths). Its effect is to
regularise the instability and this much can be shown simply from the form of
equation (1.35) (see Schekochihin et al., 2010). For the gyrothermal instability
the role of G is even more fundamental because it (the GTI) turns out to be an
FLR instability. That is, in its absence, the GTI is not captured at all.
In short, equation (1.35) is a minimal description for Ps that both captures the
pressure anisotropies inherent in a magnetized plasma whilst remaining well posed
in the face of some of the microscale instabilities that they drive4. It is also the
minimal description that captures the unstable interactions between heat fluxes
and FLR terms. It is still necessary to determine the functional forms of the
pressure anisotropy (we do so now) but at least we have some very crude idea of
its role. (For a fully self-consistent theory it is also necessary to determine the
form of the heat fluxes, but we side step that challenge here.)
1.5 Determining the pressure anisotropy
1.5.1 The CGL equations
The earliest expressions for anisotropic pressures are the CGL or double adiabatic
equations of Chew et al. (1956). Although their equations are limited (they are
for a collisionless plasma that neglects parallel Landau damping and does not
include heat fluxes) they remain a good conceptual guide to the physical processes
the growth rates of these two pressure anisotropy driven instabilities requires closures of varying
degrees of sophistication. For the firehose, a fluid closure is sufficient, whereas for the mirror,
the importance of resonant particles makes a kinetic approach necessary (Kulsrud, 1983).
4It is not clear yet whether the mirror instability is regularised by the gyroviscous correction
given by equation (1.28) and we neglect detailed discussion of this issue in this work.
20 1. Introduction
at work (Kulsrud, 1983; Snyder et al., 1997; Sharma et al., 2006). Since it is not
particularly difficult to modify these equations to include the heat fluxes and an
approximate form for collisions, we now use them to explain the origin and form
of the pressure anisotropy that is so important for what is to follow.
Determining the modified CGL equations is easiest to do by taking moments
of the Kinetic-MHD equation (Kulsrud, 1962, 1983). This equation is entirely
equivalent to taking the gyroaverage, (1/2pi)
∫ 2pi
0 dϑ, of equation (1.21) and noting
that the left hand side vanishes and, therefore, the gyroaverage of the right hand
side must be zero (this is effectively a solvability condition that enforces the
gyrophase periodicity of the left hand side).
The resultant equation, as given by Schekochihin et al. (2010), is
df (0)s
dt
+ v‖b0 · ∇f
(0)
s +
v2⊥
2
(∇ · b)
∂f (0)s
∂v‖
+ b · as
(
v‖
v
∂f (0)s
∂v
+
∂f (0)s
∂v‖
)
+
bb : ∇us
[(
v2⊥
2
− v2‖
)
1
v
∂f (0)s
∂v
− v‖
∂f (0)s
∂v‖
]
−∇ · us
v2⊥
2
1
v
∂f (0)s
∂v
= C[f (0)s ],
(1.36)
where, because v‖ = v · b(t, r) is a function of space and time, we have used the
following identities to obtain equation (1.36):
(
df (0)s
dt
)
v
=
(
df (0)s
dt
)
v,v‖
+
db
dt
· v
(
∂f (0)s
∂v‖
)
v
, (1.37)
(∇f (0)s )v = (∇f
(0)
s )v,v‖ + (∇b) · v
(
∂f (0)s
∂v‖
)
v
. (1.38)
To obtain evolution equations for the parallel and perpendicular pressures it is
not actually necessary to solve equation (1.36). Rather, as for equations (1.24)
and (1.27), it is simply a matter of taking its
∫
d3vv2‖ and
∫
d3vv2⊥/2 moments.
1.5 Determining the pressure anisotropy 21
Doing so yields,
dp⊥,s
dt
= −∇·q⊥,s − q⊥,s∇·b+ p⊥,s (bb :∇us − 2∇·us)− νs(p⊥,s − p‖,s),
(1.39)
dp‖,s
dt
= −∇·q‖,s+ 2q⊥,s∇·b− 2p‖,s (bb :∇us+∇·us)− 2νs(p‖,s − p⊥,s),
(1.40)
where we have neglected inter-species collisional heating and approximated the
full Landau collision operator by the Lorentz pitch-angle scattering operator
C[fs] = νs∂/∂ξ((1 − ξ2)/2)∂fs/∂ξ where ξ = v‖/v (see e.g. Helander & Sigmar,
2002; Rosin et al., 2010b).
Writing the shear and divergence of the flow in terms of ns and B5, equations
(1.11) and (1.16), yields the modified version of the CGL equations:
p⊥,s
d
dt
ln
p⊥,s
nsB
= −∇ · q⊥,s − q⊥,s∇ · b− νs(p⊥ − p‖), (1.41)
p‖,s
d
dt
ln
p‖,sB2
n3s
= −∇ · q‖,s + 2q⊥,s∇ · b− 2νs(p‖ − p⊥). (1.42)
These are the two equations for the components of the pressure we referred to in
section 1.3.
For the purposes of considering a collisionally dominated plasma, it is also useful
to recast them in terms of ps,
3
2
dps
dt
= (p⊥,s− p‖)bb : ∇us +
1
2
(4p‖,s + p⊥,s)∇ ·us−∇ · (q⊥,s + q‖,s), (1.43)
and the pressure anisotropy,
d
dt
(
p⊥,s − p‖,s
)
= (p⊥,s + 2p‖,s)
d lnB
dt
− (3p‖,s − p⊥,s)
d lnns
dt
−∇ · (q⊥,s − q‖,s)− 3q⊥,s∇ · b− 3νs(p⊥,s − p‖,s). (1.44)
In this form, the source terms for evolution of ps are almost recognisable from
5Equation (1.16) describes the evolution of B in terms ui not ue. However, recalling that
electron Lorentz force is the only term that appears at lowest order in the electron force balance,
equations (1.13) and (1.15), an identical electron version of equation (1.16) readily follows.
22 1. Introduction
neutral fluid theory as, from left to right, viscous heating, compressive heating
and the heat flux.
With the mathematical expressions for the evolution of the pressure anisotropy
in place, it is worthwhile considering the associated physical mechanisms.
1.5.2 Physical origins
It is a fundamental property of a Hamiltonian system that the action, defined as
∮
dqˆ pˆ around a loop, is adiabatically invariant. Here pˆ and qˆ are the canonical
momentum and position variables (not to be confused with the p the pressure).
Neglecting collisions and heat fluxes, and assuming slowly varying fields in space
and time, equation (1.18), the first adiabatic invariant of a charged particle
executing Larmor orbits in a magnetic field is given by its magnetic moment
J1 = msv2⊥/2B. Here pˆ = msv⊥ρs is its angular momentum and qˆ = ϑ its gy-
rophase angle. Since J1 is conserved for each particle individually, multiplying
the distribution function by J1 and carrying out a weighted average over velocity
space yields the macroscopically conserved quantity,
∫
d3vJ1fs
∫
d3vfs
=
p⊥,s
nsB
(Conserved). (1.45)
It follows that if B changes (at constant ns) then to conserve J1, p⊥,s must change
too, and one can even cast the equation for the magnetic moment density, p⊥,s/B
in conservative form (Sharma et al., 2006).
Similarly, if the magnetic field exhibits a local well, there is a second adiabatic
invariant associated with the particle motion, the bounce frequency J2 = msv‖lM
where lM is the distance between mirror points at which charged particles are
reflected. In this case pˆ = msv‖ + e/cA‖ where A‖ is the parallel component
of the vector potential and qˆ is a coordinate measuring the distance along the
magnetic field. By conservation of fluid mass and magnetic flux lM ∝ B/(msns)
and so, multiplying the distribution function by msJ22 and, again, integrating
1.5 Determining the pressure anisotropy 23
over velocity space yields
ms
∫
d3vJ22fs∫
d3vfs
=
p‖,sB2
n3s
(Conserved), (1.46)
which is also a macroscopically conserved quantity. Here, changes in B at con-
stant ns cause changes in p‖,s and the rate at which p⊥,s and p‖,s change with
respect to B differ, leading to pressure anisotropies (see e.g. Hazeltine & Meiss,
2003; Gurnett & Bhattacharjee, 2005).
The left hand sides of equations (1.41) and (1.42) simply represent the conserva-
tion of these two constants of particle motion, and the right hand sides, departures
from them.
The heat flux contributions q⊥,s and q‖,s may be simply understood as the parallel
fluxes of the perpendicular and parallel pressures, p⊥,s and p‖,s. Because we
have restricted ourselves to the case where ρs is much less than all other length
scales, the heat fluxes perpendicular to the magnetic field are suppressed from
our equation. For example, in a collisionally dominated plasma, the coefficient
of perpendicular thermal conductivity is a factor (νs/Ωs)2 = (ρs/λmfp,s)2 smaller
than the parallel component (Braginskii, 1965).
As well as the heat fluxes, collisions affect the pressure anisotropy and, because of
the important role they will play in this work, they require a little more discussion.
Let us consider the simple case where the heat fluxes are set to zero and the
density is constant. In this case, equation (1.41) can be written as,
1
B
dB
dt
=
dp⊥,s
dt
+ νs∆s, (1.47)
where
∆s ≡
p⊥,s − p‖,s
p⊥,s
, (1.48)
is a dimensionless measure of the pressure anisotropy. Treating the left hand side
of equation (1.47) as a prescribed drive for the right hand side, it is clear that
as p⊥,s changes, the size of the collisional term changes too. Taking the heuristic
view that p‖,s is constant (of course it in fact changes, but in the “right” way for
24 1. Introduction
this argument to hold), the steady state must be given by
1
B
dB
dt
= νs∆s. (1.49)
The implication of equation (1.49) is that for a given drive, it is collisions that
set the level of the anisotropy. This is true whether the source of the anisotropy
is flows, heat fluxes, compression or some combination of the three.
The role of collisions is to isotropise the pressure. In their complete absence, the
changing magnetic field will increase the magnitude of the anisotropy without
bound (or, actually, until other processes come into play). The velocity-space
dependency of a kinetic plasma makes the role of collisions especially impor-
tant, relative to a fluid. For example, both kinetic and fluid systems exhibit a
turbulent cascade of energy from some injection scale to a smaller, phase-space
dissipation scale where it is irreversibly turned into heat. In both cases this re-
lies on the existence of a non-vanishing dissipation mechanism, e.g. viscosity,
resistivity, collisions, but what makes collisions especially important in a kinetic
plasma, and therefore here, is their role in relaxing velocity-space structures as
well as real-space structures (Abel et al., 2008; Schekochihin et al., 2009). This
is because, fundamentally, pressure anisotropies are velocity space structures in
the distribution function.
The effect of collisions on pressure anisotropies can be formalised rigourously
using Boltzmann’s H-theorem (briefly explored in Appendix B.0.3 and see Boltz-
mann, 1872; Chapman & Cowling, 1970; Hazeltine & Meiss, 2003). The upshot of
the theorem is that the presence of collisions, even weak ones, irreversibly drives
the distribution function towards an isotropic Maxwellian in velocity space (the
state of maximum entropy)
fs =
ns
(
piv2th,s
)3/2 e
−v2/v2th,s . (1.50)
Since the pressure associated with a Maxwellian is always isotropic, i.e. p⊥,s −
p‖,s = 0, this explains why collisions limit the size of the anisotropy for a given
1.6 Summary 25
drive as shown in equation (1.49).
This, and the other processes described in this section, account for the form of
the CGL equations (1.41) and (1.42), or alternatively equations (1.43) and (1.44).
Along with the FLR corrections given by equation (1.28), these expressions rep-
resent the conceptual basis for much of the remainder of our work. Although we
will not always work with them directly, the alternatives we employ (the Bra-
ginskii equations in chapter 2 and a set of specifically tailored kinetically-derived
equations in chapter 3) can always be understood physically by reference to them.
We have derived the CGL equations and the FLR correction to the pressure ten-
sor here to give the reader a feel for the type of calculation which, later, we will
either reference or relegate to an appendix so as not to interrupt the flow of the
thesis.
1.6 Summary
In the introduction to this dissertation we have argued that understanding the
behaviour of astrophysical magnetized plasmas constitutes an interesting and, as
of yet, largely unsolved challenge.
Starting from the kinetic equation and Maxwell’s equations we have shown that
the pressure tensor is naturally anisotropic by assuming only that the magnetic
field is strong, We have shown its general form, including the effect of FLR
terms, and in the final section we showed how magnetic fields, heat fluxes, density
variations and collisions affect it through the CGL equations, (1.41) and (1.42).
These equations, along with equations for the evolution of the density (1.11),
magnetic field (1.16) and momentum (1.17), (1.28) and (1.33), are the closed set
(neglecting heat fluxes) from which we start our next chapter.
In chapter 2 we consider the global (retaining full radial derivatives) linear be-
haviour of a magnetized plasma in the presence of the shear flow of an isothermal
accretion disc (specifically the galactic disc) and a time independent background
magnetic field. We find an instability analogous to the MRI which, instead of
being mediated by the Maxwell stress, is controlled by the anisotropic pressure
stress. We work in the long wavelength limit (kλmfp  1) so our equations are
26 1. Introduction
those of Braginskii MHD and, locally, the instability is the MVI of Balbus (2004);
Islam & Balbus (2005).
We treat the system as a two parameter problem in the magnetic field strength
B and and the dimensionless combination SB = βΩˆ/νi where Ωˆ is the rotation
rate of the disc and νi is the ion collision frequency. For certain isolated field
configurations we show the local approximation yields incorrect results and for
more general configurations show the maximum growth rate of the instability
exceeds that of its unmagnetized counterpart by a factor 2
√
2 for the galactic
rotation profile. In several regimes we derive the asymptotic scaling of the maxi-
mum growth rate and Alfve´n frequency at which the growth occurs with respect
to B and SB.
In chapter 3 we tackle a separate, but related problem. Working in the short
wavelength limit (kλmfp  1) we show that the pressure anisotropies due to a
decreasing magnetic field lead to the fast growing firehose instability. Initially
we argue heuristically that its nonlinear evolution is expected to drive pressure
anisotropies towards marginal stability values, controlled by β.
This evolution is then worked out formally in an ab initio kinetic calculation for
the simplest analytically tractable case, that of the parallel firehose (k⊥ = 0)
instability. The detailed results from this are presented in Appendices B and C.
We use a particular physical asymptotic ordering that leads to the same expres-
sion for pressure tensor and anisotropy that we found in this chapter (but could
not be assumed to hold a priori). The result of the kinetic calculation is a closed
nonlinear equation for the firehose turbulence, which we solve numerically. We
find secular (∝ t) growth of magnetic fluctuations and a k−3‖ spectrum, starting
at scales greater than ρi. When a parallel ion heat flux is present, the parallel
firehose instability mutates into the FLR gyrothermal instability. Its nonlinear
evolution also involves secular magnetic energy growth, but its spectrum is even-
tually dominated by modes with a maximal scale ∼ ρilT/λmfp, (lT is the parallel
temperature gradient scale).
Finally, in chapter 4, we conclude with some thoughts on how our two sets of
results tie together and what further questions they prompt. We present an
example where the both the magnetized MRI and pressure anisotropy-driven-
1.6 Summary 27
instabilities may interact in an interesting way and offer some suggestions for
avenues of future work.
28
Part II
Magnetized MRI
29

Chapter 2
Magnetized MRI
2.1 Introduction
The formation of compact objects such as stars, planets and black holes from
accretion discs requires angular momentum transport. Turbulence driven by the
magnetorotational instability (MRI), and possibly the magnetoviscous instability
(MVI), offers a promising mechanism for this transport (Velikhov, 1959; Chan-
drasekhar, 1960; Balbus & Hawley, 1991)1. It has also been suggested that the
observed velocity fluctuations ∼ 6 km s−1 in the warm HI (neutral hydrogen)
parts of the interstellar medium (ISM) with low star formation rates may, in
part, arise from this process (Sellwood & Balbus, 1999; Tamburro et al., 2009).
The evidence for this comes primarily from numerical simulations and a wide
range of studies agree that weak magnetic fields and outwardly decreasing angu-
lar velocity profiles are an unstable combination (Balbus & Hawley, 1998; Balbus,
2003).
The most illuminating explanation for this comes from a shearing sheet analysis
in which the mean azimuthal flow of the differentially rotating disc is locally ap-
proximated by a constant angular velocity rotation plus a linear velocity shear
(Goldreich & Lynden-Bell, 1965; Umurhan & Regev, 2004). In the simplest pos-
1Careful reading of Velikhov (1959) reveals the following acknowledgement: “The author is
deeply grateful to S.I. Braginskii for assigning the problem and for valuable suggestions during
the course of its solution.” This is well over forty years before Braginskii’s work on anisotropic
viscosity was being evoked in connection with the MRI.
31
32 2. Magnetized MRI
sible setup, incompressible, isothermal, dissipationless, axisymmetric linear per-
turbations to a magnetic field with a weak vertical component, i.e. parallel to the
rotation axis of the disc, are unstable when the angular velocity decreases away
from the disc’s centre. Azimuthal velocity perturbations to fluid elements at dif-
ferent heights, tethered to each other by the magnetic field, increase (decrease)
their angular momentum. This causes them to move to larger (smaller) radii as
dictated by the gravitational field which sets the mean flow. In the frame rotating
at constant angular velocity, this motion deforms the tethering magnetic field,
provided it is not too strong, and this induces a prograde (retrograde) Lorentz
force on the outer (inner) element thus destabilising the system as it moves to
yet larger (smaller) radii. This mechanism is at the heart of the MRI.
This model and description captures much of the essential physics, but to fully
understand the MRI, or at least the framework in which the shearing sheet should
exist, a more nuanced approach is needed. In part this is because the shearing
sheet is formulated in a Cartesian coordinate system, where curvature terms that
arise naturally from the cylindrical geometry of an accretion disc are neglected.
Indeed, in the local approximation, the over-stabilising effects of hoop tension (a
curvature effect) associated with the azimuthal magnetic field are totally ignored.
Furthermore, the model predicts that the fastest growing, and therefore most
physically relevant, MRI modes have a homogeneous radial structure on the scale
of the shearing sheet in which the local approximation is made. This means that
the global disc structure, including boundary conditions, not captured by the
local approximation may have a significant effect on these large scale modes in
a way that cannot be determined locally. There are other limitations to this
method but they are beyond the scope of this thesis (Regev & Umurhan, 2008).
The extents to which these limitations matter have been investigated linearly by
ideal global analyses that take into account the full radial structure of the disc and
its boundaries (Dubrulle & Knobloch, 1993; Curry et al., 1994; Curry & Pudritz,
1995; Ogilvie & Pringle, 1996; Pino & Mahajan, 2008; Mahajan & Krishan, 2008),
and specifically for the galaxy, Kitchatinov & Ru¨diger (2004). The conclusions
of these investigations largely confirm the local picture of a large radial scale
instability driven by the differential rotation of the disc. In the nonlinear regime,
2.1 Introduction 33
a recent local investigation by Guan et al. (2009) used two-point correlation
functions to study turbulent MRI motions in ideal MHD. Their results, for zero
net magnetic flux and mean azimuthal magnetic field configurations, indicate that
density, velocity and magnetic fields decorrelate on local scales ∼ the disc scale
height (which is of order the shearing sheet simulation box size) in all directions2.
This combination of studies suggest that a local MRI analysis is generally correct
(especially in the nonlinear regime where modes are well confined to the shearing
sheet). Nevertheless, at least linearly, the regime of validity for these results must
be checked globally. It is the purpose of this work to do just that for the MRI
operating in a collisional, magnetized plasma (like the ISM).
As we have seen in chapter 1, the deviatoric stress tensor of such a plasma is
naturally anisotropic. In the collisional regime, the pressure anisotropy is equal
to the parallel gradient of the parallel component component of the rate of strain
tensor times a coefficient µB and so the stress tensor can be formulated as an
anisotropic viscosity (Braginskii, 1965). Although we do not consider it here,
thermal conduction also becomes anisotropic in a magnetized plasma, leading
to different values for the perpendicular and parallel heat fluxes. This last fact
is responsible for both the magnetothermal instability (MTI) and the heat flux
buoyancy instability (HBI) that arise in the presence of temperature gradients
(Balbus, 2000, 2001; Quataert, 2008). Ignoring these instabilities (our system is
isothermal), we concentrate on the interaction between the magnetic field and
the Braginskii viscosity in the presence of a (galactic) shear flow (Balbus, 2004;
Islam & Balbus, 2005).
We conduct a global linear stability analysis for three separate, galactically rele-
vant, background magnetic field orientations: purely azimuthal, purely toroidal
and pitched (magnetic field lines follow helical paths on cylinders of constant
radius). In agreement with earlier local studies, we find that when the field has
both a vertical and an azimuthal component, a linear instability with a real part
up to 2
√
2 times faster than the MRI emerges (Balbus, 2004; Islam & Balbus,
2005). In contrast to local studies we also find that it has a travelling wave
2Guan et al. (2009) also find that the largest correlation lengths exist between rotationally
modified sound waves excited by MHD turbulence. Such motions are filtered out under the
assumption of incompressibility, thereby ‘localising’ the MRI yet further.
34 2. Magnetized MRI
component and this varies with the size of the viscous coefficient.
In the presence of a vertical field, we also recover the standard inviscid MRI. Upon
introducing the Braginskii viscosity we find its growth rate is always reduced, and
significantly so when Re . 1. A similar effect is found for a purely azimuthal
field where we find that the Braginskii viscosity damps modes that are, inviscidly,
neutrally stable. In the case of vertical and pitched fields, the instability we find
is the Braginskii-modified-MRI, or MVI in the limit of vanishing Lorentz force,
(closely related to the collisionless MRI of Quataert et al. (2002); Sharma et al.
(2003)).
The structure of this chapter is as follows. In section 2.2 we summarise the con-
ditions in our physical example, the ISM, and introduce the model and governing
equations we use to describe it. (For the reader primarily interested in our math-
ematical treatment of the instability, rather than an astrophysical example where
it is relevant, we suggest skipping directly to section 2.2.3.) In section 2.3, we
introduce and perturb a series of equilibrium solutions which form the basis of
our global stability analysis. We combine the perturbed equations into a single
ODE which we analyse in the following sections. In sections 2.4 – 2.6 we consider
the stability of a purely azimuthal field, a purely vertical field, and a slightly
pitched field, respectively. For each case we compare and contrast the inviscid
behaviour to the behaviour in the presence of the Braginskii viscosity. In section
2.7 we provide a physical explanation of the instability and its dependence on
the magnetic field orientation. We also examine the similarities and differences
between the global and local results. After this we discuss some potential issues
in modelling the magnetized MRI within the uniform disc approximation and
fluid regimes. Finally, in section 2.8, we summarise our findings.
2.2 Galactic discs
2.2.1 Conditions in the ISM
The study of magnetized accretion discs has attracted increasing attention in
recent years in both the collisionless (Quataert et al., 2002; Sharma et al., 2003,
2.2 Galactic discs 35
2006, 2007) and collisional regimes (Balbus, 2004; Islam & Balbus, 2005; Ferraro,
2007; Mikhailovskii et al., 2008; Devlen & Peku¨nlu¨, 2010). However, a number of
fundamental questions, like those mentioned in section 1.1, remain unanswered.
Primarily, what is the non-linear fate of the MRI in a collisional magnetized
plasma, i.e. one where the pressure anisotropy is described by the Braginskii
viscosity? Does it transport angular momentum and if so, is the transporting
stress primarily viscous, Maxwell or Reynolds3 (Balbus & Hawley, 1998)? On
what scales do the most unstable modes emerge and how does this vary with
the system parameters (Curry & Pudritz, 1995; Sharma et al., 2003; Islam &
Balbus, 2005)? In what regime will local analysis become untenable and global
effects (either radial or vertical) become important (Gammie & Balbus, 1994;
Guan et al., 2009)? What effect does the presence, or absence, of a net vertical
field have given Cowling’s anti-dynamo theorem and the dissipative properties of
the Braginskii viscosity (Moffatt, 1978; Balbus & Hawley, 1998; Lyutikov, 2007)?
Could viscous heating from the Braginskii viscosity lead to secondary magnetized
or unmagnetized thermal instabilities (Balbus, 2001; Piontek & Ostriker, 2005,
2007; Quataert, 2008; Kunz et al., 2010)? Do channel solutions, or something
approaching them, emerge (Goodman & Xu, 1994)? If they do, the associated
field growth will generate pressure anisotropies that could feed new parasitic
instabilities such as the mirror. What would their effects be at this stage and in
the inevitable turbulence where the firehose instability will also arise (see chapter
3)?
Addressing these questions will require a two-pronged approach involving both
numerical and analytic studies. It may transpire that much of the existing work
on the unmagnetized MRI is directly applicable, but determining this requires
a rigorous treatment and as such, the subject of this work is a global stability
analysis for the magnetized MRI. The mathematical results in this chapter should
be applicable to any magnetized accretion disc with a radially decreasing angular
velocity profile. However, it is helpful to provide a physical example where our
results are expected to hold.
3Simulations in the collisionless regime by Sharma et al. (2006) shows that there is angular
momentum transport and the anisotropic pressure constitutes a significant portion of the total
stress (∼ Maxwell and  Reynolds).
36 2. Magnetized MRI
We consider the ISM of a spiral galaxy, a system for which a range of analytic and
numerical unmagnetized MRI studies already exist (Kim et al., 2003; Kitchatinov
& Ru¨diger, 2004; Dziourkevitch et al., 2004; Piontek & Ostriker, 2005, 2007;
Wang & Abel, 2009). This choice is motivated by the fact that our theory is
developed for a disc with an angular velocity profile that varies inversely with
radius, and this closely resembles a galactic rotation profile. Further motivation
comes from Sellwood & Balbus (1999) who have argued that in the H I region
outside the optical disc of a spiral galaxy, the velocity dispersion measurements
of ∼ 5−7 km s−1 may be driven by the differential rotation of the disc, mediated
by the MRI. (Inside the optical disc, supernovae and other processes drive the
same velocity dispersions.) Since the ISM is a varied multiphase system (cold
atomic and molecular clouds, warm atomic and ionized gas, and hot ionized gas)
both their study and ours does not pertain to much, & 1/2 by mass, of the
medium. Whether this should be rectified (to include the remaining phases)
seems unnecessary, at least for the magnetized MRI, since the interstellar clouds
are too cold to be magnetized and the hot ionized gas too buoyantly unstable for
the MRI to have any obviously significant effect (Ferrie`re, 2001).
On this basis, we concentrate on the warm ISM in the quiescent regions of a
typical spiral galaxy where the plasma is subject to a weak magnetic field and
an outwardly decreasing angular velocity profile. Generally it is magnetized and
so, on large scales, the naturally occurring pressure anisotropy can be described
in terms of the Braginskii viscosity. In order to show this and to determine the
equations that govern the system, we must set the ion parameters. We adopt
the set of fiducial values found in (Binney & Tremaine, 1988; Beck et al., 1996;
Ferrie`re, 2001). Our numbers are similar to those of NGC1058, the well studied
face-on disc galaxy considered by Sellwood & Balbus (1999). These are:
• Particle number density (the same for ion and electron)
n ∼ 0.3 cm−3. (2.1)
• Assuming ions and electrons are in thermal equilibrium viz. equation (1.14),
2.2 Galactic discs 37
the temperature is
T ∼ 5× 104 K, (2.2)
and consequently the ion thermal speed is
vth,i =
(
2Ti
mi
)1/2
∼ 3× 106 cm s−1, (2.3)
where mi is the mass of hydrogen and Ti is in erg. The ion Debye length is
λDi =
vth,i
ωDi
= vth,i
(
4pie2ni
mi
)−1/2
∼ 4× 104 cm. (2.4)
• The ion collision frequency (in seconds, assuming ni in cm−3, Ti in K and
the Coulomb logarithm Cl = 25) is4
νi ∼ 1.5nT
−3/2
i ∼ 4× 10
−8 s−1, (2.5)
and consequently, the mean free path is
λmfp =
vth,i
νi
∼ 7× 1013 cm. (2.6)
• The observed mean magnetic field strengths vary between galaxies but on
the lower end of the scale are at the present time (Beck et al., 1996),
B ∼ 2× 10−6 G (present). (2.7)
However, if the MRI is the dominant turbulence generating mechanism
in the ISM, this figure must represent the saturated state of the magnetic
field. One must assume that present field strengths have been amplified over
time; it follows that at some earlier time they were weaker (e.g. Malyshkin
& Kulsrud, 2002; Kitchatinov & Ru¨diger, 2004).
To ensure our theory is fluid-like (see section 2.2.2) the most unstable modes
4In a thermalised plasma νi is dominated by the ion-ion collision frequency (ion-electron
collisions are sub-dominant) and the full expression for this is given by νi = 4
√
pine4Cl/3m2iT 3/2
where e is the elementary charge and T is in erg (Spitzer, 1962).
38 2. Magnetized MRI
must exist above λmfp. Accordingly, we adopt the following value for the
historical “initial” field strength:
B ∼ 2× 10−10 G (initial). (2.8)
The origin of this initial value is beyond the scope of this thesis but see
the following for interesting discussion: Kulsrud (1999); Brandenburg &
Subramanian (2005).
For a plasma in this “initial” era, or the ISM in a galaxy where the magnetic
field is not saturated, the ion plasma beta is
βi =
8piniTi
B2
∼ 1.3× 109, (2.9)
the ion cyclotron frequency is
Ωi =
eB
mic
∼ 2× 10−6 s−1, (2.10)
the ion Larmor radius is
ρi =
vth,i
Ωi
∼ 1.5× 1012 cm, (2.11)
and the plasma is magnetized Ωi  νi (and will become even more so if the
magnetic field becomes stronger).
• The typical rotation rate of a spiral galaxy is
Ωˆ ∼ 5× 10−16 s−1. (2.12)
A typical radius for the outer edge of the optical disc, or indeed the coro-
tation radius between spiral arms where magnetized shear instabilities may
be important (private correspondence: Shukurov, 2010), is
r ∼ 3× 1022 cm. (2.13)
2.2 Galactic discs 39
Outside the optical disc the vertical scale-height is
H ∼ 1021 cm, (2.14)
and so the disc is thin. The measured system-scale rotation (not turbulent)
velocity is
U ∼ 2× 107 cm s−1. (2.15)
• The magnetic Prandtl number Pm, the ratio of the viscous coefficient (in
this case µB for the Braginskii viscosity) and the resistivity η, is5
Pm =
µB
η
' 7.5× 10−6
T 4
n
∼ 7.5× 1013, (2.16)
where T is in K.
• The dimensionless parameter SB that characterises the relative effectiveness
of kinetic magnetization and dynamic magnetic fields will be important for
what follows6. It can be estimated as
SB =
µBΩˆ
B2
=
βiΩˆ
2.88 νi
∼ 2.3× 10−15
ΩˆT 5/2
B2
∼ 15, (2.17)
for the conditions above (although we consider larger values later).
2.2.2 Model assumptions
On the basis of the parameters given in section 2.2.1, our galactic disc model is
as follows. Working in cylindrical polar coordinates (r, φ, z) we consider a thin,
gravitationally constrained, differentially rotating disc made up of an isothermal,
5The coefficient of the Braginskii viscosity is given by µB = 2.88 pi/minνi. The expression
for the Spitzer resistivity is given by η = 3
√
2me/pic2Cle2T−3/2, where me is the electron mass
and T is in erg (Spitzer, 1962; Braginskii, 1965).
6In the collisionless regime, the dependency of the MRI growth rate on the relative impor-
tance of kinetic magnetization and dynamic magnetic fields was first noted by Quataert et al.
(2002) and Sharma et al. (2003). Amongst other things they showed the transition between the
collisionless and collisional MRI in a local setting as a function of this parameter. We recover
their results in the local limit, see Figures 3 and 4 of Quataert et al. (2002) and Figure 2 of
Sharma et al. (2003).
40 2. Magnetized MRI
incompressible, magnetized plasma fluid whose motion is restricted by an inner
boundary at some finite radius r0. We treat the structure in the plane of the disc
globally but ignore the vertical structure perpendicular to the plane by consider-
ing scales much less than the vertical scale height of the disc H ∼ cs/Ωˆ ∼ where cs
is the isothermal sound speed (see Gammie & Balbus (1994) for global-in-z anal-
ysis). Throughout we treat the ion mass density nimi as constant and absorb it
into our definitions of the thermal pressure and magnetic field B/
√
4pinmi → B
so the latter is now equivalent to the Alfve´n speed. On the basis of equation
(2.16), we neglect η and take the Braginskii viscosity to be the only dissipative
process in our system.
Within this framework we allow for a steady equilibrium state which varies on spa-
tial scales of order r0 and perturbations to it that vary with a vertical wavenumber
k. Physically, we expect the growth rate γ ∼ Ωˆ and this will be justified by our
subsequent results. Formally these model assumptions can be expressed as a
hierarchy of time-scales which are well satisfied for our set of parameters,
1
Ωi

1
νi

1
kcs

1
Ωˆ

1
Ωˆ
r0
H
. (2.18)
From left to right, these inequalities correspond to: the plasma being magnetized,
the plasma being collisional (i.e., a fluid), the disc being uniform in the vertical
direction, and, finally, the disc being thin. The first two inequalities are illustrated
in Figure 2.1.
Whilst the uniformity-in-z ordering in equation (2.18) gives a lower bound on
the order of k we must look to the physics of the MRI to find an upper bound.
First we split the magnetic field into a vertical and azimuthal component (so as
to construct a time independent equilibrium, we neglect radial magnetic fields7).
That is, B = Bzeˆz + Bφeˆφ = B(sin θeˆz + cos θeˆφ) where θ = arctan(Bz/Bφ) is
the pitch angle of the magnetic field. The vertical Alfve´n frequency is σ = kBz =
7Although magnetic field configurations vary from galaxy to galaxy, they are commonly
found tracing the spiral arms and therefore, in the plane of the disc, predominantly azimuthal.
Beck et al. (1996) gives values for the mean in-plane field Br/Bφ ∼ 0.25 thereby justifying, to
some degree, our neglect of the radial field.
2.2 Galactic discs 41
Figure 2.1: In the long wavelength (collisional) regime plasma, 1/k  λmfp  ρi
(or equivalently 1/kcs  1/νi  1/Ωi), the pressure anisotropies that occur
naturally in a magnetized plasma can be directly related to the rate of strain
tensor, viz. equations (2.24) and (2.27).
kB sin θ, and locally we have
σ2n,m = −
1
4
dΩˆ2
dln r
(
1 +
κ2
4Ωˆ2
)
, σ2c = −
dΩˆ2
dln r
. (2.19)
The growth rate of the vertical field instability peaks for the maximizing Alfve´n
frequency σn,m and is suppressed for the critical Alfve´n frequency σc. Here,
κ2 = 2Ωˆ(2Ωˆ+ dΩˆ/d ln r) is the square of the epicyclic frequency.
This implies that for fixed Ωˆ, as B decreases, the range of unstable wavenumbers
increases. It follows that at some critically weak field, there will be an unstable
wavenumber such that kλmfp & 1 which would invalidate the fluid approximation.
Therefore our theory describes unstable modes only in the regime where
Ωˆ
cs
 k <
Ωˆ
B sin θ
and
1
λmfp
. (2.20)
This assumption will be justified later when we show the scalings between σc and
42 2. Magnetized MRI
Ω also hold globally.
Similarly, for small values of B and SB, the scaling of maximizing wavenumber
with B, holds globally and later we make use of this to guide our asymptotic
ordering.
2.2.3 Governing equations
In the long wavelength regime (kλmfp  1), the set of closed equations found
in chapter 1 can be further simplified; in particular the pressure tensor P given
by equation (1.35). Its gyrophase-dependent component G can be neglected as a
factor kus/Ωs smaller than the remaining terms (see section 1.4.2) and so:
Ps = psI + D,
= psI +
1
3
p⊥,s∆s(I− 3bb), (2.21)
and the pressure anisotropy ∆s itself can be related to the ion rate of strain tensor
as we now demonstrate.
The fact that all the frequencies, barring Ωi, are small compared to νi in the long
wavelength regime has two consequences (which are also true for electrons). First,
as shown by equation (1.49), collisions are strong enough to bring the system into
steady state faster than the changing magnetic field can bring it away. Second,
the size of the anisotropy ∆s  1 will be “small” (although as we will see later,
large enough to have dramatic effects).
In this case the ion pressure anisotropy, as given by equation (1.44) (or equation
(1.49)) yields the expression8
∆i =
1
νi
dlnB
dt
=
1
νi
bb : ∇ui ' 3
µB
pi
bb : ∇ui, (2.22)
where we have used equation (1.16) to re-write the evolution of B in terms of
the strain and the factor of 3 in the definition of µB comes from the full kinetic
calculation (Braginskii, 1965). A similar equation exists for electrons.
8Working in the slow-flow regime M 1 Mikhailovskii & Tsypin (1971, 1984) have calcu-
lated, kinetically, the analogous equation including the contribution from heat fluxes.
2.2 Galactic discs 43
Now, recall that Ti ∼ Te, ui ∼ ue and mi  me, viz. equation (1.14). From
this it follows that νi ∼ (mi/me)1/2νe and so the electron pressure anisotropy is
massively sub-dominant to the ion equivalent, i.e. |∆i|  |∆e|, but the isotropic
pressures are comparable, i.e. pi ∼ pe, (Spitzer, 1962; Helander & Sigmar, 2002).
As a a result, the total pressure tensor is given by
P =
∑
s=i,e
psI +
1
3
p⊥,s∆s(I− 3bb),
= (pi + pe) I + µB (I− 3bb) bb : ∇ui. (2.23)
This expression for the deviatoric component of the pressure tensor is simply the
lowest order contribution (in the small parameter νi/Ωi  1) of the ion Braginskii
viscosity (Braginskii, 1965). Substituting this into the momentum equation (1.17)
the equations describing the system are those of ideal MHD with the Braginskii
viscosity (e.g. Lifshitz et al., 1984; Schekochihin et al., 2005). Explicitly they
are the momentum equation (1.17) including the Braginskii stress tensor and,
for the first time, gravity; the induction equation (1.16); the continuity equation
in the incompressible limit, equation (1.11); and the viscous stress tensor itself.
Collecting them together, we have:
∂ui
∂t
+ ui · ∇ui = −∇Π +B · ∇B −∇ΦD −∇ ·D, (2.24)
∂B
∂t
+ ui · ∇B = B · ∇ui, (2.25)
∇ · ui = 0 , (2.26)
D = µB (I− 3bb) bb : ∇ui. (2.27)
Recall that the mass density is constant and has been absorbed into our defini-
tions. Here, Π = pi + pe +B2/2 is the total gas plus magnetic pressure and ΦD is
the gravitational potential we now use to set the equilibrium flow in our global
stability analysis.
44 2. Magnetized MRI
2.3 Global stability analysis
2.3.1 Equilibrium solutions
In this section we introduce equilibrium solutions to equations (2.24)-(2.27) that
describe a differentially rotating global shear flow constrained by gravity and
threaded by a magnetic field that lies on cylinders of constant radius. In these
solutions, by virtue of the scalings given in equation (2.18), it is gravity ∇ΦD
that dictates the rotation profile of the equilibrium flow Ωˆ = Ωˆ0(r/r0)−q. Ωˆ0 is
the rotation frequency at the inner boundary and q is a dimensionless measure of
the shear. We are interested in the case of q = 1, which is analytically treatable.
Physically, this corresponds to a galactic disc where, unlike the Keplerian case
of q = 3/2, the gravitational potential of the dark matter halo sets the rotation
profile (Rubin & Ford, 1970; Sofue & Rubin, 2001). In modelling this we set
the dark matter mass distribution (whose sole purpose is to ultimately set the
rotation profile) to be inversely proportional to r and, via Poisson’s equation,
the potential to be ΦD ∝ r(ln r − 1) (Mestel, 1963). In this case, the flow
ui = Ωˆ(r)reˆφ = Ωˆ0r0eˆφ does not vary with radius.
At this stage we do not specify the pitch angle θ at which the magnetic field
lies. However, for mathematical simplicity, we demand that both Bφ and Bz
are independent of radius so θ remains constant. Whilst this implies a vertical
current ∝ 1/r is associated with Bφ, a simple super-galactic magnetic field can
account for Bz. If the Bz is also bound by equation (2.20), we can ensure the
system will be unstable to the inviscid MRI (but only treatable by fluid theory if
the λmfp condition holds too). Our equilibrium flow and magnetic fields therefore
take the form
ui = Ω0r0eˆφ, B = Bzeˆz +Bφeˆφ, (2.28)
and are constant in space in and time.
It is a mathematically convenient feature of our equilibrium solutions that there is
no evolution of the magnetic field strength and therefore no pressure anisotropy.
This means that pressure anisotropy driven instabilities such as the mirror are
2.3 Global stability analysis 45
not present (Schekochihin et al., 2005). Additionally, this implies that the viscous
stress tensor D is zero, and equations (2.24)-(2.27) admit an ideal MHD solution.
This is in contrast to the case where Laplacian viscosity (or indeed resistivity) is
present, and hence we can construct an ideal MHD solution independent of any
radial flows (Kersale´ et al., 2004).
2.3.2 Perturbed equations
To determine the stability of this system we introduce small axisymmetric per-
turbations to our equilibrium solutions to equations (2.24)-(2.27). We allow the
velocity to vary as δui = δui(r) exp[ikz + γt] and similarly for the magnetic and
pressure fields. As before k is the wavenumber in the z direction and γ the growth
rate. We retain curvilinear terms arising from the cylindrical geometry but ne-
glect self-gravity and terms quadratic in the perturbed quantities. This leads to
a set of coupled equations for the components of the flow and magnetic field.
Substituting in our equilibrium plus perturbed solutions, non-dimensionalising
with respect to time-scales Ωˆ0 and length-scales r0, and writing µB = SBB2 from
equation (2.17), the components are as follows
γδur,i = 2
δuφ,i
r
−
dδΠ
dr
+ iσδBr−2
B cos θδBφ
r
−
dδZ
dr
− 3
cos2 θ
r
δZ, (2.29)
γδuφ,i = −
δur,i
r
+
B cos θδBr
r
+ iσδBφ + 3ik
sin 2θ
2
δZ, (2.30)
γδuz,i = −ikδΠ + iσδBz − ik(1− 3 sin
2 θ)δZ, (2.31)
γδBr = iσδur,i, (2.32)
γδBφ = iσδuφ,i −
δBr
r
+
B cos θδur,i
r
, (2.33)
γδBz = iσδuz,i, (2.34)
ikδuz,i = −
1
r
d(rδur,i)
dr
, (2.35)
δZ = SBB
2
[(
cos2 θ − i
k
γ
sin 2θ
2
)
δur,i
r
+
ik sin2 θδuz,i + ik
sin 2θ
2
δuφ,i
]
, (2.36)
46 2. Magnetized MRI
where, once again, σ = kB sin θ is the Alfve´n frequency that measures the tension
of the vertical field (it is always zero for θ = 0) and we have written Z = SBB2bb :
∇ui. Equations (2.29)-(2.36) form a closed set which we can combine into a single
differential equation.
Eliminating the perturbed magnetic fields, stress, pressure and the vertical com-
ponent of the velocity field yields two coupled ordinary differential equations for
δur,i and δuφ,i,
C0
dδuφ,i
dr
+ C1δuφ,i = C2
d2δur,i
dr2
C3
dδur,i
dr
+ C4δur,i, (2.37)
D0δuφ,i = D1
dδur,i
dr
+D2δur,i, (2.38)
where
C0 =
1
r
k2
(
2− 2i
γ
σ
B cos θ
)
− 3ikSBσ
2 cos2 θ cot θ,
C1 = −3ikSBσ
2 sin
2 θ
2
,
C2 = −
1
γ
(γ2 + σ2)− 3SBσ
2 sin2 θ,
C3 =
1
r
[
−
1
γ
(γ2 + σ2) + 3SBσ
2(E0 − cot
2 θ)
]
,
C4 =
1
γ
(γ2 + σ2)
(
k2 +
1
r2
)
+
1
r2
[
2k2
γ
B cos θ
(
B cos θ − i
σ
γ
)
+ 3SBσ
2E0(cot
2 θ − 1)
]
,
and
D0 = γ
2 + σ2 + 3γSBσ
2 cos2 θ,
D1 = −
i
k
3SBγσ
2 cos θ sin θ,
D2 =
1
r
[
σ
(
2iB cos θ +
σ
γ
−
γ
σ
)
+
i
k
3SBγσ
2 cot θE0
]
,
E0 = −i
k
γ
sin(2θ)
2
+ cos2 θ − sin2 θ.
2.3 Global stability analysis 47
2.3.3 Bessel’s equation and its solutions
Equations (2.37) and (2.38) can then readily be combined into a single complex
second order ordinary differential equation for δur,i, the modified Bessel equation,
d2δur,i
dr2
+
dδur,i
dr
−
(
w2 −
v2
r2
)
δur,i = 0, (2.39)
where
w2 ≡
k2
E1
(σ2 + γ2)(σ2 + γ2 + 3SBσ
2γ cos2 θ), (2.40)
v2 ≡ −
1
E1
(γ4 + a3γ
3 + a2γ
2 + a1γ + a0), (2.41)
and
E1 = (σ
2 + γ2)(σ2 + γ2 + 3SBγσ
2),
a3 = 3SBσ
2[cos2 θ(cot2 θ − 1) + sin2 θ],
a2 = 2[σ
2(1 + cot2 θ) + k2] + 6iSBσ
2 sin 2θk(1− cot2 θ),
a1 = 3SBσ
4
[
cos2 θ
(
3− 2
k2
σ2
)
+ 1− cot2 θ
]
− 8iσ2k cot θ
a0 = σ
2[σ2(1− 2 cot2 θ)− 2k2].
To obtain solutions to equation (2.39), we choose boundary conditions: an im-
penetrable wall at r0 so δur,i(r0) = 0, and the requirement that r
1
2 δur,i decays at
infinity (see Furukawa et al. (2007) for an inviscid treatment that includes the
shear singularity at the origin). In this case, its eigenfunctions are the modi-
fied Bessel functions of the second kind Kiv(wr), of argument wr and order v,
(Watson, 1944; Abramowitz & Stegun, 1964). The spectrum of solutions is dis-
crete and we index v (and the corresponding γ) with n: vn and γn9. In the
special case when w (or γn from equation (2.40)) is real (in general it is complex)
the problem is Sturm-Liouville and vn is an infinite ordered set of eigenvalues
9For notational simplicity we retain the notation γn for the growth rate even in the degen-
erate limits where there is no dependence on n.
48 2. Magnetized MRI
v0 < v1 < v2 . . . < v∞. To determine vn and therefore γn, it is necessary to solve,
Kivn(w) = 0. (2.42)
From solutions to this and equations (2.53) and (2.54), the full set of flow, mag-
netic and pressure fields can be constructed. Although they can easily be found
from equations (2.29)-(2.36) we list the functional form of these fields here for
completeness,
δur,i = Kivn(wr) exp[ikz + γnt], (2.43)
δuφ,i =
D1
D0
dδur,i
dr
+
D2
D0
δur,i, (2.44)
δuz,i =
i
k
1
r
d(rδur,i)
dr
, (2.45)
δBr = i
σ
γn
δur,i, (2.46)
δBφ = i
σ
γ
D1
D0
dδur,i
dr
+
[
i
σ
γ
D2
D0
+
1
r
(
B cos θ
γ
− i
σ
γ2
)]
δur,i, (2.47)
δBz = −
σ
γn
1
kr
d(rδur,i)
dr
, (2.48)
δΠ =
(γ2 + σ2)
k2γ
1
r
d(rδur,i)
dr
− SB
σ2
k2
(csc2 θ − 3)×
[(
D2
D0
− sin2 θ
)
dδur,i
dr
+
(
E0
r
+ ik
sin 2θ
2
D1
D0
)
δur,i
]
. (2.49)
In Figure 2.2 we consider the case θ = pi/2 for which γn, and therefore vn, is real
and show three particular Kivn(wr) which satisfy equation (2.42).
2.3.4 Specific configurations
Equations (2.40)-(2.42) provide a complete description for γn as a function of k
and the system parameters SB and σ,B, θ. (Although one of these three quan-
tities is redundant, it is both notationally and conceptually convenient to retain
them all.) Whilst it is entirely possible to determine γn for any set of parameters,
in general it must be done numerically.
However, for the most physically relevant magnetic field configurations, the prob-
2.3 Global stability analysis 49
2 3 4 5
r
0.2
0.4
0.6
0.8
1.0
K
iv
n
HwrL
Figure 2.2: Radial component of the flow, equation (2.43), for the fastest grow-
ing n = 0 branch MRI modes with θ = pi/2, B = 0.02 and SB = 10, (solid),
100 (dashes), 1000 (dots/dashes). The associated (purely real) growth rates
are γ0,m = 0.36, 0.26, 0.14 and the Alfve´n frequency at which they occur are
σn,m = 0.67, 0.52, 0.32. The mode width is of order 1/w, so as SB increases, so
does the mode width, viz. equation (2.40).
lem becomes, in part, analytically tractable. These cases, a purely azimuthal field
θ = 0 (in the galactic plane), a purely vertical field θ = pi/2 (a super-galactic
field) and a slightly pitched field, θ  1 (very slightly out of the galactic plane),
exhibit categorically different behaviour because θ → 0, pi/2 are singular limits.
In the first case, linear perturbations are damped. In the second, the system is
unstable to the MRI but the Braginskii viscosity can reduce the maximum growth
rate γn,m below the Oort-A value maximum (|d ln Ωˆ/d ln r|)/2 = 1/2 (Balbus &
Hawley, 1992). In the third case, the system is unstable to the MVI with a growth
rate approaching γn,m =
√
|d ln Ωˆ2/d ln r| =
√
2, even for asymptotically small θ
(Balbus, 2004).
To show this in the azimuthal case we use a variational energy principle a` la
Chandrasekhar (1961), whereas in the vertical field case we make use the asymp-
totic properties of Kivn(w). In the pitched field case we make use of the smallness
of the pitch angle to order the system parameters and variables and by doing so,
we reduce equations (2.40) and (2.41) to a form that can be treated in the same
manner as the vertical case. We present the details of these calculations now.
50 2. Magnetized MRI
2.4 Azimuthal field, θ = 0
The stability of inviscid axisymmetric perturbations to a purely azimuthal mag-
netic field in the presence of a shear flow are well known. When the shear flow has
q < 2 and a magnetic field of the form B = Br−deˆφ with d > −1 the system is
always stable (as ours is). When only one criterion is met (depending on the form
of the fields) the system may still remain stable by the modified Rayleigh crite-
rion (Rayleigh, 1916; Michael, 1954; Chandrasekhar, 1961; Coppins, 1988). (It
is worth noting however that global non-axisymmetric perturbations are unsta-
ble, Ogilvie & Pringle (1996).) Do these results for our inviscidly stable system,
R(γn) = 0, persist in the presence of the Braginskii viscosity? The answer is no.
Setting θ = 0 in equations (2.39)-(2.41), whereby σ = 0, and making a change of
variables, δur,i → r−1/2δur,i, we obtain the simple expression,
γ2n
d2δur,i
dr2
−
(
γ2nQ1 + γnQ2 +Q3
)
δur,i = 0 (2.50)
where
Q1 =
3
4
1
r2
+ k2, Q2 = 3SB
B2k2
r2
, Q3 =
2k2
r2
(1 +B2). (2.51)
We now multiply equation (2.50) by the complex conjugate of the perturbed
radial velocity δur,i† and integrate between the inner boundary r0 and infinity.
Noting that the first term in equation (2.50) can be integrated by parts and that
boundary terms vanish, we find
γ2n
((
dδur,i
dr
)2
+Q1|δur,i|2
)
+ γnQ2|δur,i|2 +Q3|δur,i|2 = 0, (2.52)
where |δur,i|2 =
∫∞
r0
dr δur,iδur,i† > 0 is positive definite (and similarly with
(dδur,i/dr)2).
Equation (2.52) is a quadratic in γn whose roots depend crucially on SB. When
SB = 0 (the inviscid limit), γn is purely imaginary (neutrally stable, travelling
waves), whereas when SB > 0 the result is quite different. In this case R(γ) <
0 ∀ SB and the only question is whether perturbations are purely damped, or
2.5 Vertical field, θ = pi/2 51
damped and travelling.
It follows that if the system is stable in the absence of the Braginskii viscosity,
it remains so in its presence. From a modelling perspective this suggests that
any axisymmetric simulation of a magnetized accretion disc in the presence of
a purely azimuthal field will remain laminar and therefore not accrete. We will
now show the same is not true for a vertical field.
2.5 Vertical field, θ = pi/2
We now consider the case of a purely vertical magnetic field, B = Bzeˆz i.e.
θ = pi/2. Equations (2.40) and (2.41) again reduce to a relatively simple form10:
w2 ≡ k2
γ2n + σ
2
γ2n + σ2 + 3SBγnσ2
, (2.53)
v2n ≡ −
[
2k2
γ2n + σ2 + 3SBγnσ2
(γ2n − σ
2)
(γ2n + σ2)
+ 1
]
. (2.54)
In section 2.5.2 we determine the values of γn that satisfy our boundary condi-
tions. We find that for each n there are four branches, one of which is a purely
growing mode with a growth rate proportional to the shear of the disc. This is
the MRI in the presence of Braginskii viscosity (two of the other roots are com-
plex and describe damped Alfve´n waves whilst the final root is a purely damped
mode). However, before we detail these results, let us first consider the inviscid
case.
In this case and for the subsequent configurations we adopt the following format.
After stating the governing equations we first present our numerical result, and
then, our analytic results in three asymptotic limits: the weak field limit, the
highly oscillatory limit and, where relevant, the highly viscous limit.
52 2. Magnetized MRI
Figure 2.3: Top panel: The fastest growing mode γ0 of the inviscid global MRI as
a function of σ for a several values of B (recall, length-scales and time-scales are
non-dimensionalised with respect to the radius of the inner boundary r0 and the
frequency at it Ωˆ0). In the weak field limit, we recover the behaviour of the local
MRI (dotted line). Bottom panel: Maximum growth rate γ0,m as a function of
B. As B → 0 the fastest accessible growth rate asymptotes to the local Oort-A
maximum = 1/2 (dotted line) in a galactic rotation profile. The dashed line is the
weak B asymptotic solution for the fastest growing mode, as given by equation
(2.63). Inset: The Alfve´n frequency σ0,m at which γ0,m occurs as a function of
B; its maximum
√
3/4 (dotted line) and its asymptotic scaling (dashed line and
equation (2.63)). Note: In the weak field limit, any magnetic field with θ 6= 0
is unstable to perturbations with the same γn as perturbations to a vertical field
with the same strength as the vertical component of the pitched field, see section
2.6.3
2.5 Vertical field, θ = pi/2 53
2.5.1 Vertical inviscid MRI
The case of the inviscid global MRI with a vertical magnetic field has been ex-
amined in earlier works and our results are in agreement with these studies e.g.
(Dubrulle & Knobloch, 1993; Curry et al., 1994).
When SB = 0, equation (2.53) simplifies to w = k and therefore equation (2.54),
which also simplifies, becomes real too. The problem is reduced to11
Kivn(k) = 0 (2.55)
with
v2n ≡ −
[
2k2
γ2n + σ2
(γ2n − σ
2)
(γ2n + σ2)
+ 1
]
. (2.56)
For v2n < 0 we note that Kivn(k) is a sign definite function of monotonically
decreasing magnitude with respect to k and therefore it has no nontrivial zeros.
For equation (2.55) to be satisfied it follows that v2n > 0; thus explaining our sign
convention for v2n in equation (2.39). A simple examination of equation (2.56)
then reveals that γ2n is bounded from above by σ
2. Further detail requires a more
in depth investigation.
Numerical results
We use Newton’s method (implemented through Mathematica’s root finding al-
gorithm) to determine vn, and therefore γn. We find a real instability, the global
MRI, with γn ∼ the shear rate. Its fastest growing branch of unstable modes γ0,
shown in Figure 2.3, are those that correspond to the lowest eigenvalue v0, i.e.
10Although we do not show it here, the entire analysis that follows can be reproduced if
compressive effects, i.e. sound waves, are included. In the limit of infinite sound speed the
incompressible results we present here are recovered.
11Equations (2.39) and (2.56) correspond to equations (3.4) and (3.5) of Curry et al. (1994)
with a = 1 and equations (30) and (31) of Dubrulle & Knobloch (1993).
54 2. Magnetized MRI
n = 0, the first zero of Kivn(k). This branch peaks at
γ0,m =
1
2
(2.57)
σ0,m =
√
3
4
. (2.58)
Even for strong magnetic fields we find, in agreement with equation (2.19), that
σ0,m ∼ 1 and so, when B . 1, unstable wavenumbers are of order k ∼ 1/B & 1
(physically, smaller scale modes are suppressed by magnetic tension). Since the
mode width is of order 1/k (see Figure 2.2), curvature effects in this regime that
occur on this scale become pronounced. The result is that the maximum occurs
for k  1 when the mode is localised on the inner boundary (Curry et al., 1994).
By making use of the asymptotic properties of the modified Bessel function it is
relatively easy to recover this result for B  1 analytically and we do so now.
Weak field limit
The fastest growing MRI mode has a wavenumber k ∼ 1/B and so we find, k  1
in the weak field limit. It has been shown by Cochran (1965) and Ferreira & Sesma
(1970, 2008) and references therein, that in this asymptotic limit, |k|  1, the
zeros of Kivn(k) are given by
vn ∼ k + an2
− 13k
1
3 + . . . , (2.59)
where an is the modulus of the nth real negative zero of the Airy function Ai
(Abramowitz & Stegun, 1964) and omitted terms are of the form kb with b <
1/3. This result can be used to find the asymptotic-in-k form of γn by inverting
equation (2.56) to form a bi-quadratic
γ4n + 2
(
σ2 +
k2
1 + v2n
)
γ2n + σ
2
(
σ2 − 2
k2
1 + v2n
)
= 0, (2.60)
into which we substitute equation (2.59). Retaining the first two terms of the
large k limit of vn yields
γ4n + 2
(
σ2 + 1− snk
− 23
)
γ2n + σ
2
(
σ2 − 2
(
1− ssnk
− 23
))
= 0, (2.61)
2.5 Vertical field, θ = pi/2 55
where sn = an2
2
3 . To draw an analogy with the local approximation, the n that
indexes the number of zeros in the domain is somewhat like a radial wavenumber,
section 2.7.2.
Solving equation (2.61) for the fastest growing mode and its associated Alfve´n
frequency σn,m, we find
γn,m = 1/2
(
1− snk
− 23/2
)
, (2.62)
σn,m =
√
3/4
(
1− snk
− 23/2
)
, (2.63)
which is valid for snk−
2
3  1 and, as expected, peaks at γ0,m = 1/2. For
k ∼ 1/B > 10, there is excellent agreement between these results and the full
numerical solution for n = 0, Figure 2.3.
For fixed k, increasing n decreases γn,m, confirming the numerical results of the
previous section. The mode with the least radial structure – the n = 0 mode–
is the most unstable. Extrapolating from this, we would expect γn,m to always
decrease with n. However, the results of this section are limited to values of
snk−
2
3  1. To determine γn,m outside this regime, it is appropriate to take the
large n limit of the governing ODE.
Highly oscillatory limit
Let us consider solutions to the governing ODE, equation (2.39), in the limit
vn  kr = x (corresponding to large n since vn is an unbounded monotonically
increasing series in n). Here we have introduced the scaled coordinate x (this is
effectively a WKB approach to solving equation (2.39) using the small parameter
x/vn, (Dubrulle & Knobloch, 1993; Ogilvie, 1998)). In this case, the expansion
given by equation (2.59) no longer applies (n is large and so higher order terms
must be included). However, values of vn satisfying our boundary conditions do
still exist. In this limit, the equation governing these solutions is
d2δur,i(x)
dx2
+
v2n
x2
δur,i(x) = 0. (2.64)
56 2. Magnetized MRI
Demanding that δur,i is real, this has solutions
δur,i =
√
x cos(vn lnx), (2.65)
and to ensure cos(vn lnx) = 0 at r = 1, it must satisfy the boundary condition
vn ln k =
(
n+
1
2
)
pi. (2.66)
This constraint on vn determines the spectrum of solutions just as equations
(2.42) and (2.59) do. Neglecting the factor of 1/2 since n is large, this can be
inverted to yield a dispersion relation,
γ4n + 2
[
σ2 +
(
k ln k
npi
)2
]
γ2n + σ
2
[
σ2 − 2
(
k ln k
npi
)2
]
= 0, (2.67)
valid in the limit vn  kr. The maximum growth rate associated with this
equation decreases as n increases, and the Alfve´n frequency at which it occurs
are given by
γn,m = k/2vn = k ln k/2npi, (2.68)
σn,m =
√
3/4 k/vn, (2.69)
so modes with more radial structure are less unstable.
In the large n limit, on small enough scales, the mode structure given by equation
(2.65) is especially simple. Reverting to r as our radial coordinate, we consider
a small region at some radius r1 of order 1 and departures from it such that
|r − r1| < δr where δr  1. In this limit we can expand equation (2.65) about
r = r1 and retaining the leading order terms in δr we find,
δur,i = A
(
1 +
1
2
r
r1
)
cos
(
vn
r1
r + ξ
)
, (2.70)
where A =
√
kr1 and ξ = vn ln kr1.
Since vn  kr1 (our initial assumption) equation (2.70) describes rapidly os-
cillating modes with frequency vn and a slowly varying amplitude ∝ A. For
2.5 Vertical field, θ = pi/2 57
sufficiently large vn, the variation in amplitude can be neglected and equation
(2.70) describes plane wave solutions whose growth rate decreases with n. In
section 2.7.2 we contrast these results with a local plane wave analysis. We find
that there are subtle differences associated with the spectrum of modes permitted
by our boundary conditions.
Before we do so, let us first consider the behaviour of the vertical MRI in the
presence of Braginskii viscosity.
2.5.2 Vertical MRI with Braginskii Viscosity
In the case of the Braginskii viscosity and a vertical field, our system is now
governed by equations (2.42), (2.53) and (2.54). Unlike the inviscid case, we can
no longer guarantee the reality of w as γn can, and indeed sometimes does, take
complex values. This is because the problem is generally not of Sturm-Liouville
form and so the roots of Kivn(w) are complex. However for the modes of interest
to us, this turns out to be irrelevant. We find, both numerically (next section)
and analytically (in section 2.5.2) that the only solutions with R(γn) > 0 are
purely real.
Numerical results
Again making use of Newton’s method we start from the solutions to the inviscid
problem found in section 2.5.1 and proceed to map out the two-dimensional (B
and SB) parameter space of γn for moderate w.
There are four mode branches of which three are stable and one is unstable.
Considering only the unstable branch, which can be traced from the inviscid
MRI, we again find that the growth rate is maximised for the modes the minimum
number of zeros (i.e. radial structure), γ0. Considering only these fastest growing
modes, in the limit SB  1 the effect of the Braginskii viscosity is negligible and
γ0 approaches values of the inviscid MRI. However, in the opposite limit where
SB  1, the maximum accessible γ0 is significantly reduced and the value of σ
at which it occurs is shifted to smaller values, as shown in Figure 2.2 and 2.6.
We find, as in the inviscid case, the characteristic radial scale of the modes is 1/w.
58 2. Magnetized MRI
Figure 2.4: The fastest growing mode γ0 of the global MRI with Braginskii
viscosity for a two different weak, B = 1/1000, field configurations and a range of
SB: Top panel: Vertical Field: For a given σ the growth rate is always less than
the inviscid case (dotted) which is recovered as SB → 0. For SB →∞, both the
peak growth rate and tension at which it occurs, γ0,m, σ0,m → 0. However, the
critical Alfve´n frequency σc remains unaltered. Bottom panel: Slightly pitched
field: In contrast to the purely vertical case, for a given σ the growth rate is
always greater than the inviscid case (dotted) and less than the
√
2 maximum
of the Lorentz force free limit (Balbus, 2004). When SB → 0, as expected, γ0
approaches the inviscid growth rates. In the opposite limit when SB → ∞, like
the vertical case, σ0,m → 0. However, unlike the vertical case, the corresponding
γ0,m →
√
2. In this case σc is imperceptibly modified by the effects of magnetic
hoop tension. In the formal limit that this can be neglected, B → 0, σc remains
unaltered.
2.5 Vertical field, θ = pi/2 59
In the limit where SBB2 & 1, modes that would be extremely well localised to
the inner boundary r0, were they inviscid, can exhibit significant radial structure.
This is shown in Figures 2.2 and 2.5.
Contrasting the viscous and inviscid results, we find the fastest growing inviscid
modes for weak magnetic fields with θ = pi/2 can be summarised by
SB → 0, B → 0 γn,m → 1/2, σn,m →
√
3/4, (2.71)
whereas the highly (Braginskii) viscous modes have,
SB →∞, B → 0 γn,m → 0, σn,m → 0. (2.72)
As we now show, these results can be recovered analytically.
Weak field limit
As in section 2.5.1, if B  1 the typical wavenumber at which the γn peaks is
k ∼ 1/B  1. Assuming momentarily that in this limit we also have |w|  1,
we can investigate the behaviour of vn. Fortunately, this is relatively easy to do
since the results of Cochran (1965) and Ferreira & Sesma (2008) also hold for
large complex w:
vn ∼ w + an2
− 13w
1
3 + . . . , w ∈ C, (2.73)
and so we proceed without assuming anything about the reality of w or, therefore,
γn. Here an and the terms omitted are of the same form as in equation (2.59).
As before, we invert equation (2.54) to obtain an expression for γn that takes the
form
γ4n+3SBσ
2
(
γ3n + σ
2γn
)
+2
(
σ2 +
k2
1 + v2n
)
γ2n+σ
2
(
σ2 − 2
k2
1 + v2n
)
= 0, (2.74)
where vn(w) is given by equation (2.73) and w is given by equation (2.53). The
solutions to the ensuing polynomial must be consistent with our earlier assump-
tion, namely that k  1 implies w  1.
60 2. Magnetized MRI
When SB  k2 it is simple to show that this requirement is met. Retaining only
the leading order term in equation (2.73) and substituting this into equation
(2.74) we find, neglecting factors of order unity compared to k2,
γ4n + 3
SB
k2
σ2
(
γ3n + σ
2γn
)
+ 2
(
σ2 + 1
)
γ2n + σ
2
(
σ2 − 2
)
= 0. (2.75)
The terms proportional to SB are small corrections to the inviscid result and so
γn,m ∼ σn,m ∼ 1 by equation (2.63). Therefore w2 ∼ k2/(1 + SB) from which it
follows, as necessary, w  1.
A more restrictive assumption on the size of SB allows us to analytically recover
radial structure in equation (2.74). Retaining the two leading terms in equation
(2.73) and expanding w as a Taylor series in small SB we can write
v2n ∼ k
2 (1− χ+ . . .) + snk
4
3
(
1−
4
3
χ+ . . .
)
+ . . . , (2.76)
where
χ =
3SBσ2γn
γ2n + σ2
 1. (2.77)
Assuming that χ ∼ SB ∼ snk−
2
3  1 we can neglect all terms not shown in
the first set of brackets and all but the first term in the second set. This is the
minimum needed to incorporate the radial structure into our system in the weak
field limit.
Substituting this approximation to equation (2.76) into equation (2.74) and re-
arranging yields
γ4n + 3SBσ
2snk
− 23 (σ2 + γ2n)γn + 2
(
σ2 + 1− snk
− 23
)
γ2n+
σ2
(
σ2 − 2
(
1− snk
− 23
))
= 0. (2.78)
Considering the γn solutions to this confirms the results of the numerical inves-
tigation of section 2.5.2. There are four branches to γn; two purely real (the
unstable one corresponding to the MRI in the presence of Braginskii viscosity)
and two complex ones with R(γn) < 0 corresponding to damped Alfve´n waves.
2.5 Vertical field, θ = pi/2 61
To determine the variation of γ0,m with n and SB we differentiate equation (2.78)
with respect to σ and set this to zero to obtain an equation constraining the
turning points of γn. Combining this with the equation (2.78) and assuming
a series solution for γn,m and σn,m in the small parameter k−
2
3 , yields a series
of simple algebraic equations for the coefficients of the series expansion. These
coefficients can be obtained order by order and in section 2.6.3 we present a
worked example of this procedure. However, for now we simply state the result
for the maximum growth rate and Alfve´n frequency:
γn,m =
1
2
(
1− snk
− 23
(
1
2
+
9
16
SB
))
, (2.79)
σn,m =
√
3
4
(
1− snk
− 23
(
1
2
+
33
48
SB
))
. (2.80)
Neglecting radial structure (sufficiently large k), we can also determine σc by
solving equation (2.78) and setting γn = 0. Because dissipative terms containing
SB are proportional to some non-zero power of γn, like the case of a local Laplacian
viscosity, σ2c = d ln Ωˆ
2/d ln r (as in the inviscid case), thereby justifying the
assumptions in equation (2.20) (Pessah & Chan, 2008). These results are in
complete agreement with the numerical analysis preceding this.
In expanding equation (2.74) we assumed SB  k2 initially (and then SB ∼ snk−
2
3
when we introduced radial structure). What happens when SB & k2 is less clear
because k  1 no longer necessarily implies w  1 and so equation (2.73) need
not hold. In fact, as we now show, assuming it does hold leads to a scaling of σn,m
with SB that does not match the full numerical solution (although the scaling we
find for γn,m does).
Large SB limit
Assuming that vn ∼ w, equation (2.75) governs the behaviour of γn. In the large
SB limit, the dominant balance for the unstable mode is between the γn and the
constant terms so
γn =
2− σ2
3SBB2
, SB  1 (2.81)
62 2. Magnetized MRI
1 2 3 4 5 6
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
rr0
zr
0
-4 -2 0 2 4
-4
-2
0
2
4
cosΦ rr0
rr
0s
inΦ
Figure 2.5: The flow fields associated with unstable Braginskii-MRI modes for
a vertical magnetic field as given by equations (2.43)-(2.45) and the numerical
results of section 2.5.2. The magnetic field is weak B = 0.02 and SB = 1000 is
large. The mode shown is the fastest growing mode for this set of parameters,
it has γ0,m = 0.14 and σ0,m = 0.32 (significantly less than if the system were
inviscid). Left panel: Spanning the disc midplane and projected onto the r − z
plane: As required by the impenetrable inner boundary condition at r = 1, the
radial component of the flow becomes zero here. Relative to the flow field, the
r − z components of the magnetic field (not shown) are a factor σ0,m/γ0,m larger
and displaced vertically down by half a wavelength = pi/k but otherwise identical
(see equations (2.46) and (2.48)). Right panel: Face-on view of the disc, r − φ
plane, at the z/r0 = 0 midplane: At the inner boundary (black ring) the radial
component of the flow vanishes. Compared to the flow field, the φ component of
the magnetic field (not shown) is a factor 1/2(σ0,m/γ0,m− γ0,m/σ0,m) smaller and
displaced vertically down by half a wavelength = pi/k (see equation (2.47)).
2.5 Vertical field, θ = pi/2 63
where we have written k = σ/B. This implies γn is maximised as σ → 0.
However, this limit is inconsistent with the assumption k  1 implicit in vn ∼ w.
To determine the fastest growing mode we differentiate equation (2.75) with
respect to σ and set this equal to zero, yielding an equation governing the turning
points of γn. This equation is
γ2n +
3
2
SBB
2γn + σ
2 − 1, (2.82)
and eliminating σ between it and equation (2.75) gives a quadratic from which
we determine:
γn,m =
2
4 + 3SBB2
=
2
4 + 3/Re
, (2.83)
σn,m =
√
12 (1 + SBB2)
(4 + 32SBB2)
2 =
√
12 (1 + 1/Re)
(4 + 3/Re)2
. (2.84)
where SBB2 = 1/Re is the inverse of the Reynolds number. For SBB2  1 the
dominant contribution to γn,m and σn,m is the same as the inviscid case (ignoring
snk−
2
3 as we have done here, see equation (2.63)). Furthermore, in full agree-
ment with the numerical solution, it is only when SBB2 & 1 that the damping
effects of the Braginskii viscosity become pronounced. It is hardly surprising that
significant damping only occurs when Re . 1.
In Figure 2.7 the scalings given by equation (2.83) are plotted against the full
numerical solutions. Our analytic expression for γn,m is in good agreement, but
in the large SBB2 regime, the numerical solution for σn,m appears to decrease
more like with S−3/4B than the scaling of S
−1/2
B predicted here. This discrepancy
indicates that it was not correct to assume vn ∼ w in the large SBB2 regime,
despite the fact that γn is correctly captured. Unfortunately, as a result, we are
unable to produce a better analytic estimate for σn,m.
Leaving the large SB limit aside for now (we return to it in section 2.6.3), let us
again consider the highly oscillatory limit, his time with Braginskii viscosity.
64 2. Magnetized MRI
Highly oscillatory limit
Our numerical results suggest that, like the inviscid case, as n increases the (real)
growth rate of the MRI in the presence of the Braginskii viscosity and a vertical
field, decreases. We investigate the large n limit here by following the methods
developed for the inviscid case in Section 2.5.1. Here, however, we replace k by
w and recall that vn is now given by equation (2.54).
In the limit SB  1 (so equation (2.77) holds) we can write w = k(1−χ/2 + . . .)
and the ensuing dispersion relation is
γ4n + 3SBσ
2(σ2 + γ2n)γn + 2
[
σ2 +
(
k ln k
npi
)2
]
γ2n+
σ2
[
σ2 − 2
(
k ln k
npi
)2
]
= 0, (2.85)
valid in the limit vn  kr  1. As in section 2.5.2 (and detailed in 2.6.3), we
can obtain a series solution for γn,m and σn,m, this time using the small expansion
parameter k/vn = k ln k/npi. The result of such an expansion is
γn,m =
1
2
k
vn
(
1−
9
16
SB
)
, (2.86)
σn,m =
√
3
4
k
vn
(
1−
33
48
SB
)
, (2.87)
and in the limit SB → 0, we recover the inviscid results of section 2.5.1.
That is, in the highly oscillatory limit, the effect of SB is to decrease both γn,m
and σn,m (as is the effect of increasing n). Furthermore, in a small enough region
about r1, δur,i the modes take on the same sinusoidally varying form as the
inviscid modes (as do all the other field and flow components; the pressure,
however, does not).
2.5.3 Vertical field summary
For a vertical magnetic field, the inviscid and weakly (Braginskii) viscous MRI
are very similar. In both cases a real instability bounded by the Oort-A value of
2.6 Pitched field, θ  1 65
1/2 emerges. Its growth rate is maximised for the n = 0, and the mode structure
(apart from the pressure) is the same.
Outside this regime, where SBB2 ≡ 1/Re & 1, the damping effects of the Bragin-
skii viscosity become pronounced. We found that in this regime γn,m, σn,m → 0
as SB → ∞. Although this is not surprising, since ultimately the Braginskii
viscosity is a dissipative process, it is somewhat novel since in the unperturbed
state, the system behaves ideally for all values of SB. That is, there is no direct
Laplacian viscosity equivalent. We discuss this further in section 2.7.
The overall picture is of a large (radial) scale instability whose growth rate is
maximised when B is weak and reduced when B or SBB2 is strong. For the ISM
parameters of section 2.2.1 this reduction is unlikely to be particularly important.
However, at an earlier, hotter epoch when the collision frequency, and therefore
Re, was less, this effect may have been more significant.
From the numerical modelling perspective, one should certainly be aware of the
damping effects of the Braginskii viscosity for the vertical field MRI since, in the
linear stages, the growth rate of the instability will be reduced. Of course, after
the initial stages the instability will deform the background field so it acquires
an azimuthal component. In this case, as we now show, the behaviour of the
instability is categorically different.
2.6 Pitched field, θ  1
When the magnetic field has both a vertical and an azimuthal component, the
perturbed Braginskii stress tensor (equation (2.36)) exerts an azimuthal “tension”
force on separating plasma elements, equation (2.44). For arbitrary θ, the system
is neither Sturm-Liouville (equation (2.39) is complex) nor amenable to the kind
of polynomial inversion used in sections 2.5.1 and 2.5.2. If we assume that θ  1,
matters simplify considerably. Physically, this choice of pitch angle represents
the most realistic non-isolated galactic magnetic field configuration (Beck et al.,
1996). Mathematically, as we now show, it is a singular limit that constitutes the
stability threshold between actively damped modes (section 2.4) and an unstable
configuration that grows even faster than the inviscid global MRI (to which the
66 2. Magnetized MRI
following stability threshold also applies):
Damped : θ = 0, Unstable : θ ∼ . (2.88)
Here   1 is a small parameter with respect to which we order the pitch angle
and the remaining quantities in equation (2.39): γ, k, d/dr, B and SB.
There are many different, self-consistent ways of ordering the parameters of the
system, each corresponding to a different physical regime. In this work we con-
centrate on the order 1 magnetic field limit but, so as to compare our results
with the vertical field case, restrict its values to B ≤ 0.2 which we took to be the
maximum field strength consistent with our uniform disc assumption (equation
(2.18)).
In this section, we first introduce our small angle orderings and equations be-
fore examining, first the pitched inviscid MRI and then the pitched MRI in the
presence of the Braginskii viscosity.
2.6.1 Ordering assumptions
Letting θ ∼  we can retain only the first few terms in a series expansion of our
trigonometric functions,
cos θ = 1−
θ2
2
+O(θ4), sin θ = θ −
θ3
6
+O(θ5), (2.89)
and we assume the magnetic field strength B ∼ 1 which allows us to include the
effects of magnetic tension. Similarly, we order SB ∼ 1 to incorporate the effects
of kinetic magnetization (the Braginskii viscosity).
To retain the effects of magnetic tensions σ = kB sin θ to lowest order (failing to
do would remove the high wavenumber cutoff and leave the equations ill-posed),
we set k ∼ 1/. Then, in anticipation of unstable modes that grow at the shear
rate, we order γ ∼ 1 too.
It now simply remains to order the radial scale of variations d/dr. Examination
of equation (2.40), in conjunction with the orderings given above, indicates that
w (the boundary layer width of an unstable mode) is dominated by k. This
2.6 Pitched field, θ  1 67
implies that the scale of the radial derivative must be comparable, and so we
order d/dr ∼ k ∼ 1/.
In all, the relevant scalings are,
θ ∼ , γ ∼ σ ∼ B ∼ SB ∼ 1, k ∼
d
dr
∼ −1. (2.90)
Applying these scalings to equations (2.39)-(2.41) and retaining only the lowest
order terms in , we find δur,i is still governed by Bessel’s equation and γn is
determined by
w2 ≡ k2 (2.91)
v2 ≡ −
1
E1
(γ4 + b3γ
3 + b2γ
2 + b1γ + b0), (2.92)
where now
E1 = (σ
2 + γ2)(σ2 + γ2 + 3SBγσ
2),
b3 = 3SB
σ2
θ2
,
b2 = 2
(
k2 +
σ2
θ2
)
− 12iSBσ
2k
θ
,
b1 = −3SBσ
4
(
2
k2
σ2
+
1
θ2
)
− 8iσ2
k
θ
,
b0 = −2σ
4
(
k2
σ2
+
1
θ2
)
.
To determine the behaviour of the pitched field MRI in the presence of Bragin-
skii viscosity, we solve equations (2.91) and (2.92) using a combination of basic
numerics and further asymptotic analysis. To provide a basis for comparison we
briefly consider the inviscid case first.
2.6.2 Pitched inviscid MRI
When SB = 0 and B → 0 (but σ ∼ 1), equation (2.92) for vn reduces exactly
to that of the inviscid vertical field instability γn,m = 1/2 ∈ <. In this case the
magnetic field influences the growth rate, through its vertical component only, by
68 2. Magnetized MRI
setting the range of k at which growth occurs. The radial length-scale is therefore
also ∝ 1/k.
The most unstable wavenumbers will inevitably be large k ∼ 1/Bθ and so the
curvature effects, like those shown in figure 2.3, for k & 1 will be precluded.
For a vertical magnetic field of the same (necessarily weak) value as the vertical
component of a weak pitched field, the systems (growth rates and mode structure)
are identical. However, for a weak vertical field, an unstable perturbation that
satisfies the fluid condition kλmfp  1 may fail to satisfy it if the same strength
field becomes pitched (equation (2.20)) and so collisionless effects will come into
play (Quataert et al., 2002; Sharma et al., 2003). This aside though, the growth
rates of a weak vertical and a weak pitched magnetic field are the same, Figure
2.3.
If the magnetic field is not weak, inverting equation (2.92) and letting w ∼ k,
viz. equation (2.59), yields the complex quartic
γ4n + 2
[
σ2
(
1 +
1
k2θ2
)
+ 1
]
γ2n − 8i
σ2
kθ
+
[
σ2
(
1− 2
1
k2θ2
)
− 2
]
= 0. (2.93)
Solutions to this polynomial are considered in detail in Curry & Pudritz (1995)
(and further discussion can be found in Knobloch (1992)). For our purposes, is
is sufficient to consider solutions in the limit B  1 but still finite. In this case
we can expand γn as a power series in B. Substituting this into equation (2.93)
and solving up to first order, we find
γn = 2i
Bσ
√
4σ2 + 1
+
√(√
4σ2 + 1− σ2 − 1
)
, (2.94)
with a maximum value
γn,m =
1
2
+ i
√
3
4
B, (2.95)
which occurs at σn,m =
√
3/4. The imaginary part of γn is of order B and, as
can be seen in the SB = 0 limit of Figure 2.6, this is in excellent agreement with
the full numerical solution for values of the magnetic field as large as B = 0.2. In
the inviscid case, the MRI is an over-stability whose travelling wave component
2.6 Pitched field, θ  1 69
arises from the finite azimuthal hoop tension (∝ B) absent in the vertical case.
This much is already well known. However, as we now show, both this strong
field result and the weak field result change dramatically when the Braginskii
viscosity is introduced. A new faster instability emerges whose behaviour we
now describe.
2.6.3 Pitched MRI with Braginskii viscosity
When the Braginskii viscosity is not zero, the structure of vn admits a complex
instability whose real part and imaginary part exceeds that of the inviscid case.
To show this we invert equation (2.92) to form a polynomial for γn containing
vn. This takes the form
γ4n + c3γ
3
n + c2γ
2
n + c1γn + c0 = 0, (2.96)
where
c3 = 3SBσ
2
(
1 +
1
v2nθ2
)
,
c2 = 2
[
σ2 +
1
v2n
(
k2 +
σ2
θ2
)]
− 12iSBσ
2k
θ
,
c1 = 3SBσ
4
[
1−
1
v2n
(
2
k2
σ2
+
1
θ2
)]
− 8i
σ2
v2n
k
θ
,
c0 = σ
4
[
1− 2
(
k2
σ2
+
1
θ2
)]
.
In the weak field limit, this equation is very similar to equation (2.60) for the
vertical Braginskii-MRI, but rather than solving equation (2.42) to determine vn,
we must now solve equation (2.55). The only other difference is the second term
in c1. This term constitutes the additional stress (proportional to the shear) that
arises from the viscous stress tensor, equation (2.36), and is responsible for the
enhanced growth rate we will find. When the finite magnetic field strength is
not neglected, new terms appear, including the imaginary term in c2 that, in the
large SB limit, enhances =(γn). We investigate these effects now.
70 2. Magnetized MRI
Figure 2.6: Maximum real, and when they exist, corresponding imaginary (inset)
growth rates γ0,m as a function of SB for two different field orientations and sets
of magnetic field strengths, B = 0.001 (solid), B = 0.01 (dashed), B = 0.1 (dash-
dot) and B = 0.2 (dash-dot-dot). Also shown, (inset) is the σ0,m at which they
occur. Top panel: Vertical field: The maximum growth rates γ0,m (which are
real) and the corresponding values of σ0,m are bounded from above by the weak
field limits of 1/2 and
√
3/4 (dotted lines) respectively. Bottom Panel: Slightly
pitched case, θ = 0.001: The maximum real part of the growth rate <(γ0,m)
(which is now complex) and the corresponding σ0,m (top inset) are bounded from
above, respectively, by the weak field limit of
√
2 (main panel, top dotted line)
and the inviscid limit
√
3/4 (top inset, dotted line). They are also bounded from
below by, again respectively, the inviscid limit 1/2 (main panel, bottom dotted
line) and the maximised solutions to equation (2.111). The imaginary part of the
growth rate =(γ0,m) (bottom inset) is well described in the small and large SB
limits by equations (2.95) and (2.111). See Figure 2.7 for plots of the analytic
asymptotic solutions.
2.6 Pitched field, θ  1 71
Numerical results
As in the vertical case, vn is determined by equation (2.55) and so δur,i is the
same. However, the other components of the flow, magnetic field and pressure
differ, as shown in equations (2.44)-(2.49). We will now show γn does as well.
To determine γn we have (trivially) adapted the root finding algorithm used
in the inviscid case. We find a complex instability whose peak growth rate,
<(γn) =
√
2 d ln Ωˆ/d ln r =
√
2 occurs when B → 0 and, as before, n = 0.
If we concentrate on the limit where B can be neglected entirely, γn becomes real.
Formally this requires us to re-order k ∼ d/dr ∼ −1/B, but this is a technicality
and the results are the same as simply letting B ∝ σ/θ → 0 in equation (2.96)).
In this case is useful to contrast the variation of γn with σ for a discrete set of
values of SB, which we do in Figure 2.4. In the small SB limit, γn approaches
the inviscid growth rate so, for θ  1, we have,
SB → 0, B → 0, γn,m → 1/2, σn,m →
√
3/4, (2.97)
whilst in the opposite, large SB limit (subsidiary to its  ordering) we find,
SB →∞, B → 0, γn,m →
√
2, σn,m → 0. (2.98)
So, in the weak field regime, not only do the viscous and inviscid pitched cases
differ categorically but, comparing equations (2.72) and (2.98), so do the viscous
pitched and the viscous vertical cases. This is depicted in Figure 2.7.
If the magnetic field is allowed to be finite, growing modes become over-stable
and both <(γn) and =(γn) increase above the inviscid values as seen in Figure
2.6.
As we now show, starting in the weak field limit, most of results of this section
can be derived analytically in the appropriate asymptotic limits.
Weak field limit
The equations governing the viscous pitched field MRI, (2.91) and (2.92) are
derived assuming k ∼ 1/  1. In this case the asymptotic expansion for the
72 2. Magnetized MRI
roots of the Bessel function vn, equation (2.59), still holds. Taking the first two
terms of this expansion, letting B → 0 (σ ∼ 1), and substituting into equation
(2.96) yields
γ4n + 3SBσ
2γ3n + 2
(
σ2 + 1− snk
− 23
)
γ2n+
3SBσ
2
[
σ2 − 2
(
1− snk
− 23
)]
γn + σ
2
[
σ2 − 2
(
1− snk
− 23
)]
= 0. (2.99)
If we also assume SB ∼ snk−
2
3 , as in section 2.5.2, then to first order in this
quantity we find
γn,m =
1
2
(
1 +
9
16
SB − sn
k−
2
3
2
)
, (2.100)
σn,m =
√
3
4
(
1−
SB
16
− sn
k−
2
3
2
)
. (2.101)
Unlike the vertical field case SB now appears at first order (in equations (2.79)
and (2.80) SB terms arise at second order in the expansion parameter) and here
its effect is to increase γn,m, as shown in Figure 2.7. In both the weak and strong
B regimes the maximum growth rate can only be determined in the large SB
limit which we turn to now.
Large (subsidiary) SB limit
Starting from the dispersion relation of the weak field limit, equation (2.99),
the fastest growing mode and Alfve´n frequency in the large SB limit can be
determined as follows. Treating (3SB)−1 = δ as small parameter and neglecting
k−
2
3 terms, the governing dispersion relation is
γ4n +
σ2
δ
γ3n + 2(σ
2 + 1)γ2n +
σ2
δ
(σ2 − 2)γn + σ
2(σ2 − 2) = 0. (2.102)
Differentiating this with respect to σ and setting dγn/dσ = 0, the stationary
points of γn obey the following constraint:
γ3n + 2δγ
2
n + 2(σ
2 − 1)γn + 2δ(σ
2 − 1) = 0. (2.103)
2.6 Pitched field, θ  1 73
Figure 2.7: Direct comparison of γ0,m, σ0,m behaviour of unstable modes for
θ = 0.001 (top solid line and inset) and θ = pi/2 (bottom solid line and inset)
with SB for a weak magnetic field, B = 0.001. The dotted lines represent the
inviscid limits of γ0,m = 1/2 and σ0,m =
√
3/4. The dashed lines represent the
various asymptotic solutions we have derived. For the main panel, these are
equation (2.83) for the damped θ = pi/2 field, and equations (2.100) and (2.108)
for the small and large SB limit of the θ = 0.001 field. In the top panel, equation
(2.101), (2.109) and (2.111) give the small, intermediate and large SB limits for
σ0,m. In the bottom panel, the asymptotic result given by equation (2.83) is
accurate only up to SBB2 ∼ 1. Beyond this we have plotted a line of slope S
−3/4
B
(not derived) for comparison (dash-dot).
74 2. Magnetized MRI
Considering the branch σ = σn,m that makes the stationary value of γn a maxi-
mum γn,m and introducing ζn,m = σ2n,m, we find
ζn,m =
γn,m(2− γ2n,m) + 2δ(1− γn,m)
2(γn,m + δ)
, (2.104)
and upon substituting this into equation (2.102) we obtain a polynomial whose
solutions describe the stationary points of γn including its maximum. This equa-
tion is
γ6n,m − 4γ
4
n,m − 20δγ
3
n,m + 4(1− 4δ
2)γ2n,m + 8δγn,m + 4δ = 0. (2.105)
Because it is a sixth order polynomial, there is no general formula for its roots.
However, the presence of the small parameter δ suggests that a series solution is
appropriate. We assume that γn,m and ζn,m can be written as
γn,m = γn,m,(0) + δ
1/2γn,m,(1) + δγn,m,(2) +O(δ
3/2), (2.106)
ζn,m = ζn,m,(0) + δ
1/2ζn,m,(1) + δζn,m,(2) +O(δ
3/2). (2.107)
Substituting this expression for γn,m into equation (2.105), and solving order by
order we can obtain the following set of solutions.
To lowest order we find that γn,m,(0) = ±
√
2, 0 and we take the positive root,
corresponding to the instability. At the next order we do not obtain any in-
formation about γn,m,(1), but proceeding to second order in δ1/2, we find that
γn,m,(1) = ±23/4/31/2. Guided by the full numerical solution (see Figure 2.7)
we take the negative root, and so, to first order in δ, the maximum asymptotic
growth rate is given by
γn,m =
√
2
(
1−
21/4
3
S−1/2B
)
, (2.108)
accurate up to and including terms of order δ1/2.
Now substituting our expression for γn,m into equation (2.107) and demanding
that the equation is satisfied to each order in δ1/2, we find, to lowest order,
that ζn,m,(0) = 0. However, at next order we obtain a positive result, namely
2.6 Pitched field, θ  1 75
that ζn,m,(1) = 25/2/31/2. In anticipation of writing our final answer in terms of
σn,m =
√
ζn,m we proceed to next order so that our solution for γn,m and σn,m are
accurate to the same order. Doing so reveals that ζn,m,(2) is also zero. This means
that we can write the magnetic tension associated with the fastest growing mode
as
σn,m =
25/8
√
3
S−1/4B , (2.109)
accurate up to and including terms of order δ1/2. This result and the result for
γn,m in equation (2.108) are shown in Figure 2.7 and are in good agreement with
the full numerical solution (although because of its weak dependence on SB, the
asymptotic expression for σn,m converges more slowly than γn,m). These results
confirm the findings of the previous sections, namely that γn,m increases with SB.
If we now relax our weak field assumption so γn is given by equation (2.96), we
recover the complex instability we found numerically. Assuming |γn| ∼ 1 and
writing k = σ/B, the leading order balance of terms ∝ SB is
γ2n
(
σ2 +B2
)
− 4iσBγn + σ
2(σ2 −B2 − 2) = 0, (2.110)
and the resultant growth rate is
γn =
2iBσ + σ
√
B2 (B2 − 2)− σ2 (σ2 − 2)
B2 + σ2
. (2.111)
Although analytic expressions for γn,m and σn,m can be derived from this, they
are not particularly enlightening. However, Figure 2.7 shows they agree well with
the large SB limit of σ0,m.
The results of this section confirm the destabilising properties of the Braginskii
viscosity in both the weak and strong field regimes. Of particular interest is the
difference in the =(γn) between the inviscid and highly viscous regimes, equations
(2.94) and (2.111); an effect not captured at all by local analysis and clearly visible
in Figure 2.6.
Now, finally, to compare all the limits of the pitched and vertical cases we again
consider the large vn solutions to equation (2.96).
76 2. Magnetized MRI
Highly oscillatory limit
The simplest regime in which to consider the highly oscillatory limit of the viscous
MRI is B → 0 so unstable modes will be purely real. In this case, the appropriate
mathematical treatment is formally identical to the vertical inviscid case discussed
in section 2.5.1, and so δur,i will be the same (although the remaining components
will not). The eigenvalue identity is given by equation (2.92) and so the dispersion
relation found by inverting it is
γ4n + 3SBσ
2γ3n + 2
[
σ2 +
(
k ln k
npi
)2
]
γ2n+
3SBσ
2
[
σ2 − 2
(
k ln k
npi
)2
]
γn + σ
2
[
σ2 − 2
(
k ln k
npi
)2
]
= 0, (2.112)
valid in the limit vn  kr  1 and SB ∼ 1.
Adopting a series solution approach as in the highly oscillatory limit of section
2.5.1 we can expand γn in the small parameter k/vn. Substituting this into
equation (2.112), differentiating and solving to determine the turning points of
γn, we find
γn,m =
1
2
k
vn
(
1 +
9
16
k
vn
SB
)
, (2.113)
σn,m =
√
3
4
k
vn
(
1−
1
16
k
vn
SB
)
. (2.114)
In this regime, the first order correction to γ0,m increases, rather than decreases
the growth rate viz. equation (2.86). On this, and all other counts, it appears
that the effect of the viscous stress tensor on a tilted field otherwise unstable to
the MRI, is to further destabilise it. We provide a physical explanation of this
shortly and contrast it with that of the vertical field instability, but first let us
summarise what we have found about the pitched field case.
2.6 Pitched field, θ  1 77
2.6.4 Pitched field summary
For a pitched field, the inviscid and (Braginskii) viscous MRI have some simi-
larities. As in the vertical case, when finite magnetic field effects are neglected
(B → 0, σ finite), both configurations support a purely real instability whose
growth is maximised for n = 0 modes. When they are not neglected, these modes
become over-stable and <(γn,m) is reduced.
The wide applicability of the MRI as means of generating angular momentum
transporting turbulence from magnetic fields and shear flows, in part, comes
from the fact it only requires (and is most efficient for) weak fields. This does
not change in the presence of the Braginskii viscosity.
However, some properties of the MRI do change in the presence of the Braginskii
viscosity. In the weak field limit of the inviscid case, like the vertical equivalent,
γn is bound by the Oort-A value 1/2. The viscous solution is not. Its peak
growth rate is
√
2 as SB →∞, B → 0 and its n = 0 mode is always faster than
the corresponding inviscid mode. Retaining a finite magnetic field also leads to
order of magnitude differences in =(γn) when comparing the inviscid and highly
viscous cases.
The azimuthal mode structures δuφ,i and δBφ of the viscous MRI also differ
from their inviscid counterparts. However, because k  1, in both cases the
modes are highly localised on the inner boundary and so graphing them is largely
uninformative.
The differences between the growth rates of the inviscid and viscous MRI are
important. For weak magnetic fields in the historical ISM, amplification by the
Braginskii viscous MRI to the current observed value could have occurred in
just 0.5 G yrs, a factor ∼ 3 faster in the viscous case. Potentially a significant
difference, especially in earlier periods when accounting for the existence of the
primordial galactic field is problematic (e.g. Kulsrud, 1999).
The difference between the tilted orientation and the singular vertical and az-
imuthal limits where the Braginskii viscosity damps, rather than enhances, the
growth rate of unstable modes may prove important in designing good numeri-
cal models of the ISM or other magnetized accretion discs. As such, before we
78 2. Magnetized MRI
conclude, let us discuss the physical processes that lead to such orientation depen-
dent behaviour. Let us also discuss how what we have found relates to the local
approximation, and under what conditions our model may run into difficulties.
2.7 Discussion
2.7.1 Physical mechanism
We have seen that depending on the orientation of an equilibrium magnetic field,
the evolution of magnetized shear flow can differ categorically. How is this to be
understood physically?
The most informative explanation comes in the weak field regime where the role
of the magnetic field is twofold. Firstly it facilitates the generation of pressure
anisotropies proportional to its rate of change, or equivalently (in the collisional
regime) proportional to the parallel gradient of the parallel component of the rate-
of-strain tensor. Because collisions are involved, this is a necessarily dissipative
process (Boltzmann, 1872; Chapman & Cowling, 1970). Taking the dot product
of the momentum equation, (2.24) with ui and integrating over space shows that
the energy contribution of the Braginskii viscosity is negative definite. That
is, the Braginskii viscosity damps any motions that change the magnetic field
strength. This is the first role of the magnetic field.
The second role of the magnetic field is a geometric one. Assuming anisotropies
do arise (from changes in the magnetic field strength), the field’s orientation
dictates the projection of the anisotropic stress onto the fluid elements of the
plasma, thereby affecting their dynamics. As we now explain, this fact is crucial
in determining the stability, or lack thereof, and the role global effects have on a
differentially rotating magnetized plasma.
When the field lines have both vertical and azimuthal components (i.e., the field is
pitched), fluid elements at different heights can exert an azimuthal force on each
other. The sign of this force can be either positive or negative. If the magnetic
field is unstable, so its rate of change is positive, the anisotropic stress that leads
to this force acts to oppose any azimuthal separation of the two elements. This
2.7 Discussion 79
can be identified as the fluid version of the stress responsible for the microscopic
mirror force. In this case, like the MRI, velocity perturbations to fluid elements
at different heights increase (decrease) their angular momentum causing them
to move to larger (smaller) radii. In a system with an outwardly decreasing
angular velocity profile, this leads to an azimuthal separation of the fluid elements.
The associated magnetic field growth ensures the stress is of the right sign to
oppose this separation and this transfers angular momentum between them in a
way that facilitates further (radial) separation; i.e. an instability (see Quataert
et al. (2002) for a physical explanation including a spring analogy). However,
unlike the Maxwell stress responsible for the MRI, the Braginskii stress does not
exert a radial force that opposes the fluid elements’ separation. As a result the
growth rate of the viscous instability can exceed that of the inviscid MRI. In
the limit where there is no Maxwell stress, there is no radial “tension” to limit
the instability and the growth rate is maximised. This is the second role of the
magnetic field.
In conjunction, the two roles explain our results. In isolated field geometries
where there is no projection of the stress onto fluid elements at different heights,
the Braginskii viscosity does not give rise to an instability; its only effect is
dissipative (Kulsrud, 2005; Lyutikov, 2007; Kunz et al., 2010). This accounts
for the damping of perturbations in the azimuthal configuration and the reduced
growth rate of the vertical MRI (that depends on the Maxwell, not the Braginskii,
stress). When the field is pitched its dissipative effects persist, but the free energy
contribution from the differential shear flow will always be greater, leading to an
instability. The dissipative effects cannot change the stability boundary; they
may only change the growth rate. Of course the total field energy (equilibrium
plus perturbed) will decrease if viscosity is present and viscous heating is not.
However, from the perspective of generating turbulence or transporting angular
momentum, this is a secondary consequence.
Now considering finite magnetic field strengths (and therefore hoop tension and
viscous curvature stresses) we find these modes become over-stable. The variation
of the travelling wave component of the unstable mode is simply a combination
of these two effects, to varying degrees.
80 2. Magnetized MRI
So, with this explanation in mind, let us turn to the local description of the
instability where we compare and contrast the similarities and differences between
our global analysis and those from the shearing sheet.
2.7.2 Relation to the local approximation
The local approximation, or shearing sheet analysis, is an extremely useful model
of an accretion flow. Therefore it is worth comparing how the results found using
it compare to those found here. As we now explain, for the most part they are in
agreement and any discrepancies that arise are the result of neglected curvature
terms or different boundary conditions. The parts of the following discussion not
specific to the Braginskii viscosity can be found in existing literature but, for
completeness, we reproduce it (Dubrulle & Knobloch, 1993; Curry et al., 1994;
Curry & Pudritz, 1995).
In the local approximation, the coordinate system is Cartesian, the shear is mod-
elled linearly, and the perturbed quantities are assumed to vary rapidly in the ra-
dial direction (with respect to the background variations) so they may be written
as δui(r) = δui exp[ilr], lr  1. Under this assumption, a WKB-style analysis
ignores terms proportional to 1/r and replaces d/dr by il. This is, conceptually,
the same as the highly-oscillatory approximation of sections 2.5.1, 2.5.2 and 2.6.3.
As given by Islam & Balbus (2005); Ferraro (2007), the resultant dispersion
relation is
γ4 + d1γ
3 + d2γ
2 + d3γ + d4 = 0, (2.115)
where
d1 =3SBσ
2 l
2 + k2 cos2 θ
l2 + k2
,
d2 =2
(
σ2 +
k2
l2 + k2
)
,
d3 =3SBσ
2
(
σ2
l2 + k2 cos2 θ
l2 + k2
− 2 cos2 θ
k2
l2 + k2
)
,
d4 =σ
2
(
σ2 − 2
k2
l2 + k2
)
.
2.7 Discussion 81
We now compare this with our global results in the relevant limits.
Similarities
Equation (2.115) supports both the inviscid MRI and, for θ 6= 0, pi/2, the Bragin-
skii facilitated MRI. Taking the limit k  l, we can compare it to the weak field,
large k limit of equations (2.60), (2.78) and (2.99), appropriate when the system
is, respectively, vertical and inviscid; vertical and viscous; and pitched and vis-
cous. We find the equations, and therefore the all-important growth rates, are
identical.
For the same setups, the large l limit of equation (2.115) differs from the large
n-limit of the highly oscillating solutions found in equations (2.67), (2.85) and
(2.112). The local wavenumber takes continuous or, in the case of periodic bound-
ary conditions, discrete values that are independent of k, whereas the global
equivalent is given by vn = (n+ 1/2)pi/ lnw which is not independent of k. How-
ever, for sufficiently large values of l (and vn) this difference is unimportant and
when the values of l and vn coincide, the growth rates are identical. Both analyses
show that the presence of radial structure decreases the effect of the destabilising
shear and increases the effect of viscous dissipation.
Discrepancies
In the strong, pitched magnetic field limit, the local and global results diverge.
Whilst the local results predict unstable modes are always real, the global model
finds that they are always complex. In the inviscid limit, =(γn) ∼ B and so
this component can often be reasonably neglected when B is weak. In the very
viscous limit =(γn) ∼ σB/(σ2 + B2) and, as shown in Figure 2.6, this can be
over an order of magnitude greater than the inviscid result. This difference may
prove important (from a modelling perspective) for viscous systems which could
have been treated locally, were they inviscid.
Another discrepancy occurs for isolated field configurations. It is easy to see that
when θ = 0 and θ = pi/2, l = 0 the terms A1, A3 = 0 and the local dispersion
relation reduces to that of the inviscid case (i.e. the effects of the Braginskii
82 2. Magnetized MRI
viscosity are absent). That is, axisymmetric perturbations are neutrally stable
for θ = 0 and lead to the MRI for θ = pi/2. This is in disagreement with sections
2.4 and 2.5. This discrepancy can be understood as follows.
In the azimuthal case, the relevant components of perturbed stress tensor bb :
∇ui (associated with Braginskii viscosity, equation (2.27)) are given by
δ(bb : ∇ui) = eˆφ · (eˆφ · ∇(δur,ieˆr)) =
δur,i
r
global, (2.116)
= 0 local, (2.117)
and so locally there is no change in the magnetic field (equation (2.25)). This
means that there is no Braginskii viscosity and therefore no dissipation, which
thereby explains the different results.
In the vertical case there are two differences between the local and global analysis.
The incompressibility condition, equation (2.26), is
∇ · δui = ikδuz,i +
dδur,i
dr
+
δur,i
r
global, (2.118)
= ikδuz,i + ilδur,i local, (2.119)
and so in the local case, assuming l = 0, we find δuz,i = 0. However, in a global
flow with an impenetrable inner boundary, radial derivatives cannot be ignored
as dδur,i/dr = 0 implies δur,i = 0 (this is not true in a local flow with periodic
boundary conditions). Because any radial structure reduces γn (be it n 6= 0 or a
finite mode width; see Figure 2.2 and sections 2.5.1 and 2.5.1) the global MRI,
both with and without the Braginskii viscosity, will always be slower than its
local l = 0 counterpart.
A second difference, specific to the Braginskii viscosity, is related to the existence
of vertical motions and their effect in a vertical field. Again, l = 0 locally implies
δ(bb : ∇ui) ∝ δuz,i = 0 whereas, globally, vertical motions are always allowed
and so δ(bb : ∇ui) = ikδuz,i 6= 0. This means that the fastest growing local
mode (l = 0) is not damped by the Braginskii viscosity, whereas the equivalent
global mode is.
A final point is worth making. The local model predicts that the fastest growing
2.7 Discussion 83
unstable modes (θ 6= 0) occur as l → 0. Formally this is inconsistent with the
assumption l  1/r 6= 0 that went into deriving equation (2.115). The local
solutions obtained under the WKB-style approximation cannot be considered
self-consistent and in fact should be described by the type of global solution we
have constructed here. However, this fact turns out to be unimportant because,
as we have found, it is indeed the modes with the least radial structure that grow
the fastest and so the local approach turns out to be correct in this respect.
Overall, aside from the limitations of the local approximation in these two isolated
field geometries, there is good agreement between it and our global model in the
appropriate limits. Although it would be interesting to consider whether this
similarity holds for arbitrary θ, this is outside the scope of the current study.
Before we conclude let us consider a few modelling implications of our results.
2.7.3 Modelling implications
Uniform discs
In a magnetized accretion disc, the scale at which the fastest growing linear MRI
modes emerge depends on the angle and strength of the background magnetic
field. This much is known in the unmagnetized (or inviscid) case and the scaling
we find here does not differ qualitatively when collisions are strong (viscosity is
small), k ∼ 1/(Bθ). However, for a pitched field (the most generic configura-
tion) and weak collisions (large viscosity) the scale of the fastest growing mode
acquires an additional parametric dependence on the collision frequency so that
k ∝ S−1/4B ∝ ν
1/4
i . This implies that for a sufficiently collisionless plasma, but still
one described by fluid theory, a local-in-z analysis is inappropriate (our choice
of θ  1 renders this consideration unimportant for the choice of parameters
we consider in section 2.2.1). Physically, density and temperature gradients in
the background state will need to be considered as the mode extends over multi-
ple disc scale heights. What this implies in a magnetized plasma is unclear but
Gammie & Balbus (1994) have conducted such a study in the inviscid regime and
found the presence of conducting boundaries to be important.
84 2. Magnetized MRI
Finite pressure anisotropies and instabilities
The nonlinear fate of the MRI in the presence of Braginskii viscosity cannot be
determined without a full numerical study and the emergence, or not, of channel
modes has not been established here. However, if it is anything like the inviscid
MRI, we can reasonably assume that on macroscopic scales far greater than λmfp
coherent structures that grow at a rate ∼ Ω will emerge. When these reach finite
amplitudes they will give rise to pressure anisotropies which, as we discuss in
detail in chapter 3, drive microscale instabilities. If modelled kinetically (as they
should be) the peak growth rates of these instabilities are, in general, faster than
the rate of change of the anisotropy generating magnetic fields and they occur
on scales far less than λmfp. In this case the long wavelength description given
by equation (2.24) should be used with care and one must ask: Under what con-
ditions, if any, can pressure anisotropy driven micro-instabilities be neglected?
In general they cannot be. However, in certain specific magnetized parameter
regimes, like the one we consider here, they can. As such, let us briefly con-
sider the mirror instability that occurs for ∆ ∝ dB/dt > 0 (Hasegawa, 1969;
Southwood & Kivelson, 1993).
It can been shown that for ∆ > 1/βi equations (2.24)-(2.27) support the slow
wave polarised mirror instability. Although a full kinetic calculation is necessary
to accurately capture its growth rate, its existence can be determined even in the
long wavelength regime (Schekochihin et al., 2005). In this regime modelling the
mirror is potentially less problematic than one might expect because its growth
rate asymptotes to a finite value12 (Private correspondence: Kunz, 2010). The
fluid version of the mirror instability is therefore not necessarily inconsistent,
but it is not a true representation. This is because replacing equation (2.27) by
the appropriate kinetic expression of the pressure anisotropy (as one must for
kλmfp  1) yields a growth rate
γ ∼ kvth,i
(
∆−
1
βi
)
(Mirror Instability), (2.120)
12We deliberately leave discussion of the firehose that can occur when dB/dt < 0 to the next
chapter. Unlike the mirror, in the fluid regime its growth rate does not asymptote to a finite
value and its description is therefore ill-posed even there.
2.8 Conclusion 85
so the fastest growing mode peaks, unphysically, at infinitesimal scales and the
kinetic dispersion relation describing it is ill posed. This indicates that confining
oneself to the fluid regime where the growth rate is finite, is missing something
very important. This fact requires attention, or at the least acknowledgement,
because fluid simulations that ignore this feature are missing what may turn
out to be an important kinetic process. Although it remains an open question
how to best incorporate its effects, in lieu of a microphysical transport theory,
one possibility is to append to the governing equations, the finite Larmor radius
corrections given by the hot plasma dispersion relation (Hellinger, 2007; Rincon
et al., 2010).
In the linear regime it is known that this regularises the mirror instability. So,
for the historic values of the ISM we considered in section 2.2.1, the peak kinetic
growth rate of the mirror is now given by
γ ∼ |∆|2 Ωi ∼ 10
−22 s−1  Ωˆ (Peak Mirror Instability), (2.121)
where ∆ ∝ bb : ∇ui ∼ Ωˆ for the MRI. So in this one case, the growth rate of
the mirror is very slow compared to the MRI, and so its effect can be neglected.
This is what we meant when we said that in certain parameter regimes microscale
instabilities can be neglected. However, it is often the case (especially in a mag-
netized plasma) that microscale instabilities cannot be neglected. This cannot
be stressed strongly enough. In the next chapter we will consider the firehose,
the sister instability to the mirror that occurs when dB/dt < 0 and show, in the
context of galaxy clusters, that its effects are significant and dramatic. However,
before we do so let us summarise what we have learnt here.
2.8 Conclusion
Over time the nature of the ideal MRI has been well established but how it is
modified in various non-ideal settings remains an active topic of research (Blaes
& Balbus, 1994; Balbus & Terquem, 2001; Kunz & Balbus, 2004; Ferraro, 2007;
Pessah & Chan, 2008; Devlen & Peku¨nlu¨, 2010). The subject of this study has
been the global nature of the MRI in a collisional magnetized (Braginskii MHD)
86 2. Magnetized MRI
plasma.
We found that for an azimuthal magnetic field with an asymptotically small
vertical component, like the MRI, a singular linear instability emerges. The
growth rate of this instability depends on SB, a dimensionless combination of the
temperature, shear rate and (mass density normalised) magnetic field strength
and is up to a factor of 2
√
2 faster than the MRI. We determined the scaling of
the maximum growth rate γn,m =
√
2
(
1− 21/4/3S−1/2B
)
and Alfve´n frequency (or
equivalently wavenumber) at which the growth occurs. We also found that the
presence of a finite magnetic field strength causes the unstable mode to become
over-stable with an imaginary component that can be over an order of magnitude
greater than the equivalent over-stable inviscid mode.
In addition to the destabilising effects of the Braginskii viscosity we showed that
dissipative effects are important in the isolated field geometries considered here:
purely vertical or azimuthal fields. In contradiction to local theory we found
that it damps all modes, even the fastest growing ones, and for these it does so
significantly when SBB2 ≡ 1/Re & 1. We showed these effects occur without
modifying the stability boundaries of the system.
Associated with both its dissipative and growth enhancing properties, we found
the Braginskii viscosity pushes the fastest growing unstable mode towards longer
vertical wavelengths as a function of its strength. This may have implications
for the resolution and accessible parameter regime of any future numerical study
that includes it.
Whilst we motivated our investigation by considering the historic conditions in
the ISM, we expect our results to apply to magnetized accretion discs with non-
galactic rotation profiles, e.g. Keplerian. In modelling all such magnetized sys-
tems, the effects of the Braginskii viscosity should be included. Since anisotropic
stresses can now be included in the Athena MP code, it should be relatively easy
to simulate the effects of the Braginskii viscosity locally, and hopefully soon,
globally as well (Stone et al., 2008; Skinner & Ostriker, 2010). Our results should
be of help in benchmarking or at least constraining the appropriate parameter
regime of such simulations.
Part III
Firehose instability
87

Chapter 3
Firehose instability
3.1 Introduction
It has recently been realised in various astrophysics and space physics contexts
that pressure anisotropies (with respect to the direction of the magnetic field)
occur naturally and ubiquitously in magnetized weakly collisional plasmas. They
lead to very fast microscale instabilities such as the firehose and mirror, whose
presence is likely to fundamentally affect the transport properties and, therefore,
both small- and large-scale dynamics of astrophysical plasmas — most interest-
ingly, the plasmas of galaxy clusters and accretion discs (Hall & Sciama, 1979;
Schekochihin & Cowley, 2006; Schekochihin et al., 2008, 2005; Sharma et al.,
2006, 2007; Lyutikov, 2007). These instabilities occur even (and especially) in
high-beta plasmas and even when the magnetic field is dynamically weak. The
current state of theoretical understanding of this problem is such that we do not
in general even have a set of well-posed macroscopic equations that govern the
dynamics of such a plasma. This is because calculating the dynamics at long
spatial scales l  ρi and slow time scales corresponding to frequencies ω  Ωi
requires knowledge of the form of the pressure tensor and the heat fluxes, which
depend on the nonlinear evolution and saturation of the instabilities triggered by
the pressure anisotropies and temperature gradients. Since this is not currently
understood, we do not have an effective mean-field theory for the large-scale
dynamics.
89
90 3. Firehose instability
In the absence of a microphysical theory, it is probably sensible to assume that
the instabilities will return the pressure anisotropies to the marginal level and to
model large-scale dynamics on this basis, via a suitable closure scheme (Sharma
et al., 2006, 2007; Schekochihin & Cowley, 2006; Lyutikov, 2007). This approach
appears to be supported by the solar wind data (Gary et al., 2001; Kasper et al.,
2002; Bale et al., 2009)
However, a first-principles calculation of the nonlinear evolution of the instabil-
ities remains a theoretical imperative, in order to construct the correct closure,
we must understand the mechanism whereby the instabilities control the pres-
sure anisotropy: do they scatter particles? do they modify the structure of the
magnetic field? The calculation presented in this chapter will lead us to conclude
that the latter mechanism is at work, at least in the simple case we are consider-
ing, and indeed a sea of microscale magnetic fluctuations excited by the plasma
instabilities will act to pin the plasma to marginal stability.
In this chapter, we present a theory of the nonlinear evolution of the simplest of
the pressure-anisotropy-driven instabilities, the parallel (k⊥ = 0) firehose insta-
bility and the gyrothermal instability (Schekochihin et al., 2010). To be specific,
we consider as our main application a plasma under physical conditions char-
acteristic of galaxy clusters: weakly collisional, fully ionized, magnetized and
approximately (locally) homogeneous. We will explain at the end the extent to
which our results are likely to be useful in the context of the solar wind (section
3.7).
The plan of exposition is as follows. In section 3.2, we give an extended, qual-
itative, mostly low-analytical-intensity introduction to the problem, explain the
relevant properties of the intracluster plasma (section 3.2.1), sketch the linear the-
ory of the firehose instability (section 3.2.2), the main principle of its nonlinear
evolution (section 3.2.3), and show that a more complicated theory is necessary
to work out the spatial structure of the resulting “firehose turbulence” (section
3.2.4). In section 3.3, a systematic such theory is developed via asymptotic expan-
sions of the electron and ion kinetics (the basic structure of the theory is outlined
in the main part of the chapter, while the detailed derivation is relegated to Ap-
pendices B and C), culminating in a very simple one-dimensional equation for
3.2 Galaxy Clusters 91
the nonlinear evolution of the firehose fluctuations (section 3.4.1), the study of
which is undertaken in section 3.4. The results are a theoretical prediction for the
nonlinear evolution and spectrum of the firehose turbulence (section 3.4.3) and
some tentative conclusions about its effect on the momentum transport (section
3.4.4). In section 3.5, we extend this study to include the effect of parallel ion
heat flux on the firehose turbulence: in the presence of a parallel ion temperature
gradient, a new instability emerges (the gyrothermal instabilty recently reported
by Schekochihin et al. 2010 and recapitulated in section 3.5.2) — which, under
some conditions, can take over from the firehose. For it as well, we develop a one-
dimensional nonlinear equation (section 3.5.1), solve it to predict the nonlinear
evolution and spatial structure of the gyrothermal turbulence (section 3.5.3) and
discuss the implications for momentum transport (section 3.5.4). A discussion
of differences from previous work on firehose instability in collisionless plasmas
is given in section 3.6 and a brief survey of astrophysical implications in the so-
lar wind and galaxy clusters in section 3.7. Finally, section 3.8 contains a very
concise summary of our findings and of the outlook for future work.
A reader not interested in the technicalities of kinetic theory is advised to ignore
section 3.3, and Appendices B and C. A reader only interested in the formal
derivation may skip section 3.2, as section 3.3 (supplemented by Appendices B
and C) and the sections that follow it can be read in a self-contained way.
3.2 Galaxy Clusters
3.2.1 Conditions in the ICM
Galaxy clusters have long attracted the interest of both theoreticians and ob-
servers both as dynamical systems in their own right and as cosmological probes
(Bahcall, 2000; Peterson & Fabian, 2006). While gravitationally they are dom-
inated by dark matter, most of their luminous matter is a hot, diffuse, fully
ionized, X-ray emitting hydrogen plasma (Sarazin, 2003) known as the intraclus-
ter medium, or ICM (the galaxies themselves are negligible both in terms of their
mass and the volume they occupy). Crudely, we can think of an observable galaxy
cluster as an amorphous blob of ICM about 1 Mpc across, sitting in a gravita-
92 3. Firehose instability
tional well, with a density profile peaking at the center and decaying outwards.
Observationally, on the crudest level, we know what the overall density and tem-
perature profiles in clusters are (e.g., Piffaretti et al., 2005; Vikhlinin et al., 2005).
Recent highly resolved X-ray observations reveal the ICM to be a rich, compli-
cated, multiscale structure displaying ripples, bubbles, filaments, waves, shocks,
edges etc. (Fabian et al., 2003a,b, 2006; Sanders & Fabian, 2008; Markevitch &
Vikhlinin, 2007), temperature fluctuations (Markevitch et al., 2003; Simionescu
et al., 2007; Million & Allen, 2009; Lagana et al., 2009) and most probably also
broad-band disordered turbulent motions (Churazov et al., 2004; Schuecker et al.,
2004; Rebusco et al., 2006, 2008; Sanders et al., 2010b). Radio observations tell us
that the ICM also hosts tangled magnetic fields, which are probably dynamically
strong (Carilli & Taylor, 2002; Vogt & Enßlin, 2005; Govoni, 2006)
These and other observations motivate a number of questions about the ICM like
the ones posed in chapter 1. Principally, can we explain the observed ICM tem-
perature profiles, in particular the apparent lack of a cooling catastrophe at the
cluster core predicted by fluid models (Fabian, 1994; Peterson & Fabian, 2006;
Parrish & Quataert, 2008; Bogdanovic´ et al., 2009)? This requires modelling
various heating processes involving conversion of the energy of plasma motions
(turbulent or otherwise) into heat via some form of effective viscosity (e.g., Fabian
et al., 2005; Dennis & Chandran, 2005; Kunz et al., 2010), the dynamical effect of
thermal instabilities arising in the magnetized ICM (Parrish et al., 2008; Sharma
et al., 2008, 2009; Bogdanovic´ et al., 2009; Parrish & Quataert, 2008; Parrish
et al., 2010), and the effective thermal conductivity of this medium with ac-
count taken of the tangled magnetic field (Chandran & Cowley, 1998; Narayan &
Medvedev, 2001; Zakamska & Narayan, 2003). Which, if any, of these processes
dominates and how are its effects best incorporated?
Addressing these questions requires a theoretically sound mean-field theory for
the ICM dynamics, i.e., a set of prescriptions for its effective transport properties
(viscosity, thermal conductivity), which depend on the unresolved microphysics.
Without such a theory, all we have is numerical simulations based on fluid models
(see references above). While they can often be tuned to produce results that are
visually similar to what is observed, they are not entirely satisfactory because
3.2 Galaxy Clusters 93
they lack a solid plasma-physical basis. Additionally, refining the numerical res-
olution often breaks the agreement with observations and requires retuning. A
satisfactory transport theory is lacking because any plasma motions in the ICM
that change the strength of the magnetic field trigger microscale plasma insta-
bilities (see section 3.2.2) and we do not know what happens next. The type
of arguments employed in section 2.7 to neglect their effects must be seen as a
special case rather than the norm.
How some of these instabilities arise and evolve is discussed in greater detail
below. In order to make this discussion more quantitative, we need to fix a few
physical parameters that characterise the ICM. In reality, these parameters vary
considerably both between different clusters and within any individual cluster (as
a function of radius: from the cooler, denser core to the hotter, more diffuse outer
regions). However, for the purposes of this discussion, it is sufficient to adopt
a set of fiducial values. Let us consider the plasma in the core of the Hydra A
cluster where the parameters are (David et al., 2001; Enßlin & Vogt, 2006)
• Particle (ion and electron) number density
ni = ne ∼ 6× 10
−2 cm−3. (3.1)
• The measured electron temperature is
Te ∼ 3× 10
7 K, (3.2)
and although the ion temperature is unknown, it is assumed to be compa-
rable, Ti ∼ Te. The ion thermal speed is then
vth,i =
(
2Ti
mi
)1/2
∼ 7× 107 cm s−1, (3.3)
and the ion Debye length is
λDi =
vth,i
ωDi
= vth,i
(
4pie2ni
mi
)−1/2
∼ 2× 105 cm. (3.4)
• The ion collision frequency (in seconds, assuming ni in cm−3 and Ti in K)
94 3. Firehose instability
is
νi ∼ 1.5niT
−3/2
i ∼ 5× 10
−13 s−1, (3.5)
and consequently the mean free path is
λmfp =
vth,i
νi
∼ 1.3× 1020 cm. (3.6)
• The rms magnetic field strength is (Vogt & Enßlin, 2005)
B ∼ 7× 10−6 G, (3.7)
and consequently the plasma (ion) beta is
βi =
8piniTi
B2
∼ 130, (3.8)
the ion cyclotron frequency is
Ωi =
eB
mic
∼ 0.07 s−1, (3.9)
and the ion Larmor radius is
ρi =
vth,i
Ωi
∼ 109 cm. (3.10)
Note that the magnetized-plasma condition ρi  λmfp is satisfied extremely
well.
• The typical velocity of the plasma motions is (cf. Sanders et al., 2010a,b,
who place an upper bound on the velocities in a range of clusters including
Hyrda A)
U ∼ 2.5× 107 cm s−1, (3.11)
while the typical length scale of these motions is
L ∼ 2× 1022 cm. (3.12)
3.2 Galaxy Clusters 95
Consequently the Mach number is
M =
U
vth,i
∼ 0.3, (3.13)
(so the motions are subsonic, hence approximately incompressible on scales
smaller than that of the mean density variation) and the Reynolds number
based on collisional parallel viscosity is
Re =
LU
λmfpvth,i
∼ 60. (3.14)
Assuming Kolmogorov scalings for turbulence, the viscous cutoff scale is
l ∼ LRe−3/4 ∼ 1021 cm, (3.15)
and the typical velocity at this scale is
u ∼ URe−1/4 ∼ 107 cm s−1, (3.16)
so the approximate rms rate of strain (assuming a viscous cutoff for the
motions) is
γ0 ∼
u
l
∼
U
L
Re1/2 ∼ 10−14 s−1. (3.17)
3.2.2 Firehose instability
If we consider length scales greater than both ρi and λmfp and time scales longer
than both Ωi and νi then the moments governing the plasma evolution are those
found in chapter 1, namely the ion momentum equation (1.16), and the induction
equation (1.17). For clarity, these are
mini
dui
dt
= −∇
(
p⊥ +
B2
8pi
)
+ ∇ ·
[
bb
(
p⊥ − p‖ +
B2
4pi
)]
, (3.18)
dB
dt
= B ·∇ui −B∇ · ui. (3.19)
96 3. Firehose instability
As in chapter 2, we have used equation (1.35) for the pressure tensor and neglected
Gs at these long wavelengths and slow frequencies. As in our work on the MRI, we
have restricted ourselves to collisional scales. This means the pressure anisotropy
is
∆ =
1
νi
1
B
dB
dt
=
1
νi
bb : ∇ui ∼
γ0
νi
, (3.20)
where γ0 is the typical rate of strain of the plasma motion and again we have
dropped the subscript i from ∆ because electrons contribute only small fraction
of the anisotropy1. Thus, the pressure anisotropy is regulated by the ratio of the
collision frequency to the typical rate of strain or, equivalently, the rate of change
of the magnetic-field strength.
Substituting the numbers from section 3.2.1, we find that |∆| ∼ 0.02 in the core
of Hydra A. Is this a large number? It turns out that it is a huge number because
such anisotropies will make the plasma motion violently unstable.
While the full description of the plasma instabilities triggered by pressure anisotropies
requires kinetic treatment, it is extremely straightforward to deduce the presence
of the firehose instability directly from equation (3.18).
Consider some “fluid” solution (u0,i,B0, p0⊥, p0‖) of equations (3.18) and (3.19)
that varies on long time and spatial scales — that can be thought of as the turbu-
lence and/or some regular magnetofluid motion caused by global dynamics. Let
us now examine the linear stability of this solution with respect to high-frequency
(ω  |∇u0,i|), short-scale (k  |∇u0,i|/u0,i) perturbations (δu, δB, δp⊥, δp‖).
Mathematically, this is simply equivalent to perturbing a straight-magnetic-field
1 If a Kolmogorov-style turbulence is assumed to exist in the ICM, the typical rate of strain
γ0 will be dominated by the motions at the viscous cutoff scale. However, as we saw in section
3.2.1, the Reynolds-number estimates for ICM do not give very large values and one might
wonder whether calling these motions turbulence is justified (Fabian et al., 2003b). However,
for our purposes, it is not important whether the rate of strain is provided by the viscous cutoff
of a turbulent cascade or by a single-scale motion because either can change the magnetic field
and thus cause pressure anisotropy (Schekochihin & Cowley, 2006).
3.2 Galaxy Clusters 97
equilibrium of equations (3.18) and (3.19):
−miniωδu = −k⊥
(
δp⊥ +
B0δB‖
4pi
)
+ k‖δb
(
p0⊥ − p0‖ +
B20
4pi
)
−
k‖b0
[
δp‖ +
(
p0⊥ − p0‖
) δB‖
B0
]
, (3.21)
−ω
δB
B0
= k‖δu− b0 (k · δu) , (3.22)
where δb = δB⊥/B0, we have used k · δb = −k‖δB‖/B0 (from ∇ ·B = 0), and ⊥
and ‖ are with respect to the unperturbed magnetic field direction b0. Pressure
perturbations can only be calculated from the linearised kinetic equation (see,
e.g., Schekochihin et al., 2005), but even without knowing them, we find that for
the Alfve´nically polarised modes, δu ∝ b0 × k, the dispersion relation is
ω = ±k‖
(
p0⊥ − p0‖
mini
+ v2A
)1/2
= ±k‖cs
(
∆ +
2
β
)1/2
, (3.23)
here, vA = B0/
√
4pimini, cs = (p0⊥/mini)1/2, ∆ = (p0⊥ − p0‖)/p0⊥ and β =
8pip0⊥/B20 . (This is the type of procedure one carries out to show the existence
of the fluid mirror instability viz. section 2.7.3.)
Equation (3.23) is simply the dispersion relation for Alfve´n waves with a phase
speed modified by the pressure anisotropy. If the pressure anisotropy is negative,
∆ < 0, the associated stress opposes the Maxwell stress (the magnetic tension
force), the magnetic-field lines become more easily deformable, the Alfve´n wave
slows down and, for ∆ < −2/β, turns into a nonpropagating unstable mode: the
firehose instability (Chandrasekhar et al., 1958; Parker, 1958; Rosenbluth, 1956;
Vedenov & Sagdeev, 1958). Its growth rate can, in general, be almost as large as
the ion cyclotron frequency as k‖ρi approaches finite values (see section 3.2.4).
For the ICM parameters given in section 3.2.1, the instability is, therefore, many
orders of magnitude faster than either the large-scale dynamics (typical turnover
rate ∼ |∇u0| ∼ γ0) or collisions (typical rate νi). Unlike the effects of the mirror
in ISM, it cannot be ignored.
Thus, any large-scale motion that leads to a local decrease in the strength of
98 3. Firehose instability
the magnetic field2 gives rise to a negative pressure anisotropy, which, in turn
triggers the firehose instability. This produces Alfve´nically polarised fluctuations
at small parallel scales unless the plasma beta is sufficiently low (magnetic field is
sufficiently strong) for the magnetic tension to stabilise these fluctuations. Using
the typical size of ∆ estimated earlier in this section, we find that the typical
beta below which the firehose is stable is β ∼ 100, which is quite close to the
measured value (see section 3.2.1) — perhaps not a coincidence?
Positive pressure anisotropies also lead to instabilities (most importantly, the mir-
ror that we discussed in section 2.7), but they involve resonant particles and are
mathematically harder to handle. We will not discuss them here (see Schekochi-
hin et al., 2005, 2008; Rincon et al., 2010).
3.2.3 Nonlinear evolution of the firehose instability
A nonlinear theory of the firehose instability can be constructed via a quasilin-
ear approach, in which the unstable small-scale (perpendicular) fluctuations of
the magnetic field on the average change the local magnetic-field strength and
effectively cancel the pressure anisotropy (Schekochihin et al., 2008). In equation
(3.20), let us treat the changing magnetic field as the sum of the large-scale field
and the small-scale firehose fluctuations: B = B0 +δB⊥. Then the field strength
averaged over small scales is
B ∼ B0
(
1 +
1
2
|δB⊥|2
B20
)
, (3.24)
where the overbar denotes the average. The contribution from δB⊥ is small, but
for large enough k‖, it is growing at a greater rate than the rate of change of the
large-scale field, so its time derivative can be comparable to the time derivative
of B0. As B0 is assumed to be decreasing, the growth of the fluctuations can
then cancel this decrease and drive the total average pressure anisotropy to the
2While turbulence on the average is expected to lead to the growth of the magnetic field
(the dynamo effec), locally there will always be regions where the field strength (temporarily)
decreases. Decrease of the field and, consequently, negative pressure anisotropy can also result
from expanding motion, which decreases the density of the plasma — as, e.g., in the solar wind.
3.2 Galaxy Clusters 99
marginal level, ∆ = −2/β. From equation (3.20), we get
∆ ∼
1
νi
(
1
B0
dB0
dt
+
1
2
d
dt
|δB⊥|2
B20
)
= −
2
β
. (3.25)
The rate of change of B0 is the typical rate of strain of the (large-scale) motion,
(1/B0)dB0/dt ∼ −|γ0|. The firehose growth rate γ = −iω is given by equation
(3.23). As long as the firehose fluctuations are smaller than the critical level
|δB⊥|2
B20
∼
|γ0|
γ
, (3.26)
they cannot enforce the marginality condition expressed by equation (3.25) and
will continue growing until they reach the required strength (which is still small
compared to the large-scale field because |γ0|/γ  1 for sufficiently large k‖).
After that, their evolution becomes nonlinear and is determined by equation
(3.25), whence we find that their energy has to grow secularly:
|δB⊥|2
B20
∼
(
|γ0| −
2νi
β
)
t. (3.27)
As long as the large-scale field keeps decreasing, the small-scale fluctuation energy
cannot saturate because if it did, its time derivative would vanish, the anisotropy
would drop below marginal and the instability would come back.
The secular growth given by equation (3.27) leads to δB⊥/B0 ∼ 1 after roughly
one turnover time (∼ |γ0|−1) of the large-scale background motion that produces
the anisotropy in the first place — thus, the magnetic field can develop order-
unity fluctuations before this background motion decorrelates. What all this
means for the large-scale dynamics on longer timescales, we do not know.
In what follows, we will be guided by the simple ideas outlined above in con-
structing a more rigorous kinetic theory of the nonlinear firehose instability.
3.2.4 Effect of finite Larmor radius
Although we have discussed it elsewhere, in this chapter we have so far carefully
avoided discussing the magnitude of the wavenumber k‖ of the firehose fluctu-
100 3. Firehose instability
ations. We simply referred to them as “small-scale,” with the implication that
their scale would be smaller than that of the background fluid dynamics that
cause the instability. Examining the dispersion relation (3.23), we see that the
growth rate of the instability is proportional to k‖, so the smaller the scale the
faster the instability. This ultraviolet catastrophe cannot be resolved within the
long-wavelength approximation (by which we mean kρi  1 as opposed to colli-
sional) in which equation (3.18) is derived. Instead Finite-Larmor-radius (FLR)
corrections like those encapsulated in Gs, the gyrophase dependent component of
the pressure tensor given by equation (1.28), must be brought in.
Introducing G to the momentum equation yields a well posed dispersion relation,
(Davidson & Vo¨lk, 1968, this result will emerge in section 3.4.2) with a peak
growth rate and scale
γmax ∼
∣
∣
∣
∣∆ +
2
β
∣
∣
∣
∣Ωi ∼ 10
−3 s−1  γ0, (3.28)
k‖ ∼
∣
∣
∣
∣∆ +
2
β
∣
∣
∣
∣
1/2
ρ−1i ∼ 10
−10 cm−1  λ−1mfp, (3.29)
for the parallel (k⊥ = 0) firehose. The numerical values are based on the ICM
parameters of section 3.2.1 and the estimate of ∆ from section 3.2.2. We have
also neglected the heat fluxes for simplicity. For the oblique firehose with k⊥ 6= 0
in general we find γmax ∼ |∆ + 2/β|1/2Ωi ∼ 10−2 s−1 and k ∼ 1/ρi (Yoon et al.,
1993; Hellinger & Matsumoto, 2000).
There are two conclusions to be drawn from this. First, the linear instability
is enormously fast compared with the large-scale dynamics that cause it, so its
nonlinear behaviour must be fundamentally important at all times. Second, in
order to understand the spatial structure of the firehose fluctuations, we need a
theory that takes the FLR effects explicitly into account because it is the FLR
that sets the scale and the growth rate of the fastest-growing mode as shown in
Figure 3.1. We now proceed to construct such a theory for the simplest case —
the parallel (k⊥ = 0) firehose instability.
3.2 Galaxy Clusters 101
Figure 3.1: Microscale firehose fluctuations in a magnetized plasma are regu-
larised by FLR effects so the fastest growing modes occur on scales λmfp  1/k &
ρi, as given by equation (3.29). Macroscopic fluid motions on scales  box size,
induce changing magnetic fields and therefore pressure anisotropies which provide
the free energy source to drive the growing modes.
102 3. Firehose instability
3.3 Kinetic theory
3.3.1 Outline of the derivation
We start from the fundamental kinetic equation (1.2) and Maxwell’s equations
(1.3)-(1.6). The tactic is to expand them to obtain a series solution in much the
same manner as in section 1.4.1, but in this case however our ordering is more
sophisticated and differs for ions and electrons. To aid the reader, we start by
quickly restating some key moment equations from the start of chapter 1. The
procedure that follows can be thought of as an expansion tailored specifically to
the problem of anisotropy driven instabilities.
Electron kinetics: Induction equation
The electron kinetic equation can be expanded in the square root of the electron-
ion mass ratio (me/mi)1/2 ≈ 0.02, a natural small parameter for plasma. This
expansion is carried out in Appendix B, where we also explain what assumptions
have to be made in order for it to be valid. The outcome of the mass-ratio
expansion is that electrons are Maxwellian, isothermal (Te = const), and the
electric field can be determined in terms of ue, B and ne, again, via a generalised
Ohm’s law:
E +
ue ×B
c
= −
∇pe
ene
= −
Te∇ne
ene
. (3.30)
This can now be combined with Maxwell’s equations to form the standard induc-
tion equation, this time with a Hall term:
∂B
∂t
=∇×
[(
ui −
c
4piZeni
∇×B
)
×B
]
. (3.31)
Ion kinetics: continuity and momentum equations
To close this set of equations, we must determine ni and ui. Taking moments
of the kinetic equation (1.2) we find that ni and ui satisfy the continuity and
3.3 Kinetic theory 103
momentum equations
∂ni
∂t
= −ni∇ · ui, (3.32)
∂ui
∂t
= −
∇ · Pi
mini
−
ZTe∇ni
mini
+
(∇×B)×B
4pimini
, (3.33)
(3.34)
where the second term on the right-hand side is the electron pressure gradient
and the ion pressure tensor is
Pi = mi
∫
d3v vvfi. (3.35)
We must solve the ion kinetic equation to calculate Pi in terms of ui and B. We
do this by means of an asymptotic expansion in a physical small parameter.
3.3.2 Asymptotic ordering
The small parameter we will use to solve the ion kinetic equation is expressed in
terms of the Mach and Reynolds numbers (Schekochihin et al., 2008):3
 =
M
Re1/4
∼ 0.1, (3.36)
where we used the ICM parameters of section 3.2.1. This is the natural small
parameter for the plasma motions because, using equations (3.11)-(3.17), it is
easy to see that
u
vth,i
∼
λmfp
l
∼ , (3.37)
where l is the viscous scale and u the typical flow velocity at this scale. The
typical rate of strain γ0 ∼ u/l is the relevant parameter for determining the
size of the pressure anisotropy because, even though the viscous cutoff we are
using is based on the parallel collisional viscosity and so motions can exist below
this scale, these motions do not change the strength of the magnetic field (see
3As already pointed out in footnote 1, our considerations do not depend on Re being large.
If a single-scale flow is considered, our expansion is simply an expansion in Mach number.
104 3. Firehose instability
Schekochihin & Cowley, 2006).4 Thus, the pressure anisotropy is (from equation
(3.20))
∆ ∼
γ0
νi
∼
u
vth,i
λmfp
l
∼ 2. (3.38)
We solve the ion kinetic equation by asymptotic expansion in . All ion quantities
are expanded in , so
fi = f0i + f1i + f2i + f3i + · · · , (3.39)
ni = n0i + n1i + n2i + n3i + · · · , (3.40)
ui = u0i + u1i + · · · , (3.41)
B = B0 +B1 + · · · . (3.42)
The lowest-order quantities n0i, u0i, B0 are associated with the motions that
produce the pressure anisotropy and have the length scale l and time scale γ0, so
we order
u0i ∼ vth,i, ∇u0i ∼ γ0 ∼ 2νi. (3.43)
Since the instability parameter is ∆ + 2/βi, we must order B0 so that
2
βi
∼ ∆ ∼ 2 ⇒
B0√
4pimin0i
∼ vth,i. (3.44)
The perturbations n1i, u1i, B1 around this slow large-scale dynamics are assumed
to be excited by the parallel (k⊥ = 0) firehose instability and have much shorter
spatial and time scales. Their typical wavenumber is the one at which the insta-
bility’s growth rate peaks and their time scale is set by this maximum growth
4This statement applies to macroscopic motions: for example, Alfve´nic turbulence below the
parallel viscous scale that can occupy a wide range of scales all the way down to the ion Larmor
scale (e.g., Schekochihin et al., 2009). The fast, microscale plasma fluctuations triggered by
plasma instabilities, including the firehose fluctuations that will be considered in this chapter,
will, on the average, change the field strength (see section 3.2.3). Accordingly, their ordering
[equation (3.46)] will be arranged in precisely such a way that they are able to have an effect
comparable to the macroscale motions that produce γ0.
3.3 Kinetic theory 105
rate (see section 3.2.4):
k‖ρi ∼
∣
∣
∣
∣∆ +
2
βi
∣
∣
∣
∣
1/2
∼ , γ ∼
∣
∣
∣
∣∆ +
2
βi
∣
∣
∣
∣Ωi ∼ 
2Ωi. (3.45)
In order to be able to proceed, we must order the time scales of the lowest-order
(“equilibrium”) fields and of the fluctuations with respect to each other. Physi-
cally, they depend on different things and are not intrinsically related. However,
our a priori consideration of the nonlinear evolution of the instability (section
3.2.3) suggests that for the nonlinearity to become important, we must have (see
equation (3.26))
B1
B0
∼
(
γ0
γ
)1/2
. (3.46)
Since B1 ∼ B0, this tells us that we must order
γ0 ∼ 
2γ ∼ 4Ωi ⇒ νi ∼ 
2Ωi, ρi ∼ 
2λmfp. (3.47)
These relations are, of course, not strictly right in the quantitative sense — the
Larmor radius is grossly overestimated here if we take the value of  for the
ICM given by equation (3.36) and compare it with the observational value given
by equation (3.10). However, ordering ρi this way allows us to capture all the
important physics in our formal expansion. We will also argue in section 3.4.3
that this ordering of the finite Larmor radius physics gets quantitatively better
as the nonlinear regime proceeds (see footnote 8).
Let us summarise our ordering of the relevant time and spatial scales compared
to k‖vth,i and k‖, respectively: using equations (3.47) and (3.45), we have
γ0 ∼ 
3k‖vth,i, γ ∼ νi ∼ k‖vth,i, Ωi ∼ 
−1k‖vth,i, (3.48)
l−1 ∼ 2k‖, λ
−1
mfp ∼ k‖, ρ
−1
i ∼ 
−1k‖. (3.49)
106 3. Firehose instability
3.3.3 Firehose fluctuations
The ordering we adopted, inasmuch as it concerns the properties of the firehose
fluctuations, applies to the parallel firehose only, so we now explicitly restrict
our consideration to the case of ∇⊥ = 0 for all first-order perturbations. Since
∇ ·B = 0, this immediately implies
B‖1 = 0, (3.50)
so B1 = B⊥1 . Here and in what follows, ‖ and ⊥ refer to directions with respect
to the unperturbed field B0.
The induction equation (3.31), taken to the lowest order in , gives
d
dt
B⊥1
B0
= ∇‖u
⊥
1i (3.51)
(all terms here are order 2k‖vth,i; see section 3.3.2; note that the Hall term in
equation (3.31) is subdominant by two orders of  and so, as promised in section
1.3.2, irrelevant). Here d/dt = ∂/∂t + u0 ·∇ is the convective derivative, but,
since ∇u0 ∼ 3k‖vth,i, the shearing of the perturbed field due to the variation of
u0 is negligible and we can replace d/dt by ∂/∂t by transforming into the frame
moving with velocity u0.
In the continuity equation (3.32) taken to the lowest order in , setting ∇⊥ = 0
gives
d
dt
n1i
n0i
= −∇‖u
‖
1i (3.52)
(all terms are order 2k‖vth,i). Anticipating the form of the unstable perturbation,
we will set
n1i = 0, u
‖
1i = 0 (3.53)
without loss of generality. In Appendix C.0.11, we will explicitly prove that
n1i = 0. In Appendix C.0.14, we will learn that n2i = 0 as well.
Consider now the ion momentum equation (3.34). In the lowest order of the 
3.3 Kinetic theory 107
expansion (terms of order k‖v2th,i), it gives, upon using equation (3.53),
∇ · P1i = 0. (3.54)
We will learn in Appendix C.0.13 that this can be strengthened to set
P1i = 0. (3.55)
In the next order (2k‖vth,i), we get (using n2i = 0)
∇ · P0i +∇ · P2i + ZTe∇n0i = 0. (3.56)
Averaging this over small scales eliminates the perturbed quantities, so we learn5
∇ · P0i + ZTe∇n0i = 0 (3.57)
and, therefore, from equation (3.56), also
∇ · P2i = 0 (3.58)
(confirmed in Appendix C.0.14). Finally, in the third order (3k‖v2th,i), the per-
pendicular part of equation (3.34) determines the perturbed velocity field:
du⊥1i
dt
= −
(∇ · P3i)⊥
min0i
+ v2A∇‖
B⊥1
B0
, (3.59)
where vA = B20/4pimin0i. There is no ZTe∇⊥n3i term in equation (3.59) because
we assume that the only small-scale spatial variations of all quantities are in the
parallel direction. The ion pressure term (∇ · P3i)⊥ is to be calculated by solving
the ion kinetic equation (see section 3.3.5).
To summarise, we are looking for perturbations such that ∇⊥ = 0, n1i = 0,
B‖1 = 0, u
‖
1i = 0, while B
⊥
1 and u
⊥
1i satisfy equations (3.51) and (3.59). Physi-
cally, this reflects the fact that the parallel (k⊥ = 0) firehose perturbations are
5This is simply the pressure balance for the large-scale dynamics, an expected outcome for
a system with low Mach number. In Appendix C.0.10, we will show that the zeroth-order
distribution is Maxwellian, so the pressure associated with it is a scalar, p0i = n0iT0i, and
equation (3.57) becomes (T0i +ZTe)∇n0i +n0i∇T0i = 0. Further discussion of the role played
by the ion temperature gradient can be found in section 3.5.
108 3. Firehose instability
Alfve´nic in nature (have no compressive part). That it is legitimate to consider
such perturbations separately from other types of perturbations is not a priori
obvious, but will be verified by our ability to obtain a self-consistent solution of
the ion kinetic equation, which will satisfy equations (3.55), (3.57), and (3.58)
(see Appendix C).
3.3.4 Large-scale dynamics
In section 3.3.3, equations for the first-order fields, u⊥1i and B
⊥
1 emerged after
expanding the induction equation (3.31) and the continuity equation (3.32) to
lowest order in  and the momentum equation (3.34) up to the third order. If,
using the ordering of section 3.3.2, we go to the next order and average over
small scales to eliminate small-scale perturbations, we recover the equations for
the large-scale (unperturbed) fields: the induction equation
dB0
dt
= B0 ·∇u0i −B0∇ · u0i (3.60)
(all terms are order 3k‖vth,iB0), the continuity equation
dn0i
dt
= −n0i∇ · u0i (3.61)
(all terms are order 3k‖vth,in0i), and the momentum equation
min0i
du0i
dt
= −∇ · P2i −∇
B20
8pi
+
B0 ·∇B0
4pi
(3.62)
(all terms are order 4min0ik‖v2th,i). The divergence of the second-order ion pres-
sure tensor here is with respect to the large-scale spatial variation (according to
equation (3.58), it has no small-scale dependence). Again, P2i is calculated from
ion kinetics.
Equations (3.60)-(3.62) are precisely the kind of mean-field equations that are
needed to calculate the large-scale dynamics of astrophysical plasmas. They look
just like the usual fluid MHD equations, the only nontrivial element being the
pressure term in the momentum equation (3.62). The goal of kinetic theory is
to calculate this pressure, which depends on the microphysical fluctuations at
3.3 Kinetic theory 109
small scales. In this thesis, we only do this for the parallel (k⊥ = 0) firehose
fluctuations. The implications of our results for the ion momentum transport
will be discussed in section 3.4.4.
3.3.5 Solution of the ion kinetic equation
We now proceed to use the ordering established in section 3.3.2 to construct an
asymptotic expansion of the ion kinetic equation. This procedure, while analyti-
cally straightforward, is fairly cumbersome and so its detailed exposition is exiled
to Appendix C. The results are as follows.
In the expansion of the ion distribution function (equation (3.39)), f0i is found to
be a Maxwellian (Appendix C.0.10), with density n0i and temperature T0i that
have to satisfy the equilibrium pressure balance constraint (see equation (3.57)
and Appendix C.0.13).
The first-order perturbed distribution function, f1i, is proportional to b0 ·∇T0i
and is responsible for the large-scale collisional ion heat fluxes (Appendix C.0.13).
The second-order perturbed distribution function f2i contains the pressure anisotropy.
The corresponding second-order pressure tensor is diagonal:
P2i = p2iI + (p⊥2i − p
‖
2i)
(
1
3
I− b0b0
)
, (3.63)
where p2i is the perturbed isotropic pressure and p⊥2i − p
‖
2i is the lowest-order
pressure anisotropy. The isotropic part of the pressure is determined from the
large-scale equations for density n0i, temperature T0i and velocity u0i of the fluid
— this is explained in detail in Appendix C.0.17, but here let us just assume for
simplicity that the zeroth-order density and temperature are constant (∇n0i = 0,
∇T0i = 0), in which case the continuity equation (3.61) reduces to∇·u0i = 0 and
p2i then follows from enforcing this incompressibility constraint on the momentum
110 3. Firehose instability
equation (3.62). The pressure anisotropy is calculated in Appendix C.0.18:
∆(t) ≡
p⊥2i − p
‖
2i
p0i
=∆0 +
3
2
∫ t
0
dt′e−3νi(t−t
′) ∂
∂t′
|B⊥1 (t′)|2
B20
, (3.64)
where p0i = n0iT0i is the equilibrium pressure, the overbar denotes the averaging
over small scales of the nonlinear feedback on the anisotropy from the firehose
fluctuations, and ∆0 is the pressure anisotropy arising from the large-scale mo-
tions. In general, it contains contributions from changes in the magnetic field
strength (because of the approximate conservation of the first adiabatic invari-
ant, as discussed in section 1.5.2), compression and heat fluxes (see equation
(C.57)). When n0i and T0i are constant, only the anisotropy induced by the
changes in field strength survives,6 which is the case we will consider here:
∆0 =
b0b0 :∇u0i
νi
=
1
νi
1
B0
dB0
dt
=
γ0
νi
. (3.65)
This is exactly what was anticipated qualitatively — see equation (3.20). Since
we are interested in the firehose instability, we assume ∆0 < 0.
Finally, the third-order perturbed distribution function f3i is responsible for the
third-order pressure tensor that appears in the perturbed ion momentum equa-
tion (3.59). The relevant part of that tensor is calculated in Appendix C.0.20.
Assuming constant density and temperature (otherwise, there is again a contri-
bution from the heat fluxes; see Appendix 3.5), it may be written as follows
(∇ · P3i)⊥
p0i
= −∇‖
[
∆(t)
B⊥1
B0
+
∇‖u⊥1i
Ωi
× b0
]
, (3.66)
where ∆(t) is given by equation (3.64).
Let us now use these results to study the firehose turbulence (sections 3.4.1–3.4.3)
and its effect on the large-scale dynamics (section 3.4.4).
6The effect of heat fluxes on the firehose turbulence is considered in section 3.5.
3.4 Firehose turbulence 111
3.4 Firehose turbulence
3.4.1 Firehose turbulence equation
Using the results derived in Appendix C and summarised in section 3.3.5, we
find that the ion momentum equation (3.59), which describes the evolution of
the perturbed ion velocity, is, in the reference frame moving with u0i,
∂u⊥1i
∂t
=
v2th,i
2
∇‖
[(
∆(t) +
2
βi
)
B⊥1
B0
+
∇‖u⊥1i
Ωi
× b0
]
, (3.67)
where we have used equation (3.66) for the pressure term in equation (3.59).
Three forces appear on the right-hand side of this equation. First, there is the
stress due to the anisotropy ∆ of the ion distribution, given by equation (3.64).
This is is the quantitative form of the expression for ∆(t) that we guessed in
equation (3.25): the first term in equation (3.64) is due to the slow decrease of
the large-scale magnetic field, the second to the average effect of the growing
small-scale fluctuations, which strive to cancel that decrease. The second term
in equation (3.67), proportional to 1/βi = v2A/v
2
th,i, is the magnetic tension force,
which resists the perturbation of the magnetic-field lines and, therefore, acts
against the pressure-anisotropy driven instability. The instability is marginal
when ∆ + 2/βi → −0. Finally, the third term is the FLR effect, which, as was
promised in section 3.2.4 and as will shortly be demonstrated, sets the scale of
the most unstable perturbations.
Let us now combine equation (3.67) with the induction equation (3.51) for the
perturbed magnetic field, also taken in the reference frame moving with u0. After
differentiating equation (3.51) once with respect to time, we get
∂2B⊥1
∂t2
=
v2th,i
2
∇2‖
[(
∆ +
2
βi
)
B⊥1 +
1
Ωi
∂B⊥1
∂t
× b0
]
. (3.68)
In the second term on the right-hand side, we have used equation (3.51) to express
∇‖u⊥1i in terms of the time derivative of B
⊥
1 . Equation (3.68) with ∆(t) defined
by equation (3.64) is a closed equation for the perturbed magnetic field with
nonlinear feedback (last term in equation (3.64)). This is the equation for the
112 3. Firehose instability
-0.2 0 0.2
-0.02
0
0.02
Figure 3.2: Frequencies (thin lines) and growth rates (bold lines) of the unstable
firehose modes (red/solid: the “+” mode; blue/dashed: the “−” mode) given by
equation (3.70). The instability parameter here is ∆ + 2/βi = −0.01. Dotted
vertical lines indicate the wavenumber of fastest growth kp = 0.2 (equation (3.72))
and the dotted horizontal lines the corresponding maximum growth rate γmax =
Imωp = 0.01 (equation (3.73)).
one-dimensional (k⊥ = 0) firehose turbulence. It represents the simplest nonlinear
model for this kind of turbulence available to date.7
3.4.2 Linear theory
In the linear regime, we may neglect the second term in equation (3.64), so
∆ = ∆0. The linear dispersion relation for equation (3.68) is
[
ω2 −
k2‖v
2
th,i
2
(
∆ +
2
βi
)]2
=
k4‖v
4
th,i
4
ω2
Ω2i
. (3.69)
7The essential difference with the equation derived in Schekochihin et al. (2008) is the FLR
term, which removes the ultraviolet catastrophe of the long-wavelength firehose and thus allows
equation (3.68) to handle non-monochromatic (multiscale) solutions. In section 3.4.3, we will
see that this produces a much more complex behaviour than was seen in Schekochihin et al.
(2008), justifying the term “firehose turbulence.”
3.4 Firehose turbulence 113
This has four roots out of which two are unstable when ∆ + 2/βi < 0:
ω
Ωi
= ±
k2
4
+ i
|k|
√
2
∣
∣
∣
∣∆ +
2
βi
∣
∣
∣
∣
1/2
√
1−
k2
k20
, (3.70)
where k = k‖ρi and
k0 = 2
√
2
∣
∣
∣
∣∆ +
2
βi
∣
∣
∣
∣
1/2
. (3.71)
Unlike in the long-wavelength limit (k‖ρi → 0), there is now a real frequency
(so the firehose perturbation propagates while its amplitude grows exponentially
and the vector B⊥1 rotates; see section 3.4.3) and the growth rate has its peak at
kp = k0/
√
2, so
kp = 2
∣
∣
∣
∣∆ +
2
βi
∣
∣
∣
∣
1/2
, (3.72)
ωp = (±1 + i)
∣
∣
∣
∣∆ +
2
βi
∣
∣
∣
∣ , (3.73)
where the complex peak frequency ωp is in units of Ωi. At k‖ρi > k0, there is
no growth and the firehose perturbations turn into purely propagating Alfvee´n
waves (modified by pressure anisotropy and dispersive FLR corrections).
The dependence of the frequencies and growth rates of the two unstable modes
on wavenumber is given by equation (3.70) and plotted in Figure 3.2 for a repre-
sentative value of the instability parameter ∆ + 2/βi = −0.01 (this is the value
used in the numerical solution of section 3.4.3).
It should be pointed out here that in this theory, there is no dissipation of the
magnetic fluctuations excited by the firehose. The most unstable wavenumber is
set by dispersive effects; the stable modes are undamped.
114 3. Firehose instability
Figure 3.3: Top panel: evolution of the magnetic energy |B⊥1 |2/B
2
0 =
∑
k |Ak|
2
with time in a numerical solution of equations (3.77) and (3.78) with parameters
(3.89); the time here is normalised using the collision frequency νi, not the cy-
clotron frequency Ωi; the two horizontal lines show the “collisional” (lower line)
and “collisionless” (upper line) estimates for the energy at which the nonlinear
feedback turns on: equations (3.80) and (3.81), respectively; the red dotted line
shows the nonlinear asymptotic given by equation (3.82). Bottom panel: evo-
lution of the instability parameter (pressure anisotropy) ∆ + 2/βi in the same
numerical solution. Inset: log-log plot of the evolution of |∆ + 2/βi|; the red line
shows the slope corresponding to 1/t (see equation (3.86)).
3.4 Firehose turbulence 115
3.4.3 Nonlinear evolution and spectrum
Firehose turbulence equation in scalar form
Since the nonlinearity involves the spatially averaged perturbed magnetic energy,
the firehose turbulence is compactly described in Fourier space not just in the
linear but also in the nonlinear regime: this amounts to replacing ∇2‖ → −k
2
‖ in
equation (3.68) and |B⊥1 |2 =
∑
k‖
|B⊥1 (k‖)|
2 in equation (3.64). A simple ansatz
can now be used to convert equation (3.68) into scalar form. Without confusing
Ak with the vector potential, let
B1x
B0
= Ak(t) cos
(
k2
4
t+ φk
)
, (3.74)
B1y
B0
= Ak(t) sin
(
k2
4
t+ φk
)
, (3.75)
where the axes (x, y) in the plane perpendicular to b0 are chosen arbitrarily and
we have non-dimensionalised wavenumbers and time:
k‖ρi → k, Ωit→ t. (3.76)
This ansatz amounts to factoring out the rotation of the vector B⊥1 (k) (the first
term in equation (3.70)). The wavenumber-dependent but time-independent
phase φk is determined by the initial condition. We assume φk = φ−k, so
A∗k = A−k must be satisfied to respect the fact that B
⊥
1 is a real field. The
fluctuation amplitude Ak(t) satisfies
∂2Ak
∂t2
=
k2
2
[
−
(
∆ +
2
βi
)
−
k2
8
]
Ak, (3.77)
∆(t) = ∆0 +
3
2
∫ t
0
dt′e−3ν∗(t−t
′) ∂
∂t′
∑
k
|Ak(t
′)|2, (3.78)
where ν∗ = νi/Ωi = ρi/λmfp and we remind the reader that ∆0 < 0. It is manifest
in the form of equation (3.77) how the dispersion relation (3.70) (without the first
term) is recovered. Note that there is no coupling between different wavenumbers
modes in the sense that if a mode is not initially excited, it is never excited. The
only effect that modes have on each other is via the sum over k in equation (3.78),
116 3. Firehose instability
to which they all contribute.
Qualitative picture
Using only linear theory and the qualitative considerations of section 3.2.3, we
can construct a fairly clear picture of the evolution of the firehose turbulence.
Assuming a broad-band infinitesimal initial perturbation in k space, at first,
∆ = ∆0 and all modes with k < k0 (see equation (3.71)) will go unstable,
with the fastest-growing one given by kp = k0/
√
2. Eventually the amplitude
in this mode reaches the level at which the back-reaction becomes important:
approximating equation (3.78) by
∆(t) ' ∆0 +
1
2ν∗
∂
∂t
∑
k
|Ak|
2, (3.79)
we find that the nonlinear contribution is comparable to |∆0 + 2/βi| when (cf.
equation (3.26)):
∑
k
|Ak|
2 ' 2
∣
∣
∣
∣∆0 +
2
βi
∣
∣
∣
∣
ν∗
γmax
=
2νi
Ωi
=
2ρi
λmfp
, (3.80)
where γmax is the imaginary part of ωp given by equation (3.73). Equation (3.80)
only gives a good estimate of the critical amplitude if 3ν∗ is larger or not too much
smaller than γmax (collisions are sufficiently strong). If ν∗  γmax (as a subsidiary
limit within our  ordering), then a better approximation than equation (3.79) is
to replace the collisional relaxation exponent in equation (3.78) by unity, which
gives
∑
k
|Ak|
2 '
2
3
∣
∣
∣
∣∆0 +
2
βi
∣
∣
∣
∣ . (3.81)
Once the nonlinear feedback becomes active, exponential growth must cease and
secular growth starts because the anisotropy must be kept close to marginal:
using equation (3.79), we find, to dominant order,
∆ ' −
2
βi
⇒
∑
k
|Ak|
2 ' 2
∣
∣
∣
∣∆0 +
2
βi
∣
∣
∣
∣ ν∗t. (3.82)
3.4 Firehose turbulence 117
This is valid regardless of which of the two estimates (3.80) or (3.81) of the am-
plitude at the onset of nonlinearity was appropriate. This is because the effective
growth rate associated with the secular growth decreases with time and so we will
always eventually end up in the regime where the collisional relaxation exponent
in equation (3.78) is faster than the magnetic energy growth and equation (3.79)
gives a good approximation of equation (3.78).
The evolution of the fluctuation spectrum must be consistent with equation
(3.82). As the magnitude of the total pressure anisotropy ∆ approaches the
marginal value, both the cutoff wavenumber k0(t) and the most unstable wavenum-
ber kp(t) decrease, as they can still be estimated by equations (3.71) and (3.72)
with ∆ = ∆(t). The modes whose growth has been thus switched off become
oscillatory: from equation (3.77), it is obvious that for k  k0(t),
Ak = c1e
ik2t/4 + c2e
−ik2t/4, (3.83)
where c1 and c2 are integration constants and c∗1 = c2 because A
∗
k = A−k (note
that this oscillation of the amplitude is superimposed on the oscillation with the
same frequency that was factored out in equations (3.74)-(3.75)). Since these
modes oscillate in time at a rate that is much larger than the rate of change of
the anisotropy, they no longer contribute to the feedback term in equation (3.78).
Thus, as the range of growing modes, peaked at kp(t) and cut off at k0(t), sweeps
from large to small wavenumbers, they leave behind a spectrum of effectively
passive oscillations, whose amplitude no longer changes. Since there is no fixed
special scale in the problem (except initial most unstable wavenumber), one ex-
pects the evolution to be self-similar and the spectrum a power law. It is not
hard to determine its exponent. Let |Ak|2 ∼ k−α. Since the total energy must
grow linearly (equation (3.82)), we have
∑
k
|Ak|
2 ∼ k1−αp ∼ t ⇒ kp ∼ t
−1/(α−1) (3.84)
This is valid if α > 1; the extra power of k comes from the integration over
wavenumbers. On the other hand, for the fastest-growing mode, we must have,
118 3. Firehose instability
Figure 3.4: Top panel: spectrum of the magnetic fluctuation energy at three
specific times (tνi = 1, 4, 16) during the evolution shown in Figure 3.3; red short-
dashed lines show the k−3 slope (equation (3.87)); blue long-dashed lines show the
firehose growth rate (see equation (3.70)) for the instantaneous values of ∆+2/βi
at the times the spectra are plotted. Bottom panel: magnetic fluctuations in real
space (B1x/B0 vs. z) at the same times as the spectra in the left panel.
3.4 Firehose turbulence 119
assuming secular growth,
1
Akp
∂Akp
∂t
∼
1
t
∼ γmax ∼
∣
∣
∣
∣∆ +
2
βi
∣
∣
∣
∣ , (3.85)
where the last relation follows from equation (3.73). This gives us a prediction for
the time evolution of the residual pressure anisotropy and, via equation (3.72),
of the most unstable wavenumber (the infrared cutoff of the spectrum):
∣
∣
∣
∣∆ +
2
βi
∣
∣
∣
∣ ∼
1
t
, kp ∼
1
√
t
. (3.86)
The only way to reconcile equations (3.84) and (3.86) is to set α = 3. Thus, we
expect the one-dimensional firehose turbulence spectrum to scale as
|Ak|
2 ∼ k−3. (3.87)
The secular growth of the firehose fluctuations will continue until our asymp-
totic expansion becomes invalid, i.e., when the fluctuation amplitude is no longer
small.8 From equation (3.82), this happens at t ∼ (νi|∆0 + 2/βi|)
−1 ∼ |γ0|−1,
where dimensions have been restored. This is the time scale of the large-scale dy-
namics. Thus, as we have already explained in section 3.2.3, there is no saturation
of the firehose fluctuations on any faster time scale. Unsurprisingly, at the same
time as the fluctuation amplitude becomes large enough to break our ordering,
the scale of the fluctuations also breaks the ordering: substituting the above time
scale into equation (3.86) gives k‖ρi ∼ (|γ0|/Ωi)1/2 ∼ 2, or k‖λmfp ∼ 1, while our
original ordering assumption was k‖ρi ∼ , or k‖λmfp ∼ 1/ (see equation (3.45)).
Numerical solution
The firehose turbulence equation (3.77) is one-dimensional, so it is very easy to
solve numerically; equation (3.78) is most conveniently solved in a differential
8 Note that while the amplitude grows and thus eventually breaks the ordering introduced
in section 3.3.2, the stability parameter |∆ + 2/βi| decreases, so the approximation of small
Larmor radius gets quantitatively better with the growth of the firehose fluctuations moving
to larger scales (equation (3.72)) — equivalently, our ordering of ρi introduced in section 3.3.2
(equation (3.47)) is quantitatively better satisfied.
120 3. Firehose instability
Figure 3.5: Top panel: contour plot of the time evolution of the firehose turbu-
lence for the numerical solution discussed in section 3.4.3 — the vertical axis is
space (the middle fifth of our entire periodic domain), horizontal axis is time,
the colours respresent the value of B1x/B0. Bottom panel: the same plot for the
gyrothermal turbulence discussed in section 3.5.3. Note that, as explained in
the text, the firehose turbulence exhibits a gradual coarsening of the dominant
structure with time, while the gyrothermal turbulence ends up dominated by a
single scale.
3.4 Firehose turbulence 121
form:
∂
∂t
(
∆−
3
2
∑
k
|Ak|
2
)
= −3ν∗(∆−∆0) (3.88)
with the initial condition ∆(0) = ∆0. Here we describe the results obtained from
such a numerical calculation with the following parameters:
∆0 = −0.02,
2
βi
= 0.01, ν∗ =
ρi
λmfp
= 0.0001. (3.89)
This means that the maximum wavenumber at which firehose fluctuations can be
excited is k0 ' 0.28 (equation (3.71); see Figure 3.2). We solve equation (3.77)
for 1024 wavenumbers in a periodic domain of size λmfp, so the smallest and the
largest wavenumbers are (still normalised to ρi) kmin = 2piρi/λmfp ' 0.00063 and
kmax ' 0.32. The initial conditions are random amplitudes in each wavenumber
(satisfying the reality condition A−k = A∗k). Note that with the parameters (3.89),
our ordering parameter is  ∼ 0.1, so we have chosen a spatial scale separation
between collisions and the Larmor motion that substantially exceeds 1/2 formally
mandated by our ordering (section 3.3.2). This does not break anything and is
in fact more realistic for the physical parameters in weakly collisional plasmas of
interest (section 3.2.1). It also widens the scale interval available to the firehose
turbulence spectrum and ensures that even deep in the nonlinear regime, when
the wavenumber of the firehose fluctuations drops substantially, there is still a
healthy scale separation between them and the collisional dynamics.
The evolution of the total magnetic energy, |B⊥1 |2/B
2
0 =
∑
k |Ak|
2, is shown in
Figure 3.3 (left panel). Initially it grows exponentially at the (normalised) rate
γmax = Imωp (see equation (3.70); this part of the evolution is trivial and so
not shown). The exponential growth is followed by a secular, linear in time,
growth of the energy in accordance with equation (3.82). The energy at which
this nonlinear regime starts is closer to the estimate given by equation (3.81)
than by equation (3.80) because, as discussed above, we have taken a very small
value of ν∗. Note that in this and all subsequent figures, we have normalised
time using the collision frequency νi, not the cyclotron frequency Ωi — this is
indicated explicitly in the figures.
122 3. Firehose instability
Figure 3.3 (bottom panel) shows the time evolution of the instability parame-
ter ∆ + 2/βi. As expected it is tending to the marginal stability value (zero).
The inset shows that this approach to zero is consistent with the 1/t prediction
(equation (3.86)).
The evolution of the spectrum of firehose fluctuations is illustrated by Figure
3.4 (top panel). As anticipated in section 3.4.3, the spectral peak moves to
smaller wavenumbers in the nonlinear regime. The spectrum extending from
this moving peak to the original wavenumber of the fastest linear growth (kp =
0.2; see equation (3.72)) is statistically stationary and consistent with the k−3
power law predicted by equation (3.87). The instantaneous firehose growth rate is
overplotted on the spectra in Figure 3.4 (top panel) and confirms that the position
of the spectral peak closely follows the wavenumber of the fastest instantaneous
growth of the firehose instability.
Figure 3.4 (bottom panel) shows snapshots of one of the components (B1x) of the
perturbed magnetic field corresponding to the spectra in Figure 3.4 (top panel).
The emergence of increasingly larger-scale fluctuations is manifest. Perhaps a
better illustration of this real-space evolution of the firehose turbulence is Figure
3.5 (top panel), which is the space-time contour plot for the middle fifth of the
domain.
3.4.4 Implications for momentum transport
Substituting the second-order pressure tensor calculated in section 3.3.5 into the
large-scale momentum equation (3.62), we get
min0i
du0i
dt
= −∇Π˜ +∇ ·
[
p0ib0b0
(
∆ +
2
βi
)]
, (3.90)
where, in the absence of density and temperature gradients, the total isotropic
pressure Π˜ = p⊥2i +B
2
0/8pi is set by the condition ∇ ·u0i = 0,
9 while the pressure
anisotropy ∆ is given by equation (3.64). A remarkable feature of equation (3.90)
is that all of the effects of the magnetic field appear in the term proportional
9More generally, Π˜ adjusts in such a way as to reconcile the pressure balance with the
continuity and heat conduction equations; see Appendix C.0.17.
3.5 Gyrothermal turbulence 123
to ∆ + 2/βi, which is precisely the instability parameter that the small-scale
firehose turbulence described in section 3.4.3 contrives to make vanish. In the
marginal state that results, the tension force (the 2/βi term) is almost entirely
cancelled by the combined pressure anisotropy due to large- and small-scale fields.
This suggests that in regions of the plasma where the firehose is triggered (i.e.,
where the magnetic field is locally decreased by the plasma motion), the plasma
motions become effectively hydrodynamic, with magnetic-field lines unable to
resist bending by the flows.
Since the cancellation of the second term in equation (3.90) by the firehose tur-
bulence also removes the (parallel) viscosity of the plasma, these hydrodynamic
motions are not dissipated. In a turbulent situation, this should enable a cascade
to ever smaller scales. Obviously, once this happens, the original motion that
caused the negative pressure anisotropy to develop is supplanted by other, faster
motions on smaller scales. The theory developed in this work eventually breaks
down because the scale separation that formed the basis of our asymptotic ex-
pansion is compromised: while the fluid motions penetrate to smaller scales, the
firehose fluctuations move to larger scales (see section 3.4.3).
Note also that the fluid motions produced by the turbulent cascade can give rise
to both positive and negative pressure anisotropies — and to have a full descrip-
tion of their further evolution, we must know the effect on momentum transport
not just of the firehose but also of the mirror and other instabilities triggered by
positive pressure anisotropies (locally increasing magnetic field strength). This
remains to be determined. Another important adjustment to the viscous-stress
reduction argument above has to do with the modification of the firehose insta-
bility by the parallel ion heat fluxes — this we now proceed to investigate.
3.5 Gyrothermal turbulence
3.5.1 Firehose turbulence equation with heat fluxes
As we briefly mentioned in section 3.3.5, allowing a non-zero ion temperature
gradient along the unperturbed magnetic field leads to substantial modifica-
124 3. Firehose instability
tions. These are of two kinds. First, as shown in Appendix C.0.18, the pressure
anisotropy ∆0 caused by the large-scale dynamics contains contributions from the
collisional parallel heat fluxes (proportional to b0 ·∇T0i) and from compressive
motions (as we pointed out in footnote 5, the presence of a temperature gradi-
ent automatically implies a density gradient as well because of the requirement
that pressure balance should be maintained; see equation (C.25) and Appendix
C.0.17). Instead of equation (3.65), valid in the incompressible case, we must
use the more general equation (C.57). This, however, does not change much: the
unstable firehose fluctuations will grow in the manner described in section 3.4.3,
first exponentially, then secularly, to compensate whatever pressure anisotropy is
set up by the large-scale dynamics. The only change is the physical interpreta-
tion of the origin of the pressure anisotropy: as long as ion temperature gradients
are present, the anisotropy is not tied exclusively to the change in the magnetic
field. Physically, the heat-flux contributions to the anisotropy have to do with
the fact that “parallel” and “perpendicular” heat flows along the magnetic-field
lines somewhat differently and so imbalances between p⊥ and p‖ can occur —
this can be seen already from the CGL equations (see Appendix C.0.19).
The second heat-flux-related modification of the theory developed thus far is more
serious. It involves an additional contribution to the FLR term in the third-order
pressure tensor (equation (3.66)) and, therefore, to the firehose turbulence equa-
tion (3.68). This contribution was derived in Appendix C.0.20, but suppressed in
our previous discussion. It is given by equation (C.63) and consequently equation
(3.68) now reads
∂2B⊥1
∂t2
=
v2th,i
2
∇2‖
[(
∆ +
2
βi
)
B⊥1 +
1
Ωi
(
∂B⊥1
∂t
− ΓT
vth,i∇‖B⊥1
B0
)
× b0
]
.(3.91)
We have introduced a dimensionless parameter measuring the magnitude of the
parallel heat flux:10
ΓT =
1
2
vth,i
νi
b0 ·∇T0i
T0i
=
1
2
λmfp
lT
, (3.92)
10We stress that we are discussing the effect of the ion heat flux as the electrons are assumed
isothermal at the scales we are considering (see Appendix B). We also stress that these heat-
flux effects enter through the FLR terms in the plasma pressure tensor and are absent in, e.g.,
the lowest-order Braginskii (1965) equations.
3.5 Gyrothermal turbulence 125
where lT is the parallel length scale of the ion temperature variation. We see
that the functional form of the firehose turbulence equation is changed. We now
proceed to study the effect of this change.
-0.4 -0.2 0 0.2 0.4-0.04
-0.02
0
0.02
0.04
-0.2 0 0.2-0.004
-0.002
0
0.002
0.004
Figure 3.6: Left panel: frequencies (thin lines) and growth rates (bold lines) of
the unstable firehose modes (red/solid: the “+” mode; blue/dashed: the “−”
mode) given by equation (3.94); the parameters here are ∆ + 2/βi = −0.01 and
ΓT = 0.02, so the instability parameter is Λ = 0.0054 (equation (3.95)). Right
panel: same, but for ∆ + 2/βi = 0.00075, so Λ = 0.000025 (close to marginal sta-
bility); dotted vertical lines indicate the wavenumber of fastest growth kp = 0.085
(equation (3.100)) and the dotted horizontal lines the corresponding maximum
growth rate γmax = Imωp = 0.0004 (equation (3.101)).
3.5.2 Linear theory: the gyrothermal instability
The linear dispersion relation for equation (3.91) is
[
ω2 −
k2‖v
2
th,i
2
(
∆0 +
2
βi
)]2
=
k4‖v
4
th,i
4
(
ω + k‖vth,iΓT
)2
Ω2i
. (3.93)
Like in the case of equation (3.69), there are four roots of which two are potentially
unstable:
ω
Ωi
= ±
k2
4
+ i
|k|
√
2
√
−
(
∆ +
2
βi
)
∓ kΓT −
k2
8
, (3.94)
126 3. Firehose instability
where k = k‖ρi. Instability occurs at wavenumbers for which the expression under
the square root is positive. There is an interval of such unstable wavenumbers if
and only if
Λ ≡ Γ2T −
1
2
(
∆ +
2
βi
)
> 0. (3.95)
If this condition is satisfied, the “+” mode is unstable for
−4
(
ΓT +
√
Λ
)
< k < −4
(
ΓT −
√
Λ
)
, (3.96)
and the “−” mode for
4
(
ΓT −
√
Λ
)
< k < 4
(
ΓT +
√
Λ
)
. (3.97)
where we have assumed, without loss of generality, that ΓT > 0. When ∆+2/βi <
0, these two intervals intersect, so all modes with |k| < k0 = 4
(
ΓT +
√
Λ
)
are unstable (others are pure propagating waves). When ∆ + 2/βi > 0, the
intervals are separated and there is an interval of stability at long wavelengths,
viz., |k| < 4
(
ΓT −
√
Λ
)
.
What is remarkable about all this is that not only the stability conditions and
specific expressions for the firehose growth rate are modified by heat flux, but
the presence of the heat flux allows for instability even when firehose is stable,
∆ + 2/βi > 0 (but positive pressure anisotropy not too large and βi not too
small, subject to equation (3.95)). This gyrothermal instability (GTI), leads to
the growth of Alfve´nically polarised fluctuations in the parameter regime in which
they are otherwise stable (Schekochihin et al., 2010).11
The formulae for the wavenumber of the fastest-growing mode and the maximum
growth rate for the combined firehose-GTI are straightforward to write down.
They are not particularly illuminating in the general case, but are interesting in
various asymptotic limits. When the firehose instability parameter ∆ + 2/βi < 0
and its magnitude is much larger than Γ2T , the effect of the heat flux is a small cor-
rection to the firehose instability already described in section 3.4.2. Conversely,
11Note that for ∆− 1/βi > 0, the mirror mode is unstable as well.
3.5 Gyrothermal turbulence 127
when |∆ + 2/βi|  Γ2T , the GTI is dominant and, for the fastest growing mode,
kp ' ∓6ΓT , (3.98)
ωp ' 9Γ
2
T
(
±1 +
i
√
3
)
, (3.99)
where ωp is normalised to Ωi. Finally, close to the marginal state, Λ → +0, we
have
kp ' ∓4ΓT
(
1 +
Λ
Γ2T
)
, (3.100)
ωp ' 4Γ
2
T
(
±1 +
i
√
Λ
ΓT
)
. (3.101)
Note that, unlike the firehose, the GTI has a definite preferred wavenumber that
does not change as marginal stability is approached.
Figure 3.6 (left panel) shows the dependence of the frequencies and growth rates
of the two unstable modes on wavenumber for a set of parameters for which the
instability is a hybrid of firehose and GTI (these are the parameters used in the
numerical solution of section 3.5.3). Figure 3.6 (right panel) shows the same for
the case in which the firehose is stable (∆ + 2/βi > 0) and that is very close
to marginal stability: we see that the instability only exists in the immediate
neighbourhood of the last unstable wavenumber given by equation (3.100).
3.5.3 Nonlinear evolution and spectrum
Firehose-GTI turbulence equation in scalar form
As happened in section 3.4.3, equation (3.91) can be reduced to one equation
for a scalar field, although it is now a slightly more complicated transformation.
Let us again non-dimensionalise time and space according to equation (3.76)
and introduce new fields A±k (t) (not to be confused with the vector potential) as
follows:
B1x
B0
± i
B1y
B0
= A±k exp
[
∓i
(
k2
4
t+ φk
)]
. (3.102)
128 3. Firehose instability
Figure 3.7: Top panel: evolution of the magnetic energy |B⊥1 |2/B
2
0 =
∑
k |Ak|
2
with time in a numerical solution of equations (3.103) and (3.104) with parame-
ters (3.109); the red dotted line shows the nonlinear asymptotic given by equation
(3.105); this figure is the GTI analog of Figure 3.3 (left panel). Bottom panel:
evolution of the instability parameter Λ (pink) and the pressure anisotropy pa-
rameter ∆ + 2/βi (black) in the same numerical solution. Inset: log-log plot of
the evolution of |Λ|; the black line shows the slope corresponding to 1/t2 (see
equation (3.107)).
3.5 Gyrothermal turbulence 129
With the ansatz (3.102), equation (3.91) becomes
∂2A±k
∂t2
=
k2
2
[
−
(
∆ +
2
βi
)
∓ kΓT −
k2
8
]
A±k , (3.103)
∆(t) = ∆0 +
3
2
∫ t
0
dt′e−3ν∗(t−t
′) ∂
∂t′
∑
k
|A+k (t
′)|2 + |A−k (t
′)|2
2
. (3.104)
It is now manifest how the dispersion relation (3.94) emerges from equation
(3.103). Unlike in the case of pure firehose turbulence (ΓT = 0), the evolu-
tion of the mode now depends on the sign of its real frequency — that is why
we have two scalar equations. However, these equations have a symmetry: if
we arrange initially that A+k = A
−
−k (which we can always do by an appropriate
choice of the phases φk), then this relation will continue to be satisfied at later
times. This also means that A±k are real because, in order for B
⊥
1 to be a real
field, we must have (from equation (3.102)) (A+k )
∗ = A−−k (we assume the phases
satisfy φk = φ−k). The conclusion is that it is enough to solve just one of the two
equations (3.103) — either for the + or the − mode. The total energies of the
two modes that enter equation (3.104) are equal.12
Qualitative picture
The evolution of the firehose-GTI turbulence is easy to predict arguing along the
same lines as we did in section 3.4.3. Let us consider the case when initially
the pressure anisotropy is negative and −(∆0 + 2/βi) 2Γ2T , i.e., the instability
parameter Λ0 > 0 (given by equation (3.95) with ∆ = ∆0). In this regime, the
heat flux does not matter and the evolution proceeds as in the case of the firehose
turbulence: magnetic fluctuations grow and eventually the nonlinear feedback in
equation (3.104) starts giving an appreciable positive contribution to the pressure
anisotropy (estimates (3.80) and (3.81) for the fluctuation amplitude at which
this happens are still valid). A k−3 spectrum will then form, with the infrared
cutoff (wavenumber of maximum growth) moving to larger scales and |∆ + 2/βi|
12The same approach could have been taken in section 3.4.3: instead of solving equation
(3.103) for a complex function Ak subject to A∗k = A−k, we could have solved for one of two
real functions A±k subject to A+k = A−−k. The magnetic field is then recovered via equation
(3.102).
130 3. Firehose instability
Figure 3.8: Top panel: spectrum of the magnetic fluctuation energy at three
specific times (tνi = 1, 4, 16) during the evolution shown in Figure 3.7; red short-
dashed lines show the k−3 slope; blue long-dashed lines show the firehose/GTI
growth rate (see section 3.5.2) for the instantaneous values of ∆ + 2/βi at the
times the spectra are plotted. Bottom panel: magnetic fluctuations in real space
(B1x/B0 vs. z) at the same times as the spectra in the left panel. This figure is
the GTI analog of Figure 3.4.
3.5 Gyrothermal turbulence 131
decreasing (i.e., ∆ + 2/βi increasing and thus becoming less negative).
The evolution of the gyrothermal fluctuations starts to differ from the pure fire-
hose case after |∆ + 2/βi| becomes comparable to Γ2T . The GTI is now the dom-
inant instability mechanism. Since the fluctuations continue growing, ∆ + 2/βi
continues to increase and will become positive, tending eventually to 2Γ2T , so as
to push the instability parameter Λ (equation (3.95)) to zero and the GTI to its
marginal state. As Λ → +0, the growth is concentrated in a shrinking neigh-
bourhood of the wavenumber kp = 4ΓT (see equation (3.100)). This means that
the spectrum stops spreading towards lower wavenumbers and its infrared cutoff
stabilises at kp. All the growth of magnetic energy is now provided by the growth
of the one mode associated with kp, which will soon tower over the rest of the
spectrum.
The growth is still secular: using equation (3.104) and the marginality condition
Λ = 0, we find to dominant order, analogously to equation (3.82),
∆ ' 2Γ2T −
2
βi
⇒
∑
k
|Ak|
2 ' 4Λ0ν∗t. (3.105)
Finally, we can calculate the evolution of the residual Λ. Analogously to equation
(3.85), the growing mode satisfies
1
Akp
∂Akp
∂t
∼
1
t
∼ γmax ∼ 4ΓT
√
Λ, (3.106)
where we used equation (3.101) for γmax. Therefore,
Λ ∼
1
t2
. (3.107)
As in the case of the firehose turbulence, the secular growth will continue until the
fluctuation amplitude is no longer small: t ∼ (νiΛ0)−1 ∼ |γ0|−1 (time scale of the
large-scale dynamics). The key difference from the pure firehose case is that the
fluctuations are now stuck at a microscopic spatial scale given by equation (3.100):
restoring dimensions and using equation (3.92), the corresponding wavenumber
132 3. Firehose instability
is
k‖ρi ∼
λmfp
lT
(3.108)
(this scale is collisionless, k‖λmfp  1, provided lT  λ2mfp/ρi; for galaxy clusters,
this is always true as is easy to ascertain by using the numbers from section
3.2.1). Thus, the gyrothermal turbulence is essentially one-scale, in the sense
that fluctuations at this one scale become energetically dominant as marginal
stability is approached at late stages of the nonlinear evolution.
Numerical solution
We have solved equations (3.103) and (3.104) in a manner completely analogous
to that described in section 3.4.3. The parameters we used are
∆0 = −0.02,
2
βi
= 0.01, ΓT = 0.02, ν∗ =
ρi
λmfp
= 0.0001.
This implies that the instability parameter in the linear regime is Λ = 0.0054
(equation (3.95)) and so the maximum unstable wavenumber is k0 = 4(ΓT +
√
Λ) ' 0.37 (equations (3.96) and (3.97); see Figure 3.6 (left panel)). Our nu-
merical solution now has 2048 wavenumbers, so kmin ' 0.00063 and kmax ' 0.64.
As expected, the evolution of the total magnetic energy is similar to the case
of pure firehose turbulence discussed in section 3.4.3: exponential, then secular
growth (see equation (3.105)) — this is shown in Figure 3.7 (left panel). The evo-
lution of the instability parameter Λ (equation (3.95)) towards its zero marginal
value is given in Figure 3.7 (right panel). The inset shows that this approach
to zero is consistent with the 1/t2 prediction (equation (3.107)). Also shown in
Figure 3.7 (right panel) is the evolution of the pressure anisotropy parameter
∆ + 2/βi, which for the pure firehose used to be the instability parameter. Since
Λ→ 0, it should tend to 2Γ2T = 0.0008 and it indeed does.
Finally, Figure 3.8 (left panel) illustrates the evolution of the spectrum of fire-
hose/gyrothermal fluctuations. It follows the scenario outlined in section 3.5.3.
At first it is similar to the firehose turbulence spectrum with the spectral peak
moving towards larger scales leaving behind a k−3 spectrum. As the wavenum-
3.5 Gyrothermal turbulence 133
ber of fastest growth kp approaches the value corresponding to the near-marginal
GTI, kp = 4ΓT = 0.08 (see equation (3.100)), the peak stays there and continues
growing, eventually dominating all other modes. The emergence of a one-scale
sea of gyrothermal fluctuations is further illustrated by Figure 3.5 (right panel),
which shows what these fluctuations look like in real space as time progresses.
The difference between them and the pure firehose fluctuations in Figure 3.5 (left
panel) is manifest: the gyrothermal ones stay at the same scale while the firehose
ones become larger-scale as time progresses.
3.5.4 Implications for momentum and heat transport
Let us now revisit the discussion of the effect of plasma instabilities on the mo-
mentum transport modification attempted for the pure firehose in section 3.4.4.
As before, the combined large-scale viscous and Maxwell stress is contained in
the second term on the right-hand side of equation (3.90). However, with parallel
ion heat fluxes present, the nonlinear evolution of the GTI pushes the quantity
∆ + 2/βi not to zero but to a positive value 2Γ2T , corresponding to the marginal
state Λ = 0 (equation (3.105)). Since any smaller value of ∆ + 2/βi is GTI un-
stable, this leads to a curious conclusion that the momentum transport is now
effectively determined by the ion heat flux:
min0i
du0i
dt
= −∇Π˜ +∇ ·
(
p0ib0b02Γ2T
)
= −∇Π˜ +∇ ·
[
b0b0
n0i (b0 ·∇T0i)
2
miν2i
]
,
where we used equation (3.92) for ΓT . This equation has to be supplemented
with the transport and pressure-balance equations for n0i, T0i and Π˜ as explained
in Appendix C.0.17.
Equation (3.109) probably merits a careful study (which is a subject for future
work), but we would like to accompany it with a very important caveat. Since
pressure anisotropy in the nonlinear state of the GTI can be positive, other
plasma instabilities may be triggered. Thus, if ∆ > 1/βi, i.e., if Γ2T > 3/(2βi),
the plasma will be mirror unstable. The magnetic fluctuations that the mirror
instability produces are different from the GTI both in polarization (δB‖, not
δB⊥) and scale (k⊥  k‖, k⊥ρi ∼ (∆− 1/βi)1/2, k‖ρi ∼ ∆− 1/βi for the mirror,
134 3. Firehose instability
whereas for the GTI we had k⊥ = 0, k‖ρi ' 4ΓT ). How they saturate and
what they do to the effective pressure anisotropy is a matter under active current
investigation (e.g. Rincon et al., 2010) — and it is completely unknown how
mirror and gyrothermal fluctuations might coexist.
The key question is whether the pressure anisotropy will be set by the GTI or the
mirror marginal condition. If it is set by the latter (∆ = 1/βi, as, e.g., seems to be
indicated by the solar wind data; see Bale et al. 2009), then one may ask whether a
turbulent plasma has a way of suppressing the GTI by adjusting not the pressure
anisotropy but the heat flux to the marginal condition: Γ2T = 3/(2βi). This
raises the possibility that not only the pressure anisotropy but also the (ion) heat
fluxes are determined by the marginal stability conditions of the firehose/GTI
and mirror. Thus, plasma instabilities may be the crucial factor in setting both
the momentum and heat transport properties of a weakly collisional plasma. We
stress, however, that under the assumptions adopted in this work, we have not
produced a nonlinear mechanism for changing the ion heat flux and this remains
a subject for future work.
3.6 Comparison with previous work and a col-
lisionless plasma
The theory developed above is basically quasilinear in that the fluctuation ampli-
tude is assumed small and it is found that such small fluctuations can drive the
instability to a marginal state. There have been a number of previous quasilinear
treatments of the firehose instability (Shapiro & Schevchenko, 1964; Kennel &
Sagdeev, 1967; Davidson & Vo¨lk, 1968; Gary & Feldman, 1978; Quest & Shapiro,
1996), so it is perhaps useful to explain why they do not obtain similar results.
The approach in such theories is to consider a collisionless plasma with some
initial distribution that has a negative pressure anisotropy (let us call it ∆0 < 0)
and work out how it relaxes. The result is that a fluctuation level builds up,
3.6 Comparison with previous work and a collisionless plasma 135
with13
|B⊥1 |
2
B20
∼
∣
∣
∣
∣∆0 +
2
βi
∣
∣
∣
∣ , (3.109)
which is small when the instability parameter ∆0 + 2/βi is small. This satu-
rated fluctuation level suffices to marginalize the instability. This result is easily
recovered in our theory if we formally set νi = 0 in equation (3.64). This gives
∆(t) = ∆0 +
3
2
|B⊥1 (t)|2
B20
(3.110)
and assuming saturation in the marginal state ∆ = −2/βi, we recover equation
(3.109).
The difference in our approach is to include weak collisions and consider the case
when the pressure anisotropy is constantly driven by the large-scale dynamics
(which is physically how it appears). The steady level of the anisotropy is then
set by the competition between collisions and the drive (equation (3.65)) and to
offset this anisotropy and keep the instability marginal, the fluctuation level has
to keep growing secularly (the earlier quasilinear theories can then be interpreted
to describe correctly what happens before one collision time has elapsed).
It is interesting to inquire what would happen if the anisotropy were driven (rather
than just initially imposed) but collisions were not strong enough to balance the
drive and impose a steady anisotropy. Formally speaking, our theory breaks down
in this case because the equilibrium distribution cannot be proved Maxwellian.
However, that is a technical issue and one could, in fact, reformulate our theory as
a near-marginal expansion in the instability parameter |∆+2/βi|. We expect that
equation (C.52) (or, equivalently, equation (C.59)), with the collisional relaxation
term removed, would still describe the evolution of the anisotropy:
∂∆
∂t
= 3γ0 +
3
2
∂
∂t
|B⊥1 (t)|2
B20
, (3.111)
13Hall (1981) argues qualitatively for a similar saturation level, but due to trapping of parti-
cles in firehose fluctuations. As the systematic kinetic calculation presented above shows, the
trapping effect does not play a role at these amplitudes, at least not under the assumptions we
adopted (k⊥ = 0 and relatively high collision frequency).
136 3. Firehose instability
where γ0 (assumed negative) is the drive — it contains all the terms in equation
(C.52) due the large-scale dynamics. Under these conditions, the driven part of
anisotropy is constantly increasing and so again the fluctuations will have to grow
secularly in order to keep it at the marginal level:
3
2
|B⊥1 (t)|2
B20
=
∣
∣
∣
∣3
∫ t
0
dt′γ0(t
′) +
2
βi
∣
∣
∣
∣ . (3.112)
This will, of course, break down once the fluctuation level is no longer small or
collisions catch up.
There exist a number of numerical studies of the nonlinear evolution of the fire-
hose instability (Quest & Shapiro, 1996; Gary et al., 1998; Hellinger & Mat-
sumoto, 2001; Horton et al., 2004a,b; Matteini et al., 2006). They mostly adopted
the same relaxation-of-initial-anisotropy approach as the quasilinear theories dis-
cussed above and the results they report are broadly consistent in that the mag-
netic fluctuation energy saturates at a level scaling with the size of the initial
anisotropy (see equation (3.109)). A notable exception is the recent work of
Matteini et al. (2006) who consider the anisotropy driven by the expansion of
the solar wind — the fluctuation levels they see are probably well described by
equation (3.112).
The spectrum of the firehose fluctuations has not previously been addressed an-
alytically. However, the results of section 3.4.3 are not in disagreement with
some of the numerical evidence which does point to the growing predominance of
smaller wavenumbers in the nonlinear regime (Quest & Shapiro, 1996; Matteini
et al., 2006).
Finally, to our knowledge, the effect of heat fluxes on the nonlinear behaviour
of the firehose instability, studied in section 3.5, has not been specifically con-
sidered before. Note that although the heat fluxes are present in the numerical
simulations of Sharma et al. (2006, 2007), their momentum equation does not
have the gyroviscous and gyrothermal terms that regularise the firehose at small
scales and give rise to the gyrothermal instability. The appropriate modification
to the fluid equations suggested by Schekochihin et al. (2010) should in principle
enable one to study the spectrum of the firehose and GTI fluctuations numeri-
3.7 Astrophysical implications 137
cally (although, it is not clear whether it will provide the necessary regularisation
for the mirror that one would also expect).
3.7 Astrophysical implications
3.7.1 Solar wind
Much of the observational evidence about the firehose instability comes from the
measurements of pressure anisotropies and fluctuation levels in the solar wind.
Since the wind is expanding, both the local density and the local magnetic-
field strength are dropping, so one expects a negative pressure anisotropy to
develop. This can be described by equation (3.111), where the drive is roughly
γ0 ∼ −Vsw/R (solar wind speed divided by the distance from the Sun) and the
collisional relaxation is neglected. The evidence for this trend, negative pressure
anisotropy developing with increasing distance from the Sun, is given by Matteini
et al. (2007); a number of other papers also document the fact that the measured
pressure anisotropies are bounded from below by the firehose marginal stability
condition (Kasper et al., 2002; Bale et al., 2009).
Bale et al. (2009) found increased levels of ion-Larmor-scale magnetic fluctuations
close to this stability boundary — presumably due to the firehose instability.
There are also indications of an injection of energy into parallel wavenumbers
just above the ion Larmor scale (Wicks et al., 2010) — again, conceivably by the
firehose instability. It is unclear how these firehose fluctuations coexist with the
solar wind inertial- and dissipation-range turbulence. This is one of the contexts
in which the absence of a complete microphysical theory of the firehose turbulence
and its effect on the plasma motions is particularly acutely felt.
3.7.2 Galaxy clusters
In section 3.2, we discussed at length the basic properties of the galaxy cluster
plasmas, the inevitability of pressure anisotropies and, therefore, plasma instabil-
ities arising in a turbulent ICM, as well as the fundamental theoretical questions
that this poses. We cannot answer these questions here as we have analysed only
138 3. Firehose instability
one of several plasma instabilities that must be understood. However, just like in
the case of the solar wind and the accretion flows, an impatient astrophysicist can
conceivably glimpse the contours of the eventual theory by constraining pressure
anisotropies and possibly also heat fluxes by the marginal stability conditions
of the plasma instabilities — with all the same caveats and acknowledgments of
uncertainty as discussed above.
Let us discuss how far this approach can take us in answering some of the problems
posed in sections 1.1 and 3.2.1.
Regulation of cooling flows
The apparent refusal of the galaxy cluster cores to exhibit a cooling catastrophe
(e.g., Peterson & Fabian, 2006) has long evaded a satisfactory theoretical expla-
nation. A comprehensive review of the relevant literature is outside the scope
of this brief discussion. It is probably fair to summarise the two main physical
mechanisms invoked to explain the relatively weak drop in the ICM temperature
between the bulk and the core as thermal conduction and some form of viscous
conversion into heat of the mechanical energy injected into the ICM by the cen-
tral active galactic nuclei (probably in a self-regulating way). It is clear that
the latter mechanism cannot be ignored because the thermal conductivity of the
ICM is unlikely to be sufficiently large (e.g., Voigt & Fabian, 2004) and at any
rate, thermal conduction is a thermally unstable mechanism of balancing radia-
tive cooling. Kunz et al. (2010) recently proposed that if a sufficient amount of
turbulent power is assumed to be available, the viscous heating, regulated by the
pressure anisotropy and, therefore, by the marginal stability of the mirror and/or
firehose instabilities, can balance the cooling in a thermally stable way. They also
found that assuming such a balance leads to reasonable predictions of the mag-
netic field strength, magnitude of the turbulent velocities and the outer scale of
the turbulence in the ICM (interestingly, both for the cool-core and non-cool-core
clusters).
3.8 Conclusion 139
Temperature fluctuations and the GTI
While detailed simulations of the turbulent ICM, bubble dynamics etc. similar to
those of Sharma et al. (2006, 2007) for accretion flows have not been attempted,
the marginal stability condition for the GTI (equation (3.95)) could perhaps be
used to impose a lower bound on the typical scale of temperature fluctuations in
the ICM (Schekochihin et al., 2010). Indeed, if the magnitude of the ion heat
flux is limited so as to prevent the GTI from being unstable (see section 3.5.4),
then from equations (3.95) and (3.92), we get
lT & β
1/2
i λmfp ∼ 5× 10
−4 T
5/2
i
n1/2i B
∼ 1.4× 1021 cm, (3.113)
where ni is in cm−3, Ti is in K, B is in G, and the numerical value has been com-
puted for the plasma parameters in the core of Hydra A discussed in section 3.2.1
(see equations (3.6) and (3.8)). Interestingly, kpc-scale temperature fluctuations
are indeed observed in cool-core clusters (e.g. Fabian et al., 2006; Lagana et al.,
2009). Furthermore, if we use in equation (3.113) the physical parameters appro-
priate for the bulk of the cluster plasma, rather than the cores (say, Ti ∼ 108 K
and ni ∼ 10−3 cm−3) we would get much larger scales — in the 100 kpc range,
which is also consistent with reported observational values for the cluster bulk
(Markevitch et al., 2003).
3.8 Conclusion
Let us recapitulate what we have discovered and how it relates to what was
known previously. It has been appreciated for some time that macroscale tur-
bulence of magnetized weakly collisional plasma (exemplified by the ICM) will
naturally produce pressure anisotropies, which will in turn trigger firehose and
mirror instabilities at spatial and temporal microscales (Schekochihin et al., 2005,
see sections 3.1 and 3.2). Since the pressure anisotropies are essentially due to
local temporal change of the magnetic field strength, it is qualitatively intuitive
that the nonlinear evolution of the instabilities is governed by the tendency to
cancel this change on average; hence it follows that in a driven system (see dis-
140 3. Firehose instability
cussion in section 3.6) the fluctuations must continue growing in the nonlinear
regime, albeit secularly rather than exponentially (see Schekochihin et al., 2008).
In this thesis, we have constructed a full ab initio (weakly) nonlinear kinetic
theory of this process for the parallel (k⊥ = 0) firehose instability, which is the
simplest analytically tractable case. Not only the evolution of the fluctuation
energy but also of the full spectrum of the resulting firehose turbulence has been
worked out, including the effect of gradual spreading of the fluctuations to ever
larger scales as the nonlinearly compensated pressure anisotropy approaches its
marginal-stability value (section 3.4.3). We have also extended our kinetic calcu-
lation to include the effect of ion temperature gradients parallel to the magnetic
field (parallel heat fluxes). As was pointed out recently, they lead to a new in-
stability, the GTI, of parallel Alfve´nic fluctutions (Schekochihin et al., 2010, see
section 3.5.2). Here we have constructed a nonlinear theory of its evolution, fea-
turing again a secular growth of magnetic fluctuations, but this time developing
a spectrum heavily dominated by a particular scale (section 3.5.3).
While a speculative discussion of the implications of these results for transport in
a general magnetized plasma (sections 3.4.4–3.5.4) and for particular astrophys-
ical systems (section 3.7) is possible, a full transport theory has to await, at the
very least, the completion of similar ab initio kinetic investigations of the nonlin-
ear evolution of the mirror instability and of the oblique (k⊥ 6= 0) firehose. Only
then can one attempt to devise an effective mean field theory for the macroscale
dynamics of cosmic plasmas based on solid microphysical foundations. One goal
of this research has been to set forth a template for building these microphysical
foundations.
In the meanwhile, it appears sensible to rely on (or at least consider reasonable)
the semiquantitative closure approach to the macroscale dynamics based on the
assumption that average pressure anisotropies and, probably, also heat fluxes,
are set by the marginal stability conditions of the microscale plasma instabilities.
This approach has found strong observational support in the solar wind mea-
surements (e.g. Bale et al., 2009) and has already yielded nontrivial and possibly
sensible physical predictions for the evolution of cosmic magnetism (Schekochihin
& Cowley, 2006), accretion disk dynamics (Sharma et al., 2006, 2007), and the
3.8 Conclusion 141
turbulence and heating in the intracluster medium (Kunz et al., 2010).
142
Part IV
Conclusion
143

Chapter 4
Conclusions
4.1 Outlook
Before we summarise what we have found let us speculate on an example where
both the types of magnetized processes considered in this thesis might come
together.
Consider a laminar, isothermal, magnetized, differentially rotating disc with a
decreasing angular velocity profile, perhaps the early ISM or the hot accretion
flows of Quataert et al. (2002); Sharma et al. (2003). We saw in chapter 2 that
in the collisional regime, even if the disc is initially laminar, it cannot remain so
for long if the ambient magnetic field has a vertical component.
The magnetized MRI destabilises the flow leading, in the linear regime, to mag-
netic field amplification at a rate proportional to the disc’s shear (and in excess
of its unmagnetized counterpart). As the field increases in strength the Larmor
radius decreases (so the plasma becomes more magnetized) and purely grow-
ing modes mutate into over-stabilities (provided the background field has an
azimuthal component). Both of these factors will presumably affect the turbu-
lence that eventually arises. However, even before fully developed turbulence
has arisen, the growing magnetic field will generate pressure anisotropies. These
will in turn generate micro-instabilities with growth rates that are, if properly
treated, in general well in excess of the MRI. These must be accounted for.
The numerical models of Sharma et al. (2006, 2007) employ an ad hoc closure,
145
146 4. Conclusions
constraining the pressure anisotropy to lie within the marginal stability bound-
aries. They argue that this is justified if it can be shown microphysically that
plasma instabilities (in their case, ion cyclotron and firehose) produce fluctu-
ations at the ion Larmor scale, where pitch-angle scattering of particles off the
fluctuation “foam” isotropizes pressure. (The same view was taken by Schekochi-
hin & Cowley (2006) in their model of the dynamo action in a weakly collisional
plasma.)
This is not the only possibility and our results for the parallel (k⊥ = 0) firehose
do not support this picture as the fluctuations are not quite at the Larmor scale
even in the linear regime and move to larger scales in the nonlinear stage. It is
possible that the oblique firehose (which is much harder to treat analytically than
the parallel one considered here) might produce fluctuations at the ion Larmor
scale. However, it is not obvious that the validity of a closure based on the average
pressure anisotropy being maintained at the marginal level must be predicated
on the possibility of pitch-angle scattering. As we have shown above, a sea of
secularly growing magnetic fluctuations far above the Larmor scale can produce
the same effect.
What we do not know is the fate of these firehose fluctuations over long (trans-
port) time scales and the eventual structure of the tangled magnetic field that
results — a crucial question for accretion theories because they require knowledge
of the Maxwell and Braginskii stresses in order to estimate the rate of the angular
momentum transport (Shakura & Sunyaev, 1973). However, let us assume that
somehow the firehose (and other microscale instabilities such as the mirror) pin
the pressure anisotropy at the marginal value for the microscale instabilities. In
this case the viscous stress generated by the Braginskii viscosity will proceed to
heat the plasma at a rate ∝ ∆2 (Kunz et al., 2010).
Spatial inhomogeneities in the growth rate of the magnetic field, as one would
expect in a turbulent system, will lead to inhomogeneities in the local pressure
anisotropy, both in magnitude and sign. In regions of decreasing field strength
the firehose will pin the anisotropy at |∆| = 2/β and in regions of increasing
field strength the mirror will pin it at |∆| = 1/β. The implication of this is that
heating in regions of increasing and decreasing magnetic field strength could differ
4.1 Outlook 147
by a factor of ∼ four. If differential heating of this nature does occur then one
would expect temperature gradients and therefore, possibly, the GTI to develop.
It is difficult to even conjecture what happen next without numerical evidence,
but a number of possibilities present themselves. The GTI could regulate the
pressure anisotropy in a way that encouraged more evenly distributed heating,
thereby keeping the disc isothermal. It could facilitate the transport of heat
directly, thereby eradicating temperature gradients. Alternatively, as suggested
by Schekochihin et al. (2010), it could distort the magnetic field δb ∼ 1 to
minimise the parallel ion temperature gradient thereby switching itself off. In
magnetized plasmas on fluid scales, such behaviour has been found for the fluid
instability, the HBI (Bogdanovic´ et al., 2009; Parrish & Quataert, 2008).
What all this means for the magnetic field structure is entirely unclear too. How
a turbulent dynamo might operate in such a plasma remains a completely open
problem. Schekochihin & Cowley (2006) have made a speculative attempt to
leapfrog the detailed microphysical derivations and model the large-scale dy-
namics based on the idea that microscale instabilities would always isotropize
pressure towards marginal stability values. This led to a rather dramatic con-
clusion that the ICM might support self-accelerating, explosive dynamos. While
this conclusion remains to be tested by more rigorous analytical approaches, it
does illustrate the general conjecture that plasma instabilities are likely to result
in radical changes to, rather than merely small corrections to, the large-scale
dynamics of cosmic plasmas.
One thing is clear: On the basis of our findings and the literature cited in this
thesis, the default assumption should be that magnetized plasmas are categori-
cally different to their unmagnetized counterparts (and also ubiquitous). Filling
in the missing pieces represented by the crude multi-scale picture of Figure 1.2
constitutes an important and well posed scientific problem for which we now
reiterate what we think are potentially fruitful lines of attack.
First, we suggest a suite of global and local nonlinear simulations to investigate
the magnetized MRI. Prescriptions for handling the firehose and GTI could come
from the gyroviscous corrections found in chapter 3 and the mirror could, if
feasible, be corrected by the FLR term from linear theory or simply left as it
148 4. Conclusions
is. It would be especially interesting to include a viscous heating term too as
per the discussion above. Second, the development of a transport theory for the
mirror and oblique firehose is absolutely imperative (see section 3.8). Until such
theories are developed any magnetized plasma theory that constrains the pressure
anisotropy to the marginal stability boundary by fiat, will be introducing an
arbitrary step. Both projects would contribute significantly to our understanding
of astrophysical plasma dynamics.
4.2 Summary
The principal topic of this thesis has been the evolution and nature of two particu-
lar magnetized plasma instabilities and, where relevant, their effect on transport.
These instabilities are the MRI and the firehose instability.
In chapter 1 we explained how magnetized plasmas differ from unmagnetized
plasmas and neutral fluids, and therefore why they require a separate treatment.
We introduced the basic formalism to do this and showed, assuming the cyclotron
frequency and Larmor radius are the smallest parameters in the system, that the
pressure tensor is naturally anisotropic. The same analysis showed that this
tensor can be naturally decomposed into gyrophase-dependent and gyrophase-
independent terms, and we explained the role each component was to have for
the rest of our work. For the former, which constitutes the pressure anisotropy, we
provided a physical description. The theory in this chapter laid the groundwork
from which the next two chapters followed.
In chapter 2 we considered the global linear stability of a magnetized, collisional
plasma (the pressure anisotropy was described by the Braginskii viscosity) in the
presence of a galactic shear flow and a variety of equilibrium magnetic field con-
figurations. We treated the system as a two parameter problem in the magnetic
field strength, and SB a dimensionless combination of the collision frequency,
rotation rate and magnetic field strength that is large when magnetized effects
are important and small when they are not. We found a global version of the
MRI in the presence of Braginskii viscosity, and showed its growth rate and the
scale of the most unstable mode varies as a function of both parameters. Moti-
4.2 Summary 149
vated, in part, by understanding the existence of turbulence in quiescent regions
of the ISM, we considered three galactically relevant background magnetic field
configurations: purely azimuthal, purely vertical and slightly pitched. We found
that the first two cases are singular limits in which the effects of the Braginskii
viscosity are stabilising for all modes (whch is in disagreement with local anal-
ysis). In particular, we found numerically that for a vertical field the growth
rate varies ∝ 1/(SBB2) = Re for small values of the Reynolds number. This was
in agreement with the accompanying global asymptotic analysis. In the pitched
field case we found that instead of the Braginskii viscosity being a stabilising
influence it leads to an enhanced growth rate. In the limit SB  1 the magne-
tized MRI tended to a value 2
√
2 times faster than its unmagnetized counterpart.
In the weak magnetic field regime we determined (numerically and asymptoti-
cally) the scaling of its (purely real) growth rate '
√
2(1 − 21/4/3 S−1/2B ) and
the Alfve´n frequency at which this most unstable mode occurs ∝ S −1/4B . For
finite magnetic field strengths we found these modes became over-stable and the
frequency of the travelling wave component also increases with the SB and, in
the limit SB  1 it can be up to an order of magnitude greater than the unmag-
netized over-stability that also occurs for finite magnetic field strengths. We also
provided a physical explanation of this behaviour, compared our results to local
analysis and discussed some implications of our results for modelling magnetized
accretion discs.
In chapter 3 we considered the local evolution of the firehose instability that is
driven by the pressure anisotropy that naturally arises when the magnetic field
strength decreases. We showed that in the long-wavelength limit (e.g. lowest
order Braginskii theory) the governing equations are ill-posed as a result of the
instability. Generally, and in the specific context of the ICM of galaxy clusters,
we argued that the instability can dramatically affect the large-scale ( λmfp) dy-
namics. We argued that its nonlinear evolution should drive pressure anisotropies
towards marginal stability values, controlled by the plasma beta βi. As such,
we developed a nonlinear theory for the instability in an first-principles kinetic
calculation for the parallel (k⊥ = 0) firehose in a high-beta plasma. We used
a particular physical asymptotic ordering to determine the pressure anisotropy,
and therefore close our equations. We obtained a single nonlinear equation for
150 4. Conclusions
the firehose turbulence. (The lengthy details of this derivation were relegated
to Appendices B and C). In the nonlinear regime, both analytical theory and
the numerical solution show that the instability leads to secular (∝ t) growth
of magnetic fluctuations. The fluctuations develop a k−3‖ spectrum, extending
from scales somewhat larger than ρi to the maximum scale that grows secularly
with time (∝ t1/2); the relative pressure anisotropy tends to the marginal value
−2/βi. When a parallel ion heat flux is present, the parallel firehose instability
mutates into the GTI, which continues to be unstable up to values of pressure
anisotropy that can be positive and are limited by the magnitude of the heat flux.
Its nonlinear evolution also involves secular growth of the magnetic energy, but
the fluctuation spectrum is eventually dominated by modes around a maximal
scale ∼ ρilT/λmfp, where lT is the typical scale of the parallel temperature varia-
tion. Implications for momentum and heat transport were discussed, as were the
implications for the solar wind and, our principle example, galaxy clusters.
Throughout we have attempted to highlight areas where our results are related to
other magnetized plasma phenomena and in this chapter we presented a highly
speculative account of how the two main processes we consider in this thesis could
interact.
We hope that our findings have provided some insight, if not sparked an interest,
in magnetized plasma physics.
Part V
Appendices
151

Appendix A
Gyrophase dependent pressure
Following Hazeltine & Meiss (2003) and Schekochihin et al. (2010) we determine
G, the gyrophase dependent part of P(1)s =
∫
d3vvvf (1)s .
The results in this section are entirely general in the sense that they do not
depend on the species type, and therefore we suppress the species subscript on
all quantities barring the distribution function, f (0)s , f
(1)
s .
Using suffix notation we can write the T that appears in the gyrophase dependent
part of vv, equation (1.23), as
Tij =
1
4
Mijklvkvl, (A.1)
where
Mijkl = (δik + 3bibk)jlnbn + ilnbn(δjk + 3bjbk). (A.2)
Substituting this expression for Tij into equation (1.28) and integrating by parts
yields
Gij = −
Mijkl
4
∫
d3vvkvl
∂f (1)s
∂ϕ
= −
Mijkl
4Ω
∫
d3vvkvl
[
df (0)s
dt
+ v ·∇f (0)s − v · ∇u ·
∂f (0)s
∂v
− C[f (0)s ]
]
,
(A.3)
153
154 A. Gyrophase dependent pressure
where we have substituted in equation (1.21) for ∂f (1)s /∂ϕ in the second line and
eliminated the term proportional to a after integration by parts and the use of
∫
d3vvf (0)s = 0 by definition of the peculiar velocity. The full expression for Gs is
therefore
Gij = −
Mijkl
4Ω
[
dP (0)kl
dt
+∇mQ
(0)
mkl+
(
δmnP
(0)
kl + δknP
(0)
ml + δlnP
(0)
mk
)
∇mun − C
(0)
kl
]
(A.4)
where C(0)kl = m
∫
d3vvkvlC[f
(0)
s ] andQ
(0)
mkl = m
∫
d3vvmvkvlf
(0)
s . Determining their
specific form relies on knowing that f (0)s is gyrotropic and so the ϕ component
of the velocity space integral acts only on vi, vivj and vivjvk. Performing these
integrals yields
〈vi〉 = v‖bi,
〈vivj〉 = v
2
‖bibj +
v2⊥
2
(δij − bibj),
〈vivjvk〉 = v
3
‖bibjbk + v‖
v2⊥
2
[(δij − bibj)bk + (δik − bibk)bj + (δjl − bjbl)bi] ,
where we denote the gyroaveraging operation (1/2pi)
∫ 2pi
0 by angle brackets. Com-
bining Mijkl with these quantities yields, eventually, the following expressions for
the pressure and collisional terms in the square bracket of equation (A.4):
MijklP
(0)
kl ∇muk = MijklC
(0)
kl = 0, (A.5)
Mijkl
dP (0)kl
dt
= −4(p(0)⊥ − p
(0)
‖ )
(
b
db
dt
× b+
db
dt
× bb
)
, (A.6)
MijklPml∇muk = −p
(0)
⊥
[
b× (∇u)·(I + 3bb)− (I + 3bb)·(∇u)T × b
]
, (A.7)
MijklPmk∇mul = −p
(0)
⊥
[
b× (∇u)T · (I + 3bb)− (I + 3bb) · (∇u)× b
]
− 4(p⊥ − p‖) [b(b · ∇u× b) + (b · ∇u× b)b] . (A.8)
Also,
Qmkl = q
(0)
⊥ (bmδkl + δmkbl + δmlbk)− (3q
(0)
⊥ − q
(0)
‖ )bmbkbl, (A.9)
155
from which it follows
Mijkl∇mQ
(0)
mkl = (I + 3bb) · V × b− b× V · (I + 3bb)
− 4(3q⊥ − q‖) [b(b · ∇b× b) + (b× b · ∇b)b] , (A.10)
where V = ∇q(0)⊥ +(∇q
(0)
⊥ )
T. Now substituting equations (A.5)-(A.10) into equa-
tion (A.4) and simplifying leads to our final expressions, equations (1.28)-(1.30).
156
Appendix B
Kinetic theory: Electrons
B.0.1 Mass-ratio ordering
The kinetic equation (1.2) for electrons is (recall that v is the peculiar velocity)
dfe
dte
+ v ·∇fe −
[
e
me
(
Le +
v ×B
c
)
+
due
dte
+ v ·∇ue
]
·
∂fe
∂v
= C[fe],
1
(
me
mi
)− 12
(
me
mi
)− 12
(
me
mi
)−1 (
me
mi
) 1
2
1
(
me
mi
)− 12
(B.1)
where Le = E+ue×B/c (we use this notation sparingly) and d/dte = ∂/∂t+ue ·
∇ is the convective derivative with respect to the electron flow. We have labelled
all terms in equation (B.1) according to their ordering in powers of (me/mi)1/2,
while taking Te ∼ Ti as per equation (1.14). The ordering has been done relative
to kvth,ife and we have assumed
∂
∂t
∼ ω ∼ kvth,i, ∇ ∼ k ∼ ρ−1i ∼
(
me
mi
)1/2 eB
mec vth,e
, ue ∼ vth,i, (B.2)
v ∼ vth,e ∼
(
me
mi
)−1/2
vth,i, E ∼
ue ×B
c
, νi ∼ ω ∼
(
me
mi
)1/2
νe. (B.3)
We stress that these are formal orderings with respect to the mass-ratio expan-
sion, not statements about the exact size of various quantities and their deriva-
tives: thus, some of the quantities ordered as unity within the mass-ratio expan-
157
158 B. Kinetic theory: Electrons
sion (e.g., kρi or ue/vth,i) will be ordered small in the subsidiary  expansion to
be used in solving the ion kinetics (see section 3.3.2).
We now expand the electron distribution function in powers of (me/mi)1/2: fe =
f (0)e +f
(1)
e +· · · . (This is not be confused with the notationally identical expansion
of section 1.4.1 where the small parameter was proportional to Ωs). It turns out
that we can learn all we need to know from just the two lowest orders in the
expansion of equation (B.1). Note that we do not expand any of the fields —
exact E and B are kept.
B.0.2 Order (me/mi)−1: gyrotropic electrons
To this order, equation (B.1) is
−
e
me
v ×B
c
·
∂f (0)e
∂v
= −Ωe
∂f (0)e
∂ϑ
= 0. (B.4)
(Recall Ωe = −eB/mec and ϑ is the gyroangle variable). Thus, in this order, we
have learned that the lowest-order electron distribution function is gyrotropic.
B.0.3 Order (me/mi)−1/2: Maxwellian electrons
To this order, equation (B.1) is
v ·∇f (0)e −
e
me
(
E +
ue ×B
c
)
·
∂f (0)e
∂v
− Ωe
∂f (1)e
∂ϑ
= C[f (0)e ]. (B.5)
Let us multiply this equation by 1 + ln f (0)e and integrate over the entire phase
space. This gives
∫ ∫
d3rd3vC[f (0)e ] ln f
(0)
e = 0 (B.6)
because the left-hand side of equation (B.5) is an exact divergence in the phase
space. Let us recall that, according to the Boltzmann (1872) H-theorem,
d
dt
∫ ∫
d3rd3vfe ln fe =
∫ ∫
d3rd3vC[fe] ln fe ≤ 0, (B.7)
159
where the inequality becomes equality only for a local Maxwellian distribution
(the proof for plasmas can be found in, e.g., Chapman & Cowling 1970; Hazeltine
& Meiss 2003). Therefore, equation (B.6) implies that f (0)e is a local Maxwellian:
f (0)e =
ne
(
piv2th,e
)3/2 e
−v2/v2th,e , (B.8)
Since v is peculiar velocity, the mean flow ue has already been accounted for.
Note that the perturbation expansion of fe can always be constructed in such a
way that ne and Te in equation (B.8) are the exact density and temperature of
the electron distribution.
B.0.4 Isothermal electrons
More can be learned about the electrons without going to higher orders. Let us
now substitute the expression (B.8) for f (0)e into equation (B.5) and gyroaverage
this equation, (1/2pi)
∫
dϑ, to eliminate the term containing f (1)e :
v‖b·∇f
(0)
e +
e
me
E‖
2v‖
v2th,e
f (0)e =
[
b ·∇ne
ne
+
(
v2
v2th,e
−
3
2
)
b ·∇Te
Te
+
eE‖
Te
]
v‖f
(0)
e = 0,
(B.9)
where E‖ = E ·b. Since equation (B.9) must hold for all v, it follows from it that
E‖ = −
Teb ·∇ne
ene
, (B.10)
b ·∇Te = 0. (B.11)
The second equation means that electrons (to lowest order) are isothermal along
the magnetic-field lines ∼ λmfp(mi/me)1/2. For our fiducial ICM parameters, we
have λmfp(mi/me)1/2 ∼ 6×1021 cm, which is larger than the scale l of the motions
that have the highest rate of strain (see section 3.2.1). For turbulent plasmas,
this implies globally isothermal electrons (Te = const) because the field lines are
stochastic. We will adopt this assumption of globally isothermal electrons in all
our calculations.
160 B. Kinetic theory: Electrons
B.0.5 Generalised Ohm’s law
Let us again go back to equation (B.5), multiply it by mev and integrate over the
velocity space. The result is the electron momentum equation to lowest order in
the mass-ratio expansion:
ene
(
E +
ue ×B
c
)
= −∇ ·
∫
d3vmevvf (0)e = −∇pe = −Te∇ne, (B.12)
where the electron pressure is isotropic because the distribution is Maxwellian,
pe = neTe, and the gradient only affects ne because Te = const (section B.0.4).
Note that equation (B.10) is simply the parallel part of equation (B.12). Equation
(B.12) is the generalised Ohm’s law, equation (3.30). We have thus arrived at
the starting point of the derivation in section 3.3.1.
Appendix C
Kinetic theory: Ions
C.0.6 Ordering
The kinetic equation (1.2) for ions is
dfi
dti
+ v ·∇fi +
[
Ze
mi
(
Li +
v ×B
c
)
−
dui
dti
− v ·∇ui
]
·
∂fi
∂v
= C[fi],
Equil. 3 2 2 −1 4 3 
Pert. 2   1 3 2 2
(C.1)
whereLi = E+ui×B/c and d/dti = ∂/∂t+ui·∇ is the convective derivative with
respect to the ion flow. We have labeled all terms according to their ordering in
powers of . The ordering has been done relative to k‖vth,ifi using the assumptions
explained in section 3.3.2. The first row of orderings in equation (C.1) applies
to the equilibrium (lowest-order) quantities and their gradients. The second row
gives the lowest order in which perturbed quantities appear in each term of the
kinetic equation.
C.0.7 Expansion of the Lorentz force
We require an explanation of the ordering and the expansion of the Lorentz
force. The Lorentz force is given in terms of ni and B by eliminating the ion
161
162 C. Kinetic theory: Ions
flow. Expanding this equation in , we have to three lowest orders
Ze
mi
Li = −
ZTe∇n1i
mi (n0i + n1i + n2i)
−
ZTe∇ (n0i + n2i)
mi (n0i + n1i)
−
ZTe∇n3i
min0i
+ v2A∇‖
B⊥1
B0
,
 2 3 3
(C.2)
where we have used B‖1 = 0 (see equation (3.50)). The ordering of the Lorentz
force in equation (C.1) follows from equation (C.2). Note that, in order to keep
3 precision, we have to keep perturbed densities in the denominators of the first
two terms on the right-hand side. However, as promised in section 3.3.3, we will
see in section C.0.11 that n1i = 0, so the contributions to the Lorentz force will
start at order 2 and equation (C.2) will simplify to read
Ze
mi
Li = −
ZTe∇ (n0i + n2i)
min0i
−
ZTe∇n3i
min0i
+ v2A∇‖
B⊥1
B0
+ · · · (C.3)
In section C.0.14, we will find that n2i = 0 as well.
C.0.8 Order −1: gyrotropic equilibrium
We now proceed to expand the ion kinetic equation (C.1). To lowest order, −1,
we get (cf. section B.0.2)
Ze
mi
v ×B0
c
·
∂f0i
∂v
= −Ωi
∂f0i
∂ϑ
= 0. (C.4)
Thus, the ion equilibrium distribution is gyrotropic. We will express the fact that
f0i is independent of the gyroangle ϑ by writing f0i as a function of two velocity
variables, v = |v| and v‖ = v · b0. In the derivation that follows these variables
are more convenient than the perhaps more intuitive pair (v⊥, v‖). Thus,
f0i = f0i(t, r, v, v‖). (C.5)
163
Hence follows an identity that will be useful shortly both for f0i and other gy-
rotropic functions:
∂f0i
∂v
=
v
v
(
∂f0i
∂v
)
v‖
+ b0
(
∂f0i
∂v‖
)
v
. (C.6)
C.0.9 Order 0
In the next order, equation (C.1) is
Ze
mi
v ×B⊥1
c
·
∂f0i
∂v
− Ωi
∂f1i
∂ϑ
= 0, (C.7)
where we have again used (Ze/mic)(v×B0) · ∂/∂v = −Ωi∂/∂ϑ. Using equation
(C.6), we get
∂f1i
∂ϑ
=
1
Ωi
Ze
mi
v ×B⊥1
c
· b0
(
∂f0i
∂v‖
)
v
= (b0 × v⊥) ·
B⊥1
B0
(
∂f0i
∂v‖
)
v
. (C.8)
Noticing that b0 × v⊥ = ∂v⊥/∂ϑ, we integrate this equation:
f1i = v⊥ ·
B⊥1
B0
(
∂f0i
∂v‖
)
v
+ g1i(t, r, v, v‖), (C.9)
where g1i is an arbitrary function (the gyrotropic part of the first-order perturbed
distribution).
Thus, all we have learned at this order is the gyroangle dependence of f1i. This
will be a general feature of our expansion: since the gyroangle derivative in
equation (C.1) is the lowest-order term, what we learn about each perturbed
distribution function f1i, f2i, f3i, . . . , at the lowest order in which it first appears
will always be its dependence on ϑ.
164 C. Kinetic theory: Ions
C.0.10 Order 1: Maxwellian equilibrium
At this order, equation (C.1) is
v·∇f1i+
Ze
mi
(
E +
ui ×B
c
)
1
·
∂f0i
∂v
−Ωi
∂f2i
∂ϑ
+
Ze
mi
v ×B⊥1
c
·
∂f1i
∂v
︸ ︷︷ ︸
I1
= C[f0i], (C.10)
where, using equation (C.9) and other tricks already employed in the two previous
sections, we can express the last term on the left-hand side of equation (C.10) as
follows
I1 = Ωi (b0 × v⊥) ·
B⊥1
B0
[
v⊥ ·
B⊥1
B0
(
∂2f0i
∂v2‖
)
v
+
(
∂g1i
∂v‖
)
v
]
, (C.11)
= Ωi
∂
∂ϑ
[
1
2
(
v⊥ ·
B⊥1
B0
)2
(
∂2f0i
∂v2‖
)
v
+ v⊥ ·
B⊥1
B0
(
∂g1i
∂v‖
)
v
]
. (C.12)
Collisions have made their first appearance at this order and we can now prove
that f0i is a Maxwellian. The proof is similar to the one for electrons in section
B.0.3: we multiply equation (C.10) by 1 + ln f0i and integrate over the entire
phase space. All terms on the left-hand side vanish because, to the order at
which we are computing them, they are all full derivatives with respect to the
phase-space variables. Thus,
∫ ∫
d3rd3vC[f0i] ln f0i = 0 ⇒ f0i =
n0i
(
piv2th,i
)3/2 e
−v2/v2th,i , (C.13)
is a Maxwellian.
C.0.11 Order 1 continued: more information about f1i
The fact that f0i is a Maxwellian allows us to uncover three important additional
pieces of information. First, from equation (C.9), we learn that f1i is gyrotropic:
f1i = g1i(t, r, v, v‖). (C.14)
165
Second, we can now prove that n1i = 0, as promised in sections 3.3.3 and C.0.7.
Using equations (C.12) and (C.13) in equation (C.10), gyroaveraging this equa-
tion, (1/2pi)
∫
dϑ, and substituting for the Lorentz force the lowest-order expres-
sion from equation (C.2), we get
v‖
(
∇‖f1i +
ZTe
T0i
∇‖n1i
n0i
f0i
)
= 0 ⇒
(
1 +
ZTe
T0i
)
∇‖n1i = 0, (C.15)
where the second equation has been obtained by cancelling v‖ in the first equation
and integrating it over velocities. We have used the shorthand ∇‖ = b0 ·∇, which
henceforth will be employed wherever fast parallel variation of the perturbed
quantities is involved (for slow parallel gradients, we will continue writing b0 ·∇
explicitly to emphasize that b0 is curved on the large scales). Equation (C.15)
implies that we may set
n1i = 0 (C.16)
(q.e.d.; see equation (3.53)) and absorb whatever slow-varying density perturba-
tion may arise into n0i. After eliminating n1i from equation (C.15), we get
∇‖f1i = 0, (C.17)
so f1i has no small-scale spatial variation at all. Note that this confirms equation
(3.55), which was derived from the ion momentum equation in the 1 order (i.e., it
is the velocity moment of equation (C.10)) and restricted the fast spatial variation
of the first-order pressure tensor. Equation (C.3) is also now confirmed.
C.0.12 Order 1 continued: gyroangle dependence of f2i
Finally, we go back to equation (C.10) to determine the gyroangle dependence
of f2i. Since f1i does not have a small-scale part, the first two terms drop out.
Using the fact that f0i is a Maxwellian and equation (C.12), we integrate equation
(C.10) with respect to the gyroangle and get
f2i = v⊥ ·
B⊥1
B0
(
∂f1i
∂v‖
)
v
+ g2i(t, r, v, v‖), (C.18)
166 C. Kinetic theory: Ions
where g2i is the gyrotropic part of f2i (so far arbitrary).
C.0.13 Order 2: role of equilibrium density and temper-
ature gradients
At this order, equation (C.1) becomes, upon substitution of the Maxwellian f0i
and the lowest-order (2) expression for the Lorentz force from equation (C.3)
∂f1i
∂t
+ v ·∇f0i + v‖∇‖f2i +
ZTe
T0i
v ·∇n0i + v‖∇‖n2i
n0i
f0i − Ωi
∂f3i
∂ϑ
+
Ze
mi
(
v ×B2
c
·
∂f1i
∂v
+
v ×B⊥1
c
·
∂f2i
∂v
)
+
2v‖
(
∇‖u⊥1i
)
· v⊥
v2th,i
f0i
︸ ︷︷ ︸
I2
= C[f1i],
(C.19)
where we have explicitly enforced the assumption that perturbed quantities have
no fast perpendicular spatial dependence. Analogously to equation (C.12), upon
using equations (C.14) and (C.18) and noticing that v⊥ = −∂(b0 × v)/∂ϑ, we
find that the last two terms on the left-hand side of equation (C.19) are a full
gyroangle derivative:
I2 = Ωi
∂
∂ϑ
[
v⊥ ·
B⊥2
B0
(
∂f1i
∂v‖
)
v
+
1
2
(
v⊥ ·
B⊥1
B0
)2
(
∂2f1i
∂v2‖
)
v
+ v⊥ ·
B⊥1
B0
(
∂g2i
∂v‖
)
v
−
2v‖∇‖u⊥1i
v2th,i
·
b0 × v⊥
Ωi
f0i
]
. (C.20)
In view of equation (C.20) and of the gyroangle independence of f1i (equation
(C.14)), the gyroaverage of equation (C.19) is
∂f1i
∂t
+ v‖
[
b0 ·∇f0i +∇‖g2i +
ZTe
T0i
b0 ·∇n0i +∇‖n2i
n0i
f0i
]
= C[f1i]. (C.21)
Since f1i has no fast spatial gradients (equation (C.17)), averaging equation
(C.21) over small scales gives
∂f1i
∂t
+ v‖
[
b0 ·∇f0i +
ZTe
T0i
b0 ·∇n0i
n0i
f0i
]
= C[f1i]. (C.22)
167
This equation determines f1i purely in terms of the equilibrium density and tem-
perature gradients. Since the time variation of the equilibrium is slow, it is clear
that f1i will converge to a steady solution after a few collision times. Then ∂f1i/∂t
in equation (C.22) can be neglected and the solution obtained by inverting the
linearised collision operator. Since we are not interested in exact collisional trans-
port coefficients here, instead of the full Landau collision operator, we will use a
very simple model one — the Lorentz pitch-angle scattering operator (see, e.g.,
Helander & Sigmar, 2002), so equation (C.22) becomes in steady state
C[f1i] = νi
∂
∂ξ
1− ξ2
2
∂f1i
∂ξ
= ξv
[(
1 +
ZTe
T0i
)
b0 ·∇n0i
n0i
+
(
v2
v2th,i
−
3
2
)
b0 ·∇T0i
T0i
]
f0i,
(C.23)
where ξ = v‖/v and, once again, the dependency of νi on v has been suppressed
for simplicity. The solution of equation (C.23) that satisfies equation (C.16) is
f1i = −
v‖
νi
[(
1 +
ZTe
T0i
)
b0 ·∇n0i
n0i
+
(
v2
v2th,i
−
3
2
)
b0 ·∇T0i
T0i
]
f0i
= −
v‖
νi
(
v2
v2th,i
−
5
2
)
b0 ·∇T0i
T0i
f0i. (C.24)
The second, simplified, expression above is obtained by noticing that the tem-
perature and density gradients are, in fact, related by the equilibrium pressure
balance, equation (3.57), which was obtained in section 3.3.3 from the ion momen-
tum equation in the 2 order. It is easily recovered by taking the velocity moment
of equation (C.19) and averaging out the small scales. Since f0i is a Maxwellian,
the pressure balance takes the form (previewed in footnote 5, chapter 3)
(
1 +
ZTe
T0i
)
∇n0i
n0i
+
∇T0i
T0i
= 0, (C.25)
whence immediately follows the final expression for f1i in equation (C.24).
Let us note two useful properties of the solution (C.24). First, f1i makes no
168 C. Kinetic theory: Ions
contribution to the pressure tensor:
P1i = mi
∫
d3v vvf1i = 0, (C.26)
a result we promised in section 3.3.3 (equation (3.55)). Second, the derivative of
f1i with respect to v‖ is isotropic:
(
∂f1i
∂v‖
)
v
= −
1
νi
(
v2
v2th,i
−
5
2
)
b0 ·∇T0i
T0i
f0i,
(
∂2f1i
∂v2‖
)
v
= 0, (C.27)
which leads to vanishing of one of the terms in equation (C.20).
We will see in Appendix C.0.17 that f1i encodes the ion collisional heat flux
(equation (C.24) is the standard form of the appropriate contribution to the
perturbed distribution function; see, e.g., equation (D16) of Schekochihin et al.
2009). We will carry the ion-temperature-gradient effect contained in f1i through
to the end of this calculation because it will interesting and instructive to see
how contributions from the ion heat flux arise in the problem. However, this is
not the main effect we are after and an impatient reader attempting to follow
this derivation may find the following simplification useful. If one assumes by
fiat that ∇T0i = 0, then ∇n0i = 0 as well (from equation (C.25)) and in all the
calculations that follow one may set f1i = 0 and f0i = const, which substantially
reduces the amount of algebra.
C.0.14 Order 2 continued: more information about f2i
Staying at this order, we can learn more about f2i and f3i. Subtracting equation
(C.22) from equation (C.21), we get
v‖
(
∇‖g2i +
ZTe
T0i
∇‖n2i
n0i
f0i
)
= 0 ⇒
(
1 +
ZTe
T0i
)
∇‖n2i = 0, (C.28)
analogously to equation (C.15). We have used the fact that, as follows from
equation (C.18), n2i =
∫
d3vf2i =
∫
d3vg2i. Similarly to the argument in section
169
C.0.11, this implies that n2i has no fast spatial variation and so we can set
n2i = 0. (C.29)
Equation (C.28) then implies
∇‖g2i = 0, (C.30)
i.e., g2i has no small-scale spatial dependence. Since the first term in equa-
tion (C.18) does not contribute to the second-order pressure tensor (because the
derivative of f1i is a function of v only), we have
P2i = mi
∫
d3v vv g2i, (C.31)
and so, in view of equation (C.30), P2i has no small-scale dependence. This con-
firms equation (3.58), derived in section 3.3.3 from the ion momentum equation.
Note that the tensor P2i will be needed in the large-scale momentum equation
(3.62). It will contain the lowest-order pressure anisotropy.
C.0.15 Order 2 continued: gyroangle dependence of f3i
Subtracting equation (C.21) from equation (C.19) and using equation (C.18), we
get
Ωi
∂f3i
∂ϑ
= I2 +v⊥ ·
[
∇f0i +
ZTe
T0i
∇n0i
n0i
f0i
]
+
v‖
(
∇‖B⊥1
)
· v⊥
B0
(
∂f1i
∂v‖
)
v
, (C.32)
where I2 is given by equation (C.20) (note that the second derivative of f1i van-
ishes there; see equation (C.27)). Using again the fact that v⊥ = −∂(b0×v⊥)/∂ϑ,
we integrate equation (C.32) and get
f3i = −
b0 × v⊥
Ωi
·
[
2v‖∇‖u⊥1i
v2th,i
f0i +
v‖∇‖B⊥1
B0
(
∂f1i
∂v‖
)
v
+
(
v2
v2th,i
−
5
2
)
∇T0i
T0i
f0i
]
+ v⊥ ·
[
B⊥2
B0
(
∂f1i
∂v‖
)
v
+
B⊥1
B0
(
∂g2i
∂v‖
)
v
]
+ g3i(t, r, v, v‖), (C.33)
170 C. Kinetic theory: Ions
where g3i is the gyrotropic part of f3i (so far arbitrary) and we have used equation
(C.25) to simplify the terms that contain equilibrium gradients.
We will see in section C.0.20 that we do not need to know either g3i orB2 in order
to calculate the third-order ion pressure tensor P3i and close the ion momentum
equation (3.59) for the firehose perturbations. The only remaining quantity we
do need is g2i — we will now derive the equation for it by going to next order in
the  expansion.
C.0.16 Order 3
At this order, equation (C.1) is, upon substitution of the Maxwellian f0i, gy-
rotropic f1i, and equation (C.3) for the Lorentz force,
df0i
dt0i
+
df2i
dt0i
+ v ·∇f1i + v‖∇‖f3i +
(
ZTe
T0i
v‖∇‖n3i
n0i
−
2
βi
(
∇‖B⊥1
)
· v⊥
B0
)
f0i
− Ωi
∂f4i
∂ϑ
+
Ze
mi
[
v ×B⊥1
c
·
∂f3i
∂v
+
v ×B2
c
·
∂f2i
∂v
︸ ︷︷ ︸
I3
+
v ×B3
c
· b0
(
∂f1i
∂v‖
)
v
]
−
ZTe
mi
∇n0i
n0i
·
[
v
v
(
∂f1i
∂v
)
v‖
+ b0
(
∂f1i
∂v‖
)
v
]
+
2vv :∇ (u0i + u2i)
v2th,i
f0i
+
du⊥1i
dt0i
·
2v⊥
v2th,i
f0i − v‖
(
∇‖u
⊥
1i
)
·
v⊥
v
(
∂f1i
∂v
)
v‖
= C[f2i] + C[f1i, f1i],
(C.34)
where d/dt0i = ∂/∂t + u0i ·∇ is the convective derivative (with respect to the
large-scale flow). Note that ∇f1i in the above equation is with respect to slow
spatial variation. Note also that the collision operator at this order has two parts:
the linearised operator describing interaction of f2i with the Maxwellian equilib-
rium f0i and the nonlinear operator, denoted C[f1i, f1i], describing interaction of
f1i with itself.
We will only ever need the gyroaverage of equation (C.34). Many terms then
vanish or simplify. What happens is mostly straightforward: the gyroaverages of
171
∂/∂ϑ are zero, the gyroaverages of the velocities are done using the identities
〈v〉 = v‖b0, 〈vv〉 =
v2⊥
2
(I− b0b0) + v2‖b0b0 (C.35)
(as before angle brackets denote (1/2pi)
∫
dϑ). There are a few terms that are
perhaps not obvious and so require explanation.
The first is essentially a variation on equations (1.37) and (1.38). To recap; we
showed above that f1i is a function of v, v‖, and r. However, the spatial gradient
here is still taken at constant v. Since one of the new velocity variables v‖ = v ·b0
is a function of v and r, we have
v · (∇f1i)v = v · (∇f1i)v,v‖ + (vv :∇b0)
(
∂f1i
∂v‖
)
v
, (C.36)
and so
〈v · (∇f1i)v〉 = v‖b0 ·∇f1i + (∇ · b0)
v2⊥
2
(
∂f1i
∂v‖
)
v
. (C.37)
In the final expression, ∇f1i is now understood to be at constant v and v‖.
Since ∇ · b0 = − (b0 ·∇B0) /B0, the additional term that has emerged is readily
interpreted as the mirror force associated with the large-scale variation of the
magnetic field.
Now let us turn to the two terms in equation (C.34) denoted by I3: the second
of these terms gives
〈
Ze
mi
v ×B2
c
·
∂f2i
∂v
〉
=
〈
Ze
mi
v ×B2
c
·
[
B⊥1
B0
(
∂f1i
∂v‖
)
v
+ b0
(
∂g2i
∂v‖
)
v
]〉
(C.38)
=Ωiv‖
(
b0 ×
B⊥2
B0
)
·
B⊥1
B0
(
∂f1i
∂v‖
)
v
, (C.39)
where we have used equation (C.18) and the fact that
(
∂f1i/∂v‖
)
v
only depends
172 C. Kinetic theory: Ions
on v (equation (C.27)); the first term, upon substitution of equation (C.33), gives
〈
Ze
mi
v ×B⊥1
c
·
∂f3i
∂v
〉
=
〈
Ze
mi
v ×B⊥1
c
·
{
B⊥2
B0
(
∂f1i
∂v‖
)
v
+ b0 v⊥ ·
B⊥1
B0
(
∂2g2i
∂v2‖
)
v
+
b0
Ωi
×
[
2v‖∇‖u⊥1i
v2th,i
f0i +
v‖∇‖B⊥1
B0
(
∂f1i
∂v‖
)
v
+
(
v2
v2th,i
−
5
2
)
∇T0i
T0i
f0i
]
− b0
b0 × v⊥
Ωi
·
[
2∇‖u⊥1i
v2th,i
f0i +
∇‖B⊥1
B0
(
∂f1i
∂v‖
)
v
]
+ b0
(
∂g3i
∂v‖
)
v
}〉
(C.40)
which becomes
〈
Ze
mi
v ×B⊥1
c
·
∂f3i
∂v
〉
=
Ωiv‖
(
b0 ×
B⊥1
B0
)
·
B⊥2
B0
(
∂f1i
∂v‖
)
v
+ v‖
(
b0 ×
B⊥1
B0
)
·
{
b0×
[
2v‖∇‖u⊥1i
v2th,i
f0i +
v‖∇‖B⊥1
B0
(
∂f1i
∂v‖
)
v
+
(
v2
v2th,i
−
5
2
)
∇T0i
T0i
f0i
]}
−
B⊥1
B0
· 〈(b0 × v⊥) (b0 × v⊥)〉 ·
[
2∇‖u⊥1i
v2th,i
f0i +
∇‖B⊥1
B0
(
∂f1i
∂v‖
)
v
]
. (C.41)
The first term in equation (C.41) exactly cancels when equation (C.39) is added
to it. Simplifying the double vector product in the second term and noticing that
the gyroaverage in the third term is equal to (v2⊥/2) (I− b0b0), we have
〈I3〉 =
B⊥1
B0
·
{(
v2‖ −
v2⊥
2
)[
2∇‖u⊥1i
v2th,i
f0i +
∇‖B⊥1
B0
(
∂f1i
∂v‖
)
v
]
+ v‖
(
v2
v2th,i
−
5
2
)
∇T0i
T0i
f0i
}
=
2v2‖ − v
2
⊥
v2th,i
[
1
2
d
dt
|B⊥1 |
2
B20
f0i +
v2th,i
4
∇‖|B⊥1 |
2
B20
(
∂f1i
∂v‖
)
v
]
+ v‖
(
v2
v2th,i
−
5
2
)
B⊥1
B0
·
∇T0i
T0i
f0i, (C.42)
173
where we have used the perturbed induction equation (3.51) to express ∇‖u⊥1i in
terms of B⊥1 . Finally, the gyroaverage of equation (C.34) is
df0i
dt
+
∂g2i
∂t
+v‖
(
b0 ·∇f1i +∇‖g3i
)
+(∇ · b0)
v2⊥
2
(
∂f1i
∂v‖
)
v
+
ZTe
T0i
v‖∇‖n3i
n0i
f0i
+ 〈I3〉+
[
2v2‖ − v
2
⊥
v2th,i
b0b0 :∇ (u0i + u2i) +
v2⊥
v2th,i
∇ · (u0i + u2i)
]
f0i
−
ZTe
mi
b0 ·∇n0i
n0i
[
v‖
v
(
∂f1i
∂v
)
v‖
+
(
∂f1i
∂v‖
)
v
]
= C[g2i] + C[f1i, f1i], (C.43)
where 〈I3〉 is given by equation (C.42). Note that g2i does not have fast spatial
variation (equation (C.30)), so, to the order we are keeping, dg2i/dt = ∂g2i/∂t.
The next step is to average this equation over small scales: again many terms
vanish (in particular, all terms where the fast-varying perturbed quantities enter
linearly) and we get
df0i
dt
+
∂g2i
∂t
+ v‖b0 ·∇f1i + (∇ · b0)
v2⊥
2
(
∂f1i
∂v‖
)
v
−
ZTe
mi
b0 ·∇n0i
n0i
b0 ·
∂f1i
∂v
+
[
2v2‖ − v
2
⊥
v2th,i
(
b0b0 :∇u0i +
1
2
∂
∂t
|B⊥1 |2
B20
)
+
v2⊥
v2th,i
∇ · u0i
]
f0i
= C[g2i] + C[f1i, f1i], (C.44)
where the overline denotes the small-scale average and the derivatives of f1i have
been written in a compact form that will prove useful momentarily. Remarkably,
the contribution of the perturbations has survived in equation (C.44) in the form
of a single quadratic term — in section C.0.18, we will see that it gives rise to
precisely the nonlinear feedback on the pressure anisotropy that was anticipated
qualitatively in section 3.2.3.
By averaging out the gyroangle- and small-scale-dependent parts of the third-
order kinetic equation (C.34), we have eliminated f4i, g3i, B2, B3, and u2i, which
are unknown and potentially very cumbersome to calculate. As we are about to
see, in order to calculate the ion pressure tensor to the relevant orders, we do not,
in fact, need to know any of these quantities, so we will neither have to revisit the
unaveraged equations (C.34) or (C.43) or go to higher orders in the  expansion
174 C. Kinetic theory: Ions
of equation (C.1). Equation (C.44) determines g2i, which is all that we require
to calculate P2i (section C.0.18) and P3i (section C.0.20). Knowing these tensors
will then allow us to close the ion momentum equations describing the plasma
motion at large (equation (3.62)) and small (equation (3.59)) scales.
C.0.17 Order 3 continued: transport equations
Since g2i is gyrotropic, the tensor P2i is diagonal:
P2i = p⊥2i (I− b0b0) + p
‖
2ib0b0 = p2iI + (p
⊥
2i − p
‖
2i)
(
1
3
I− b0b0
)
, (C.45)
where the scalar pressures are
p⊥2i =
∫
d3v
miv2⊥
2
g2i, p
‖
2i =
∫
d3vmiv2‖g2i, p2i =
2
3
p⊥2i +
1
3
p‖2i. (C.46)
Then the ion momentum equation (3.62) becomes1
min0i
du0i
dt
= −∇Π˜ +∇ ·
[
b0b0
(
p⊥2i − p
‖
2i +
B20
4pi
)]
, (C.47)
where Π˜ = p2i + (p⊥2i − p
‖
2i)/3 + B
2
0/8pi and we have recovered the familiar mo-
mentum equation in the long-wavelength limit (equation (3.18)). Let us first
explain how p2i (or, equivalently, Π˜) is determined and then calculate the pres-
sure anisotropy p⊥2i − p
‖
2i (section C.0.18).
First, let us integrate equation (C.44) over velocities. Since
∫
d3v g2i = 0 and
∫
d3v v‖f1i = 0, we get
dn0i
dt
= −n0i∇ · u0i, (C.48)
an unsurprising result (the continuity equation was already obtained in section
3.3.4; see equation (3.61)). Now multiply equation (C.44) bymiv2/3 and integrate
1Formally, this equation is the result of taking the velocity moment of the kinetic equation
(C.1) at the order 4. We do not write explicitly equation (C.1) at this order because it is not
needed for anything except the momentum equation, the form of which we already know.
175
over velocities:
dp0i
dt
+
∂p2i
∂t
= −
2
3
∇ ·
(
b0q1i
)
−
5
3
p0i∇ · u0i, (C.49)
where p0i = n0iT0i and the parallel collisional heat flux is, using equation (C.24),2
q1i =
∫
d3v
miv2v‖
2
f1i = −
∫
d3v
miv2v2‖
2νi
(
v2
v2th,i
−
5
2
)
f0i
b0 ·∇T0i
T0i
= −
5
4
n0i
v2th,i
νi
b0 ·∇T0i. (C.50)
The heat flux term in equation (C.49) arises from the third and fourth terms on
the left-hand side of equation (C.44) (in the fourth term, write v2⊥ = v
2− v2‖ and
integrate by parts with respect to v‖ at constant v).
Since formally all terms in equation (C.49) except the one involving p2i have slow
time dependence, we may assume that so does p2i and, therefore, ∂p2i/∂t can
be dropped from this equation (this can be formalised via averaging over short
timescales). Using now equations (C.48) and (C.50), we can rewrite equation
(C.49) as an evolution equation for the equilibrium temperature:
n0i
dT0i
dt
=∇ · (n0iκib0b0 ·∇T0i)−
2
3
n0iT0i∇ · u0i, κi =
5
6
v2th,i
νi
, (C.51)
where κi is the ion thermal conductivity. The first term on the right-hand side
represents collisional heat transport, the second compressional heating.
Equations (C.48) and (C.51) evolve n0i and T0i. However, the equilibrium density
and temperature can only change in such a way that pressure balance, equation
(C.25), is maintained. This means that if we know the spatial distribution of
T0i, we also know that of n0i, or vice versa. Compressive motions will develop to
make the density and temperature distributions adjust to each other and preserve
the pressure balance. These motions must be consistent with the momentum
equation (C.47) and the isotropic pressure perturbation p2i will adjust to make
2Since we used a very simplified collision operator in our calculation of f1i in section C.0.13,
the numerical prefactor in the expression for the heat flux should not be regarded as quanti-
tatively correct. This is not a problem for our purposes. The same caveat applies to equation
(C.54) and equation (C.64).
176 C. Kinetic theory: Ions
it so. Thus, if we provide the expression for the pressure anisotropy p⊥2i− p
‖
2i, the
other 5 equilibrium quantities — u0i, n0i, T0i, p2i, and B0 — are determined by
the closed set of 5 equations:3 momentum equation (C.47), continuity equation
(C.48), heat conduction equation (C.51), pressure balance equation (C.25), and
induction equation (3.60).
C.0.18 Order 3 continued: pressure anisotropy
In order to calculate the pressure anisotropy p⊥2i−p
‖
2i, we multiply equation (C.44)
by mi(v2⊥ − 2v
2
‖)/2 and integrate over velocities. All isotropic terms vanish, as
does the term containing ∂f1i/∂v, which generally cannot contribute to pressure
(after integration by parts, it is zero because
∫
d3v vf1i = 0). The result is
∂
∂t
(p⊥2i − p
‖
2i) = 3p0i
(
b0b0 :∇u0i −
1
3
∇ · u0i +
1
2
∂
∂t
|B⊥1 |2
B20
)
−∇ ·
[
b0(q⊥1i − q
‖
1i)
]
− 3q⊥1i∇ · b0 − 3νi(p
⊥
2i − p
‖
2i), (C.52)
where q⊥1i and q
‖
1i are parallel fluxes of perpendicular and parallel heat, respec-
tively: using equation (C.24),
q⊥1i =
∫
d3v
miv2⊥v‖
2
f1i = −
1
2
n0i
v2th,i
νi
b0 ·∇T0i, (C.53)
q‖1i =
∫
d3vmiv3‖ f1i,= −
3
2
n0i
v2th,i
νi
b0 ·∇T0i, (C.54)
and we note that q⊥1i+q
‖
1i/2 = q1i (see equation (C.50)). To work out the collision
term in equation (C.52), we have again resorted to brutal simplification by using
the Lorentz operator (see equation (C.23)) and dropping the nonlinear collision
term C[f1i, f1i] in equation (C.44) (the consequences of retaining this term, which
are mostly small and irrelevant for our purposes, have been explored by Catto &
3If, as discussed in section C.0.13, one takes the easy option and assumes ∇T0i = 0 and
∇n0i = 0, then no equations are needed for the constant density and temperature, while p2i
in equation (C.47) is determined by the incompressibility condition ∇ · u0i = 0 (which follows
from equation (C.48) with n0i = const).
177
Simakov 2004). The solution of equation (C.52) is
∆(t) ≡
p⊥2i − p
‖
2i
p0i
(C.55)
= ∆0 +
3
2
∫ t
0
dt′e−3νi(t−t
′) ∂
∂t′
|B⊥1 (t′)|2
B20
, (C.56)
where ∆0 is the part of the anisotropy due to the large-scale dynamics:
∆0 =
1
νi
{
b0b0 :∇u0i −
1
3
∇ · u0i −
∇ ·
[
b0(q⊥1i − q
‖
1i)
]
+ 3q⊥1i∇ · b0
3p0i
}
(C.57)
and we have assumed ∆(0) = ∆0. The first two terms in equation (C.57) are
the well known collisional contributions to the pressure anisotropy calculated by
Braginskii (1965). The heat-flux terms did not occur in Braginskii’s calculation
because they were small in his assumed sonic-flow ordering (u0i ∼ vth,i). They
occur here because our ordering is subsonic (u0i ∼ vth,i; see equation (3.43)) —
that heat fluxes appear in the pressure tensor under such assumptions is also a
known fact (Mikhailovskii & Tsypin, 1971, 1984; Catto & Simakov, 2004). The
new, nonlinear part of the anisotropy is the second term in equation (C.56), which
is due to the firehose fluctuations. This result was predicted on heuristic grounds
in section 3.2.3.
Equation (C.56) completes the set of transport equations derived in section
C.0.17, but we still need to calculate |B⊥1 |2. This is done via equations (3.51) and
(3.59). The third-order pressure term in the latter equation will be calculated in
section C.0.20.
C.0.19 CGL equations with nonlinear feedback
It is certainly useful to compare what we have found to the modified CGL equa-
tions found in chapter 1. They were the conceptual starting point for our work.
Let us first recall that the induction equation (3.60) implies
1
B0
dB0
dt
= b0b0 :∇u0i −∇ · u0i. (C.58)
178 C. Kinetic theory: Ions
Using equation (C.48) for ∇ · u0i, we may, therefore, rewrite equation (C.52) as
follows
∂
∂t
(p⊥2i − p
‖
2i) = 3p0i
(
1
B
dB
dt
−
2
3
1
n0i
dn0i
dt
)
−∇ ·
[
b0(q⊥1i − q
‖
1i)
]
− 3q⊥1i∇ · b0 − 3νi(p
⊥
2i − p
‖
2i), (C.59)
where B includes both the large-scale magnetic field B0 and the averaged firehose
fluctuations (see equation (3.24)). This is the same as equation (1.44). Equiva-
lently, we can combine equations (C.49) and (C.59), we get
d
dt
ln
p⊥i
n0iB
= −
∇ ·
(
b0q⊥1i
)
+ q⊥1i∇ · b0
p0i
− νi
p⊥i − p
‖
i
p0i
, (C.60)
d
dt
ln
p‖iB
2
n30i
= −
∇ ·
(
b0q
‖
1i
)
− 2q⊥1i∇ · b0
p0i
− 2νi
p‖i − p
⊥
i
p0i
. (C.61)
These are equations (1.39) and (1.40), the original equations of Chew et al. (1956)
(without neglecting the heat fluxes or collisions). They are also the equations that
form the basis for recent numerical studies by Sharma et al. (2006, 2007) of the
effects of pressure anisotropies as related to the MRI. We have derived, from first-
principles, the equations we started with chapter 1. We might have guessed these
would emerge exactly, but this had to be demonstrated rather than assumed.
As pointed out in Sharma et al. (2006), equation (C.60) can be rewritten in a form
that makes explicit the conservation of the first adiabatic invariant (including by
the heat-flux terms):
∂
∂t
p⊥i
B
+∇ ·
(
p⊥i
B
u0i + b0
q⊥1i
B
)
= −νi
p⊥i − p
‖
i
p0i
. (C.62)
The nonlinear feedback in equations (C.60)-(C.61) is provided by the small-scale
magnetic fluctuations in B. Sharma et al. (2006, 2007) do not have the small-scale
fluctuations and model their effect by introducing strong effective damping terms
in equations (C.60)-(C.61) that limit the pressure anisotropies to the marginal
state of the plasma instabilities. The calculation carried out here attempts to
derive this feedback from first principles.
179
C.0.20 Pressure tensor for the firehose turbulence
In order to close the small-scale momentum equation (3.59), we must calculate
the divergence of the third-order ion-pressure tensor or, more precisely, the per-
pendicular part thereof: since the fast spatial variation is only in the parallel
direction, we have, from equation (C.33)
(∇ · P3i)⊥ = ∇‖
∫
d3vmiv‖v⊥f3i
= ∇‖
∫
d3v
miv2⊥
2
{
v‖
(
∂g2i
∂v‖
)
v
B⊥1
B0
+ v2‖
b0
Ωi
×
[
2∇‖u⊥1i
v2th,i
f0i +
∇‖B⊥1
B0
(
∂f1i
∂v‖
)
v
]}
= −∇‖
{
(p⊥2i − p
‖
2i)
B⊥1
B0
+
1
Ωi
[
p0i∇‖u
⊥
1i − (2q
⊥
1i − q
‖
1i)
∇‖B⊥1
B0
]
× b0
}
, (C.63)
where the last formula was obtained via integration by parts with respect to v‖
(at constant v). Note that the terms in equation (C.33) containing B2 and g3i do
not contribute, so, as announced at the end of section C.0.16, we do not need to
compute these quantities. The pressure anisotropy p⊥2i − p
‖
2i is given by equation
(C.56) and the heat fluxes q⊥1i and q
‖
1i by equation (C.54), whence
4
ΓT ≡
2q⊥1i − q
‖
1i
p0ivth,i
=
1
2
vth,i
νi
b0 ·∇T0i
T0i
. (C.64)
The second term in equation (C.63) is recognizable as the collision-independent
“gyroviscosity” (Braginskii, 1965) and the third term as the collisional heat-flux
contribution to it that arises for subsonic flows (Mikhailovskii & Tsypin, 1971,
1984; Catto & Simakov, 2004). It is the gyroviscous term that will limit the range
of wavenumbers susceptible to the firehose instability (see section 3.4.2), while
the heat-flux term will lead to substantial modifications of the firehose turbulence
4As explained in footnote 2, the numerical prefactor here should not be taken literally
because we have used a very simplified collision operator. The correct prefactors can be found,
e.g., in Catto & Simakov (2004).
180 C. Kinetic theory: Ions
and even give rise to an additional source of unstable behaviour (the gyrothermal
instability; see section 3.5).
Appendix D
Commonly used symbols
Throughout a subscript s = i, e denotes ions or electrons (except for cs as stated
below). A numbered subscript or parenthetical superscript usually denotes a term
an asymptotic series, e.g. f2. However, in chapter 2, a subscript is the order (or
branch index) of the mode. A linearly perturbed quantity is prefaced by the
symbol δ and an equilibrium quantity is usually subscripted with a “0”. In other
contexts δ is also a small expansion parameter. The symbols ‖ and ⊥ denote the
“orientation” of the associated quantity with respect to the magnetic field and
an over-line denotes a spatial average e.g. B (in chapter 3 it is an average over
scales less than λmfp).
181
182 D. Commonly used symbols
Vectors & Tensors Expression
a Particle force (see equation (1.2))
b Magnetic field unit vector
B Magnetic field (in chapter 2 it is also the Alfve´n speed)
E Electric field
j Current
q‖, q⊥ Heat flux (parallel, perpendicular)
u Fluid velocity
P Pressure tensor
D Deviatoric stress tensor
I Identity
G Gyrophase dependent component of the pressure tensor
Roman symbols Expression
cs Sound speed (subscript indicates “sound” not species)
fs Number density distribution function
k wavenumber
K Modified Bessel function (second kind)
M Mach number
ms Mass
ns Number density
n Mode index (chapter 2 only)
vth,s Thermal speed
vA Alfve´n speed
p, p‖, p⊥ Isotropic pressure (total, parallel, perpendicular)
r Radius (cylindrical polar coordinates)
Re Reynolds number
SB Measure of Braginskii viscosity (see equation (2.17))
t Time
T Temperature
v, v‖, v⊥ Particle velocity variable (total, parallel, perpendicular)
vn Order of the modified Bessel function (second kind)
w Argument coefficient of the modified Bessel function (second kind)
z Vertical direction (in chapter 3 it is aligned with b)
183
Greek symbols Expression
Ωs Cyclotron frequency
Ωˆ Galactic rotation frequency
β Plasma beta
∆ Pressure anisotropy (without a subscript read ions)
γ Growth rate (general field)
ΓT Magnitude of the parallel heat flux
λmfp Inter-particle mean free path
µB Coefficient of the Braginskii viscosity
νs Collision frequency
ω Frequency (general field)
σ Alfve´n frequency
φ Azimuthal angle (cylindrical polar coordinates)
ρs Larmor radius
θ Pitch angle: arctan(Bz/Bφ)
ϑ Gyrophase angle
184
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