Tidal interactions between planets
and stars
Adrian John Barker
March 2011
Selwyn College University of Cambridge
THIS DISSERTATION IS SUBMITTED
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

Tidal interactions between planets and stars
A.J. Barker, Selwyn College
Since the first discovery of an extrasolar planet around a solar-type star, observers
have detected over 500 planets outside the solar system. Many of these planets have
Jovian masses and orbit their host stars in orbits of only a few days, the so-called
“Hot Jupiters”. At such close proximity to their parent stars, strong tidal interactions
between the two bodies are expected to cause significant secular spin-orbit evolution.
This thesis tackles two problems regarding the tidal evolution of short-period extrasolar
planets.
In the first part, we adopt a simple model of the orbit-averaged effects of tidal
friction, to study the tidal evolution of planets on inclined orbits. We also analyse the
effects of stellar magnetic braking. We then discuss the implications of our results for
the importance of Rossiter-Mclaughlin effect observations.
In the second part, we study the mechanisms of tidal dissipation in solar-type stars.
In particular, internal gravity waves are launched at the interface of the convection and
radiation zones of such a star, by the tidal forcing of a short-period planet. The fate of
these waves as they approach the centre of the star is studied, primarily using numerical
simulations, in both two and three dimensions. We find that the waves undergo
instability and break above a critical amplitude. A model for the tidal dissipation
that results from this process is presented, and its validity is verified by numerical
integrations of the linear tidal response, in an extensive set of stellar models. The
dissipation is efficient, and varies by less than an order of magnitude between all solar-
type stars, throughout their main-sequence lifetimes, for a given planetary orbit. The
implications of this mechanism for the survival of short-period extrasolar planets is
discussed, and we propose a possible explanation for the survival of all of the extrasolar
planets currently observed in short-period orbits around F, G and K stars.
We then perform a stability analysis of a standing internal gravity wave near the
centre of a solar-type star, to understand the early stages of the wave breaking process
in more detail, and to determine whether the waves are subject to weaker parametric
instabilities, below the critical amplitude required for wave breaking. We discuss the
relevance of our results to our explanation for the survival of short-period planets
1
presented in the second part of this thesis.
Finally, we propose an alternative mechanism of tidal dissipation, involving the
gradual radiative damping of the waves. Based on a simple estimate, it appears that
this occurs even for low mass planets. However, it is in conflict with current observa-
tions since it would threaten the survival of all planets in orbits shorter than 2 days.
We discuss some hydrodynamic instabilities and magnetic stresses which may prevent
this process.
2
Preface
This thesis describes work done between October 2007 and December 2010 in the
Astrophysical Fluid Dynamics & Nonlinear Patterns research group at the Department
of Applied Mathematics & Theoretical Physics, University of Cambridge, under the
supervision of Dr Gordon Ogilvie. No part of this thesis has been submitted for
any qualification other than the degree of Doctor of Philosophy at the University of
Cambridge.
Chapter 1 constitutes an introduction and review of relevant background material.
Subsequent chapters contain original material, from which the following have been
published in, submitted to, or in preparation for submission to, peer-reviewed journals:
Chapter 2: A.J. Barker & G.I. Ogilvie, “On the tidal evolution of Hot Jupiters on
inclined orbits”, MNRAS, 2009, 395, 2268-2287
Chapter 3: A.J. Barker & G.I. Ogilvie, “On internal wave breaking and tidal dissi-
pation near the centre of a solar-type star”, MNRAS, 2010, 404, 1849-1868
Chapter 4: A.J. Barker, “Three-dimensional simulations of internal wave breaking
and the fate of planets around solar-type stars”, MNRAS, accepted
Chapter 6: A.J. Barker & G.I. Ogilvie, “Stability analysis of a tidally excited internal
gravity wave near the centre of a solar-type star”, in preparation for submission to
MNRAS
The remaining chapters represent work that is a reorganisation and refinement of some
material contained in these publications.
This thesis is my own work and contains nothing which is the outcome of work done
in collaboration with others, except as specified in the text and acknowledgements.
Cambridge, March 2011
3
Acknowledgements
Firstly, and most importantly, I would like to thank my PhD supervisor, Gordon
Ogilvie, for invaluable advice and many useful suggestions, that have sustained me
throughout the three years of my PhD, and made this research possible. In our collab-
oration in two published papers, Gordon contributed Figs. 2.9 & 5.8, as well as some
of the original calculations in §3.4, 3.5 & 4.2.2, which are present in this thesis. I am
also grateful for him taking the time to carefully read and helpfully criticise a draft of
this thesis, and several of its chapters in earlier forms.
I would like to thank the Astrophysical Fluid Dynamics research group, for many
stimulating discussions over coffee, or during seminars. Of those, I would particularly
like to thank Geoffroy Lesur, for providing a version of his SNOOPY code, before it
became publicly available, and for help with modifying the code, and visualisation
of the results. I also have him to thank for correcting some of my coding errors,
which were only spotted by me being reminded that his code “never fails”. For many
fascinating discussions, and for coping with being pestered about “simple” computer
problems, as well as proof-reading several documents over the years, I would like to
thank my fellow office resident, Hanno Rein.
For kindly taking the time to compute a set of stellar interior models, and assisting
with their interpretation, I am grateful to Jørgen Christensen-Dalsgaard. For several
extremely useful references, albeit to technical JFM papers from the 1960’s and 70’s,
I would like to thank Michael McIntyre.
This work would not have been possible without the financial support of STFC,
who funded my studentship1 for three years, and the Cambridge Philosphical Society,
who funded a studentship for the final three months. I have also received financial
support for travel to various conferences and workshops from DAMTP, Selwyn College,
Cambridge Philosophical Society, and the International Astronomical Union, to all of
which I am grateful.
1Unfortunately, their postdoctoral fellowships are currently lacking such support...
4
Contents
1 Introduction 10
1.1 Observations of short-period extrasolar planets . . . . . . . . . . . . . . 11
1.2 Formation and early migration of giant planets . . . . . . . . . . . . . . 14
1.3 Introduction to tidal friction . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Simple calculation of the tidal torque . . . . . . . . . . . . . . . . . . . 18
1.5 Tidal quality factors and empirical estimates . . . . . . . . . . . . . . . 20
1.6 Mechanisms of tidal dissipation . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Effects of tidal interactions on short-period planets . . . . . . . . . . . 25
1.8 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
I Constant lag time model of tidal friction 28
2 Tidal evolution of Hot Jupiters on inclined orbits 29
2.1 Model of tidal friction adopted . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Magnetic braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Analysis of the effects of MB on tidal evolution in a simplified system . 37
2.3.1 Circular, coplanar orbit with MB . . . . . . . . . . . . . . . . . 37
2.3.2 Extending the analysis to eccentric coplanar orbits . . . . . . . 42
2.3.3 Extending the analysis to inclined circular orbits . . . . . . . . . 44
2.4 Tidal evolution timescales . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Numerical integrations of the full equations for an inclined orbit . . . . 48
2.5.1 Inclusion of MB and the importance of coupled evolution of a
and i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.2 Inclined and eccentric orbits . . . . . . . . . . . . . . . . . . . . 51
5
2.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.6 Application to an observed system: an explanation for the misaligned
spin and orbit of XO-3 b . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
II Internal gravity waves and tidal dissipation in solar-type
stars 61
3 Internal wave breaking near the centre of a solar-type star: Boussinesq-
type model and 2D simulations 62
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Internal gravity waves: elementary properties, wave breaking and criti-
cal layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Basic description of the problem . . . . . . . . . . . . . . . . . . . . . . 69
3.3.1 Tidal potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.2 Central regions of a star . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Derivation of a Boussinesq-type system of equations . . . . . . . . . . . 71
3.5 Linear theory of IGWs approaching the stellar centre . . . . . . . . . . 74
3.5.1 Linear solution steady in a frame rotating with the pattern speed
of forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5.2 Properties of the (non-)linear solution . . . . . . . . . . . . . . . 76
3.6 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.8 Low-amplitude forcing: coherent reflection . . . . . . . . . . . . . . . . 84
3.9 High-amplitude forcing: wave breaking and critical layer formation . . . 88
3.9.1 Growth of different azimuthal wavenumbers in the flow . . . . . 93
3.9.2 Discussion of wave reflection from the critical layer . . . . . . . 95
3.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Internal wave breaking near the centre of a solar-type star: 3D sim-
ulations 102
4.1 Tidal potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6
4.2 Linear theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.1 Criterion for overturning isentropes . . . . . . . . . . . . . . . . 106
4.2.2 Weakly nonlinear theory . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Numerical set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4.1 Low-amplitude simulations . . . . . . . . . . . . . . . . . . . . . 109
4.4.2 High-amplitude simulations . . . . . . . . . . . . . . . . . . . . 113
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5 Implications 119
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Numerical computation of the linearised tidal response throughout the
star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2.2 Semi-analytical calculation of the ingoing energy flux . . . . . . 126
5.2.3 Variation between different stellar models . . . . . . . . . . . . 131
5.2.4 Are 1D linear hydrodynamic calculations reasonable? . . . . . . 134
5.3 When does wave breaking occur? . . . . . . . . . . . . . . . . . . . . . 135
5.4 Q′? for solar-type stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5 Orbital evolution of the planetary companion . . . . . . . . . . . . . . 138
5.6 Long-term evolution of the RZ . . . . . . . . . . . . . . . . . . . . . . . 140
5.7 Tidal dissipation in F-stars . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.8 An explanation for the survival of all (currently observed) short-period
planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6 Stability analysis of a standing internal gravity wave near the centre
of a solar-type star in 2D 148
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 Internal gravity wave stability analysis . . . . . . . . . . . . . . . . . . 151
6.2.1 Exact primary wave solution . . . . . . . . . . . . . . . . . . . . 152
6.2.2 Infinitesimal perturbations . . . . . . . . . . . . . . . . . . . . . 154
7
6.2.3 Derivation of the evolutionary equations . . . . . . . . . . . . . 155
6.2.4 Linear solutions (in the absence of the primary wave) . . . . . . 156
6.2.5 Nonlinear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.6 Diffusive terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.3.1 Numerical computation of table of integrals . . . . . . . . . . . 160
6.4 Kinetic and potential energy equations . . . . . . . . . . . . . . . . . . 161
6.5 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.5.1 Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.5.2 Weakly nonlinear . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.6 Results for waves with A < 1: parametric instabilities . . . . . . . . . . 164
6.6.1 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.6.2 Energetics of the instabilities . . . . . . . . . . . . . . . . . . . 171
6.7 Results for waves with A > 1: the initial stages of wave breaking . . . . 172
6.7.1 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.7.2 Energetics of the instabilities . . . . . . . . . . . . . . . . . . . 179
6.8 Summary and discussion of results . . . . . . . . . . . . . . . . . . . . 180
6.8.1 Wave breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.8.2 Parametric instabilities . . . . . . . . . . . . . . . . . . . . . . . 181
6.8.3 Comparison with the plane IGW problem . . . . . . . . . . . . 182
6.8.4 Implications for tidal dissipation . . . . . . . . . . . . . . . . . . 183
6.8.5 Comparison with Kumar & Goodman . . . . . . . . . . . . . . . 185
6.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7 Critical layer formation by radiative diffusion 189
7.1 Order-of-magnitude estimate . . . . . . . . . . . . . . . . . . . . . . . . 189
7.2 IGWs subject to viscous and radiative damping in 2D . . . . . . . . . . 193
7.2.1 Angular momentum flux . . . . . . . . . . . . . . . . . . . . . . 198
7.3 2D simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.4.1 Magnetic fields in the RZ . . . . . . . . . . . . . . . . . . . . . 203
8
7.4.2 Goldreich-Schubert-Fricke instability . . . . . . . . . . . . . . . 205
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8 Conclusion 207
8.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
III Appendices 213
A 214
A.1 Additional secular perturbing accelerations . . . . . . . . . . . . . . . . 214
A.2 Stability of the synchronous state without MB . . . . . . . . . . . . . . 215
A.3 Tidal potential valid for any stellar obliquity . . . . . . . . . . . . . . . 216
A.4 Ingoing/outgoing wave decomposition . . . . . . . . . . . . . . . . . . . 218
A.5 ZEUS comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
A.6 Analytic calculation of F in the launching region . . . . . . . . . . . . 223
A.7 Toy model: parametric instability of primary wave . . . . . . . . . . . . 226
A.8 Acronyms and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . 227
Bibliography 238
9
Chapter 1
Introduction
It has now been more than a decade since the discovery of the first extrasolar planet
orbiting a solar-type star (Mayor & Queloz 1995). The detection of this first Jovian-
like planet 51 Peg b, and its 4 day orbit, certainly came as a surprise to the astrophysics
community. To put this discovery in context, the closest planet in our solar system,
Mercury, completes a single orbit of the Sun in 88 days, and so 51 Peg b is more than
seven times closer to its host star than Mercury is to the Sun (and 100 times closer
than Jupiter).
The discovery of 51 Peg b rapidly spurred further research into the detection,
formation and evolution of planetary systems, making this one of the most exciting
fields in astrophysics today. Since 1995, more than 500 planets have been detected
around stars other than the Sun, and this total is increasing on an almost weekly
basis1. This data set shows that the existence of planets in extraordinarily tight orbits
(< 0.1 AU), such as 51 Peg b, is rather common. Indeed, from the current statistics,
nearly a third of planets orbit within 0.1 AU of their host stars, making an explanation
of the existence and properties of such planets an important goal of planet formation
and evolution studies.
For such close-in planets, the stellar insolation that they receive could lead to the
surface temperature of these worlds attaining values as high as 1500 K, which together
with their roughly Jovian-masses, has led to them being dubbed “Hot Jupiters” (HJs).
It is the purpose of this thesis to describe and understand some of the surprising
1see http://exoplanet.eu/ or http://www.exoplanet.hanno-rein.de/ for the latest updates
10
properties of these HJs, and their evolutionary history.
1.1 Observations of short-period extrasolar planets
We should perhaps not be surprised that the first planets detected be close-in and of
Jovian mass, if we consider the observational biases of the currently favoured detection
techniques. 51 Peg b, as with over 400 of the planets detected so far, was discovered
using the radial velocity (RV) method (Mayor & Queloz 1995). This involves measuring
the Doppler shift of stellar spectral lines, which are slightly blue-shifted and red-
shifted by the orbital motion of the star around the centre of mass of the star-planet
system, throughout the orbit. The RV method gives a measurement of the line-of-sight
orbital velocity of the star (the velocity semi-amplitude K). In addition, the period
of the Doppler shift fluctuations gives the orbital period (P ), and the shape of the RV
curve allows a determination of the eccentricity (e) of the orbit. These data allow a
determination of Mp sin Ip, which is a lower bound on the mass of the planet (Mp),
where Ip is the angle of inclination of the orbital plane to the plane of the sky. The
velocity semi-amplitude for a Keplerian orbit of eccentricity e is given by (Papaloizou
& Terquem 2006)
K =
(
2piG
P
) 1
3 Mp
(Mp +M?)
2
3
1
√
1− e2
, (1.1.1)
where M? is the stellar mass. We see that a close-in, massive planet produces a larger
velocity semi-amplitude, which is easier to detect above stellar jitter. It must be noted
that most RV surveys concentrate on late-type, relatively chromospherically inactive
stars which tend to be slow rotators, since stellar activity can cause significant jitter
in RV data, which hinders the detection of planets.
In the last decade, the RV method has been supplemented by an additional tech-
nique for the detection of extrasolar planets, which when combined with RV spec-
troscopy, can provide much more information about a system. This is transit pho-
tometry, which involves looking for a reduction in stellar flux caused by the passage
of a planet across the stellar disc in the plane of the sky (Charbonneau et al. 2000;
Charbonneau et al. 2007). A Jupiter-mass planet on a short-period orbit can cause a
11
fractional dimming of the starlight by as much as 1% (see Fig. 1.1), and this can be
readily observed. This technique gives us unprecedented access to the physical prop-
erties of the planet, such as a direct estimate of its mass and radius. These are both
important for determining the interior structure of these planets, thereby constraining
theoretical models (e.g. Fortney et al. 2007).
Another advantage to combining RV and photometric transit data is the possibility
of detecting the Rossiter-McLaughlin (RM) effect (Rossiter 1924; McLaughlin 1924).
The RM effect is a spectral distortion of the RV data that results from the planet
occulting a spot on the rotating surface of the star as it transits the stellar disc. High-
precision RV data during a transit allow a determination of the angle (λ) between the
sky-projected angular momentum vectors of the planetary orbit (h) and stellar spin
(Ω), through the RM effect. This measured value λ is not necessarily the same as
the inclination i (which is also called the stellar obliquity, given by i = arccos(Ωˆ · hˆ))
which is the angle between the equatorial plane of the star and orbital plane of the
planet, since λ is just a sky-projection of this angle. These angles are related by
cos i = cos I? cos Ip + sin I? sin Ip cosλ, where Ip and I? are the angles of inclination of
the planetary orbital plane and stellar equatorial plane, to the plane of the sky. For a
transit, the orbit must be close to edge-on, so Ip ≈ 90◦, giving cos i ' sin I? cosλ. In
this case, λ gives a lower bound on the angle between these two vectors.
The RM effect has now been used to measure the degree of spin-orbit alignment
in more than 30 systems (Triaud et al. 2010), and this number is expected to increase
significantly over the next few years. The first systems were all consistent with λ
being zero, with the exception of XO-3 b (He´brard et al. 2008). We discuss a model
for the survival and remnant misalignment of this planet in §2.6. In the past year,
the misalignment of XO-3 b has been confirmed by Winn et al. (2009), and significant
misalignments have been found in several additional systems: HD 80606 b (Moutou
et al. 2009) WASP-14 b (Johnson et al. 2009), and possible retrograde orbital motion
has been identified for HAT-P-7 b (Winn et al. 2009; Narita et al. 2009) and WASP-2
b, WASP-15 b, and WASP-17 b (Triaud et al. 2010), amongst others.
12
Figure 1.1: Left: Transit lightcurve for HD 209458, as a function of transit phase (defined
to be zero when the planet passes the centre of the stellar disc). Right: Close-up of RV
data throughout a transit for the same system, showing the RM effect as an “anomalous”
spectral distortion. As expected for a prograde orbit, we first see a net red-shift, followed
by a net blue-shift. The RM effect allows a determination of the sky-projected spin-orbit
misalignment (λ) in a transiting system, which is found to be λ = −4◦.4 ± 1◦.4 for HD
209458. This figure was taken from Winn et al. (2005).













































































































































































































































































































































































































































































































i
Ip
I?
λ
Ω
h
Figure 1.2: Angles involved in the calculation of the inclination i, from the sky-projected
spin-orbit misalignment angle λ, given that Ω is inclined to the line-of-sight by I? degrees,
and h is inclined to the line-of-sight by Ip degrees. λ is detectable through the RM effect, Ip
can be determined for a transit (and is ≈ 90◦). For transiting planets λ gives a lower bound
on the true spin-orbit misalignment i.
13
1.2 Formation and early migration of giant planets
Observations of the solar system show that the orbits of the planets all approximately
lie within a common plane, whose normal is oriented only 6◦ from the solar rotation
axis (Beck & Giles 2005). This observed coplanarity is what originally inspired Kant
(1755)2 and Laplace (1796) to propose that the Sun and its planets condensed from a
spinning, flattened nebula, which is the so-called Nebular Hypothesis.
Since the time of Kant and Laplace our theories have complicated considerably
(see Papaloizou & Terquem 2006 for a comprehensive review), but the main idea of
the Nebular Hypothesis has remained unchanged. It is now generally believed that
planets form out of the discs of gas and dust observed around young T-Tauri stars.
These stars and their circumstellar discs are thought to have themselves formed from
the collapse of molecular cloud material with appreciably nonzero angular momentum,
in which centrifugal forces hinder collapse perpendicular to the rotation axis, resulting
in the formation of a gaseous disc. These protoplanetary discs around T-Tauri stars
last for up to 10 Myr, after which the discs appear to dissipate (Beckwith 1996).
Planet formation theory must therefore explain the formation of the planets in the
solar system, and their extrasolar analogues, within this window of opportunity.
There are two competing theories for the formation of giant planets. The first
involves the coagulation and accumulation of small particles of ice and rock to form
a solid core (Safronov 1969), which rapidly accretes gas from the protoplanetary disc.
This is the core-accretion scenario (Pollack et al. 1996). The second relies on the
gravitational instability of the gaseous disc (see Durisen et al. 2007, for a recent review)
to produce the giant planet directly.
Both of these formation scenarios necessitate the formation of planets such as 51
Peg b in colder regions of the disc, much further out (with semi-major axes of several
AU), before a migratory process that brings the planet in towards the star and to its
present location (Lin et al. 1996). Planetary migration in gaseous protoplanetary discs
had already been predicted prior to the 1980’s (Goldreich & Tremaine 1979; Lin &
Papaloizou 1979), and is thought to be driven by the dynamical interaction between
2The Nebular Hypothesis was first proposed in 1734 by Emanuel Swedenborg (Swedenborg 1734),
though Immanuel Kant developed the theory further in 1755.
14
the planet and its gaseous protoplanetary disc. This migration occurs because the
planet produces inward and outward propagating spiral density waves in the disc,
which both interact gravitationally with the planet. These waves carry negative and
positive angular momentum fluxes respectively, and a corresponding tidal torque acts
on the orbit of the planet. This is referred to as type I migration, and is relevant for
terrestrial planets and giant planet cores. When the mass of the planet grows, the
density perturbation becomes non-linear and the planetary wake turns into a shock
in its vicinity. This results in the formation of a gap in the disc, with the subsequent
inward migration of the planet occurring on the disc viscous timescale. Research into
both of these migration regimes is still underway, and there remain many unsolved
problems (see Papaloizou et al. 2007 for a recent review), though a combination of
both processes during the early stages of planet formation could potentially produce
a population of HJs.
The formation of systems of giant planets can be thought of as occurring in two
oversimplified stages (Juric´ & Tremaine 2008).
• During stage 1 the cores of the giant planets are formed, they accrete gas and
undergo migration, driven by the dynamical interaction between the planets
and the gaseous protoplanetary disc. This stage lasts a few Myr until the gas
dissipates, by which time a population of gas giants may exist. If these form
sufficiently closely packed then stage 2 follows.
• Stage 2 lasts from when the disc has dissipated and continues until the present,
and primarily involves gravitational interactions and collisions between the plan-
ets. Recent studies into stage 2 (Juric´ & Tremaine 2008; Chatterjee et al. 2008;
Ford & Rasio 2008) have shown that this is a chaotic era, in which planet-planet
scatterings force the ejection of all but a few (∼ 2− 3) planets from the system,
in a period of large-scale dynamical instability lasting . 108yr.
Planet-planet scatterings can excite the eccentricities of the planets to levels re-
quired to explain observations. They tend also to excite the inclinations of the planets
with respect to the initial symmetry plane of the system, though this has been found
to be less efficient than the excitation of eccentricity. This could potentially lead to
observable consequences via the RM effect.
15
It is possible for HJs orbiting a host star which has a distant and inclined stellar
companion, or massive inclined outer planetary companion, to undergo another type
of migration. This is Kozai migration (Wu & Murray 2003). The presence of such
an outer companion to an exoplanet host star could cause Kozai oscillations, which
produce periods of extreme eccentricity in the planet orbit, if various conditions are
satisfied (e.g. see section 1.2 of Fabrycky & Tremaine 2007). The subsequent tidal
dissipation that occurs during the periods of small pericentre distance leads to grad-
ual inward migration of the planet. It has even been proposed that a combination of
planet-planet scattering, tidal circularisation and the Kozai mechanism in outer plan-
ets, can produce HJs around single stars (Nagasawa et al. 2008). HJs produced from
these processes generally have their orbital angular momentum vector misaligned with
respect to the stellar spin axis by large angles – occasionally larger than 90◦ (Fabrycky
& Tremaine 2007; Nagasawa et al. 2008).
Misaligned orbits are not predicted from stage 1 alone, so if λ is measured to be
appreciably nonzero in enough systems, then it could be seen as evidence for planet-
planet scattering or Kozai migration. This is because gas-disc migration does not
seem able to excite orbital inclination (Lubow & Ogilvie 2001; Cresswell et al. 2007).
Alternatively, if observed planets are all found with λ consistent with zero, this could
rule out planet-planet scattering or Kozai migration as being of any importance.
One important consideration is that at such close proximity to their parent stars,
strong tidal interactions between the star and planet are expected to cause significant
long-term spin-orbit evolution, including changes to the value of λ (actually the true
spin-orbit misalignment angle i) over time. If tides can change λ since the time of
formation, then we may have difficulty in distinguishing migration caused by planet-
planet scattering and Kozai oscillations, from gas-disc migration. The tidal evolution
of such inclined orbits must therefore be considered an important goal in planetary
evolution studies. In Part I, we study the effects of tidal friction on such inclined
orbits.
16
1.3 Introduction to tidal friction
It has been known since the time of Newton that the gravitational interaction between
the Earth and the Moon produces a tide in the seas. An early analysis of the nature
of this tidal interaction was introduced by Darwin (1880)3. In his model, the Earth is
treated as a homogeneous spherical fluid body, which continually adjusts to maintain a
state of quasi-hydrostatic equilibrium in the time-varying gravitational potential of the
Moon. Darwin calculated that viscosity would introduce a phase lag in the response of
the Earth to the forcing potential of the Moon. This leads to a misalignment between
the tidal bulge and the line connecting the centres of the Earth and Moon, resulting
in the action of torques on the rotation of the Earth and on the orbit of the Moon.
This concept is referred to as the equilibrium tide model (see Fig. 1.3). Significant
spin-orbit evolution can occur over secular timescales through these tides, as can be
inferred from Lunar Laser Ranging observations (Chapront et al. 2002), which show
that the Moon is receding from the Earth at a current rate of 3.8 cm per year as the
Earth spins down at a rate of 23µs per year.
































































































α
m1
m2
n
Ω
Figure 1.3: Illustration of the tidal torque on an asynchronously rotating primary. When
the primary rotates faster than the orbital motion (Ω > n), its mass distribution is shifted
by an angle α from the line joining the centres of the two components, due to dissipation of
energy. Since the forces applied to the two tidal bulges are not equal, a torque is exerted on
the star, which tends to synchronise its rotation with the orbital motion (Ω → n). A similar
picture emerges when Ω < n; see §1.4 for a derivation of the torque in this case.
The tidal interaction between two orbiting bodies acts to continually change the
orbital and rotational system parameters, and continually dissipates mechanical en-
ergy. Ultimately – in the absence of angular momentum loss from the system – either
3who also had happened to have a somewhat influential father...
17
an equilibrium state is asymptotically approached, or the two bodies spiral towards
each other at an accelerating rate, and eventually collide. The equilibrium state is
characterised by coplanarity (the equatorial planes of the bodies coincide with the
orbital plane), circularity and corotation (rotational frequencies of each body match
the orbital frequency) (Hut 1980).
Tidal forces are also thought to be important elsewhere in the solar system (see
Goldreich & Soter 1966 for an old review and Peale 1999 for a more modern one), and
are responsible for producing the spectacular vulcanism on Jupiter’s moon Io. There
also exist close-binary stars whose orbits appear to have been circularised, which shows
that tidal friction is not only important for planets and their satellites in our solar
system (e.g. Meibom & Mathieu 2005). In addition, a close-in planet with an orbital
period shorter than the stellar spin period is subject to tidally induced inspiral into
the star (the opposite evolution to that of the Earth-Moon system). The problem
tackled in Part II is to explain the survival of the observed short-period planets, by
understanding the efficiency of this process.
1.4 Simple calculation of the tidal torque
In this section we derive the torque on an asynchronously rotating star with a com-
panion planet on a coplanar, circular orbit following the approach of Zahn (2008),
for illustrative purposes. In the equilibrium tide model we assume that the star is in
hydrostatic equilibrium, and that in the absence of dissipation its shape would adjust
instantaneously to the time-dependent perturbing potential of its orbiting companion.
We further assume that the star rotates slower than the planet revolves around the
star, i.e., Ω < n. Fig. 1.3 sketches the system under consideration, except that it
shows the case Ω > n, appropriate for the Earth-Moon system.
The relative height of the tidal bulge is given approximately by the ratio of the
differential force exerted on the bulges (ftide) to the surface gravity of the body (fbind)
δR1
R1
≈
ftide
fbind
≈
Gm2m1R1/r3
Gm21/R
2
1
≈
m2
m1
(
R1
r
)3
. (1.4.1)
In these expressions, r is the star-planet separation and R1 is the stellar radius. If we
18
further assume that the star has a constant density, then its tidal bulges would have a
mass δm1 ≈ δR1R1 m1. This is not strictly correct for a realistic stellar model, due to the
strong central condensation of the star (which can be taken into account by defining
a Love number k < 1; it is also not exact for a homogeneous body), but it suffices for
the present purpose.
When Ω 6= n, dissipation of energy stored in the tide by any process will introduce
a slight lag of the tidal bulges relative to the lines joining the centres of the star and
the planet. The star therefore experiences a torque which acts to synchronise the spin
of the star with the orbit. Due to conservation of total angular momentum, this is
equal in magnitude and opposite in sign to the torque which acts on the orbit of the
planet. In the weak friction approximation this lag is represented by a lag angle α¿ 1,
which is assumed to be proportional to the tidal frequency ωˆ = 2(n − Ω). The tidal
frequency is the Doppler shifted forcing frequency that is seen in the frame rotating
with the star (with angular velocity Ω), and is twice the orbital angular frequency (n)
because the dominant tide is quadrupolar, i.e., has two high tides and two low tides
(see e.g. Murray & Dermott 1999). A common non-dimensional parametrisation of
the lag angle is
α =
ωˆ
tdiss
R31
Gm1
, (1.4.2)
where we have introduced the dynamical timescale (R31/Gm1)
1/2 to make α dimension-
less. This gives a tidal torque acting to synchronise the star with the orbital motion
of magnitude
Γ ≈ δm1ftideR1 sinα ≈
ωˆ
tdiss
m22
m1
R21
(
R1
r
)6
, (1.4.3)
when α ¿ 1. This expression is only valid if the dissipation is relatively inefficient,
with large tdiss (and therefore Q′ À 1, which we define in the next section). A torque
of equal magnitude, but of opposite sign, acts on the orbit of the planet, which results
in a transfer of angular momentum from the planetary orbit to the stellar spin. The
consequences of this torque are that the star spins up as the planet falls into the star,
until either the star is spun up to rotate with the angular frequency of the orbit, or
the planet has plunged into the star.
19
1.5 Tidal quality factors and empirical estimates
The efficiency of tidal dissipation in a body is often parametrised by a dimensionless
quality factor Q, which reflects the fact the body undergoes a forced oscillation and
dissipates a small fraction of the associated energy during each oscillation period. This
is analogous to the quality factor in a forced, damped harmonic oscillator (Murray &
Dermott 1999; Feynman 1963), and is defined by
Q = 2piE0
(∮
−E˙ dt
)−1
,
where E0 is the maximum energy stored in an oscillation and the integral represents
the energy dissipated over one cycle. This is related to the time-lag τ in the response
of the body to tidal forcing of frequency ωˆ by Q−1 = ωˆτ . We further find it convenient
to define Q′ = 3Q2k , where k is the Love number of the body, since this combination
always appears together in the evolutionary equations. The factor k < 1 if the body
is centrally condensed, as is the case for stars and giant planets. Q′ reduces to Q for
a homogeneous fluid body, where k = 32 .
Typically assumed values of Q′ ∼ 106 for stars are roughly consistent with ob-
servational data regarding the circularisation periods of binary stars (Ogilvie & Lin
2007, hereafter OL07). This value can explain the observed circularity of the orbits of
close binary stars with orbital periods shorter than the circularisation period (which
increases with system age), if tidal friction is responsible for the circularisation. In ad-
dition, the magnitude of Q′ for HJs is often assumed to be similar to that for Jupiter,
which has been inferred to be in the range 2 × 105 − 8 × 106 (Yoder & Peale 1981),
since k ≈ 0.38 for Jupiter (Gavrilov & Zharkov 1977). This estimate is based on
a model of the tidal origin of the Laplace resonance among the Galilean satellites;
however, it has been argued that even if the origin of the resonance is primordial, the
average Q′ cannot be far from these bounds (Peale & Lee 2002). This estimate also
appears consistent with Jackson et al. (2008), who found that one can reproduce the
outer planet (with semi-major axes wider than 0.2 AU) eccentricity distribution from
integrating the tidal evolution equations backwards in time for the observed close-in
planets (with semi-major axes smaller than 0.2 AU) quite well if Q ∼ 105.5 for stars
20
and Q ∼ 106.5 for HJs. However, the resulting stellar Q′ is difficult to reconcile with
the existence of the planets on the tightest orbits, such as WASP-12 b (Hebb et al.
2009) and OGLE-TR-56 b (Sasselov 2003), since it would imply that the inspiral time
for these planets would be much less than the age of the system. That several planets
have been found with similarly short periods makes this seem unlikely on probabilistic
grounds. Additionally, Jackson et al. (2008) assume that Q′ is the same for all exo-
planet host stars. This may be a major oversimplification, which we discuss in Part II
of this thesis.
In the model of § 1.4, the reciprocal of the tidal quality factor of the star is given
by
1
Q′
∝
1
tdiss
(
R31
Gm1
)
|ωˆ|. (1.5.1)
If the dissipation of the equilibrium tide is solely due to an effective viscosity associated
with turbulent convection, then tdiss can be calculated by integrating the turbulent
viscosity (νt) over convective regions of the star. In the mixing length prescription,
this turbulent viscosity is defined to be νt = vl, where v is the eddy velocity and l
is the mixing length. When the tidal period is shorter than the convective turnover
time, i.e.,
2pi
ωˆ
≡ Ptide < tconv ≡
l
v
, (1.5.2)
the turbulent viscosity should be reduced by some factor, over which there is uncer-
tainty (see e.g. Zahn 1966; Goldreich & Nicholson 1977; Goodman & Oh 1997; Penev
et al. 2007). Zahn reduces the turbulent viscosity by the following prescription with
p = 1:
νt =
1
3
vl [1 +B (|ωˆ| tconv)
p ]−1 , (1.5.3)
where B is a constant numerical factor. This prescription gives tdiss ∝ |ωˆ|. If
this reduction is correct, then modelling the effects of tidal friction by a frequency-
independent Q′ would be reasonable. If on the other hand, νt is assumed to be in-
dependent of ωˆ, then tdiss is frequency-independent. This is the constant time-lag
21
approach (Alexander 1973; Hut 1981), which is justified if the dissipative process acts
instantaneously, i.e., the convective turnover time, which is the relaxation time of the
turbulent stress, is zero (Ivanov & Papaloizou 2004). In Part I we outline a derivation
of the constant lag-time model of tidal friction, in a model which is valid for arbitrary
eccentricity and stellar and planetary obliquity.
1.6 Mechanisms of tidal dissipation
The problem of theoretically determining the efficiency of tidal dissipation, and there-
fore quantifying the evolution of a system, amounts to calculating Q′ factors for each
body. Q′ is in principle a function of the tidal forcing frequency and (in nonlinear
cases) the amplitude of the tidal disturbance, and is a result of complex dissipative
processes in each body (Zahn 2008). For solid bodies such as the Earth, Q′ is thought
to be approximately constant over a wide range of frequency (Munk & MacDonald
1960), which together with computational simplicity, has led to most studies of tidal
evolution for solar system bodies assuming a frequency-independent Q′. This assump-
tion for gaseous stars and giant planets is questionable, and the nature of the response
of such a body to tidal forcing remains under investigation.
The tidal disturbance can generally be decomposed into two parts: an equilibrium
tide, and a dynamical tide. The equilibrium tide is the quasi-hydrostatic ellipsoidal
tidal bulge. In the frame corotating with the fluid, the time-dependence of the equi-
librium tide, or its associated velocity field, is dissipated through its interaction with
turbulent convection. However, the damping rate is uncertain, particularly when the
convective time exceeds the tidal period, as we discussed in the previous section. The
dynamical tide consists of internal waves that are excited by low-frequency tidal forc-
ing, and has received much recent interest with regard to its possible contribution to
Q′ (Witte & Savonije 2002; Ogilvie & Lin 2004, hereafter OL04; Wu 2005; Papaloizou
& Ivanov 2005; Ivanov & Papaloizou 2007; OL07; Goodman & Lackner 2009). This is
because if these waves have short wavelength, then they are more easily damped than
the large-scale equilibrium tide by radiative diffusion (Zahn 1975; Zahn 1977), convec-
tive viscosity (Terquem et al. 1998; hereafter T98), or nonlinear breaking (Goodman
& Dickson 1998; hereafter GD98). For rotating fluid bodies, calculations of the exci-
22
tation and dissipation of internal waves have indicated that Q′ varies in a complicated
way with the tidal frequency (Savonije et al. 1995a; Savonije & Papaloizou 1997a;
Papaloizou & Savonije 1997a; OL04; OL07) – for an illustration see Fig. 1.4.
Figure 1.4: Q′ as a function of the ratio of tidal frequency to spin frequency ωˆ/Ω. These
results refer to tidal forcing by the l = m = 2 spherical harmonic in a model of the Sun with
a spin period of 3 days. Left: Q′ from the viscous dissipation of inertial waves in the CZ.
Right: Q′ from the excitation of internal inertia-gravity waves in the RZ. The dashed lines
show the effect of omitting the Coriolis force, which has been found to be of great importance
when |ωˆ| < 2|Ω|. The efficiency of tidal dissipation depends in a highly erratic way on the
tidal forcing frequency. This figure was taken from OL07.
The tidal frequency is typically much lower than the dynamical frequency of the
body, so the relevant internal waves must be approximately incompressible, restored
not by pressure, but by buoyancy or rotation. OL04 found that the dissipation of
tidally excited inertial waves, whose restoring force is the Coriolis force, can contribute
significantly to the dissipation rate in a giant planet, whose interior is mostly convective
(see also Wu 2005 and Ivanov & Papaloizou 2007). These waves can also contribute
to the dissipation rate in convection zones (CZs) of stars (OL07). They are excited by
tidal forcing of frequency ωˆ, if this is less than the Coriolis frequency (2Ω), and this
is true for many astrophysically relevant circumstances. However, these waves are not
excited if the tidal frequency exceeds the Coriolis frequency, so this process is then not
effective at dissipating the tide, and contributing to Q′.
In slowly rotating stars (with Ω < n), such as those that host most of the short-
period planets, the waves are primarily excited by tidal forcing in radiative regions,
and are restored by buoyancy. These are commonly referred to as internal gravity
23
waves (IGWs). IGWs are damped by radiative diffusion, but this is only efficient if
their frequencies are sufficiently low compared with the damping rate, which occurs if
they have a short radial wavelength. However, in solar-type stars, the waves excited by
planets with several-day orbital periods or less, are of sufficiently large wavelength that
radiative diffusion is negligible in their attenuation (see § 7.1; GD98). It is therefore
appropriate to ask whether there are any other mechanisms that can contribute to their
dissipation (and to the stellar Q′), which would have implications for the survival of
these short-period planets.
One such mechanism was proposed by GD98, and involves nonlinear effects be-
coming important near the centre of a solar-type star with a radiative core, where
these waves become geometrically concentrated, and may attain large amplitudes. If
nonlinear effects result in the absorption of the waves that propagate towards the
centre, then this would be a relatively efficient mechanism of tidal dissipation, which
could dominate over convective damping of the equilibrium tide. It is important to
understand whether this process actually occurs, and if indeed it does, how efficient
it is at affecting the orbits of the short-period planets. In Part II we study this mech-
anism in detail. We perform the first simulations of IGWs approaching the centre of
a solar-type star, in both two and three dimensions, with the aim of studying how
the reflection of waves from the centre is modified by nonlinearities as the amplitude
is increased. We find that the waves break near the centre if they attain sufficient
amplitude, and that this has important implications for the survival of short-period
extrasolar planets.
We now summarise several mechanisms for dissipating energy stored in tidal oscil-
lations in gaseous bodies, which have been found to be of considerable importance.
• Dissipation of the equilibrium tide by the turbulent viscosity of convective eddies
in the convective envelopes of solar-type stars and the convective interiors of giant
planets.
• The excitation and subsequent dissipation by turbulent viscosity of inertial waves
in convective regions. These are excited when tidal frequencies are smaller than
twice the spin frequency (|ωˆ| < 2|Ω|).
• The excitation of internal inertia-gravity waves – essentially IGWs modified
24
through Coriolis accelerations by rotation – in radiation zones (RZs) of stars
and radiative envelopes of highly irradiated giant planets. These are dissipated
by radiative diffusion, which is particularly strong near the surface of a giant
planet or star. In addition, nonlinear effects may be important near the centre
of a solar-type star, which could result in wave breaking and additional dissipa-
tion. This latter process is studied in detail in Part II.
1.7 Effects of tidal interactions on short-period plan-
ets
Following the discovery of 51 Peg b, it was realised that tidal interactions between a HJ
and its host star could result in significant spin-orbit evolution over secular timescales
(Rasio et al. 1996; Marcy et al. 1997). Rasio et al. (1996) showed that 51 Peg b is
subject to orbital decay due to tides dissipated in the star, but has an inspiral time
slightly longer than the main-sequence lifetime of the star in their model (though note
that this timescale depends on the tidal quality factors of the star and planet). It
is also expected that tides dissipated in the planet will synchronise the rotation of
the planet with the orbital motion in a few Myr. In addition, Rasio et al. (1996)
suggested that tidal dissipation in the planet may explain the negligible eccentricities
of the shortest-period planets, and indeed observed planets with orbital periods shorter
than 6 days primarily have circular orbits. This is illustrated in Fig. 1.5.
Dobbs-Dixon et al. (2004) proposed an explanation for the coexistence of both
circular and eccentric orbits for the planets in the period range 7-21 days as the result
of the variation in spin-down rates of young stars. Planets with orbits within this
range are expected to be moderately affected by tides, whereas planets with orbital
periods longer than 21 days are expected to be only negligibly affected, due to the
strong dependence of the tidal force on separation. Recently Pont (2009) discussed
empirical evidence for tidal spin-up of exoplanet host stars, and found indications of
such a process occurring in the present sample of transiting planets. These works
highlight the importance of stellar spin evolution on tidal evolution.
25
Figure 1.5: Orbital period-eccentricity distribution for the currently observed extrasolar
planets. Orbits with P . 6 days have a smaller spread in e than those with larger orbital
periods. This could be a result of circularisation due to dissipation of tides raised in the
planet. This figure was taken from exoplanet.hanno-rein.de.
Jackson et al. (2008) found from considering coupled evolution of the eccentricity
and semi-major axis that simple timescale considerations of circularisation may not
accurately represent the true evolution. In addition, they found that it is inaccurate
to neglect the combined effects of both the stellar and planetary tides. In Part I we
study the accuracy of simple timescale estimates for tidal evolution, when coupled
evolution of the orbital and rotational elements is considered, in a more general model
of the long-term effects of tidal friction than Jackson et al. (2008) consider. We include
stellar spin-down, and study its effects in a simplified system. In particular, we study
the tidal evolution of inclination, since this has not been done in previous work.
1.8 Outline of this thesis
In Part I we study the tidal evolution of inclined orbits, using a variant of the simple
“constant time-lag” model of tidal friction. We study the inclusion of stellar rotational
spin-down and the accuracy of simple timescale estimates of tidal evolution. These
26
results have important consequences for the interpretation of observations of spin-orbit
alignment angles using the RM effect, which will be discussed.
In Part II, we turn to the larger problem of understanding the mechanisms of
tidal dissipation in solar-type stars. It is of particular interest to study the efficiency
of tidal dissipation in these stars, since many have been found to harbour close-in
planets, whose survival is determined by the stellar Q′. This is because a planet with
an orbital period shorter than the stellar spin period is subject to tidally induced
orbital decay, with an inspiral time that depends linearly on the stellar Q′. We study
and extend a nonlinear model of tidal dissipation, initially proposed by GD98. In
this model IGWs are excited at the interface of the convection and radiation zones of
a solar-type star, which then propagate to the centre, where they can break if their
amplitude is sufficiently large. We model the dynamics of the central regions of the
star to study this process, primarily through the use of direct numerical simulations, in
both two and three dimensions. Particular emphasis is made on analysing the resulting
ingoing wave absorption which takes place near the centre. We then perform a detailed
stability analysis of an IGW near the centre of a solar-type star to understand the
breaking process in more detail, and to determine the existence and properties of any
instabilities that onset for lower amplitude waves than were observed to break in the
simulations. We also outline an additional mechanism of tidal dissipation involving
the gradual attenuation of the waves by radiative diffusion, and discuss whether it
is likely to operate in reality. The implications of these processes for the survival of
short-period planets around solar-type stars is discussed throughout. This work allows
us to propose a possible explanation for the survival of all short-period planets around
F, G and K stars.
27
Part I
Constant lag time model of tidal
friction
28
Chapter 2
Tidal evolution of Hot Jupiters on
inclined orbits
In this chapter, we use a simple model of the orbit-averaged effects of tidal friction
to study the evolution of close-in planets on inclined orbits, due to tides. We first
analyse the effects of the inclusion of stellar rotational spin-down due to magnetic
braking (hereafter MB). A phase-plane analysis of a simplified system of equations,
including the MB torque is performed. We find that the inclusion of MB can lead
to qualitatively different spin-orbit evolution; its neglect can result in a very different
system history. We then present the results of numerical integrations of the tidal
evolution equations, where we find that it is essential to consider coupled evolution of
the orbital and rotational elements, including dissipation in both the star and planet,
to accurately model the evolution. The main result of these integrations is that for
typical HJs, tidal friction aligns the stellar spin with the orbit on a similar time as it
causes the orbit to decay. This tells us that if a planet is observed to be aligned, then
it probably formed coplanar. This reinforces the importance of Rossiter-McLaughlin
effect observations in determining the degree of spin-orbit alignment in transiting
systems. Finally, we apply these results to the first observed system with a spin-orbit
misalignment, XO-3, and constrain the efficiency of tidal dissipation (i.e. the modified
tidal quality factors Q′) in both the star and the planet in this system.
29
2.1 Model of tidal friction adopted
In light of the uncertainties in Q′, and the difficulty in calculating the spin-orbit
evolution when Q′ is a complicated function of ωˆ, we adopt a simplified model, based
on a frequency-independent lag time (with one modification discussed below). We
adopt the model of Eggleton et al. (1998), which is based on the equilibrium tide
model of Hut (1981). In this formulation, we calculate the evolution of the specific
angular momentum of the planetary orbit
h = r× r˙ = na2
√
1− e2 hˆ, (2.1.1)
together with its eccentricity vector e, and the stellar and planetary spin vectors Ω1
and Ω2. The eccentricity vector has the magnitude of the eccentricity, and points in
the direction of pericentre, and is defined by
e =
r˙× h
Gm12
− rˆ, (2.1.2)
where m12 = m1 +m2 is the sum of the stellar and planetary masses.
The evolutionary equations are derived by starting with the equation of relative
motion of a planet of mass m2 and its host star of mass m1
d2r
dt2
= −
Gm12
r3
r+ f , (2.1.3)
where r is the separation and f represents a perturbing acceleration. The solution of
the unperturbed equation (with f = 0) is the ellipse
|r| =
a(1− e2)
1 + e cos f
, (2.1.4)
where f is the angle referred to pericentre (the true anomaly). This is the Newtonian
two-body problem (Murray & Dermott, 1999). Both h and e are conserved for an
unperturbed Keplerian orbit; therefore under weak external perturbations their com-
ponents vary slowly compared with the orbital period. This allows averaging of the
effects of the tidal perturbation over a Keplerian orbit, resulting in a set of secular
evolution equations for the rotational and orbital elements.
30
In Eggleton & Kiseleva-Eggleton (2001), f contains many contributions: the effects
of general relativity, perturbing accelerations due to other planets and tidal and spin
distortions of the star and planet, as well as tidal friction. In this chapter we solely
consider the perturbing effects of tidal friction (see Appendix A.1 for the inclusion of
the additional contributions) and set f = ftf = f1tf + f
2
tf , where
f1,2tf = −3τ1,2k1,2n
2
(
m2,1
m1,2
)(
R1,2
a
)5 (a
r
)8
[3(rˆ · r˙)rˆ+ (rˆ× r˙− r Ω1,2)× rˆ] . (2.1.5)
Here n =
√
Gm12
a3 is the orbital mean motion, k1,2 are the second-order potential
Love numbers for the star and planet respectively (which is twice the apsidal motion
constant), and τ1,2 is the effective tidal lag time for each body. This form of the
dissipative force of tidal friction is that derived under the assumption of a constant
lag time in the equilibrium tide model (Eggleton et al., 1998). In this model, we
assume that the body quasi-hydrostatically adjusts to the perturbing potential of its
companion, but delayed by some small lag time (τ1 for the star, τ2 for the planet)
that is proportional to the dissipation. Thus for each body, Q = 1ωˆτ is assumed to be
inversely proportional to the tidal frequency ωˆ, so that the lag time τ is independent
of tidal frequency, thus the same for all components of the tide.
In the resulting equations we have chosen to parametrise the efficiency of tidal
dissipation in each body by redefining Q = 1nτ , and adopt a constant Q which does
not change during the evolution, i.e., we assume effectively that the lag time scales
with the orbital period (then Eq. 2.1.5 matches Eq. 4 in Mardling & Lin 2002, where
this assumption was not made explicit). We also introduce the definition Q′ = 3Q2k .
This allows us to discuss “Q′ values” for particular bodies, which do not change as
the orbital and rotational elements vary. For the purposes of this work we define
Q′ = 32knτ . This is equivalent to assuming that the relevant tidal frequency ωˆ = n.
Note that this may not give identical numerical factors in the resulting equations to
other formulations of tidal friction (e.g. Goldreich & Soter 1966; Zahn 1977; Hut 1981),
but we feel that this is the best way to study the general effects of tidal friction, given
the uncertainties in the value of Q′, and its dependence on ωˆ, for realistic giant planets
and stars.
31
The planet’s specific orbital angular momentum h changes at a rate
dh
dt
= r× (f1tf + f
2
tf), (2.1.6)
with a corresponding rate of angular momentum transfer between the orbit and spin
of each body given by
J˙1,2 = I1,2Ω˙1,2 = −µr× f
1,2
tf , (2.1.7)
since total angular momentum is conserved, and where µ = m1m2m12 is the reduced mass
of the system. The eccentricity vector evolves as
de
dt
=
[2(ftf · r˙)r− (r · r˙)ftf − (ftf · r)r˙]
Gm12
. (2.1.8)
Eqs. 2.1.6 and 2.1.8 are time-averaged over the orbit, and the resulting differential
equations are given below. Numerical integration of these equations gives the secular
evolution of the orbital elements. Note that we have written these equations so that
they are regular at e = 0, unlike those in Eggleton & Kiseleva-Eggleton (2001) and
Mardling & Lin (2002). To do this we have eliminated reference to eˆ, which is undefined
for a circular orbit, whereas e is perfectly well defined and equal to zero. The resulting
equations read
dh
dt
= −
1
tf1
[
Ω1 · e
2n
f5(e
2)he−
Ω1
2n
f3(e
2)h+
(
f4(e
2)−
Ω1 · h
2n
1
h
f2(e
2)
)
h
]
−
1
tf2
[
Ω2 · e
2n
f5(e
2)he−
Ω2
2n
f3(e
2)h+
(
f4(e
2)−
Ω2 · h
2n
1
h
f2(e
2)
)
h
]
=
(
dh
dt
)
1
+
(
dh
dt
)
2
(2.1.9)
h
de
dt
= −
1
tf1
[
Ω1 · e
2n
f2(e
2)h+ 9
(
f1(e
2)h−
11
18
Ω1 · h
n
f2(e
2)
)
e
]
−
1
tf2
[
Ω2 · e
2n
f2(e
2)h+ 9
(
f1(e
2)h−
11
18
Ω2 · h
n
f2(e
2)
)
e
]
(2.1.10)
dΩ1
dt
= −
µ
I1
(
dh
dt
)
1
+ ω˙mb (2.1.11)
=
µ
I1tf1
[
Ω1 · e
2n
f5(e
2)he−
Ω1
2n
f3(e
2)h+
(
f4(e
2)−
Ω1 · h
2n
1
h
f2(e
2)
)
h
]
+ ω˙mb
dΩ2
dt
= −
µ
I2
(
dh
dt
)
2
(2.1.12)
32
=
µ
I2tf2
[
Ω2 · e
2n
f5(e
2)he−
Ω2
2n
f3(e
2)h+
(
f4(e
2)−
Ω2 · h
2n
1
h
f2(e
2)
)
h
]
We also need to define an inverse tidal friction timescale t−1f for each body (here
for body 1, change 1 → 2 to get the corresponding expression for body 2), and the
functions of the eccentricity (first derived in a similar form by Hut 1981).
1
tf1
=
(
9n
2Q′1
)(
m2
m1
)(
R1
a
)5
=
√
Gm12
(
9
2Q′1
)(
m2
m1
)
R51 a
− 132 (2.1.13)
f1(e
2) =
1 + 154 e
2 + 158 e
4 + 564e
6
(1− e2)
13
2
(2.1.14)
f2(e
2) =
1 + 32e
2 + 18e
4
(1− e2)5
(2.1.15)
f3(e
2) =
1 + 92e
2 + 58e
4
(1− e2)5
(2.1.16)
f4(e
2) =
1 + 152 e
2 + 458 e
4 + 516e
6
(1− e2)
13
2
(2.1.17)
f5(e
2) =
3 + 12e
2
(1− e2)5
(2.1.18)
f6(e
2) =
1 + 312 e
2 + 2558 e
4 + 18516 e
6 + 2564e
8
(1− e2)8
(2.1.19)
In the absence of MB (ω˙mb = 0) the total angular momentum is conserved (dLdt = 0).
With the inclusion of MB (see §2.2), the total angular momentum of the system
decreases at a rate
dL
dt
=
d
dt
(µh+ I1Ω1 + I2Ω2) = I1ω˙mb = −αmbI1Ω21Ω1. (2.1.20)
Tidal dissipation and MB result in a loss of energy from the system. The total rate
of energy dissipated can be expressed as follows:
E˙tot =
1
2
Gm1m2
a
a˙
a
+ I1Ω1Ω˙1 + I2Ω2Ω˙2 (2.1.21)
= −H1 −H2 − αmbI1Ω
4
1, (2.1.22)
where the final term in the last expression is the energy dissipated by MB and
H1 =
µh
ntf1
[
1
2
(
Ω21f3(e
2) +
(Ω1 · h)2
h2
f2(e
2)− (Ω1 · e)2f5(e2)
)
− 2n
(Ω1 · h)
h
f4(e
2) + n2f6(e
2)
]
,
33
H2 =
µh
ntf2
[
1
2
(
Ω22f3(e
2) +
(Ω2 · h)2
h2
f2(e
2)− (Ω2 · e)2f5(e2)
)
− 2n
(Ω2 · h)
h
f4(e
2) + n2f6(e
2)
]
,
are the tidal heating rates in the star and planet, respectively. It has been proposed
that H2 can result in planetary inflation, if the energy is deposited deep in the interior
of the planet (e.g. Bodenheimer et al. 2001). Note that these expressions approach
zero as e→ 0, Ω1,2 · e → 0 and Ω1,2 · hˆ → n, i.e. tidal dissipation continues until the
orbit becomes circular, and the spin of each body becomes coplanar and synchronous
with the orbit.
This formulation is beneficial because it can treat arbitrary orbital eccentricities
and stellar and planetary obliquities, unlike other models which are only valid to a
given order in the eccentricity, or for small (or zero) orbital inclinations (Goldreich
& Soter 1966; Hut 1981). Using the secular evolution equations allows us to perform
integrations quickly that represent dynamical evolution over billions of years. The full
set of equations are written in a form which is straightforward to numerically integrate.
2.2 Magnetic braking
Observations of solar-type stars have shown that the mean stellar rotational velocity
decreases with time, following the relation Ω ∝ t−1/2 (Skumanich, 1972). This is
the empirical Skumanich relation, and can be interpreted as telling us that solar-
type stars have been undergoing continuous spin-down since they first started on the
main sequence. MB by a magnetised outflowing wind has long been recognised as
an important mechanism for the removal of angular momentum from rotating stars
(Weber & Davis, 1967), and such a mechanism seems able to explain most of the
observed stellar spin-down (Barnes, 2003).
The surface layers of solar-type stars are convectively unstable, and turbulent mo-
tion and rotation can generate magnetic fields via a dynamo action, which heats the
corona and produces a magnetically controlled stellar wind. Magnetic stresses impart
angular momentum to the stellar wind, which forces the plasma to corotate with the
star up to the Alfve´n radius (rA), which is where the kinetic and magnetic energy den-
sity of the outflowing plasma balance. This enhances the angular momentum loss via
the stellar wind considerably over a non-magnetic case (by a factor (rA/R¯)2 ∼ 100,
34
for the Sun). This mechanism is referred to as MB, and can be attributed to Schatz-
man (1962), though the first formulations of such a steady, magnetised, thermally
driven outflow in the ideal-MHD limit were due to Weber & Davis (1967) and Mestel
(1968).
The simplest models of MB, such as Weber & Davis (1967), involve a thermally-
driven wind flowing out on magnetic field lines, which are essentially radial close
to the star (split monopole). Magnetic stresses impart angular momentum to the
outflowing gas, but the field weakens with the inverse square of distance from the star
(see Fig. 2.1). This model gives an angular momentum loss rate of
dJ
dt
=
8pi
3µ0
(
B0R
2
1
)2 Ω
vA(rA)
, (2.2.1)
where J = IΩ is the spin angular momentum of the star, B0 is the magnetic field at
the surface of the star, R1 is the radius of the star and vA(rA) is the Alfve´n speed
evaluated at the Alfve´n radius (given by B(rA)√
µ0ρ(rA)
). If we assume that the open-field
magnetic flux scales linearly with the rotation rate, i.e. 4piB0R21 ∝ Ω, then we recover
the commonly-used relation
dΩ
dt
∝ −Ω3. (2.2.2)
Integration of this equation leads asymptotically to the Skumanich relation (Collier
Cameron, 2002).
Although the Skumanich law is well established for stars with rotational velocities
in the range 1 − 30 km s−1, it overestimates the spin-down rates of stars younger
than 108 yrs, and thus cannot explain the presence of fast rotators in the Pleiades
(Ivanova & Taam, 2003). As a resolution to this problem it has been suggested that
the angular momentum loss rate for high rotation rates could be reduced, as a result of
the saturation of the stellar dynamo (MacGregor & Brenner, 1991), or alternatively due
to a reduction in the number of open field lines in a complex magnetic field topology
(Taam & Spruit, 1989). These and similar approaches lead to modified models of the
MB torque for fast rotators, and several such models have been proposed (e.g. Ivanova
& Taam 2003; Holzwarth & Jardine 2005). Nevertheless, Barnes (2003) find that
the Skumanich relation is remarkably accurate at modelling the spin-down of Sun-like
35
Co−rotating frame
Field line / streamline
rA
Figure 2.1: Sketch of the magnetic field lines and direction of flow of outflowing plasma
in the equatorial plane of a stellar wind. Charged stellar wind particles flow out these on
magnetic field lines, which are essentially radial close to the star. Magnetic stresses enforce
corotation of the outflowing plasma up until rA, which enhances the angular momentum loss
rate over a non-magnetic case by ∼ (rA/R¯)2.
stars that are not rapid rotators, so to a first approximation, a MB torque based on
the empirical Skumanich law is best for our purposes.
In this chapter, we include the effects of MB in the equations written down in §2.1
in the tidal evolution equations, through the inclusion of the Verbunt & Zwaan (1981)
braking torque, with the particular coefficients of Dobbs-Dixon et al. (2004),
ω˙mb = −αmb Ω21 Ω1, (2.2.3)
where αmb = 1.5× 10−14γ yrs. γ is a correction factor for an F-star, which takes the
value 0.1 for an F star, but is unity for a G or K star. We can also define a MB
timescale τmb ≡ Ω1ω˙mb =
1
αmb
1
Ω21
, which is approximately 1010 yrs for the Sun.
36
2.3 Analysis of the effects of MB on tidal evolution
in a simplified system
2.3.1 Circular, coplanar orbit with MB
We first study the effects of MB on a simplified system of a circular, coplanar orbit
under the influence of only the tide that is raised on the star by the planet, and
MB. We have neglected the tide in the planet here since the moment of inertia of
the planet is much smaller than that of star and the orbit (i.e. I2 ¿ I1 ∼ m2a2), so
to a first aproximation we can neglect the effects of planetary spin; in any case the
planetary spin is expected to synchronize rapidly with the orbit. The following set of
dimensionless equations can be derived from the full set of equations in §2.1:
dΩ˜
dt˜
= n˜4
(
1−
Ω˜
n˜
)
− A Ω˜3, (2.3.1)
dn˜
dt˜
= 3 n˜
16
3
(
1−
Ω˜
n˜
)
, (2.3.2)
where we have normalised the stellar spin frequency Ω1 and orbital mean motion n to
the orbital frequency at the stellar surface, together with a factor C
3
4 . C is the ratio
of the orbital angular momentum of a mass m2 in an orbit with semi-major axis equal
to the stellar radius R1, to the spin angular momentum of an equally rapidly rotating
star of radius R1, mass m1 and dimensionless radius of gyration rg1. C is important
for classifying the stability of the equilibrium curve Ω˜ = n˜ in the absence of MB, and
it can be shown from energy and angular momentum considerations (see Appendix
A.2), that this equilibrium is stable if n˜ ≤ 3−
3
4 . This is equivalent to the statement
that no more than a quarter of the total angular momentum can be in the form of
spin angular momentum for stability (Hut, 1980). We have thus defined the following
dimensionless quantities:
Ω˜ = Ω1
(
R31
Gm12
) 1
2
C−
3
4 , (2.3.3)
n˜ = n
(
R31
Gm12
) 1
2
C−
3
4 , (2.3.4)
37
C =
µR21
I1
=
µ
r2g1 m1
, (2.3.5)
t˜ =
(
Gm12
R31
) 1
2
(
9
2Q′1
)(
m2
m1
)
C
13
4 t, (2.3.6)
A = αmb
(
Gm12
R31
) 1
2
(
2Q′1
9
)(
m1
m2
)
C−
7
4 . (2.3.7)
There is only one parameter (A) that completely characterises the solution in the
(n˜, Ω˜)-plane, and its value may be estimated as
A ' 100 γ
(
Q′1
106
)
, (2.3.8)
for a Jupiter-mass planet orbiting a Sun-like star undergoing MB (with standard αmb
and with Q′ = 106). The size of this term shows that in general MB dominates the
stellar spin evolution. Note that in the absence of MB (A = 0), Eqs. 2.3.1 and 2.3.2 do
not contain reference to the masses of the star and planet or to the tidal Q′ of the star.
The parameter A, together with the initial conditions (n˜0, Ω˜0), completely determines
the evolution.
We plot some solutions on the (n˜, Ω˜)-plane in Fig. 2.2 by integrating Eqs. 2.3.1
and 2.3.2 for various initial conditions. Fig. 2.2 shows two phase portrait plots, which
show the general qualitative behaviour of the solutions to Eqs. 2.3.1 and 2.3.2, for
a given value of the parameter A. The arrows on each curve show the direction of
time evolution from the initial state. For prograde orbits we restrict ourselves to
studying the region, 0 ≤ n˜ ≤ 10, 0 ≤ Ω˜ ≤ 10, in the (n˜, Ω˜)-plane. This is because
Ω1 =
√
Gm12/R31 corresponds to stellar breakup velocity and n ≥
√
Gm12/R31 means
that the planet would be orbiting at, or beneath, the stellar surface. For a HJ with
a mass of MJ orbiting a star of mass M¯, C ∼ 0.01, so n˜ ' 10 corresponds to an
orbital semi-major axis of a ' 0.01 AU, and n˜ ' 0.1 corresponds to a ' 0.2 AU, so
these plots represent the full range of orbits of the HJs. The top panel of Fig. 2.2 is
for A = 100, which corresponds to canonical MB for a G/K star (γ = 1) and Q′1 of
106.
In the absence of MB (A = 0) we recover the standard tidal evolution equations for
a coplanar, circular orbit. These have been well studied in the literature (e.g. Coun-
selman 1973; Hut 1981). These equations have an equilibrium of coplanarity and
38
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
n˜
Ω˜
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
n˜
Ω˜
 
 
Figure 2.2: (n˜, Ω˜)-plane with A = 100 for a HJ orbiting a Sun-like star. The diagonal
dashed line in each plot corresponds to corotation (Ω˜ = n˜). Top: MB spins the star down so
that the planet finds itself inside corotation, where the sign of the tidal torque changes, and
planet is subject to tidally induced orbital decay. For an initially high n˜ outside corotation
tidal friction efficiently transfers angular momentum from spin to orbit, which pushes the
planet outwards. Bottom: Solutions with the same initial conditions are plotted with and
without MB for a HJ around a solar-type star, with dot-dashed lines having A = 0 and
solid lines have A = 100. The dot-dashed lines are also curves of constant total angular
momentum. This shows that the inclusion of MB is extremely important in determining
the secular evolution of the system, and its absence results in a very different evolutionary
history unless Ω˜ ¿ n˜ in the initial state.
39
corotation, i.e. i = 0 and Ω˜ = n˜, where the orbital inclination (or stellar obliquity) i is
defined by cos i = Ωˆ1 · hˆ and i ≥ 0. The system will approach this equilibrium if both
the spin angular momentum is less than a quarter of the total angular momentum, and
the total angular momentum exceeds some critical value (Greenberg 1974; Hut 1980).
With no braking, orbits initially outside corotation (Ω˜ > n˜) are not subject to tidally
induced orbital decay, and asymptotically approach a stable equilibrium Ω˜ = n˜, with
n˜ ≤ 3−
3
4 . Orbits initially inside corotation can evolve in two different ways, depending
on the stability of the equilibrium state on the solution’s closest approach to Ω˜ = n˜.
If dΩ˜
dt˜
> dn˜
dt˜
> 0 near corotation, then n˜ ≤ 3−
3
4 , and the equilibrium state is locally
stable (though no such curves are plotted in Fig. 2.2, since they occur only in the far
bottom left of the plot, near the origin). This is when the corotation radius moves
inwards faster than the orbit shrinks due to tidal friction, which can result in a final
stable equilibrium state for the system if the corotation radius “catches up” with the
planet. On the other hand, orbits inside corotation for which this condition is not
satisfied are subject to tidally induced orbital decay, since tidally induced angular
momentum exchange enhances the difference between Ω˜ and n˜, which leads to further
orbital evolution, and the spiralling in of the planet. This evolution can be seen from
the dot-dashed lines in the bottom panel of Fig. 2.2.
Including MB (A 6= 0) means that Ω˜ = n˜ is no longer an equilibrium state, and
the total angular momentum of the system is not conserved. For an orbit initially not
subject to spiralling into the star via tidal transfer of angular momentum from orbit
to spin (Ω˜ ≥ n˜) we see from the top panel in Fig. 2.2, that MB will spin the star down
so that the planet finds itself inside the corotation radius of the star. Passing through
corotation changes the sign of the tidal torque and causes the planet to spiral into the
star. Note that, if we ignore the age of the system, any bound orbit will eventually
decay in a finite time since the system has no stable equilibrium. The effect of MB
is to increase the minimum semi-major axis at which the orbit is not subject tidally
induced orbital decay over the nuclear lifetime of the star.
This means that an initially rapidly rotating G-type star hosting a close-in Jupiter
mass companion will lose significant spin angular momentum through MB (over a time
∼ τmb). During this stage of spin-down the spin frequency of the star may temporarily
40
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
n˜
Ω˜
Figure 2.3: (n˜, Ω˜)-plane showing the effects of reducing the stellar mass fraction partic-
ipating in angular momentum exchange (by a factor ²? = 10−2) with the orbit, while the
braking rate is unchanged, with A = 100. This has been proposed to apply to F stars like τ
Boo.
equal the orbital frequency of its close-in planet, but the rate of angular momentum
loss through MB will exceed the tidal rate of transfer of angular momentum from
orbit to spin. The stellar spin continues to drop well below synchronism until the
efficiency of transfer of tidal angular momentum from orbit to spin can compensate or
overcompensate for the braking. If dΩ˜
dt˜
> dn˜
dt˜
> 0 inside corotation, then tides will act
to spin up the star, though the timescale for this to cause significant spin-up may be
much longer than the stellar lifetime, and this only occurs if the orbit has sufficient
angular momentum to noticeably spin up the star. Otherwise, the planet continues to
spiral inwards once it moves inside corotation, and Ω˜ ' const.
So far we have considered the whole star to participate in tidal angular momentum
exchange with the orbit. For an F-star (like τ Boo), it has been proposed that only
the outer convective envelope (of mass fraction ²?) participates in angular momentum
exchange with the orbit (Marcy et al. 1997; Dobbs-Dixon et al. 2004; Donati et al.
2008). If the core and envelope of such a star can decouple, then tides would only have
to spin up the outer layers of the star, which would reduce the spin-up time by ²?. In
41
this case the system could remain in a state with Ω˜ ' n˜ just inside corotation, with
the resulting torque on the orbit small. This may explain the spin-orbit synchronism
of stars such as τ Boo, as noted by Dobbs-Dixon et al. (2004). Fig. 2.3 shows the
phase plane for a simplified system in which the moment of inertia of the star acted
on by tides is reduced by a factor ²? = 10−2, but the braking rate is unchanged, i.e.,
we multiply the first term on the right-hand side of Eq. 2.3.1 by ²−1? . Note that this
may be too simple a model to describe such core-envelope decoupling, and we have
ignored associated changes to the braking rate.
2.3.2 Extending the analysis to eccentric coplanar orbits
We can extend the simplified system in §2.3.1 to arbitrary eccentricity. If the orbit is
eccentric but still coplanar, then we have three coupled nonlinear ODEs:
de
dt˜
= 9e n˜
13
3
[
11
18
Ω˜1
n˜
f2(e
2)− f1(e
2)
]
(2.3.9)
dΩ˜
dt˜
= n˜4
[
f4(e
2)−
Ω˜
2n˜
(
f3(e
2) + f6(e
2)
)
]
− A Ω˜3 (2.3.10)
dn˜
dt˜
= 3n˜
16
3
[
f4(e
2)−
Ω˜
2˜n
(
f3(e
2) + f6(e
2)
)
]
(2.3.11)
The polynomials in e2 are as before, and we have used the dimensionless quantities
defined in §2.3.1.
The solutions to these equations in the absence of MB have been studied in Hut
(1981), so we move on to analyse the effects of MB. Fig. 2.4 we plot the phase plane
for A = 100 with e = 0.2 initially (top), and A = 10 with e = 0.6 initially (bottom)
to represent cases of moderate and highly eccentric orbits. A smaller value of A is
chosen in order to show the effects of tides more clearly, since reducing A is equivalent
to reducing Q′. Tidal effects become more important for smaller n˜ as e is increased
since this reduces the pericentre distance rp = a(1 − e). This is shown in Fig. 2.4.
The evolution is otherwise similar to that for a circular orbit. Eq. 2.3.9 exhibits the
eccentricity instability, where e is excited if Ω˜ ≥ 1811
f1(e2)
f2(e2)
n˜. The evolution of e is not
plotted, and will in general be different for different curves, but is found to decay once
Ω˜ < 1811
f1(e2)
f2(e2)
n˜.
42
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
n˜
Ω˜
 
 
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
n˜
Ω˜
 
 
Figure 2.4: Top: (n˜, Ω˜)-plane with A = 100 and e = 0.2 initially, where the dashed line
corresponds to Ω˜ = n˜. This is qualitatively similar to Fig. 2.2, showing that moderate
eccentricity does not affect the evolution significantly. This is partly due to the decay of the
eccentricity after the star has spun down so that Ω˜ ≤ 1811
f1(e2)
f2(e2)
n˜. Bottom: (n˜, Ω˜)-plane with
A = 10 and e = 0.6 initially. This differs from the top panel since the eccentricity remains
appreciably nonzero until after corotation is passed (eccentricity evolution not shown). Finite
eccentricity increases the slope of dn˜
dt˜
over the circular case, since tidal friction becomes much
more important with decreasing pericentre distance. In addition, the eccentricity is excited
when Ω˜ ≥ 1811
f1(e2)
f2(e2)
n˜.
43
2.3.3 Extending the analysis to inclined circular orbits
We can extend the simplified system of equations analysed in §2.3.1 to arbitrary incli-
nation (i) of the orbital plane with respect to the equatorial plane of star:
dΩ˜
dt˜
= n˜4
[
cos i−
Ω˜
2n˜
(
1 + cos2 i
)
]
− A Ω˜3, (2.3.12)
dn˜
dt˜
= 3 n˜
16
3
[
1−
Ω˜
n˜
cos i
]
, (2.3.13)
di
dt˜
= −n˜4Ω˜−1 sin i
[
1−
Ω˜
2n˜
(
cos i− n˜
1
3 Ω˜
)
]
, (2.3.14)
For small inclination, Eq. 2.3.14 reproduces Eq. 13 from Hut (1981), with the exception
that we have used a constant Q′1 rather than a constant time-lag in the equations (i.e.,
replace time lag τ by 12
3
2k1nQ′1
, and note that k1 is twice the apsidal motion constant
of the star).
From Eq. 2.3.13 the orbit begins to decay if
Ω˜ cos i < n˜, (2.3.15)
which is always satisfied for a retrograde orbit (i ≥ 90◦). This is just a generalisation
of the corotation condition Ω˜ = n˜, to a non-coplanar orbit. The inclination grows if
Ω˜ > Ω˜crit = 2n˜
(
cos i− n˜
1
3 Ω˜
)−1
, (2.3.16)
where we have assumed the quantity in brackets is positive, i.e., cos i > n˜
1
3 Ω˜. This
agrees with the condition from Hut (1981) when i ∼ 0. For sufficiently close-in orbits
that tidal friction is important, MB will rapidly spin down the star such that this
condition is not satisfied, so we can safely conclude that the inclination is not likely to
grow appreciably by tidal friction. When this condition is not satisfied, the inclination
decays to zero, on a timescale τi (defined in the next section). Note that i = 180◦ is
an unstable equilibrium value of the inclination.
In the absence of MB we recover the evolution considered by Greenberg (1974), so
we move on to concentrate on the inclusion of MB. The top panel in Fig. 2.5 plots
the (n˜, Ω˜)-plane for A = 10 for an initial inclination of i = 90◦. The evolution of i
44
is not plotted (and will in general be different for each curve), but is found to decay
once the stellar spin decays sufficiently that Eq. 2.3.16 is not satisfied. The bottom
panel in Fig. 2.5 shows the effect of increasing the inclination in steps to illustrate
the behaviour as i is increased, on various curves with otherwise the same initial
conditions, with A = 10. The orbit begins to decay for smaller n˜, and decays at a
faster rate as i is increased. This peaks for a perfectly retrograde orbit (i = 180◦),
with anti-parallel spin and orbit, where the rates of change of spin and orbital angular
frequencies are maximum. Fig. 2.5 shows that the orbit generally begins to decay
outside corotation, once Ω˜ cos i < n˜. This has implications for the tidal evolution of
close-in planets on inclined orbits, in that if this condition is satisfied, the planet will
be undergoing tidally induced orbital decay – though the inspiral time may be longer
than the expected stellar lifetime, depending on the value of Q′1.
In this section we have seen that MB can only be reasonably neglected for a coplanar
orbit when Ω ¿ n. For an inclined orbit, this condition must be generalised to
Ω cos i¿ n – obvious from Eq. 2.3.13. Neglecting ω˙mb for stars for which this condition
is not satisfied can result in a qualitatively different evolution, as already seen in the
bottom panel of Fig. 2.2 for a coplanar orbit.
2.4 Tidal evolution timescales
It is common practice to interpret the effects of tidal evolution in terms of simple
timescale estimates. The idea behind these is that if the rate of change of a quantity X
is exponential, then X˙/X will be a constant, so we can define a timescale τX = X/X˙.
If X˙/X 6= const, then these may not accurately represent the evolution. Here we
reproduce the timescales that can be derived from the equations in §2.1.
A tidal inspiral time can be calculated from the equation for a˙, by considering
only the effects of the tide raised on the star by the planet (not unreasonable since
I2 ¿ I1 ∼ m2a2). Here a˙/a ∼ a−13/2 6= const, so a more accurate estimate of the
inspiral time for a circular coplanar orbit is
τa ≡ −
2
13
a
a˙
(2.4.1)
45
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
n˜
Ω˜
 
 
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
n˜
Ω˜
 
 
Figure 2.5: Top: (n˜, Ω˜)-plane for an orbit with an initial i = 90◦, with A = 10, where
the dashed line corresponds to Ω˜ = n˜. This is similar to Fig. 2.2, except that the orbit
decays once Ω˜ cos i < n˜, which can occur above the dashed line, in contrast with the circular
case. The evolution of i is not plotted, and is in general different for each curve, but is
found to decay once the stellar spin drops below that given by Eq. 2.3.16. Bottom: various
initial inclinations, with A = 10, where the dashed line corresponds to Ω˜ = n˜. The bottom
trajectory has i = 0, and the inclination is increased in steps towards the top curve, which
has i = 180◦, to illustrate the behaviour. Note that orbits with larger initial i decay for
smaller n˜, once Ω˜ cos i < n˜ is satisfied.
46
' 12 Myr
(
Q′1
106
)(
m1
M¯
) 8
3
(
MJ
m2
)(
R¯
R1
)5( P
1d
) 13
3
(
1−
P
P?
)−1
, (2.4.2)
Here P and P? are the orbital and stellar spin periods, respectively. We have already
seen in §2.3 that it is unreasonable to assume that Ω is fixed unless Ω ¿ n, due to
MB.
If the orbit is inside corotation, angular momentum will be transferred from the
orbit to the spin of the star, giving a tidal spin-up time of
τΩ1 ≡ −
Ω1
Ω˙1
'
13τa
2 α
, (2.4.3)
where α = µhIΩ is the ratio of orbital to spin angular momentum. For the HJ problem,
τΩ1 ≥ τa since α = O(1), though this neglects the spin-down effects of MB. The
planetary spin Ω2 will tend to synchronise much faster, since the moment of inertia of
the planet is much less than that of the orbit (by ∼ 105), and will not be considered
further, i.e., we assume Ω2 = n.
A circularisation time can be derived from the equation for e˙, and is given for a
coplanar orbit by
τe ≡ −
e
e˙
(2.4.4)
' 17 Myr
(
Q′1
106
)(
m1
M¯
) 8
3
(
MJ
m2
)(
R¯
R1
)5( P
1d
) 13
3
×
[(
f1(e
2)−
11
18
P
P?
f2(e
2)
)
+ β
(
f1(e
2)−
11
18
f2(e
2)
)]−1
, (2.4.5)
where we have included both the stellar and planetary tides, as these have been shown
to both contribute to the tidal evolution of e (Jackson et al., 2008). Note that e˙/e ∼
const only if P ∼ const and e¿ 1, where f1,2(e2) ' 1. The factor
β =
Q′2
Q′1
(
m1
m2
)2(R2
R1
)5
∼ 10
Q′2
Q′1
, (2.4.6)
for the HJ problem.
If the orbital and stellar equatorial planes are misaligned, then dissipation of the
47
tide raised on the star by the planet would align them on a timescale
τi ≡ −
i
di
dt
(2.4.7)
' 70 Myr
(
Q′1
106
)(
m1
M¯
)(
MJ
m2
)2(R¯
R1
)3( P
1d
)4
×
(
Ω1
Ω0
)[
1−
P
2P?
(
1−
1
α
)]−1
, (2.4.8)
where we have assumed that the orbit is circular and made the small i approximation.
We take Ω0 = 5.8× 10−6s−1, which corresponds to a spin period of ∼ 12.5 d.
The validity of these timescales to accurately represent the tidal evolution of the
orbital and rotational elements is an important subject of study, since these timescales
are commonly applied to observed systems. In a recent paper, Jackson et al. (2008)
found that it is essential to consider the coupled evolution of e and a in order to
accurately model the tidal evolution, and that both the stellar and planetary tides must
be considered. They showed that the actual change of e over time can be quite different
from simple circularisation timescale considerations, due to the coupled evolution of
a. In the following we will consider the validity of the spin-orbit alignment timescale
to accurately model tidal evolution of i.
2.5 Numerical integrations of the full equations for
an inclined orbit
We perform direct numerical integrations of the equations in §2.1 with a 4th/5th order
Runge-Kutta scheme with adaptive stepsize control, using a scheme with Cash-Karp
coefficients similar to that described in Press et al. (1992). Our principal aim is to
study inclination evolution and to determine the accuracy of the spin-orbit alignment
timescale τi for close-in planets.
We choose a “standard” system of a HJ in orbit around an FGK star. We have
m1 = M¯,m2 = MJ , R1 = R¯, R2 = RJ , and we modify1 a, i, e. We choose the
initial ratio Ω1/n = 10, and include MB in all simulations, setting γ = 1 unless stated
1We take Ω1 · e = 0, though this need not be assumed. We find negligible difference between
integrations for which Ω1 · e = 0 and Ω1 · e 6= 0
48
otherwise. We also take Q′1 = Q
′
2 = 10
6, and Ω2/n = 1. The dimensionless radii of
gyration are chosen to be r2g1 = 0.076 and r
2
g2 = 0.261, which are values appropriate
for polytropic stellar and planetary models with respective indices 3 and 1.
2.5.1 Inclusion of MB and the importance of coupled evolu-
tion of a and i
For a prograde orbit (i < 90◦) initially outside corotation, MB rapidly spins the star
down sufficiently to ensure that inclination is not excited through tidal friction (so
that Eq. 2.3.16 is not satisfied), and the inclination begins to decay. Subsequent spin-
down moves the orbit-projected corotation radius beyond the orbit of the planet (so
that Ω cos i < n), and the resulting tidal inspiral accelerates as the difference between
Ω cos i and n is enhanced. The associated reduction in a increases the rate of stellar
spin-orbit alignment. Thus the inclusion of ω˙mb increases the rate of alignment, and
reduces τi from the simple estimate, which ignores MB and coupled a and i evolution.
The effect of MB on a retrograde orbit (i ≥ 90◦) is qualitatively different. A
retrograde orbit is always subject to tidally induced inspiral since Ω cos i < n for all
i ≥ 90◦. MB acts to reduce Ω, thereby reducing the difference |Ωcos i−n|, making the
tidal torque smaller. This acts to increase the timescale for alignment of the stellar
spin and orbit, though the effect is found to be small.
The most important effect of including MB is simply that of reducing the stellar
spin sufficiently so that Ω cos i < n, where the semi-major axis can then decay through
tidal friction. As a subsequently decreases, the tidal torque increases, resulting in a
faster tidal evolution. MB can only be neglected if |Ωcos i| ¿ n. If the orbit is already
highly inclined, such as i ≥ 90◦, then the inclusion of MB is not so important, since
Ω cos i < n regardless of the spin rate.
For an orbit initially at a = 0.05 AU, the simple estimate of the stellar spin-orbit
alignment timescale gives τi ' 2× 1010 yrs. Fig. 2.6 shows the evolution of i and a for
various initial inclinations for an orbit at a = 0.05 AU and a = 0.08 AU respectively.
The outer orbit a changes only slightly over 10 Gyr, and i evolves as expected from
the simple estimate of τi ' 3×1011 yrs. The inner orbit, on the other hand, is subject
to tidally induced orbital decay, with an inspiral time of 1− 3 Gyr, and this reduction
49
105 106 107 108 109 10100
0.010.02
0.030.04
0.050.06
t/yr
a/AU
 
 
(a) a evolution for a = 0.05 AU
105 106 107 108 109 1010−0.8−0.6
−0.4−0.2
00.2
0.40.6
0.81
t/yr
cos(i)
 
 
(b) cos i evolution for a = 0.05 AU
105 106 107 108 109 101000.01
0.020.03
0.040.05
0.060.07
0.08
t/yr
a/AU
 
 
(c) a evolution a = 0.08 AU
105 106 107 108 109 1010−0.8−0.6
−0.4−0.2
00.2
0.40.6
0.81
t/yr
cos(i)
 
 
(d) cos i evolution a = 0.08 AU
Figure 2.6: Tidal evolution for a circular, inclined orbit at a = 0.05 AU and a = 0.08 AU,
with various initial inclinations: i = 6◦, 26◦, 45◦, 90◦, 180◦. (a) and (c) show their respective
semi-major axis evolutions (with the highest inclination orbit decaying first – the bottom
curve – and the lowest inclination orbit decaying last – the top curve), and (b) and (d)
show the respective inclination evolution for these systems. The outer orbit a changes only
slightly over 10 Gyr, and i evolves as expected from simple estimates of τi. The inner orbit,
on the other hand, is tidally shrunk, and this reduction in a means that the true evolution
is much faster than the simple timescale estimates predict. This highlights the importance
of considering coupled evolution of a and i.
50
in a increases the rate of inclination evolution. This highlights the importance of
considering coupled evolution of a and i, especially for large initial inclinations, where
inspiral occurs for higher stellar spin rates. The difference between the simple estimate
of τi and the true timescale can be up to an order of magnitude different for orbits
whose a changes appreciably.
2.5.2 Inclined and eccentric orbits
Jackson et al. (2008) highlighted the importance of coupled a and e evolution for a
coplanar orbit. We will now consider how e might affect i evolution for a non-coplanar
orbit, which is the subject of this section.
A nonzero eccentricity reduces the pericentre distance rp = a(1−e), which increases
the tidal torque over a circular orbit, since the tidal torque ∼ r−6. Although the planet
spends less time near pericentre, the torque there is much greater, so dominates the
orbit-averaged torque. We therefore expect the stellar spin-orbit alignment time to
be reduced as we increase e. In addition, we expect that an orbit at large e would
more strongly affect the rate of alignment over a circular orbit, than one at large i
would over a coplanar orbit, because e reduces rp, whereas i only changes the difference
(Ω cos i − n). The tidal torque ∼ r−6(Ω cos i − n), which depends more strongly on
r than Ω. This behaviour can be seen in Figs. 2.6(d) and 2.7(b) which shows that
the ratios of stellar spin-orbit alignment times for an orbit with small i and large i is
∼ O(1), whereas the ratios of stellar spin-orbit alignment times for an orbit with small
e and large e can be up to several orders of magnitude.
From Fig. 2.7 we can compare the simple estimate of τi ' 3.0 × 1011 yrs for
an orbit at a = 0.08 AU, with the coupled evolution of the orbital and rotational
elements from integration of the full equations. We see that the simple estimate gives
a misleadingly long stellar spin-orbit alignment time compared with that obtained from
integrating the full equations in cases where e is initially non-negligible – as a result of
the strong functions of the eccentricity in this model of tidal friction. This confirms the
conclusion of Jackson et al. (2008), in that it is essential to consider coupled evolution
of a and e in order to determine an accurate system history. We also find that the
associated changes in semi-major axis strongly affect stellar spin-orbit alignment. A
51
105 106 107 108 109 10100
0.010.02
0.030.04
0.050.06
0.070.08
0.09
t/yr
a/AU
 
 
105 106 107 108 109 10100.7
0.75
0.8
0.85
0.9
0.95
1
t/yr
cos(i)
 
 
105 106 107 108 109 10100
0.10.2
0.30.4
0.50.6
0.70.8
t/yr
e
 
 
Figure 2.7: Top: Semi-major axis evolution for an inclined orbit with initial i = 45◦ at
a = 0.08 AU for various initial e, with e = 0, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8. Middle and bottom:
Inclination and eccentricity evolution for the same systems. Solutions with the smallest initial
e have the smallest change in a and i. Solutions with the largest initial e undergo much more
rapid tidal evolution (note that for e = 0.8, e decays to less than 0.7 within 105 yrs) – curves
can be distinguished by noting that the curves corresponding to the fastest evolution have
the largest initial e. Increasing the eccentricity can be seen to reduce the inspiral time by up
to several orders of magnitude over the circular case. In contrast, increasing the inclination
in Fig. 2.6 only reduces the inspiral time by a factor ∼ O(1) over a coplanar orbit. Also note
that τe < τi for all integrations.
52
marginally better estimate for τi can be made by replacing the orbital period with the
orbital period around periastron (equivalent to replacing a by rp), though this is still
inadequate since it neglects evolution of a.
2.5.3 Discussion
For typical HJs, we find that the stellar spin-orbit alignment time is comparable to
the inspiral time, i.e. τi ∼ τa. This means that if we observe a planet, then its survival
implies that tides are unlikely to have aligned its orbit. For planets on an accelerating
inspiral into the star, the rate of inclination evolution will have been much lower in
the past. Therefore if we observe a planet well inside corotation (Ω cos i < n), with a
roughly coplanar orbit, we can assume that it must have started off similarly coplanar
– unless we are lucky enough to be observing a planet on its final rapid inspiral into the
star after it has undergone most of the evolution, where it is now in a very short-period
orbit, close to being consumed.
We expect τi ∼ τa when α ¿ 1 – which is true for close-in terrestrial planets –
since Ω can be considered fixed, with the inclination changing only due to changes in
h. For typical values of α = O(1) for HJs, the inclination changes due to rotations of
both Ω and h, so the timescales are not exactly the same, but would be expected to
be of the same order of magnitude.
Hut (1981) showed by considering only the tide in the star, that the stellar spin-
orbit alignment timescale is longer than the circularisation timescale (τi > τe) unless
α > 6. This was based on exponential decay estimates for small e and i, but nev-
ertheless holds for the systems integrated in this work, as can be seen from Fig. 2.7.
This makes intuitive sense, since α = O(1) means that spin angular momentum is im-
portant, and circularisation involves only a property of the orbit, whereas alignment
involves both the spin and the orbit. For typical HJs α < 6; we therefore expect
τe < τi, especially when the eccentricity damping effect of the tide in the planet is
taken into account, which further enhances this inequality. The tide in the planet is
completely negligible in changing the stellar obliquity since I2 ¿ I1 ∼ m2a2.
This means that if an orbit is initially inclined and eccentric as a result of planet-
planet scattering or Kozai migration into a short-period orbit, we would expect the
53
orbit to become circular before it aligns with the spin of the star. We should therefore
observe fewer eccentric orbits than inclined orbits, if those systems start with a uniform
distribution in e, i–space. This also means that if we observe a close-in planet on a
circular orbit with non-zero i, we cannot rule out a non-negligible eccentricity in the
past.
2.6 Application to an observed system: an expla-
nation for the misaligned spin and orbit of XO-
3 b
The first system observed with a spin-orbit misalignment was XO-3 (He´brard et al.
2008; Winn et al. 2009), which has a sky-projected spin-orbit misalignment angle of
λ = 37.3◦ ± 3.7◦ (after being revised down from λ ' 70◦). This system has a very
massive m2 = 12.5MJ planet on a moderately eccentric e = 0.29, P = 3.2 d orbit
around an F-type star of mass m1 = 1.3M¯. Its age is estimated to be τ? ' (2.4−3.1)
Gyr. Note that even if the star is rotating near breakup velocity (P? ∼ 1 d), the planet
is still subject to tidal inspiral, since P? > P cos i (where we henceforth assume i = λ,
which may slightly underestimate i). If we assume that the angle of inclination of the
stellar equator to the plane of the sky is ∼ 90◦, then P? = 3.3 d ∼ P , i.e., Ω ∼ n.
He´brard et al. (2008) quote a stellar spin-orbit alignment timescale of ∼ 1012 yr
for this system, but we find that this is in error by ∼ 105. We believe that the reason
for this discrepancy is that their estimate was based on assuming that the spin-orbit
alignment time for XO-3 b is the same as for HD17156 b (Narita et al. 2008; Cochran
et al. 2008), which is a less massive planet on a much wider orbit. We find τi ∼ 30
Myr (using the expression in §2.4) assuming Q′1 = 10
6 to align the whole star with
the orbit. Circularisation time of τe ∼ 10 Myr and the inspiral time is estimated to be
τa ∼ 16 Myr from simple estimates.
Integrations for this system are given in Fig. 2.8 for a variety of stellar and planetary
Q′ values. These integrations again highlight the importance of considering coupled
evolution of the orbital and rotational elements, since timescales for tidal evolution can
be quite different from the simple estimates. Indeed, the actual spin-orbit alignment
54
 
0
 
0.01
 
0.02
 
0.03
 
0.04 1.0
e+05
1.0e
+06
1.0e
+07
1.0e
+08
1.0e
+09
1.0e
+10
a/AU
t/yr
log(Q1
’) =  6
, log(Q
2’) = 6
log(Q1
’) =  7
, log(Q
2’) = 6
log(Q1
’) =  8
, log(Q
2’) = 6
log(Q1
’) =  9
, log(Q
2’) = 6
log(Q1
’) = 10
,log(Q
2’) = 6
 
0
 
0.2
 
0.4
 
0.6
 
0.8 1 1.0e
+05
1.0e
+06
1.0e
+07
1.0e
+08
1.0e
+09
1.0e
+10
cos(i)
t/yr
log(Q1
’) =  6
, log(Q
2’) = 6
log(Q1
’) =  7
, log(Q
2’) = 6
log(Q1
’) =  8
, log(Q
2’) = 6
log(Q1
’) =  9
, log(Q
2’) = 6
log(Q1
’) = 10
,log(Q
2’) = 6
 
0
 
0.05 0.1
 
0.15 0.2
 
0.25 0.3 1.0
e+05
1.0e
+06
1.0e
+07
1.0e
+08
1.0e
+09
1.0e
+10
e
t/yr
log(Q1
’) =  1
0, log(
Q2’) =
 6
log(Q1
’) =  1
0, log(
Q2’) =
 7
log(Q1
’) =  1
0, log(
Q2’) =
 8
log(Q1
’) =  1
0, log(
Q2’) =
 9
log(Q1
’) = 10
,log(Q
2’) = 1
0
Figure 2.8: Tidal evolution of XO-3 b taking current values for the orbital properties of the
system, except that i = 90◦. MB is included with γ = 0.1, and Ω1/n = 2 initially (results do
not depend strongly on this choice). From the top and middle plots we require Q′1 ≥ 10
10
for the planet to survive for several Gyr, and maintain its high inclination. From the bottom
plot we see that if Q′1 ≥ 10
10, we require Q′2 ≥ 10
8 to maintain the current eccentricity for
a few Gyr. Tidal dissipation in both the planet and star must therefore be weak to explain
the current configuration of the system.
55
time from integrating the full set of equations is about an order of magnitude smaller
than that from the simple decay estimate, due to the semi-major axis evolution. We
choose an initial i = 90◦ to make the evolution clearer (this is not unreasonable since
λ, in any case, gives a lower bound on i, and the results do not depend strongly on
the non-zero value of i).
For the cases considered, the system can only survive and remain with its current
inclination for ∼ 3 Gyr if Q′1 ≥ 10
10. An explanation for the survival and remnant
orbital inclination of XO-3 b could therefore be the inefficiency of tidal dissipation in
the host star. The host star is an F-star of mass m1 = 1.3± 0.2M¯, so it will contain
a small convective core and a very thin outer convection zone (OCZ) separated by a
RZ. Dissipation in the convective core will only weakly affect the tide, and dissipation
in the RZ will also be weak (Zahn, 2008). This is because internal inertia-gravity
waves excited at the interface between convective and radiative regions cannot reach
the photosphere, where they can damp efficiently, as supposed for high-mass stars.
In addition, nonlinear effects due to geometrical concentration of the waves near the
centre of the star (studied in detail in Part II) cannot occur because the waves will
reflect from the outer boundary of the inner convection zone well before they become
nonlinear.
We expect that most dissipation occurs in the OCZ of the star. A calculation of
Q′ for the dissipation of the equilibrium tide and inertial modes, in the thin OCZ,
using a stellar model2 appropriate for this star (note that the metallicity of the star
is subsolar, with Z ∼ 0.01), was performed (see OL04 for details of the numerical
method). It must be noted that these calculations involve uncertainties regarding the
effective viscosity of turbulent convection, though the general trends in the results
below (and those discussed in §5.7) are likely to be quite robust. The results for the
m = 0, 1, 2 components of the tide are plotted in Fig. 2.9. In Appendix A.3, we
show that a combination of the m = 0, 1, 2 components of the tide are relevant for
spin-orbit alignment and inspiral for an inclined orbit, and so we must calculate Q′
for all components of the l = 2 (quadrupolar) tide. The relevant tidal frequencies,
assuming Ω ∼ n currently, would be those of integer ωˆ/Ω. However, since the angle
of inclination of the stellar equatorial plane to the plane of the sky has not been
2for which we use EZ Web at http://shayol.bartol.udel.edu/∼rhdt/ezweb/
56
Figure 2.9: Q′ as a function of the ratio of tidal frequency to spin frequency ωˆ/Ω, from
dissipation of the l = 2, m = 0, 1, 2 components respectively, of the equilibrium tide and
dissipation of inertial modes in the OCZ of an F-type star (see OL07 for details of this
calculation). This used a stellar model appropriate for XO-3 to model the convection zone.
The dashed lines represent the effect of omitting the Coriolis force, and therefore inhibiting
inertial waves. The prominent features in each figure (which occur for ωˆ/Ω = −1 and −1/6
for m = 1 and ωˆ/Ω = −1/3 for m = 2) are Rossby wave resonances, which probably occur
for tidal frequencies not relevant for the XO-3 system. For most tidal frequencies Q′ ≥ 1010,
which could explain the survival and remnant orbital inclination of XO-3 b.
57
determined, the relevant tidal frequencies cannot be calculated with any certainty.
Nevertheless, Q′ ≥ 1010 for most tidal frequencies for the host star XO-3. This can
explain the survival and remnant inclination of XO-3 b, since both τa and τi would be
longer than the age of the system.
In addition, the remnant eccentricity could be maintained due to weak damping of
the tide in the star for the same reasons. However, we must also explain the inefficient
damping of the tide in the planet, if indeed the reason for the eccentricity is that
τe > τ?. The planet in this system is massive, and may be a low-mass brown dwarf.
If it formed without a core, then the dissipation of inertial modes may be reduced if
they are able to form global modes, as found in OL04.
2.7 Conclusions
In this chapter we have investigated the long-term tidal evolution of HJs in a simple
model of tidal friction. We have studied the effects of MB on tidal evolution in a
simplified system, and then performed numerical integrations for a variety of HJ sys-
tems, with particular emphasis on inclination evolution. We now summarise the main
results of this work.
MB moves the corotation radius of the star outwards such that any close-in planets
will eventually orbit inside corotation (Ω cos i < n), and be subject to orbital decay due
to tides. MB is found to be important for the tidal evolution of HJs unless Ω cos i¿ n,
in which case the orbit is already well inside the orbit-projected corotation radius. MB
is particularly important when it comes to interpreting the tidal evolution of observed
systems from formation to the present day, since the star may have been rotating
much more rapidly in the past. Nevertheless, it is probably not a bad approximation
to neglect MB for calculating the future tidal evolution of most observed HJs, if the
star has already spun down so that Ω ¿ n.
Combining our results with Jackson et al. (2008), we find that coupled evolution
of the orbital and rotational elements is essential to accurately model tidal evolution.
This is especially true for highly eccentric orbits.
We find that the timescale for stellar spin-orbit alignment is comparable to the
inspiral time for HJs, therefore the orbits of most close-in planets have probably not
58
aligned with the spin of the star. Observed inclinations are likely to be a relic of
the migration process. This means that RM observations of transiting planets can
potentially distinguish between migration caused by planet-planet scattering or Kozai
oscillations combined with tidal dissipation in the star, and that produced by tidal
interaction with the gas disc. If the majority of candidates are found with λ ∼ 0, this
strongly disfavours planet-planet scattering or Kozai migration, since they are expected
to produce significantly inclined orbits, and we have found that tides are unlikely to
have aligned orbits without causing inspiral. Alternatively, if systems are found with
significantly nonzero λ, then some planet-planet scattering or Kozai migration could
have occurred to produce these orbital inclinations.
The population of planets whose spin-orbit alignments have been observed with the
RM effect can be statistically analysed to give a distribution of i for HJs (Fabrycky
& Winn, 2009). Collecting all RM observations together, this indicates the surprising
result that most HJs appear to be misaligned, possessing a wide variety of spin-orbit
angles (Triaud et al., 2010). This distribution of i appears consistent with that pre-
dicted by the process of Kozai cycles with tidal friction (Fabrycky & Tremaine, 2007)
(see §1.2). Together with our result that the spin-orbit angle is not likely to change
appreciably over the lifetime of the system due to tides, if future observations can
confirm this i distribution, then it would indicate that Kozai migration may indeed be
an important formation mechanism of HJs. However, it is unlikely to work in all cases.
These observations do appear to disfavour disk migration as the sole mechanism for
producing HJs.
For most HJs, tides tend to circularise the planet’s orbit before spin-orbit alignment
or inspiral occurs. Therefore, if we observe a planet on an inclined, circular orbit, we
cannot rule out a non-negligible eccentricity in the past. This means that we should
observe fewer eccentric orbits than inclined orbits if those systems start with a uniform
distribution of points in (e, i)-space, due to tidal friction. This should be considered
when comparing the observed (e, i) distribution with those predicted from theoretical
work on Kozai migration or planet-planet scattering, before we can further constrain
these theories (Fabrycky & Tremaine 2007; Juric´ & Tremaine 2008).
The misaligned spin and orbit of the XO-3 system could potentially be explained
59
in terms of inefficient tidal dissipation inside the host star. The required stellar Q′1 &
1010 required for the survival and remnant misalignment is predicted from theoretical
calculations of tidal dissipation in the OCZ of an F-star. In addition, the remnant
eccentricity poses constraints on the planetary Q′2 & 10
8, in the absence of perturbing
forces that could excite the eccentricity.
In this part of the thesis we have adopted a simple model of the effects of tidal
friction and examined some of its consequences. In Part II we turn to studying in
detail the mechanisms of tidal dissipation in solar-type stars, to try to understand the
importance of tides in the evolution of short-period planets and their host stars.
60
Part II
Internal gravity waves and tidal
dissipation in solar-type stars
61
Chapter 3
Internal wave breaking near the
centre of a solar-type star:
Boussinesq-type model and 2D
simulations
We study the fate of IGWs approaching the centre of an initially non-rotating solar-
type star, primarily using two-dimensional numerical simulations based on a cylindrical
model. A train of IGWs is excited by tidal forcing at the interface between the
convection and radiation zones of such a star. We derive a Boussinesq-type model
of the central regions and obtain a nonlinear wave solution. We then use spectral
methods to integrate the system of equations numerically, with the aim of studying at
what amplitude the wave is subject to instabilities. These instabilities lead to wave
breaking whenever the amplitude exceeds a critical value. Below this critical value,
the wave reflects perfectly from the centre of the star. Wave breaking leads to mean
flow acceleration, which corresponds to a spin up of the central region of the star, and
the formation of a critical layer, which acts as an absorbing barrier for subsequent
ingoing waves. As these waves continue to be absorbed near the critical layer, the
star is spun up from the inside out. This has implications for the fate of short-period
planets orbiting solar-type stars, which will be discussed in Chapter 5.
62
3.1 Introduction
IGWs are thought to account for the efficient tidal dissipation inferred from the cir-
cularisation of early-type binary stars (Zahn 1975; Zahn 1977; Zahn 2008; Savonije
& Papaloizou 1983; Papaloizou & Savonije 1985; Savonije et al. 1995b; Savonije &
Papaloizou 1997b; Papaloizou & Savonije 1997b), which are massive enough to have
a convective core and an exterior radiative envelope. In these stars, IGWs are ex-
cited near the boundary between these two regions, where the buoyancy frequency
(or Brunt-Va¨isa¨la¨ frequency, see § 3.2) matches the tidal forcing frequency. These
propagate outwards into the stably stratified RZ, towards the surface, where they are
fully or partially damped by radiative diffusion. In this picture, these stars are tidally
synchronized from the outside in, since angular momentum is deposited in the regions
of the star where these waves damp. Goldreich & Nicholson (1989) propose that a
corotation resonance (or critical layer, which we define in the next section) forms near
the surface, which absorbs outward propagating waves. This is analogous to the pic-
ture that we will describe later in this chapter, except that our model has the star
spinning up from the inside out.
The above model only works for stars with an exterior RZ, which is unlike that
of the Sun and other stars of solar type, which have radiative cores and convective
envelopes. It is of particular interest to study the efficiency of tidal dissipation in
these stars, since many have been found to harbour close-in planets, whose survival is
determined by the stellar Q′. This is because a planet with an orbital period shorter
than the stellar spin period is subject to tidally induced orbital decay, with an inspiral
rate that depends linearly on dissipation in the star. In these stars, a train of IGWs
are again excited at the interface between the convective and radiative regions, but
here it propagates towards the stellar centre. If they can coherently reflect from the
centre, global standing modes can form in the RZ. In this case, tidal dissipation is
efficient only when the tidal frequency matches that of a global standing mode (which
are commonly referred to as g-modes; Terquem et al. 1998; Savonije & Witte 2002).
When this occurs, the dissipation can be very large, leading to rapid evolution of the
system out of resonance. This would not contribute appreciably to the time-averaged
Q′ because the transition through resonance is relatively fast, unless the system gets
63
locked into resonance by evolutionary processes, such as stellar evolution or MB (Witte
& Savonije 1999; Witte & Savonije 2001). On the other hand, if these waves do not
reflect coherently from the centre, and are either strongly dissipated there, or are
reflected with a perturbed phase, then efficient dissipation is possible over a broad
range of tidal frequencies (GD98; OL07). The extent of nonlinearity in the waves
near the centre is likely to be the factor that determines whether these waves reflect
coherently, and this is controlled by the amplitude and frequency of the tidal forcing,
as well as the properties of the stellar centre.
In this chapter we study the problem of IGWs approaching the centre of a solar-
type star, primarily using two-dimensional numerical simulations. We first derive
a Boussinesq-type system of equations appropriate for the stellar centre, which are
ideal for integrating numerically using spectral methods. An exact solution for tidally
forced waves in 2D is derived, and some of its properties are discussed. Our numerical
set-up is described and results are presented for both linear and nonlinear forcing
amplitudes, including an analysis of the reflection coefficient and a study of the growth
of different azimuthal wavenumbers in the disturbance. The simulations in this chapter
are extended to three dimensions in Chapter 4. This is followed by a discussion of
the results, especially their relevance to Q′ for solar-type stars, and to the survival of
close-in giant planets in orbit around such stars, in Chapter 5. We study the stability
of the nonlinear wave derived in this chapter in Chapter 6. Finally, in Chapter 7 we
present and discuss an alternative mechanism of tidal dissipation involving the gradual
radiative damping of the waves being able to produce a critical layer.
3.2 Internal gravity waves: elementary properties,
wave breaking and critical layers
IGWs are a family of dispersive waves that are ubiquitous in nature. They propagate
in any fluid with a stable density stratification, due to the restoring force of buoy-
ancy. Their influence can be observed in the oceans and atmosphere of the Earth on
a range of spatial and temporal scales, from the visual undulations of striated cloud
structures, to the complex interplay between these waves and shearing flows, which
64
produces the large-scale Quasi-Biennial Oscillation in the equatorial stratosphere. It is
widely recognised that IGWs play a prominent role in the transport of energy and an-
gular momentum in geophysical and astrophysical flows (McIntyre 2000; Bu¨hler 2009).
IGWs are thought to be important in stably stratified RZs of stars. When excited by
turbulent convection, they were at one stage put forward as potential explanations for
maintaining the solid body rotation of the radiative interior of the Sun (Schatzman
1993; Zahn et al. 1997). However, it was pointed out that the “antidiffusive” nature
of IGWs tends to enhance local shear rather than reduce it (Gough & McIntyre 1998).
IGWs are still thought to produce angular velocity variations in the RZ (Rogers &
Glatzmaier 2006). They have also been invoked to explain the Li depletion problem
in F-stars (Garcia Lopez & Spruit 1991), affecting solar neutrino production (Press
1981), and possibly having an effect on the solar cycle (Kumar et al. 1999).
Observations of oscillations on the solar surface are able to provide information
about the interior properties of the Sun (Christensen-Dalsgaard 2002). IGWs in the
radiative interior of the Sun can form global standing modes, commonly referred to as
g-modes, if their frequency matches that of a free mode of oscillation. These are known
to have their amplitude largest close to the centre (see § 3.5 & 5.2) and would therefore
seem ideal probes of the deep interior. Unfortunately for observers, the standing
g-modes are effectively trapped in the radiative interior, where the stratification is
stable, and are evanescent in the CZ, and so are unlikely to be visible at the solar
surface. Nevertheless, modes of sufficiently low degree, with high enough amplitude,
may have already been observed at the surface by Garc´ıa et al. (2007), though it must
be noted that thus far there is no undisputed evidence for observations of g-modes
(Appourchaux et al. 2010).
The frequencies of the largely incompressible internal waves lie in ranges controlled
by the buoyancy frequency (or Brunt-Va¨isa¨la¨ frequency) N , and the Coriolis frequency
2Ω. The square of the buoyancy frequency in a spherically symmetric star is defined
by
N2(r) = −
1
ρ
dp
dr
(
1
Γ1
d ln p
dr
−
d ln ρ
dr
)
, (3.2.1)
where ρ, p are the density and pressure, and Γ1 =
(
∂ ln p
∂ ln ρ
)
s
, is the first adiabatic
65
exponent (derivative taken at constant specific entropy s and constant chemical com-
position). The local dispersion relation for linear noncompressive internal waves in a
fluid body rotating with angular velocity Ω is
ω2 = N2 sin2 α+ 4Ω2 cos2 β, (3.2.2)
where α is the angle between the wavevector k and the gravitational acceleration g,
and β is the angle between k and Ω. The frequency of these waves is independent
of wavelength (in the absence of viscosity or thermal conduction), and only depends
on the direction of the wavevector. This is different from waves whose restoring force
is due to compressibility, which have frequency inversely proportional to wavelength.
When N = 0, Eq. 3.2.3 describes inertial waves, which have frequencies in the range
(0, 2Ω). If the body is non-rotating (Ω = 0), then these waves are IGWs, and possess
frequencies in the range (0, N). In the presence of nonzero Ω and N , these waves are
intermediate between inertial waves and IGWs and are referred to as inertia-gravity
waves. For waves in a spherical star, at a given latitude there is a minimum frequency
for inertia-gravity wave propagation. Near the equator, waves can propagate with
arbitrarily low frequency. From here on we neglect the bulk rotation, and assume that
Ω = 0, i.e., we consider only IGWs. The local dispersion relation for IGWs can be
rewritten
ω2 = N2
k2h
k2r + k
2
h
, (3.2.3)
where kh and kr are the horizontal and radial wavenumbers.
The phase and group velocities of these waves can be calculated from Eq. 3.2.3 to
give
cp =
ω
|k|
k
|k|
=
Nkh
(k2r + k
2
h)
3
2
(krer + kheh) (3.2.4)
≈
Nkh
k3r
(krer + kheh) , (3.2.5)
cg = ∇kω = −
Nkr
(k2r + k
2
h)
3
2
(kher − kreh) (3.2.6)
≈ −
N
k2r
(kher − kreh) , (3.2.7)
66
in the tidally relevant limit that the radial wavelength of the waves is much shorter
than the horizontal wavelength, i.e., kh ¿ kr (which is true except near a turning
point, or within the last few wavelengths from the centre of a star). In this limit,
cg ·er = −N
kh
k2r
= −cp ·er, i.e., the radial wave pattern moves in the opposite direction
to the radial energy flux. Since ω is independent of |k|, cg · k = 0, meaning that the
energy in IGWs propagates along surfaces of constant phase.
For waves on a non-zero horizontal background shear flow U, Eq. 3.2.3 still applies
if the Richardson number Ri = N
2
|∂U/∂r|2 À 1, if we replace ω by the Doppler shifted
frequency ωˆ, and similarly for the phase and group velocity of the waves,
ωˆ = ω − k ·U, cˆp = cp −U, cˆg = cg −U. (3.2.8)
We now have the possibility of the wave frequency being Doppler shifted upwards to
N , in which case cˆg · er reverses, giving total internal reflection.
The other extreme, ωˆ → 0, occurs when the horizontal velocity in the shear matches
the horizontal phase velocity. This occurs at a so-called “critical layer”, which is
defined as the layer in which the wavelength of the waves would be Doppler shifted to
zero, if they were ever to reach it. Note though, that cˆg · er → 0 as ωˆ → 0, so in linear
theory the waves never reach the critical layer in a finite time. Early work on IGWs
in a background shear, including a study of critical layers, can be found in Booker
& Bretherton (1967) and Hazel (1967). They find that an IGW propagating through
a critical layer is attenuated by a factor ∼ exp
(
−2pi(Ri− 1/4)1/2
)
. If Ri À 1/4,
the wave is fully absorbed, and irreversibly transfers its energy to the mean flow.
However, it must be noted that at the critical layer, linear theory predicts that the
wave steepness
s =
max(u · eh)
cˆp · eh
, (3.2.9)
where u is the velocity perturbation, of these waves goes to infinity, as the Doppler
shifted horizontal phase speed cˆp · eh goes to zero, i.e., the waves become strongly
nonlinear at the critical layer (which McIntyre 2000, refers to as linear theory predicting
its own breakdown). Since wave breaking is expected to occur whenever s > 1, the
waves are likely to break before they reach the critical layer. Nonlinear effects have
67
been studied in early simulations by Winters & D’Asaro (1994), who find that the
of initial wave energy on encountering a critical layer, roughly one third reflects, one
third results in mean flow acceleration, and the remainder cascades to small scales
where it is dissipated. This implies that wave absorption by the mean flow need not
be complete, in contrast with the prediction from linear theory. This is discussed
further in § 3.9.2, in relation to the results of our simulations.
Wave breaking is defined as any wave-induced process that leads to the rapid
and irreversible deformation of “otherwise wavy” material contours (McIntyre 2000),
and it leads to the production of turbulence and irreversible energy dissipation. The
breaking process results from the growth of an instability upon a basic state composed
of a wave with s > 1. The susceptibility of a wave to breaking can be enhanced by
resonant triad interactions, in which a primary wave resonantly interacts with a pair
of low-amplitude secondary waves. This process transfers energy to the secondary
waves, whose steepness can then grow beyond the critical value required for breaking
to occur, even though the primary steepness may not be sufficient for breaking on its
own (Staquet & Sommeria 2002). Previous work has shown that the process leading
to the wave steepening is two-dimensional, but that breaking is a three-dimensional
process (Klostermeyer 1991; Winters & D’Asaro 1994). Nevertheless, the mechanisms
responsible for breaking and the final outcome of the breaking process are likely to be
similar in 2D.
Here we are interested in studying what happens when IGWs excited by tidal
forcing approach the centre of a star with an inner RZ, i.e. G-type stars such as
the Sun, which do not possess a convective core. We are primarily concerned with
the efficiency of tidal dissipation for solar-type stars, resulting from breaking of these
waves near the centre. Throughout the rest of this chapter we restrict our problem to
2D, and postpone study of any 3D effects until Chapter 4. We now describe our basic
problem in more detail.
68
3.3 Basic description of the problem
3.3.1 Tidal potential
Consider a star and a planet in a mutual Keplerian orbit (though we make no assump-
tions about the relative masses at this stage). The tidal potential experienced by the
star can be written as a sum of rigidly rotating spherical harmonics (e.g. OL04). For
the simplest case of a planet on a circular orbit, that is coplanar with the equatorial
plane of the star, we can consider a two-dimensional restriction of the problem to this
plane. This allows us to write the time-dependent part of the quadrupolar (l = 2)
tidal potential in the equatorial plane of the star as
Ψ(r, φ, t) = −
3
4
Gmp
a3
r2 cos(2φ− ωˆt), (3.3.1)
in spherical polar coordinates (r, θ, φ) with origin at the centre of the star, in the plane
θ = pi/2. Here mp is the planet mass, a is the orbital semi-major axis, and n > 0 is
the mean motion. The relevant tidal frequency is ωˆ = 2n − 2Ω, for a star rotating
with angular velocity Ω. From here on we assume that the star is non-rotating, i.e.,
we assume that Ω = 0, which is a reasonable assumption if the star is spinning much
slower than the orbit. This is appropriate if a short-period planet orbits a solar-type
star, that has been spun down by MB for the duration that it has spent on the main-
sequence, to rotate with a spin period of several tens of days.
We consider a restriction of the full three-dimensional problem, in which the tidal
potential is composed of many different spherical harmonic components for an orbit of
arbitrary eccentricity and inclination, to instead consider a simplified two dimensional
model of a star, forced by this single component of the tide. In Chapter 4 we will
carry out fully three-dimensional simulations of this problem, which find that the two
dimensional approximation is indeed reasonable.
69
0 0.2 0.4 0.6 0.80
1
2
3
4
5
6
r/R⊙
N/ω d
yn
Figure 3.1: Buoyancy frequency N normalised to dynamical frequency ωdyn versus radius,
based on Model S of the current Sun. Also plotted is the frequency 2pi/0.5day ∼ 0.1ωdyn,
corresponding with the tidal frequency for a one-day orbit. Note that ωˆ ¿ N throughout
the bulk of the RZ. Only near the centre and at the CZ/RZ interface, does ωˆ ∼ N .
3.3.2 Central regions of a star
The buoyancy frequency is real and comparable with the dynamical frequency of the
star
ωdyn =
(
Gm?
R3?
) 1
2
, (3.3.2)
throughout the bulk of the RZ, where m? and R? are the stellar mass and radius,
respectively. In Fig. 3.1 we plot N normalised to ωdyn in the RZ for Model S of the
current Sun, which is a high-resolution standard solar model developed for helioseis-
mology (Christensen-Dalsgaard et al. 1996). For our problem, the tidal frequencies of
interest ωˆ ¿ N , which implies kr À 2pi/r, except near the centre. IGWs are excited
at the top of the RZ by a combination of tidal forcing in that region, together with
the pressure of inertial waves acting at the interface if |ωˆ| < 2Ω (OL07). It is within
this transition region that N increases linearly with distance into the RZ, so there is
a point at which N ∼ ωˆ, and IGWs are efficiently excited. These propagate towards
the centre with radial wavelengths λr ∼ 10−3 − 10−2R¯, for typical tidal frequencies.
70
Expanding the standard equations of stellar structure about r = 0, we obtain the
density stratification ρ ≈ ρ0 + ρ2r2. For sufficiently small r, g is linear in r, and
N ≈ Cr, where C is a constant that varies with stellar model, and main-sequence age.
For the current Sun, C ≡ C¯ ≈ 8.0 × 10−11m−1s−1. This is valid throughout only
the inner . 3% of the Sun. However, even considering the largest radial wavelength
produced by a HJ, this region contains multiple wavelengths.
3.4 Derivation of a Boussinesq-type system of equa-
tions
The ideal compressible fluid equations in 2D plane polar (r, φ) coordinates are
Dur −
u2φ
r
= −
1
ρ
∂rp− ∂rΦ, (3.4.1)
Duφ +
uruφ
r
= −
1
ρr
∂φp−
1
r
∂φΦ, (3.4.2)
Dρ+ ρ
[
1
r
∂r(rur) +
1
r
∂φuφ
]
= 0, (3.4.3)
Dp−
γp
ρ
Dρ = 0, (3.4.4)
∇2Φ = 4piGρ, (3.4.5)
D = ∂t + ur∂r +
uφ
r
∂φ. (3.4.6)
The basic state is static and circularly symmetric. Near the centre, we pose the
expansion
ρ = ρ0 + ρ2r
2 + ρ4r
4 +O(r6), (3.4.7)
and similarly for p and Φ. The coefficients are related by the condition for the back-
ground to be in hydrostatic equilibrium, together with Poisson’s equation, giving
p2 = −ρ0Φ2, p4 = −ρ0Φ4 +
p2ρ2
2ρ0
, (3.4.8)
Φ2 = piGρ0, Φ4 =
piGρ2
4
, (3.4.9)
and so on for the coefficients of higher-order terms.
71
We are interested in a region where r/R? = O(²), where ²¿ 1, so let r = ²x. Now
introduce a slow time τ = ²t, then the solution including the basic state, and a slow
nonlinear density perturbation, has the form
ρ = ρ0 + ²
2ρ2x
2 + ²4ρ4x
4 + ...
︸ ︷︷ ︸
basic state
+ ²2ρ′2(x, φ, τ) + ...︸ ︷︷ ︸
nonlinear perturbation
, (3.4.10)
p = p0 + ²
2p2x
2 + ²4p4x
4 + ...
︸ ︷︷ ︸
basic state
+ ²4p′4(x, φ, τ) + ...︸ ︷︷ ︸
nonlinear perturbation
, (3.4.11)
Φ = Φ0 + ²
2Φ2x
2 + ²4Φ4x
4 + ...
︸ ︷︷ ︸
basic state
+ ²4Φ′4(x, φ, τ) + ...︸ ︷︷ ︸
nonlinear perturbation
, (3.4.12)
ur = ²
2ur2(x, φ, τ) + ...
︸ ︷︷ ︸
nonlinear perturbation
, (3.4.13)
uφ = ²
2uφ2(x, φ, τ) + ...
︸ ︷︷ ︸
nonlinear perturbation
, (3.4.14)
In the Boussinesq approximation, the fractional pressure perturbation is small com-
pared with the fractional density perturbation because this is a low-frequency dis-
turbance, with ω ¿ ωdyn, for which acoustic effects are negligible. For such short-
wavelength perturbations, the gravitational perturbation is also small (Christensen-
Dalsgaard 2002). These two points explain the absence of terms in the above solutions
proportional to ²2 for p and Φ in the nonlinear perturbation. This corresponds to
looking for small density constrasts and weak material accelerations compared with
gravity.
Substituting these expansions into the basic equations, and subtracting terms that
arise only in the basic state, we obtain at leading order
D1ur2 −
u2φ2
x
= −
1
ρ0
∂xp
′
4 +
2xp2ρ′2
ρ20
− ∂xΦ
′
4, (3.4.15)
D1uφ2 +
ur2uφ2
x
= −
1
ρ0x
∂φp
′
4 −
1
x
∂φΦ
′
4, (3.4.16)
ρ0
[
1
x
∂x(xur2) +
1
x
∂φuφ2
]
= 0, (3.4.17)
2xp2ur2 −
γp0
ρ0
(D1ρ
′
2 + 2xρ2ur2) = 0, (3.4.18)
∇2Φ′4 = 4piGρ
′
2, (3.4.19)
D1 = ∂τ + ur2∂x +
uφ2
x
∂φ. (3.4.20)
72
We can rewrite these equations in a more natural notation, removing the asymp-
totic scalings to find
Dur −
u2φ
r
= −∂rq + rb, (3.4.21)
Duφ +
uruφ
r
= −
1
ρr
∂φq, (3.4.22)
1
r
∂r(rur) +
1
r
∂φuφ = 0, (3.4.23)
Db+ C2rur = 0, (3.4.24)
D = ∂t + ur∂r +
uφ
r
∂φ, (3.4.25)
where
b =
2p2
ρ20
ρ′2, (3.4.26)
q =
1
ρ0
p′4 + Φ
′
4, (3.4.27)
are a buoyancy variable and a modified pressure variable, and
C2 = 4Φ2
(
p2
γp0
−
ρ2
ρ0
)
, (3.4.28)
is related to the buoyancy frequency N by N = Cr, with C2 > 0 for a stable strati-
fication. Note that Poisson’s equation is no longer required, since we have separated
out the gravitational potential perturbation in this approximation.
We can write these equations in the vector-invariant form (which can be used in
two or three dimensions)
Du = −∇q + rb, (3.4.29)
Db+ C2r · u = 0, (3.4.30)
∇ · u = 0, (3.4.31)
D = ∂t + u · ∇. (3.4.32)
These equations are similar to the standard Boussinesq system for a slab of fluid in
Cartesian geometry (see e.g. Bu¨hler 2009, Ch. 6) with a uniform stratification, with
the exception that our problem is in cylindrical geometry, with g and N proportional
73
to r. Note also that the buoyancy variable defined here is related, but not identical,
to that of the standard Boussinesq approximation used in atmospheric sciences and
oceanography (see e.g. Bu¨hler 2009). The buoyancy variable is proportional to the
density and entropy perturbation.
An energy equation for our system can be derived by contracting Eq. 3.4.29 with
u:
∂t
(
1
2
|u|2 +
b2
2C2
)
+∇ ·
[(
1
2
|u|2 +
b2
2C2
+ q
)
u
]
= 0. (3.4.33)
Thus E = 12ρ0|u|
2 + ρ0 b
2
2C2 is the energy density per unit volume, and
FE = ρ0
(
1
2 |u|
2 + b
2
2C2 + q
)
u is the energy flux density.
If the fluid is at rest, with b = 0, then the stratification surfaces are circles (spheres
in 3D), and C2 measures the strength of the stable stratification. If we disturb the fluid
from rest, then a positive (negative) buoyancy is associated with an inward (outward)
radial displacement of particles, resulting in an outward (inward) acceleration of the
fluid due to buoyancy, to restore the system to equilibrium. From the energy equation
Eq. 3.4.33, the state b = 0 is seen to be that of minimum gravitational potential energy,
since the available potential energy density ρ0 b
2
2C2 is minimised. This makes sense, since
this corresponds to having a background state with no wave-like disturbance.
3.5 Linear theory of IGWs approaching the stellar
centre
3.5.1 Linear solution steady in a frame rotating with the pat-
tern speed of forcing
If the RZ is forced from above then this will excite waves which propagate to the centre
of the star. If the incoming wave has frequency ω and azimuthal wavenumber m, then
we can assume that the response is steady in a frame rotating with the angular pattern
speed Ωp = ω/m, in the absence of instabilities. The dependence on φ and t is then
74
only through the combination
ξ = m(φ− Ωpt). (3.5.1)
We can choose Ωp/C as a unit of length, and Ω−1p as a unit of time
1, to allow us to
write the equations in the dimensionless form
Dur −
u2φ
r
= −∂rq + rb, (3.5.2)
Duφ +
uruφ
r
= −
m
ρr
∂ξq, (3.5.3)
1
r
∂r(rur) +
m
r
∂ξuφ, (3.5.4)
Db+ rur = 0, (3.5.5)
D = ur∂r +m
(uφ
r
− 1
)
∂ξ. (3.5.6)
The energy equation (Eq. 3.4.33) allows us to infer that the radial energy flux
FEr = ρ0
∫ 2pi
0
[
1
2
(
|u|2 + b2
)
+ q
]
rurdξ, (3.5.7)
is independent of r for disturbances steady in this frame of reference, since the solutions
are periodic in ξ with period 2pi.
The radial angular momentum flux is
FLr =
m
ω
FEr . (3.5.8)
We can obtain a solution to these equations by linearization, assuming that the
solution is proportional to eiξ, obtaining (where real parts are assumed to be taken)
−imur = −∂rq + rb, (3.5.9)
−imuφ = −
imq
r
, (3.5.10)
1
r
∂r(rur) +
im
r
uφ = 0, (3.5.11)
−imb+ rur = 0. (3.5.12)
The incompressibility constraint allows expression of the velocity in terms of the
1Note that these units are only used in this section, §4.2, §7.2, and in Appendix A.4.
75
streamfunction ψ(r, φ), defined by
u = ∇× (ψez) =
(
1
r
∂φψ
)
er + (−∂rψ) eφ, (3.5.13)
so we can write
ur = Re
[
im
r
ψ
]
, (3.5.14)
uφ = Re [−∂rψ] . (3.5.15)
This enables us to reduce the system to Bessel’s equation of order m,
Lmψ = ∂r(r∂rψ) + r
(
1−
m2
r2
)
ψ = 0 (3.5.16)
with solution regular at the origin ψ ∝ Jm(r). This represents a wave that approaches
from infinity, reflects perfectly from the centre and goes out to infinity. Pure ingoing
and outgoing wave solutions are described by Jm(r) ± iYm(r), with the appropriate
sign explained below.
3.5.2 Properties of the (non-)linear solution
The general solution can be written in the form of a sum of ingoing and outgoing
waves, with complex amplitudes Ain and Aout, as follows:
ψin(r, ξ) = [Jm(r) + iYm(r)]e
iξ, (3.5.17)
ψout(r, ξ) = [Jm(r)− iYm(r)]e
iξ, (3.5.18)
ψ(r, ξ) = Ainψin(r, ξ) + Aoutψout(r, ξ). (3.5.19)
We can check that ψin corresponds to an ingoing wave by calculating its phase and
group velocities. A simple calculation shows that the radial phase velocity is directed
outward if we adopt the convention that ω = mΩp > 0, and the group velocity is
directed inward. This highlights one of the peculiarities of IGWs – that the phase and
group velocities are oppositely directed, as discussed in § 3.2. For these linear waves,
we have
FEr = ρ0pimrIm[ψ∂rψ
∗] = 2mρ0
(
|Aout|
2 − |Ain|
2
)
. (3.5.20)
76
The solution for a wave that perfectly reflects from the centre is
ψ(r, ξ) = 2AinJm(r)e
iξ, (3.5.21)
and has FEr = 0, since Ain = Aout.
If we take the curl of Eq. 3.4.29, we obtain
∂t(∇× u) = ∇× (rb)−∇× (u · ∇u), (3.5.22)
which has eliminated the modified pressure perturbation q. The z-component of this
equation, expressed in terms of the streamfunction, is
∂t(−∇
2ψ) = −∂φb+ J(ψ,−∇
2ψ), (3.5.23)
and the buoyancy equation is
∂tb = −∂φψ + J(ψ, b). (3.5.24)
The nonlinear terms take the form of Jacobians,
J(A,B) =
1
r
∂(A,B)
∂(r, φ)
= (∂rA)(
1
r
∂φB)− (
1
r
∂φA)(∂rB). (3.5.25)
For the solution above, we have
J(ψ,−∇2ψ) = J(ψ, b) = 0, (3.5.26)
which expresses the surprising result that the solutions derived are exact nonlinear
solutions of the system. This follows from the fact that −∇2ψ = b = ψ for these
waves. This arises because although the nonlinear terms u ·∇u 6= 0, they are balanced
by the modified pressure term in the equations of motion. We also have u · ∇b = 0.
This is distinct from, but analogous to, the result that a single propagating plane
IGW in a uniform stratification is a nonlinear solution of the standard Boussinesq
system (Drazin 1977; Klostermeyer 1982). This is a consequence of the fact that
k · u = 0 for these waves, which implies that the advective operator u · ∇ annihilates
any disturbance belonging to the same plane wave. A useful consequence of this is that
77
a stability analysis can be performed on finite-amplitude propagating IGWs, allowing
detailed understanding of the initial stages of the breaking process for these waves
(Drazin 1977; Klostermeyer 1982; Klostermeyer 1991). Such studies have shown that
a single propagating IGW solution is always unstable whatever its amplitude, since
it undergoes resonant triad interactions. One important difference in our problem
is that the nonlinearity is spatially localized to the innermost wavelengths, whereas
the nonlinearity is present everywhere in the plane IGW problem. We postpone a
detailed study of the stability of our nonlinear standing wave until Chapter 6. However,
we expect waves of sufficiently large amplitude to be unstable if they overturn the
stratification.
The amplitude required to overturn the stratification can be derived using Eq. 3.5.21.
The entropy (or more precisely, a quantity proportional to the entropy) is s = b +
(1/2)r2, in these units. Overturning the stratification means that the entropy profile,
perturbed by a wave with buoyancy perturbation b, must satisfy ∂rs < 0, which im-
plies (1/r)∂rb < −1. Since b = ψ for these nonlinear waves, this can be expressed in
terms of the streamfunction as (1/r)∂rψ < −1. Reintroducing dimensional variables,
and substituting for uφ, modifies this criterion to (uφ/r) > Ωp, i.e., overturning occurs
if the angular velocity in the wave exceeds the angular pattern speed. Equivalently,
wave breaking occurs if
max(uφ) &
ω2
4C
, (3.5.27)
whose largest value occurs where the amplitude of ∂rJ2(r) is largest, which is one
wavelength from the centre.
3.6 Numerical methods
We solve the system of equations 3.4.29-3.4.32 using a Cartesian spectral code, SNOOPY
(Lesur & Longaretti 2005; Lesur & Longaretti 2007). It is advantageous to use a Carte-
sian code over one in the more natural (for the problem) cylindrical geometry, because
of the absence of a coordinate singularity at the origin, near to which is the region
of the flow that we are most interested in. We also avoid the timestep issues close
78
to the centre that would be present in a time-explicit cylindrical code. These arise
from the Courant–Friedrichs–Lewy (CFL) condition, because the grid spacing becomes
very small near the origin. In addition, using a Boussinesq-type model has the ma-
jor advantage, over a fully compressible calculation, that it does not resolve sound
waves. Since the ratio of the sound speed to a typical IGW radial group velocity is
cs/cg,r ∼ 104 À 1, fully compressible simulations would spend most of their time re-
solving sound waves, and are therefore impractical for use on this problem. We found
that this was the case from preliminary low-resolution calculations using ZEUS-2D
(Stone & Norman 1992), which nevertheless confirm our main results, and which are
described in Appendix A.5, for completeness.
This code is a 3D Eulerian spectral Fourier code based on FFTW libraries, which
was written by G. Lesur, for studying hydrodynamic and magnetohydrodynamic tur-
bulence in accretion discs in the shearing box. The code has been parallelized using
both the MPI protocol and OPEN-MP. The algorithm is a pseudo-spectral method
which will now be briefly described. Spatial derivatives are computed in Fourier space,
which means that they are extremely accurate down to the grid scale, which is one
of the principle advantages of spectral methods (Boyd 2001). However, the nonlinear
terms, and those in which the coefficients have spatial dependence, are computed in
real space, since a real space product is a convolution in Fourier space. This proce-
dure is more efficient than a direct convolution product since the FFT computation
time scales as N logN , where N is the number of grid cells, whereas a convolution in
Fourier space scales as N2. The only drawback with this procedure is that the finite
resolution generates a numerical artifact commonly known as the aliasing effect, which
is the apparition of non-physical waves near the Nyquist frequency. This effect can
be handled through a de-aliasing procedure, in which the non-linear terms (and those
with spatially variable coefficients) are computed with a resolution 3/2 higher than
the effective resolution used in the source terms (Peyret 2002). This is implemented
by setting components of the solution with wavenumbers larger than 2/3 times the
maximum wavenumber to zero during every timestep, using a de-aliasing mask. Time
evolution is computed using a 3rd-order Runge-Kutta scheme.
Since this is a Fourier spectral code, the problem must be periodic in space. We
79
solve our non-periodic problem using this code by setting up a region near the outer
boundary, in which the fluid variables are smoothed to zero as we approach the bound-
ary, using a parabolic smoothing function. We find it is quite acceptable to do this over
a region about 1/10 of the total box size. For this value there is negligible interaction
between neighbouring boxes. This approach is one that may be useful in many appli-
cations which would benefit from the use of spectral methods, but have non-Cartesian
geometry and/or non-periodic boundary conditions. The obvious drawback of such
an approach is the slight increase in computational cost, since the smoothing region
is additional to the flow in the region of interest. Interior to this we have a thin ring
in which we implement a forcing term in the radial momentum equation of the form
fr cos(2φ − ωt). This is designed to excite IGWs with m = 2, but is not designed to
accurately describe the excitation of IGWs at the top of the RZ, since we are only
interested in the dynamics of the central region. Our forcing is non-potential, which
reflects the fact that the tidal forcing of waves is indirect (e.g. see Chapter 5; OL04).
A potential force would be absorbed in this model by a hydrostatic adjustment of q.
Note that for uniform rotation Ω = Ωez, 2Ω× u = ∇(2Ωψ), so the resulting Coriolis
force is also a potential force, and would have no effect.
We solve the equations
Du = −∇q + rb+ ν∇2u+
{
0, 0 ≤ r < rforce,
f , rforce ≤ r < rsmooth,
(3.6.1)
Db+ C2r · u = 0, (3.6.2)
∇ · u = 0, (3.6.3)
D = ∂t + u · ∇, (3.6.4)
where f = −fr cos(2φ − ωt) er. We use a parabolic smoothing function d(r) =
(
r−rsmooth
rbox−rsmooth
)2
, to instantaneously smooth ur, uφ and b to zero as we approach the outer
boundary. We do this by multiplying the variables in the region rsmooth ≤ r < rbox by
d(r) during every timestep. We also solve a Poisson equation for the modified pressure
during each timestep.
Our choice of units for length and time are arbitrary. We study a region −1.5 <
x < 1.5, −1.5 < y < 1.5 and set rforce = 0.85 rbox, rsmooth = 0.9 rbox, with rbox = 1.5.
80
We choose a typical IGW radial wavelength of λr = 0.1, so that we are resolving
∼ 12 wavelengths within the box. The radial wavelength of these waves is not strictly
constant, but its variation for large r is small, and can be reasonably approximated by
a constant value in that region. We choose a forcing frequency ω = 1. These choices
are arbitrary, and are made to ensure that we are resolving a sufficient number of
wavelengths within the box.
Explicit viscosity is added to Eq. 3.4.29, since the code has no intrinsic dissipation.
This is necessary for stability – to ensure that we have no unphysical growth of energy
at small scales. The value of the viscosity is chosen such that it dissipates disturbances
on the grid scale, and a value of ν = 2 × 10−6, is chosen for all simulations. Viscous
terms are implemented in a time-implicit manner. We do not include thermal diffusion
in the buoyancy equation since this was found to be unnecessary for stability (however,
thermal diffusion is considered in §7.3).
The velocity components are normalised with respect to a typical radial phase
velocity of the wave ωλr/2pi,
ur,φ = u˜r,φ
(
ωλr
2pi
)
, (3.6.5)
in which u˜φ is equivalent to the wave steepness s, and is a measure of the nonlinearity
in the wave. This condition for overturning the stratification in Eq. 3.5.27, is then
u˜φ >
1
2
. (3.6.6)
Since we have chosen to specify the radial wavelength λr and frequency ω of the
waves that we wish to study, we have already constrained the stratification
C =
piω
λr
. (3.6.7)
There is now only one further parameter, fr to fully specify the problem. We set
fr = f˜r
(
ωλr
2pi
)
ω (3.6.8)
and vary the normalised amplitude f˜r, to model the effects of different tidal forcing
amplitudes. From preliminary investigation, it is found to be appropriate to choose
81
values between 10−2 and 103, since these result in central amplitudes that range from
u˜r ¿ 1 to u˜r = O(1) (higher central amplitudes are not observed, as is described in
the results, owing to wave breaking above a critical amplitude). This represents a
vast range of amplitudes of tidal forcing, from cases in which the secondary body is a
low-mass planet to a solar-mass binary companion, in a close orbit.
Our background is a hydrostatic equilibrium with no wave, with b = 0 in the
initial state. We use a resolution of 512 × 512 for most simulations, though higher
resolution runs have been performed using 1024× 1024 and 1536× 1536. We confirm
that the results are not dependent on the numerical method (and that our system
Eqs. 3.4.29-3.4.32 correctly describes the relevant physics) by reproducing the basic
results using ZEUS-2D. We describe our implementation of the problem in this code in
Appendix A.5. ZEUS reproduces the same basic results as the SNOOPY code, which
indicates that the effects of nonzero compressibility are unimportant. In light of this,
we only discuss the SNOOPY results below.
3.7 Numerical results
We use the set-up described in § 3.6 for a set of simulations with a variety of forcing
amplitudes f˜r. The typical radial group velocity and wave crossing time are, respec-
tively,
cg,r =
Cλ2r
2pi2
, (3.7.1)
tc =
rbox
cg,r
. (3.7.2)
For the initial conditions described in the previous section, tc ∼ 90. We define a
“wave” to be a non-axisymmetric oscillatory flow represented by a single azimuthal
wavenumber m 6= 0, whereas a “mean flow” is an axisymmetric azimuthal flow with
m = 0.
We perform several different quantitative analyses of the results. We separate the
amplitudes of the waves into an ingoing wave (IW) and an outgoing wave (OW), and
calculate a reflection coefficient, using the method described in Appendix A.4. The
reflection coefficient R is defined as the ratio of the absolute amplitudes of the OWs
82
(Aout) and IWs (Ain) for a given radial ring,
R =
∣
∣
∣
∣
Aout
Ain
∣
∣
∣
∣ , (3.7.3)
and measures the amplitude decay for a wave travelling from r to the centre and back
to r. We can relate it to the phase change on reflection (∆φ) by
∆φ = i
[
lnR− ln
(
Aout
Ain
)]
(3.7.4)
For perfect standing waves, Ain = Aout, and R = 1. If the IW is entirely absorbed
at the centre, then R = 0. Thus, R is a measure of how much the wave has been
attenuated on reflection from the centre.
We also Fourier analyse the solution, to study the temporal evolution of different
azimuthal wavenumbers in the flow. This is done by selecting a ring of cells in the
grid at a particular radius, which is chosen to be at r = 0.1, since this is probably
close enough to the centre to detect the effects of nonlinear wave couplings, if they
occur. Since this is a Cartesian grid, we do this by selecting all cells within a particular
radial ring to within a tolerance width comparable with the size of a grid cell. We
then compute the Discrete Fourier Transform of the velocity components, and from
this calculate the power spectral density,
Pm =
1
N
∣
∣
∣
∣
∣
N−1∑
k=0
ur,k exp
(
−imk
(
2pi
N
))∣∣
∣
∣
∣
2
(3.7.5)
and similarly for uφ, where N is the number of grid points in the ring – which depends
on r and the resolution, though 102 < N < 103, for all adopted resolutions at r = 0.1.
From this we can determine which components of the solution grow or decay as a result
of viscous damping, instabilities or nonlinear wave-wave interactions. Note that this
is only a rough approximation to the azimuthal power spectral density because the
points are irregularly spaced around the ring, yet we have assigned an even weighting
to each point. Nevertheless, this is justified in practice because this method works well
when tested on low-amplitude solutions that are well described by the standing wave
in § 3.5, for which we know that the solution is an m = 2 wave for both ur and uφ.
Note also that Pm = P−m since u is real, so we cannot distinguish between waves with
83
wavenumbers m and −m, without also observing the time-dependence of the flow.
The main result that will be discussed in more detail below is that we find that
there exists a critical wave amplitude beyond which wave breaking occurs near the
centre. Below this amplitude, the waves reflect coherently from the centre, and a
steady state is reached in the reference frame rotating with Ωp, consisting of an m = 2
standing wave solution. This is the outcome inferred from linear theory (see §3.5),
and we will discuss these cases first, followed by those in which nonlinear effects start
to become important. We refer to the former as “low-amplitude” cases, and the latter
as “high-amplitude” cases.
3.8 Low-amplitude forcing: coherent reflection
When the simulations are started, transients are excited by the forcing at many dif-
ferent frequencies (and radial wavelengths), centred around ω = 1 in frequency space.
As more inward propagating waves are excited by the forcing, an IW train propagates
toward the centre. At this stage in the time evolution, the solution is composed of
many different frequencies, so our decomposition of the solution into a single IW and
OW does not work well. As transients escape the region and are damped, the primary
response of the fluid is in the form of waves with frequency ω = 1 and azimuthal
wavenumber m = 2.
As the waves approach the centre and reflect, an OW is produced. As this process
continues, the amplitude of the OW matches that of the IW near the centre. For
these low-amplitude cases, the phase change on reflection is negligible. This means
that we have coherent reflection from the centre, which allows standing waves to be
produced. These waves are stationary in a frame rotating with Ωp, as expected from
linear theory. We confirm that our simulations produce the correct standing wave
solution, by plotting an example of a comparison between the simulation and the
wave solution in Fig. 3.2. We plot the velocity components in two dimensions for an
example simulation in which standing waves have formed, with max(u˜φ) ∼ 0.3, in
Fig. 3.3.
After a few wave crossing times, the reflection coefficient increases to values ap-
proaching unity throughout the grid, though its value decreases with radius, shown in
84
0 0.5 1 1.5−0.1
0
0.1
0.2
0.3
u˜ r
r
Figure 3.2: Radial velocity along the x-axis for a simulation with forcing amplitude in-
sufficient to cause breaking. The amplitude of this wave is largest within the innermost
wavelength. Also plotted is the corresponding analytic standing wave Eq. 3.5.21, converted
into a radial velocity using Eq. 3.5.14. This illustrates that our simulations accurately de-
scribe the waves for the case in which the waves reflect coherently from the centre.
Fig. 3.4 for a low-amplitude case, with max(u˜φ) ∼ 0.3. In this figure we also plot the
results of our IW/OW decomposition in a small-amplitude simulation with a resolu-
tion 1536 × 1536. Our reconstructed solutions match the data well except very close
to the centre, thus showing that our decomposition works well for these cases.
The decay in R with radius is a result of the nonzero viscosity, which results in
a decay of wave amplitude with time (and therefore distance from where they are
excited). The OW has been damped for longer, which results in the amplitude of the
OW being smaller than that of the IW. A simple estimate of the amplitude decay due
to viscosity with propagation from radius r and then reflected back to r again gives
ur
ur,0
∝ exp
(
−2
∫ r
0
νk2
cg,r
dr
)
≈ exp
(
−
16pi3ν
ωλ3r
r
)
, (3.8.1)
since k ∼ kr, except near the centre, and cg,r ' ωλr/2pi, throughout the box. This
roughly matches the amplitude decay between Ain and Aout at r = 1.2, implying that
the decay in amplitude is indeed due to viscous damping of the waves. In addition,
our inviscid exact solution is not exact in the presence of viscosity (see Chapter 7), so
85
Figure 3.3: 2D plot of the radial (top) and azimuthal (bottom) velocity in the equatorial
plane of the star for small-amplitude waves. This is at a time t = 36tc, once standing waves
have formed, in a simulation with max(u˜φ) ∼ 0.3, in which we have coherent reflection from
the centre. In the outer part of the grid, the solution is smoothed to zero to satisfy periodic
boundary conditions.
86
0 0.5 10
0.5
1
|A|
r 0 0.5 10
0.5
1
1.5
R
r
0 0.5 1−0.2
−0.10
0.10.2
r
u˜ r
0 0.5 1−0.5
0
0.5
r
u˜ φ
Figure 3.4: Radial velocity along the y-axis (bottom left) and azimuthal velocity along the
line y = x (bottom right) in a small-amplitude simulation, after standing waves have been
set up at t = 36tc, with max(u˜φ) ∼ 0.3 (solid blue lines). Also plotted is the reconstructed
solution using Ain and Aout obtained using the method described in Appendix A.4 (dashed
red lines). These are well matched everywhere except near the centre, showing that our
decomposition works well for these cases. In the top left we plot |Ain| (solid blue line) and
|Aout| (dashed red line) versus radius, together with a linear fit to each curve. The waves
damp as they propagate due to viscosity. In the top right we plot R versus radius.
87
nonlinear terms do not vanish for this wave, though their effects are found to be small.
We verified that this amplitude decay was due to viscosity by running a low-amplitude
simulation with ν = 0, in which case Ain and Aout no longer decay with radius, once
standing waves have formed.
We find that the wave reflects coherently from the centre when u˜φ . 0.5. Long-term
simulations (t ∼ several hundred tc) do not show the development of any instabilities
that act on waves with u˜φ . 0.5, though there is a slow growth of m = 0 components
of u˜φ in the solution, as can be seen in Fig. 3.5. In this figure, we plot Pm for the
first few even wavenumbers in the flow from an example low-amplitude simulation.
Negligible growth in odd m-values is observed, which is consistent with the symmetry
of the basic wave, and the quadratic nonlinearities of the Boussinesq-type system. The
growth in m = 0 is a result of viscosity, which acts to damp the waves and transfer
angular momentum from m = 2 to the mean flow. This can distinguished from a
process resulting from nonlinear interactions, because it is found to depend on ν. To
conclude, no instability is observed for waves with u˜φ . 0.5.
3.9 High-amplitude forcing: wave breaking and crit-
ical layer formation
If we increase the value of f˜r, then the above picture changes considerably when a
critical wave amplitude is exceeded. Once u˜φ > u˜φ,crit ≈ 0.5, wave breaking occurs
near the centre within several wave periods (a few 2pi/ω), and the outcome of the
simulations is very different from the small-amplitude case. This occurs when the
wave overturns the stratification – see Eqs. 3.5.27 and 3.6.6. In Fig. 3.6 we plot the
2D velocity components after wave breaking has occurred, in a simulation with f˜r = 15.
For highly nonlinear forcing, for example with f˜r > 20, the waves break as they
reach the centre with sufficient amplitude before there has been any significant reflec-
tion. For f˜r ∼ 10, the amplitude of the IW alone is insufficient to cause breaking, and
we must wait for reflection at the centre to produce an OW of comparable amplitude
before u˜φ > u˜φ,crit. Once this critical value is exceeded, the waves break. This is an
irreversible deformation of the otherwise wavy material contours.
88
0 200 400 600 800 1000 120010−10
10−810
−6
10−410
−2
100
t
P m
 
 |m|=0|m|=2|m|=4|m|=6
0 200 400 600 800 1000 120010−10
10−810
−6
10−410
−2
100
t
P m
 
 
|m|=0|m|=2|m|=4|m|=6
Figure 3.5: Temporal evolution of the power spectral density Pm at r = 0.1, in the lowest
four azimuthal wavenumbers m in the solution for ur (top) and uφ (bottom), in a low-
amplitude simulation with u˜φ ∼ 0.3 near the centre, for grid resolution 512 × 512. The
solution is in the form of m = 2 waves, and reaches a steady state in the frame rotating
with Ωp. Growth of m = 0 is nonzero as a result of viscous damping of the waves. No wave
breaking occurs because u˜φ < 0.5 in the solution. The m = 6 components are most likely
due to errors in the Fourier analysis.
89
Figure 3.6: Radial velocity (top) and azimuthal velocity (bottom) in a high-amplitude
simulation with f˜r = 15 at t = 7tc, after wave breaking has occurred, from a simulation with
a resolution 1536× 1536.
90
0 0.5 1 1.5−0.1
−0.05
0
0.05
0.1
u˜ r
r
0 0.5 1 1.5−1
−0.50
0.51
1.52
u φ rΩ p
r
Figure 3.7: The top panel shows the radial velocity along y = x for the same case as
Fig. 3.6. The bottom panel shows the angular velocity of the fluid normalised to the angular
pattern speed of the forcing Ωp along the x-axis, at the same time in the simulation. This
shows that the central regions after wave breaking are spun up to slightly exceed Ωp. The
critical layer occurs where uφ/(rΩp) = 1.
91
Once breaking occurs, we observe consequent mean flow acceleration (i.e. growth of
m = 0 components of uφ), as the angular momentum of the waves is deposited locally
where the wave breaks. This acts to spin up (if Ωp > 0) the central regions, which at
this stage contain a sufficiently small fraction of the angular momentum of the star
that their spin can be readily affected by these waves. Once this process has begun, the
central regions spin up to ∼ Ωp (which corresponds with the orbital angular frequency
of the planet), and a critical layer is formed, at which the Doppler shifted frequency
of the waves goes to zero. At this location, the azimuthal phase velocity of the waves
would equal that of the local rotation of the fluid, if they were ever to reach it intact.
In reality, as subsequent IWs approach the critical layer, nonlinearities dominate, and
the waves undergo breaking before they reach it – though see the discussion in § 3.9.2.
We plot the angular velocity of the fluid normalised to Ωp in the bottom panel of
Fig. 3.7.
As IWs approach the critical layer, their radial wavelength decreases, and they
slow down, i.e. cˆg,r → 0 as ωˆ → 0. This causes a buildup of wave energy just above
the critical layer, in which nonlinearities become important. In this thin region, the
quadratic nonlinearities produce higher wavenumber disturbances with even m-values
– see § 3.9.1. These are produced by the self-nonlinearity of the primary IW (m = 2)
as it approaches the critical layer – these self-nonlinearities vanish in the absence of a
mean flow. Daughter waves are damped faster than the primary wave because they
have lower frequencies and therefore shorter radial wavelengths, which is a result of
the theorem proved by Hasselmann (1967). Thus the IW is irreversibly deformed, and
transfers its angular momentum to either the mean flow or to daughter waves that
are then more easily dissipated by viscosity. The angular momentum carried by these
waves must be given to the mean flow when they are dissipated. This process acts to
spin up the fluid just above the critical layer to ∼ Ωp. As subsequent IWs are absorbed
by the critical layer, the spatial extent of the mean flow expands outwards, i.e. the star
is spun up from the inside out. We envisage that this process will continue until the
mean flow encompasses the bulk of the RZ, or the planet plunges into the star – this
is discussed further in Chapter 5. Long-term simulations, lasting for several hundred
tc, show that this appears to be the case.
92
This picture is analogous to Goldreich & Nicholson (1989), who propose that early-
type stars in close binaries would spin down (if Ω > n, or spin up if Ω < n) from the
outside in, once a critical layer has formed near the surface as a result of strong
radiative damping of the waves. In our problem, an instability of the primary wave,
which occurs once the wave overturns the stratification, causes wave breaking. This
results in angular momentum deposition and spin-up (for the case in which Ω < n,
spin-down if Ω > n) of the central regions, which causes the formation of a critical
layer near the centre of a solar-type star. The rate of expansion of the spatial extent of
this region depends on the forcing amplitude; for larger amplitudes it expands faster.
The critical layer moves outwards when the dissipation of subsequent IWs deposits
sufficient angular momentum to spin up the fluid to ∼ Ωp. In this picture, there is a
front of synchronisation which gradually moves outwards.
3.9.1 Growth of different azimuthal wavenumbers in the flow
We now discuss the results of a spectral analysis of the simulation data, to study the
growth of the mean flow (m = 0), and daughter waves produced by breaking (other
|m| 6= 2 wavenumbers). We plot Pm for the first few even wavenumbers for a set of
examples in Figs. 3.8 and 3.9. Negligible growth in odd m-values is observed, which
is consistent with the symmetry of the basic wave and the quadratic nonlinearities
(though it is possible in principle for the wave to be unstable to odd-m perturbations).
At the beginning of the simulations m = 2 dominates until the primary wave
breaks and transfers angular momentum to the mean flow. When subsequent waves
approach the critical layer, the primary IW transfers angular momentum to higher
m-value disturbances – this can be seen from Fig. 3.8 prior to t ∼ 100, after which
the ring r = 0.1 is enveloped by the mean flow. After this, m = 0 dominates uφ. On
the other hand, ur is then primarily in the form of |m| = 2 disturbances, which from
examination of simulation output, counter-rotate with the forcing, and have angular
pattern speed −Ωp. The excitation of these waves could explain the counter-intuitive
effect of the most central regions spinning slightly faster than Ωp, since they carry
negative angular momentum. These waves appear to reflect from both the m = 2
critical layer (though note that these waves do not see this as a critical layer) and
93
0 50 100 150 20010−6
10−4
10−2
100
102
t
P m
 
 |m|=0|m|=2|m|=4|m|=6
0 50 100 150 20010−6
10−4
10−2
100
102
104
t
P m
 
 |m|=0|m|=2|m|=4|m|=6
Figure 3.8: Temporal evolution of the power spectral density Pm at r = 0.1, in the lowest
four even azimuthal wavenumbers m in the solution for ur (left) and uφ (right), in a simula-
tion with f˜r = 15 and resolution 1536×1536. The solution is primarily in the form of m = 2
waves, until growth of even-m disturbances occurs as the wave breaks. Note that m = 0
grows strongest for uφ, i.e., angular momentum is transferred from the primary waves to the
mean flow. Once r = 0.1 is located inside the rotating region, ur is primarily composed of
|m| = 2.
94
the centre. As these waves approach the m = 2 critical layer, since they are counter-
propagating waves, their frequency is Doppler shifted upwards towards N , and they
undergo total internal reflection. These waves appear to reflect back and forth from
the critical layer and the centre.
We also note the appearance of oscillations in the energy in m = 6 in the solution.
This is most likely due to errors in the Fourier analysis, because we are not sampling
the solution with evenly spaced points. Oscillations in m = 4 could be due to nonlin-
earitesthe resulting from the presence of viscosity, since they are also present to some
degree in the low-amplitude simulations. The amplitude of these oscillations is much
smaller than that of the m = 2 or m = 0 waves, so should not affect any conclusions
drawn from these results.
We experimented with the forcing amplitude to study cases in which the waves
were of insufficient amplitude to cause breaking, and ran the simulation for ∼ 100tc.
We find that, although breaking does not occur, a critical layer can eventually form
through the gradual spin-up of the central regions by viscous damping of the primary
waves. The results of our spectral analysis of the results of such a simulation are
plotted in Fig. 3.9. We clearly see evidence for viscous damping in producing growth
of energy in m = 0. This can be distinguished from the sudden growth at t ≈ 2200
which results once a critical layer forms within the first wavelength. We discuss this
mechanism further in Chapter 7.
3.9.2 Discussion of wave reflection from the critical layer
Once the critical layer has formed, we find that a large fraction of the IW angular
momentum is absorbed near the centre. First, we confirmed this naively, by watching
animations of the time dependence of the velocity components. For both components,
the wave pattern moves outwards, which corresponds to inward propagating IGWs, so
at least a significant fraction of the solution is in the form of IWs. This is quantified by
performing our IW/OW decomposition. The results of this for a typical simulation are
plotted in Fig. 3.10, where f˜r = 15. The time has been chosen after the critical layer
has formed, and the mean flow has been accelerated near the centre. The IW/OW
wave decomposition does not work well near the centre, as we might expect, since
95
0 1000 2000 3000 400010−10
10−810
−6
10−410
−2
100
t
P m
 
 
|m|=0|m|=2|m|=4|m|=6
0 1000 2000 3000 400010−8
10−610
−4
10−210
0
102
t
P m
 
 
|m|=0|m|=2|m|=4|m|=6
Figure 3.9: Temporal evolution of the power spectral density Pm at r = 0.1, in the lowest
four even azimuthal wavenumbers m in the solution for ur (top) and uφ (bottom), in a
simulation with f˜r = 2.5 and resolution 512 × 512. Until t = 2200, viscous damping acts
on the waves and transfers angular momentum to the mean flow. A critical layer forms at
t ≈ 2200, resulting in a jump in the growth of m = 0, and a drop of energy in |m| = 2.
96
here the disturbance is primarily the mean flow (m = 0), though there are also other
components. Several wavelengths from the critical layer, in the region 0.3 < r < 1.2,
the reconstructed solution matches the simulation output quite well. The matching is
much noisier than in Fig. 3.4, since the solution contains contributions from m 6= 2
wavenumbers and ω 6= 1 frequencies, in addition to m = 2, ω = 1 waves.
Fig. 3.10 shows that the amplitude of the IW decays as it propagates towards the
centre. There is significant absorption of wave angular momentum near the critical
layer, as the IW propagates through the mean shear. This results in |Aout| ¿ |Ain|,
though the reflection is nonzero, so |Aout| 6= 0. This leads toR ¿ 1, in the region where
the decomposition works well, which implies that most of the angular momentum in
IWs is absorbed near the centre – also note that the energy flux ratio ∝ R2 ¿ 1. In
addition, the phase of the OW is perturbed with respect to the IW, which inhibits the
formation of standing waves. R oscillates with radius, mainly because the solution
is composed of some ω 6= 1 and m 6= 2 components, which are not filtered by our
IW/OW decomposition.
When a single propagating wave approaches a critical layer, the outcome has pre-
viously been found to depend on the ratio of the strength of nonlinear wave-wave
couplings to linear viscous and radiative damping. Nonlinear wave-wave couplings
occur over a timescale (Booker & Bretherton 1967)
tNL = O
(
k
− 23
h |∂rUh|
− 13U
− 23
r
)
, (3.9.1)
and linear viscous and radiative damping occur over a timescale
tL = O
(
k
− 23
h χ
− 13 |∂rUh|
− 23
)
. (3.9.2)
The ratio of these terms define the parameter (Maslowe 1986; Koop 1981)
λ ≡
tNL
tL
∼
(
χ|∂rUh|
U2r
)1/3
. (3.9.3)
where kh is the horizontal wavenumber, ∂rUh is the typical shear in the mean flow, Ur
is a typical radial velocity in the wave, and χ is a diffusivity – which for the centre
of a star is likely to be primarily radiative diffusion rather than viscosity, so χ will be
97
0 0.5 10
2
4
6
|A|
r 0 0.5 10
0.5
1
R
r
0 0.5 1−0.2
−0.10
0.10.2
r
u˜ r
0 0.5 1−1
−0.50
0.51
r
u˜ φ
Figure 3.10: Radial velocity along the line y = x (bottom left) and azimuthal velocity along
the x-axis (bottom right) in a large-amplitude simulation after wave breaking, at t = 7tc,
with f˜r = 15. These are plotted (blue solid lines) together with the reconstructed solution
using Ain and Aout obtained from the method described in Appendix A.4 (red dashes lines)
– these are quite well matched for r ∈ [0.3, 1.2]. The top left panel shows Ain (top line) and
Aout (bottom line) versus radius. There is significant absorption of IWs near the critical
layer at r ∼ 0.1, resulting in |Aout| ¿ |Ain|. The top right panel shows R versus radius.
R ¿ 1 in the region where the decomposition works well.
98
primarily the thermal conductivity κ.
Nonlinearity acts to promote energy transfer away from the critical layer through
the generation of daughter waves, and linear damping acts to suppress this resonant
wave production. If λ À 1, then the time required for nonlinear effects to become
important is long compared with that for linear damping to become important. In
this limit, there is negligible wave reflection, and nearly all incoming wave energy is
absorbed by the critical layer, as predicted by Hazel (1967). In the opposite limit,
when λ ¿ 1, nonlinear effects become manifest prior to the time when they become
suppressed by linear damping. In this case, the flow can be extremely complicated, and
nonlinearity in the critical layer region can lead to wave reflection, with ampltiudes
O
(
exp
(
−pi(Ri− 1/4)1/2
))
or less (Breeding 1971). In this limit, nonlinear wave-wave
couplings lead to the generation of many smaller-scale daughter waves, some of which
propagate away from the critical layer, carrying a fraction of the wave energy (Fritts
1979). Experiments of internal waves approaching a critical layer have been performed
by Koop (1981) and Koop & McGee (1986), in which λ = O(1). For this value, the
effects of both nonlinearity and linear damping become manifest at approximately the
same time. Their laboratory experiments show that wave reflection, in the form of
daughter waves produced by nonlinear couplings, that propagate away from the critical
layer, is suppressed by viscosity for this value of λ.
Relating these results to our simulations, we find typical values of the parameter
λ ∼ 10−1 near the critical layer, due to the explicit viscosity (ν = 2×10−6) added in the
code, since |∂rUh| ∼ O(101) and Ur = u˜r ∼ 0.1. In this limit nonlinearities are likely
to become important before viscous diffusion. Since this is the limit in which we would
be expected to find wave reflection, if any occurs at all, and we find little reflection
of waves from the critical layer, it is likely that most of the IWs are absorbed near to
the centre, and not reflected. In any case, the reflected waves will not have the same
frequency and horizontal wavenumber as the primary wave. Instead, reflected waves
will be in the form of disturbances with smaller frequencies, and therefore shorter radial
wavelengths, as well as higherm-values, as a result of wave-wave coupling (Hasselmann
1967). Such disturbances will be more easily dissipated by radiative diffusion, since
the rate of energy dissipation ∼ m3/ω4.
99
We conclude that if wave breaking and critical layer formation occurs, it is probably
reasonable to assume that the IWs are entirely absorbed in the RZ, primarily near to
the critical layer. This is inferred from our simulations, as can be seen in the example
in Fig. 3.10, in which |Aout| ¿ |Ain|. The result of this is that if wave breaking and
critical layer formation occurs, it is not possible for global standing modes to develop
in the RZ. It would then be appropriate to calculate the tidal dissipation rate using
the method of GD98. This is discussed in Chapter 5.
3.10 Conclusions
In this chapter, we have presented a study of the fate of IGWs approaching the centre
of a solar-type star, primarily using two-dimensional numerical simulations. A train
of IGWs is excited which propagates towards the centre. These waves break if they
reach the centre with steepness sufficient to overturn the stratification, which in 2D
corresponds to u˜φ & 0.5. Once this occurs, nonlinear effects cause the subsequent
formation of a critical layer, as the waves transfer their angular momentum to the
mean flow, bringing the central regions of the star into corotation with the tidal forcing.
This acts as an absorbing barrier for subsequent IWs, which continue to be absorbed
near the critical layer, resulting in an expansion of the spatial extent of the mean flow.
By decomposing the numerical solutions into an IW and OW, we find that if critical
layer formation occurs, most of the angular momentum of the IW is absorbed near the
centre, and is not reflected. This has very important implications for tidal dissipation
in solar-type stars. These issues will be discussed in detail in Chapter 5. If critical
layer formation, by wave breaking, does not occur, the wave is approximately perfectly
reflected from the centre, and global standing modes can be set up in the RZ.
The general picture of this process is that the star spins up (or down) from the
inside out, until either the planet has plunged into the star, or the RZ of the star has
spun up (or down) to match that of the evolving orbit. This is discussed further in
Chapter 5. In Chapter 6 we study the stability of the nonlinear gravity wave derived
in § 3.5 to understand the instability that leads to wave breaking in more detail, and
to determine what instabilities may set in for waves with insufficient amplitudes to
overturn the stratification. In the next chapter, we extend these simulations to three
100
dimensions.
101
Chapter 4
Internal wave breaking near the
centre of a solar-type star: 3D
simulations
In the previous chapter, we derived a Boussinesq-type system of equations that is rele-
vant for describing the dynamics of IGWs approaching the centre of a solar-type star.
We then performed numerical simulations, solving these equations in two dimensional
cylindrical geometry. In this chapter, we extend these simulations to three-dimensional
spherical geometry, and confirm that the most important results of the previous chap-
ter are not affected by this extension.
4.1 Tidal potential
The tidal potential experienced by a star hosting a short-period planet can be written
in standard spherical polar coordinates (r, θ, φ) as a sum of rigidly rotating spherical
harmonics
Re
[
Ψl,mr
lY ml (θ, φ)e
−iωt
]
, (4.1.1)
in a non-rotating (but non-inertial) reference frame centred on the star, where Y ml is
a spherical harmonic (normalised such that the integral of |Y ml |
2 over solid angles is
unity) and Ψl,m is an amplitude. Here ω is the frequency in that frame, related to the
102
tidal frequency by ωˆ = ω −mΩ, where Ω is the spin angular frequency of the star. In
this paper we consider the waves excited by planets on circular, coplanar orbits, as in
Chapter 3. In this case the dominant component of the tidal potential is quadrupolar
(l = 2), with m = 2, and takes the form
Ψ(r, θ, φ, t) = −
√
6pi
5
Gmp
a3
r2Y 22 (θ, φ−
ωˆ
2
t), (4.1.2)
where mp is the planetary mass and a is its orbital semi-major axis. Since most short-
period planets orbit faster than their stars spin, as a result of stellar magnetic braking,
we take the star to be (initially) non-rotating, so that ωˆ = ω = 2n, where n is the
orbital angular frequency of the planet.
4.2 Linear theory
In this section we derive a linear wave solution, starting from the Boussinesq-type
system Eqs. 3.4.29–3.4.32, that was derived in 2D but is equally valid in 3D. Using the
non-dimensionalisation of §3.5, we can then linearise about hydrostatic equilibrium in
3D spherical geometry, seeking solutions steady in the frame rotating at the angular
rate Ωp, proportional to eimξ, where ξ = φ − Ωpt is the azimuthal coordinate in this
frame. This leads to the following equations:
−imur = −∂rq + rb, (4.2.1)
−imuθ = −
1
r
∂θq, (4.2.2)
−imuφ = −
im
r sin θ
q, (4.2.3)
−imb = −rur, (4.2.4)
1
r2
∂r(r
2ur) +
1
r sin θ
∂θ(sin θuθ) +
im
r sin θ
uφ = 0. (4.2.5)
To obtain the linear solution, we expand scalar quantities (i.e. q and b) in spherical
harmonics Y ml (θ, ξ), since the problem is separable in both angular coordinates. In
Eqs. 4.2.1–4.2.5 we have already included the ξ-dependence of these functions (eimξ).
103
We thus take
q = qˆ(r)Y ml (θ, 0), (4.2.6)
so that the total functions are expanded onto Y ml (θ, ξ), and similarly for b and ur,
where from now on we drop the hats on the radial functions. The remaining velocity
components are expanded onto angular functions as appropriate to satisfy Eqs. 4.2.1–
4.2.5. The relation
q =
im
l(l + 1)
∂r(r
2ur), (4.2.7)
follows from incompressibility and the result
[
1
sin θ
∂θ sin θ∂θY
m
l +
1
sin2 θ
∂2φY
m
l
]
= −l(l + 1)Y ml . (4.2.8)
This enables us to derive the linear differential equation
∂2r (r
2ur)−
l(l + 1)
m2
(m2 − r2)ur = 0, (4.2.9)
whose solutions can be written in terms of Bessel functions of half-integer order (alter-
natively these can be written as spherical Bessel functions, or they can be reduced to
elementary functions). The corresponding (total) linear solution for a standing wave
in 3D can be written (where real parts are assumed to be taken)
ur(r, θ, ξ) = Br
− 32Jl+ 12 (kr)Y
m
l (θ, ξ), (4.2.10)
uθ(r, θ, ξ) =
B
l(l + 1)
1
r
∂r
[
r
1
2Jl+ 12 (kr)
]
∂θY
m
l (θ, ξ), (4.2.11)
uφ(r, θ, ξ) =
imB
l(l + 1)
1
r
∂r
[
r
1
2Jl+ 12 (kr)
] 1
sin θ
Y ml (θ, ξ), (4.2.12)
b(r, θ, ξ) = −
iB
m
r−
1
2Jl+ 12 (kr)Y
m
l (θ, ξ), (4.2.13)
where B ∈ C is an amplitude, and
k =
1
m
√
l(l + 1). (4.2.14)
104
IW and OW solutions can be obtained by replacing the Bessel function of the first
kind by equivalent Hankel functions of the first (Jν + iYν) and second kinds (Jν − iYν),
respectively. Note also that
1
r
∂r
[
r
1
2
(
Jl+ 12 (kr)± iYl+ 12 (kr)
)]
= r−
3
2
[
(1 + l)
(
Jl+ 12 (kr)± iYl+ 12 (kr)
)
−kr
(
J(l+1)+ 12 (kr)± iY(l+1)+ 12 (kr)
)]
. (4.2.15)
Starting from Eq. 3.4.33, we can calculate a conserved energy flux. Integrating this
equation over ξ from 0 to 2pi, eliminates the terms containing derivatives in t and φ
due to periodicity in ξ, as a result of the fundamental theorem of calculus. This allows
the definition of a conserved quantity proportional to the energy flux,
F =
∫ pi
0
∫ 2pi
0
r2 sin θur(E + q)dξdθ. (4.2.16)
For linear waves, terms involving products are small, so E ¿ q, to this order. This
leaves
F = pir2
∫ pi
0
Re [urq
∗] sin θdθ. (4.2.17)
Whether this is positive or negative depends on whether the wave is ingoing or out-
going.
Substituting the linear solution Eq. 4.2.10 into Eq. 4.2.17 provides a simple expres-
sion for the flux of a single l,m wave:
F =
m
pil(l + 1)
(|Aout|
2 − |Ain|
2), (4.2.18)
with corresponding energy flux FEr = ρ0F , and angular momentum flux F
L
r =
ρ0m
ω F .
This follows from the orthonormality of spherical harmonics and the Wronskian of the
Hankel functions
∫ 2pi
0
∫ pi
0
Y ml (θ, ξ)[Y
m′
l′ (θ, ξ)]
∗ sin θdθdξ = δl
′
l δ
m′
m , (4.2.19)
W [Jν(kr) + iYν(kr), Jν(kr)− iYν(kr)] =
4
ipir
. (4.2.20)
105
In particular, the l = m = 2 wave has the components
ur(r, θ, ξ) = Br
− 32J 5
2
(kr)Y 22 (θ, ξ), (4.2.21)
uθ(r, θ, ξ) =
B
l(l + 1)
1
r
∂r
[
r
1
2J 5
2
(kr)
]
∂θY
2
2 (θ, ξ), (4.2.22)
uφ(r, θ, ξ) =
imB
l(l + 1)
1
r
∂r
[
r
1
2J 5
2
(kr)
] 1
sin θ
Y 22 (θ, ξ), (4.2.23)
b(r, θ, ξ) = −
iB
m
r−
1
2J 5
2
(kr)Y 22 (θ, ξ), (4.2.24)
where k =
√
6/2.
4.2.1 Criterion for overturning isentropes
A condition in 3D for isentropic overturning can be derived from considering when
the radial gradient of the entropy s = b + 12r
2 becomes negative. This occurs when
|1r∂r(rur)| > |m|. To correlate our notation with the appendix of OL07, we define the
dimensionless nonlinearity parameter A, such that overturning occurs if |A| > 1. This
is defined such that the radial velocity is the real part of
ur = 40Ar
−4
[
1
√
6
(
1−
1
2
r2
)
sin kr −
1
2
r cos kr
]
sin2 θeimξ, (4.2.25)
in the dimensionless units that we have been using in this section (this is equivalent
to Eq. 4.2.21).
Overturning is achieved at the centre when the radial velocity in the wave takes
a maximum value ur = 1.27 at its innermost peak at r = 2.04. This value is used to
compare it to the magnitude of the radial velocity achieved in numerical simulations at
the onset of wave breaking. In these dimensionless units, we similarly require uφ & 0.99
or b & 0.38. If we instead normalise the dimensional fluid velocity with the constant
(asymptotic) radial group velocity, then we obtain a different non-dimensionalisation1,
that we previously used in Chapter 3. In those units, the criterion is modified to
u˜φ & 0.49. Note that there is not such a simple interpretation of the criterion on uφ
as in 2D, though the value is quantitatively very similar. From the results of the 2D
simulations in the previous chapter, we expect the waves to undergo instability and
1In the figures below, we also use this nondimensionalisation. This is identical to Chapter 3, and
gives values exactly 1/2 of those in the units of this section.
106
break within several wave periods after these criteria begin to be satisfied.
4.2.2 Weakly nonlinear theory
The linear solution written down in Eqs. 4.2.10–4.2.13 is not a nonlinear solution,
unlike the equivalent in 2D. This can be shown by computing the nonlinear terms in
the full Boussinesq-type system using the linear solution, in Mathematica, for example.
We find that u · ∇b 6= 0, in general, and similarly for the nonlinear terms in the
momentum equation. This means that the reflection of the waves from the centre
could be different than in 2D, since nonlinearities do not vanish for this wave. In this
section we perform a weakly nonlinear analysis to determine the dominant nonlinear
effects for small amplitudes. Since these nonlinearities do not vanish, this highlights
the importance of numerical simulations for these waves approaching the centre. We
describe the results of such simulations in § 4.4.
We propose a weakly nonlinear solution of the form
ur(r, θ, ξ) =
²
2
{ur1(r, θ)e
iξ + u∗r1(r, θ)e
−iξ}
+²2{ur20(r, θ) +
1
2
(
ur22(r, θ)e
2iξ + u∗r22(r, θ)e
−2iξ
)
} (4.2.26)
+²3ur3(r, θ, ξ) +O(²
4),
and similarly for the other variables, where ² ¿ 1. Here we write ur1(r, θ)eiξ =
ur(r, θ, ξ) from Eq. 4.2.21, for the l = m = 2 wave above, and similarly for other
variables. We substitute these expansions into the Boussinesq-type system and equate
powers of ². At each order we also equate coefficients of einξ. At leading order only
one mode is present, and we obtain the previously derived linear solution. The incom-
ing wave generates harmonics through the quadratic nonlinearities. These additional
waves (other thanm = 0) will escape to infinity and carry away a portion of the energy
flux. After some algebra the solution at O(²2) can be computed to give
ur22(r, θ, ξ) = A22r
− 32
[
J9/2
(√
5/6kr
)
+ iY9/2
(√
5/6kr
)]
Y 44 (θ, ξ), (4.2.27)
which is an l = m = 4 wave with complex amplitude A22, which has been computed
using Mathematica and given in terms of A.
107
For the wave described by Eq. 4.2.27, F can be computed. The ratio of the energy
flux in the outgoing l = m = 4 wave to the ingoing l = m = 2 wave can be shown
to be approximately 1.2× 10−5|A|2, so the reflection coefficient for reflection from the
centre, for a weakly nonlinear l = m = 2 wave, is
|R|2 ≈ 1− 1.2× 10−5|A|2. (4.2.28)
This means that a weakly nonlinear primary wave (with |A| ¿ 1), will reflect approx-
imately perfectly from the centre, with a reflection coefficient that is close to unity.
However, a small fraction of the IW energy flux is transferred to waves with higher
l and m-values, reinforcing the fact that Eqs. 4.2.10–4.2.13 is not an exact solution,
contrary to the analogous solution in 2D.
4.3 Numerical set-up
We solve the Boussinesq-type system in three dimensions using the Cartesian pseu-
dospectral code SNOOPY, as in Chapter 3. However, we modify the forcing and
damping to take into account the z-direction, and instead of forcing an m = 2 wave
in the equatorial plane, we now have
f = −frRe
[
Y 22 (θ, φ−
ω
2
t)
]
er,
= −fr
1
4
√
15
2pi
1
r2
{(
x2 − y2
)
cosωt− 2xy sinωt
}
er, (4.3.1)
in Cartesian coordinates (with r2 = x2 + y2 + z2), which is applied in the region
0.85rbox ≤ r ≤ 0.9rbox. We study a region x, y, z ∈ [−rbox, rbox], where rbox = 1.5, in
arbitrary units (not the same as § 4.2). For r > 0.9rbox the solution is damped to zero
by using a parabolic smoothing function.
We primarily use a resolution of 2563, for which the simulations were possible to
run on a single Intel Core i7 machine, utilising all 8 cores, with a typical run time of
several weeks to resolve a hundred wave crossing times. We set ω = 1, λr = 0.15, ν =
4 × 10−6, κ = 0. An otherwise identical setup is used as in Chapter 3, except using
spherical geometry instead of cylindrical geometry. A typical IGW wavelength of
108
λr = 0.15 is chosen, which is slightly larger than that for most of the 2D calculations.
This increases the number of grid points within a wavelength of the primary wave,
partially offsetting the reduction in resolution that results from using a smaller number
of grid points per dimension than in 2D. Choosing larger wavelengths than this is found
to result in unwanted effects from the proximity of the forcing region, which modifies
the linear solution. We have confirmed that the linear solution is well reproduced with
this numerical setup.
4.4 Numerical results
In this section we describe the results of the numerical simulations. We find that
f˜r ≡ fr2pi/(ω2λr) & 0.2 is required for breaking, so a variety of simulations are per-
formed with forcing amplitudes either side of this value. The basic results of these
simulations are that the wave reflects approximately perfectly from the centre of the
star if the amplitude of the wave is smaller than a certain critical value, which is found
to correspond with that required for isentropic overturning. Above this value, wave
breaking and critical layer formation occur. This picture is identical to that in 2D.
4.4.1 Low-amplitude simulations
For a low-amplitude simulation, with max(u˜r) below the critical value for isentropic
overturning (using f˜r = 0.1), we plot the variation in amplitudes of the IWs and
OWs, and also the reconstructed solutions in Fig 4.1. We analyse the results using
the method described in Appendix A.4, choosing a time once transients have been
sufficiently damped and standing waves have formed. The decay in radius is roughly
(though slightly smaller than) that which would be expected from viscous damping,
by the fractional amount
ur
ur,0
≈ exp
(
−2
∫ r
0
νk2
cg,r
dr
)
, (4.4.1)
where cg,r = Cλ2r/(2pi
2) is the (constant asymptotic) radial group velocity, for a wave
of the given wavelength. As in 2D, we have confirmed this explanation by running
simulations without viscosity, which are found to not exhibit this decay (though these
109
simulations eventually become numerically unstable if ν = κ = 0). Using a smaller
viscosity is also found to reduce the wavelength-scale oscillations around the mean
slope. These result from the fact that the linear solution to the forced wave problem
is no longer exact in the presence of viscosity. We find that increasing the number of
grid points within each shell (by reducing the values of istep, jstep and kstep) slightly
reduces the vertical extent of these oscillations, because this averages out the errors
that result from the assumption that the inviscid linear solution is exact. However,
increasing the number of grid points within each shell has negligible effect on the mean
slope.
0 0.5 12
2.53
3.5
|A|
r 0 0.5 10
0.51
1.5
R
r
0 0.5 1−0.1
0
0.1
r
u˜ r
0 0.5 1−0.1
0
0.1
r
u˜ θ
0 0.5 1−0.2
0
0.2
r
u˜ φ
0 0.5 1−0.2
0
0.2
r
b
Figure 4.1: In the top left panel we plot Ain (blue) and Aout (red) vs. r, from a low-
amplitude simulation in which the primary wave approximately perfectly reflects from the
centre, with R displayed in the top right panel. Below, are the velocity components and
the buoyancy variable (blue), plotted together with the reconstructed linear solution (red),
against radius. This is taken from a simulation with f˜r = 0.1 at t = 250, once standing
waves have formed. The amplitude decay with radius can be explained as due to viscous
damping.
The spatial structure of the solutions in three dimensions in the xy-plane is very
similar to that in two dimensions, as can be seen in Fig. 4.2 (which can be compared
110
Figure 4.2: 2D plots of u˜r (top) and u˜φ (bottom) on the xy-plane for a simulation in which
the wave perfectly reflects from the centre, for which f˜r = 0.1. This can be qualitatively
compared with Fig. 3.3.
111
with Fig. 3.3). In Fig. 4.3, we plot u˜φ on the xz-plane. This shows that the magnitude
of the azimuthal velocity peaks at θ = pi/2, due to the latitudinal form of Y 22 .
Figure 4.3: 2D plot of u˜φ on the xz plane for a simulation in which the wave perfectly
reflects from the centre, for which f˜r = 0.1. u˜φ is of largest magnitude in the equatorial
plane, where |Y 22 | peaks.
From the calculation in § 4.2.2, we expect the effects of nonlinearity to be much
weaker than the effects of viscosity for small-amplitude waves which do not cause
isentropic overturning. Since the effects of weak nonlinearity are very small, it is
difficult to quantitatively confirm the results in § 4.2.2, using, for example, an extension
of the method described in Appendix A.4 for multiple l andm values. Nevertheless, we
have qualitatively confirmed the result that the reflection is coherent and nearly perfect
(in that R ≈ 1) for amplitudes below that required for overturning the stratification.
As in 2D we do not observe any instabilities that act on the waves when they have
insufficient amplitude to overturn the stratification. In this case, the waves can form
global modes in the RZ.
112
4.4.2 High-amplitude simulations
In high-amplitude simulations, in which the wave amplitude exceeds the overturning
criterion, the wave overturns the stratification during part of its cycle and a rapid
instability acts on the wave, which leads to wave breaking within 1− 3 wave periods.
This causes the rapid (within several wave periods) deposition of primary wave angular
momentum, which spins up the mean flow to Ωp and produces a critical layer. This
critical layer acts as an absorbing barrier for IWs, as is shown from Fig. 4.4, which
plots the variation in amplitude of the IW and OW, and also the reconstructed wave
solutions (which can be contrasted with Fig. 4.1). Once the critical layer has formed,
we find |Aout| ¿ |Ain|. The central regions are not well described by the linear model,
as we would expect. However, the region outside of r ≈ 0.25 is well described by the
linear solution, with |Aout| ¿ |Ain|. In this region, R ¿ 1, so it is reasonable to
assume that the IWs are efficiently absorbed near the centre. This picture is identical
to that in 2D.
The picture in 3D in the xy-plane is very similar to that in 2D, as can be seen
in Fig. 4.5 (to compare with Fig. 3.6). However, one noticeable difference is that the
primary wave preferentially transfers its angular momentum at low latitudes, close to
the equatorial plane. This can be seen in Fig. 4.6, where we plot the angular frequency
of the flow normalised to Ωp once a critical layer has formed, in both the xy and xz
planes. This is a consequence of the latitudinal form of Y 22 , whose magnitude peaks
at θ = pi/2, as is illustrated in Fig. 4.3. The critical layer absorption is observed
to continue as the wave forcing is ongoing, so this differential rotation is continually
reinforced by the absorption of l = m = 2 IWs. Since there are wave motions in the
region of fluid interior of the critical layer, parts of these regions spin slightly faster
than Ωp (this is not seen in Fig. 4.6 due to the adopted colour scale). This was also
observed in the 2D simulations.
4.5 Conclusions
In this chapter, we have demonstrated that the fate of IGWs approaching the centre of
a solar-type star is unaffected by the extension to three dimensions. We first derived
113
0 0.5 10
20
40
|A|
r 0 0.5 10
0.51
1.5
R
r
0 0.5 1−0.5
0
0.5
r
u˜ r
0 0.5 1−0.5
0
0.5
r
u˜ θ
0 0.5 1−0.5
0
0.5
r
u˜ φ
0 0.5 1−0.5
0
0.5
r
b
Figure 4.4: In the top left panel we plot Ain and Aout vs. r in a high-amplitude simulation
in which the primary wave breaks (f˜r = 1) at t = 450, with R in the top right panel.
Below these are the velocity components and the buoyancy variable (blue), plotted together
with the reconstructed linear solution (red), against radius. The central regions are not well
described by the linear model, as expected.
114
Figure 4.5: 2D plot of u˜r (top) and uφ (bottom) on xy-plane for a simulation in which
breaking occurs with f˜r = 1, at t = 450.
115
Figure 4.6: 2D plot of the angular velocity uφ/R of the central regions in the xy (left) and
xz (right) planes, normalised to the angular pattern speed (i.e., orbital angular frequency),
in a large-amplitude simulation with f˜r = 1, at t = 450. Latitudinal differential rotation is
produced by absorption of ingoing l = m = 2 waves.
116
a linear wave solution in 3D, and found that nonlinearities do not vanish for this
wave, unlike the 2D solution, which is exact. Nevertheless, these waves are found
to reflect approximately perfectly for moderate amplitudes, a result which we have
qualitatively confirmed in numerical simulations of moderate-amplitude tidal forcing.
In this case global modes may form in the RZ, and an appreciable contribution to
the tidal dissipation will only occur when the system enters a resonance with a global
mode of oscillation.
The general picture for high-amplitude forcing is that IGWs break within the
innermost wavelengths of a star, if they reach the centre with sufficient amplitude to
overturn the stratification. If this occurs, they form a critical layer, which we have
confirmed from the simulations, efficiently absorbs ingoing wave angular momentum.
This results in the star being spun up to the orbital angular frequency of the planet,
from the inside out. This could be very important to the survival of massive planets
in short-period orbits around solar-type stars, which we will discuss in the following
chapter.
One noticeable difference in 3D is that the absorption of l = m = 2 IWs results
in the formation of latitudinal differential rotation. This is perpetually reinforced by
critical layer absorption. Instabilities may act on this rotation profile, which could
homogenise the horizontal angular momentum distribution. These include shear in-
stabilities, which can be linear (Watson, 1981) or nonlinear instabilities, that set in
at a critical Reynolds number (Richard & Zahn, 1999). These have growth times
comparable to the tidal period, and could transfer angular momentum latitudinally.
There are also doubly diffusive instabilities (Goldreich & Schubert 1967; Knobloch &
Spruit 1982), or magnetic instabilities, such as the magnetorotational instability (Bal-
bus & Hawley, 1994). However, these mechanisms are unlikely to be able to prevent
the critical layer absorption, and thus prevent the tidal engulfment of a short-period
planet.
As in Chapter 3, our simulations do not show any instabilities to act on the waves
when they have insufficient amplitude to overturn the stratification. However, it may
be that weaker parametric instabilities operate for waves with lower amplitudes (sug-
gested by GD98). A detailed stability analysis of the 2D exact wave solution written
117
down in Chapter 3 is presented in Chapter 6, and will shed some light on this matter.
118
Chapter 5
Implications
In Chapters 3 and 4, we have studied a mechanism of critical layer formation due
to wave breaking at the centre of a solar-type star. If such a critical layer forms,
then ingoing wave angular momentum is efficiently absorbed at the centre. In this
chapter we examine the consequences of this process for the orbital evolution of a
close planetary companion. We determine the amplitude of the waves at the centre
required to cause wave breaking, in terms of the star and planet parameters. We also
calculate the tidal quality factors Q′? that result, in a range of solar-type star models.
This involves a discussion of the launching region at the top of the RZ, for which we
present numerical calculations of the linearised adiabatic tidal response, together with
a semi-analytic approximation for the angular momentum flux transported in IGWs.
We discuss the relevance of this mechanism to the survival of short-period extrasolar
planets, such as WASP-18 b, WASP-12 b and OGLE-TR-56 b. These results allow us
to propose a possible explanation for the survival of all currently observed short-period
planets around F, G and K-type stars.
5.1 Introduction
The main motivation for our work is to study Q′? for solar-type stars, and in particular
to connect this with the survival of close-in extrasolar planets. We have demonstrated
in Chapters 3 & 4 that if IGWs approach the centre of a solar-type star with sufficient
amplitude to break and form a critical layer, wave absorption is efficient, and global
119
modes (of any frequency very similar to the orbital frequency) are prevented from being
set up in the RZ. The tidal torque can then be computed from assuming that the IWs
are entirely absorbed. In this case a calculation along the lines of GD98 for the ingoing
angular momentum flux of the waves excited at the top of the RZ is required. This
estimates the tidal torque, and thus the orbital evolution of the planetary companion.
In this section, we perform numerical integrations of the linearised tidal response
in an extensive set of stellar models of solar-type stars with masses in the range
0.5 ≤ m?/M¯ ≤ 1.1, throughout their main sequence lifetimes. We aim to determine
the tidal torque numerically, and compare it with a simple model of the launching
region at the top of the RZ, previously derived in GD98.
If the ingoing energy flux in IGWs is F , then converting to an angular momentum
flux, and assuming all wave angular momentum is deposited in the star, the torque is
Γ =
m
ω
F =
9
4
1
Q′?
(
mp
m? +mp
)2 m?R2?
ω2dyn
n4. (5.1.1)
Given a value of F , we can compute the tidal quality factor
Q′? =
9
4
ω
mF
(
mp
m? +mp
)2 m?R2?
ω2dyn
n4. (5.1.2)
In addition, if we are sufficiently far from resonance with a global mode, the amplitude
at the centre required for wave breaking is
A =
(
72
√
6F
5pi
C5
ρcω8
) 1
2
. (5.1.3)
Note, however, that if we are close to a resonance, then A may be much larger than
this estimate would predict. In this chapter we will discuss calculations of F in models
of solar-type stars, and then discuss the implications of our results in Part II for the
survival of short-period extrasolar planets.
120
5.2 Numerical computation of the linearised tidal
response throughout the star
In this section we solve the linearised equations governing the adiabatic tidal response
throughout the star, computing the excitation of both the equilibrium and dynam-
ical tides numerically. This allows us to determine the ingoing energy and angular
momentum fluxes in IGWs launched at the top of the RZ, and to check the valid-
ity of approximate semi-analytic formulae for these quantities, presented in the next
section. This is important because the orbital evolution of a planetary companion is
determined by the ingoing angular momentum flux absorbed at the critical layer.
We solve the following coupled ODEs for the radial and horizontal displacements:
dξr
dr
= −
[
2
r
+
N2
g
+
d ln ρ
dr
]
ξr +
[
l(l + 1)
r
−
ω2rρ
Γ1p
]
ξh +
fr2ρ
Γ1p
, (5.2.1)
dξh
dr
=
[
1
r
−
N2
rω2
d ln p
d ln ρ
]
ξr −
[
1
r
−
N2
g
]
ξh −
fN2r
ω2g
. (5.2.2)
An outline of the derivation of these equations is presented in T98. Note that we are
ignoring the self-gravity of the entire tidal response, which is reasonable because most
of the mass of the star is concentrated near the centre. This assumption is certainly
valid for the dynamical tide, and is approximately valid for the equilibrium tide. In
these equations, we take the tidal potential in the frame rotating with Ωp to be equal
to Eq. 4.1.2 with φ = ξ and ω = 0, so that1
f = −
√
6pi
5
mp
(m? +mp)
n2. (5.2.3)
This is the amplitude of the largest tide for a circular orbit. In this frame, the dis-
placement field is separated into radial and horizontal (non-radial) components
ξ = ξrY ml (θ, ξ)er + ξhr∇Y
m
l (θ, ξ). (5.2.4)
We can further decompose the tidal response of a fluid body, which in this section
we take to mean a solar-type star, into an equilibrium and a dynamical tide, defined
1Note that we are defining spherical harmonics in a standard manner, normalised so that the
integral of |Y ml |
2 over solid angles is unity, unlike T98.
121
such that the total displacement is the sum of these two displacements, i.e, that
ξ = ξd + ξe. The equilibrium tide is a quasi-hydrostatic bulge defined by
ξer = −
Ψ
g
, and ∇ · ξe = 0, (5.2.5)
in stratified regions (Goldreich & Nicholson 1989), where Ψ is the tidal gravitational
potential experienced by the body and g is the gravitational acceleration. The total
displacement is not well described by Eqs. 5.2.5 in convective regions (GD98; T98;
OL04). This is because in a barotropic flow (with adiabatic stratification) vorticity is
conserved, so we must have ∇× ξ = 0, whereas ∇× ξe 6= 0, in general. The presence
of a convection zone (hereafter CZ) thus implies that a dynamical tide must exist. The
dynamical tide ξd is defined as the residual response that results from the equilibrium
tide not being the exact (linearised) solution to the problem, when the tidal frequency
is nonzero.
The equations governing the adiabatic equilibrium and dynamical tides are
0 = −
1
ρ
∇δpe +
δρe
ρ2
∇p−∇Ψ, (5.2.6)
−ω2ξd = −
1
ρ
∇δpd +
δρd
ρ2
∇p+ ω2ξe
︸︷︷︸
Forcing
, (5.2.7)
from which it is clear that the dynamical tide is not forced directly by the tidal
potential, only by the inertial terms in the equation of motion.
We impose a free upper boundary, i.e., take the Lagrangian pressure perturbation
∆p = 0 at r = R?, so that the Eulerian pressure perturbation δp = −ξr
dp
dr . Since the
relation
ξh =
1
ω2r
(
δp
ρ
+ fr2
)
, (5.2.8)
follows from the non-radial equation of motion, this relates our variables at the surface
of the star. We take an IW BC at the inner boundary at r ≈ 0.02R?, where we match
ξr and ξh onto the analytic solution for an IW derived in §4.2,
ξr(r) = Aξr
− 32 (J 5
2
(kr) + iY 5
2
(kr)), (5.2.9)
122
ξh(r) =
Aξ
6
r−
3
2
[
3(J 5
2
(kr) + iY 5
2
(kr))
−kr(J 7
2
(kr) + iY 7
2
(kr))
]
, (5.2.10)
where k = mCω . We take Aξ to be a free parameter, so that these equations relate
the ratio of ξr and ξh. This solution is quite accurate when r/R? . 5% since ξ
e is
negligible in this region. This BC is meant to represent the IW absorption at a critical
layer.
We solve Eqs. 5.2.1–5.2.2 using data interpolated from a stellar model at points
required by a 4th/5th order adaptive step Runge-Kutta integrator, using a cubic spline
interpolation. In particular, the coefficients of ξr, ξh, in Eqs. 5.2.1–5.2.2 are singular
at the origin, so we first multiply these quantities by r before interpolating their values
to the locations required by the ODE integrator, using the stellar model parameters.
We then divide by r, after the interpolation. This is done to get the correct behaviour
for small r.
Our method of solution is a shooting method to an intermediate fitting point (Press
et al. 1992), which we take to be the CZ/RZ interface, where we enforce continuity
of the solution. The freely specifiable initial conditions for each ODE integration are
chosen to be ξr at the surface, and Aξ at the inner boundary. We use ξr = ξer as our
starting “freely specifiable” estimate at the surface, which is an accurate approximation
since |ξdr | ¿ |ξ
e
r | at r = R?.
The Eulerian pressure perturbation for the dynamical tide is
δpd = ρω2r
(
ξeh + ξ
d
h
)
, (5.2.11)
from the horizontal component of Eq. 5.2.7. The radial energy flux at each radius is
F =
ωr2
2
Im
[
(δpd)∗ξdr
]
, (5.2.12)
which follows from manipulating Eqs. 5.2.6 and 5.2.7 to derive an energy equation.
From Eq. 5.2.7, we derive the relation
∇ · Im
{
δpd(ξd)∗
}
= ρω2Im
{
(ξd)∗ · ξe
}
, (5.2.13)
123
0 0.2 0.4 0.6 0.8 1−5
0
5
ωξ
r
Figure 5.1: Typical values of the real parts of the radial (ωξr, in blue solid lines) and
horizontal (ωξh, in red dashed lines) velocity components, in m s−1. We use Model S of the
current Sun, and consider the tidal perturber to be a P = 1 d, mp = 1MJ , planet orbiting
the current Sun.
which can be integrated over solid angle to give
Im
{
∂r
(
r2δpd(ξdr )
∗
)}
= ρr2ω2Im
{
(ξdr )
∗ξer + l(l + 1)(ξ
d
h)
∗ξeh
}
. (5.2.14)
This is telling us that the equilibrium tide does work to excite the dynamical tide. As
part of the validation of our numerical code, we have confirmed that this is accurately
satisfied from the numerical solutions computed with an ingoing BC. This should
adequately convince ourselves that the code is able to accurately compute the energy
flux in the dynamical tide.
5.2.1 Results
As an illustration, we present the results of our integrations for a fiducial case with
a P = 1 d, mp = 1MJ planet orbiting the current Sun (for which we use Model S,
described in Christensen-Dalsgaard et al. 1996) in Fig. 5.1. We plot the real parts
of ωξr (solid blue) and ωξh (dashed red) throughout the star, which represent typical
124
0 0.2 0.4 0.6 0.8 10
1
2
3
4
F F√ x
r
Figure 5.2: Numerically computed F , normalised to the prediction from the analytic theory
in Eq. 5.2.24 (solid blue line) and Eq. 5.2.18 (red dashed line). Eq. 5.2.24 overestimates F
by less than 10%, and is thus a good approximation of the ingoing energy flux.
values of the radial and horizontal velocity components. The radial wavelength of the
waves decreases as the waves propagate deeper into the RZ, where N2 increases. The
large increase in the velocity amplitude near the centre is evident, as predicted from
the linear solutions in § 4.2. This can be compared with a similar calculation in T98,
displayed in their Fig. 1, which our code correctly reproduces for the given planetary
orbital period when a regularity condition in applied at the centre. The only difference
between our calculations and theirs is that we use an IW BC, whereas they allow the
waves to perfectly reflect from the centre.
In Fig. 5.2 we plot the ingoing energy flux, normalised to both the semi-analytic
prediction of GD98 (red dashed lines), and a revised expression derived in Appendix
A.6 (blue solid lines), which will be discussed in the next section. This illustrates that
the ingoing energy flux oscillates about its final asymptotic value, which it eventually
approaches deep in the RZ.
125
5.2.2 Semi-analytical calculation of the ingoing energy flux
In this section we compare the numerically computed F with the semi-analytic estimate
of GD98, and present a slight refinement which is found to be appropriate for solar-type
stars.
In the launching region at the top of the RZ, there is a location at which N = 0 at
r = rb ≈ 0.71R¯, near to which it follows from the dispersion relation that the radial
wavenumber of gravity waves vanishes. Near this turning point we can approximate
the solution in this region by assuming a functional form for N2. GD98 take N2 ∝
rb − r ≡ x, in which case the problem in the launching region reduces to the solution
of Airy’s differential equation for ξdr :
d2ξdr
dx2
− xξdr ≈ 0, (5.2.15)
if we define κx = r − rb, where
κ =
(
l(l + 1)
ω2r2b
dN2
dx
) 1
3
. (5.2.16)
The reciprocal Llaunch = κ−1 is the radial lengthscale of the launching region, and is
roughly equal to the first wavelength of the gravity waves launched. This approxima-
tion is valid only if 1κ2r2 ¿ 1, otherwise Eq. 5.2.15 must be forced on the right hand
side. In this model the wave is assumed to be excited in a thin shell of radial extent
Llaunch, at a radius r ∼ rb, within which N2 ∝ rb − r.
If the solution is an IW, then the relevant solution of Airy’s equation is
ξdr (x) ∝ Bi(x)− iAi(x). (5.2.17)
The energy flux F in IWs excited at the interface (r = rb) in a non-rotating star, is
(see Appendix A.6 for a similar derivation)
F ≡ Fx =
3
2
3
8pi
[
Γ
(
1
3
)]2
[l(l + 1)]−
4
3
[
ρbr
5
b
∣
∣
∣
∣rb
dN2
dx
∣
∣
∣
∣
− 13
∣
∣
∣
∣
∂ξr
∂x
∣
∣
∣
∣
2
]
ω
11
3 , (5.2.18)
where ω = 2n is the frequency of IGWs excited by a planet in a circular, nonsyn-
126
chronous orbit, and
rb
dN2
dx
≈ −100ω2dyn, (5.2.19)
for the current Sun. The quantity
∂ξr
∂x
≈ σc
Ψ
ω2dyn
, (5.2.20)
and σc is a constant whose value depends primarily on the thickness of the convection
zone, and is equal to -1.2 for the current Sun. The values of these quantities depend
on the stellar model. To calculate σc, we numerically solve GD98 Eq. 3, which we
reproduce below, for the dynamical tide in the CZ using a linear shooting method,
with N2 = 0. We use the BCs that ξdr = 0 at the top and bottom of the CZ. The
motivation behind these choices is that the surface is roughly an equipotential, and
we want to enforce continuity of the solution at the top of the RZ, where ξdr ∼ 0 (from
the perspective of the solution in the CZ). The equation we solve is
Lξdr (r) = f(r), (5.2.21)
with the linear differential operator acting on the dynamical tide
Lξdr (r) =
d2
dr2
(
r2ξdr
)
+
d
dr
(
d ln ρ
dr
r2ξdr
)
+
l(l + 1)
ω2
(
N2 − ω2
)
ξdr , (5.2.22)
and the forcing by the equilibrium tide given by
f(r) = l(l + 1)ξer −
d2
dr2
(
r2ξer
)
. (5.2.23)
We find that a slightly better approximation is to take N2 ∝
√
x over the launching
region2, since this is valid over ∼ 0.15R¯ from the interface, as we illustrate in Fig. 5.3.
The slope of the curve
√
x can be obtained through fitting to the profile of N2 in the
stellar model. This allows a slightly more accurate calculation of F based on the stellar
model than is obtained through direct application of Fx (where the gradient dN2/dx
is not uniquely defined). Nevertheless, the differences are only a factor of two at most.
2However, no simple physical arguments for such a profile have been found, which is perhaps to
be expected since stellar models contain complicated combinations of physics.
127
0 0.2 0.4 0.6 0.8−5
0
5
10
15
20
25
N2 /ω
2 dyn
r
Figure 5.3: N2 normalised to ω2dyn (solid blue line), together with our local approximation
N2 ∝ (rb − r)1/2 (red dashed line). This approximation is reasonable over a region of size
∼ 0.15R¯. We also plot an arbitrarily scaled profile of F , to compare its radial profile with
the profile of N2, for P = 1 d.
128
In Appendix A.6 we calculate the solution in the launching region and the resulting F ,
for any power law profile of N2 with a positive exponent, using the framework of OL04.
Using our approximation, the radial extent of the launching region Llaunch ∼ 0.03R¯.
For the case N2 ∝
√
x, we obtain, for a non-rotating background,
F ≡ F√x =
(
2
5
) 1
5 pi
2
[
Γ
(
3
5
)]2 [l(l + 1)]
− 75 ω
19
5
[
ρbr
5
b
∣
∣
∣
∣
√
rb
dN2
d
√
x
∣
∣
∣
∣
− 25
∣
∣
∣
∣
∂ξr
∂x
∣
∣
∣
∣
2
]
. (5.2.24)
This matches the numerically computed asymptotic value of F for the current Sun
quite well, to within 10% (see Fig. 5.2). The remaining discrepancy is due to the slight
variation in background density, and the magnitude of the equilibrium tide, over the
launching region. We note that the main change in the energy flux from modifying
the profile of N2 in the transition region is to change its frequency dependence.
To illustrate the refinement for short-period forcing, in the top panel of Fig. 5.4 we
compare the numerically computed dynamical tide in the launching region (black solid
lines) with Eq. 5.2.17 and Eq. A.6.16. It is clear that the Airy function approximation
(blue dashed lines), using Eq. 5.2.19 for the slope at the interface, does not correctly
represent the solution in first wavelength, whereas the solution derived in Appendix
A.6 (red dashed lines), with a best fit
√
rb dN
2
d
√
x = 15ω
2
dyn, matches the solution in this
region quite well. However, this discrepancy can be rectified by choosing a shallower
slope at the interface for the Airy function approximation, as can be seen in the
bottom panel, where both solutions match the numerically computed dynamical tide
quite well. In the bottom panel we have used a slope rb dN
2
dx = 20ω
2
dyn, for computing
the blue dashed line using Eq. 5.2.17.
As can be seen from Fig. 5.2, the numerically computed asymptotic value of F
differs from Fx by a factor . 2, even in the case of short-period forcing. We would
expect Fx to be correct to within a factor ∼ 2 even if the slope varies by an order of
magnitude, since it is only raised to the power −13 . However, our slight refinement
improves this estimate by ∼ 50% for P = 1 d.
We have performed numerical integrations for several different forcing frequencies
(planetary orbital periods), for which we find F ∝ P−8.06 for fixed stellar and planetary
properties (other than the orbital period). If we take into account the fact that ∂xξdr ∼
Ψ ∼ P−2, and then consider a fixed tidal potential, we find that F ∝ ω4.06, which is
129
0.5 0.55 0.6 0.65 0.7 0.75−1000
−500
0
500
1000
1500
ξd r
r
0.5 0.55 0.6 0.65 0.7 0.75−1000
−500
0
500
1000
1500
ξd r
r
Figure 5.4: Top: ξdr from numerical integrations (black solid line), compared with the
Airy function approximation (blue dashed line) and the Bessel function approximation of
Appendix A.6 (red dashed line). Bottom: Same, but with dN
2
dx reduced by a factor of 5 for
Airy function approximation (blue dashed line).
130
0 0.5 114
16
18
20
22
24
26
logF
log P
Figure 5.5: Power law dependence of F on P , for fixed stellar and planetary properties
(other than the orbital period). This has slope −8.06.
slightly different from the power law dependence in Eq. 5.2.24. The discrepancy most
likely results from the variation in the background density, and the magnitude of the
equilibrium tide (which forces the dynamical tide), within this region.
For P . 2− 3 d, which is the relevant regime for which this process is potentially
important for the survival of close-in planets, as we discuss later, there is a few per-cent
variation in background parameters over a lengthscale Llaunch. This means that our
calculation of F√x differs from the numerically computed value of F , by an amount that
increases as P is made smaller to a maximum of 20% when P = 0.5 d. Nevertheless,
this discrepancy is small, and our semi-analytic estimate F√x is a good approximation
to F for all cases that we have modelled. This is used to provide an estimate of Q′?,
and thus the tidal torque, in § 5.4.
5.2.3 Variation between different stellar models
Eq. 5.2.24, matches the numerically computed value to within a few per-cent for a
variety of solar-type stars. This is because it generally arises that N2 ∝ xα, with
α ∼ 0.5, when the launching region is a few percent of the stellar radius. This occurs
131
for the waves excited by planets in short-period orbits. We have confirmed this by
computing F in a number of stellar models with masses in the range 0.5 ≤ m?/M¯ ≤
1.1, and ages that represent the range of main-sequence ages expected for these stars.
These were computed using ASTEC (Christensen-Dalsgaard 2008).
We collect all terms in Eq. 5.2.24, that depend on the properties of the launching
region, into a single parameter
G = ρbr
5
b
∣
∣
∣
∣
dN2
d ln
√
x
∣
∣
∣
∣
− 25
σc. (5.2.25)
This parameter has been computed in each of the stellar models in our study. This
involved integrating Eq. 5.2.21 throughout the CZ in each of these models, where
N2 ∼ 0, using a linear shooting method, to determine σc. The result of this study
is that G ∼ G¯ ≈ 1.2 × 1048kgm2s2/3, to within a factor of 5 for all solar-type stars,
throughout their main-sequence lifetimes. This is true even taking into account the
evolution of the position of the CZ/RZ interface, and the resulting change in the
density of the star at the interface.
The main uncertainty in these models is the profile of N2 within the launching
region, especially since the slope within 0.02R¯ of the interface is not well constrained
by theory or observations. Helioseismic observations are not yet able to constrain
the stratification within this region, owing to the lack of observed g-modes (Ellis
1984; Appourchaux et al. 2010). In addition, the relevant physics included in the
stellar models is also uncertain, particularly as a result of changes to the compositional
gradient from convective overshoot. The inclusion of helium settling tends to make
the interface profile sharper, and the inclusion of turbulent diffusion, that results
from convective overshoot, and often parameterised using a simplified 1D model of
this process, tends to smooth out the profile near the interface (Jørgen Christensen-
Dalsgaard, private communication). However, these changes are small and occur only
within a region smaller than the launching region for P . 3 d. This means that
uncertainties in the observations, and the physics, at the interface, are unlikely to
significantly change G (and therefore F ) for P . 3 d, which are the planets whose
survival could be threatened by our mechanism (as we discuss below). In addition,
∣
∣
∣
dN2
d
√
x
∣
∣
∣ is only raised to the -2/5 power in our model.
132
0.6 0.65 0.7 0.75 0.8−1
01
23
45
6
N2 /ω
2 dyn
r
0.5 0.55 0.6 0.65 0.7 0.75−2000
−1000
0
1000
2000
ξd r
r
Figure 5.6: In the top panel we plot N2 in a 1.0M¯ star at t = 6.95 Gyr, with our fitted
solutions in the bottom panel. This model has a “bump” in the N2 profile, but this occurs
over a region smaller than Llaunch, and so does not reduce the accuracy of our analytic model
in the launching region (or the corresponding energy flux), shown in the bottom panel.
133
For Model S, the compositional gradient (∇µ) is unimportant in the launching
region, and N2 primarily results from temperature gradients. As the star evolves, the
settling of elements heavier thanH, produces a compositional gradient at the top of the
RZ, which can produce “bumps” in the profile of N2. In Fig. 5.6 we plot an example of
the profile of N2 for a 1M¯ model (with Z = 0.02) at t = 6.95 Gyr, together with the
best fit solution in the launching region. For P . 3 d, Llaunch is generally much larger
than these “bumps”, so the wave launching process does not notice such departures
from a smooth stratification profile, and Eq. 5.2.24 remains a good approximation for
the energy flux. However, if the frequency is sufficiently low (P & 6 d), the radial
extent of the launching region can become comparable with the size of these “bumps”,
and the numerical solution can depart appreciably from our analytical model. Planets
in such orbits are very unlikely to be affected by stellar tides at such orbital distances,
so we do not consider such effects worthy of further consideration. To summarise, we
have confirmed that Eq. 5.2.24 is a good approximation for the energy flux for planets
on orbits of a few days throughout the range of solar-type stars in our study.
5.2.4 Are 1D linear hydrodynamic calculations reasonable?
The calculations done in § 5.2 are performed under the assumption of linearity. We
have demonstrated in Chapters 3 and 4, that this is not a valid assumption near the
centre of a star. A simple estimate of the nonlinearity in IGWs in the launching
region, for want of a better measure, compares the radial displacement to the radial
wavelength, for which ξdr/λr ∼ 10
−4, for a hot Jupiter orbiting the Sun on a one-day
orbit (this estimate can be obtained from Fig. 5.1). This value increases linearly with
the mass of the planet, but we are still in the linear regime even if we have a close-
binary perturber, which indicates that linearity is likely to be a good approximation
in the launching region (and throughout the RZ, except for the central regions).
However, it remains to be seen whether F for the 1D calculations is the same as
that in 2D or 3D numerical simulations of realistic tidal forcing in a model including
both a CZ and a RZ. Such simulations as Rogers et al. (2006) could be performed of
the whole star subject to tidal forcing. The turbulent convection in these simulations
will produce a spectrum of waves at the top of the CZ, in addition to those excited by
134
tidal forcing. It may be that the interaction of these waves reduces the ingoing energy
flux. Alternatively, the profile of N2 in the transition region could be modified by
realistic modelling of convective overshoot, altering the strength of the stratification
within a few percent of a pressure scale height from the interface. This would affect
F if the overshoot region is comparable with the size of the launching region, though
probably not significantly, as we discussed in the previous section.
A toroidal magnetic field in the launching region could also affect the amplitudes of
these waves, and the value of F . Rogers & MacGregor (2010) find that when the IGW
frequency is approximately equal to the Alfve´n frequency (ωA), strong wave reflection
occurs. This behaviour follows from the dispersion relation for IGWs in the presence
of a magnetic field (e.g. Kumar et al. 1999), and could have important consequences
for F , if the magnetic field is sufficiently strong. However, for ω ≈ ωA in the launching
region, we require Bφ = 2pi
√
µ0ρbrb/P & 6 MG, when P = 1 d, which is close to being
ruled out on empirical grounds, from measurements of the solar oblateness (Friedland
& Gruzinov 2004). If ωA < ω, little attenuation of wave energy over the non-magnetic
case is found by Rogers & MacGregor (2010), so this seems unlikely to affect F in our
case.
In the innermost wavelength, a strong magnetic field would be able to reflect IGWs
before they reach the centre, if ω ≈ ωA in this region. This requires a toroidal field
of strength Bφ & 6 MG, or a poloidal (radial) field of strength Br & 28 MG. Since
it is unlikely that such fields could exist in the RZ (Friedland & Gruzinov, 2004), a
magnetic field will probably not affect the reflection of IGWs (excited by short-period
planets) in the RZ. It is therefore appropriate to ask whether the waves will break on
reaching the centre.
5.3 When does wave breaking occur?
In this section we estimate the orbital and stellar properties required to excite waves
that are sufficiently nonlinear near the centre for breaking to occur, using the linear
solution in Chapter 4. The nonlinearity parameter is defined so that for |A| > 1, the
wave overturns the stratification during part of its cycle. Assuming we are not close
to resonance with a global g-mode, we can estimate the amplitude near the centre,
135
assuming standing waves have formed, by using Eq. 5.2.24, to give
|A| =
3
10
(
3
2
) 1
20 pi
2
5
5
1
10Γ(35)
G
1
2
ω2dyn
(
C5
ρc
) 1
2
(
mp
m? +mp
)
P
1
10 (5.3.1)
≈ 0.4
(
G
G¯
) 1
2
(
C
C¯
) 5
2
(
mp
MJ
)(
M¯
m?
)(
P
1 day
) 1
10
, (5.3.2)
for the current Sun, where G¯ ≈ 1× 1048kgm2s2/3, and C¯ ≈ 8× 10−11m−1s−1.
We can therefore write the following criterion for wave breaking:
(
G
G¯
) 1
2
(
C
C¯
) 5
2
(
mp
MJ
)(
M¯
m?
)(
P
1 day
) 1
10
& 3.6. (5.3.3)
A Jupiter-mass planet in a one-day orbit around the current Sun would not raise tides
of sufficient amplitude near the centre for breaking to occur, and would presumably
survive, because it does not satisfy this criterion. Indeed, all of the HJs currently
observed around G or K type stars3 do not satisfy this criterion, so this could be an
important explanatation for their survival.
Note, however, that sufficiently long-period planets may be able to satisfy this
criterion around the current Sun, due to the (weak) period dependence of the nonlin-
earity. The waves excited by such planets would be of very low amplitude, but they
are also of very low frequency. This means that their wavelength is extremely short, so
the energy of these waves would be concentrated into an extremely small volume near
the centre of the star, if they were to reach it. However, radiative diffusion is certain
to damp these waves before they reach the centre, since they are of such short wave-
length. This process would, in any case, contribute negligibly to the orbital evolution
of such long-period planets.
The variation in |A| is primarily dependent on C, since |A| ∝ G
1
2C
5
2 . The strong
dependence on C means that as a star evolves the tide could become nonlinear at a
critical age, since C increases with evolution on the main-sequence. In Fig. 5.7 we plot
the variation in C versus main-sequence age, normalised to its value in the current Sun,
from a sequence of solar models that pass through Model S. The increase in C with
age is due both to the increasing central condensation, and the build-up of a gradient
in the hydrogen abundance (there is a small drop around 8 Gyr when hydrogen is
3Contained in the catalogue at http://exoplanet.eu/catalog.php
136
0 2 4 6 8 100
12
34
56
7
t/Gyr
C/C ⊙
Figure 5.7: Coefficient C in the expansion N ≈ Cr near the centre of the Sun normalised to
its current value versus main-sequence age, for a sequence of solar models that pass through
Model S of the current Sun. The stratification steepens as the star evolves.
nearly used up and the contribution from the composition gradient decreases, after
which the central density increases rapidly).
For a given age, C is larger in more massive stars, due to their greater central
condensation. In addition, stars with lower metallicity also have a greater central
condensation for a given age, and so have larger C values over stars with higher
metallicity. Over the range of stars considered in this study, C is found to take values
between 0.1 − 10 C¯. This leads to a large variation in A values, for fixed orbital
parameters. As a result, C is critical in determining whether wave breaking occurs at
the centre. We have stated that a short-period Jupiter-mass planet does not satisfy
Eq. 5.3.3 around the current Sun. However, such a planet around a similar-age 1.0M¯
star with a metallicity Z = 0.01, will cause wave breaking at the centre, since C is
larger by a factor of 3. Thus, there is a strong dependence of the breaking criterion
on the stellar model, primarily through the parameter C.
137
5.4 Q′? for solar-type stars
The main motivation for our work is to study Q′? for solar-type stars, and to connect
this with the survival of close-in extrasolar planets. If a critical layer forms by wave
breaking (or radiative diffusion, as we discuss in Chapter 7), then we can assume that
the entire ingoing angular momentum flux is absorbed in the RZ. In that case, we use
Eq. 5.1.2, together with Eq. 5.2.24, to write
Q′? =
9
32
(
3
2
) 2
5
[
Γ(−25)
]2
5
4
5pi
24
5
1
G
m?R
2
?ω
2
dynP
14
5 (5.4.1)
≈ 0.9× 105
[
P
1 day
] 14
5
. (5.4.2)
Note that the exact value depends on the stellar model adopted – in particular the
value of G, which is determined from the stellar properties in the launching region, and
the thickness of the convection zone. The given value applies to a stellar model of the
current Sun. However, we stated in §5.2.3 that G varies by only a factor of 5 throughout
the range of main-sequence stars in our study. Taken together with changes in stellar
mass and radius, our estimate of Q′? varies by only a similar amount, thus Eq. 5.4.2 can
be considered quite robust, if critical layer absorption occurs near the centre. This is
approximately true for all stars within the mass range 0.5 ≤ m?/M¯ ≤ 1.1, throughout
their main-sequence lifetimes.
As the star evolves on the main-sequence, the position of the interface rb moves
inwards towards higher density material, which slightly increases F . This means that
Q′? tends to decrease with main-sequence age, though by only a factor of 5 or so. The
dissipation that results from wave absorption at a critical layer thus becomes more
effective as the star evolves.
5.5 Orbital evolution of the planetary companion
Tidal dissipation of the quadrupolar tide raised in the star leads to evolution of the
semi-major axis at the rate
a˙
a
= −
9
2
mp
m?
(
R?
a
)5
n
[
sgn(2n− 2Ω)
Q′?
]
. (5.5.1)
138
The inspiral time for a planet into the current Sun is then
τa = −
10
107
a
a˙
≈ 1.03 Myr
(
MJ
mp
)(
m?
M¯
) 8
3
(
R¯
R?
)5( P
1 day
) 107
15
, (5.5.2)
since a˙/a ∝ a−107/10. Note that the strong frequency dependence of Q′? means that
this mechanism could be very important for short-period systems. This predicts that
a planet spiralling into its star will undergo rapid acceleration as it migrates inwards.
This is a consequence not only of the reduction in semi-major axis, but also the
decrease in Q′?, as the tidal frequency increases with the inspiral. It must be noted
that simple timescale estimates do not accurately reflect the evolution if the orbit is
eccentric, inclined, or if the stellar spin is not much slower than the orbit (see Chapter
2), but this estimate shows that this mechanism can be very efficient in contributing
to the tidal evolution of hot Jupiters on the tightest orbits. Indeed, τa . O(1) Gyr
for a Jupiter-mass planet in an orbit of less than about three days around the current
Sun, if a critical layer were to form near the centre.
We can crudely estimate the maximum orbital period of a planet that can be pulled
into the star by this process, by equating the moments of inertia of the RZ, to that of
the orbit µa2 = r2gm?, giving
P ' 1.8 d
(
m?
M¯
) 1
2
(
rb
0.7R¯
)(
MJ
µ
) 1
2
, (5.5.3)
where µ = mpm?mp+m? is the reduced mass, and r
2
g = 0.076 is the squared dimensionless
radius of gyration for a polytrope of index 3.
A Jupiter-mass planet on a ∼ 1 day orbit will spin up a substantial fraction of
the RZ on infall. If the ratio of the orbital moment of inertia to the spin moment of
inertia of the RZ & 1, then the above process alone will be unable to cause the planet
to spiral into the star. However, if this ratio is smaller than unity for such a short-
period planet, the whole of the RZ must be spun up to cause the planet to completely
spiral into the star. Once the entire RZ has spun up to the orbital frequency, this
process becomes ineffective, and the corresponding tidal torque will vanish. However,
the tidal torque is nonzero due to dissipation of the equilibrium tide by turbulent
convection, and so even if this process stops, it does not guarantee the survival of the
139
planet. In addition, magnetohydrodynamic coupling between the CZ and RZ could
act to partially counteract the spin-up of the interior, if the CZ is spinning slower than
the orbit. Magnetic braking of the star through the interaction of its magnetic field
with a stellar wind acts to spin down the CZ, which would also gradually spin down
the RZ through these couplings. If the coupling between the CZ and RZ of the star
is efficient, then our mechanism would again become effective. In the case that the
coupling timescale balances the timescale for spin up of the RZ due to IGW absorption
at a critical layer, then the planet would migrate into the star on the magnetic braking
timescale. A detailed study of these effects is not currently possible, since there are
many uncertainties, but this is worthy of future consideration.
5.6 Long-term evolution of the RZ
As the planet migrates inwards due to the IGWs it excites being absorbed at a critical
layer, ω increases, since n˙ > 0. If we assume that the tidal frequency increases from
ω1 at a time t1 to ω2 > ω1 at a slightly later time t2, then since ω2 > ω1, the waves
excited at t2 no longer see the critical layer for ω1 waves. If the change in frequency
is sufficiently small, we would still expect significant attenuation by the shear as the
waves approach the critical layer for ω1 waves. This would transfer angular momentum
from the waves to the mean flow, and may spin up the region near the original critical
layer to the pattern speed for ω2 waves, hence producing a critical layer for these waves.
If this process occurs, then a weak radial differential rotation profile could be set up
in the RZ, with dΩ(r)dr > 0. Similarly, if the planet migrates outwards for the case of
a rapidly rotating star, then a profile with dΩ(r)dr < 0 could be set up (which would be
unstable). If the flow does not have time to adjust to the change in frequency of the
forcing, and cannot spin up sufficiently to produce a critical layer for ω2 waves, then
the dissipation rate may be reduced. However, it seems plausible that the change in
the orbit will be gradual enough so that IGWs reaching the centre will be significantly
attenuated. This is because their radial wavelengths get Doppler-shifted downwards
by the shear, making them more susceptible to radiative damping.
We have so far restricted our investigation to m = 2 waves. If the orbit of the
planet is eccentric or inclined with respect to the stellar equator, then IGWs with
140
other m-values could be excited. If these break, then they could each have their own
critical layer. It would be interesting to study the effects of tidal forcing with several
different frequencies and m-values to see how this would affect the reflection of the
different waves. In addition, Rogers & Glatzmaier (2006) found that IGWs excited
by turbulent convection at the top of the RZ, which had 1 < m < 15 and typical
frequencies ω ∼ 20µHz, could reach the centre with sufficient amplitude to undergo
breaking and spin up the central regions. If this commonly occurs in solar-type stars,
then we might expect the core to be differentially rotating, even in the absence of
tidal forcing. For most m-values the resulting spin frequency of the central regions is
probably slower than that of the relevant pattern speed for tides raised by a close-in
planet, so we expect that this would minimally attenuate any tidally excited low-
m waves. Nevertheless, the interactions of multiple waves near the centre warrants
further study.
5.7 Tidal dissipation in F-stars
In this chapter we have discussed the implications of the formation of a critical layer
near the centre of a solar-type star, which can occur due to wave breaking. However,
many planets orbit F-type stars, which have convective cores and thinner OCZs. Our
calculation of Q′? for XO-3 in Chapter 2 indicates that it would be worthwhile to study
the range of Q′? expected for F-stars, and how these may differ from the solar-type
stars that we have considered so far in Part II of this thesis.
We have performed calculations of tidal dissipation in the OCZs of a variety of F-
stars between the masses of 1.2−1.6M¯, using the numerical method of OL04/OL07,
and the stellar models of EZ-Evolution. These stars contain convective cores sur-
rounded by a RZ and an OCZ. Since these stars are not stably stratified near the
centre, the process of wave breaking and critical layer formation will not occur in such
stars, because IGWs reflect from the core before they become nonlinear. This removes
the possibility of the efficient mechanism of tidal dissipation that we have so far dis-
cussed, with damping of the dynamical tide in the RZ being due only to inefficient
radiative diffusion (which is discussed further in §7.1). For this reason, we expect that
tidal dissipation in the OCZ will dominate the dissipation.
141
Figure 5.8: Q′? as a function of the ratio of tidal frequency to spin frequency ωˆ/Ω, from
dissipation of the l = m = 2 component (we find the behaviour of the m = 0 and 1
components is similar in magnitude) of the equilibrium tide, and dissipation of inertial modes,
in the OCZ of two F-type star models. The dashed lines represent the effect of omitting the
Coriolis force, and therefore inhibiting inertial waves. The top figure shows a high-mass
1.5M¯ F-star, with solar metallicity and an age of 0.7 Gyr, assumed to be spinning with
a period of 3 d. This has a very thin OCZ, and tidal dissipation is extremely weak. The
bottom figure shows a low-mass 1.2M¯ F-star, with solar metallicity and an age of 1.0 Gyr,
assumed to be spinning with a period of 3 d. Tidal dissipation is much stronger in this
star, though still smaller than that in Fig. 6 from OL07 for a 1.0M¯ G-star. Together with
Fig. 2.9 this shows that Q′? can vary considerably between stars, even within the range of
F-stars.
142
We consider the range of OCZ properties for F-stars, as the mass and metallicity
are varied. A selection of illustrative examples will now be presented, which represent
the range of properties expected for F-stars. Fig. 5.8 shows Q′? as a function of tidal
frequency for two F-stars, with different OCZ properties. The top panel in Fig. 5.8
shows that tidal dissipation in more massive F-stars, with very thin OCZs, is found
to be extremely weak, with Q′? & 10
12 for most tidal frequencies. This implies that
tidal dissipation in such stars is probably negligible in contributing to the spin-orbit
evolution of HJ systems. The bottom panel in Fig. 5.8 shows that tidal dissipation in
lower mass F-stars with thicker convective envelopes, is similar to but slightly weaker
than that for solar-type stars (OL07), since Q′? & 10
8 for most tidal frequencies (even
those in the range |ωˆ| < 2|Ω|).
A model with 1.3M¯ star similar to XO-3, except that we choose supersolar metal-
licity (Z = 0.03), is found to have a similar OCZ to the 1.2M¯ model in the bottom
panel of Fig. 5.8, and has a very similar Q′?. The metallicity of the star affects the
thickness of the OCZ, and therefore the efficiency of tidal dissipation.
From these examples, it is clear that assuming a single Q′? applies for all stars is
incorrect. Even within the mass range of F-stars there is considerable variation (for a
given ωˆ) in Q′? ∼ 10
8 − 1012 in our calculations, primarily as a result of the variation
in the mass fraction contained in the OCZ. Lower mass stars, and those with higher
metallicity, tend to have thicker OCZs than higher mass, low-metallicity stars. In
addition, higher mass stars are more centrally condensed, so the mass fraction in their
outer regions will be lower. This results in low-mass, high-metallicity stars having
lower Q′? then high-mass, low-metallicity stars. F-stars, in general, also have different
Q′? than solar-type stars, with radiative cores.
5.8 An explanation for the survival of all (currently
observed) short-period planets
Our most important result is that this mechanism can potentially explain the survival
of all short-period extrasolar planets around solar-type stars4 with masses in the range
4Contained in the catalogue at http://www.exoplanet.hanno-rein.de/ or
http://exoplanet.eu/catalog.php
143
0.5 ≤ m?/M¯ ≤ 1.1. From using the closest fit stellar model to each of these stars,
we find that no planet clearly satisfies Eq. 5.3.3 with a period P . 3 d (planets much
further out may satisfy the criterion, though this tidal effect is then unimportant). All
of these planets are insufficently massive (or orbit sufficiently young stars, with low
values of C/C¯) that they are unlikely to cause wave breaking at the centre of their
hosts. The dominant mechanism of tidal dissipation in these stars is therefore likely
to be damping of the equilibrium tide by turbulent convection. This is likely to be
relatively inefficient for these periods, since the turbulent viscosity must be reduced
when the orbital period is shorter than the convective timescale, as we discussed in
§ 1.4 (see e.g. Zahn 1966; Goldreich & Nicholson 1977; Goodman & Oh 1997; Penev
et al. 2007). This could be an important explanation for the survival of these planets.
As an example, WASP-19 b (Hebb et al. 2010) is the latest candidate for the
shortest period massive transiting planet. This planet has mass mp = 1.15MJ , in an
orbit of P = 0.78 d, around a star of mass m? ≈ 0.95M¯, and hence will contain
a radiative core. The stellar age is poorly constrained, but we find that the IGWs
excited by this planet will not have sufficient amplitude to cause wave breaking at the
centre of the star, for all reasonably-aged stellar models of a similar mass star. This
means that the planet will not be subject to accelerating tidal decay through critical
layer absorption, perhaps explaining its survival. Constraints on the tidal Q′? for this
system would then give us information on alternative mechanisms of tidal dissipation,
such as the dissipation of the equilibrium tide by turbulent convection. The existence
of this planet on its observed orbit can be explained through the absence of wave
breaking at the centre of its star.
The results of this thesis also allow us to propose an explanation for the survival
of planets around F-type stars, with convective cores. This is because in these stars,
the mechanism outlined in Part II of this thesis will not work in such stars, regardless
of the mass of the planet. The reason for this is that IGWs do not propagate in
convective regions, and will reflect from the top of the convective core before they
reach the centre. Together with our results in § 5.7, this allows us to explain the
survival of planets such as WASP-18 b (Hellier et al. 2009), WASP-12 b (Hebb et al.
2009) and OGLE-TR-56 b (Sasselov 2003). Taking the current values for the stellar
144
properties of the host stars in these systems, we find that they are each likely to have
a convective core, which will prevent IGWs from reaching the centre of each star. In
addition, these stars are slowly rotating, and so the relevant tidal frequencies are likely
to be outside the range of inertial waves (|ωˆ| > 2|Ω|), which would imply that such
waves are not excited by tidal forcing. This means that the resulting Q′? is due only to
damping of the equilibrium tide, which from Fig. 5.8 (in the limit in which |ωˆ| À |Ω|)
is seen to be much larger than its value when inertial waves are excited. Therefore it
is likely that the relevant Q′? & 10
10 in these stars, which would imply that tidally-
induced inspiral will not occur within the age of these systems. Weak dissipation in
the star could therefore explain the survival of these planets, which all orbit F-stars.
Our mechanism can also explain why the most massive short-period planets (which
still have smaller or comparable moments of inertia in the orbit as the RZ of the
star), such as WASP-18 b (Hellier et al., 2009), WASP-14 b (Joshi et al., 2009) or
CoRoT-14 b, are exclusively found around F-stars, which have convective cores, and
in which critical layer formation induced by wave breaking at the centre is unable to
operate. Note, however, that very massive perturbers around solar-type stars may have
sufficient orbital angular momentum to be able to synchronise their stars and reach an
approximate tidal equilibrium state, neglecting stellar magnetic braking, which would
prevent orbital decay, even for planets in orbit around a solar-type star. This may be
an explanation for the synchronisation of close-binary stars (e.g. Mazeh 2008).
Our results make several predictions, which will be tested by ongoing and future
studies of transiting planets, such as WASP and Kepler. In particular, we make the
prediction that fewer massive planets in the range 3 . MJ . 20, in orbits with
P . 2− 3 d, that satisfy Eq. 5.3.3, will be found around solar-type stars.
5.9 Conclusion
In Part II of this thesis, a mechanism of tidal dissipation has been studied in which
tidally excited IGWs break at the centre of a solar-type star, and form a critical
layer. In this chapter, we considered the implications of this process for the survival
of short-period planets around solar-type stars. We computed numerically the lin-
earised adiabatic tidal response, in order to calculate the angular momentum flux of
145
the waves launched at the CZ/RZ interface. A semi-analytic model of this quantity
was presented, and its validity was verified by numerical integrations.
We computed the tidal Q′? that results from this process in an extensive set of
stellar models. We found that Eq. 5.4.2 is a quite robust estimate, which was found
to vary by no more than a factor of 5 throughout the range of G and K stars, within
the mass range 0.5− 1.1M¯, at any stage in their main-sequence lifetime, for a given
planetary orbit. This results in a relatively rapid inspiral for planets on short-period
orbits, if a critical layer is produced.
One important result of this work is to highlight the importance of the internal
structure of the star on the magnitude of the tidal dissipation. We briefly discussed
the tidal dissipation expected in F-stars, which contain convective cores. There are
differences between the dissipation in G and F stars, due to differences in their internal
structure – most notably the presence of a radiative core in a G-type star, which
support IGWs, enabling more efficient dissipation if wave breaking (and critical layer
formation) occurs. Taken together with a result of OLO7, that the spin period of the
star also affects Q′?, we can conclude that it is certainly incorrect to assume Q
′
? is
identical for all stars.
We provided an explanation for the survival of all currently observed short-period
planets around F, G or K stars, as a result of weak tidal dissipation in the star. Around
G and K-type stars, planets can survive if their masses are insufficient to cause wave
breaking at the centre (and therefore to satisfy Eq. 5.3.3). The stars are also slowly
rotating, so it is likely that damping of the equilibrium tide by turbulent convection is
the dominant mechanism of tidal dissipation, which has previously been found to be
inefficient in the appropriate period range (e.g. Goldreich & Nicholson 1977; Goodman
& Oh 1997). Around F-stars, planets can survive because the process of wave breaking
will be prevented by reflection of the tidally excited IGWs from the convective core
before they become nonlinear, together with weak dissipation of the equilibrium tide
in the thin OCZ, as long as the stars are slowly rotating.
We predict that there will be very few massive planets (with masses . 20MJ)
orbiting G-stars (which satisfy Eq. 5.3.3) in orbits of less than 2 days, due to the rapid
inspiral that results from critical layer absorption near the centre. We also predict that
146
the population of short-period HJs around G-stars will decrease with main sequence
age, as a result of the increasing central condensation of the star, making breaking
more likely. A corollary of these predictions is that there should be more massive
planets in short-period orbits observed around F-stars. This appears to be the case
from the current sample of extra-solar planets.
147
Chapter 6
Stability analysis of a standing
internal gravity wave near the
centre of a solar-type star in 2D
In Chapters 3 and 4 we have presented the results of numerical simulations of IGWs
approaching the centre of a solar-type star, in both two and three dimensions. We
found that the waves break if they overturn the isentropes near the centre, but they
were not observed to undergo instability for smaller amplitudes. It has been proposed
that parametric instabilities could be important for smaller amplitude waves (e.g. Ku-
mar & Goodman 1996; GD98). Our simulations may not have the spatial resolution or
have a sufficiently long run time for these instabilities to be observed. In this chapter
we perform a stability analysis of our 2D exact wave solution. This work has two aims:
to determine any instabilities that set in for small amplitude waves, and to understand
the breaking process in more detail. Our results are compared with the stability of a
plane IGW in a uniform stratification, and with previous work by Kumar & Goodman
(1996) on a similar problem to our own. The astrophysical relevance of our results is
then discussed.
148
6.1 Introduction
Many stability analyses of a plane IGW in Cartesian geometry with a uniform stratifi-
cation have been performed (e.g. McEwan & Robinson 1975; Meid 1976; Drazin 1977;
Klostermeyer 1982). These indicate that a monochromatic propagating plane IGW
is always unstable to parametric instabilities, whatever its amplitude, in the absence
of diffusion. In that problem such analyses were made possible for finite-amplitude
(in addition to infinitesimal amplitude) waves because the solution is exact. This is a
consequence of the fact that k · u = 0, implying that the advective operator u · ∇ an-
nihilates any disturbance belonging to the same plane wave. These stability analyses
allow a detailed understanding of the initial stages of the breaking process for these
waves (e.g. Drazin 1977; Klostermeyer 1982; Lombard & Riley 1996).
When a small perturbation is added to a basic plane wave, the resulting evolu-
tionary equations have periodic coefficients. This allows the possibility for parametric
instability to occur. The first study of this problem was by McEwan & Robinson
(1975), who considered perturbations with length scales much smaller than the pri-
mary IGW wavelength, in which case the problem can be reduced to the solution of
Mathieu’s equation. The motion of the fluid in the basic wave gives rise to unstable
modes, just as parametric oscillations of a pendulum are excited by periodic changes
of its length. The growth rates of these parametrically unstable modes increases (lin-
early) with the amplitude of the basic wave.
Subsequent analyses expanded the perturbation onto a Floquet basis, and relaxed
the small-scale assumption. These studies all find that in a dissipationless fluid, the
disturbances with the largest growth rates have the smallest spatial scales (e.g. Drazin
1977; Klostermeyer 1982). In viscous or radiative fluids, dissipative effects scale with
the inverse square of the length scale of a given mode. This means that the most
unstable wavelengths will no longer be those of the smallest spatial scale, but will be
those for which the competing effects of dissipation and (nonlinear) growth favour the
latter, and this will depend on the Reynolds number (also the Prandtl number when
radiative diffusion is included).
Lombard & Riley (1996) performed a detailed stability analysis of a plane IGW
which demonstrated that the instability that contributes to wave breaking is driven by
149
a combination of wave shear and wave entropy gradients. They find that wave-wave
resonance interactions are the primary mode of instability for small amplitude waves,
with the picture being much more complicated near overturning amplitudes. However,
no difference in the type of instability is found for waves that do and do not overturn
the stratification for some wave phase.
In our problem we have obtained an exact 2D standing wave solution representing
IGWs near the centre of a solar-type star. This enables us to perform a stability
analysis of this wave for any amplitude. This is the subject of the present chapter. One
important difference between our problem and previous studies is that the nonlinearity
is spatially localised in the innermost wavelengths, whereas the nonlinearity is present
everywhere in the plane IGW problem.
In the centre of a star, viscous (molecular) damping is negligible, and the dominant
linear dissipation mechanism is radiative diffusion, which is too inefficient to prevent
the excitation of waves with scales not much smaller than the primary wavelength.
This is true for the waves excited by planets orbiting solar-type stars with several-
day periods. This means that parametrically excited modes with scales much shorter
than the primary wave could be excited. These will then be damped by diffusion but
not before they can draw energy the primary wave, and possibily contribute to wave
breaking.
We have already demonstrated through direct numerical simulations that a wave
with sufficient amplitude to overturn the stratification undergoes a rapid instability
(with a growth time on the order of a wave period) which leads to wave breaking. We
found that the wave overturns the stratification during part of its cycle if the angular
velocity in the wave exceeds the angular pattern speed of the forcing, i.e. uφ/r & Ωp.
In these 2D simulations, the wave reflects perfectly from the centre of the star if its
amplitude is insufficient to satisfy this criterion, and long-term integrations do not
show that any instabilities act on the waves. The picture in 3D is very similar. In
this chapter we perform a weakly nonlinear stability analysis of our 2D wave solution
using a Galerkin spectral method. This work has two main aims:
• to better understand the early stages of the breaking process for large amplitude
waves,
150
• to determine what (if any) instabilities may set in for waves that are unable to
overturn the isentropes at any location in the wave.
The motivation for this study is that if the waves are subject to parametric insta-
bilities, whatever their amplitudes, the reflection of waves from the centre of the star
will not be perfect. This would stand in contrast to the prediction from linear theory,
and the results of our numerical simulations. The simulations performed thus far may
not have the spatial resolution or have long enough run time to be able to capture
small-scale parametric instabilities. If they indeed occur, and the tidally excited waves
are weakly nonlinearly damped by parametric instabilities, this could contribute to the
tidal dissipation, and have implications for the survival of short-period planets with
insufficient masses to cause breaking.
6.2 Internal gravity wave stability analysis
We start with the dimensionless Boussinesq-type system Eqs. 3.4.29–3.4.32, previously
derived in Chapter 3. We consider a circular region with r ∈ [0, rout], taking rout =
1, which is an impermeable outer boundary at constant entropy, i.e., ψ(1, φ, t) =
b(1, φ, t) = 0, to confine the modes. Our inner boundary condition is a regularity
condition1 at r = 0, which is implicitly enforced through our choice of basis functions
for the perturbation to the wave, as we describe below.
We use dimensionless units such that the unit of length [L] = rout, the unit of time
[T ] = Nout = C−1r
−1
out, and take C = 1. We then express the velocity field in terms of
a (real) streamfunction ψ, defined by
ur =
1
r
∂φψ, (6.2.1)
uφ = −∂rψ, (6.2.2)
which automatically enforces the solenoidality constraint on the velocity. We consider
a stationary, stably stratified background containing a nonlinear wave (denoted by
1However, in the computation of the table of integrals described below, this is replaced by an inner
boundary at rin = 10−4, to avoid the coordinate singularity at the origin.
151
subscript w), subject to a perturbation (denoted by primes). That is, we expand
b = bw + b
′, (6.2.3)
ψ = ψw + ψ
′. (6.2.4)
To eliminate the modified pressure variable q, we take the curl of the momentum
equation
∂t(∇× u) = ∇× rb−∇× (u · ∇u), (6.2.5)
which can be expanded to first order in the perturbed quantities. The z-component
of this linearised equation expressed in terms of the streamfunction, together with the
buoyancy equation, is
∂t(−∇
2ψ′) + ∂φb
′ = J(ψw,−∇
2ψ′) + J(ψ′,−∇2ψw) (6.2.6)
∂tb
′ + ∂φψ
′ = J(ψw, b
′) + J(ψ′, bw), (6.2.7)
which is two equations for two unknowns (ψ′, b′). The vorticity perturbation is ζ ′ =
−∇2ψ′. The nonlinearities in this system provide coupling between different waves.
We neglect the terms J(ψ′,−∇2ψ′) and J(ψ′, b′), which is consistent with our weakly
nonlinear approach.
6.2.1 Exact primary wave solution
In the frame in which the wave is steady and φ is the azimuthal coordinate, our primary
wave is
ψw = Re
[
4
k32,np
AJ2(k2,npr)e
2iφ
]
=
2
k32,np
[
AJ2(k2,npr)e
2iφ + A∗J2(k2,npr)e
−2iφ
]
, (6.2.8)
which is an m = 2 wave with np radial nodes (to be chosen later), where A ∈ C, in
general, and is time-independent in this frame. From here on, we take A ∈ R, without
loss of generality. Note that bw = k2,npψw and ∇
2ψw = −k22,npψw. We define k2,np
such that J2(k2,np) = 0. This is equivalent to confining the primary wave in a circular
152
region of unit radius with an impermeable outer boundary at constant entropy. We
plot this wave with np = 4 in Fig. 6.1. Note that we define np so that np = 2 when
there are no zeroes except at the boundaries, which is a nonstandard definition.
φ
r
ψ
 
 
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
−0.02−0.015
−0.01−0.005
00.005
0.010.015
0.02
Figure 6.1: Streamfunction of the primary wave ψw for np = 4, with an arbitrary amplitude
for illustration. Contours of constant ψw are the streamlines of the primary wave flow. The
flow goes clockwise around the red streamlines and anticlockwise around the blue streamlines.
Stagnation points are located at the radial nodes r where J2(k2,npr) = 0, at azimuthal
locations φ = (2n+1)4 pi for n ∈ Z.
This wave overturns the stratification when ∂rs < 0, where s = (1/2)r2 + bw. This
is equivalent to 1r∂rbw < −1. Note that overturning occurs only when A > 1, and is
more likely for waves with large n and small m values. The size of the convectively
unstable region can be illustrated for a given A and np, by calculating when
N2 = g∂rs = r(r + ∂rb) (6.2.9)
= r2 + Re
[
2A
k2,np
r
(
J1(k2,npr)− J3(k2,npr)e
2iφ
)
]
< 0, (6.2.10)
(where we take cp = 1 for convenience). For illustration, we plot the 2D region that is
convectively unstable for several A values when np = 4 in Fig. 6.2. An approximate
size for the overturning region for small r when A > 1 is
rov ≈
√
6
k22,np
(
1−
1
A
)
, (6.2.11)
153
at the most unstable wave phase. When A = 1 the overturning region is the point
r = 0, with the region expanding for larger A. If the instabilities that cause wave
breaking are convectively driven, we would expect them to be strongly localised within
the convectively unstable regions of the primary wave.
φ
r
log10 Re[√ N 2]
 
 
0 1 2 3 4 5 600.2
0.40.6
0.81
−4−3.5
−3−2.5
−2
φ
r
log10 Re[√ N 2]
 
 
0 1 2 3 4 5 600.2
0.40.6
0.81
−3.5−3
−2.5−2
−1.5−1
−0.5
Figure 6.2: Spatial exent of the region that is made convectively unstable by the primary
wave for A = 1.1 (top) and 10 (bottom), with np = 4. This regions expands from the point
r = 0 when A = 1 to encompass the innermost few wavelengths for larger A.
6.2.2 Infinitesimal perturbations
We consider linear pertubations to this finite-amplitude primary wave, which we ex-
pand as (dropping the primes from now on)
ψ =
∞∑
m=−∞
∞∑
n=2
ψm,n(t)Jm(km,nr)e
imφ, (6.2.12)
b =
∞∑
m=−∞
∞∑
n=2
bm,n(t)Jm(km,nr)e
imφ, (6.2.13)
154
where km,n is chosen such that the solutions for each azimuthal wavenumber m, and
radial node number n ≥ 2, satisfy the outer boundary condition at r = 1, for which
Jm(km,n) = 0. (6.2.14)
This condition forces km,n ∈ R, for |m|, n ∈ Z+. The above expansion automatically
enforces a regularity condition on the perturbations at r = 0. Eqs. 6.2.12 and 6.2.13
define our Galerkin basis. This basis is adopted for two reasons: the linear solutions
(A = 0) take the same form, and it automatically ensures that the total energy of the
perturbations is conserved, since the energy flux through the boundaries is always zero
(because ur = b = 0 at r = 0, 1).
Note that our spectral-space amplitudes ψm,n(t), bm,n(t) ∈ C, so we must take the
real part at the end of the calculation to obtain physical quantities. For each m,
there is an infinite number of components with different values of n. In our spectral
representation of the solution, we truncate these infinite series at 1−M ≤ m ≤M−1,
where M is an odd number, and n = N . This truncation is chosen so that we have
an exactly equal number either side of m = 0 (and a similar number either side of the
primary wave m = 2) which ensures that our mathematical realisation of the problem
has the symmetry property that we discuss in § 6.5.3 below.
6.2.3 Derivation of the evolutionary equations
Evolutionary equations for the amplitudes ψm,n(t) and bm,n(t) can be derived by pro-
jection through integration onto the Galerkin basis. To do this we substitute the
above expansions into the weakly nonlinear system defined by Eqs. 6.2.6–6.2.7. An
important orthogonality relation is
∫ 2pi
0
∫ 1
0
rJm(km,n′r)Jm(km,nr)e
i(m−m′)φdrdφ = pi [Jm+1(km,n)]
2 δn,n′δm,m′ , (6.2.15)
where δ is the Kronecker delta. Note that this results in different normalisation factors
for each m and n wave. Also note that
−∇2
(
Jm(km,nr)e
imφ
)
= k2m,nJm(km,nr)e
imφ. (6.2.16)
155
6.2.4 Linear solutions (in the absence of the primary wave)
Consider J(. . . , . . . ) = 0, which is equivalent to having a hydrostatic background
with no primary wave flow. If we substitute the expansions Eq. 6.2.12–6.2.13 into
Eqs. 6.2.6–6.2.7, and then multiply by rJm′(km′n′r)e−im
′φ, and finally integrate over
φ ∈ [0, 2pi] and r ∈ [0, 1], we obtain
k2m,nψ˙m,n + imbm,n = 0 (6.2.17)
b˙m,n + imψm,n = 0 (6.2.18)
for each m,n, after redefining m′ → m, and n′ → n after the integration. This system,
together with the boundary conditions, can be solved to give
ψm,n = Am,ne
−iωm,nt, (6.2.19)
which are oscillations if Am,n ∈ C, with frequency ωm,n = m/km,n in the inertial frame.
This is the dispersion relation for modes confined to a circular region of unit radius,
and is the frequency of a non-interacting wave which is able to exist in the container in
the absence of any primary wave flow, which can take any value in the range [0, 1] (if we
do not truncate the Galerkin basis at some finite values of N andM , then these modes
would be dense in this frequency interval), since the maximum buoyancy frequency
Nmax = 1. In the frame rotating with the pattern speed of the primary wave with
angular frequency ωp/2 = 1/k2,np , the Doppler-shifted frequency is ωˆm,n = ωm,n − ωp.
Note that km,n increases with both m and n, but ωm,n decreases with n and increases
with m. When substituting the above solution back into Eq. 6.2.12–6.2.13, we obtain
the linear solutions of the system, which are Bessel functions of a given order m with
n nodes in the radial direction. This motivated our choice of Galerkin basis.
6.2.5 Nonlinear terms
We obtain our weakly nonlinear system from Eqs. 6.2.6–6.2.7 through the same ap-
proach as in the previous section, to obtain for each m and n,
k2m,nψ˙m,n + imbm,n =
1
pi [Jm+1(km,n)]
2
∫ 2pi
0
∫ 1
0
{
rJm(km,nr)e
−imφ
156
×
[
J(ψw,−∇
2ψ) + J(ψ,−∇2ψw)
]
}
drdφ, (6.2.20)
b˙m,n + imψm,n =
1
pi [Jm+1(km,n)]
2
∫ 2pi
0
∫ 1
0
{
rJm(km,nr)e
−imφ
× [J(ψw, b) + J(ψ, bw)]
}
drdφ, (6.2.21)
where the Jacobians contain sums over n′ and m′. The sum over m′ is eliminated to a
pair of terms through the φ integration, using Eq. 6.2.15. A set of coupling integrals of
triple products of Bessel functions also results, for which there is a sum of such terms
over n′, i.e., an m,n wave is coupled through nonlinear terms to waves with m±2 and
(in principle) all node numbers n ∈ {2, . . . ,∞}. The system reduces to
k2m,nψ˙m,n + imbm,n =
∞∑
n′=2
{
αm,n,n′
(
k2m−2,n′ − k
2
2,np
)
A˜ψm−2,n′
+βm,n,n′
(
k2m+2,n′ − k
2
2,np
)
A˜∗ψm+2,n′
}
, (6.2.22)
b˙m,n + imψm,n =
∞∑
n′=2
{
αm,n,n′A˜
(
bm−2,n′ − k2,npψm−2,n′
)
+βm,n,n′A˜
∗
(
bm+2,n′ − k2,npψm+2,n′
)
}
, (6.2.23)
where
A˜ =
2
k32,np
A. (6.2.24)
The coupling coefficients are
αm,n,n′ =
2
[Jm+1(km,n)]
2
︸ ︷︷ ︸
normalisation
(
i(m− 2)I1m,n,n′ − 2iI
2
m,n,n′
)
, (6.2.25)
βm,n,n′ =
2
[Jm+1(km,n)]
2
︸ ︷︷ ︸
normalisation
(
i(m+ 2)I3m,n,n′ + 2iI
4
m,n,n′
)
, (6.2.26)
with the integrals
I1m,n,n′ =
∫ 1
0
Jm(km,nr)
[
∂rJ2(k2,npr)
]
Jm−2(km−2,n′r)dr, (6.2.27)
I2m,n,n′ =
∫ 1
0
Jm(km,nr)J2(k2,npr) [∂rJm−2(km−2,n′r)] dr, (6.2.28)
157
I3m,n,n′ =
∫ 1
0
Jm(km,nr)
[
∂rJ2(k2,npr)
]
Jm+2(km+2,n′r)dr, (6.2.29)
I4m,n,n′ =
∫ 1
0
Jm(km,nr)J2(k2,npr) [∂rJm+2(km+2,n′r)] dr. (6.2.30)
Note that these are related by
I3m,n,n′ = I
1
m+2,n′,n , (6.2.31)
I4m,n,n′ = I
2
m+2,n′,n − I
1
m+2,n′,n . (6.2.32)
For use in the derivation of the spectral space energy equation in a subsequent section,
we find it convenient to define
α˜m,n,n′ = −ipi [Jm+1(km,n)]
2 αm,n,n′ , (6.2.33)
and similarly for β˜m,n,n′ . This is because we then have the relation
β˜m,n,n′ = α˜m+2,n′,n. (6.2.34)
6.2.6 Diffusive terms
In the presence of viscosity and radiative diffusion (or hyperdiffusion) Eqs. 6.2.6–6.2.7
become
∂t(−∇
2ψ) + ∂φb = J(ψw,−∇
2ψ) + J(ψ,−∇2ψw) + (−1)
2+αν∇2+2αψ, (6.2.35)
∂tb+ ∂φψ = J(ψw, b) + J(ψ, bw) + (−1)
1+ακ∇2αb, (6.2.36)
where α is chosen to give the standard diffusive operator (α = 1), or hyperdiffusion
(α = 2, 3). In this case we obtain the (linearised) system
k2m,nψ˙m,n + imbm,n = −νk
2+2α
m,n ψm,n, (6.2.37)
b˙m,n + imψm,n = −κk
2α
m,nbm,n, (6.2.38)
instead of Eq. 6.2.17–6.2.18. The dispersion relation is then
(
ωm,n + iνk
2α
m,n
) (
ωm,n + iκk
2α
m,n
)
=
m2
k2m,n
, (6.2.39)
158
indicating that the frequencies of the allowed solutions are modified in the presence of
ν, κ. The growth rate expected in the presence of weak diffusion is therefore modified
to be Im [ω]− 12(ν+κ)k
2α
m,n, with Im [ω] being the appropriate inviscid growth rate. To
obtain growing modes, diffusion must be sufficiently weak so that it does not overcome
nonlinear growth. Hyperdiffusion with α = 3 is adopted since it better restricts the
dissipation to the highest wavenumbers. This enables numerical convergence in the
eigenvalue problem discussed below, but does not significantly perturb the growth rates
of the lower wavenumber eigenmodes with the values of ν, κ that we adopt. From here
on, we also take ν = κ.
With the inclusion of diffusive terms we require 1+2α additional boundary condi-
tions at each boundary. These are regularity conditions at the centre, and at the outer
boundary we can consider a variety of boundary conditions of the form ∇2nψ = 0
for n = 0, 1, . . . , α and ∇2nb = 0 for n = 0, 1, . . . , α − 1. These are automatically
satisfied by our Galerkin basis Eq. 6.2.12–6.2.13 and the definition of km,n. With these
boundary conditions (which force km,n to be real), the Galerkin basis is therefore ex-
act for the single-wave diffusive problem. Note that this is also true if hyperdiffusion
is adopted, due to property Eq. 6.2.16. In the numerical solution of the eigenvalue
problem in the following sections we include these diffusive terms to achieve numerical
convergence. This is necessary because in the absence of diffusion, the most unstable
modes are found to prefer the smallest spatial scales.
One important difference between this problem and that discussed in Chapter 7 is
that in that case the solution with viscous and radiative damping is no longer exact.
This makes sense physically because we are damping the wave, but the difference can
be mathematically explained as follows. In that problem we force a particular wave,
i.e., choose a wave with ω ∈ R. With diffusion km,n ∈ C, so that Jm(km,nr) ∈ C, in
general. Since the real part of this is taken before computing the nonlinear terms, the
property Eq. 6.2.16 no longer implies that the nonlinear terms vanish. In this section,
we are forcing km,n ∈ R through the boundary conditions, thus each component of
our Galerkin basis is exact with viscous and radiative damping, and the corresponding
solutions will have a nonzero decay rate as a result of these diffusive processes. This
further supports our choice of the Galerkin basis Eq. 6.2.12–6.2.13.
159
6.3 Method of solution
Our weakly nonlinear system Eqs. 6.2.20–6.2.21 can be written in the form of a gen-
eralised eigenvalue problem of the form
AU = ωBU, (6.3.1)
where U is the column vector whose components are the quantities (ψm,n, bm,n) for
each m and n. A is the block tridiagonal matrix representing the system. This is done
by seeking normal mode solutions of the form ψm,n(t) ∝ e−iωt for each m and n, and
similarly for bm,n(t). B is the matrix that can be represented as
B = −i


k2m,n 0
0 1

 , (6.3.2)
for a single m and n. We solve this problem using standard generalised eigenvalue
solver routines, such as ZGGEV in the LAPACK libraries. This returns the eigen-
values {ω}, and the spectral space eigenfunctions {ψm,n, bm,n} corresponding to each
eigenvalue. The real space eigenfunctions can be reconstructed from these, using
Eqs. 6.2.12–6.2.13.
We choose N = 50 and M = 27. With our choice of α = 3 hyperdiffusion, it has
been found from preliminary investigation that 10−11 ≤ ν ≤ 10−14 is appropriate. This
hyperdiffusion is found to give the numerical junk eigenvalues, whose eigenfunctions
oscillate at the smallest scales, a large decay rate, and allows our growing modes to
be adequately converged for the values of A and np that we consider.
6.3.1 Numerical computation of table of integrals
The tables of integrals defined by Eqs. 6.2.27–6.2.30 for each value of m,n, n′ are
computed using a 4th/5th order adaptive step Runge-Kutta integrator. To enable
efficient computations, the Bessel functions are computed simultaneously with the
integrals. This is desirable because using built-in Mathematica or Matlab routines for
computing these functions is extremely slow when they are evaluated at every step.
160
Note that Bessel’s equation
∂r(r∂rψ) + r
(
k2m,n −
m2
r2
)
ψ = 0, (6.3.3)
can be rewritten as the coupled set of first order ODEs
dξ
dr
= r
(
m2
r2
− k2m,n
)
ψ, (6.3.4)
dψ
dr
=
ξ
r
. (6.3.5)
We also need the derivatives of various Bessel functions, so we compute
d2ψ
dr2
=
(m2 − r2k2m,n)ψ − ξ
r2
, (6.3.6)
for each r. To compute the integrals, we solve
dIi
m,n,n′
dr for i ∈ {1, 2, 3, 4}. This involves
solving a system of 15 ODEs in total for each m,n, n′. Using our chosen resolution
of N = 50, M = 27, this involves the computation of M(N − 1)2 ∼ 105 integrals in
total. These are computed for a given number of radial nodes in the primary wave
in the range 2 ≤ np ≤ 12. We impose an inner boundary at rin = 10−4 to avoid
the coordinate singularity at the origin, and use initial conditions appropriate from
considering the asymptotic behaviour of the Bessel functions. For small N and M
values the numerical integrals have been checked to agree with those computed from
Mathematica, and for large N and M , several integrals containing the highest n and
m values were also checked. We use a relative error tolerance of 10−13, which has been
found to compute the most oscillatory integrals (corresponding to the highest n and
m value) accurately (compared with Mathematica) to within at least 6 decimal places.
6.4 Kinetic and potential energy equations
The kinetic and potential energies can be computed from either the real-space or
spectral-space eigenfunctions. This will enable a determination of the dominant source
of free energy driving the instability (e.g. Lombard & Riley 1996), and will also pro-
vide an independent calculation of the growth rate, which can be used to check our
numerical code. We derive an energy equation in spectral space, and compute the
161
volume-integrated terms using the numerically computed eigenfunctions, without con-
verting to real space. This has been found to reduce numerical errors, resulting from
large numerical cancellations in the most oscillatory Bessel functions, when the energy
equations are instead computed in real space. In addition, it is simpler to construct
the hyperdiffusion terms in spectral space, so that they can be fully taken into account
in the energy budget.
We define K = 12 |u|
2, P = 12b
2 and E = K + P , as the kinetic, potential and total
energy densities of the disturbance, respectively. We define a volume (strictly an area,
since we are in 2D) integral by
〈X〉 =
∫ 2pi
0
∫ 1
0
Xrdrdφ. (6.4.1)
Note that
〈K〉 =
1
2
∞∑
m=−∞
∞∑
n=2
k2m,n|ψm,n|
2pi [Jm+1(km,n)]
2 , (6.4.2)
〈P 〉 =
1
2
∞∑
m=−∞
∞∑
n=2
|bm,n|
2pi [Jm+1(km,n)]
2 , (6.4.3)
〈E〉 = 〈K〉+ 〈P 〉. (6.4.4)
Evolutionary equations for the volume-integrated energy can be obtained from Eq. 6.2.22
& 6.2.23. After some rearrangement, these can be written
˙〈K〉 = 〈Nsw〉+ 〈Fb〉+ 〈Fν〉, (6.4.5)
˙〈P 〉 = 〈Nbw〉 − 〈Fb〉+ 〈Fκ〉, (6.4.6)
˙〈E〉 = 〈Nsw〉+ 〈Nbw〉+ 〈Fν〉+ 〈Fκ〉, (6.4.7)
where
〈Nsw〉 = Re
{
∞∑
m=−∞
∞∑
n=2
∞∑
n′=2
iα˜m,n,n′
(
k2m−2,n′ − k
2
m,n′
)
A˜ψ∗m,nψm−2,n′
}
, (6.4.8)
〈Nbw〉 = Re
{
∞∑
m=−∞
∞∑
n=2
∞∑
n′=2
iα˜m,n,n′A˜k2,np
(
b∗m,nψm−2,n′ + ψ
∗
m,nbm−2,n′
)
}
, (6.4.9)
〈Fb〉 = Re
{
∞∑
m=−∞
∞∑
n=2
imψ∗m,nbm,npi [Jm+1(km,n)]
2
}
, (6.4.10)
162
〈Fν〉 = Re
{
∞∑
m=−∞
∞∑
n=2
−νk2+2αm,n |ψm,n|
2pi [Jm+1(km,n)]
2
}
, (6.4.11)
〈Fκ〉 = Re
{
∞∑
m=−∞
∞∑
n=2
−κk2αm,n|bm,n|
2pi [Jm+1(km,n)]
2
}
. (6.4.12)
〈Nsw〉 represents the production of perturbation kinetic energy from the primary wave
shear. 〈Nbw〉 represents the production of perturbation potential energy from the
primary wave entropy gradients. Whichever of 〈Nsw〉 or 〈Nbw〉 is dominant tells us
whether this instability is driven by the wave shear or the wave entropy gradients.
〈Fb〉 is the buoyancy flux term, representing conversion between kinetic and potential
energies of the disturbance.
After truncation at |m| = M and n = n′ = N , each of these terms are computed
from the spectral space eigenfunctions, together with the numerically computed table
of integrals. The growth rate can then be computed from
Im[ω′] =
˙〈E〉
2〈E〉
=
˙〈K〉
2〈K〉
=
˙〈P 〉
2〈P 〉
. (6.4.13)
We have checked that each of these equations are accurately satisfied to within at
most a few percent for each of the unstable modes discussed in this chapter. This is
an excellent check of our analytical derivations and numerical calculations, and should
convince ourselves that our results are consistent.
6.5 Numerical tests
In this section we briefly mention several numerical tests which we have performed to
validate our numerical code. Following this section, in § 6.6 and 6.7 we discuss the
results of our stability analysis for waves with A < 1 and A > 1, respectively.
6.5.1 Linear
In the absence of nonlinear couplings (A = 0), we obtain a set of non-interacting
modes with eigenfreqencies ω ∈ {ωm,n}, where ωm,n = m/km,n (in the inertial frame),
as we predicted in § 6.2.4. In the absence of diffusion, these have zero growth rate,
i.e., Im[ω] = 0, for all eigenmodes. When hyperdiffusion is included, the eigenmodes
163
each have a nonzero decay rate determined by the values of ν and κ, which is very
accurately (to more than 10 decimal places) computed from considering only the terms
〈Fν〉 and 〈Fκ〉 in the energy equation. The real-space eigenfunctions that result are
what is predicted from linear theory, in that they are Bessel functions of order m with
n nodes in the radial direction.
6.5.2 Weakly nonlinear
We followed a typical eigenmode as A is gradually increased from zero, and found
that for |A| ¿ 1, the shift in the eigenfrequency Re[δω] ∝ A2, as we would expect for
modes not undergoing parametric resonance (e.g. Landau & Lifshitz 1969).
6.5.3 Symmetries
The real-space solutions can be represented in the form
Re
[
∑
m
∑
n
cm,ne
i(mφ−ωt)
]
. (6.5.1)
This is symmetric under the transformations cm,n → c∗m.n, m → −m and ω → −ω
∗.
This symmetry should exist for all primary wave amplitudes, and results from the fact
that only the sign of the pattern speed ω/m has meaning, and not the sign of the
wavenumber or frequency. This means that when the eigenvalues are plotted on the
complex frequency plane, they should be symmetric about Re [ω] = 0.
6.6 Results for waves with A < 1: parametric in-
stabilities
We examine the unstable modes that exist when 0 < A < 1, which is when the primary
wave does not overturn the stratification at any location in the wave. The instability is
a parametric instability, for which a simple model is briefly reviewed in Appendix A.7.
When A 6= 0, the fraction of eigenmodes that are growing is nonzero (above a critical
A set by the values of np and ν, which can be understood from Eq. A.7.7) and increases
with A, as nonlinear growth starts to dominate over the decay due to diffusion for a
164
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80
0.5
1
1.5
2
2.5x 10−3
Re[ω] /ωp
Im[ω]/ω
p
 
 1e−121e−131e−14
Figure 6.3: Distribution of unstable eigenvalues on the complex frequency plane forA = 0.1,
np = 4 and ν = 10−12, 10−13, 10−14.
larger number of modes. For small A, the eigenvalues of the unstable modes displayed
on the complex plane are distributed in two curves that are approximately equidistant
from the origin at each location. This is illustrated in Fig. 6.3, and is a result of the
symmetry described in § 6.5.3.
In the limit that A→ 0, the unstable modes can be identified as the parametrically
excited free-wave modes. These are a pair of free wave modes that exist when A = 0,
which undergo modifications to their complex frequencies at O(A) that reinforce each
other. As A is increased, the unstable modes consist of gradually more complicated
superpositions of free wave modes, until for A & O(1), the eigenfunctions become
localised in the convectively unstable regions. This will be studied in § 6.7. For A . 1,
the eigenfunctions exist because of their confinement by the boundaries, though they
interact quite strongly with the primary wave, and are generally not simply free wave
modes with a nonzero growth rate. However, we have verified that the modes consist
of a pair whose frequencies approximately add up to ωp in the inertial frame, with a
detuning |∆|/ωp . 10−2 for most pairs of modes.
The number of unstable modes that exist in this amplitude range depends quite
strongly on viscosity. This is illustrated in Fig. 6.3. The number is also found to
decrease as np is increased. These two behaviours are related by the fact the wave
165
−4 −3 −2 −1 0 1 2 3 40
0.005
0.01
0.015
0.02
Re[ω] /ωp
Im[ω]/ω
p
 
 A=0.1A=0.3A=0.7A=1.0
Figure 6.4: Distribution of unstable eigenvalues on the complex frequency plane for various
values of A, with np = 4 and ν = 10−14.
should undergo parametric instabilities, which have largest growth rates when the
resonant tuning is good, which is more likely to occur for perturbations with larger
wavenumbers. However, these large wavenumber components are strongly damped by
diffusion. Increasing np means that the “effective resolution” available to capture the
unstable modes decreases. This is the same as increasing ν, hence the same trends
exhibited in increasing np and ν.
The neat distribution of eigenvalues into two curves does not persist as A is in-
creased, as illustrated in Fig. 6.4. The frequencies (in the rotating frame) are primarily
smaller than the primary wave frequency in the inertial frame. However, a small num-
ber of modes exist for A & 0.3 which have frequencies larger than ωp. Nevertheless, in
each case the frequencies of the unstable modes are always smaller than the maximum
buoyancy frequency in the flow (which corresponds with 1/ωp ∼ 6 in the units of this
figure, for np = 4). This makes sense if these are gravity wave-like disturbances, which
are parametrically excited by the primary wave.
In Fig. 6.5, the growth rate for the most unstable mode is shown to scale approxi-
mately linearly with A for A ≤ 1. A slope of 1 in this figure is predicted for A¿ 1 if
the instability is due to a parametric resonance.
Our most important result of this section is illustrated in Fig. 6.6. This shows that
166
−1 −0.8 −0.6 −0.4 −0.2 0−3
−2.5−2
−1.5−1
−0.50
log 10{Im
[ω]/ω p}
log10 A 
 1e−131e−14
Figure 6.5: Im[ω] vs. A for the most unstable mode with np = 4 and ν = 10−13, 10−14. The
solid red line has slope 1. This shows that the instability approximately scales linearly with
A for small A, indicating that the instability when A < 1 is due to a parametric resonance.
the growth rate scales inversely with the number of wavelengths within the domain
(note that this is after normalising by ωp). In this figure we plot the logarithm of
the growth rate versus log10 np for A = 0.5 and 0.8, respectively. The slope is always
approximately equal to −1 for low np. The tail-off at larger np is due to diffusion, and
arises because modes with smaller spatial scales parametrically excite modes with even
smaller spatial scales (as a result of the theorem proved by Hasselmann 1967), which
are then more easily damped by diffusion. As we would expect from this interpretation,
the value of np at which diffusion dominates moves to smaller np as ν is increased.
The inverse dependence on np that is present when diffusion is unimportant is a key
result. This suggests that although parametric instabilities exist for any amplitude in
the absence of diffusion, in a sufficiently large domain they become unimportant. We
discuss the relevance of this result in § 6.8.
6.6.1 Eigenfunctions
In the top panels of Figs. 6.7 & 6.8 we plot the real (blue solid lines) and imaginary
parts (red dashed lines) along φ = 0 of the spatial eigenfunctions of the most unstable
mode for A = 0.1 and 0.5, respectively. We have taken np = 4 and ν = 10−13. These
167
0.2 0.4 0.6 0.8 1 1.2−4
−3.5−3
−2.5−2
−1.5
log 10{Im
[ω]/ω p
}
log10 np 
 1e−131e−14
0.2 0.4 0.6 0.8 1 1.2−3.5
−3−2.5
−2
−1.5−1
log 10{Im
[ω]/ω p
}
log10 np 
 1e−131e−14
Figure 6.6: log10Im[ω/ωp] vs. log10 np for the most unstable mode when A = 0.5 and 0.8
respectively, for ν = 10−13 and 10−14. The red line has a slope −1. The tail-off at larger np
is due to diffusion.
168
0 0.5 1−0.010
0.010.02
r
ψ
0 0.5 1−0.5
0
0.5
r
b
0 0.5 1−0.5
0
0.5
r
u r
0 0.5 1−0.5
0
0.5
r
u φ
5 10 15 20 25 30 35 40 45 50−20
−100
1020
n
m
ψm,n
 
 
−5−4
−3−2
−10
5 10 15 20 25 30 35 40 45 50−20
−100
1020
n
m
bm,n
 
 
−5−4
−3−2
−10
Figure 6.7: Top: Real (blue solid lines) and imaginary (red dashed lines) parts along φ = 0
of the spatial eigenfunctions for the most unstable mode for A = 0.1, np = 4, ν = 10−13.
The eigenfrequency is ω/ωp = 0.378 + 0.001i. Bottom: Spectral space eigenfunction of the
same mode. The colour scale represents log10 |ψm,n/max{ψm,n}|, and similarly for bm,n.
169
0 0.5 1−0.02
0
0.02
r
ψ
0 0.5 1−0.5
0
0.5
r
b
0 0.5 1−0.5
0
0.5
r
u r
0 0.5 1−0.5
0
0.5
r
u φ
5 10 15 20 25 30 35 40 45 50−20
−100
1020
n
m
ψm,n
 
 
−5−4
−3−2
−10
5 10 15 20 25 30 35 40 45 50−20
−100
1020
n
m
bm,n
 
 
−5−4
−3−2
−10
Figure 6.8: Top: Real (blue solid lines) and imaginary (red dashed lines) parts along φ = 0
of the spatial eigenfunctions for the most unstable mode with A = 1, np = 4, ν = 10−13.
The eigenfrequency is ω/ωp = 0.540 + 0.017i. Bottom: Spectral space eigenfunction of the
same mode. The colour scale represents log10 |ψm,n/max{ψm,n}|, and similarly for bm,n.
170
modes exist throughout the box and are confined by the boundary. They become more
distorted from the free wave modes as A is increased towards unity (and in fact also
for A > 1, until the localised modes appear), especially in the innermost wavelengths
of the primary wave. Note that the amplitude of the eigenfunctions is arbitrary since
we are solving an eigenvalue problem. The bottoms panels of Figs. 6.7 & 6.8 show the
spectral space eigenfunctions of the same unstable modes. We have normalised ψm,n
and bm,n to their maximum absolute values and taken the base 10 logarithm of each
component to produce the figures. These show that growing modes for the chosen
values of A are not simply a pair of free wave modes which are excited by the primary
wave. They contain many n and m values localised around a particular n and m, and
are therefore interacting strongly with the primary wave. Multiple n and m values are
involved even when A = 0.1. For the value of ν adopted, these modes are well resolved,
as is shown from the amplitude decay of the spectral space eigenfuntion, which occurs
before the resolution limit is reached in n and m.
6.6.2 Energetics of the instabilities
When A < 1, the isentropes are never overturned by the primary wave, so a pure radial
convective instability is not possible. However, these instabilities could be driven
by the free energy resulting from the primary wave shear or entropy gradients, or
a combination of the two. In this section, we compute the spectral space energy
contributions outlined in § 6.4, for a sample of growing modes in this amplitude range.
We have confirmed that the growth rate is accurately computed from Eq. 6.4.13, to
within a few percent at most. In Table 6.1 we outline the contributions to the growth
rate from each term in Eqs. 6.4.5–6.4.7 for the most unstable mode for A = 0.1, 0.5
and 1 with np = 4 and ν = 10−13. The eigenfunctions for two of these are plotted in
Figs. 6.7 & 6.8. These examples are illustrative of every unstable mode that exists
when A < 1 (and also the non-localised modes that exist when A > 1).
Firstly, we note that the integrated kinetic and potential energy of the modes are
in approximate equipartition. A single wave can be shown to be in exact equipartition,
so we would expect 〈K〉 ≈ 〈P 〉 if these modes are parametrically excited gravity waves
with a single n and m. That they are in approximate equipartition and include many
171
A = 0.1 A = 0.5 A = 1
Re [ω] 0.065 0.080 0.093
Im [ω] 1.86× 10−4 1.97× 10−3 3.00× 10−3
〈K〉 3.28× 10−2 3.82× 10−2 3.70× 10−2
〈P 〉 3.25× 10−2 3.87× 10−2 4.16× 10−2
〈Nsw〉 −3.20× 10−7 −3.98× 10−5 −9.55× 10−5
〈Nbw〉 3.57× 10−5 3.76× 10−4 6.62× 10−4
〈Fb〉 1.81× 10−5 2.04× 10−4 3.51× 10−4
〈Fν〉 −5.58× 10−6 −1.35× 10−5 −3.29× 10−5
〈Fκ〉 −5.55× 10−6 −2.01× 10−5 −6.03× 10−5
Table 6.1: Energy components of the most unstable mode for A = 0.1, 0.5 and 1, with
np = 4 and ν = 10−13. Note that ωp ≈ 0.17.
n and m components indicates that these modes are the larger A generalisations of
the parametrically excited free wave modes. The source of free energy driving these
modes is entirely the potential energy resulting from primary wave entropy gradients.
Somewhat surprisingly, the primary wave shear contribution is much smaller, and
actually stabilises the modes. This instability converts primary wave potential energy
to disturbance potential energy, and then converts approximately half of this input
energy to the disturbance kinetic energy through the buoyancy flux term. This process
results in approximate equipartition between 〈K〉 and 〈P 〉. Note that the entropy
gradients in the primary wave are insufficient to cause convective instability. These
modes are driven by weaker entropy gradients in radius and azimuth.
6.7 Results for waves with A > 1: the initial stages
of wave breaking
When A > 1, the primary wave overturns the stratification during part of its cycle. Our
simulations have shown that an instability breaks the wave within a few wave periods
once this first occurs. The initial stages of this breaking process are examined in this
section by choosing A > 1. To resolve convectively unstable modes with the adopted
values of N and M , we require the size of the overturning region to be sufficiently
large. Since overturning occurs only at the point r = 0 when A = 1, this necessitates
choosing values of A larger than unity to capture such instabilities. We are interested
172
in instabilities which act to break the waves in an (effectively) unbounded domain (the
central regions of the RZ of a solar-type star), therefore the appropriate unstable mode
should not rely on the boundaries for confinement, and should be localised within the
innermost wavelength of the primary wave. This is because the presence of confining
boundaries is artificial, and is imposed to specify the problem. With this in mind, we
now discuss the results of our stability analysis for waves with A > 1.
−30 −20 −10 0 10 20 300
0.5
1
1.5
2
Re[ω] /ωp
Im[ω]/ω
p
 
 A=5A=8
Figure 6.9: Distribution of unstable eigenvalues on the complex frequency plane for np = 4
and various A with ν = 10−13.
The eigenvalues of the unstable modes displayed on the complex plane are shown
in Fig. 6.9 for A = 5 and A = 8, both with np = 4 and ν = 10−13. The most
unstable modes are located on distinct branches, which stand above the continuation
of the modes that exist when A < 1. From studying their eigenfunctions, we find that
the modes on the branches are localised disturbances, unlike those below the main
branches. The modes on the branches could therefore represent the type of mode
that breaks the primary wave (this is discussed further in the next subsection). These
branches extend further from the origin the greater the value of A. In Fig. 6.10 we
plot the unstable modes for A = 5 for two values of np. This shows that the unstable
modes on the branches do not move around significantly as np is varied. They therefore
depend only weakly on the location of the outer boundary. Note, however, that the
growth rate becomes smaller as we go to larger np because of the increasing importance
173
−20 −15 −10 −5 0 5 10 15 200
0.2
0.4
0.6
0.8
1
Re[ω] /ωp
Im[ω]/ω
p
 
 np=2np=4
Figure 6.10: Distribution of unstable eigenvalues on the complex frequency plane for A =
10 and various np, with ν = 10−13.
of diffusion.
Note that the largest frequency of some growing modes is larger than the maximum
buoyancy frequency Nmax = 1 (which corresponds with 1/ωp ∼ 6 in the units of this
figure, for np = 4). The nonzero frequencies of the modes in this frame indicate that
they are oscillatory, and are non-steady. In addition, the growth rates of the most
unstable modes are sufficiently fast compared with the primary wave frequency that
the instability grows within several wave periods after onset.
The growth rate of the most unstable modes for a given np increases with A as
illustrated in Fig. 6.11 for np = 4, where curves for ν = 10−13 and 10−14 have been
plotted. There is an approximate square root dependence for A & 5. If the instability
is driven by convectively unstable entropy gradients, then we might expect
Im [ω′] .
√
max [−N2]
=
(
max
{
−r2 − Re
[
2A
k2,np
r
(
J1(k2,npr)− J3(k2,npr)
)
e2iφ
]})1/2
. (6.7.1)
Thus, for large A the growth rate should scale with the square root of the primary wave
amplitude. This behaviour is not observed when 1 . A . 5. In this range, the square
root dependence may not be exhibited partly because there is insufficient resolution to
accurately capture the modes that contribute to breaking since the overturning region
174
0 5 10 15 200
0.51
1.52
2.53
Im[ω]/
ω p
A 
 1e−131e−14
Figure 6.11: Im[ω] vs. A for the most unstable mode for np = 4, ν = 10−13 and 10−14.
The instability grows within a primary wave period when A & 5, which is when the localised
modes begin to appear.
is small compared with the box size. We have noticed that the growth rate does not
significantly depend on ν (and therefore the resolution), for the most unstable modes,
except in the range 1 . A . 5, which supports this explanation.
The behaviour of the growth rate on np is illustrated in Fig. 6.12 for the most
unstable mode when A = 5, for two values of ν. For large np, the unstable modes
have sufficiently small spatial scales for diffusion to become important, so we expect a
tail-off at large np. The important point that can be taken from this figure is that the
(normalised) growth rate of the localised modes on the branches does not depend on
the number of wavelengths within the domain, for modes that are not strongly affected
by diffusion. This means that the instability can be important in a large domain, such
as the RZ of a solar-type star, which contains many primary wavelengths.
6.7.1 Eigenfunctions
In the top panel of Fig. 6.13 we plot the real (blue solid lines) and imaginary parts
(red dashed lines) along φ = 0 of the spatial eigenfunctions of the most unstable mode
for A = 5, np = 4 and ν = 10−13. The spectral-space eigenfunction of this mode is
plotted in the bottom panel. The mode is strongly nonlinearly interacting with the
175
0.2 0.4 0.6 0.8 1 1.2−0.8−0.6
−0.4−0.2
00.2
0.40.6
log 10{Im
[ω]/ω p
}
log10 np 
 1e−131e−14
Figure 6.12: Im[ω] vs. np for the most unstable localised modes when A = 5 and ν = 10−13
and 10−14.
primary wave, as shown by the number of different n and m values that appreciably
contribute. The contribution to the eigenfunction is nonzero, but not maximal, near
|m| = M − 1, and is negligible at n = N , which indicates that this mode is only
just adequately resolved. The eigenfunction is spatially localised within the innermost
wavelengths of the primary wave. Each of the several most unstable modes in the
range 5 ≤ A ≤ 10 which lie on the branches in Fig. 6.9 are localised modes, and have
qualitatively different form to the type of modes that exist below the branches, which
are a continuation of the modes that exist when A < 1. As we go to larger A for the
same value of ν, the most unstable mode utilises an increasing number of n and m
values up to the resolution limit. This means that to adequately resolve the modes we
would have to either increase the resolution or the value of ν.
The components of the spatial eigenfunction of the most unstable mode when A =
5, np = 4 and ν = 10−13, is plotted in Fig. 6.14 on the (φ, r)-plane, to further illustrate
the spatial dependence of this mode. In Fig. 6.15 we plot the region of negative
N2 for the same primary wave. A comparison of these figures makes clear that the
eigenfunction is primarily localised within the regions made convectively unstable by
the primary wave entropy perturbation. This adds further evidence to the conjecture
that the instability is convective. We also find that any unstable mode on the branches
176
0 0.5 1−0.04−0.02
00.02
r
ψ
0 0.5 1−0.50
0.51
r
b
0 0.5 1−0.50
0.51
r
u r
0 0.5 1−1
0
1
r
u φ
5 10 15 20 25 30 35 40 45 50−20
−100
1020
n
m
ψm,n
 
 
−5−4
−3−2
−10
5 10 15 20 25 30 35 40 45 50−20
−100
1020
n
m
bm,n
 
 
−5−4
−3−2
−10
Figure 6.13: Top: Real (blue solid lines) and imaginary (red dashed lines) parts along φ = 0
of the spatial eigenfunctions for the most unstable mode with A = 5, np = 4, ν = 10−13.
The eigenfrequency is ω/ωp = 9.68 + 0.302i. Bottom: Spectral space eigenfunction of the
same mode. The colour scale represents log10 |ψm,n/max{ψm,n}|, and similarly for bm,n.
177
Figure 6.14: 2D Spatial eigenfunction of the most unstable mode for A = 10, np = 4 and
ν = 10−13, plotted on the (φ, r)-plane.
178
φr
log10 Re[√ N 2]
 
 
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
−3.5−3
−2.5−2
−1.5−1
Figure 6.15: Unstable region for A = 10, np = 4. This can be compared with Fig. 6.14.
of Fig. 6.9 that are excited when 5 ≤ A ≤ 10 are similarly localised and have a
qualitatively similar appearance to the eigenfunction plotted in Fig. 6.14.
6.7.2 Energetics of the instabilities
In this section, we compute the spectral space energy contributions outlined in § 6.4
for a representative sample of the localised growing modes that exist when A > 1.
We have confirmed that the growth rate is accurately computed from Eq. 6.4.13, to
within a few percent for the modes considered in this analysis. However, it must be
noted that the most unstable mode when A & 5 is typically not fully resolved with
our adopted resolution and ν, in that there is nonzero power in the highest n and m
values. This can lead to errors in the energy analysis typically of order 10 − 30%, so
we leave these modes out of this analysis, and only choose those that are adequately
resolved for Table 6.2. In this table, we outline the contributions to the growth rate
from each term in Eqs. 6.4.5–6.4.7 for several unstable modes that exist when A & 5,
each with np = 4 and ν = 10−13. The eigenfunction corresponding to the first of these
is plotted in Figs. 6.13 & 6.14.
As in the case of the modes that exist when A < 1, the instability is driven by the
free energy associated with primary wave entropy gradients, as is shown by the fact
that 〈Nbw〉 is the dominant contribution to the growth. This is indeed what would
179
A = 5 A = 7
Re [ω] 1.67 2.40
Im [ω] 0.052 0.134
〈K〉 6.95× 10−2 3.37× 10−2
〈P 〉 1.73× 10−2 1.02× 10−2
〈Nsw〉 −5.50× 10−3 −1.57× 10−3
〈Nbw〉 2.05× 10−2 1.55× 10−2
〈Fb〉 1.82× 10−2 1.22× 10−2
〈Fν〉 −2.10× 10−3 −1.33× 10−3
〈Fκ〉 −0.517× 10−3 −0.80× 10−3
Table 6.2: Energy components of the most unstable mode for A = 5 and 7, with np = 4
and ν = 10−13. Note that ωp ≈ 0.17.
be expected of a convectively driven instability. In addition, the primary wave shear
is much weaker and tends to stabilise the modes. Unlike the modes that exist when
A < 1, we do not necessarily have 〈K〉 ≈ 〈P 〉, and examples have been found that
do and do not satisfy approximate equipartition, so these modes do not appear to be
gravity wave-like, unlike the parametrically excited modes that exist when A < 1.
The growth rates are always . Re
[√
−N2
]
, which is expected to be an upper
limit if the instability is convective. The negative contribution of shear, as well as
hyperdiffusion mean that the modes that we have calculated have somewhat smaller
growth rates that this simple estimate would predict. The route of energy transfer
that drives the instability is the same as for the parametric instabilities discussed in
the previous section.
6.8 Summary and discussion of results
In the previous two sections we have analysed the instabilities that exist when A < 1
and A > 1.
6.8.1 Wave breaking
When A > 1 we have identified a class of localised modes that are driven by con-
vectively unstable entropy gradients in the primary wave. These modes exist in the
absence of an outer boundary, and are very likely to have initialised the wave break-
180
ing process in the simulations presented in Chapters 3 & 4. Subsequent stages in
the breaking process are not studied using this stability analysis because these would
involve nonlinear interactions between the perturbations to the wave, which we ne-
glected in our weakly nonlinear approach. The instability growth time is of the order
of a primary wave period, which is in agreement with the wave breaking times observed
in our simulations.
6.8.2 Parametric instabilities
When A < 1, there exist pairs of parametrically excited modes driven by (convectively
stable) primary wave entropy gradients, with wave shear playing a subordinate stabil-
ising role. These modes exist because of their confinement by the outer boundary. Our
most important result regarding these modes is the inverse dependence of the growth
rate on np. This can be explained by considering the relative time the primary wave
spends in the innermost regions, where its nonlinearity is strongest. The fraction of
the total wave propagation time spent in the innermost regions, where growing modes
are excited by the nonlinearity, scales with k−12,np ∝ n
−1
p , so the growth rate should also
scale with n−1p . Combining this with our observation that the growth rate increases
approximately linearly with wave amplitude, we can write
Im [ω] ∝
A
np
, (6.8.1)
for the modes excited when A < 1.
Our simulations in Chapters 3 & 4 did not show any instabilities acting on the
waves when A < 1. This can be neatly explained from Eq. 6.8.1 in the limit as
np → ∞, where the growth rate tends to zero. This limit is appropriate since the
waves have effectively no outer boundary in the simulations, because we damp the
waves before they reach the boundaries of the computational domain. The fact that
we observed no instabilities in the simulations is therefore consistent with this stability
analysis, and is not a consequence of limited run time or insufficient spatial resolution.
181
6.8.3 Comparison with the plane IGW problem
This problem has some important differences with the case of a plane IGW in a uniform
stratification. For that problem, as we discussed in the introduction to this chapter,
parametric instabilities act for any A, and always result in instability in the absence
of diffusion. In addition, Lombard & Riley (1996) find that the presence or absence of
isentropic overturning does not seem to play a dominant role in, and is not the cause
of, the instability2. This is different from our problem, where we find that overturning
results in the presence of a different class of localised modes, that are excited in the
convectively unstable regions of the wave. However, we do find that the source of free
energy driving the instability is the same whether A < 1 or A > 1. It is true that
parametric instabilities exist for any A in our problem, like in the plane IGW problem,
but these become unimportant in a large domain because the nonlinearity is spatially
localised in the innermost wavelengths. This is different from the plane IGW problem,
in which the nonlinearity is important everywhere in the wave.
The importance of overturning in our case could be because the primary wave shear
does not drive the modes, and in fact typically acts to stabilise them. In the plane
IGW problem, instabilities for any A are driven by a combination of 〈Nsw〉 and 〈Nbw〉,
whereas in our problem instabilities are always driven solely by 〈Nbw〉. Shear therefore
does not play as important a role as for the instability of plane IGWs.
From the results of their stability analyses, Lombard & Riley (1996) and Sonmor &
Klaassen (1997) state that wave stability is a three-dimensional problem. This might
suggest that the picture we have outlined could differ in 3D. However, the simulations
performed in Chapter 4 show a strong similarity with the 2D results. Performing a
similar stability analysis in 3D would be somewhat involved, and would be restricted
to studying the stability of small amplitude waves, because the wave solution is not
exact in 3D. It would be possible to calculate higher order terms to the solution, which
would make it valid for larger A, and then perform a stability analysis of this wave.
Without performing such an analysis, it is difficult to quantify the importance of three-
dimensional effects on the wave stability. Nevertheless, the excellent correspondence
2However, it is possible that in their calculations they have insufficient resolution to be able to
resolve any localised convectively unstable modes. If they go to larger A, than the maximum they
consider of 1.1, and/or consider larger resolutions, such localised modes may start to appear.
182
between the results of the simulations in 2D and 3D suggest that for our problem the
inclusion of a third dimension would be unimportant, with regards to wave stability.
6.8.4 Implications for tidal dissipation
In Chapter 5, we discussed the implications of the wave breaking process for tidal
dissipation in solar-type stars, and therefore to the survival of short-period planets.
What more can we say in light of the results of this chapter?
One result is that parametric instabilities exist for waves with A < 1. These
do not occur in an unbounded domain (the limit as np → ∞), but will be present
in the RZ of a solar-type star, since this does have an outer boundary, albeit many
wavelengths from the centre of the star. We can roughly fit Eq. 6.8.1 to the results of
our stability analysis, allowing us to give an upper bound to the expected growth rate
of the strongest instability of a tidally excited gravity wave with A < 1. We write
Im
[
ω
ωp
]
= K
A
np
, (6.8.2)
and calculate a value of K from the solutions to our eigenvalue problem, where we
typically find K ≈ 0.1. In the RZ of a solar-type star, tidally excited gravity waves
have 102 . np . 103, for orbital periods in the range 1 . P . 3 days. We can therefore
calculate an upper bound on the expected growth rate of a parametric instability in
a real star from taking A = 1 and np = 102, giving Im [ω/ωp] ≈ 10−3, so that the
resulting growth time,
tgrow =
1
Im [ω]
≈ 2.7 yrs. (6.8.3)
It is important to note that this estimate is likely to be an approximate lower bound on
tgrow, and will not be strongly affected by the inclusion of the rest of the RZ, because
the the amplitude of the waves, and therefore the nonlinearity, is much smaller away
from the centre. (In addition, note that our Boussinesq-type model is only valid where
N ∝ r, which is only true near the centre of the star.) These calculations constrain
the effects of nonlinear wave-wave interactions in the innermost regions, but do not
take into account the rest of the RZ.
183
It is important to estimate the magnitude of the resulting tidal dissipation, so
that we can evaluate its role in the evolution of short-period planets. Instead of
considering the problem of continual forcing of the primary wave by the planet, we
consider initialising the primary wave and ask how long it takes to be attenuated (and
its energy dissipated), calling this timescale tnl (this is similar to the highly eccentric
binary problem discussed in Kumar & Goodman (1996), which we discuss in the next
section). The torque on the star due to the gradual attenuation of the wave due to the
combined action of these parametric instabilities at nonlinearly damping the wave, is
m
ω
F
(
1− e−α
)
, (6.8.4)
with the attenuation factor α = tgroup/tnl, and F being computed as outlined in
Chapter 5. We define the global group travel time tgroup = 2
∫ rb
0 (1/cg,r)dr ≈ 25 d, from
a numerical calculation for the waves excited by a planet in a one-day orbit around
the current Sun. The next question is: what is tnl? To calculate this accurately is a
very difficult problem, and involves many uncertainties, particularly those involving
the saturation process for these nonlinear couplings. However, we note that a lower
bound on tnl can be obtained by the growth rate of the fastest growing parametric
instability tgrow. This is because this will act as a bottleneck for the nonlinear cascade
of energy from the primary wave, and so will limit the maximum decay rate of the
primary wave. This is probably also true if we are continually forcing the wave. We
can then estimate
α .
tgroup
tgrow
≈ 0.025. (6.8.5)
This gives an upper bound on the torque resulting from the nonlinear damping of the
primary wave. Using Eq. 5.1.2, this can be used to give a lower bound on the tidal
quality factor resulting from nonlinear damping of the primary wave in the A < 1
regime,
Q′? &
105
1− e−α
≈ α−1105 ≈ 5× 106, (6.8.6)
in the weak damping limit. The efficiency of this process is less than critical layer
184
absorption by a factor α−1 À 1. Note that this gives a lower bound on Q′?, because
tnl is likely to be somewhat larger than tgrow (e.g. Kumar & Goodman 1996 take
tnl = 10tgrow). The resulting Q′? may therefore be one or several orders of magnitude
larger than this lower bound. Furthermore, it is interesting to note that this bound
may not be sensitive to the number of wavelengths in the RZ, and therefore to the
orbital period, because the dependences of tgroup and tgrow on np cancel at leading
order.
The parametric instabilities that exist when A < 1 are much slower than the rapid
instabilities that onset when A > 1. The nonlinear outcome of the A > 1 instabilities
is that the wave breaks and forms a critical layer, which then absorbs subsequent
ingoing waves, and results in astrophysically efficient tidal dissipation. The estimate
of this section indicates that the parametric instabilities that exist when A < 1 are
much less efficient at dissipating energy in the tide, by several orders of magnitude.
This important result supports the explanation outlined in Chapter 5 for the survival
of short-period planets around solar-type stars.
6.8.5 Comparison with Kumar & Goodman
We can qualitatively compare our results with previous work by Kumar & Goodman
(1996), who studied nonlinear damping of tidal oscillations in highly eccentric solar-
type binaries. They used a truncated Hamiltonian approach to study parametric
instabilities of tidally excited f and g-modes. In their model, stellar eigenmodes are
coupled together from terms that exist at third order in displacement in the expansion
of the Lagrangian density, i.e., they adopt a weakly nonlinear approach. They consider
the evolution of a mode that has been tidally excited, but is no longer subject to forcing,
due to nonlinear coupling with a large number of g-modes that are present in the RZ
of the star (and exist because they have already been excited by turbulent convection,
for example). Their result indicates that high order and high degree g-modes can be
parametrically excited by low order quadrupolar f and g-modes, and can draw energy
from the primary mode on a timescale that is much shorter than the radiative damping
time of the primary mode.
A direct comparison of our work with theirs is not possible for several reasons.
185
Firstly, in their numerical work they mainly consider a primary f-mode coupling to
many g-modes in the RZ. The f-mode eigenfunction has its largest magnitude at the
surface and decays rapidly inwards, in contrast to the primary g-modes that we are
considering, so the coupling strengths are likely to be different. Secondly, we only
consider the nonlinear interactions in the central regions of the star, where they are
likely to be most important for g-modes, whereas they consider these interactions
throughout the whole star. Thirdly, our model is 2D, whereas their eigenfunctions
are valid in 3D for a spherically symmetric background. This last point, however, is
probably not going to be important.
One important point is that they neglect the possibility of wave breaking, which
would provide an upper limit to the amplitude of a given mode. This would prevent
modes with large amplitudes from coupling with the primary wave, and the nonlinear
outcome of the breaking (most notably critical layer formation) would significantly
modify the strength of tidal dissipation. Their results will therefore not be valid for
primary or daughter waves that satisfy a breaking criterion, since weakly nonlinear
theory is insufficient in this case. Indeed, the concept of parametric instability is no
longer valid if the daughters break and cannot form standing modes.
Keeping in mind the differences between our approach and theirs, we now directly
apply their results to our problem, and quantitatively compare the growth time of
parametric instabilities with those found in this chapter. The growth time in their
work
tgrow ≈ 4
(
Ep,0
1035J
)−1/2
, (6.8.7)
where Ep,0 is the initial energy in the primary wave. For the g-modes that we consider,
Ep,0 =
∫ ∫ ∫
Er2 sin θdrdθdφ = F
∫
(1/cg,r)dr = Ftgroup. (6.8.8)
This can be computed to give Ep,0 ≈ 2× 1029J for a Jupiter-mass planet on a one-day
orbit around the current Sun, which has A ≈ 0.3. This means that tgrow ≈ 3 yr when
A = 1, which happens to agree surprisingly well our calculation in the previous section,
given the differences in our approach. The total number of daughter modes which
simultaneously interact with the primary in their model is ∼ 1010
(
Ep,0
1035J
)5/4
∼ 102
186
for our fiducial case. We also find that there are many growing modes for a given
set of parameters in our stability analysis, so these statements appear qualitatively
consistent. They find that collectively, these modes absorb most of the energy of the
primary wave after a time ∼ 10tgrow (this is equivalent to assuming α ∼ 10−3 in the
previous section). This predicts Q′? ∼ 5×10
7 in their approach. We therefore conclude
that our results are broadly consistent with Kumar & Goodman (1996).
6.9 Conclusions
In this chapter we have performed a stability analysis of the 2D exact wave solution
derived in Chapter 3. The equations governing the evolution of the perturbations were
written down in spectral space using a Galerkin spectral method, and then solved as
an eigenvalue problem. This required the imposition of an artificial impermeable outer
boundary.
We have identified the modes that initiate the breaking process when the wave
overturns the stratification. This type of mode is strongly localised in the convectively
unstable regions of the primary wave, and is driven by unstable entropy gradients. Its
growth time is comparable with the primary wave period, which is consistent with the
breaking time observed in the simulations of Chapters 3 and 4.
We have also studied the instabilities which exist for waves with insufficient am-
plitudes to overturn the stratification. We find that these are parametric instabilities
driven by (convectively stable) entropy gradients in the primary wave. The growth
rate of these modes scales inversely with the number of wavelengths within the domain,
so they become less important for a real star than for the small container considered
here. It is estimated that their growth times in a real star would be of the order of
3 yr, which is much longer than the orbital period of a short-period planet, though
many such modes are excited. Rough estimates are made that provide a upper bound
on the magnitude of the resulting tidal dissipation, for which we find Q′? & 10
7 from
this process. This is clearly much weaker than the dissipation resulting from critical
layer absorption, and so is unlikely to change the picture outlined in Chapter 5 for the
survival of short-period planets.
We discussed several differences between our problem and the stability of a plane
187
IGW in a uniform stratification (e.g. Lombard & Riley 1996). We have confirmed
that when the wave is confined in a container with an outer boundary it is unstable
whatever its amplitude, in the absence of diffusion. However, the inverse dependence
of the growth rate on the number of wavelengths within the container is quite different,
and results from the finite time of nonlinear interaction being much shorter than the
group travel time across a large container.
We compared our results to Kumar & Goodman (1996), who studied the nonlinear
damping of tidally excited oscillations in highly eccentric binaries, and found some
agreement. They predict that many (∼ 102) modes collectively draw energy from the
primary wave, which we have qualitatively confirmed from our stability analysis. The
growth rates of parametric instabilities for the same problem in both of our approaches
when A < 1 are of the same order of magnitude. They therefore predict a similar lower
bound for Q′? resulting from this process. This is promising, given the differences in our
approach. It would be interesting to extend their numerical calculations by studying
the parametric instabilities of g-modes including continual tidal forcing of the primary
wave and nonlinear couplings involving many daughter and granddaughter modes, as
well as taking into account the amplitude limiting effects of wave breaking. Weakly
nonlinear theories such as ours and theirs are likely to be valid when considering the
initial stages of the breaking process, and in studying whether any instabilities exist
for suboverturning waves, which were the topics of study in this chapter. However,
they should not be used to determine long-term behaviour for waves which overturn
the stratification (such as for the close binary problem whenever A > 1).
It would be worthwhile to confirm the results of this chapter using 2D numeri-
cal simulations with SNOOPY. An artificial impermeable outer boundary could be
implemented in the code, and the resulting instabilities then studied. Of particular
importance is to determine the rate at which energy is lost from the primary wave due
to the parametric instabilities for suboverturning waves that we studied in this chapter
(i.e., to numerically calculate tnl). This would enable a more accurate calculation of
the magnitude of Q′? and would provide a useful independent check of our results. We
defer such calculations to future work.
188
Chapter 7
Critical layer formation by
radiative diffusion
In Part II of this thesis, we have studied critical layer formation by wave breaking. This
occurs if the tidally excited IGWs have sufficient amplitude when they reach the centre
to overturn the stratification. This process only operates if the wave amplitudes exceed
a critical value. In this section we consider a mechanism which could operate even if the
waves are of too low amplitude to initiate breaking. This mechanism is critical layer
formation through radiative damping of the tidally forced primary waves. This process
could provide efficient tidal dissipation in solar-type stars perturbed by less massive
companions. However, this mechanism results in a very slow deposition of angular
momentum (∼ Myr), in comparison to the relatively rapid deposition which occurs
when the wave breaks (∼ few days). This means that this process may be prevented by
effects that resist the development of differential rotation, of which several are briefly
discussed.
7.1 Order-of-magnitude estimate
Near the centre of a star, radiative diffusion is the dominant linear dissipation mech-
anism. If the waves are gradually attenuated by radiative diffusion on their reflection
from the centre, over a sufficiently long time, they may be able to spin up the inner-
most regions of the star to the angular pattern speed of the tide Ωp, hence producing
189
a critical layer. Radiative diffusion is much weaker near the centre than at the top of
the RZ, but the central regions are more easily spun up by this gradual deposition of
angular momentum, because of their small moment of inertia. We have already ob-
served this process occuring in our simulations in Fig. 3.9, albeit with viscosity and not
radiative diffusion acting on the waves. Later in this chapter we perform simulations
with radiative diffusion, and confirm that this effect is present in our simulations. In
this section we provide an order-of-magnitude estimate of the timescale for radiative
diffusion to spin up a region, of size of the order of the central wavelength of the
primary wave, to Ωp.
If the wave is attenuated as it propagates from a radius R, to the centre, and back
to R again, by a factor e−α/2, i.e., such that the outgoing energy flux is Fe−α, then
the torque on this region of the star is given by the angular momentum transferred to
the mean flow, and is
Γ =
m
ω
F (1− e−α). (7.1.1)
Here we assume that F is that given in Eq. 5.2.24, which is reasonable when the
response is non-resonant. The wave attenuation
α ≈ 2
∫ R
0
ηk2
cg,r
dr, (7.1.2)
where η is the thermal diffusivity and k is the wavenumber. We have k ∼ kr, except
within the innermost wavelength, and the radial group velocity cg,r ≈ ωkr . The thermal
diffusivity can be calculated from the properties of the appropriate stellar model from
η =
16σT 3
3κρ2cp
, (7.1.3)
where σ is the Stefan-Boltzmann constant, κ is the opacity, and cp is the specific heat
at constant pressure. This assumes N2 is due only to entropy gradients, rather than
a mixture of entropy and composition gradients, so the attenuation may be weaker
in practice. However, the stratification at the stellar centre is composed of these in
roughly equal parts, so this is unlikely to affect the attenuation significantly. We can
take η to be constant over the inner ∼ 5% of a star, to a first approximation, with a
190
value ≈ 16.7m2s−1, in the case of the Sun. In addition, kr is roughly constant with
radius within this region, though formally diverges as r → 0. We can reasonably
estimate
α ≈ 2× 6
3
2η
C3
ω4
R ≈ 4× 10−4
(
R
R¯
)(
P
1d
)4
, (7.1.4)
where R is the size of the region spun up by this process. This is because kr ∼
√
l(l+1)N
rω ,
if ω ¿ N , and N = Cr. We also note that the tidal frequency ω = 2
(
2pi
P
)
. This means
that the attenuation by radiative diffusion within the innermost few percent of the
star is small for waves excited by planets on one-day orbits. However, the wavelength
of the waves becomes shorter for longer period orbits, so their attenuation by radiative
diffusion is more efficient. Note that α & 1 only when P & 8 days, even when radiative
diffusion over the entire RZ is considered (in fact GD98, who made a more accurate
calculation by including the radial dependence of the wavenumber and diffusivity,
found that α & 1 for P & 11.6 days). This means that the waves excited by planets
on short-period orbits with P . 3 days, whose survival could be threatened by the
process of wave breaking, will not be significantly attenuated in traversing the RZ,
according to linear theory. It is therefore appropriate to ask whether they will break
on reaching the stellar centre.
To a first approximation, the central ∼ 3% of a star can be modelled as a uniform
density sphere, with the central density ρc. Its moment of inertia is I = 45piR
5ρc. Using
Eq. 7.1.1, we have the following differential equation describing the spin evolution of
the central regions:
I
dΩ
dt
=
m
ω
F (1− e−α). (7.1.5)
Assuming that the system evolves slowly, which is probably true until breaking occurs,
we can take F , ω, and α, to be constant in time as the central regions are spun up.
This allows a straightforward solution, giving the resulting timescale to spin up the
region of the central wavelength to Ωp, in the case of Jupiter orbiting the Sun with a
191
one-day period (in which R ∼ 0.01R¯), of
tSU =
4
5
piR5ρcΩ2p
F (1− e−α)
≈ 3 Myr. (7.1.6)
Note that this timescale strongly depends on the size of the region that is spun up.
When α¿ 1, which is valid for P . 10 days, tSU ∝ P 5/3.
In this estimate we have used Model S of the current Sun, so this value only applies
to our star. However, we have repeated this calculation using the same set of stellar
models discussed in Chapter 5, with masses in the range 0.5 ≤ m?/M¯ ≤ 1.1, and
find that tSU . 1 Gyr for each of these models, for Jupiter orbiting the star with a
one-day period.
This striking estimate indicates that all gas giants on short-period orbits around
G or K stars could eventually cause the formation of a critical layer near the centre
of the star, given sufficient time . O(1) Gyr. Once this has formed, we have found in
our simulations in Chapters 3 & 4 that it is reasonable to assume that the IW angular
momentum flux is entirely absorbed near the centre. Hence, our estimate of Q′? in the
previous chapter could apply to all slowly rotating G and K stars.
Unlike the mechanism of nonlinear wave breaking, which has so far been the main
subject of this thesis, the formation of a critical layer by radiative diffusion requires
the progressive spin-up of the region of the central wavelength by a gradual deposition
of angular momentum. This angular momentum deposition is much slower than the
rapid deposition that results from wave breaking (spin-up time of several days), so this
process could be interrupted by mechanisms that resist the development of differential
rotation, such as hydrodynamic instabilities or magnetic stresses. We briefly consider
several such effects later in this chapter.
This section has only presented an order-of-magnitude estimate of the timescale
for this process. One question that must be answered is whether a more detailed
calculation will affect this process. In the next section, we study the effects of viscosity
and radiative diffusion on our nonlinear primary wave in 2D.
192
7.2 IGWs subject to viscous and radiative damping
in 2D
In this section, we perform a 2D calculation to study the effects of viscosity and
radiative diffusion on the nonlinear wave derived in Chapter 3. Our aim is to check
whether a more detailed calculation of the wave damping is consistent with our simple
estimate of the previous section.
We are interested in the limit in which the wave reflects approximately perfectly
from the centre of the star, which is appropriate for low-amplitude waves. In this case,
global standing modes form in the RZ, which are damped over a long time by radiative
diffusion. We consider a circular region of radius rout, and analyse the effects of both
viscosity and radiative diffusion on a primary wave
ψw(r, ξ) = Re
[
AJ2(r)e
iξ
]
, (7.2.1)
by calculating the perturbed solutions
ψ = ψw + δψ, b = bw + δb. (7.2.2)
We start from the vorticity and buoyancy equations (in the dimensionless units of
§ 3.5)
∂t(−∇
2ψ) + ∂φb = J(ψ,−∇
2ψ) + F [ψ], (7.2.3)
∂tb+ ∂φψ = J(ψ, b) +G[b]. (7.2.4)
The functionals F [ψ] and G[b] represent the effects of viscosity and radiative diffusion,
respectively, and evaluate straightforwardly in Cartesian coordinates as
F [ψ] = (∇× ν∇2u) · ez = −ν∇4ψ, (7.2.5)
G[b] = κ∇2b. (7.2.6)
A forced wave equation can be derived by combining the vorticity and buoyancy equa-
193
tions, to give
∂2t (−∇
2ψ)− ∂2φψ = ∂tJ(ψ,−∇
2ψ)− ∂φJ(ψ, b) + ∂tF [ψ]− ∂φG[b]. (7.2.7)
We neglect nonlinear coupling Jacobians, i.e., we consider the perturbation to be
produced only from viscous and radiative damping of the primary wave. This means
that we assume J(δψ,−∇2ψw) = J(ψw,−∇2δψ) = J(δψ, bw) = J(ψw, δb) = 0. These
nonlinearities would couple an m = 2 wave with m = 0 and m = 4 waves, though
to first order, δψ has the same azimuthal form as the primary m = 2 wave. Since
|δψ| is assumed small, we can neglect these couplings, to O(A). Other wavenumbers
grow due to these couplings at O(A2), competing with radiative and viscous damping
of the perturbed solution, which we also neglect, i.e., we assume F [δψ] = G[δb] =
0. Note that ∂2t (−∇
2ψw) − ∂2φψw = ∂tJ(ψw,−∇
2ψw) − ∂φJ(ψw, bw) = 0, since the
adiabatic primary wave solution is exact. We make the assumption of linearity, that
J(δψ,−∇2δψ) = J(δψ, δb) = 0. Substituting the perturbed solutions, under these
assumptions, gives
∂2t (−∇
2δψ)− ∂2φδψ = ∂tF [ψw]− ∂φG[bw], (7.2.8)
which can be rewritten as (factoring out eiξ)
L˜mδψ = f(r), (7.2.9)
where the linear differential operator
L˜m ≡
m2
r
Lm ≡
m2
r
[
∂rr∂r − r(1−
m2
r2
)
]
, (7.2.10)
and after simplification, we obtain the forcing
f(r)eiξ ≡ ∂tF [ψw]− ∂φG[bw], (7.2.11)
= 2iA(ν + κ)J2(r)e
iξ, (7.2.12)
which has its maximum near the centre, and oscillates with radius with an amplitude
decaying asymptotically like r−
1
2 . This can be obtained easily using the fact that
194
∇2ψw = −ψw = −bw. It makes sense that this has the same symmetry in azimuth and
time as the primary wave. We also expect the forcing to have a similar form to the
primary wave, but to be out of phase by pi/2, i.e., to cause decay in wave amplitude
at every location.
Note that radiative diffusion (κ) and viscosity (ν) have the same effects on the
primary wave. This is results from the fact that ψw = bw for these waves, and that
the φ and t dependence only comes through ξ (so the φ and t dependence is the same
except for a phase shift of pi). The waves also satisfy equipartition of kinetic and
potential energies, when integrated over a multiple of half-wavelengths, as we will now
prove.
If we rewrite Bessel’s equation in the form
1
r
∂r(r∂rJm)−
m2
r2
Jm = −Jm, (7.2.13)
multiply by rJm, and integrate over radius from r1 to r2, we obtain
∫ r2
r1
[
(∂rJm)
2 +
m2
r2
J2m
]
rdr =
∫ r2
r1
J2mdr + [rJm∂rJm]
r2
r1
. (7.2.14)
Since 2P = J2m and 2K = (∂rJm)
2 + m
2
r2 J
2
m, where K and P are the kinetic and po-
tential energies, respectively, this statement is telling us that equipartition holds if we
integrate over a range where Jm or ∂rJm are zero at the end points. Equipartition
therefore holds when the energies are integrated over integer multiples of half a wave-
length, which is r2 − r1 = n2λr for n ∈ Z, in the WKB limit. This equipartition of
kinetic and potential energies is only satisfied in an averaged sense, and not pointwise,
as we can see from Fig. 7.1.
In the form of Eq. 7.2.9, we can solve the problem by finding Green’s function,
which is the solution of
L˜mG(r, r
′) = δ(r − r′). (7.2.15)
This can be found by imposing the two-point BCs
δψ(rin) = 0, (7.2.16)
195
2 4 6 8 10 12 14
r
0.05
0.10
0.15
0.20
E
Figure 7.1: Kinetic (blue line) and potential (red line) energies in the primary wave as a
function of radius. K increases first, and then the remaining part of the first wavelength has
P > K. Equipartition holds in an averaged sense, but is not satisfied pointwise ∀r.
δψ(rout) ∝ Jm(rout)− iYm(rout), (7.2.17)
which correspond to the physical condition of regularity at the centre (rin = 0), and
an OW BC at r = rout. Note that we are assuming the solution is proportional to,
and not equal to, the given OW, otherwise we would be forcing the solution at the
outer boundary, and fixing the amplitude of the solution. Hence, we are looking for
waves with an outgoing energy/angular momentum flux. We are required to choose
rout to be much closer than infinity, since the wave would entirely damp if rout = ∞;
δψ would also increase to no longer satisfy the requirement that |δψ/ψw| ¿ 1 for a
linear analysis. We choose rin = 0, and rout to be several wavelengths from the centre.
Two linearly independent solutions of the homogeneous (unforced) wave equation
are the standing wave and OW solutions, Jm and Jm − iYm, respectively. Green’s
function is then
G(r, r′) =
{
ipir
2 Jm(r) [Jm(r
′)− iYm(r′)] , rin ≤ r ≤ r′,
ipir
2 [Jm(r)− iYm(r)] Jm(r
′), r′ ≤ r ≤ rout.
(7.2.18)
The solution is the real part of
δψ(r, ξ) =
∫ rout
rin
G(r, r′)f(r′)dr′eiξ
196
= [Jm(r)− iYm(r)] e
iξ
∫ r
rin
ipir′
2
Jm(r
′)f(r′)dr′ (7.2.19)
+Jm(r)e
iξ
∫ rout
r
ipir′
2
[Jm(r
′)− iYm(r
′)] f(r′)dr′.
Note that as r → rout, the solution is an OWwith amplitude δAout =
∫ rout
rin
ipir′
2 Jm(r
′)f(r′)dr′,
and as r → rin, the solution is a standing wave with amplitude
∫ rout
rin
ipir′
2 [Jm(r
′)− iYm(r′)] f(r′)dr′. The relevant part for the angular momentum
transport is the part which is an OW, i.e., the first integral only. Note that the
OW decays asymptotically like r−
1
2 e−ir, so at infinity, this wave decays to zero.
Using Mathematica to compute the integrals, we obtain the closed-form solution
(for m = 2), which is the real part of
δψ(r, ξ) =
A
2
(ν + κ)eiξ
{
piroutJ2(r)
(
rout[J1(rout)]
2 − 4J1(rout)J2(rout) + rout[J2(rout)]
2
)
+ipirY2(r)
(
r[J1(r)]
2 − 4J1(r)J2(r) + r[J2(r)]
2
)
−i
√
piJ2(r)G
2,2
1,3

r,
1
2
1, 32 ,
1
2
1, 3 , −1, 0, 12

 (7.2.20)
+i
√
piJ2(r)G
2,2
1,3

rout,
1
2
1, 32 ,
1
2
1, 3 , −1, 0, 12


}
.
Note that along ξ = npi2 , for n ∈ Z, Re[δψ] ∝ J2(r). The “G” functions above are
Meijer-G functions (e.g. Lozier et al. 2010), which result from the second integral.
They are written in Mathematica as, e.g.,
G2,21,3

rout,
1
2
1, 32 ,
1
2
1, 3 , −1, 0, 12

 ≡ MeijerG[{{1,
3
2
}, {
1
2
}}, {{1, 3}, {−1, 0,
1
2
}}, r,
1
2
].
Some care is needed to ensure that there are no large numerical cancellations when
dealing with the Meijer-G function. This has been found to occur when the difference
of the final two terms on top and bottom are integers (here, 12 and any of −1, 0,
1
2),
particularly for large r & 20. This is likely to be a numerical instability from the
algorithm that evaluates these complicated functions in Mathematica.
197
2 4 6 8 10 12 14
r
-15
-10
-5
5
10
dpsiHrL
Figure 7.2: δψ along ξ = 0 for the perturbed solution, showing both the real (blue line)
and imaginary (red line) parts. These are oscillatory functions of radius, out of phase by
pi/2. |Re[δψ]| is largest in the centre. Further out, the amplitudes of oscillation of the real
and imaginary parts are comparable.
7.2.1 Angular momentum flux
Since our aim is to determine whether our simple estimate of the spin-up due to
radiative diffusion is confirmed by this calculation, we turn to compute the torque on
the central regions. The perturbed angular momentum flux is
δFLr =
(mρ0
ω
)
pimrIm [(∂rδψ)ψ
∗
w + δψ
∗(∂rψw)] , (7.2.21)
which is proportional to ν + κ since δψ ∝ ν + κ, as expected. For the above solution
this evaluates (when m = 2) to
δFLr (r) =
(ρ0
ω
)
2pi2|A|2(κ+ ν)r
{
r2[J1(r)]
3Y2(r)− rJ2(r)[J1(r)]
2 (rY1(r) + 4Y2(r))
+r[J2(r)]
2J1(r) (4Y1(r) + rY2(r))− r[J2(r)]
3 (rY1(r) + 2Y2(r)) (7.2.22)
+2pi−1/2[J2(r)]
2G2,12,4

r,
1
2
1 , 0
1
2 ,
5
2 ,
3
2 , 0


}
.
The main contribution to this is from δAout(r), because this corresponds to the am-
plitude of the OW. In fact, it can be shown that δFLr ≈ −m
2δAout(r), which is easily
confirmed by plotting these functions. The radial flux at a particular radius does not
198
(and should not) depend on the location of the outer boundary (which only comes into
the second integral).
2 4 6 8 10 12 14
r
20
40
60
80
100
120
dFLrHrL
Figure 7.3: δFLr for the perturbed solution. This is equivalent to the total torque on the
region inside a ring of radius r. This has approximately the same form as the first integral
δAout. Note that δFLr ∼ r
6 as r → 0, but δFLr ∼ r as r → ∞.
In Fig. 7.3 we plot the perturbed angular momentum flux. This is equivalent to the
torque on the region inside a ring of radius r. It makes physical sense for the torque to
increase with radius from the centre. This is because the OWs have travelled further,
and experienced more dissipation due to ν, κ, the larger the value of r.
The corresponding torque density per unit radius, defined such that the total torque
Γ ≡ δFLr ≡
∫ rout
0
T (r)dr, (7.2.23)
which results from viscous and radiative damping of the primary wave, is
T (r) = ∂r(δF
L
r ). (7.2.24)
The torque density T is plotted in Fig. 7.4. The torque density oscillations have
roughly constant amplitude, outside of the innermost wavelength. This is what we
would expect from the a simple calculation in the WKB approximation. In this ap-
proximation, the OW can be expressed as (J2 − iY2)eiξ ∼
√
2
pire
−i(r+ξ− 5pi4 ), for which
the torque scales linearly with radius. In fact, it is only for the innermost wavelength
that geometrical factors become important, where the solution is modified from that
199
2 4 6 8 10 12 14
r
5
10
15
THrL
Figure 7.4: Torque density for the perturbed solution. This has roughly constant amplitude
as a function of radius, as expected from a simple calculation in the WKB approximation.
This is largest within the first wavelength, but not significantly, marking the only departure
from WKB.
expected from this approximation.
The torque in the order-of-magnitude estimate of § 7.1 scales linearly with radius
when α¿ 1, for which α ∼ R. More precisely, we can express
Γ = FLr (1− e
−α) ≈ FLr α1R +O(R
2), (7.2.25)
where α ≈ α1R = 16ηC
3
ω4R in 2D, and F
L
R is the angular momentum flux in IWs. In
dimensionless units we can express α = 2 mω2 (ν + κ)r, if we write η = (ν + κ)
Ωp
C2 . We
then have
T
FLr
≈ 4
(ν + κ)
ω2
r. (7.2.26)
This allows us to compare the magnitude of the torque with that from the calculation
of this section.
For the detailed calculation, we find that δAout(r) ≈ 2r outside of the first half-
wavelength. This means that
δFLr ≈ 2rm
2
(ρ0
ω
)
|A|2(ν + κ). (7.2.27)
200
If we normalise this with respect to the angular momentum flux in the ingoing part of
the primary standing wave solution, taking FLr =
(mρ0
ω
)
2m|Ain|2, with |Ain| = |A|/2
(to get the correct scaling for the IW), then
δFLr
FLr
≈ 4
(ν + κ)
ω2
r. (7.2.28)
This demonstrates that the torque has a similar magnitude and scaling with radius as
predicted from our simple estimate.
To summarise this section, we have performed a two-dimensional analysis, which
has confirmed that our simple order-of-magnitude calculation of the torque on the
central regions of the star due to radiative damping of the waves, has the correct
magnitude and scaling with radius. We have also found that the effects of viscosity
and radiative diffusion on the waves are the same, and would both be able, in principle,
to spin up the central regions.
The resulting torque density is (slightly) largest within the innermost wavelength,
and is oscillatory. Coupled with the fact that the moment of inertia of the central
regions ∼ r5, because the density is approximately uniform in the innermost wave-
lengths, this implies that the central wavelengths are the most likely to be spun up
by radiative diffusion (or viscosity). Note, that the torque is always positive, and only
acts to spin up (and not down) the central regions, as expected when Ω < n.
7.3 2D simulations
Motivated by the simulation presented in Fig. 3.9, in which viscosity acts to gradually
transfer angular momentum from the wave to the mean flow, eventually producing a
critical layer, we will now present similar simulations with radiative diffusion. We use
the two-dimensional set-up of Chapter 3, except with ν = 0, and choose various values
of κ, ranging from 10−4 − 10−7. Neglecting viscosity is observed to result in numerical
instability if any wave breaking occurs, but here we only follow the evolution until
a critical layer forms by radiative diffusion. We use a forcing amplitude f˜r = 2.5,
which is insufficient to cause breaking, appropriate for considering the tidal evolution
of lower-mass tidal perturbers.
201
We find that the waves are significantly attenuated on propagation to the centre,
from the launching region, if κ & 10−4. This situation is that appropriate for waves
excited by planets on orbits of around ten days, for which the attenuation α due to
radiative diffusion is close to unity. The waves are gradually damped as they propagate,
but in this case have deposited most of their angular momentum before they approach
the central regions, which they reach with low amplitude. The result is that the central
regions are negligibly spun up by the waves excited by planets on orbits of P & 8− 11
days, and therefore will not produce a critical layer. Regions farther out have larger
moment of inertia, and so are harder to spin up by this process.
In Fig. 7.5 we plot the temporal evolution of the dominant Fourier components,
m = 2 and m = 0, in the solution, in a simulation with κ = 10−5. For this simulation,
the attenuation by radiative difusion on propagation from the launching region is
small to moderate, and the waves still have reasonable amplitudes near the centre.
Over many wave crossing times, the gradual deposition of angular momentum results
in appreciable spin-up of the central regions. Thus radiative diffusion has the same
result as viscosity (see Fig. 3.9) acting on the waves, as predicted by our analysis in
the previous section. In this simulation, a critical layer is eventually formed within the
innermost wavelength. The resulting evolution would then be similar to that described
in Chapter 3, except that in these simulations numerical instabilities set in since we
neglect viscosity.
These simulations confirm the picture so far outlined in this chapter, that radiative
diffusion can be responsible for the production of a critical layer near the centre. This
outcome could occur for tidal perturbers with insufficient masses to be able to cause
wave breaking, if they have short enough orbital periods for the waves to reach the
centre with sufficient amplitude, i.e., for P . 8 days. Radiative diffusion preferentially
spins up the central regions because these have smaller moment of inertia than regions
farther out. Given the similarities between the two and three-dimensional problems,
as observed in Chapter 4, we expect that this picture would be the unchanged in 3D.
202
0 500 1000 1500 200010−3
10−2
10−1
100
101
t
P m
 
 |m|=0|m|=2
Figure 7.5: Temporal evolution of the power spectral density Pm, for m = 0 and m = 2,
for a simulations with κ = 10−5. Together with Fig. 3.9, this shows that radiative diffusion
has the same effects on these waves as viscosity, and is able to gradually spin up the mean
flow, to eventually produce a critical layer.
7.4 Discussion
The process that we have outlined in this chapter could provide efficient tidal dissi-
pation in solar-type stars perturbed by less massive companions than wave breaking
would require. However, this mechanism results in a very slow deposition of angular
momentum (∼ Myr), in comparison to the relatively rapid deposition which occurs
when the wave breaks (∼ few days). This process is therefore more likely to be pre-
vented by effects that resist the development of differential rotation. In this section
we discuss several of these possible mechanisms.
7.4.1 Magnetic fields in the RZ
It is well known that large-scale magnetic fields have the ability to hold fluids in uni-
form rotation. In particular, in ideal MHD, Ferraro’s Law of Isorotation holds, which
states that an axisymmetric steadily rotating system has constant angular velocity
along each field line. Any shear in the component of the velocity transverse to the
magnetic field generates a transverse field, which grows until the Lorentz force acts
203
back on the shear. This process generates Alfve´n waves, leading to a torsional os-
cillation of the field with a period determined by the Alfve´n travel time across the
disturbance (Mestel & Weiss 1987):
τA =
R
vA
=
R
√
µ0ρ
B0
, (7.4.1)
where vA is the Alfve´n speed, R is the spatial scale of the shear (i.e., the region forced
into differential rotation by radiative diffusion or wave breaking), B0 is the background
magnetic field, ρ is the fluid density, and µ0 is the magnetic permeability.
Indeed, the presence of a magnetic field in the RZ of the Sun has been invoked
to explain why the differential rotation profile of the CZ has not been imposed on
the interior (Gough & McIntyre 1998). It is not currently possible to measure the
magnetic field beneath the solar surface, though calculations indicate that a poloidal
field of only 10−2 G (with global Alfve´n travel time ∼ 106 years) would be sufficient
to bring the Sun’s RZ into uniform rotation. If such a field exists in the Sun, and in
other solar-type stars, then magnetic stresses could inhibit the gradual development of
differential rotation, and prevent the formation of a critical layer by radiative diffusion.
The Alfve´n travel time across a region of radial extent R, is
τA ≈ 10
4yr
(
R
R¯
)(
1G
B0
)(
ρ
ρc
) 1
2
, (7.4.2)
where we have used the central density of the star, to give an upper limit, for a given
field strength. Mestel & Weiss (1987) suppose that the strength of the field lies in the
range 10−2 G . Bp . 103 G, so the global τA could be several orders of magnitude
shorter than this estimate. In any case, this indicates that magnetic effects could be
very important, since this timescale is probably much shorter than Eq. 7.1.6.
If the central ∼ 1% of the star is rotating with angular speed Ωp, then the ratio of
the energy in the magnetic field to the energy in the shear is
B20/(2µ0)
(1/2)R2Ω2p
≈ 8× 10−9
(
B0
1G
)(
0.01R¯
R
)2( P
1d
)2
. (7.4.3)
This tells us that unless we have a magnetic field much larger than the proposed upper
bound (which can probably be ruled out on empirical grounds; Friedland & Gruzinov
204
2004), then we are in the weak-field limit of Spruit (1999), in which the differential
rotation winds up the poloidal field to produce a strong toroidal field. This toroidal
field may then be subject to instabilities, such as the Tayler instability (Tayler 1973),
which would set in on the Alfve´n timescale to produce turbulence. However, the final
nonlinear outcome of such instabilities on angular momentum redistribution is unclear.
7.4.2 Goldreich-Schubert-Fricke instability
Throughout the RZ, η/ν À 1 (taking a value ∼ 106 just below the convection zone).
In this limit, a powerful hydrodynamic instability enabled by thermal diffusion results
if the specific angular momentum decreases outwards from the rotation axis. In both
of the mechanisms that we have outlined in this part of the thesis, a region in which
∂RR
2Ω(R) ≤ 0, (7.4.4)
is produced, where R is the cylindrical radius. This is unstable to the Goldreich-
Schubert-Fricke (GSF) instability (Goldreich & Schubert 1967; Fricke 1968; also see
Menou et al. 2004 for the inclusion of a weak magnetic field). The instability is
axisymmetric, and the maximum growth rate is comparable with the local rotation
period, which is approached for wavelengths λ . 2pi (η/N)1/2, where thermal diffusion
can overcome the buoyant restoring force. In our simulations with thermal diffusion,
λ . 2pi
(
κΩp
piC3R
)1/2
∼ 10−4, so we would not be able to resolve such instabilities. (In the
Sun, we expect λ . 10−7R¯ near the centre.) The nonlinear outcome of this instability
with regards to angular momentum redistribution and entropy mixing is unclear. In
addition, the presence of compositional gradients may be able to stabilise the flow
against such instabilities, so it is unclear how effective they will be at preventing the
gradual spin-up by radiative diffusion that we have presented in this chapter.
7.5 Conclusions
In this chapter we have identified a further process by which efficient dissipation of
IGWs could occur. If the waves are of too low amplitude to initiate breaking, but
reach the centre without significant attenutation on one passage of the RZ, then the
205
weak deposition of angular momentum through radiative damping of the waves, can
spin up the region of the central wavelength over a timescale of millions of years, until
a critical layer is formed. The subsequent evolution of the system is that discussed in
previous chapters of Part II of this thesis. This process could provide efficient tidal
dissipation in solar-type stars perturbed by less massive companions. However, it
may be prevented by hydrodynamic instabilities or magnetic stresses, that resist the
development of differential rotation. Further work is required to study whether this
remains a viable mechanism of tidal dissipation, in a detailed study including these
effects.
206
Chapter 8
Conclusion
In this thesis we have studied two problems regarding tidal interactions between short-
period extrasolar planets and their host stars.
In the first part, we adopted a simple model of tidal friction to understand the
secular spin-orbit evolution of HJs on inclined orbits, due to tides. We also analysed
the influence of stellar spin-down due to magnetic braking on the tidal evolution. We
found that simple timescale estimates can be misleading, and it is essential to consider
coupled evolution of the orbital and rotational elements, including dissipation in both
the star and planet, to accurately model the evolution.
The most important result of Part I is that the timescale for stellar spin-orbit align-
ment is comparable with the inspiral time for HJs. This result was obtained using the
constant lag time model, and is true if the ratio of the moments of inertia of the orbit
to that of the fraction of the star which tidally exchanges angular momentum with
orbit, is of order unity or smaller. This means that observed inclinations are likely
to be a relic of the migration process that produced these extraordinary planets, thus
highlighting the importance of RM effect observations of transiting planets, which can
determine the sky-projected angle between the stellar spin and the orbit. Since differ-
ent formation scenarios produce different distributions of this angle, these observations
have the potential to determine the formative histories of HJs. In particular, they can
distinguish between migration caused by planet-planet scattering or Kozai oscillations
combined with tidal dissipation in the star, and that produced by tidal interaction
with the gas disk.
207
The population of planets whose spin-orbit alignments have been observed with the
RM effect can be statistically analysed to give a distribution of i for HJs (Fabrycky
& Winn 2009). Such work indicates the surprising result that most HJs appear to be
misaligned, possessing a wide variety of spin-orbit angles (Triaud et al. 2010). This
distribution of i appears consistent with that predicted by the process of Kozai cycles
with tidal friction (Fabrycky & Tremaine 2007). Together with our result that the
spin-orbit angle is not likely to change appreciably over the lifetime of the system due
to tides, if future observations can confirm this i distribution, this would be consistent
with the hypothesis that Kozai migration may indeed be a primary formation mech-
anism of HJs. However, it is unlikely to be able to account for all of these planets.
These observations do seem to reduce the importance of disk migration at producing
HJs.
In the second part of this thesis, we studied the mechanisms of tidal dissipation in
solar-type stars. In particular, IGWs are launched at the CZ/RZ interface of a such
a star, by the tidal forcing of a short-period planet. The fate of these waves as they
approach the centre of the star was studied, primarily using numerical simulations.
In Chapter 3 we first derived a Boussinesq-type model of the central regions of a
solar-type star, together with a wave solution, which is exact in 2D. We then integrated
this system of equations in 2D, using spectral methods, studying the instabilities of
the wave as the amplitude of forcing was varied. We found that the waves undergo
instability and break above a critical amplitude, below which the waves reflect perfectly
from the centre of the star. Wave breaking leads to the formation of a critical layer,
which absorbs subsequent ingoing waves. This process can spin the star up from the
inside out, as the planet is pulled into the star.
In chapter 4 we extended these simulations to 3D. The wave solution valid near the
centre is not exact in 3D, and nonlinearities, though small, do not vanish. Nevertheless,
we confirmed that this process in 3D gives the same general picture as that in 2D.
One exception, however, is that the deposition of angular momentum by the waves
results in latitudinal differential rotation, though this is unlikely to significantly affect
the critical layer absorption.
We studied the implications of the production of a critical layer by wave breaking,
208
for tidal dissipation in solar-type stars, in Chapter 5. We calculated the tidal quality
factors Q′? that result from this process, improving a model by Goodman & Dickson
(1998). The validity of this model was verified by numerical integrations of the lin-
earised tidal response, in an extensive set of solar-type stellar models. We found that,
for a given planetary orbit, the dissipation resulting from critical layer absorption is
very efficient, with
Q′? ≈ 10
5
(
P
1 day
) 14
5
, (8.0.1)
which varies by less than an order of magnitude between all solar-type stars, in the
mass range 0.5 ≤ m?/M¯ ≤ 1.1, throughout their main-sequence lifetimes.
The critical amplitude required for wave breaking has been obtained, as a function
of the mass of the planet, and the strength of the stable stratification at the stellar
centre. This varies significantly between different stars, with breaking being more
likely in massive and more highly evolved stars, with stronger central condensations.
The implications of this mechanism for the survival of short-period extrasolar planets
has been discussed. In particular, if wave breaking occurs, a close-in planet will rapidly
spiral into its host star in only a few million years. However, this process only occurs
if the planet is sufficiently massive, or the star is sufficiently evolved. In fact, none of
the planets currently observed orbiting a solar-type star is likely to cause breaking at
the centre. This could be an important explanation for their survival.
We studied the tidal dissipation in F-type stars, which have masses in the range
1.1 . m?/M¯ . 1.8. This is in general weaker than in solar-type stars, due to
differences in their internal structure, most importantly the presence of a convective
core prevents the wave breaking mechanism that we have described in Part II of this
thesis. In addition, the convective envelopes of these stars are generally thinner than
in solar-type stars, so the dissipation of the equilibrium tide (and inertial waves if they
are excited) by turbulent convection is weaker. Contrary to common assumptions, Q′?
is found to vary with the internal structure of the star, with tidal frequency, and for
solar-type stars, exhibits a dependence on the amplitude of the tide. It is therefore
inappropriate to consider Q′? as a constant, uniform across all stars, except in making
qualitative statements about tidal evolution, or in constraining the magnitude of the
209
dissipation (such as we did in Chapter 2).
The results of this thesis were combined to propose a possible explanation for the
survival of all of the extrasolar planets currently observed in short-period orbits around
F, G and K stars. We provided several predictions, which will be tested in ongoing
studies of transiting planets, such as WASP and Kepler. In particular, we predict
that fewer massive planets will be found around G-stars in periods of less than 2 days,
i.e., those massive enough to satisfy Eq. 5.3.3. We also predict that the fraction of
short-period HJs in orbits shorter than 2 days around G-stars will decrease with stellar
age.
Our simulations in Chapters 3 & 4 indicate that nonlinearities do not play an
important role in preventing the reflection of the waves from the centre, unless the
wave has sufficient amplitude to overturn the stratification. In Chapter 6 we performed
a stability analysis of our 2D standing internal gravity wave near the centre of a
solar-type star, to determine whether the waves are subject to weaker parametric
instabilities below the critical amplitude required for wave breaking. We found that
although the waves undergo parametric instabilities, driven by entropy gradients in
the primary wave, at any amplitude, because the nonlinearity is spatially localised
in the innermost wavelengths, the growth rate scales inversely with the number of
wavelengths contained within the radiation zone. This means that the growth rates of
these instabilities tend to be sufficiently small that they do not result in astrophysically
important tidal dissipation. These results support our explanation for the survival of
short-period planets that we have presented in this thesis. We also identified the
instabilities that contribute to the early stages of wave breaking, and found that they
are driven by the free energy associated with primary wave entropy gradients, and are
strongly localised within the convectively unstable regions.
Finally, in Chapter 7, we proposed an alternative mechanism of tidal dissipation.
This involves the formation of a critical layer by the gradual damping of the waves
by radiative diffusion. Based on an order-of-magnitude estimate, which we followed
up with analytical calculations and numerical simulations, we found that this process
could operate for lower-mass planets. However, this mechanism is in conflict with
current observations, since it would pose a threat to the survival of all short-period
210
planets in orbits of less than 2 days. We discussed some hydrodynamic instabilities
and magnetic processes which may prevent this mechanism.
8.1 Future work
The subject of resonances has been avoided so far in this thesis. The probability of
the tidal frequency exactly matching some normal mode of the star at any one time
is tiny. However, evolution of the tidal frequency, due to changes in the stellar spin or
the orbit, or evolution of the stellar g-mode eigenfrequencies, can result in the system
passing through resonance. Simple estimates indicate that it is most likely for at least
one resonance to be passed in the lifetime of a system. It is not yet understood whether
this will result in the amplitude of the tide to grow sufficiently for wave breaking to
occur, when both radiative damping and the gradual evolution of the frequencies is
considered. I am currently performing some calculations to try to understand this
problem.
Part II of this thesis has studied a promising mechanism of tidal dissipation, which
may operate in stars with radiative cores. However, we have so far neglected the influ-
ence of magnetic fields and rotation. Future work is certainly required to understand
in more detail the effects of magnetic fields. In particular, a strong toroidal magnetic
field at the top of the RZ may be able to reduce the amplitude of the waves that are
launched. Large-scale magnetic fields may also be able to prevent the spin-up of the
interior by the process outlined in Chapter 7, though probably not the more rapid
spin-up due to wave breaking. In addition, they could affect the picture that we have
outlined for the long-term evolution of the RZ due to critical layer absorption. It is
also of interest to consider the effects on the wave amplitude by differential rotation
in the RZ.
Orbits with nonzero eccentricities, and inclinations with respect to the stellar equa-
tor, excite IGWs with different frequencies and azimuthal wavenumbers. In this thesis,
we only considered the waves excited by planets in circular, coplanar orbits, which have
m = 2. Future work is required to see whether the same process is efficient at circu-
larising the orbits of short-period planets, or of aligning the stellar spin with the orbit.
This could be studied in a straightforward extension of the simulations described in
211
this thesis.
Another route for further work regards core-envelope coupling in stars, and how
this can be produced, and will affect, tidal evolution of HJs. It has been proposed by
Winn et al. (2010), that the current population of misaligned planets can be explained
as a result of inefficient tidal dissipation in their host stars. This is because most of
the misaligned planets orbit F-stars, and we found in Chapter 5 that tidal dissipation
in those stars is likely to be weaker than in solar-type stars. Such stars may thus
be less likely to realign with the orbit than G-stars, which have stronger dissipation.
However, it is uncertain whether core-envelope coupling can occur in such stars, and
this is certainly an area which requires further study, since it is not yet understood,
even for the Sun.
To conclude, tidal dissipation in stars requires much further work before we can say
that we fully understand its effects in the evolution of short-period planetary systems.
This thesis has presented a mechanism that is likely to be important in solar-type
stars, but much further work is required. Hopefully myself and others will be able to
continue to explore this avenue in the next few years.
212
Part III
Appendices
213
Appendix A
A.1 Additional secular perturbing accelerations
In Chapter 2.1, we ignored the contribution of any perturbing accelerations except
tidal friction. In this section we list the remaining terms from Eggleton & Kiseleva-
Eggleton (2001), which we have rewritten without reference to eˆ, so that they are
regular for a circular orbit. The resulting contributions to f are listed in Mardling &
Lin (2002). Here we write down the additional contributions to the secular evolution
of the system due to binary (outer body) perturbations (b), quadrupolar distortion
of the star due to their spins and tidal bulges (q1), and general relativistic apsidal
precession (GR):
(
dh
dt
)
b
= −3Cbh
[
(1− e2)
h2
(n · h)(n× h)− 5(n · e)(n× e)
]
, (A.1.1)
(
dh
dt
)
q1
= −
α1
h(1− e2)2
(Ω1 · h) (Ω1 × h) , (A.1.2)
h
(
de
dt
)
b
= 3Cb(1− e
2) [2(h× e)− (n · h)(n× e) + 5(n · e)(n× h)] , (A.1.3)
h
(
de
dt
)
q1
=
α1
(1− e2)2
[
1
2
(
3
h2
(Ω1 · h)2 − Ω21
)
+
15Gm2
a3
f2(e
2)(1− e2)2
]
(h× e)
+
α1
h2(1− e2)2
(Ω1 · h) (Ω1 · h× e)h, (A.1.4)
h
(
de
dt
)
GR
=
3Gm12n
ac2(1− e2)
(h× e) , (A.1.5)
214
n is a unit vector normal to the binary (outer body) orbital plane, c is the speed of
light, and
α1,2 =
R21,2k1,2m2,1
2µna5
, (A.1.6)
Cb =
m3
(m12 +m3)
n2out
n
1
4(1− e2)1/2(1− e2out)3/2
. (A.1.7)
Similar terms for the quadrupolar distortion of the planet (q2) can be obtained by
including additional terms identical to Eqs. A.1.2 and A.1.4, with 1 → 2 (and m2 →
m1). These terms can be included on the right hand side of Eqs. 2.1.9–2.1.13. The
integrator used in Part I has been extended with these additional contributions, and
tested on the Kozai migration of HD-80606 b using the initial conditions of Wu &
Murray (2003). The quantity Cb gives the magnitude of the binary (outer body)
perturbation, and is related to the period of a Kozai oscillation (if one is induced),
which is on the order of (1/3Cb)(1− e2)−1/2.
A.2 Stability of the synchronous state without MB
Here the stability of the equilibrium state Ω˜ = n˜ is analysed, in the absence of MB.
We need to consider the energy (E) and total angular momentum (L) of the system,
which we write in the dimensionless form
L˜ = n˜−
1
3 + Ω˜, (A.2.1)
E˜ = −n˜
2
3 + Ω˜2, (A.2.2)
where
L˜ =
L
(Gm12R1)
1
2 µ
C
1
4 , (A.2.3)
E˜ =
2R1E
Gm1m2
C−
1
2 , (A.2.4)
are dimensionless measures of the total angular momentum and total energy of the
system, respectively. L˜ and E˜ represent the values that the total angular momentum
and total energy would take for a Keplerian orbit at radius R1, together with a factor
215
C involving the ratio of orbit to spin angular momentum at the stellar surface. These
definitions allow the equations to be written in a very simple form.
It is straightforward to see from Eqs. 2.3.1 & 2.3.2 that there exists a curve of
equilibrium points corresponding to spin-orbit synchronism (Ω˜ = n˜). This can also be
shown by minimising the energy at constant total angular momentum
∂E˜
∂ n˜
∣
∣
∣
Ω˜=n˜
constL˜
= 0. (A.2.5)
The stability of the equilibrium curve as a function of n˜ follows from
∂2E˜
∂ n˜2
∣
∣
∣
Ω˜=n˜
constL˜
=
2
9
n˜−
4
3
[
n˜−
4
3 − 3
]
. (A.2.6)
A synchronous state is stable if n˜ ≤ 3−
3
4 , which is equivalent to the criterion of Coun-
selman (1973) & Hut (1980), who proved that an equilibrium state reached through
tidal friction is stable if IΩµh ≤
1
3 , i.e., no more than a quarter of the total angular
momentum is in the form of spin angular momentum.
A.3 Tidal potential valid for any stellar obliquity
In this section we expand the tidal potential into its separate Fourier components.
This is required to study which tidal frequencies are relevant for significantly inclined
orbits, such as that of XO-3 b. We consider two bodies in mutual Keplerian orbit
with semi-major axis a and mean motion n. Adopt a coordinate system with origin
at the centre of body 1, and let this represent the star, and body 2 the planet. The
tidal potential experienced at an arbitrary point P in body 1 is the nontrivial term of
lowest order in r,
Ψ =
Gm2
2R5
[
R2r2 − 3(R · r)2
]
, (A.3.1)
where position vector of the point P in body 1 is r, and the position vector of the
centre of mass of body 2 is R(t). Body 2 is treated as a point mass, of mass m2.
We consider an inclined, circular orbit. Without loss of generality, we consider body
2 to orbit in a plane inclined to the (x, y)-plane by an angle i, so that its Cartesian
216
coordinates are
R = a (cosnt cos i, sinnt, cosnt sin i) , (A.3.2)
while the Cartesian coordinates of a point P are
r = r (sin θ cosφ, sin θ sinφ, cos θ) , (A.3.3)
where (r, θ, φ) are the usual spherical polar coordinates. Then
Ψ =
Gm2
2a3
r2 [1− 3 (cosnt cos i sin θ cosφ
+ sinnt sin θ sinφ+ cosnt sin i cos θ)2
]
. (A.3.4)
Let
P˜ml (cos θ) =
[
(2l + 1)(l −m)!
2(l +m)!
] 1
2
Pml (cos θ), (A.3.5)
where 0 ≤ m ≤ l and l ∈ Z+, denote an associated Legendre polynomial normalised
such that
∫ pi
0
[
P˜ml (cos θ)
]2
sin θdθ = 1. (A.3.6)
The tidal potential correct to arbitrary order in the stellar obliquity i can be expanded
as a series of rigidly rotating spherical harmonics of second degree,
Ψ =
Gm2
a3
[
8∑
j=1
Aj(i)r
2P˜mj2 (cos θ) cos(mjφ− ωjt)
]
, (A.3.7)
where the azimuthal order mj, frequency ωj, tidal (Doppler shifted) forcing frequency
ωˆj = ωj −mjΩ, and the obliquity dependent amplitude Aj(i) of each component are
m1 = 0, ω1 = 0, ωˆ1 = 0, A1 =
√
1
10
(
1−
3
2
sin2 i
)
, (A.3.8)
m2 = 2, ω2 = 2n, ωˆ2 = 2n− 2Ω, A2 = −
√
3
5
cos4
i
2
, (A.3.9)
m3 = 0, ω3 = 2n, ωˆ3 = 2n, A3 = −
3
2
√
1
10
sin2 i, (A.3.10)
217
m4 = 1, ω4 = 0, ωˆ4 = −Ω, A4 =
√
3
5
cos i sin i, (A.3.11)
m5 = 1, ω5 = 2n, ωˆ5 = 2n− Ω, A5 =
1
2
√
3
5
sin i (cos i+ 1) , (A.3.12)
m6 = 1, ω6 = −2n, ωˆ6 = −2n− Ω, A6 =
1
2
√
3
5
sin i (cos i− 1) , (A.3.13)
m7 = 2, ω7 = 0, ωˆ7 = −2Ω, A7 =
1
2
√
3
5
sin2 i, (A.3.14)
m8 = 2, ω8 = −2n, ωˆ8 = −2n− 2Ω, A8 = −
√
3
5
sin4
i
2
. (A.3.15)
There are eight components of the tide that contribute for arbitrary stellar obliquity.
For small inclination, the terms that are of first order in the obliquity are for m = 1
and are the j = 4 and j = 5 components above. For a coplanar orbit, only j = 1 and
j = 2 components above contribute, and these reduce to the first two components in
OL04.
For an orbit with significantly nonzero i, many components contribute to the tidal
force. Thus, for XO-3, for example, it is important to calculate the resulting Q′ for each
of the m = 0, m = 1 and m = 2 components of the tide. The relevant tidal frequencies
for this system cannot be calculated with any certainty, since the stellar spin period
has not been accurately determined. If we assume that the angle of inclination of the
stellar equatorial plane to the plane of the sky is ∼ 90◦, then we have Ω ∼ n currently.
The relevant tidal frequencies would then be ωˆ = 0,±Ω,±2Ω,±3Ω,±4Ω.
A.4 Ingoing/outgoing wave decomposition
Since we have an analytic solution to the Boussinesq-type problem (see §3.5 & §4.2),
we can deconstruct the numerical solution into a sum of IWs and OWs. Doing this
enables us to quantify the amount of angular momentum absorbed as these waves
approach and/or reflect from the centre. The approach we use in 3D is now described
(we use an analogous method in 2D).
At each location in the grid, i.e. every point with spatial coordinates (x, y, z),
we have four pieces of information, namely, ur, uθ, uφ and b. Thus, it is possible
to compute the (complex) IW and OW amplitudes Ain and Aout for the l = m = 2
218
wave at a single point (note that this is not true in 2D, where we require two or more
points for this calculation, because we only have the variables ur, uφ and b). This is,
however, computationally expensive, since the routines for computing Bessel functions
are relatively slow (typically several hundred times slower than a square root). In
addition, this method would be subject to potentially significant round-off errors. A
more efficient method, which also (in principle) allows a simple extension to calculate
the amplitudes Almin and A
lm
out, for several values of l and m, is described in this section.
We aim to fit the simulation output to a linear model, corresponding to an IW and
an OW. Hence, our problem is an overdetermined system of linear equations in terms
of the unknown wave amplitudes. Let us write our system as
yi =
4M∑
i=1
Aijxi, (A.4.1)
where y = (ur, uθ, uφ, b) is a vector of data variables at each grid point, of size 4N ,
where N = NxNyNz is the total number of grid points. M is the number of spherical
harmonics for which we compute the wave amplitudes, which is usually taken to be
one. x = (Ain, Aout) is a vector of size 4M < 4N , whose components are the (complex)
IW/OW amplitudes (for each l and m value), whose values are to be determined. The
matrix A contains the IW and OW radial functions and spherical harmonic functions,
evaluated at the selected grid points, and has size 4N × 4M . Its components at a
single point are
A =








Re[uinr (r, θ, ξ)] Im[u
in
r (r, θ, ξ)] Re[u
out
r (r, θ, ξ)] Im[u
out
r (r, θ, ξ)]
Re[uinθ (r, θ, ξ)] Im[u
in
θ (r, θ, ξ)] Re[u
out
θ (r, θ, ξ)] Im[u
out
θ (r, θ, ξ)]
Re[uinφ (r, θ, ξ)] Im[u
in
φ (r, θ, ξ)] Re[u
out
φ (r, θ, ξ)] Im[u
out
φ (r, θ, ξ)]
Re[bin(r, θ, ξ)] Im[bin(r, θ, ξ)] Re[bout(r, θ, ξ)] Im[bout(r, θ, ξ)]








, (A.4.2)
where each component uses the solution from §4.2 for an IW/OW (before taking real
parts). Taking only a single point, this matrix is non-singular for all radii, and is
related to the Wronskian. M is the number of spherical harmonics for which we
compute the wave amplitudes, which is usually taken to be one.
We use a method of least squares to fit our model to the data (Press et al., 1992).
This finds the best fit between the linear model data and simulation data, and com-
219
putes the solution for which the sum of the squared residuals is its least value. We
compute the coefficients x, which fit the model to minimise the least squares residuals,
i.e., find xˆ such that
||Axˆ− y||2 ≤ ||Ax− y||2, ∀x. (A.4.3)
This minimisation problem has a unique solution if the 4N columns of the matrix A
are linearly independent, which is true in our case, since W [Jν(kr)+ iYν(kr), Jν(kr)−
iYν(kr)] = 4ipir 6= 0, ∀r. The solution is found by solving the normal equations
xˆ = (ATA)−1ATy. (A.4.4)
We take into account radial variations in the amplitudes by splitting up the region
inside the forcing region into a set of concentric spherical shells of thickness δr ∼ λr/2,
after removing an inner region of a few grid cells. This approach assumes the solution
is locally independent of r, hence we can ignore radial derivatives of the amplitudes
within each shell. This is not valid in regions where the solution varies rapidly. In
addition, we speed up computation by stepping over the grid points in each Cartesian
direction, by factors istep, jstep, kstep, chosen to take values between 1 and 10. We always
ensure that sufficient grid points are available in each shell to accurately compute the
amplitudes.
We checked our method with the analytic wave solutions using Mathematica, and
wrote a Matlab routine, which reads in SNOOPY/ZEUS output and solves the linear
least squares problem for the wave amplitudes. We also compute the reflection coef-
ficient R, which is defined in Eq. 3.7.3. For perfect standing waves, Ain = Aout, and
R = 1. If the IW is entirely absorbed at the centre, then R = 0.
The main disadvantage of our approach is that ω 6= 1, l 6= 2,m 6= 2 components also
contribute to the amplitudes. This problem can be ignored if we trust the computed
values of R only where the solution is well described by the linear solution, i.e., far
from the wave breaking and forcing regions. We therefore reconstruct ur, and plot it
along the x-axis, uθ along the line y = z, uφ along the line y = x, and b along the
line y = x. All components of the solution are not zero for all radii along these lines
when ξ = 0 (this requires shifting the phase of the linear model solution when the
220
wave phase is nonzero in the simulation data). Comparing the reconstructed solution
with the simulation data allows us to check whether the deconstruction has worked.
A.5 ZEUS comparison
We confirm that the results are not dependent on the numerical method by reproduc-
ing the basic results using a stripped down version of ZEUS–2D1 (Stone & Norman,
1992). ZEUS solves the equations of ideal compressible hydrodynamics, using a simple
Eulerian method based on finite-differences, implemented using a covariant formalism,
enabling the use of non-cartesian orthogonal coordinate systems. For our problem we
solve the problem using cylindrical (r, φ) coordinates, which are the most natural to
use for our problem. However, the coordinate singularity at the origin requires that
we cut out a small region at the centre, on which we impose reflecting boundary con-
ditions. What may seem an advantage of this coordinate system, that is the higher
resolution at the centre, which is automatically present when we use a uniform grid in
r and φ, requires very short timesteps when the resolution is increased, as a result of
the CFL stability constraint. This becomes prohibitive as we increase the resolution
of the grid to above 100 × 150 in r, φ respectively, so only preliminary low resolution
runs were performed using this code. This is also because this code solves the com-
pressible equations, and therefore resolves sound waves. For our problem we require
a ratio of sound speed to radial group velocity of gravity waves χ = cs/cg,r ∼ 6× 103
(χ−1 is a measure of the importance of effects of compressibility), in order to repro-
duce an equivalent set-up to that used in the SNOOPY code above, so most of the
computational time is spent resolving sound waves. In the thesis we only analyse the
SNOOPY results, since they are at a much higher resolution, but here describe our
problem set-up in ZEUS, for completeness. We use a circularly symmetric parabolic
density stratification, ρ(r) = ρ0 − ρ2r2, and calculate the pressure (p) profile from
hydrostatic equilibrium. We solve the equations
Dρ = −ρ∇ · u, (A.5.1)
1which has kindly been made freely available by J.Stone at
http://www.astro.princeton.edu/∼jstone/zeus.html
221
Du = −
1
ρ
∇p+ g +
{ 0, rinner ≤ r < rforce,
f , rforce ≤ r < rdamp,
d, rdamp ≤ r < rbox,
(A.5.2)
D
(
e
ρ
)
= −
p
ρ
∇ · u, (A.5.3)
where e = (γ−1)p is the specific internal energy of the gas, f = −fr cos(2φ−ωt) er, and
d = −d(r)u. We choose γ = 5/3, as appropriate for a monatomic ideal gas. Radial
gravity has been implemented as a source term in the radial momentum equation:
g = −g1rer. Both the inner and outer boundaries have reflecting boundary conditions,
and we also implement a linear frictional damping in a region adjacent to the outer
boundary, to prevent the reflection of (most of) the outgoing wave energy. A parabolic
smoothing function d(r) =
(
r−rdamp
rbox−rdamp
)2
is used in the damping terms. We choose
rbox = 1.0, rdamp = 0.9, rforce = 0.85 and rinner = 0.01. In the code we specify
ρ0, ρ2, ω, λr & f˜r; the other relevant parameters are calculated from these. Choosing
ρ0 = 1.0, ρ2 = 0.1, ω = 1.0, λr = 0.1 and a suitable value for f˜r is sufficient to fully
specify the problem.
A minimum value of χ = 6285 is found from these initial conditions. Such a
high value is required for the wavelength of the gravity waves to be λr ' 0.1, which
allows ∼ 8 wavelengths to be resolved within the grid. This value is not much smaller
than that appropriate at the centre of a solar-type stars (χ ∼ 104 − 105). We set
up the initial conditions in such a way to minimise this value given the above input
parameters.
Calculations were performed in an inertial frame, though the results were inter-
preted in a frame rotating with the angular pattern speed of the tide, Ωp = ω/2. In
this rotating frame, the linear wave solution is steady, which allows the instability to
be easily recognised as departures from a steady state.
With this resolution, there are some numerical errors near the inner boundary.
This results from the fact that we only remove a small region near the centre, which is
comparable with the size of a grid cell. In addition, the code has no explicit viscosity
or thermal conduction, so we have less control over the scales of dissipation, than
in SNOOPY. Nevertheless, ZEUS reproduces the same basic results as the SNOOPY
222
code, which indicates both that the effects of nonzero compressibility are not important
in this problem, and that our basic results are not dependent on the numerical method.
A.6 Analytic calculation of F in the launching re-
gion
In this section, we adopt the notation of OL04, to avoid reproducing the results in
that paper, since we are simply extending the results of their §4.4. In the launching
region, we want to match the solutions of the RZ and CZ. Near the boundary r = rb,
we assume N2 ∝ (rb − r)α, where α ∈ R+, for the moment left unspecified. The
characteristic radial extent of the transition region is of order ²β, where β ∈ R+, and
² is the ratio of the spin frequency to the dynamical frequency, which is much smaller
than unity for slowly rotating bodies. We write
r = rb − ²
βx, (A.6.1)
where x is an inner variable, which is of order unity in the launching region. We can
write
N2 = ²αβDxα +O(²2αβ), (A.6.2)
where D = dN
2
dxα > 0. If we pose the perturbation expansions
u′r ∼ ²
3−β(1+α)u¯′r(x, θ), (A.6.3)
u′θ ∼ ²u¯
′
θ(x, θ), (A.6.4)
u′φ ∼ ²u¯
′
φ(x, θ), (A.6.5)
ρ′ ∼ ²2−β ρ¯′(x, θ), (A.6.6)
p′ ∼ ²2p¯′(x, θ), (A.6.7)
Φ′ ∼ ²2Φˇ′(rb, θ) + ²
2+βΦ¯′(x, θ), (A.6.8)
then we obtain the linearised system of equations 111-116 from OL04, except that we
replace x by xα in the buoyancy equation (Eq. 115). A linearised equation for the
223
modified pressure (W¯ = p¯
′
ρ0
−
p′(e)ρ2
ρ20
+ Φˇ′) in the launching region can now be derived,
for which
LW¯ +
r2
D
∂x
(
x−α∂xW¯
)
= 0, (A.6.9)
where all coefficients are evaluated at r = rb. This equation can be solved using the
separation of variables
W¯ =
∑
i
f (i)(z)wi(θ). (A.6.10)
The operator L contains only angular derivatives, and Lw¯i = λiw¯i, where λi = k
2r2
N2 ,
and k is the WKB wavenumber of the waves. We thus obtain the equation
f (i)(z) + ∂z
(
z−α∂zf
(i)(z)
)
= 0, (A.6.11)
where we have defined z = κix, and the length scale
κ−1i =
(
Dλi
r2
)− 12+α
. (A.6.12)
In the text this lengthscale is referred to as Llaunch. The solution of this equation can
be written down as a linear combination of Bessel functions of the first and second
kinds, of order α+1α+2 :
f (i)(z) = aiz
α+1
2
{
Jα+1
α+2
(
2
2 + α
z
2+α
2
)
+ siiYα+1
α+2
(
2
2 + α
z
2+α
2
)}
, (A.6.13)
so that the complete solution in the transition region is
W¯ =
∑
i
aiz
α+1
2
{
Jα+1
α+2
(
2
2 + α
z
2+α
2
)
+ siiYα+1
α+2
(
2
2 + α
z
2+α
2
)}
wi(θ) (A.6.14)
This solution should match onto the WKB solution in the RZ when xÀ 1, constraining
β as a function of α. This arises from considering the asymptotic form of the phase at
large z, whose radial derivative should match the WKB wavenumber in the RZ. From
this we find β = 2/(2 + α).
If we from now on restrict ourselves to α = 1/2, then the corresponding complete
224
solution in the transition region is the sum
W¯ =
∑
i
aiz
3
4
{
J 3
5
(
4
5
z
5
4
)
+ siiJ 3
5
(
4
5
z
5
4
)}
wi(θ). (A.6.15)
Similarly we can write the radial displacement as
ξ¯′r =
∑
i
bi
√
z
{
J 2
5
(
4
5
z
5
4
)
± itiJ 2
5
(
4
5
z
5
4
)}
wi(θ), (A.6.16)
for a superposition of wave solutions in this model.
Deep in the RZ, our solution should reduce to a wave with an inwardly directed
group velocity, for which si = ±1 (sign depending on that of the frequency). The
asymptotic forms of the radial part of this solution at large z is
z
3
4
{
J 3
5
(
4
5
z
5
4
)
± iJ 3
5
(
4
5
z
5
4
)}
∼ (−1)
9
20
√
5
2pi
z
1
8 exp
{
∓i
4
5
z
5
4
}
. (A.6.17)
Matching of W at r = rb requires the solution in the CZ to be continuous with the
solution in the transition region at z = 0. This determinines the amplitudes ai. The
inward energy flux in (inertia-)gravity waves is found from evaluating
F = pi
∫ pi
0
Re
[(ω
ωˆ
)
p′∗u′r
]
r2 sin θdθ (A.6.18)
= ²
19
5
(
2
5
) 1
5 pi2
[
Γ
(
3
5
)]2ρ0ω(1)sgn(ωˆ)
(
r2
D
) 2
5
×
∑
i
λ
3
5
i
|
∫ pi
0 w
∗
i
[
Wˇ (rb, θ)− W¯ (p)
]
sin θdθ|2
∫ pi
0 |wi|
2 sin θdθ
, (A.6.19)
which can be compared with OL04 Eq. 124. We find in general that F ∝ ω
8+3α
2+α . Thus
we see that the main change in the energy flux from modifying the profile of N2 in the
transition region is to change its frequency dependence. There are also O(1) changes
to the numerical factors.
225
A.7 Toy model: parametric instability of primary
wave
Parametric instability is a type of resonant triad interaction in which the transfer
of energy from a parent (subscript p) mode, with amplitude Ap, destabilises a pair
of daughter (subscript d1, d2) modes (which exist when Ap = 0). These can then be
damped or subject to further nonlinear interactions (to produce granddaughter modes,
and so on). The frequencies of the modes must satisfy an approximate temporal
resonance condition ωp ≈ ωd1 + ωd2, for parametric resonance to occur.
The equations governing the temporal evolution of the mode amplitudes take the
form (e.g. Dziembowski 1982; Wu & Goldreich 2001)
A˙p = γpAp − iωpAp + iωpσAd1Ad2, (A.7.1)
A˙d1 = −γd1Ad1 − iωd1Ad1 + iωd1σApA
∗
d2, (A.7.2)
A˙d2 = −γd2Ad2 − iωd2Ad2 + iωd2σA
∗
d1Ap. (A.7.3)
In these equations, γj is the linear growth/damping rate of mode j, and σ is the
nonlinear coupling strength for these three modes. Here Aj is the amplitude of mode
j, with the energy in that mode being proportional to |Aj|2.
The coupling coefficient σ is largest when ωd1 ≈ ωd2 ≡ ω, and therefore ωd1 ≈ ωp/2.
If the daughter modes have similar frequency, then we can assume that they have
similar spatial scales. Hence we can take their damping rates to be the same, i.e.,
γd1 = γd2 ≡ γ. To consider the initial stages of the breaking process we take Ap to be
approximately constant in time. In our problem the primary wave is maintained at a
constant amplitude due to forcing, and is not unstable. The evolutionary equations
reduce to
A˙d1 = −γAd1 − iωAd1 + iωσApA
∗
d2, (A.7.4)
A˙d2 = −γAd2 − iωAd2 + iωσA
∗
d1Ap. (A.7.5)
226
If we take Ad1 ∝ exp st, then the growth rate is
Re [s] = −
γ
2
+
1
2
ωσ|Ap|. (A.7.6)
The growth rate is reduced if the detuning ∆ω = ωp − ωd1 − ωd2 6= 0, by changing the
second term to (1/2)
√
ωd1ωd2σ2|Ap|2 − (∆ω)2. From this model, we expect γ 6= 0 to
simply reduce the growth rate for a given mode. In addition the growth rate scales
linearly with the amplitude of the primary (parent) mode. The threshold amplitude
for instability in this simple model is
|Ap| ≥
γ
ωσ
, (A.7.7)
which depends on the coupling strength σ.
The spatial dependence of the interaction is contained in the coupling coefficient σ,
which contains an integral of the product of the three eigenfunctions. This toy model
of parametric instability is useful as a simple model to understand some of the results
of § 6.6. It is interesting to note that in this model, ω ≈ m/km,n ∼ n−1p for A ¿ 1,
since in this limit the daughter modes have frequencies comparable with the linear
mode frequencies. This results in a growth rate scaling inversely with np.
A.8 Acronyms and abbreviations
Several acronyms and abbreviations regularly used in this thesis are listed alphabeti-
cally below.
AU: Astronomical Unit
CZ: Convection Zone
HJ: Hot Jupiter
IGW: Internal Gravity Wave
IW: Ingoing Wave
MB: Magnetic Braking
227
OCZ: Outer Convection Zone
OW: Outgoing Wave
RM: Rossiter-McLaughlin
RV: Radial Velocity
RZ: Radiation Zone
228
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