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Nonlinear tides in giant planets
Adrian Barker DAMTP, University of Cambridge (formerly at CIERA, Northwestern)
With: C. Baruteau, B Favier, P Fischer, Y Lithwick, G Ogilvie


Introduction

Motivation

·

Shortest period hot Jupiters (M ~ MJup, P < 10 d) have preferentially circular orbits => thought to be explained by tidal dissipation inside the planet (e.g. Rasio et al,1996) If so, what mechanisms are responsible?

· ·

Anomalously large radii of some HJs may partly be explained by tidal heating in the planet Orbital evolution of Jupiter/ Saturn satellites (Lainey et al, 2009,2012) due to tides
Mercury

Adrian Barker

DAMTP, Cambridge


Introduction

Tide in planet
m? = mp Rp a
3

· Tidal potential deforms the planet and

excites internal flows within (if 6= n, e 6= 0 , spin-orbit misaligned) evolution ( ! n, e ! 0 , alignment)

13/3

10

2



1d P



2

r Rp

· Dissipation of tidal flows causes spin-orbit · e.g. circularise an eccentric orbit in
circ 65 Myr e


R
p



Q 10

0 6

Porb 3d

a
?

m

?

What is

Q0 (^ , , , internal structure) !

· Linear theory contains uncertainties (e.g. Zahn, Ogilvie, Papaloizou, Ivanov, Wu).
Nonlinear effects mostly unexplored.

· Even though a tide may be "weak", nonlinear fluid effects can be important

Adrian Barker

DAMTP, Cambridge


Topic 1

· Tidally forced rotating fluid planets have

Elliptical instability
kz ! = ±2 k
r 10 Rp
2

elliptical streamlines ("equilibrium tidal bulge"), which may be subject to the elliptical instability => parametric excitation of small-scale inertial waves



P 1d 1d P

22

· Can lead to turbulence => is the resulting

turbulent dissipation sufficient to explain the circularisation of hot Jupiters?

· Not for P>2.5d, but may play a role at shorter
periods. Naive estimate: Simulations of small patch of rotating tidally deformed planet in a periodic box

· Uncertainties: presence of a core, turbulent
convection, global effects

Barker & Lithwick 2013 & 2014, MNRAS


Topic 1

· Tidally forced rotating fluid planets have

Elliptical instability
2 r 2 2 1dP 10 10 P1 d Rp
2

elliptical streamlines ("equilibrium tidal bulge"), which may be subject to the elliptical instability => parametric excitation of small-scale inertial waves

(With a weak magnetic field)

· Can lead to turbulence => is the resulting

turbulent dissipation sufficient to explain the circularisation of hot Jupiters?

· Not for P>2.5d, but may play a role at shorter
periods. Naive estimate:

· Uncertainties: presence of a core, turbulent
convection, global effects

Barker & Lithwick 2013 & 2014, MNRAS


Topic 2

Tides in rotating planets with a core

· Tidal forcing can excite small-scale inertial waves

^ (when |! | < 2|| , e.g. linear theory by Ogilvie & Lin, Papaloizou & Ivanov, Wu)

· Linear theory: strong frequency dependence of tidal
dissipation, strongly enhanced when inertial waves are excited



· What effects do fluid nonlinearities have? (Wave

breaking, generation of "mean flows", interaction with turbulent convection...)

· [Model is also relevant for terrestrial planets with

deep oceans and Neptune/Uranus-mass planets, and to convective envelopes of stars]

Ogilvie 2009

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS


Topic 2

Tides in rotating planets with a core


· Simplified model: (initially) uniformly rotating

homogeneous incompressible fluid in a spherical shell


c Rp

· Linear calculations indicate strong frequency dependence
of the dissipation (Ogilvie 2009)

· We have performed hydrodynamical numerical

r

simulations to study the effects of nonlinearities as the amplitude of forcing is increased

2 ur (r = Rp ) = A Re Y2 (, )e

i! t ^



rc = 0.5

Q0

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS


Topic 2

Tides in rotating planets with a core


· Simplified model: (initially) uniformly rotating

homogeneous incompressible fluid in a spherical shell


c Rp

· Linear calculations indicate strong frequency dependence
of the dissipation (Ogilvie 2009)

· We have performed hydrodynamical numerical

r

simulations to study the effects of nonlinearities as the amplitude of forcing is increased

2 ur (r = Rp ) = A Re Y2 (, )e

i! t ^



rc = 0.5

Q0

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS


Topic 2

Tides in rotating planets with a core
A = 10
= 10
2
5

Differential rotation develops...


Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS


Topic 2

Tides in rotating planets with a core

Differential rotation can become unstable to shear instabilities, which regulate its amplitude (occurrence depends on , A )

(x, z )

(x, y )

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS


Topic 2

Tides in rotating planets with a core

Differential rotation scaling with viscosity...

A = 10

2

Astrophysical regime...



Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS


Topic 2

Tides in rotating planets with a core
Crudely can be thought to correspond wi th:
0 6

Departure from linear theory...

Q 10

Prot 1d

Ptide 1d

0.2 rc

5

10 D

3



A = 10
= 10

2
5

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS


Topic 2

Tides in rotating planets with a core

· Angular momentum is deposited non-uniformly => planet does not spin up/down
as a solid body, instead becomes (cylindrically) differentially rotating in the process

· Departure from linear theory is observed, partly due to differential rotation and
partly to the generation of small-scale waves

· Note that simulations cannot reach the tiny molecular viscosities relevant for a
giant planet or star (e.g. 10 18 ) => may get different behaviour as viscosity is decreased (hopefully we can obtain scaling laws and extrapolate)

· Further uncertainties: effects of turbulent convection, magnetic fields, density
stratification, realistic outer boundary condition

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS


The End!

Conclusions

· Tidal interactions shape the observable properties of short-period extrasolar
planets (and close binary stars)

· Major contribution to tidal dissipation from small-scale waves in fluid layers of
planets/stars. Significant uncertainties remain.

· Nonlinear fluid effects can be important and (probably) require numerical
simulations to quantify. Two examples: 1. The elliptical instability may play a role in circularising very short-period hot Jupiters with periods <~2 days, and synchronising their spins out to ~3.5 days 2. Tidal excitation of inertial waves in planets with a core: nonlinearities generate differential rotation in the interior & departure from linear theory (probably more important than 1. for P>~2d)

Adrian Barker

DAMTP, Cambridge