Nonlinear tides in giant planets

Adrian BarkerDAMTP, University of Cambridge

(formerly at CIERA, Northwestern)

With: C. Baruteau, B Favier, P Fischer, Y Lithwick, G Ogilvie

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

Adrian Barker DAMTP, Cambridge

Mercury

Introduction

• Tidal potential deforms the planet and excites internal flows within (if , spin-orbit misaligned)

• Dissipation of tidal flows causes spin-orbit evolution ( , alignment)

• e.g. circularise an eccentric orbit in

Tide in planet

am?

Rp

⌧circ

⇡ 65 Myr✓

Q0

106

◆ ✓P

orb

3d

◆13/3

What is ?

Adrian Barker DAMTP, Cambridge

Q0(!̂,⌦, ✏, internal structure)

⌦

⌧e

✏ =m?

mp

✓Rp

a

◆3

⇠ 10�2

✓1d

P

◆2

⇠ ⇠r

Rp⌦ 6= n, e 6= 0

⌦ ! n, e ! 0

• 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

Introduction

! = ±2⌦kz

k✏ ⇠ ⇠r

Rp⇠ 10�2

✓P

1 d

◆2• Tidally forced rotating fluid planets have

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

• 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

Elliptical instabilityTopic 1

Simulations of small patch of rotating tidally deformed planet in a periodic box

Barker & Lithwick 2013 & 2014, MNRAS

⇠ 10�2

✓1d

P

◆2

✏ ⇠ ⇠r

Rp⇠ 10�2

✓P

1 d

◆2• Tidally forced rotating fluid planets have elliptical streamlines (“equilibrium tidal bulge”), which may be subject to the elliptical instability => parametric excitation of small-scale inertial waves

• 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

Elliptical instabilityTopic 1

Barker & Lithwick 2013 & 2014, MNRAS

⇠ 10�2

✓1d

P

◆2

(With a weak magnetic field)

• Tidal forcing can excite small-scale inertial waves (when , 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]

⌦

|!̂| < 2|⌦|

Topic 2

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS

Ogilvie 2009

Tides in rotating planets with a core

Q0

⇢

rcRp

ur(r = Rp) = A Re⇥Y 2

2 (✓,�)e�i!̂t⇤

⌫

• Simplified model: (initially) uniformly rotating homogeneous incompressible fluid in a spherical shell

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

• We have performed hydrodynamical numerical simulations to study the effects of nonlinearities as the amplitude of forcing is increased

Topic 2

⌦

rc = 0.5

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS

Tides in rotating planets with a core

Q0

⇢

rcRp

ur(r = Rp) = A Re⇥Y 2

2 (✓,�)e�i!̂t⇤

⌫

• Simplified model: (initially) uniformly rotating homogeneous incompressible fluid in a spherical shell

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

• We have performed hydrodynamical numerical simulations to study the effects of nonlinearities as the amplitude of forcing is increased

Topic 2

⌦

rc = 0.5

⇢

rcRp

⌫

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS

Tides in rotating planets with a core

Topic 2

⌦

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS

Tides in rotating planets with a coreDifferential rotation develops... A = 10�2

⌫ = 10�5

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

(x, y)(x, z)

Topic 2

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS

⌫

Tides in rotating planets with a core

Topic 2

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS

Tides in rotating planets with a coreDifferential rotation scaling with viscosity...

⌫Astrophysical regime...

A = 10�2

Q0 ⇠ 106✓Prot

1 d

◆✓Ptide

1 d

◆✓0.2

rc

◆5 ✓10�3

D

◆

Topic 2

⌫ = 10�5A = 10�2

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS

Tides in rotating planets with a coreDeparture from linear theory...

Crudely can be thought to correspond with:

• 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. ) => 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

⌫ ⇠ 10�18

Topic 2

Favier, Barker, Baruteau & Ogilvie, 2014, MNRAS

Tides in rotating planets with a core

• 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)

The End! Conclusions

Adrian Barker DAMTP, Cambridge