Atmospheric Dynamics
of Giant Planets Insideand Outside the SolarSystem
Adam P. Showman
University of Arizona
Collaborators: Yuan Lian, Yohai Kaspi, LorenzoPolvani, Jonathan Fortney, Nikole Lewis, Daniel
Perez-Becker, Mark Marley, Heather Knutson,

A new frontier: directly imaged planets andbrown dwarfs

Brown dwarf basics•Brown dwarfs are fluid hydrogen objects intermediate in mass between giant planetsand stars. They are often free floating, though many also orbit stars.

•Presumed to form like stars (i.e., directly collapsing from a hydrogen cloud) but havemasses too low to fuse hydrogen. Generally defined as objects with masses of 13 to~80 Jupiter masses.

•Since they cannot fuse hydrogen, they cool off over time (like Jupiter). But massive
brown dwarfs cool slowly and can still have surface temperatures >1000 K even after
many billions of years
Evolution tracks: luminosity versus time•

Over a wide mass range (~0.3 to ~80
Jupiter masses), brown dwarfs andgiant planets have radii very close toJupiter’s.

•~1000 brown dwarfs have been
discovered, mostly with hightemperature (>700 K) but now
including objects as cool as 300-400 K.

Burrows et al. (2001)

MLTJupiter

Burgasser (2009, “The brown-dwarf exoplanet connection”)

Typical brown dwarf infrared spectra

Brown dwarfs are classified according to their IR spectra into M, L, T, and Y (from hot tocold). Unlike most stars, their spectra are dominated by molecular features. Dust (i.e.,silicate clouds) affects the spectrum of M and L dwarfs, but not T dwarfs.

datamodel with clouds

Saumon et al. (2006)

T dwarfs are generally cloud free:This behavior is explained by the factthat condensate levels lie in theatmosphere for hot objects (M, Ldwarfs) but sink into the interior forcool objects (T dwarfs):Burrows et al. (2001)

model without clouds

Tsuji et al. (2004)

Brown dwarfs show evidence for condensate (dust) clouds
L dwarfs are cloudy, leading to flat spectral features:brightM
Saumon & Marley (2008)

Magnitude (brightness) in J band, which isa spectral window

Color-magnitude diagrams are useful for understanding
overall trends among brown dwarfs

LTdimblue

redColorThe change in color across the L/T transition is due to the loss of clouds, which opens
the spectral windows. This occurs better in J than K, causing a shift to the blue as theclouds disappear.

brightM
Saumon & Marley (2008)

Magnitude (brightness) in J band, which isa spectral window

Color-magnitude diagrams are useful for understanding
overall trends among brown dwarfs“J-band bump”LTdimblue

redColor

The change in color across the L/T transition is due to the loss of clouds, which opens
the spectral windows. This occurs better in J than K, causing a shift to the blue as theclouds disappear.

•Although the loss of clouds across the L/Ttransition makes sense, the details are apuzzle: the transition occurs too fast.
–

1D models of uniform cloud decks sinking
into the interior predict that the J-band flux
continually dims across the transition:–

But in reality the J-band flux actually
increases temporarily across the transition(the “J-band bump”), despite the fact that T
dwarfs are cooler than L dwarfs–

This suggests that the cloud decks are notsimply disappearing from view, butbecoming patchy or getting thin as they do so

1D models that assume the cloud deck gets
patchy across the transition do a much betterjob of reproducing the “J-band bump”

This suggests a strong role formeteorology in controlling the transition

Marley et al. (2010)•Burrows et al. (2006)

The L/T transition

Chemical disequilibrium
In cool giant planets and brown dwarfs, the equilibrium form of carbon and nitrogenat the top are CH4 and NH3. The equilibrium form at depth are CO and N2.

•In the absence of dynamics, equilibrium would prevail. But vertical mixing candredge CO-rich, CH4-poor, and NH3-poor air from depth and mix it into theatmosphere.

•This will result in an excess of CO, and a deficit of CH4 and NH3, in the atmosphere•

Just such excesses and deficits are observed, and are interpreted as the result ofvertical mixing. The observed abundances can be used to constrain the mixing rates
Saumon et al. (2006)•Thus, dynamics is required to explain the chemical disequilibrium

T2.5 brown dwarf SIMP 0136 shows weather
variability

Artigau et al. (2009); see also Radigan et al. (2012), Buenzli et al. (2012), andmany upcoming papers by Apai, Metchev, Radigan, Flateau, ….

Weather on brown dwarfs and directly imaged giant
planets
Evidence:•Clouds•

Disequilibrium chemistry (quenching of CO, CH4, NH3)•

Lightcurve variability (cloudy and cloud-free patches rotating in and out ofview)Dynamical regime:•Rapid rotation (P ~ 2-12 hours) implies rotational domination•

Vigorously convecting interior underlies stably stratified atmosphere•

No external irradiation
no imposed horizontal gradients in heating ortemperature (unlike solar system planets or transiting exoplanets)•Wave generation will play a key role⇒

Convection in a brown dwarf drives atmospheric waves

Freytag et al. (2010)

Showman & Kaspi (arXiv 1210:7573)

Rossby numbers

Convection will bedominated by rotation
at large scales

Rapidly rotating model

Slowly rotating model

Showman & Kaspi(arXiv 1210:7573)

Temperature perturbations
near top of convection zone

Showman & Kaspi(arXiv 1210:7573)

Equator-pole temperature differences at top ofconvection zone (~1 bar)Temporal variability

Wave-driven atmospheric circulation on
directly imaged EGPs and brown dwarfs

Model of wave-driven circulation•

Assume a given amplitude for the eddy acceleration, A, and solve for flow amplitudes
using primitive equations in log-pressure coordinates•

Zonal momentum balance•∂v z −z+ e (e ϖ )= 0Continuity∂yfv ≈ A
which to order-of-magnitude is vl ≈ϖH•

Thermodynamic energy: assume a balance between vertical advection and radiation,
parameterized with Newtonian heating/cooling: H 2 N 2 ∆Thoriz≈ϖτ radR•

Meridional momentum balance is thermal wind:•From this set we can derive equations for v, ϖ ,∆Thoriz ,∆U∆U ≈Rl∆Thoriz Hf∆Thoriz ≈ (ηc p T )1/ 2

NHR2lHN
1/ 2∆U ≈ (ηc p T)f(ηc p T)1/ 2
ϖ ≈ 1/ 2≈
HNτ radc p pHN

η1/ 2 gT 7 / 2σ

where η is adimensionless efficiency
giving fraction of theradiated heat flux that
goes into the wave driving

To within factors of order unity,∆Thoriz/T is η1/2 times the ratio of the gravity wave speed to the sound speed∆U over the sound speed is η1/2 times the ratio of the Rossby deformation radiusto the dominant horizontal length scale of the flow•

The time for the flow to advect vertically over a scale height is η-1/2τrad times theratio of gravity wave speed to sound speed

Efficiency

Wave-driven atmospheric circulation causes spatially coherent vertical motions
and horizontal temperature differences, helping to explain cloud patchiness andchemical disequilibrium

Showman & Kaspi(arXiv 1210:7573)

Conclusions•

Extensive evidence now exists for dynamics and weather in shaping observations ofbrown dwarfs, including the existence of clouds, the properties of the L/T transition,
chemical disequilibrium, and temporal variability•

We presented the first global models of the atmospheric dynamics on these objects.
The dynamics will be rotationally dominated (Ro << 1). At large scales, convection in
the interior will align in the direction of the axis of rotation, and the heat flux will beenhanced at the poles relative to the equator.

•The convection will generate a wealth of atmospheric waves. Just as for planets inour solar system, this will drive a large-scale circulation in the stratified atmosphereconsisting of geostrophic turbulence possibly organizing into jets and vortices.

•We presented a simple analytic theory of this circulation, suggesting the existence ofhorizontal temperature differences of ~10-100 K, wind speeds of ~10-300 m/sec, andvertical velocities that advect air vertically over a scale height in ~105-106 sec.

•This vertical motion can help explain the chemical disequilibrium, and the impliedorganization of temperature perturbations and winds suggests that patchy clouds canform near the L/T transition, helping to explain observations of variability.

Zonal (east-west) winds on the giant planetsBasic Jet Scenarios• Models for jet structure:– Shallow: Jets confined to outermost scaleheights below the clouds– Deep: Jets extend through molecularenvelope (Taylor-Proudman theorem)• Models for jet pumping:– Shallow: Turbulence at cloud level(thunderstorms or baroclinicinstabilities)– Deep: Convective plumes penetrating
the molecular envelope

These issues are among the mostimportant unsolved problems inplanetary atmospheres!

Puzzles• What causes the banded structure, with ~20 jets on Jupiter andSaturn yet only ~3 on Uranus and Neptune? What is the jetpumping mechanism?

• How deep do the jets extend?
• Why do Jupiter and Saturn have superrotating equatorial jets
whereas Uranus and Neptune do not?

• What causes the vortices? What controls their behavior? Howdo they interact with the jets?

• What is the temperature structure and mean circulation of thestratosphere and upper troposphere?

Deep convection models

Thick shell(Christensen 2001, 2002;
Aurnou & Olson 2001;
Kaspi et al. 2009,
Jones & Kuzanyan 2009,Showman et al. 2011, etc)

Thin shell(Heimpel et al. 2005;
Heimpel & Aurnou 2007;
Aurnou et al. 2008)

Lian & Showman (2010)

Jupiter

SaturnUranus/Neptune

Motivation• Convection in interiors of giant
planets has been suggested as amechanism for jet formation (Aurnouand Olson 2001; Christensen 2001,2002; Heimpel et al. 2005, etc)

• But simulations of this process canonly be performed at parametersettings far from the Jovian regime.

• Are the simulations relevant to theJupiter? What controls the trends
observed in the simulations, and how
to extrapolate them to giant planets?

Heimpel et al. (2005)

Non-Dimensional Parameters
Modified-flux Rayleigh, Ekman, and Prandtl numbers:

RaF =*αgFtotρc p Ω D
32

E=

νΩD2ν
P=
κ

On Jupiter, RaF* ~ 10-14, but published simulations generally explore
values ~10-6 – 10-2. This implies that the heat fluxes in the simulations

ρc p Ω D
*
F=RaF
αg3

are too large by a factor of 105-1010.

2

Challenges with deep convection models
simulations
10-4
10-6?
E?
Jupiter10-1510-14

10-6 10-2

RaF*
Would convection at Jupiter-like RaF* and E produce Jupiter-likewind speeds? How to extrapolate to Jupiter? This is not known!

Kaspi, Flierl, Showman (2009)

Simulations fromChristensen (2002)

Simulations from
Kaspi et al. (2009);
Showman et al. (2011)

III

Simulations fromChristensen (2002)

III?

IISimulations from
Kaspi et al. (2009);
Showman et al. (2011)

IIII?

Convective velocities are well
explained by the relation

 αgF 1/ 2w ≈  ρc p Ω which can be nondimensionalizedto yield(≡ (∆Ra )Roconv ≈ RaF − RaF** 1/ 2F)

*crit 1/ 2

Can we understand the dependence of jet speeds
on parameters? Consider Regime I.
Convection releases potential energy per mass per unit timeδρÝgw ≈ α δT g w
P≈ρNow F ≈ ρc p wδT which implies thatαgFÝP≈ρc pSuppose a fraction ε of this energy pumps the jets and is resisted by viscousdamping with a viscosity ν. Then εαgF 1/ 2
U ≈ k −1 ρc pν ⇒ Ra FRo ≈kD  E ε1/ 2* 1/ 2Thus, constant-Ro contours should have slopes of one in the RaF*-E plane!

Christensen’s (2002)
simulations

Analytic scaling
for Regime I(constant ε)Showman et al. (2010)

What about regime II? It exhibits strong
zonal/radial velocity correlations
Weaklysupercritical

Strongly
supercritical

Can we understand the jet speeds in Regime II?
Zonal momentum balance is between jet acceleration and frictional damping:∇ ⋅ (u'v s ') ≈ ν∇ u2which to order of magnitude isu'v s ' ≈ νk u

If we assume that individual eddy velocities scale with convective velocity, with acorrelation coefficient C,u'v s ' ≈ Cw
2w
Then u ≈ Cνk⇒2C Roconv
Ro ≈kD E

2

Given our previous expression for the convective velocities, we obtain finally*FC ∆RaRo ≈kD E

How to combine the two regimes?
The jet-pumping efficiency, ε, is the fraction of convective energy that goes into
pumping the jets:u∇ ⋅ (u'v s ')ε=gwα δT

Expressing numerator as ukCw2, the denominator as gαF/ρcp, andnondimensionalizing leads to2

4

Ro kD CRoconv2 Roconvε≈≈C**∆RaFE ∆RaF
Using our expression for convective velocities, we obtain*∆RaFε ≈ C2EThus, ε and C cannot simultaneously be constant!
• If C is constant, then ε must increase with increasing RaF*.
• When ε finally plateaus near its maximum value (~1), then C must decrease.
• Exactly this behavior is observed in the simulations, and it explains the transition
from Regime II to Regime I!

CεChristensen’s (2002)
simulations

Analytic scaling thatcombines
Regimes I and IIShowman et al. (2011)

Asymptotic regime?
Christensen (2002) suggested that, at sufficiently small viscosities, theconvection approaches an asymptotic regime where the wind speeds (Rossbynumbers) become independent of viscosities, empirically following(Ro = 0.53 RaF)

* 1/ 5 εαgF Can we explain this? Same scaling as before, namely U ≈ k −1 ρc ν  p 

1/ 2

but suppose damping results from an eddy (rather than molecular ornumerical) viscosity, given byν eddy ≈ wHSince w ∝ F, we have that ν eddy ∝ F

1/ 2

1/ 2Nondimensionalizing, we obtain(, which implies U ∝ FRo ≈ RaF)

* 1/ 4

1/ 4

RaF*

Christensen’s (2002)
simulations

Analytic scaling
combining
Regimes I, II, III

Showman et al. (2011)

Showman et al. (2011)

Conclusions•Current 3D simulation of convection in giant planet interiors are overforced byfactors of 105-1010. It has remained unclear how to extrapolate such simulations to
the Jovian regime, and even what processes control trends within the simulatedregime.

•We constructed a simple theory suggesting that, when the viscosity on the jet scale
dominates the damping, the mean jet speeds should scale approximately as F/ν atweakly supercritical Rayleigh numbers and (F/ν)1/2 at strongly supercritical Rayleighnumbers, where F is heat flux and ν is the numerical viscosity. This explains themean jet speeds found by Christensen (2002) and Kaspi et al. (2009) to within a factorof ~2 over a wide range of parameters.

•The relationship between the correlation coefficient C and the jet-pumping efficiency
ε naturally explains how the transition between these regimes occurs.

•If at low viscosity the mean jet speeds become independent of viscosity (as suggestedby Christensen), our simple theory predicts that mean jet speeds should scale as F1/4.
This compares favorably with an empirical fit to simulation results by Christensen,
which suggested an F1/5 dependence.

•When extrapolated to Jupiter’s heat flux, both asymptotic scalings suggest that windspeeds in Jupiter’s molecular envelope are weak. Juno will help test this prediction.