Zonal Jets

Adam P. Showman
University of Arizonaon sabbatical at Peking University

2D models appropriate to Jupiter and Saturn show that rotation causes
zonal banding

Hayashi et al. (2000)

See review in Vasavada & Showman (2005, Rep. Prog. Phys. 68, 1935)

Nozawa & Yoden (1997)

2D modelsappropriate to Jupiterand Saturn show thatjets can form inrotating atmospheres
forced by small-scale
turbulenceEmergence of banding

UbVallis and Maltrud (1993)

U2/L2

Simple scaling suggests the nonlinear term and beta term arecomparable at a scale
LR = (U/b)1/2
called the Rhines scale. Because the transition is anisotropic, banding
should emerge.

Saturn 5 mm emission illustrates scale-dependent anisotropyMechanisms of extratropical jet formation: role ofRossby waves
Grouppropagation

In the extratropics,
regions of Rossby
wave generation
correspond toeastward eddy-driven
jets. Regions ofRossby wave
damping correspondto westward flow.

Phase
propagation

Phase
propagationGroup
propagation

Mechanisms of jet formation: role of Rossby waves
Grouppropagationu’v’<0Phase
propagation

The eddy acceleration
term is positive(eastward) in theregion of Rossby
wave generation, andnegative (westward)
in the region ofRossby wave
damping/breaking.

u’v’>0Phase
propagation

Grouppropagation

Srinivasan & Young (2012)

Zonal jets require spatial organization of the turbulence•

Jet formation/maintenance requires meridional organization of the turbulence, sothat eddies that are north (south) of eastward jets transport momentum southward(northward), into the jet cores•

There are no obvious mechanisms external to the jets that could provide thisorganization. This implies that the organization of the eddies necessary to maintain thezonal jets is caused by the zonal jets themselves.

Need for feedbacks•

This idea--that jets organize the eddies to maintain the jets--suggests that eddy-meanflow interactions play a key role in jet formation. In other words, positive
feedback(s) exists in which the presence of the jets induces the eddies to develop thephase tilts necessary to maintain the jets.

•The question, then, is what are these positive feedbacks?

•In the Earth case, the baroclinic zone is at a fixed latitude. This determines whereRossby waves are generated, and since eastward jets occur in regions of Rossby wave
generation, this sets the jet latitude. But, on Jupiter and Saturn, random convective
forcing probably occurs everywhere. Then, what organizes the Rossby waves so that
preferential wave generation occurs at some latitudes and wave dissipation at otherlatitudes?

Feedback 1: shear straining

In this idea, the jets shear initially isotropic or quasi-circular eddies,
automatically giving them the phase tilts necessary to maintain the jets

Feedback 1: complications
Problem: the shear straining actually acts only on materially conserved quantities,
such as the potential vorticity (PV). Eddy velocities are not a materially conserved
quantity, and do not necessarily shear in the expected way.

Shepherd (1987); see also Holloway (2010), Showman et al. (2016)

Feedback 1: shear straining
Recent theoretical work shows that shear straining can indeed still work as a positive
feedback, despite this problem. More theoretical work is needed.
Here is an example of jet formation due to shear straining in a model where eddy-eddy
interactions have been removed:Srinivasan & Young (2012)

Feedback 2: inhomogeneous mixing of potential vorticity (PV)• Rossby waves break more easily in regions of weak meridional PV gradientthan strong meridional PV gradient• Rossby wave breaking mixes the air in latitude, locally homogenizing PV.
This reduces the mean PV gradient• This is a positive feedback: Rossby waves preferentially break in regions ofweak PV gradient, which makes the PV gradient weaker, which further
promotes Rossby wave breaking at that latitude• This results in a “staircase” pattern of constant-PV strips separated by sharp
gradients in PV…. which corresponds to a pattern of zonal jets!

• The key point is that jets can spontaneously emerge--even if the convective
stirring that generates Rossby waves is not spatially organizedsee, e.g., Dritschel & McIntyre (2008)

Earth stratosphere example of
how Rossby wave breaking
leads to PV strips with sharp
gradients in between.

Polvani et al. (1995)

Development of PV staircases in models of giant
planet atmospheres

Scott & Polvani (2007)

…and the relationship between PV and winds on rapidly rotating
planets implies that when you have PV strips, you have jets

Scott (2010)

The PV staircases in these models imply zonal jet formation

SaturnUranus/Neptune
Scott & Polvani (2007); Showman (2007);
Cho & Polvani (1996)

Jupiter

Bands and zonal jets spontaneously emerge, including an equatorial jet.
Problem is the equatorial jet is generally westward!

Superrotation can occur in shallow-water models

sin(latitude)

But the necessary ingredients that allow this to occur remain
poorly understood, as does the dynamical mechanism. Thus therelevance to Jupiter/Saturn remains unclear.

Zonal wind

Scott & Polvani (2008)

3D atmosphere models show that multiple jets can occur in
response to baroclinic instabilities in the weather layer, and thatdeep jets can arise from shallow forcing via “downward control”

Lian & Showman (2008)

Lian & Showman (2010)

3D models can
now demonstratea transition fromsuperrotation on
Jupiter/Saturn,to subrotation on
Uranus/Neptune,
with jet profiles
similar to thoseobserved on all
four planets.

Jupiter

SaturnUranus/Neptune

Storms
Observations (Galileo orbiter)

Global GCM withhydrological cycleTemperature(5 bars)

VerticalVelocity

Humidity

Vorticity(1 bar)

Gierasch et al. (2000)
Lian & Showman (2010)

Lian & Showman (2010)

Goal: determine the 3D atmospheric circulation that results fromisotropic convective forcing near the radiative-convective boundary.

•Solve global 3D primitive equations in a
stratified atmosphere with MITgcm.
Domain from 0.01-10 bars.

•Parameterize convection by adding
isotropic thermal perurbations at total
wavenumber 20 to bottom of model.

•Radiation parameterized by Newtonian
cooling:q Teq ( p) - T (l, f, p, t)=cpt rad•q

Showman et al. (in prep)

Systematically vary trad to span therange of brown dwarfs from hot (shorttrad) to cooler (long trad)

Photosphericpressures
RCB
Thermal forcing

trad = 104 sec

trad= 106 sec

trad = 105 sec

trad = 107 sec

Showman et al. (in prep)

trad = 104 sec

trad= 106 sec

trad = 105 sec

trad = 107 sec

Showman et al. (in prep)

Showman et al. (in prep)

Showman et al. (in prep)

Deep convection models
Boussinesq 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)

Many deep models now include the radial density gradientKaspi et al. (2009);
Jones & Kuzanyan (2009);
Showman et al. (2011);
Gastine & Wicht (2012);
Gastine et al. (2013);
Yadav et al. (2013)

Superrotation in convection models results from correlations
between zonal and (cylindrically) outward velocity components

Showman et al. (2011)

Models including radial
gradient in electricalconductivity

High-latitude jets are largely
suppressed, but theequatorial jet still occurs

Duarte et al. (2013)

Conclusions•Zonal jets dominate the atmospheric circulation of all known rapidly rotating planets,including Earth, Mars, and the four giant planets. Earth and Mars have a subtropical andan eddy-driven jet in each hemisphere. Jupiter and Saturn have ~10 jets in eachhemisphere; evidence indicates they are eddy driven.

•On Earth and Mars, the subtropical jet results from the Hadley circulation. The eddydriven jet results from Rossby-wave generation by baroclinic instabilities in themidlatitudes. As the Rossby waves propagate north and south away from the latitude ofwave generation, they transport momentum back into that region, driving the jet.

•On Jupiter and Saturn, several interacting feedbacks help to maintain the zonal jets.
Specifically, the jets organize the eddies in such a way that the eddies maintain the jets.
One feedback involves shear straining of eddies by the mean flow. Another involves theinhomogeneous mixing of the fluid by eddies, and its effect on the zonal jets. Bothfeedbacks cause the spontaneous emergence of zonal jets from random forcing.

•Evidence indicates that brown dwarfs have active atmospheric circulations. The above
feedbacks should occur on brown dwarfs too, suggesting they likely have zonal jets.
Nevertheless, the details may differ. The strong radiative damping (due to short radiative
time constants) may suppress zonal jets in certain cases.

Evolution of equatorial zonal-mean zonal wind

PressureTime

LatitudeLatitudeLatitudeLatitudeLatitudeEvolution of equatorial zonal-mean zonal windMechanism involves a wave-mean-flow interaction

Baldwin et al. (2001),
Plumb (1984)References1••••••••

Aurnou, J., Heimpel, M., Allen, L., King, E., Wicht, J., 2008: Convective heat transfer andthe pattern of thermal emission on the gas giants. Geophys. J. Int. 173, 793–801.
Aurnou, J.M., Olson, P.L., 2001: Experiments on Rayleigh–Bénard convection,magnetoconvection and rotating magnetoconvection in liquid gallium, J. Fluid Mech., 430,283–307.
Baldwin, W.P., Gray, L.P,, Dunkerton, T.J., Hamilton, K., Haynes, P.H., Randel, W.J.,
Holton, J.R., Alexander, M.J., Hirota, I., Horinouchi, T., Jones, D.B.A., Kinnersley, J.S.,
Marquardt, C., Sato, K., Takahashi, M., 2001: The quasi-biennial oscillation, Rev.
Geophys, 39, 179-229.
Cho, J.Y.-K., Polvani, L.M., 1996: The emergence of jets and votices in freely evolving,
shallow-water turbulence on a sphere, Phys. Fluids, 8, 1531-1552.
Christensen, U.R., 2001: Zonal flow driven by deep convection in the major planets,
Geophys. Res. Lett. 28, 2553–2556.
Christensen, U.R., 2002: Zonal flow driven by strongly supercritical convection in rotating
spherical shells, J. Fluid Mech. 470, 115–133.
Dritschel, D.G., McIntyre, M.E., Multiple jets as PV staircases: The Phillips effect and theresilience of eddy-transport barriers, J. Atoms. Sci., 65, 855-874.
Duarte, Lucia, D.V., Gastine, T., Wicht, J., 2013: Anelastic dynamo models with variable
electrical conductivity: An application to gas giants, Phys. Earth Planet. Inter., 222, 22-34.

References2
•••
•
•••
•
•

Gastine, T., Wicht, J., 2012: Effects of compressibility on driving zonal flow in gas giants,Icarus, 219, 428-442.
Gastine, T., Wicht, J., Aurnou, J.M., 2013: Zonal flow regimes in rotating anelastic
spherical shells: An application to giant planets, Icarus, 225, 156-172.
Gierasch, P.J., Ingersoll, A.P., Banfield, D., Ewald, S.P., Helfenstein, P., Simon-Miller, A.,
Vasavada, A., Breneman, H.H., Senske, D.A., Galileo Imaging Team, 2000: Observationof moist convection in Jupiter's atmosphere, Nature, 403, 628-630.
Hayashi, Y.Y., Ishioka, K., Yamada, M., Yoden, S., 2000: Emergence of circumpolar
vortex in two dimensional turbulence on a rotating sphere, In IUTAM Symposium onDevelopments in Geophysical Turbulence, 179-192, Springer Netherlands.
Heimpel, M., Aurnou, J., 2007: Turbulent convection in rapidly rotating spherical shells: Amodel for equatorial and high latitude jets on Jupiter and Saturn. Icarus, 187, 540-557.
Heimpel, M., Aurnou, J., Wicht, J., 2005: Simulation of equatorial and high-latitude jets
on Jupiter in a deep convection model. Nature, 438, 193-196.
Holloway, G., 2010: Eddy stress and shear in 2D flows, J. Turbul., 11, 1-14.
Jones, C. A., Soward, A. M., Mussa, A. I., 2000: The onset of thermal convection in arapidly rotating sphere. J. Fluid Mech., 405, 157-179.
Jones, C.A., Kuzanyan, K.M., 2009: Compressible convection in the deep atmospheres ofgiant planets, Icarus, 204, 227-238.

References3••
•
•
•

•
•

•
•
•

Kaspi, Y., Flierl, G.R., Showman, A.P., 2009: The deep wind structure of the giant
planets: Results from an anelastic general circulation model, Icarus, 202, 525-542.
Lian, Y., Showman, A.P., 2008: Deep jets on gas-giant planets, Icarus, 194, 597-615.
Lian, Y., Showman, A.P., 2010: Generation of equatorial jets by large-scale latent heatingon the giant planets, Icarus, 207, 373-393.
Nozawa, T., Yoden, S., 1997: Formation of zonal band structure in forced twodimensional turbulence on a rotating sphere, Phys. Fluids, 9, 2081-2093.
Plumb, R. A., 1984: The quasi-biennial oscillation, In Dynamics of the Middle
Atmosphere, pp.217.
Polvani, L.M., Waugh, D.W., Plumb, R.A., 1995: On the subtropical edge of thestratospheric surf zone, J. Atmos. Sci., 52, 1288-1309.
Scott, R.K., Polvani, L.M., 2008: Equatorial superrotation in shallow atmospheres,Geophys. Res. Lett., 35, L24202.
Scott, R.K., 2010: The structure of zonal jets in shallow water turbulence on the sphere,
IUTAM Symposium on Turbulence in the Atmosphere and Oceans, pp. 243-252.
Scott, R.K., and Polvani, L.M., 2007: Forced-dissipative shallow-water turbulence on thesphere and the atmospheric circulation of the giant planets, J. Atmos. Sci., 64, 3158-3176.
Shepherd, T.G., 1987: Rossby waves and two-dimensional turbulence in a large-scalezonal jet, J. Fluid Mech., 183, 467-509.

References4•••••••••

Showman, A.P., 2007: Numerical simulations of forced shallow-water turbulence: Effects
o f moist convection on the large-scale circulation of Jupiter and Saturn, J. Atmos. Sci., 64,3132-3157.
Showman, A.P., Kaspi, Y., Achterberg, R., Ingersoll, A.P., 2016: The global atmosphericcirculation of Saturn. Invited review, submitted to the book Saturn in the 21st century,
Cambridge University Press.
Showman, A.P., Kaspi, Y., Flierl, G.R., 2011: Scaling laws for convection and jet speedsin the giant planets, Icarus, 211, 1258-1273.
Showman, A.P., Polvani, L.M., 2011: Equatorial superrotation on tidally lockedexoplanets. Astrophys. J., 738, 71.
Srinivasan, K., Young, W.R., 2012: Zonostrophic instability, J. Atmos. Sci., 69, 16331656.
Vallis, G.K., Maltrud, M.E., 1993: Generation of mean flows and jets on a beta plane andover topography, J. Phys. Oceanogr., 23, 1346-1362.
Vasavada, A.R., Showman, A.P., 2005: Jovian atmospheric dynamics: an update after
Galileo and Cassini, Rep. Prog. Phys., 68, 1935-1996.
Yadav, R. K., Gastine, T., Christensen, U. R., Duarte, L. D., 2013: Consistent scaling laws
in anelastic spherical shell dynamos, Astrophys. J., 774, 6.
(p.5) https://saturn.jpl.nasa.gov/resources/3552/