Numerical Simulations of Forced Shallow-Water Turbulence: Effects of Moist Convection on the Large-Scale Circulation of Jupiter and Saturn

Abstract

To test the hypothesis that the zonal jets on Jupiter and Saturn result from energy injected by thunderstorms into the cloud layer, forced-dissipative numerical simulations of the shallow-water equations in spherical geometry are presented. The forcing consists of sporadic, isolated circular mass pulses intended to represent thunderstorms; the damping, representing radiation, removes mass evenly from the layer. These results show that the deformation radius provides strong control over the behavior. At deformation radii Ͻ2000 km (0.03 Jupiter radii), the simulations produce broad jets near the equator, but regions poleward of 15°-30°latitude instead become dominated by vortices. However, simulations at deformation radii Ͼ4000 km (0.06 Jupiter radii) become dominated by barotropically stable zonal jets with only weak vortices. The lack of midlatitude jets at a small deformation radii results from the suppression of the beta effect by column stretching; this effect has been previously documented in the quasigeostrophic system but never before in the full shallow-water system. In agreement with decaying shallow-water turbulence simulations, but in disagreement with Jupiter and Saturn, the equatorial flows in these forced simulations are always westward. In analogy with purely two-dimensional turbulence, the size of the coherent structures (jets and vortices) depends on the relative strengths of forcing and damping; stronger damping removes energy faster as it cascades upscale, leading to smaller vortices and more closely spaced jets in the equilibrated state. Forcing and damping parameters relevant to Jupiter produce flows with speeds up to 50-200 m s Ϫ1 and a predominance of anticyclones over cyclones, both in agreement with observations. However, the dominance of vortices over jets at deformation radii thought to be relevant to Jupiter (1000-3000 km) suggests that either the actual deformation radius is larger than previously believed or that three-dimensional effects, not included in the shallow-water equations, alter the dynamics in a fundamental manner.

Saturn Atmospheric Structure and Dynamics

“…The Coriolis force acting on this flow would decelerate both jets. Acceleration of the flow in this picture is instead provided by the observed convergence of the eddy momentum flux into the eastward jets and its divergence from the westward jets (Del Genio et al 2007a;Showman 2007;Lian and Showman 2008). The remarkable stability of the jets outside the equatorial region suggests that these competing influences are approximately in balance, and thus fv ~ -∂<u'v'>/∂y.…”

“…Showman (2007) and Scott and Polvani (2007) performed forceddissipative studies using the one-layer shallow-water equations, which govern the evolution of a thin layer of constant-density fluid on a sphere (intended to represent the weather-layer in the 1.5 layer formulation; see Dowling and Ingersoll 1989). Showman (2007) forced the flow by episodically injecting isolated mass pulses, intended to represent the effects of episodic thunderstorms that transport mass into the weather layer. Damping, intended to mimic radiation, consisted of Newtonian relaxation that removed available potential energy by flattening the shallow-water layer.…”

“…The reasons for the differences are unclear. As in Showman (2007), forcing by moist convection in this study is represented by the injection of mass pulses that create local negative vorticity sources; these are balanced by a positive vorticity source associated with uniform radiative cooling. Perhaps the major difference is that mass is injected randomly in space in the Showman (2007) simulations, while in Li et al (2006) mass pulses are triggered only when the thickness of the weather layer decreases below a threshold value.…”

On the energetics of a two-layer baroclinic flow

“…To gain at least a qualitative understanding of the dynamics of Jupiter and of the gas giants generally, two main modelling approaches have been taken: a shallow model and a deep model. The shallow model (typically employing the rotating shallow water equations in a single layer) suggests that jets emerge essentially through an upscale cascade of geostrophic turbulence (Williams (1978), Cho & Polvani (1996), Kaspi & Flierl (2007), Showman (2007), Scott (2007)). By contrast, the deep model suggests that the flow is organised in deep concentric rotating cylinders, with jets their external representation, (see Busse (1970), Busse (1976)).…”

Simulation of Atmospheric Vortex Flows on Jupier and Saturn

“…Атмосферные течения на Юпитере и Сатурне характеризуются турбулентностью и сложной вихревой структурой, что обусловлено значительной угловой скоростью вращения газовых гигантов [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] Гидростатическое уравнение и стандартное приближение Буссенеска для возмущений плотности заданы в виде…”