A local model for planetary atmospheres forced by small-scale convection

An equivalent-barotropic fluid on the β plane, forced at small scales by random stirring and dissipated by linear heat and vorticity drag, is considered as a local model for flow in the weather layer of internally forced planetary atmospheres. The combined presence of β, a finite deformation scale, and large-scale dissipation produce novel dynamics with possible relevance to the giant gas planets, which are apparently driven by small-scale convective stirring. It is shown that in order for anisotropy to form, one must have β(ελ⁵)^-1/3 ≳ 3.9, where ε is the (convectively driven) energy generation rate, λ is the deformation wavenumber, and β is the Coriolis gradient. The critical value above is not equivalent to the barotropic stability criterion, and numerical simulations demonstrate that anisotropic flow with average zonal velocities that are supercritical with respect to the latter can form. The formation of jets (a different matter) is not implied by the excess of zonal kinetic energy, and is instead sensitive to the relevant stability criterion for the flow geometry at hand. When β is sufficiently large that anisotropy does form, the flow scale and rms zonal velocity are set by a combination of Rossby wave cascade inhibition, the total energy constraint imposed by the large-scale dissipation, and the partitioning between available potential and kinetic energies. The resulting theory demonstrates that a relatively narrow range of parameters will allow for the formation of anisotropic flow with scale larger than the deformation scale. This is consistent with observations that indicate little separation between the jet scales and deformation scales on Jupiter and Saturn.