Scaling laws for convection and jet speeds in the giant planets

TL;DR

This paper develops scaling laws for jet speeds in giant planets based on convection theories, explaining how jet speeds depend on heat flux and viscosity, and compares these with numerical simulations.

Contribution

It introduces simple theoretical scalings for jet speeds in planetary convection, bridging the gap between simulations and planetary observations.

Findings

Jet speeds scale as F/nu in weakly nonlinear convection.

Jet speeds scale as (F/nu)^{1/2} in strongly nonlinear convection.

Potential existence of a viscosity-independent jet speed regime.

Abstract

Three-dimensional studies of convection in deep spherical shells have been used to test the hypothesis that the strong jet streams on Jupiter, Saturn, Uranus, and Neptune result from convection throughout the molecular envelopes. Due to computational limitations, these simulations must adopt viscosities and heat fluxes many orders of magnitude larger than the planetary values. Several numerical investigations have identified trends for how the mean jet speed varies with heat flux and viscosity, but no previous theories have been advanced to explain these trends. Here, we show using simple arguments that if convective release of potential energy pumps the jets and viscosity damps them, the mean jet speeds split into two regimes. When the convection is weakly nonlinear, the equilibrated jet speeds should scale approximately with F/nu, where F is the convective heat flux and nu is the viscosity. When the convection is strongly nonlinear, the jet speeds are faster and should scale approximately as (F/nu)^{1/2}. We demonstrate how this regime shift can naturally result from a shift in the behavior of the jet-pumping efficiency with heat flux and viscosity. Moreover, the simulations hint at a third regime where, at sufficiently small viscosities, the jet speed becomes independent of the viscosity. We show based on mixing-length estimates that if such a regime exists, mean jet speeds should scale as heat flux to the 1/4 power. Our scalings provide a good match to the mean jet speeds obtained in previous Boussinesq and anelastic, three-dimensional simulations of convection within giant planets over a broad range of parameters. When extrapolated to the real heat fluxes, these scalings suggest that the mass-weighted jet speeds in the molecular envelopes of the giant planets are much weaker--by an order of magnitude or more--than the speeds measured at cloud level.