Gravity and Zonal Flows of Giant Planets: From the Euler Equation to the Thermal Wind Equation

TL;DR

This paper critically examines the use of the thermal wind equation to predict gravity signals from deep zonal flows in giant planets, finding it accurate when background nonsphericity is included, especially for high-degree gravity moments.

Contribution

It demonstrates that the thermal wind equation can reliably predict gravity moments from deep zonal flows if background nonsphericity is properly accounted for.

Findings

TWE predictions agree with full Euler solutions within a few tens of percent.

Including background nonsphericity improves high-degree gravity moment accuracy.

Corrections from non-local density perturbations are negligible at low degrees and very small at high degrees.

Abstract

Any nonspherical distribution of density inside planets and stars gives rise to a non-spherical external gravity and change of shape. If part or all of the observed zonal flows at the cloud deck of Jupiter and Saturn represent deep interior dynamics, then the density perturbations associated with the deep zonal flows could generate gravitational signals detectable by the Juno mission and the Cassini Grand Finale. Here we present a critical examination of the applicability of the thermal wind equation to calculate the wind induced gravity moments. Our analysis shows that wind induced gravity moments calculated from TWE are in overall agreement with the full solution to the Euler equation. However, the accuracy of individual high-degree moments calculated from TWE depends crucially on retaining the nonsphericity of the background density and gravity. Only when the background nonsphericity of the planet is taken into account, does the TWE make accurate enough prediction (with a few tens of percent errors) for individual high-degree gravity moments associated with deep zonal flows. Since the TWE is derived from the curl of the Euler equation and is a local relation, it necessarily says nothing about any density perturbations that contribute irrotational terms to the Euler equation and that have a non-local origin. However, the predicted corrections from these density contributions to the low harmonic degree gravity moments are not discernible from insignificant changes in interior models while the corrections at high harmonic degree are very small, a few percent or less.