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

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.