  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.
