Gravity and zonal flows of giant planets: From the Euler equation to the thermal wind equation

Abstract

Any nonspherical distribution of density inside planets and stars gives rise to a nonspherical 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 the thermal wind equation (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 nonlocal 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.

Dynamo Action in the Steeply Decaying Conductivity Region of Jupiter‐Like Dynamo Models

“…While the toroidal field estimate agrees with the assessment by Cao and Stevenson (), the poloidal field estimates differ. Cao and Stevenson () consider the poloidal field produced by nonaxisymmetric (helical) flows acting on the local toroidal field, but the advective modification of turns out to be significantly larger in the SDCR of our simulations.…”

“…A more useful definition of the top of the dynamo region would be the depth where the Locally Induced Field (LIF) becomes a significant fraction of the total field. Cao and Stevenson () suggest that Rm (1) takes on a different role in the SDCR and may allow to quantify the ratio of the LIF to the background field . Using a simplified mean field dynamo model, they conclude that could reach 1% of the background field in the SDCR.…”

“…Ridley and Holme (2016) argue that the secular variation of the large-scale field, deduced from pre-Juno measurements, is so small that the zonal winds cannot contribute. The winds are thus unlikely to penetrate Journal of Geophysical Research: Planets 10.1029/2018JE005759 deeper than about 0.96 r J where the magnetic Reynolds number exceeds unity (Cao & Stevenson, 2017;Liu et al, 2008).…”

Equilibrium Tidal Response of Jupiter: Detectability by the Juno Spacecraft

“…Satisfying all such constraints requires either more complex interior thermal and compositional structure (Debras & Chabrier 2019), contributions from deep winds (Guillot et al 2018;Kaspi et al 2018) or both (Militzer et al 2020). The requisite deep wind profile decays with depth, due to interaction of the conductive fluid with the magnetic field (Cao & Stevenson 2017), and cannot be described self-consistently using a potential based theory like CMS (Militzer et al 2019).…”

“…Due to the limitations of potential theory, differential rotation can only be implemented fully consistently for cylinders extending through the deep interior of the planet (Wisdom & Hubbard 2016). It is therefore, not possible to implement the more complex 3D wind profile expected for Jupiter, in which the cylindrical flow velocities decay rapidly at depths where conductivity becomes high enough for flows to couple to the planet's magnetic field (Cao & Stevenson 2017). We therefore, cannot self-consistently test their effect on the tidal response (Militzer et al 2019).…”