Thermodynamically consistent background state and dynamo models constrained by Juno and Cassini gravity harmonics

The current Juno and recent Cassini missions have yielded unprecedented accuracy and resolution of the gravity fields of Jupiter and Saturn. The new observations of zonal harmonics through J12 have led to a new generation of interior models. Previously, interior models of two or three adiabatic layers were sufficient to satisfy gravity observations. However, to satisfy the Cassini and Juno observations, interior models require more complexity, with recent works proposing five layers, and the presence of gradients in composition and entropy. I describe the hydrostatic equations relevant to a rotating fluid planet with variable density, composition and entropy. Composition is formulated with a simple version of the additive volume law. Stability is described in terms of gradients in specific entropy and composition (mass fraction), which is assumed to be static (or slowly varying) . Relations between composition, entropy and diffusion parameters variation are described in terms of the density ratio, which is a prominent parameter of semiconvection (double diffusive convection). The resulting set of thermodynamic equations, along with gravity, are solved iteratively, calibrated by, and compared to the recent ab-initio EOS results of French et al. (2012) and Militzer et al. (2022).  To further simplify the thermodynamic formulation, non-adiabatic interior models that are polytropic where they are adiabatic are explored. Gravitational harmonics and moment of inertia of the resulting density profiles are calculated using the Theory of Figures to order 7 (Nettelman et al., 2021). Plausible and thermodynamically consistent interior models are shown to be relatively straightforward to obtain. Using the anelastic magnetohydrodynamics code MagIC (Gastine and Wicht, 2012), examples of these interior models are implemented as the background state for dynamo models of Jupiter and Saturn.