Marginal stability of almost adiabatic planetary convection

Abstract

Almost adiabatic states are typical for the deep convective interiors of all known planets and their moons, e.g., the deviations from the adiabatic state in the Earth's outer core and in the MHD dynamo region of Jupiter are about or less than 10−5%. We approximated the equations governing convection in the deep interiors of planets and their moons to obtain a system, which is more accurate than the traditional Boussinesq equations. Fortunately, our system, which adopts almost uniform entropy instead of the temperature as the basic thermodynamic states, can still be investigated by standard methods. We considered the marginal stability of well-mixed almost adiabatic states in rapidly rotating thick spherical shells, whose inner to outer radius ratio does not exceed that of the modern Earth. The critical Rayleigh-type numbers, frequencies and solution structures of the marginal states were determined by both analytical and numerical methods. Our new estimates differ from those obtained previously using the Boussinesq equations, suggesting that the earlier Boussinesq results for convection in the deep planetary interiors should be re-assessed. The small molecular Prandtl number limit was adopted to model the marginal stability of thermal planetary convection. It was found that the critical Rayleigh number for convection sharply diminished as the radius of the inner rigid core is increased. We modelled the instability of the combined compositional–thermal turbulent geo-convection for Prandtl number unity. When thermal convection is in opposition to compositional convection, extremely large critical Rayleigh numbers are possible. This might happen for a terrestrial planet during its later stage of evolution. Pure compositional convection has been investigated in the large compositional Prandtl number limit, for which the critical Rayleigh number is rather large and the variations of all critical parameters are small. The large size of the critical Rayleigh number ensures that the actual values used in numerical dynamo experiments are only moderately supercritical.