Supercritical magnetoconvection in rapidly rotating planetary cores

Abstract

Planetary magnetoconvection is highly supercritical ( R≫ R crit≈ E −1/3) in the rapidly rotating ( E≪1) planetary liquid cores where, depending on the planet, the Rayleigh R number is roughly between 10 10 and 10 22, while the Ekman E number is roughly between 10 −9 and 10 −17. In the bulk of the core, balance between Archimedean and Coriolis forces can be justified in order to determine typical hydromagnetic field values and symmetries via E, 1/ R≪1 that appear as asymptotically small parameters near the highest derivatives in the fully self-consistent planetary dynamo system. Archimedean or diffusive (thickness ∼ L/ R 1/3) boundary layers control typical velocity V *= R 2/3 κ/ L ( κ: diffusivity; L: core radius), that is about maximal velocity near the inner rigid core. Ekman–Hartmann (∼ LE 1/2) boundary layers control typical magnetic field that is B *=( SρΩ/ σ) 1/2 at core–mantle boundary if S≡ μ 0 σV * LE 1/2≤1 and B *= S( ρΩ/ σ) 1/2 otherwise. Here, ρ is the liquid core density, σ the conductivity and Ω the angular rotation rate of the mantle. Effective turbulent (hereafter burred) numbers ( Ē≈ E 1/2, R̄≈ R 1/3) and especially `secondary' turbulent numbers ( E ≈E 1/4 , R ≈R 1/9 ) have moderate values that are close to each other in different planets in contrast to the initial E, R there. That could explain close values of magnetic fields at the core–mantle boundaries in Earth, Jupiter, Saturn, Uranus and Neptune. Finally, an optimal planetary dynamo system is obtained with three (only!) initial or turbulent parameters: ( E, R, S), or ( Ē, R̄, S̄), or ( E , R , S ), those parameters are defined well via available planetary core models and the observed planetary magnetic fields. The optimal system can be easy integrated and should reproduce the observed fields in contrast to the recent very expensive numerical simulations. Those were not so effective have been using unphysical hyperdiffusivity and hyperviscosity. The conditions to gain some physical usage from the hypervalues was found here.