Spectral interactions of rapidly-rotating anisotropic turbulent viscous and thermal diffusion in the Earth’s core

Abstract

Rapidly-rotating anisotropic turbulence in the Earth’s core is modelled by the viscous and thermal diffusion tensors, D ν≔2ρν 0 I+ρν ΩΩ ΩΩ and D κ≔κ 0 I+κ ΩΩ ΩΩ , where I is the unit tensor, Ω is the angular velocity and the coefficients ν ΩΩ and κ ΩΩ are spherically-symmetric. Toroidal–poloidal spectral interactions are derived for the anisotropic parts of the body forces associated with the mean anisotropic viscous stress tensor, D ν·(∇ v ) S and with its symmetric part. The stress tensors are linear in the trace-free symmetric part of the velocity gradient. Techniques of vector and tensor spherical harmonic analysis are used to find the vector spherical harmonic components of the body forces. From these components the toroidal–poloidal field interactions, (ΩΩtt n) , (ΩΩst n) , (ΩΩts n) , (ΩΩss n) , (ΩtΩt n) , (ΩsΩt n) , (ΩtΩs n) and (ΩsΩs n) , of the toroidal ( t n ) and poloidal ( s n ) momentum equations are derived using computer algebra. The temperature spectral interactions are also derived for the mean heat flux given by the turbulent thermal diffusion tensor. These spectral interactions represent a computationally practical first step in incorporating anisotropic turbulence models into existing dynamically-consistent angular–spectral geodynamo codes.