Strong field anisotropic diffusion models for the Earth’s core

The vector spherical harmonic equations and toroidal–poloidal spectral interactions are developed for the strong azimuthal field viscous and thermal diffusion tensors given respectively by Dν=2ρν0I+ρνφφ sin 2 θ1φ1φ and Dκ=κ0I+κφφ sin 2 θ1φ1φ, where I is the unit tensor, θ is the co-latitude, 1φ is the easterly unit vector and the coefficients νφφ and κφφ are spherically symmetric. These diffusion tensors model anisotropic diffusion in the presence of a strong azimuthal magnetic field. Physically, these models represent enhanced or inhibited diffusion along the magnetic field lines in the Earth’s core. Mathematically, these models are restricted to be analytic along the rotation axis. The spectral interactions are derived for the momentum equation for the body force, ∇·τ, where τ=Dν·(∇v)s. The subscript s denotes the trace-free symmetric part of the velocity gradient. There are four viscous interaction terms, (φφttn), (φφstn), (φφtsn), (φφssn). The spectral interactions for the transpose and symmetric part of τ are also derived. The spectral interactions may be used in dynamically-consistent pseudo-spectral geodynamo codes.