Spherical anisotropic diffusion models for the Earth's core

Abstract

Estimates of the molecular values of magnetic, viscous and thermal diffusion in the Earth's core suggest turbulent small-scale velocity, magnetic and temperature fluctuations. Instability arguments (Braginsky, S.I., Meytlis, V.P., 1990. Local turbulence in the Earth's core. Geophys. Astrophys. Fluid Dynam. 55 (1990) 71–87.) indicate that the viscous and thermal diffusions of the resulting turbulence are strongly anistropic, the directions of anisotropy being determined by the mean magnetic field, the rotation of the core, the mean temperature gradient and gravity. Physical principles and invariance arguments are used to constrain the forms of the turbulent viscous stress tensor to eight types and the turbulent heat flux vector to a single type. The models are interpreted in terms of angular momentum, kinetic energy and entropy exchange between the mean and fluctuating fields. For the development of the turbulence models in spherical geometries, general non-linear spectral expansions are derived for one of the turbulent viscous stress tensors, the turbulent heat flux vector and their divergences using vector and tensor spherical harmonic methods and poloidal–toroidal representations. The spectral expansions, which are complicated and would be difficult to derive using other methods, include all invariance terms and can be programmed directly from the explicit forms given, thus laying the foundation for future computational studies of core turbulence. They are particularly suitable for linearised problems, such as anisotropic convection and magnetoconvection. The spectral techniques employed are applicable to all eight turbulent viscous stress tensors.