C.G. Phillips $^{a}$ and D.J. Ivers $^{b}$
$^{a}$ Mathematics Learning Centre , University of Sydney, Sydney, Australia. $^{b}$ School of Mathematics and Statistics , University of Sydney, Sydney, Australia.
We develop the vector spherical harmonic equations and toroidal-poloidal spectral interaction equations for the strong field viscous and thermal diffusion tensors given respectively by ${\mathbf D}_\nu = 2\rho\nu_0{\mathbf I} + \rho\nu_{\phi\phi}\sin^2\theta{\mathbf 1}_\phi{\mathbf 1}_\phi$ and ${\mathbf D}_\kappa = \kappa_0{\mathbf I} + \kappa_{\phi\phi}\sin^2\theta{\mathbf 1}_\phi{\mathbf 1}_\phi$, where ${\mathbf I}$ is the unit tensor, $\theta$ is the co-latitude, ${\mathbf 1}_\phi$ is the easterly unit vector and the coefficients 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 momentum spectral interaction equations are derived for the body force, $\nabla\cdot\tau$, where $\tau = {\mathbf D}_\nu\cdot(\nabla{\mathbf v})_s$. The subscript $s$ denotes the trace-free symmetric part of the velocity gradient. There are four viscous interaction terms, $(\phi\phi tt_n)$, $(\phi\phi st_n)$, $(\phi\phi ts_n)$, $(\phi\phi ss_n)$. The spectral interactions for the transpose and symmetric part of $\tau$ are also derived. The spectral interactions may be used in dynamically-consistent pseudo-spectral dynamo codes. We present preliminary numerical results for the effects of the above anisotropic diffusion models on thermal convection in a sphere. We use the more general vector spectral equations developed by Phillips and Ivers (Phys. Earth Planet. Int. 2000), and implemented directly in a linearised MHD stability code.