Scalar and Vector Spherical Harmonic Spectral Equations of Rotating Non-Linear and Linearised Magnetohydrodynamics

Vector spherical harmonic analyses have been used successfully to solve laminar and mean-field magnetohydrodynamic dynamo problems with interactions, such as the laminar induction term, anisotropic alpha-effects and anisotropic diffusion, which are difficult to analyse spectrally in spherical geometries. Spectral forms of the non-linear rotating Boussinesq and anelastic momentum, magnetic field and heat equations are derived for spherical geometries from vector spherical harmonic expansions of the velocity, magnetic induction, vorticity, electrical current and gravitational acceleration, and from scalar spherical harmonic expansions of the pressure and temperature. Combining the vector spherical harmonic forms of the momentum equation and the magnetic induction equation with poloidal-toroidal representations of the velocity and the magnetic field, non-linear spherical harmonic spectral equations are also derived for the poloidal-toroidal potentials of the velocity, or the momentum density in the anelastic approximation, and the magnetic field. Both compact and spectral interaction expansion forms are given. Vector spherical harmonic spectral forms of the linearised rotating magnetic induction, momentum and heat equations for a general basic state can be obtained by linearising the corresponding non-linear spectral equations. Similarly, the spherical harmonic spectral equations for the poloidal-toroidal potentials of the velocity and the magnetic field may be linearised. However, for computational applications, new alternative hybrid linearised spectral equations are derived herein. The algorithmically simpler hybrid equations depend on vector spherical harmonic expansions of the velocity, magnetic field, vorticity, electrical current and gravitational acceleration of the basic state, and scalar spherical harmonic expansions of the poloidal-toroidal potentials of the perturbation velocity, magnetic field and temperature. The spectral equations derived herein may be combined with the corresponding spectral forms of anisotropic diffusion terms derived in Phillips and Ivers (2000).