Two numerical methods to simulate flows with temperature-dependent viscosity in the spherical shell

The numerical solution of the Navier-Stokes equations in the spherical annulus is of fundamental interest for geophysical and astrophysical applications. The highly non-linear nature of the equations needs therefore special treatment. Consequently the full problem has to be calculated with numerical simulations. Due to the special geometry of the problem the class of spectral methods has become an appropriate tool to solve the equations in Boussinesq approximation. The presented method has been implemented into the numerical code developed by R. Hollerbach [1]. This code follows a spectral method, where the radial components are discretised on Chebyshev grid points. The toroidal-azimuthal components are further expanded as spherical harmonics. In a typical manner the boundaries are either stress-free or no-slip at the inner and outer sphere. The viscous components are solved with an operator splitting method. In the rst step the isoviscous, linear terms are calculated in spectral space implicitly, which guarantees high numerical stability. The temperature dependent non-linear variations are transformed in real space, followed by a real-space calculation of the tensor elements and transformed back into spectral space. Viscosity contrasts of up to T = 2 can be calculated with comparable iso-viscous numerical resolutions. The reference values are Pr=185 and Ra=20000, which correspond to the working uid 1-nonanol at 20 C and a temperature dierence