Vector Spherical Harmonic and Vector Wavelet Based Non-Linear Galerkin Schemes for Solving the Incompressible Navier-Stokes Equation on the Sphere

The spherical Navier-Stokes equation plays a fundamental role in meteorology by modelling meso-scale (stratified) atmospherical flows. In this context, this thesis is concerned with different nonlinear Galerkin methods for solving the incompressible Navier-Stokes equation on the rotating sphere. It extends the work of Debussche et al. (1995), Marion and Temam (1989), as well as Shen (1993) from one-dimensional or toroidal domains to the spherical geometry. In the first part, the method based on surface divergence free vector spherical harmonics is introduced and convergence is proven. Furthermore, it is shown that the occurring coupling terms involving three vector spherical harmonics can be expressed algebraically in terms of Wigner-3j coefficients. To improve the numerical efficiency and economy we introduce an FFTbased pseudo spectral algorithm involving vector as well was tensor spherical harmonics for computing the Fourier coefficients of the nonlinear advection term. The resulting method scales like O(N 3 ), where N denotes the maximal spherical harmonic degree.

The second part of this work introduces bandlimited as well as nonbandlimited, locally supported vector wavelets and their application to the aforementioned nonlinear Galerkin scheme. In detail, this scheme is implemented by using surface divergence free vectorial spherical wavelets, and its convergence is proven. Again we find algebraic expressions for the representation of integrals involving three vector wavelets. To improve numerical efficiency an extension of the spherical panel clustering algorithm to vectorial and tensorial kernels is constructed. This method enables the rapid computation of the wavelet coefficients of the nonlinear advection term. Thereby, we also indicate error estimates. Finally, extensive numerical simulations show the applicability of our methods. Next to some benchmark results, such as the nonlinear interaction of three vortices, we also study the forecast of a real atmospherical flow.