Syllabus
Basic principle of spectral methods. Basis functions. Fourier series. Spherical harmonic functions. Various definitions of vector and tensor spherical harmonics (SH). Approximation of geophysical quantities in terms of spherical harmonics. Products of SH series. Application of differential operators. Exercises.
Laplace-Poisson equation. Solution for gravitational potential and acceleration. Expression of centrifugal and tidal forces.
Deformation of an elastic shall with radially dependent material paremeters. Methods of including lateral variations of parameters. Elastic membrane.
Momentum and heat transport equations. Nonlinear terms. Degree 0 and 1.
Viscoelastic deformation of a spherical body, evaluation of the memory term. Compressibility and selfgravitation.
Maxwell equations. Problem of electromagnetic induction. Generation of magnetic field in the core.
Annotation
Spherical harmonic functions, vectors and tensors. Spectral approximation of data given on a sphere in terms of generalized spherical harmonics.
Application to solving PDF. Spectral solution of the following problems: Laplace-Poisson equation for gravitational potential, deformation of a spherical elastic shall, thermal convection in a mantle, viscoelastic relaxation of a spherical body, and the problem of electromagnetic induction.