Spherical Harmonic Transform Algorithms

A collection of MATLAB classes for computing and using spherical harmonic transforms is presented. Methods of these classes compute differential operators on the sphere and are used to solve simple partial differential equations in a spherical geometry. The spectral synthesis and analysis algorithms using fast Fourier transforms and Legendre transforms with the associated Legendre functions are presented in detail. A set of methods associated with a spectral field class provides spectral approximation to the differential operators ∇·, ∇×, ∇, and ∇2 in spherical geometry. Laplace inversion and Helmholtz equation solvers are also methods for this class. The use of the class and methods in MATLAB are demonstrated by the solution of the barotropic vorticity equation on the sphere. A survey of alternative algorithms is given and implementations for parallel high performance computers are discussed in the context of global climate and weather models.