Accurate and efficient Jones-Worland spectral transforms for planetary applications

Abstract

Spectral transforms between physical space and spectral space are needed for fluid dynamical calculations in the whole sphere, representative of a planetary core. In order to construct a representation that is everywhere smooth, regular and differentiable, special polynomials called Jones-Worland polynomials, based on a type of Jacobi polynomial, are used for the radial expansion, coupled to spherical harmonics in angular variables. We present an exact, efficient transform that is partly based on the FFT and which remains accurate in finite precision. Application is to high-resolution solutions of the Navier-Stokes equation, possibly coupled to the heat transfer and induction equations. Expected implementations would be in simulations with P3 degrees of freedom, where P may be greater than 103. Memory use remains modest at high spatial resolution, indeed typically P times lower than competing algorithms based on quadrature.