A conservative Fourier‐finite‐element method for solving partial differential equations on the whole sphere

Abstract

Solving transport equations on the whole sphere using an explicit time stepping and an Eulerian formulation on a latitude-longitude grid is relatively straightforward but suffers from the pole problem: due to the increased zonal resolution near the pole, numerical stability requires unacceptably small time steps. Commonly used workarounds such as near-pole zonal filters affect the qualitative properties of the numerical method. Rigorous solutions based on spherical harmonics have a high computational cost.The numerical method we propose to avoid this problem is based on a Galerkin formulation in a subspace of a Fourierfinite-element spatial discretization. The functional space we construct provides quasi-uniform resolution and high-order accuracy, while the Galerkin formalism guarantees the conservation of linear and quadratic invariants. For N 2 degrees of freedom, the computational cost is O (N 2 log N), dominated by the zonal Fourier transforms. This is more than with a finite-difference or finite-volume method, which costs O(N 2 ), and less than with a spherical harmonics method, which costs O(N 3 ). Differential operators with latitude-dependent coefficients are inverted at a cost of O(N 2 ).We present experimental results and standard benchmarks demonstrating the accuracy, stability and efficiency of the method applied to the advection of a scalar field by a prescribed velocity field and to the incompressible rotating Navier-Stokes equations. The steps required to extend the method towards compressible flows and the Saint-Venant equations are described.

Dynamical Core in Atmospheric Model Does Matter in the Simulation of Arctic Climate

“…Although uncertainty in the physical processes is declining, the effects of dynamical cores on Arctic climate modeling have been examined less thoroughly. In particular, previous studies simply addressed possible problems of pole singularity in using a latitude‐longitude grid system over the polar region (Dubos, ; Purser, ; Williamson, ), but the quantitative comparison of the quality of the Arctic climate simulation with other dynamic cores has not been addressed yet.…”

Construction of the Spherical High-Order Filter for Applications to Global Meteorological Data

“…As was discussed in the literatures (e.g., Dubos, 2009;Cheong et al, 2015), when the disturbances are given with isotropic resolution the Fourier coefficients behave near the poles as . This implies the following relationship:…”

“…When both variables ψ and F are Fourier transformed, (2) can be rewritten as following: (4) with being the Fourier transform of F. It appears in (4) that the first and second term in the parenthesis include terms divided by cosine of latitude, which causes singularity at poles due to cosθ =0 at θ = ±π/2. However, the singularity actually depends on the behavior of around poles, as discussed in Dubos (2009) andCheong et al (2015). For instance, in case of m =0, (4) becomes a function of one coordinate, i.e., the latitude, and vanishes…”

“…However, the double-Fourier series filter cannot be implemented for the reduced grid (Hortal and Simmon, 1991), while the Fourierfinite element filter can take full advantage of the reduced grid. The FFEM high-order filter has a minor point that high-order basis functions do not guarantee improved accuracy over the linear basis functions (Dubos, 2009).…”