The Pressure Term in the Anelastic Model: A Symmetric Elliptic Solver for an Arakawa C Grid in Generalized Coordinates

Abstract

For the anelastic or pseudoincompressible system, the diagnostic continuity equation is the constraint filtering sound waves. Hamiltonian fluid dynamics considers the pressure force as the reaction force to this constraint. The author emphasizes the notion of an adjoint operator, as it provides the link between the constraint and the reaction. The elliptic equation for pressure is self-adjoint. Applied to a discretized model, the author discusses the possibility to maintain this symmetry in the pressure equation. Its discretization is deduced from one of the anelastic constraints. The author takes the example of a 2D model with orography, discretized on an Arakawa C grid in generalized coordinates. A specific treatment of boundaries is necessary to prevent Gibbs-like errors in the pressure term. It is possible to solve the pressure equation by a plain conjugate gradient method. Preconditioning is achieved by the Laplacian with no orography solved by a fast direct method. Criteria for efficiency depending upon the domain geometry are given.