A Locally One-Dimensional Semi-Implicit Scheme for Global Gridpoint Shallow-Water Models

Abstract

A splitting method is presented for eliminating the need to directly solve for a two-dimensional Helmholtz-type difference equation in a semi-implicit scheme for a global gridpoint shallow-water model. In the proposed method, the model equations are split so that the gravity-oscillation terms are integrated implicitly in two locally one-dimensional steps. It is required that such splitting must preserve the irrotationality properties of the gradients of pressure in finite-difference form so that no spurious sources of vorticity and divergence are introduced into the flow. The semi-implicit scheme, thus derived, provides a locally one-dimensional method of solving the two-dimensional Helmholtz-type difference equation. This method requires the solutions of two linear tridiagonal systems of equations, which can be solved more easily and efficiently than the original two-dimensional Helmholtz-type equation. Using idealized large-scale flows on the sphere, it is shown that the scheme provides stable and accurate model solutions at considerable computational economy.