A Vorticity-Divergence Global Semi-Lagrangian Spectral Model for the Shallow Water Equations

Abstract

The shallow water equations modeling flow on a sphere are useful for the development and testing of numerical algorithms for atmospheric climate and weather models. A new formulation of the shallow water equations is derived which exhibits an advective form for the vorticity and divergence. This form is particularly well suited for numerical computations using a semi-Lagrangian spectral discretization. A set of test problems, standard for the shallow water equations on a sphere, are solved and results compared with an Eulerian spectral model. The semi-Lagrangian transport method was introduced into atmospheric modeling by Robert, Henderson, and Turnbull. A formulation based on a three time level integration scheme in conjunction with a finite difference spatial discretization was studied by Ritchie. Two time level grid point schemes were derived by Bates et al. Staniforth and Cote survey developments of the application of semi-Lagrangian transport (SLT) methods for shallow water models and for numerical weather prediction. The spectral (or spherical harmonic transform) method when combined with a SLT method is particularly effective because it allows for long time steps avoiding the Courant-Friedrichs-Lewy (CFL) restriction of Eulerian methods, while retaining accurate (spectral) treatment of the spatial derivatives. A semi-implicit, semi-Lagrangian formulation with spectral spatial discretization is very effective because the Helmholz problem arising from the semi-implicit time integration can be solved cheaply in the course of the spherical harmonic transform. The combination of spectral, semi-Lagrangian transport with a semi-implicit time integration schemes was first proposed by Ritchie. A advective formulation using vorticity and divergence was introduced by Williamson and Olson. They introduce the vorticity and divergence after the application of the semi-Lagrangian discretization. The semi-Lagrangian formulation of Williamson and Olson and Bates et al. has the property that the metric terms of the advective form are treated discretely requiring a delicate spherical vector addition of terms at the departure point and arrival point. In their formulation, the metric terms associated with the advection operator do not appear explicitly. The spherical geometry associated with the combination of vector quantities at arrival and departure points treats the metric terms and is derived in Bates et al. The formulation derived in this paper avoids this vector addition. It is possible to do this because our formulation is based entirely on a scalar, advective form of the momentum equations. This new form is made possible by the generalization of a vector identity to spherical geometry. In Section 2 the standard form of the shallow water equations in spherical geometry are given. Section 3 presents the vector identities needed to derive an advective form of the vorticity and divergence equations. The semi-implicit time integration and semi-Lagrangian transport method are described in Section 4. The SLT interpolation scheme is described in Section 5. Section 6 completes the development of the discrete model with the description of the semi-implicit spectral equations. A discussion of results on several standard test problems is contained in Section 7.