New icosahedral grid-point discretizations of the shallow water equations on the sphere

We describe the implementation of numerical models of shallow water flow on the surface of the sphere, models which include the nondivergent barotropic limit as a special case. All of these models are specified in terms of a new grid-point-based methodology which employs an heirarchy of tesselations derivative of successive dyadic refinements of the spherical icosahedron. Among the potential advantages of such methods is the O(n) complexity in operation count that can be achieved for an n degree of freedom model if multigrid techniques are employed to solve the associated elliptic problems. Currently prevalent spectral transform models are, in contrast, O(n(2)) complex due to the Legendre transform that must be performed to transform between spectral and grid-point representations of model fields at each time step. Using the new methodology, we have implemented two different formulations of each of the barotropic and shallow water dynamical systems. In one formulation, the vector velocity field is directly advanced in time; in the other, time integration is carried out entirely in terms of scaler quantities (i.e., absolute vorticity in the barotropic model and, in the more general shallow water model, height and velocity potential). We describe discretizations of the governing equations in which all calculations are performed in Cartesian coordinates in local neighbourhoods of the almost uniform icosahedral grid, a methodology that avoids potential mathematical and numerical problems associated with the poles in spherical coordinates. A number of standard numerical tests are performed with the resulting models and the results employed to compare them with each other and with previously published results obtained using other methodologies. Initial tests are performed for a standard suite that now constitutes the generally accepted benchmark for shallow water models on the sphere. The advantages and the disadvantages of the two shallow water formulations (vector and scalar) are contrasted and employed to demonstrate that the new icosahedral methodology is highly competitive with previously suggested grid-point models. The remaining results which we discuss relate to the process of erosion of a stratospheric polar vortex by a forced stationary Rossby wave disturbance, a physical problem which has previously been analyzed in detail in several well-known spectral transform simulations. It is shown that all of our models properly simulate this intensely nonlinear and computationally challenging physical process. (C) 1999 Academic Press.