Quadrati Spline Galerkin Method for the Shallow Water Equations on the Sphere

Currently in most global meteorological applications, the spectral transform method or low-order finite difference/finite element methods are used. The spectral transform method, which yields high-order approximations, requires Legendre transforms. The Legendre transforms have a computational complexity of O(N3), where N is the number of subintervals in one dimension, and thus render the spectral transform method unscalable. In this study, we present an alternative numerical method for solving the shallow water equations (SWEs) on a sphere in spherical coordinates. In this implementation, the SWEs are discretized in time using the two-level semi-Lagrangian semi-implicit method, and in space on staggered grids using the quadratic spline Galerkin method. We show that, when applied to a simplified version of the SWEs, the method yields a neutrally stable solution for the meteorologically significant Rossby waves. Moreover, we demonstrate that the Helmholtz equation arising from the discretization and solution of the SWEs should be derived algebraically rather than analytically, in order for the method to be stable with respect to the Rossby waves. These results are verified numerically using Boyd’s equatorial wave equations [2] with initial conditions chosen to generate a soliton.