Assessment of Model Error in Limited-Area Simulations of Shallow Water Test Cases on the C-Grid Plane and Sphere

Abstract

suite of limited-area test cases for the solution of the shallow water (SW) equations on the plane and sphere is collected and evaluated using the Model for Prediction Across Scales (MPAS) modeling system. Included are regional simulations of standard test cases, as well as new viscous and linearized test cases with exact analytic solutions. Four different aspects of model error are isolated and thereby assessed: 1) error generated by grid nonuniformity on the plane and sphere, 2) time-independent (balanced) error and time-dependent (propagating wave) error, 3) lateral boundary implementation error, and 4) error reduction due to a viscous equation set or an absorbing sponge layer. Results show that the nature of model error for these test cases is specific to the geophysical regime: SW flows on a rotating sphere with Froude numbers of O(0.1) and Rossby numbers also of O(0.1). For SW simulations in this context, inward reflection of gravity waves at the domain boundary does not appear to be a driver of instability or a determining factor in solution accuracy. This conclusion has important implications for idealized studies that exclude this geophysical regime, in particular one-dimensional studies of lateral boundary conditions for limited-area models and studies of the absorption of linearized gravity waves at domain boundaries using dynamical relaxation. An old debate over the efficacy of dynamical relaxation for a viscous equation set is addressed; MPAS simulations of an SW test case with and without dynamical relaxation support claims that eddy viscosity does not notably improve solution accuracy or stability.