Shallow water modeling in spherical coordinates: equation formulation, numerical implementation, and application

One class of surface water models is the shallow water models obtained by depth-averaging the microscale mass and momentum balances. Application of shallow water models to large scale problems (on the order of 1000's of km) requires the use of spherical coordinates. Traditionally, balance laws in spherical coordinates are derived by simply expanding the spatial operators in the standard depth-averaged equations. However, the equations themselves are based on an assumed planar surface so that an inconsistency exists between the derivation and the interpretation. In this article, a method is presented that properly accounts for the curvature of the Earth during the depth-averaging procedure. The derivation gives rise to new terms in both the continuity and momentum balances, terms that we refer to as curvature terms. A scaling analysis evaluates the magnitude of the terms. It is shown that the curvature term in the continuity balance is insignificant when the vertical velocity is small, i.e., at least four ...