Author(s): R. L. Kolar; W. G. Gray; J. J. Westerink; R. A. Luettich Jr.

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Abstract: 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 orders of magnitude less than lateral components. Similarly, when the vertical velocity is small, the curvature term in the momentum balance that comes from the convective terms is insignificant. The second curvature term in the momentum balance is associated with the macroscopic stress tensor; it is on the same order of magnitude as other stress terms and should be retained when momentum dissipation is modeled. Finite element discretization of the resulting balance laws utilizes the generalized wave continuity equation which, because of its monotonie dispersion relation, eliminates noise in the solution without resorting to artificial damping. A cartographic mapping simplifies finite element implementation in spherical coordinates so that Cartesian master elements can be used. Finally, an application to the Western North Atlantic and Gulf of Mexico illustrates the importance of using equations in spherical coordinates for large scale applications.

DOI: https://doi.org/10.1080/00221689409498786

Year: 1994