On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part I: Formulation and Comparison of Exact Solutions

Abstract

Motivated by the general objective of pursuing oceanographic process and data assimilation studies of the complex, nonlinear eddy and jet current fields observed over the continental shelf and slope off the west coast of the United States, we investigate the use of intermediate models for that purpose. Intermediate models contain physics between that in the primitive equations and that in the quasigeostrophic equations and are capable of representing subinertial frequency motion over the O(1) topographic variations typical of the continental slope, while filtering out high-frequency gravity-inertial waves. As an initial step, we compare and evaluate several intermediate models applied to the f-plane shallow-water equations for flows over topography. The accuracy and utility of the intermediate models are assessed by a comparison of exact analytical and numerical solutions with those of the primitive shallow-water equations (SWE) and with those of the quasi-geostrophic equations (QG). The intermediate models that we consider are based on the geostrophic momentum (GM) approximation, the derivation of Salmon (1983) utilizing Hamilton's principle (HP), a geostrophic vorticity (GV) approximation, the quasi-geostrophic momentum and full continuity equations (IM), the linear balance equations (LBE), the balance equations (BE), the related balance-type (HBE, BEM, NBE) and modified linear balance equations (LQBE), the slow equations (SE) of Lynch (1989), and the modified slow equations (MSE). In Part I, we discuss the intermediate models and develop formulations that are suitable for numerical solution in physical coordinates for use in Parts II and III. We investigate the capability of the intermediate models to represent linear ageostrophic coastally trapped waves, i.e., Kelvin and continental shelf waves, and demonstrate that they do so with accuracy consistent with standard linear low-frequency approximations. We also assess the accuracy of the models by a comparison of exact nonlinear analytical solutions to the SWE for steady flow in an elliptic paraboloid and for unsteady motion of elliptical vortices in a circular paraboloid with corresponding analytical solutions to the intermediate models and to QG. General results from the exact solution comparisons include the following. Many of the intermediate models are capable of producing more accurate solutions than QG over a range of Rossby numbers 0 < ? < 1. In some cases, the intermediate models provide accurate approximate solutions where QG is not applicable and fails to give a relevant solution. Considerable parameter-dependent variation in quality exists, however, among the different intermediate models. For the particular problems considered here, BE, HBE, BEM, NBE, and MSE reproduce the exact results of the SWE while LBE and LQBE give the same approximation as QG. The accuracy of the models is typically in the order GV, GM, IM, HP, and QG, with GV most accurate and IM and HP sometimes less accurate than QG.