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A Spectral Finite-Volume Method for the Shallow Water Equations

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  • 1 Institute of Marine and Coastal Sciences, Rutgers–The State University of New Jersey, New Brunswick, New Jersey
  • | 2 Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida
  • | 3 Institute of Marine and Coastal Sciences, Rutgers University–The State University of New Jersey, New Brunswick, New Jersey
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Abstract

A spectral finite-volume (SFV) method is proposed for the numerical solution of the shallow water equations. This is the first phase in the development of a layered (isopycnal) ocean model. Its target applications include, in particular, the simulation of the wind-driven oceanic circulation in geometrically complex basins where layer outcropping and/or isopycnal–bathymetry intersection must be handled explicitly. The present formulation is geometrically flexible and can extend accuracy to arbitrary high order with no change to the basic algorithm. A flux-corrected transport (FCT) algorithm ensures the stability of the computations in regions of vanishing layer thickness and in areas where the flow features are underresolved. The spatial discretization is based on a two-level grid: a globally unstructured elemental grid and a locally structured grid consisting of N × N quadrilateral cells within each element. The numerical solution is continuous within each element but discontinuous across elements; the discontinuity is resolved by upwinding along characteristics. The accuracy and convergence rate of the SFV method are verified on two linearized problems amenable to analytical solution; the SFV solution exhibits a convergence order of N + 1 for smooth solutions. The FCT portion of the model is tested by simulating the formation of an oblique hydraulic jump in a supercritical channel flow. The model is then applied to simulate, in reduced-gravity mode, the double-gyre and wind-driven upper-ocean circulations in a square basin. Finally, the previous experiment is repeated in the North Atlantic basin to illustrate the application of the model in a realistic geometry.

Corresponding author address: Byoung-Ju Choi, Institute of Marine and Coastal Sciences, 71 Dudley Road, New Brunswick, NJ 08901-8521. Email: bchoi@imcs.rutgers.edu

Abstract

A spectral finite-volume (SFV) method is proposed for the numerical solution of the shallow water equations. This is the first phase in the development of a layered (isopycnal) ocean model. Its target applications include, in particular, the simulation of the wind-driven oceanic circulation in geometrically complex basins where layer outcropping and/or isopycnal–bathymetry intersection must be handled explicitly. The present formulation is geometrically flexible and can extend accuracy to arbitrary high order with no change to the basic algorithm. A flux-corrected transport (FCT) algorithm ensures the stability of the computations in regions of vanishing layer thickness and in areas where the flow features are underresolved. The spatial discretization is based on a two-level grid: a globally unstructured elemental grid and a locally structured grid consisting of N × N quadrilateral cells within each element. The numerical solution is continuous within each element but discontinuous across elements; the discontinuity is resolved by upwinding along characteristics. The accuracy and convergence rate of the SFV method are verified on two linearized problems amenable to analytical solution; the SFV solution exhibits a convergence order of N + 1 for smooth solutions. The FCT portion of the model is tested by simulating the formation of an oblique hydraulic jump in a supercritical channel flow. The model is then applied to simulate, in reduced-gravity mode, the double-gyre and wind-driven upper-ocean circulations in a square basin. Finally, the previous experiment is repeated in the North Atlantic basin to illustrate the application of the model in a realistic geometry.

Corresponding author address: Byoung-Ju Choi, Institute of Marine and Coastal Sciences, 71 Dudley Road, New Brunswick, NJ 08901-8521. Email: bchoi@imcs.rutgers.edu

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