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A Conservative Formulation and a Numerical Algorithm for the Double-Gyre Nonlinear Shallow-Water Model

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Incompressible inviscid fluids

Published online by Cambridge University Press:  10 November 2015

Dongyang Kuang  and
Long Lee
Dongyang Kuang
Affiliation:
Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA
Long Lee*
Affiliation:
Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA
*
Corresponding author. Email address:dkuang@uwyo.edu (D.-Y. Kuang), llee@uwyo.edu (L. Lee)
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Abstract

We present a conservative formulation and a numerical algorithm for the reduced-gravity shallow-water equations on a beta plane, subjected to a constant wind forcing that leads to the formation of double-gyre circulation in a closed ocean basin. The novelty of the paper is that we reformulate the governing equations into a nonlinear hyperbolic conservation law plus source terms. A second-order fractional-step algorithm is used to solve the reformulated equations. In the first step of the fractional-step algorithm, we solve the homogeneous hyperbolic shallow-water equations by the wave-propagation finite volume method. The resulting intermediate solution is then used as the initial condition for the initial-boundary value problem in the second step. As a result, the proposed method is not sensitive to the choice of viscosity and gives high-resolution results for coarse grids, as long as the Rossby deformation radius is resolved. We discuss the boundary conditions in each step, when no-slip boundary conditions are imposed to the problem. We validate the algorithm by a periodic flow on an f-plane with exact solutions. The order-of-accuracy for the proposed algorithm is tested numerically. We illustrate a quasi-steady-state solution of the double-gyre model via the height anomaly and the contour of stream function for the formation of double-gyre circulation in a closed basin. Our calculations are highly consistent with the results reported in the literature. Finally, we present an application, in which the double-gyre model is coupled with the advection equation for modeling transport of a pollutant in a closed ocean basin.

Keywords

Double-gyrereduced-gravity shallow-water equationswave-propagation algorithmfractional-step algorithm

MSC classification

Secondary: 65M08: Finite volume methods 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 634 - 650
Copyright
Copyright © Global-Science Press 2015 

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