A Conservative Formulation and a Numerical Algorithm for the Double-Gyre Nonlinear Shallow-Water Model. Issue 4 (10th November 2015)

• Record Type: Journal Article
• Authors: Kuang, Dongyang; Lee, Long
• Is Part Of: Numerical mathematics. Volume 8:Issue 4(2015)
• Page Start: 634
• Page End: 650
• Publication Date: 2015-11-10
• Subjects: 65M08, -- 76B15; Double-gyre, -- reduced-gravity shallow-water equations, -- wave-propagation algorithm, -- fractional-step algorithm; Numerical analysis -- Periodicals; Numerical analysis; Periodicals; 518.05
• DOI: 10.4208/nmtma.2015.m1408
• Languages: English
• ISSNs: 1004-8979

Abstract

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.