Hostname: page-component-f554764f5-44mx8 Total loading time: 0 Render date: 2025-04-15T06:54:43.759Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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:[email protected] (D.-Y.Kuang), [email protected] (L.Lee)
Get access

Abstract

We present a conservative formulation and a numerical algorithm for thereduced-gravity shallow-water equations on a beta plane, subjected to a constantwind forcing that leads to the formation of double-gyre circulation in a closedocean basin. The novelty of the paper is that we reformulate the governingequations into a nonlinear hyperbolic conservation law plus source terms. Asecond-order fractional-step algorithm is used to solve the reformulatedequations. In the first step of the fractional-step algorithm, we solve thehomogeneous hyperbolic shallow-water equations by the wave-propagation finitevolume method. The resulting intermediate solution is then used as the initialcondition for the initial-boundary value problem in the second step. As aresult, the proposed method is not sensitive to the choice of viscosity andgives high-resolution results for coarse grids, as long as the Rossbydeformation radius is resolved. We discuss the boundary conditions in each step,when no-slip boundary conditions are imposed to the problem. We validate thealgorithm by a periodic flow on an f-plane with exactsolutions. The order-of-accuracy for the proposed algorithm is testednumerically. We illustrate a quasi-steady-state solution of the double-gyremodel via the height anomaly and the contour of stream function for theformation of double-gyre circulation in a closed basin. Our calculations arehighly consistent with the results reported in the literature. Finally, wepresent an application, in which the double-gyre model is coupled with theadvection equation for modeling transport of a pollutant in a closed oceanbasin.

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Calhoun, D. and LeVeque, R. J.. A Cartesian grid finite-volume method for the advection-diffusion equation in irregular regions. J. Comput. Phys., 156 (2002) 1–38.Google Scholar
[2]Harten, A., Enquist, B., Osher, S., and Chakravarthy, S. R.. Uniformly high-order accurate essentially non-oscillatory schemes III. J. Comput Phys., 71, (1987), 231–303.CrossRefGoogle Scholar
[3]Jiang, S., Jin, F-F., and Ghil, M.. Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model. J. Phys. Ocean., 25 (1995) 764–786.2.0.CO;2>CrossRefGoogle Scholar
[4]Jones, D .A., Poje, A. C., and Margolin, L. G.. Resolution effects and enslaved finite-difference schemes for a double gyre, shallow-water model. Theoret. Comput. Fluid Dynamics, 9 (1997) 269–280.CrossRefGoogle Scholar
[5]Lee, L. and LeVeque, R. J.. An immersed interface method for the incompressible Navier- Stokes equations. SIAM Sci. Comp., 25, (2003) 832–856.CrossRefGoogle Scholar
[6]Lee, L.. A class of high-resolution methods for incompressible flows. Compter & Fluids, 39, (2010) 1022–1032.CrossRefGoogle Scholar
[7]LeVeque, R. J.. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, August 26, 2002Google Scholar
[8]LeVeque, R. J.. Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems. SIAM, 2007.CrossRefGoogle Scholar
[9]Pedlosky, J.. Geophysical Fluid Dynamics. Springer: New York, 1987.CrossRefGoogle Scholar
[10]Poje, A. C., Jones, D .A., and Margolin, L. G.. Enslaved finite difference approximations for quasgeostrophic shallow flows. Physica D, 98 (1996), 559–573.CrossRefGoogle Scholar
[11]Salman, H., Kuzetsov, L., Jones, C. K. R. T., and Ide, K.. A method for assimilating La-grangian data into a shallow-water equation ocean model. Mon. Weather Rev., 134 (2006) 1081–1101.CrossRefGoogle Scholar
[12]Sheu, C.-W. and Osher, S.. Efficient implementation of essential non-oscillatory shock-capturing schemes. J. Comput, Physics., 77 (1988), 439–471.CrossRefGoogle Scholar
[13]Sheu, C.-W. and Osher, S.. Efficient implementation of essential non-oscillatory shock-capturing schemes II. J. Comput, Physics., 83 (1989), 32–78.CrossRefGoogle Scholar
[14]Smolarkiewicz, P. K.. A fully multidimensional positive definite advection transport algorithm with small implicit diffusion. J. Comput. Phys., 54 (1984), 325–362.CrossRefGoogle Scholar
[15]Smolarkiewicz, P. K. and Margolin, L. G.. On forward-in-time differencing for fluid. Monthly Weather Rev., 121 (1993), 1849–1859.2.0.CO;2>CrossRefGoogle Scholar
[16]Smolarkiewicz, P. K. and Margolin, L. G.. MPDATA: A finite-difference solver for geophysical flows. J. Comput. Phys., 140 (1998), 459–480.CrossRefGoogle Scholar
[17]Sweby, P. K.High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM, Numer. Anal., 21(5) (1984), 995–1011.CrossRefGoogle Scholar
[18]Xu, Z. and Shu, C.-W.. Anti-diffusion finite difference WENO methods for shallow water with transport of pollutant. J. Comput. Math., 24 (2006), 239–251.Google Scholar
[19]Yoshida, H.. Construction of higher order symplectic integrators. Physics Letters A, 150(5-7):262?268, (1990).CrossRefGoogle Scholar
[20]Zalesak, S. T.. Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys., 31 (1979), 335–362.CrossRefGoogle Scholar