AbstractWe 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.
Transparent Boundary Conditions for the Shallow-Water Equations: Testing in a Nested Environment