We present a conservation 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.
A high-order velocity-based discontinuous Galerkin scheme for the shallow water equations: local conservation, entropy stability, well-balanced property, and positivity preservation
Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation
A ghost-cell immersed boundary method for the shallow water equations on Cartesian grids: mass conservation and wetting and drying over variable topography
Variational integrator for the rotating shallow-water equations on the sphere
Memorial University of Newfoundland（纽芬兰纪念大学）；Imperial College London（伦敦帝国学院）；École Normale Supérieure/CNRS, Laboratoire de Météorologie Dynamique（巴黎高等师范学院/法国国家科学研究中心，动态气象实验室）
GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws
Modelling Engineering Risk and Complexity, Scuola Superiore Meridionale（建模工程风险与复杂性，南方高级学校）；Dept. of Electrical Engineering and Information Technology, University of Naples Federico II（电气工程与信息技术系，那不勒斯费德里科二世大学）；Institute of Science and Technology for Energy and Sustainable Mobility, Consiglio Nazionale delle Ricerche（能源与可持续交通科学与技术研究所，意大利国家研究委员会）
High order tracer variance stable transport with low order energy conserving dynamics for the thermal shallow water equations
Bureau of Meteorology（澳大利亚气象局）；Mathematical Sciences Institute, Australian National University（澳大利亚国立大学数学科学研究所）；School of Mathematics, Monash University（莫纳什大学数学学院）
Entropy analysis and entropy stable DG methods for the 1D shallow water moment equations
Bernoulli Institute, University of Groningen（格罗宁根大学伯努利研究所）；Department of Mathematics, Linköping University（林雪平大学数学系）；Department of Mathematics, Computer Science and Statistics, Ghent University（根特大学数学、计算机科学和统计系）
Weak and entropy physics-informed neural networks for conservation laws
Mohammed VI Polytechnic University（穆罕默德六世polytechnic大学）；LAGA, CNRS, UMR 7539, Université Sorbonne Paris Nord（索邦巴黎北大学LAGA实验室，法国国家科学研究中心）；Autorité de Sûreté Nucléaire et de Radioprotection (ASNR)（核安全与辐射防护管理局）；McGill University（麦吉尔大学）
Invariant discretization schemes for the shallow-water equations
Centre de recherches mathématiques, Université de Montréal（蒙特利尔大学数学研究中心）；Faculty of Mathematics, University of Vienna（维也纳大学数学系）；Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）
Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws
University of Cologne（科隆大学）；Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Center for Data and Simulation Science (CDS), University of Cologne（科隆大学数据与模拟科学中心）
A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods
University of Wisconsin–Platteville（威斯康星大学普拉特维尔分校）；Instituto Tecnológico Autónomo de México（墨西哥自治理工学院）；Iowa State University（爱荷华州立大学）；Minnesota State University Moorhead（明尼苏达摩尔黑德州立大学）；University of Florida（佛罗里达大学）
