A C⁰-discontinuous Galerkin method for the stationary quasi-geostrophic equations of the ocean

This work concerns the development of a finite-element algorithm for the stationary quasi-geostrophic equations to treat the large scale wind-driven ocean circulation. The algorithm is developed based on the streamfunction formulation involving fourth-order gradients of the streamfunction. Here, we examine the adaptation of a relatively inexpensive, nonconforming method based on C⁰ basis functions. We develop the variational form of the method and establish consistency. The method weakly enforces continuity of the normal flux across interelement boundaries and stabilization is achieved via Nitsche's method. Explicit expression for the choice of the stabilization parameter is derived. Moreover, the theoretical error estimate for the linear Stommel-Munk model is performed. Numerical results from several benchmark problems on rectangular and curved domains are provided to demonstrate the accuracy and robustness of the method. We also provide the Mediterranean sea example to demonstrate the capability of the approach to the wind-driven ocean circulation simulation.