Influence of a step-like coastline on the basin scale vorticity budget of mid-latitude gyre models

Abstract

Global vorticity budgets in C-grid shallow water (SW) and quasi-geostrophic (QG) models of winddriven ocean circulation with free-slip boundary conditions are considered. For both models, it is pointed out that the discretized vorticity equation is defined only over a subdomain that excludes boundary grid nodes. At finite resolution, this implies an advective flux of vorticity across the perimeter of the discretized vorticity domain. For rectangular basins where grid axes are aligned with the basin walls, this flux tends to zero as resolution is increased. We also consider the case in which the grid is rotated with respect to the basin, so that a step-like coastline results. Increased resolution then leads to more steps and, because the advective flux of vorticity out of the domain is particularly large at steps, it is no longer obvious that increased resolution should reduce the advective flux. Results are found to be sensitive to numerical details. In particular, we consider different formulations for the non-linear terms (for both the SW and QG models) and two formulations of the viscous stress tensor for the SW model [the conventional five-point Laplacian and the δ—ζ stress tensor suggested by Madec et al. (J. Phys. Oceanogr. 21, 1349—1371)]. For the SW model, the overall circulation and the behavior of the flux term are dependent on both the formulation of the viscous stress tensor and the non-linear terms. The best combination is found to be the δ—ζ tensor with an enstrophy-preserving advection scheme. With this combination, the circulation of the non-rotated basin is recovered in rotated basins and the advective flux tends to converge towards zero with increasing resolution. The poorest combination is the δ—ζ tensor with the conventional advective scheme. In this case, the advective flux term diverges with increasing resolution for some rotation angles and the model crashes for some others. For the QG model, the convergence order of the advective flux term of absolute vorticity is near unity (roughly the same as with the SW model). Most of the error (especially at high resolution) is related to errors in the β term (which is hidden in the advective contribution in the SW model). However, the overall circulation is less sensitive to the rotation of the grid with respect to the basin, especially when the Jacobian proposed by Arakawa (J. Comput. Phys. 1, 119—143) is used.