Search

A spectral element shallow water model with an adaptive triangular mesh is used to examine the effects of basin geometry on the single gyre Munk problem. Specifically, we consider a circular basin shape, with and without the addition of a sinusoidal undulation in the coastline. We are particularly interested in how the curviness of the coastline might effect the well-known tendency of the solution to ‘runaway’ as the lateral viscous coefficient is made small. Results were dependent on boundary conditions. We consider both no slip and three variants of the free slip condition: zero vorticity at the coast, zero normal derivative of tangential velocity and zero normal flux of tangential momentum. For cases where the coastal undulations were large in amplitude, the no slip and the third free slip condition showed the least tendency towards runaway (i.e. towards developing unrealistically large velocities as the viscous coefficient was made small). In the circular geometry, by contrast, the differences between no slip and free slip were much greater, and of the variants on free slip, the zero vorticity condition showed the least tendency towards runaway. We also show that high-frequency variability (such as Kelvin waves) can differ markedly depending on which of the three free slip variants is used. [ABSTRACT FROM AUTHOR]

ABSTRACT Global vorticity budgets in C-grid shallow water (SW) and quasi-geostrophic (QG) models of wind-driven 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... [ABSTRACT FROM AUTHOR]