On the numerical treatment of a lateral boundary layer in a shallow sea model

We consider the flow in an idealized, shallow sea, steady state, coastal boundary layer. The driving force is the along-shore slope of the sea surface. The latter is balanced by bottom stress and horizontal diffusion of momentum. The momentum equation governing the along-shore velocity is derived. Suitable assumptions are made in order to obtain a closed form solution. The governing equation is then discretized on the B- and C-grid, with slip (together with an appropriate wall function) and no-slip boundary conditions. The discrete solutions are compared with the exact one, according to different standards: the errors affecting pointwise values, as well as fluxes, are evaluated. The errors are estimated as a function of the ratio of the boundary layer width to the grid size, which ranges from 0.01 (non resolved boundary layer) to 100 (well resolved boundary layer). In general, the errors are small. However, for the B-grid with a no-slip boundary condition, the flux crossing the grid box adjacent to the coast is in error by a factor of approximate to 1/2 when the width of the boundary layer is much smaller than the grid size. The C-grid, when implemented with a slip condition, permits the highest overall accuracy.