Effect of a wavy wall on the single gyre Munk problem

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 ʹrunawayʹ 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.