We re-visit the linear model of a thermally-driven ocean in a single hemisphere, to understand dependencies on viscosity and mixing. We focus in particular on the vertical and surface velocities. In all cases, the meridional and vertical flows are intensified near the boundaries; as a result, the overturning depends on viscosity. The two types of viscosity we examine (Rayleigh damping and diffusion with the no-slip condition) yield significantly different boundary transports. This in turn can cause large changes in the surface velocities. We observe a single-gyre surface circulation with a diffusive viscosity but two gyres with the Rayleigh. We also examine how the solutions change when the vertical mixing itself is intensified near the boundaries. With (spatially) constant mixing, a significant fraction of the vertical transport occurs in the thermocline interior where viscosity is unimportant. But the localized mixing increases the boundary intensification of the upwelling, making viscosity even more important. There is evidence of similar boundary intensification and viscosity dependence in numerical models.
LaCasce, J. H.. 2004. "Diffusivity and viscosity dependence in the linear thermocline." Journal of Marine Research 62, (6). https://elischolar.library.yale.edu/journal_of_marine_research/61