In this paper, we analyze one-, two- and three-dimensional numerical solutions of a simple, inertia-less ocean circulation model. The solutions, which all approach a steady state, demonstrate that, in the limit of vanishing thermal diffusivity κ, a front of thickness κ1/2, identifiable with the thermocline, spontaneously appears at a location anticipated by simple arguments that treat the front as an "internal boundary layer." The temperature and velocity are generally discontinuous across the front, but the velocity component normal to the front is zero. In the asymptotic limit of vanishing diffusivity, the temperature has no vertical variation within the layer above the front, and the potential vorticity is correspondingly zero. The appearance of a front seems to require that the horizontal advection terms cancel in the temperature equation, i.e., that the horizontal velocity be directed along the isotherms on level surfaces. When the surface boundary conditions are specially chosen to prevent this cancellation, the front does not appear. However, in the more realistic cases in which the flow determines its own surface temperature, the cancellation occurs spontaneously and appears to be generically associated with the front.
Salmon, Rick. 1990. "The thermocline as an "internal boundary layer"." Journal of Marine Research 48, (3). https://elischolar.library.yale.edu/journal_of_marine_research/1976