Internal Boundary Layer Scaling in “Two Layer” Solutions of the Thermocline Equations

Abstract

The diffusivity dependence of internal boundary layers in solutions of the continuously stratified, diffusive thermocline equations is revisited. If a solution exists that approaches a two-layer solution of the ideal thermocline equations in the limit of small vertical diffusivity ??, it must contain an internal boundary layer that collapses to a discontinuity as ?? ? 0. An asymptotic internal boundary layer equation is derived for this case, and the associated boundary layer thickness is proportional to ?1/2?. In general, the boundary layer remains three-dimensional and the thermodynamic equation does not reduce to a vertical advective?diffusive balance even as the boundary layer thickness becomes arbitrarily small. If the vertical convergence varies sufficiently slowly with horizontal position, a one-dimensional boundary layer equation does arise, and an explicit example is given for this case. The same one-dimensional equation arose previously in a related analysis of a similarity solution that does not itself approach a two-layer solution in the limit ?? ? 0.