Time integration of diapycnal diffusion and Richardson number-dependent mixing in isopycnal coordinate ocean models

Abstract

In isopycnal coordinate ocean models, diapycnal diffusion must be expressed as a nonlinear difference equation. This nonlinear equation is not amenable to traditional implicit methods of solution, but explicit methods typically have a time step limit of order Δt ≤ h2/κ (where Δt is the time step, h is the isopycnal layer thickness, and κ is the diapycnal diffusivity), which cannot generally be satisfied since the layers could be arbitrarily thin. It is especially important that the diffusion time integration scheme have no such limit if the diapycnal diffusivity is determined by the local Richardson number. An iterative, implicit time integration scheme of diapycnal diffusion in isopycnal layers is suggested. This scheme is demonstrated to have qualitatively correct behavior in the limit of arbitrarily thin initial layer thickness, is highly accurate in the limit of well-resolved layers, and is not significantly more expensive than existing schemes. This approach is also shown to be compatible with an implicit Richardson number-dependent mixing parameterization, and to give a plausible simulation of an entraining gravity current with parameters like the Mediterranean Water overflow through the Straits of Gibraltar.