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

In isopycnal coordinate ocean models, diapyenal diffusion must be expressed as a nonlinear difference equation. This nonlinear equation is not amenabl e to traditional implicit methods of solution. but explicit methods typical ly have a rime step limit of order Delta t less than or equal to h(2)/kappa (where Delta t is the time step, h is the isopyenal layer thickness, and k appa is the diapyenal diffusivity), which cannot generally be satisfied sin ce the layers could be arbitrarily thin. It is: especially important that t he diffusion time integration scheme have no such limit if the diapyenal di ffusivity is determined by the local Richardson number. An iterative, impli cit time integration scheme of diapyenal diffusion in isopyenal layers is s uggested. This scheme is demonstrated to have qualitatively correct behavio r in the limit of arbitrarily thin initial layer thickness, is highly accur ate in the limit of well-resolved layers, and is not significantly more exp ensive than existing schemes. This approach is also shown to be compatible with an implicit Richardson number-dependent mixing parameterization. and t o give a plausible simulation of an entraining gravity current with paramet ers like the Mediterranean Water overflow through the Straits of Gibraltar.