Fourth- and Fifth-Order Finite-Difference Methods Applied to a Control-Volume Ocean Model

Abstract

A semi-implicit, control-volume, nonhydrostatic model is presented. Advection is fifth order with respect to space. Boundary conditions and molecular fluxes are also formulated at fourth order with respect to space. Computational cost is strictly proportional to the number of grid points. Barotropic and nonhydrostatic pressure gradients are calculated using fourth-order explicit calculation followed by a second-order implicit correction for the incremental pressure gradient updates. This strategy is used in the DieCAST model in order to ensure accurate treatment of geostrophy with minimal computational cost, and here it is diagnosed to be advantageous for both small- and large-scale convective flows. Anisotropic grids can result in the vertical Courant number being much larger than the horizontal Courant number, in which case it may be advantageous to use a time step?limited horizontal advection scheme with a more computationally expensive but time step?unlimited vertical advection scheme. An anisotropic grid also admits a quasi one-dimensional calculation of nonhydrostatic pressure, which is not computationally expensive. A two-scale calculation of convecting cells in the deep ocean indicates that, in this circumstance, subgrid-scale processes cannot be parameterized by any means that causes smoothing.