Improvements to Horizontal Transport in Grid Models

Abstract

The last decade has witnessed significant improvements in the quality of advection schemes in terms of reduced amounts of numerical dispersion and diffusion. The levels of numerical diffusion are now down to the point where one has to be concerned about adding back amounts of diffusion that are appropriate to the atmospheric conditions being modeled. Determination of the appropriate horizontal diffusion coefficients that should be used in a model requires that one compute them as the total desired diffusivity minus the numerical diffusivity already accompanying the advection scheme; that is, KHM = KHT - KHN Even the numerical diffusivity, KHN, known to vanish at Courant (CFL) numbers, ex = u-Δt/Δx, of 0, -1, and +1 with flux formulation algorithms and peak at CFL of +1/2 and -1/2, must first be modeled if KHM is to be properly characterized as a function of CFL. It also is important to realize that the “total” diffusivity may involve a number of different processes, such as temporal wind field variations (e.g., sub-grid-scale turbulence and grid-scale wind meander) that are unresolved or dissipated by the wind field model. In addition, there are other transport errors which begin to become evident with these higher quality transport schemes; specifically, operator splitting errors and shear flow errors, which were heretofore buried by the numerical diffusion. Some of the effects of wind shear, for example, are genuinely transportive rather than diffusive, but may be lumped into the diffusivity for expediency. The mathematical characterization of these diffusivity and error terms will be detailed in the paper along with proposed corrections.