Shear Stress Conditions on Complex Terrain Surfaces: Basic Physics and Implementation

Abstract

The behavior and dynamics of the surface stress boundary condition are explored, both in terms of the basic physics of the condition and the associated implementation in finite-difference models. Numerical experiments are presented to illustrate the impact of the stress condition on flows past a region of complex terrain, with particular emphasis on the dependence of the condition on terrain geometry. Arguments are presented to show that the surface stresses depend on the terrain geometry in two ways: (i) a dependence on slope, as represented by a normal gradient term; and (ii) a dependence on terrain curvature, which appears in the condition as a Dirichlet term. This dependence on terrain geometry is illustrated through a series of experiments in which simulations using the full form of the stress condition are compared to companion simulations using one of two widely used approximations: (a) the normal gradient condition, which accounts for the terrain slope but neglects curvature; and (b) the flat boundary assumption, which neglects both slope and curvature. The results show that for realistic flows, the terrain geometry plays an important role in the behavior of the surface stresses, and that the associated approximate conditions fail to capture important aspects of the flow over complex terrain. Previous implementations of the stress condition in numerical models (including the experiments described above) have largely relied on a direct discretization of the stresses at the boundary, ultimately resulting in a global sparse matrix inversion. However, such methods are difficult to implement in highly parallelized models, in which domain decomposition strategies are generally employed. To simplify the implementation, a new method is presented in which the drag condition is recast into a form allowing a straightforward local implementation, thus eliminating the need for a global inversion. As an example, the new approach is implemented in the context of the widely used Weather Research and Forecasting (WRF) model. Verifications are presented showing that for sufficiently high resolution, the new method as implemented in WRF produces essentially identical results to the previous matrix approach.