A Local Method for Imposing Surface Drag Conditions on Deformed Terrain Boundaries

The implementation of surface drag conditions on deformed terrain surfaces is considered, in the context of the Weather Research and Forecasting (WRF) model.

In atmospheric models, the interaction of the winds with the ground is generally represented in terms of a turbulent momentum flux across the lower boundary, referred to loosely as a surface drag. In terms of the overall model formulation, this specified drag forms part of the lower boundary condition on velocity, analogous to the partial slip condition used in other branches of fluid mechanics (e.g., rarefied gas flows, flows with moving contact lines, or high Reynolds number flows past rough surfaces). In practice, imposing the drag condition can be thought of as consisting of two parts: (i) a model giving the subgrid-scale turbulent fluxes at the boundary in terms of a given macro state of the flow (i.e., a drag parameterization); and (ii) a boundary condition matching the parameterized fluxes to the turbulent stresses in the fluid interior.

On deformed terrain surfaces, the problem of implementing the drag condition at the boundary can be somewhat complex, due in large part to the tensor nature of the stress. Previous approaches have relied on directly discretizing the boundary condition on the model grid, resulting ultimately in a global sparse matrix problem for the velocity at the boundary. However, while not incorrect, such methods have proved difficult to implement in the context of highly parallelized models using domain decomposition, due to global nature of the matrix inversion.

In the present study, a new method for implementing the drag condition is presented, in which the boundary condition is recast into a form allowing a straightforward local implementation, thus eliminating the need for a global matrix inversion. As an illustration, the new method is implemented in the context of the widely used WRF model. Verifications are presented showing that for sufficiently high resolution, the new approach produces essentially identical results to the previous sparse matrix method. Comparisons are made to the default (public) version of WRF, which effectively implements the drag as if the lower boundary flat. The results show that for sufficiently deformed topographic features, the proper handling the boundary condition leads to a significant impact on the flow.