A local implementation of the tensor shear stress condition on complex terrain surfaces

Abstract:

In atmospheric models, the exchange of momentum between the atmosphere and the ground is described in terms of a surface layer parameterization, which predicts the turbulent flux of momentum across the terrain surface. However, in current models, this flux is usually specified as if the ground were flat (horizontal), which can lead to significant errors in regions of complex terrain.

Mathematically speaking, the imposed surface layer flux amounts to a specified shear stress across the terrain surface, which acts as a boundary condition on velocity at the lower boundary of the model domain. However, in its general form, this condition can be difficult to apply, due to the tensor nature of the flux, leading most models to adopt one of two approximations: either (a) the lower boundary is assumed flat, as in current generation atmospheric models; or else (b) the terrain is approximated by its local tangent plane, which reduces the stress to a normal gradient condition. In previous work, a method for imposing the full (unapproximated) condition was developed in the finite-difference framework, in which the tensor stress condition was reduced to a sparse matrix problem at the boundary. However, while effective in principle, this approach was largely impractical for highly parallelized models, in which the decomposition of the domain into tiles complicates the matrix inversion step.

In the present study, we show that the full tensor stress condition at the boundary can be recast into a form allowing a straightforward local (or point-by-point) implementation, thus bypassing the need for a global matrix inversion. In this recast form, the condition is shown to consist of two parts: (a) a part involving the normal gradient of the tangential winds (i.e., a normal-gradient term); and (b) a part involving curvatures of the underlying terrain surface. An implementation of the condition is demonstrated in the context of the Weather Research and Forecasting (WRF) model, and evaluated by comparison to simulations using both the flat-boundary and normal-gradient approximations. It is shown that for complex terrain flows, both the flat-boundary and normal-gradient assumptions can lead to significant differences from the full tensor flux condition.