Tests of an Advanced Surface-Stress and Surface-Flux Implementation Accounting for Complex Terrain Geometry

In numerical models, the interaction of the atmosphere with the Earth's surface is typically represented in terms of surface-layer parameterizations, whose main role is to specify turbulent fluxes of heat, moisture and momentum across the lower boundary of the model domain. Various formulations can be found in the literature, with differences mainly related to the treatment of stratification effects. However, at least in current-generation models, these parameterizations all share a common limitation: namely, that the fluxes are always imposed as if the lower boundary were flat. While this is likely acceptable for relatively coarse resolutions, for high-resolution modeling over complex terrain, the assumption of a flat boundary becomes more difficult to justify.

In the present study, a method is explored for imposing turbulent fluxes across arbitrary topographic surfaces, taking full account of the complex geometry of the terrain. The emphasis is on the fluxes of momentum (i.e., the stresses), since the tensor nature of this flux makes the associated boundary conditions more difficult to impose. A series of experiments is carried out comparing simulations using the full implementation of the surface stress condition to companion runs using the conventional assumption of flat-boundary fluxes. As a test case, we consider flow past Mt.~Ngauruhoe in New Zealand, for a series of eight, essentially randomly-chosen days in 2012. It is shown that for a typical (median) day, the imposition of the full stress condition results in a roughly 10% change in the average disturbance winds over the terrain, as compared to the flat-boundary version of the model, with extreme days producing differences as high as 30%.

Further experiments show that the impact of the new stress implementation is surprisingly sensitive to the model's representation of the terrain, as affected by grid resolution and terrain smoothing. Switching from second-order to fourth-order smoothing of the terrain is shown to have a relatively strong impact, increasing the difference between the full-stress and flat-boundary versions of the model runs by nearly a factor of two in some cases. Experiments with varying grid resolution show that the full stress implementation is likely warranted only for relatively high-resolution modeling, with grid spacings of a few hundred meters or less. At lower resolutions, the complexity of the terrain is sufficiently reduced that the flat-boundary assumption for the fluxes becomes more applicable. Characterizing the results in terms of their dependence on the terrain geometry suggests that the effects of the stress condition vary mainly with the slope of the terrain, as opposed to the curvature.

It is noted that the implementation of the full, tensor stress condition at the boundary involves a global, sparse-matrix inversion, which could potentially cause problems for highly-parallelized models, such as WRF. The potential for a local approximation to the condition is considered, as implied by the dependence on the slope and curvature of the terrain.