Tests of Different Momentum Flux Boundary Implementations for High-Resolution Terrain Modeling

In numerical models, the drag exerted on the atmosphere by the Earth's surface is generally represented in terms of a turbulent momentum flux (or stress) across the lower boundary of the model domain. Due to the tensor nature of the flux, the implementation of this boundary stress can be somewhat involved for complex terrain geometries, generally requiring a sparse matrix inversion at each time step. However, in current-generation models, these complications have been avoided by adopting approximate forms of the stress condition, which are easier to implement numerically. In current mesoscale models, the flux is usually applied as if the lower boundary were simply flat (i.e., the flat-boundary approximation), while for higher-resolution CFD models, the condition is typically approximated by assuming small terrain curvature, resulting in a normal-gradient condition. The present study attempts to characterize the accuracy of the flat-boundary and normal-gradient approximations for high-resolution terrain modeling through comparison to companion simulations using an implementation of the full tensor-stress condition. As a test case, we consider flow past a stratovolcano in New Zealand (Mt. Nguaruhoe) for a series of randomly chosen days in 2012, as simulated at 90 m grid resolution. It is found that as an average over all cases considered, the flat boundary approximation produces errors in the disturbance wind speed on the order of 10-15%, with an extreme case producing errors of nearly 30%. The normal gradient condition is shown to be considerably more accurate, with wind speed errors reduced by roughly a factor of 2.5 relative to results for the flat-boundary approximation. Further experiments show that the performance of the boundary conditions is strongly sensitive to the representation of the terrain on the grid. In particular, as the grid resolution is decreased, the approximate boundary conditions become more accurate. At a grid spacing of 240 m, the mean error for the flat boundary condition is reduced to only a few percent, while the error for the normal-gradient condition is essentially negigible. The boundary conditions also show a surprising sensitivity to the degree of smoothing of the terrain. Simply adopting a fouth-order terrain smoother in place of a second-order smoother results in nearly a doubling of the wind speed errors, relative to the results cited above.