Numerical errors in flow over of steep topography: analysis and alternatives

High resolution numerical simulations are increasingly used to study topographically forced flows. As the grid resolution is refined and steeper topography is represented, it has become clear that numerical errors associated with the use of terrain-following coordinates can adversely effect the solution accuracy. Inaccuracies from the coordinate transformation are present in each spatially discretized term of the Navier-Stokes equations, as well as in the conservation equations for scalars. Errors in the computation of horizontal pressure gradients, diffusion, and horizontal advection terms have been noted in the presence of sloping coordinate surfaces and steep topography.

In this work we study the effects of these numerical errors on the flow solution for three canonical cases: idealized scalar advection in a stable atmosphere, an atmosphere at rest, and forced advection over a topographic obstacle. This study is completed using the Weather Research and Forecasting (WRF) model. Simulations with terrain-following coordinates are compared to those using a flat vertical coordinate, where terrain is represented with the immersed boundary method which was implemented in WRF in our previous work. The immersed boundary method is used as a tool which allows us to eliminate the terrain-following coordinate transformation, and thus quantify numerical errors through a direct comparison of the two solutions. Additionally, the effects of terrain slope and grid aspect ratio are studied in an effort to gain understanding of where terrain-following coordinates can successfully be used and where the solution would benefit from the use of the immersed boundary method.

Part of this work was performed under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.