Validating weather and climate models at small Rossby numbers: including a boundary layer

Abstract

Ideally, the validation of weather and climate models requires that the predictions remain close to an exact solution of the governing equations. The complexity of weather and climate models means that it is not possible to compute exact solutions except in trivial cases. However, in the limit of small Rossby number, the exact solution of the Euler equations can be shown to be close to that of a semi-geostrophic model, which can be computed. Previous studies have used the small-Rossby-number limit to validate numerical methods for a baroclinic wave without sub-grid physics. However, the method of coupling to the sub-grid physics plays an important role in the performance of weather and climate models. The aim of this article is thus to extend the previous studies to include a boundary-layer parametrization. We use a balanced model that includes a known boundary-layer parametrization, the semi-geotriptic model. We then demonstrate that the semi-geotriptic model is the appropriate small-Rossby-number limit of the solution of the Euler equations with the same boundary-layer representation. The semi-geotriptic model is then used to expose weaknesses in the numerical methods for coupling the boundary layer to the rest of the model.