Advection on cut-cell grids for an idealized mountain of constant slope

Cut cells use regular or nearly regular polygonal cells to describe fields. For a given orography, some cells may be completely under the mountain, some completely above the mountain, and some are partially filled with air. While there are reports indicating considerably improved simulations with cut cells, inaccuracies may arise with some approximations, producing noise in fields near the surface. This behavior may depend strongly on the approximations made for the advection terms near the surface. This paper investigates the accuracy of advection for numerical schemes for a nondivergent flow near a mountain surface. The schemes use C-grid staggering with densities located at cell centers or on the corners of cells. Also, a nonconserving scheme is considered, which was used in the past with real-data cut-cell simulations. Since the cut cells near the surface create an irregular resolution, the accuracy and order of some approximations may break down near the surface. The objective of this paper is to find schemes having the same accuracy for advection near the surface as in the interior of the domain. As a test problem, uniform advection by a nondivergent velocity field is used with a 45 degrees slope mountain (represented as a straight line) on a rectangular grid. Along the surface a sequence of triangular and pentagonal cells of quite different sizes are generated. Some schemes being discussed for cut cells lead to inaccurate and noisy solutions for this perfectly smooth mountain. A scheme using piecewise linear basis functions in a C grid with density points at the cell corners avoids these inaccuracies.