HPG error decreases significantly with the order of interpolation scheme: linear, quadratic and cubic spline interpolation schemes in decreasing order. For grid points with one end of the Cartesian gradient lying on or below the ground surface, HPG error due to extrapolation may vary significantly with steepness. When compared to the HPG error of sz-gradient scheme (e.g., RAMS scheme), error can be reduced further by extrapolation for slopes with height and half width ratio (h/a) smaller than 0.15. But for steep slopes (e.g., h/a>0.4) and vertically stretched grid, large error can be resulted by the extrapolation and the general sz-gradient scheme can be more stable.
Error is maintained at small value throughout the vertical layer during the 10-hour integration when cubic spline interpolation/extrapolation or cubic spline interpolation/sz-gradient schemes is used for slopes with h/a<0.3. But larger errors may occur throughout the lowest 4-km layer during the integration when linear or quadratic interpolation/extrapolation scheme is used. For slopes with h/a>0.3 and vertically stretched grid, cubic spline interpolation/sz-gradient scheme appears to be more stable. These schemes are further compared through their application to the simulation of mountain waves.
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