A constrained spectral approximation of subgrid-scale orography on unstructured grids

The representation of subgrid-scale orography is a challenge in the physical parameterisation of orographic gravity-wave sources in weather forecasting. A significant hurdle is encoding as much physical information with as simple a spectral representation as possible on unstructured geodesic grids with non-quadrilateral grid cells, such as the one used in the German Weather Service's Icosahedral Nonhydrostatic Model. Other issues include scale awareness, i.e., the orographic representation has to change according to the grid cell size. This work introduces a novel spectral analysis method approximating a scale-aware spectrum of subgrid-scale orography on unstructured geodesic grids. The dimension of the physical orographic data is reduced by more than two orders of magnitude in its spectral representation. Simultaneously, the power of the approximated spectrum is close to the physical value. The method is based on well-known least-squares spectral analyses. However, it is robust to the choice of the free parameters, and tuning the algorithm is generally unnecessary. Numerical experiments involving an idealised setup show that this novel spectral analysis performs significantly better than a straightforward least-squares spectral analysis in representing the physical energy of a spectrum. Studies involving real-world topographic data are conducted, and competitive error scores within 10% error relative to the maximum physical quantity of interest were achieved across different grid sizes and background wind speeds. The deterministic behaviour of the method is investigated along with its principal capabilities and potential biases, and it is shown that the error scores can be iteratively improved if an optimisation target is known.