A fast converging and concise algorithm for computing the departure points in semi‐Lagrangian weather and climate models

Accurate calculation of departure points is crucial for the overall accuracy of a semi‐Lagrangian advection scheme. The iterative method used in the semi‐Lagrangian advection scheme of the ECMWF model IFS (Integrated Forecast System) is based on SETTLS (Stable Extrapolation Two‐Time‐Level Scheme). The scheme converges slowly due to the long timesteps that the spectral semi‐implicit, semi‐Lagrangian formulation permits: five iterations are needed for accurate determination of the departure points. Spherical coordinates increase the expense and complexity of the relevant code while computations become sensitive to arithmetic precision and require further approximations in the polar high‐curvature region. In this article, an accurate alternative, derived by reformulating the current SETTLS scheme on a geocentric Cartesian framework, is proposed. The new scheme simplifies considerably the computations by avoiding the update of the Lagrangian trajectory equations in spherical coordinates, which involve additional expensive metric terms, eliminates approximations, and is advantageous for single precision. Methods to accelerate the convergence of the departure points were also investigated. The most accurate and efficient approach was to start the SETTLS iterations from the departure point of the previous timestep. Testing the proposed method in high‐resolution global forecast and 4D‐Var analysis experiments yields equally accurate results as the existing method, while reducing the cost of the iterative scheme. A new formulation for departure point calculation schemes in semi‐Lagrangian NWP and climate models is proposed which simplifies computations, avoids approximations and problems near the poles and is advantageous for single‐precision arithmetic. Furthermore, a simple method for accelerating the convergence of iterative departure point schemes is proposed which proves to reduce the cost of the IFS iterative scheme SETTLS by 40% while maintaining the same forecast accuracy. The plot illustrates the change in the geopotential height Anomaly Correlation Coefficient with respect to the operational 9 km forecast when the number of departure point iterations in the IFS model is reduced. The top plots refer to the existing scheme and demonstrates reduction of skill, while the bottom ones refer to the new scheme which achieves the same accuracy with a reduced number of iterations.