High-Order Generalized Lorenz N-Cycle Schemes for Semi-Lagrangian Models Employing Second Derivatives in Time

Abstract

Having recently demonstrated that significant enhancement of forecast accuracy in a semi-Lagrangian model results from the application of high-order time integration methods to the second-derivative form of the equations governing the trajectories, the authors here extend the range of available methods by introducing a class of what they call ?generalized Lorenz? (GL) schemes. These explicit GL schemes, like Lorenz?s ?N-cycle? methods, which inspired them, achieve a high formal accuracy in time for linear systems at an economy of storage that is the theoretical optimum. They are shown to possess robustly stable and consistent semi-implicit modifications that allow the deepest (fastest) gravity waves to be treated implicitly, so that integrations can proceed efficiently with time steps considerably longer than would be possible in an Eulerian framework. Tests of the GL methods are conducted using an ensemble of 360 forecast cases over the Australian region at high spatial resolution, verifying at 48 h against a control forecast employing time steps sufficiently short to render time truncation errors negligible. Compared with the performance of the best alternative semi-Lagrangian treatment of equivalent storage economy (a quasi-second-order generalized Adams?Bashforth method), our new GL methods produce significant improvements both in formal accuracy and in actual forecast skill.