HIGH-ORDER GENERALIZED LORENZ N-CYCLE SCHEMES FOR SEMI-LAGRANGIAN MODELS EMPLOYING 2ND DERIVATIVES IN TIME

Having recently demonstrated that significant enhancement of forecast accuracy in a semi-Lagrangian model results from the application of hi gh-order time integration methods to the second-derivative form of the equations governing the trajectories, the authors here extend the ran ge of available methods by introducing a class of what they call ''gen eralized Lorenz'' (GL) schemes. These explicit GL schemes, like Lorenz 's ''N-cycle'' methods, which inspired them, achieve a high formal acc uracy in time for linear systems at an economy of storage that is the theoretical optimum. They are shown to possess robustly stable and con sistent semi-implicit modifications that allow the deepest (fastest) g ravity waves to be created implicitly, so that integrations can procee d efficiently with time steps considerably longer than would be possib le in an Eulerian framework. Tests of the GL methods are conducted usi ng an ensemble of 360 forecast cases over the Australian region at hig h spatial resolution, verifying at 48 h against a control forecast emp loying 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-or der generalized Adams-Bashforth method), our new GL methods produce si gnificant improvements both in formal accuracy and in actual forecast skill.