Generalized Adams–Bashforth time integration schemes for a semi‐Lagrangian model employing the second‐derivative form of the horizontal momentum equations

We present a generic class of semi‐implicit time‐integration methods, the ‘Generalized Adams–Bashforth’ schemes, for the simultaneous treatment in a semi‐Lagrangian model of the equations of horizontal momentum and kinematics in a rotating environment. The salient feature of the approach is that it deals directly with Lagrangian parcel momentum in terms of the parcel's second time‐derivative of position. The classical Adams–Bashforth methods can be generalized to accommodate equations of second‐derivative form and, as we demonstrate, can be formulated in such a way that the further important refinement of a semi‐implicit handling of the fastest gravity modes follows in a natural way. The principal advantages expected of this unified approach over the more conventional separate semi‐Lagrangian treatment of kinematics and momentum are: (i) greater economy of storage at a given order of accuracy, (ii) smaller truncation errors at a given order of accuracy. Tests were run with a full‐physics three‐dimensional regional semi‐Lagrangian forecast model applied on a daily basis to archived operational data over a period of three months. Verifications based on the 48 hour forecasts confirm that the expected benefits of the new schemes are also realized in practice.