The development and testing of a new two-time-level semi-Lagrangian scheme (SETTLS) in the ECMWF forecast modeld

A new treatment of the two-time-level semi-Lagrangian scheme is presented which avoids extrapolation in time of the velocities used for the computation of the trajectories and for the nonlinear terms of the evolution equations. Extrapolation in time is used in the original two-time-level scheme in order to make it centred in time and therefore second-order accurate. This time-extrapolation can lead to instabilities. In the new scheme the second-order accuracy is achieved by means of a Taylor series expansion to second order around the departure point of the semi-Lagrangian trajectory thus avoiding extrapolation of the first-order term. The estimate of the second-order term in this expansion is obtained by means of a stable time extrapolation from the previous time step and therefore the scheme is named the 'Stable Extrapolation Two-Time-Level Scheme' or SETTLS. Some cases of noise in forecast fields. arising from an instability of the time extrapolation in the version of the two-time-level semi-Lagrangian scheme used operationally at the European Centre for Medium-Range weather Forecasts (ECMWF) between December 1996 and April 1998, are diagnosed it is shown that the noise can be removed by the use of the SETTLS.