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S~I M MA R Y A three-time-level semi-lagrangian global spectral model was introduced operationally at the European Centre for Medium-Range Weather Forecasts in I99 I . This paper first documents some later refinements to the three-time-level scheme, and then describes its conversion to a two-time-level scheme. Experimental results are presented to show that the two-time-level scheme maintains the accuracy of its three-time-level predecessor, while being considerably more computationally efficient. In principle, a two-time-level semi-Lagrangian scheme provides a further doubling of efficiency, through a procedure which is usually referred to as 'doubling the time step'. This is slightly misleading, as in a three-time-level leapfrog scheme the length of each time step in the usual notation is 2 A t , but successive time steps overlap by A?. More precisely, the two-time-level scheme doubles the efficiency by eliminating this overlapping, so that only half the number of time steps is needed to complete the forecast. Viewed in this light, it is clear that the time truncation error can be the same for a three-time-level scheme and for the corresponding two-time-level scheme with a 'doubled' time step. For a two-time-level scheme to be accurate as well as efficient, it is important that second-order accuracy in time be maintained in the trajectory calculations. A simple way to achieve this was independently suggested by McDonald and Bates (1 987) and Temperton and Staniforth (1987), and formed the basis for later developments. Haugen (1992, 1993) described a two-time-level semi-Lagrangian limitedarea model; subsequently, Gustafsson and McDonald (1996) presented a comparison between spectral and finite-difference versions of this model. Recent applications of two-time-level semi-Lagrangian schemes to global finite-difference medium-range forecast models have been described by Chen and Bates ( 1996) and Moorthi ( 1997). CBtC et al. ( 1998a, 1998b) describe a multi-purpose variable-resolution global finite-element model based on a two-time-level semi-Lagrangian scheme, while Qian et al. (1998) incorporate a two-time-level scheme in a global non-hydrostatic model.In this paper, a two-time-level reformulation of the semi-Lagrangian global spectral model documented in R95 is presented. This version of the two-time-level scheme was used in the operational ECMWF forecast model from December 1996 until April 1998. First, section 2 describes some modifications to the three-time-level scheme which were implemented during its operational lifetime. The conversion to a two-time-level scheme is then described in section 3. Experimental results are presented in section 4, followed

The semi-implicit semi-Lagrangian integration technique enables numerical weather prediction models to be run with much longer timesteps than permitted by a semi-implicit Eulerian scheme. The choice of timestep can then be made on the basis of accuracy rather than stability requirements. To realize the full potential of the technique, it is important to maintain second-order accuracy in time; this has previously been achieved by applying it in the context of a three-time-level integration scheme. In this paper we present a two-time-level version of the technique which yields the same level of accuracy for half the computational effort. Unlike other efficient two-time-level schemes, ours does not rely on operator splitting.We apply this scheme to a variable-resolution barotropic finite-element regional model with a minimum gridlength of 100 km, using timesteps of up to three hours. The results are verified against a control run with uniformly high resolution, and are shown to be of similar accuracy to those of a semi-implicit Eulerian integration' with a timestep of 10 minutes.

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