was defined in (2.27), L is the linearized part of R and N is the remainder or "non-linear terms". An explicit three-time-level semi-Lagrangian treatment of (3.1) is obtained by finding the approximate trajectory, over the time interval

, of a particle which arrives at each grid point

at time

. Equation (3.1) is then approximated by

(3.2)

where the superscripts

and

, respectively denote evaluation at the arrival point

, the mid-point of the trajectory

, and the departure point

. Since the mid-point and the departure point will not in general coincide with model grid points,

and

must be determined by interpolation.

It is more economical (and, as discussed later, gives better results in some circumstances; see also Tanguay et al., 1992) to evaluate the right-hand side of (3.2) as

(3.3)

since only a single interpolation (of the combined field

at the point

) is then required in order to determine

.

The right-hand sides of the time-discretized model equations also contain semi-implicit correction terms, which in the Eulerian model took the form

where the superscripts refer to time-levels, and to a single common grid point. In the semi-Lagrangian version of the model, the semi-implicit correction terms take the form

(3.4)

and again the terms to be evaluated at the departure point

can be added to other right-hand side terms before interpolation. Notice that the evaluation of

, and both ways of evaluating

, are all centred in space and time.

To obtain accurate results from a semi-Lagrangian integration scheme, it is necessary to choose the order of interpolation carefully (see for example Staniforth and Côté 1991). In practice it has been found (for the model described here) that linear interpolation is adequate for the terms evaluated at the midpoint of the trajectory, but that cubic interpolation is essential for the terms evaluated at the departure point. Cubic interpolation in three dimensions is expensive, and fortunately a `quasi-cubic' interpolation (suggested by Courtier) was found to give essentially equivalent results. The technique can be illustrated by two-dimensional interpolation on a regular grid. The target point is at

. In the first step, four interpolations are performed in the

-direction: linear (rather than the usual cubic) interpolations to the points

and

, and cubic interpolations to the points

and

. In the second step, one cubic interpolation is performed in the

-direction, to evaluate the field at the target point. The number of `neighbours' contributing to the result is reduced from 16 to 12. The generalization to three dimensions is straightforward and results in a significant saving, the number of neighbours being reduced from 64 to 32, and the computation being reduced from 21 one-dimensional cubic interpolations to 7 cubic plus 10 linear one-dimensional interpolations.

For the reduced Gaussian grid described in Subsection 2.2.3, the mesh is no longer regular. However, it is easily seen that the extra complication is relatively minor provided that the first step in the interpolation is performed in the

-direction.

The order of the interpolation in the vertical is reduced to linear when the evaluation point lies between the two highest model levels, or between the lowest two model levels. Extrapolation beyond the top or bottom levels is not allowed.

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08.04.2002

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3.1 General description