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3.2 Finding the departure point

Extending the procedure of Robert (1981) to three dimensions, the midpoint and the departure point of the trajectory for each arrival point are found by iteratively solving the equation

(3.5)

where in (3.5) is the three-dimensional wind field . Since was never explicitly required in the Eulerian version of the model (see Eqs. (2.18)-(2.19) for the Eulerian discretization of vertical advection), it is necessary to construct this field for the trajectory calculations. As is already specified at the upper and lower boundaries ( , at and at ) it would be natural to construct at the half-levels (i.e. vertically staggered with respect to and ), and indeed a preliminary version of the model was coded that way. However, it is more convenient to hold the three velocity components at the same set of points (which also coincide with the arrival points), so the formulation was changed to use at the `full' levels. Thus, the vertical velocity used in (3.5) is defined by

(3.6)

where is already defined by (2.18) and

(3.7)

In deriving (3.7) we have used (2.11) together with a formal definition of itself (which again was not required by the discretized Eulerian dynamics):

(3.8)

where is a constant pressure (chosen to be 1013.25 ).

The iterative procedure for solving (3.5) is analogous to that used by Ritchie (1991) in a -coordinate model. Given an estimate after iterations, the next iteration is given by

(3.9)

where the vertical component of the displacement is found first. The vertical component of on the right-hand side of (3.9) is then updated before the horizontal components are found taking into account the spherical geometry following Ritchie (1987, 1988). The first guess is given by

(3.10)

The calculations include approximations to the spherical geometry away from the poles, following Ritchie and Beaudoin (1994). In agreement with previous work (reviewed by Staniforth and Côté‚ 1991), little sensitivity was found to the order of interpolation used in the trajectory calculations, and linear interpolation appears to be sufficiently accurate. After providing a first guess via (3.10), a single further iteration was found to be adequate.

Once the midpoint of the trajectory has been found, the departure point is immediately obtained (in the horizontal, the backward extension of the trajectory is along a great circle). In the vertical, if the departure point is then above the first (or below the last) mode level, it is modified to lie on the first (last) level.

In solving (3.9), it is necessary to convert between a displacement in terms of the spatial coordinates and the corresponding displacement in terms of `grid lengths', in order to select the correct three-dimensional block of points for the interpolation routine. This is simple in the horizontal, since the mesh length is constant in the -direction (at a given latitude), and almost constant in the -direction. It is more difficult in the vertical, where the grid spacing changes rapidly, and the conversion algorithm for the vertical displacement makes use of an auxiliary grid defined with high uniform resolution.

At high horizontal resolutions a positive feedback mechanism, between the computation of the departure point of the trajectories and the solution of the momentum equations, can lead to instability, which results in noisy forecast fields in the winter stratosphere. In order to break the positive feedback loop, a smoothing interpolation is applied to the vertical velocity in the computation of the trajectory. This smoothing interpolation uses the same set of points around the departure point as the cubic interpolation, but the horizontal interpolations are substituted by least squares linear fits to the corresponding four points. The procedure is applied to both the arrival and the departure points of the trajectory. As the procedure is not an interpolatory procedure, the value at the arrival point is substituted by a smoothed value, as is also the case for the value at the departure point.