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3.3 `Non-interpolating' scheme in the vertical

An alternative formulation of the semi-Lagrangian scheme in three dimensions was suggested by Ritchie (1991). Equation (3.1) can be rewritten as

(3.11)

where

and is the horizontal part of the advection operator defined in (2.30). In (3.11), is defined to be a vertical velocity which would lead to the departure point of the trajectory at time lying exactly on a model level. This model level is chosen to be the one closest to the true departure point. Equation (3.11) is then approximated by

(3.12)

where the superscripts , , respectively denote evaluation at the arrival point , the midpoint and the departure point of the modified trajectory. Since the modified departure point lies by definition on a model level, no vertical interpolation is required to evaluate . As discussed in Subsection 3.1 above, it is also possible to evaluate the terms on the right-hand side of (3.12) by averaging the values at and ; in this case no vertical interpolation at all is required. Notice that a separate interpolation is required to evaluate the second term on the right-hand side of (3.12) since the quantity , defined by

(3.13)

where and are respectively the arrival and departure levels of the modified trajectory, is meaningful only at each grid point.

If the vertical velocity (or the time-step) is sufficiently small, then the modified departure point lies on the same model level as the arrival point, is zero and the treatment of vertical advection becomes purely Eulerian. In general there is an Eulerian treatment of the advection by the `residual vertical velocity' , which is small enough to guarantee that the Eulerian CFL criterion for vertical advection is respected. Thus, the `non-interpolating' scheme maintains the desirable stability properties of the `fully interpolating' scheme.

There is a subtle, but important, difference in the way the iterative scheme (3.9) is implemented to determine the modified trajectory in the non-interpolating scheme. As before, the first step at each iteration is to update the estimate of the vertical component of the displacement.The implied updated departure point is then moved to the closest model level. In the second step, the horizontal components are then updated using the winds evaluated at the midpoint of the modified trajectory. Notice that this gives a result different from that obtained by simply carrying out the trajectory calculation of the fully interpolating scheme and then projecting the departure point to the nearest model level. The modified procedure described above is easily seen to be more consistent by considering the case in which the vertical velocity is not zero, but is small enough for the modified trajectory to be horizontal ). The discretization is then equivalent to a purely two-dimensional semi-Lagrangian scheme, the trajectory being computed using the horizontal wind field evaluated on a single model level.

An incidental advantage of the `non-interpolating' scheme over the `fully interpolating' scheme is that it resolves any ambiguities about the treatment of departure points above the top model level or below the bottom model level; the modified departure points automatically lie on the top or bottom level. The treatment of vertical advection becomes Eulerian, which is well-defined at the top and bottom levels. Thus, the non-interpolating scheme removes the need for artificial `nudging' of the departure points or the extrapolation of quantities to points above or below the domain of the model levels.

Smolarkiewicz and Rasch (1991) have extended the principle of the `non-interpolating' semi-Lagrangian formulation to generate a broader class of stable and accurate advection schemes.