High-Order Numerics in an Unstaggered Three-Dimensional Time-Split Semi-Lagrangian Forecast Model

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Abstract
Traditional finite-difference numerical forecast models usually employ relatively low-order approximations on grids staggered in both the horizontal and the vertical. In a previous study, Purser and Leslie (1988) demonstrated that high-order differencing on an unstaggered horizontal grid led to improved forecast accuracy. The present investigation has two aims. The first aim simply is to extend the earlier work to a three-dimensional formulation, by using high-order horizontal numerics in a time-split, three-dimensional, semi-Lagrangian model on a grid that is unstaggered in both the horizontal and vertical. The choice of an unstaggered grid is very effective in a semi-Lagrangian model as it ensures that a single set of interpolations suffices for all variables at each advection step. The second aim specifically is to increase the accuracy of the vertical discretization of key quantities such as the vertically integrated divergence, and the computation of the geopotential from the hydrostatic equation. Errors introduced in these terms potentially can have a large impact on the forecast accuracy. The increased accuracy also serves to mitigate any possible deterioration that might result from the adoption of a vertically unstaggered grid. It is shown over a four-month period of daily 24-h forecasts that the use of vertical quadrature techniques, on the aforementioned terms, based on layer integrals of high-order interpolating Lagrange polynomials, leads to a significant reduction of about 5% in the root-mean-square errors of the geopotential and wind fields. A much greater improvement in model performance is found in the forecasts of vertical velocity and precipitation fields, as they are more sensitive to the new vertical discretization. Moreover, these gains are obtained at minimal computational cost both in time and storage.

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