A new and very accurate cell-integrated semi Lagrangian (CISL) two time level scheme has been formulated and tested for the shallow water equations in a plane channel model with realistic variation of the Coriolis-parameter and including topography. The implementation includes semi-implicit time stepping of the gravity wave terms. Furthermore, two-dimensional advection of passive tracers has been tested in idealized flows as well as in a fully general flow.

Regarding basic features, the new formulation is based on step-functions, which implicitly are advected with the flow across cell boundaries. The scheme is exact in case of passive advection by a flow that is constant in time and space. This means that the accuracy is of indefinite order. In addition to this attractive feature the scheme shows very small numerical dispersion in fully non-linear and divergent flows and in such flows the scheme is locally (and globally) mass conserving, positive definite and monotonic. The high order of accuracy is achieved by introducing two additional variables. For a given prognostic variable the memory carrying variables are: the grid-cell integrals, the values at the grid cell corner points and the functional average on the intersections between grid cell corner points in either of the two spatial directions. The penalty of running the new scheme is an increase in memory consumption by a factor of three.

In the present preliminary formulation of the scheme only the mass field is treated with the CISL scheme, while a traditional semi-Lagrangian scheme is used for the momentum equations. It is, however, possible to use the CISL scheme also for the cell-integrated form of the momentum equations. In this case also momentum is conserved.

Results of test integrations will be presented and compared to integrations based on finite difference and spectral formulations of traditional semi-implicit Eulerian and semi Lagrangian shallow water models. The very convincing tests of passive advection by idealized flows and by a fully divergent flow simulated by the shallow water model are also shown.

The presentation is completed by a discussion of the generalization of the scheme to fully three-dimensional flows on the sphere. Briefly, the semi-implicit implementation follows Machenhauer and Olk (1997) and the application on the sphere will follow the formulation presented in Nair and Machenhauer (2002) with some modifications. Following Machenhauer and Olk (1998) the generalization to three dimensions is based on two-dimensional CISL advection on model levels in combination with a diagnosing of the vertical advection under the assumption of hydrostatic balance. This formulation ensures a more consistent treatment of the vertical advection problem than is the case in the traditional hydrostatic semi-Lagrangian models based on three dimensional trajectories.