A new locally mass conserving, monotonic and accurate method for solving the continuityequations in Earth System Models

To correctly simulate the evolution and interactions of chemical constituents in Earth System Models it is important that the algorithms of the underlying dynamical core fulfil a number of properties such as high order of accuracy, inherent mass conservation, locality, positive definiteness, monotonicity, consistency, and high order of numerical stability and efficiency. Furthermore the so-called masswind inconsistency problem should be minimised in a physically consistent way. Aiming at fulfilling all these requirements simultaneously a new scheme has been introduced. It is based on two new ingredients:

The basic mass conserving SL method is relatively simple to construct from existing SL general circulation models. The trick is to introduce weights at the upstream departure points ensuring that the total mass given off by a given Eulerian grid point to the surrounding SL departure points is equal to the volume represented by that grid point. Based on the Eulerian grid point weights it is possible to obtain departure point weights representing the divergence associated with the SL trajectories as in the socalled cell-integrated semi-Lagrangian (CISL) models. The filter is simple but it efficiently ensures monotonicity and positive definiteness and surprisingly it enhances the accuracy. Furthermore it can be constructed to generally maintain constancy in non-divergent flows. So far the scheme has only been tested in traditional two dimensional advection tests and in a semi-Lagrangian, semi-implicit shallow water
model in plain geometry but with topography. Without the filter the scheme is at least as accurate as the traditional SL schemes. The filter enhances the accuracy for all tests carried out. However, it becomes particularly impressive near sharp gradients and discontinuities. The increase in numerical cost of the new scheme relative to traditional SI-SL models is small, particularly when there are several passive tracers