25 January 2000
Those of you who have followed the development of numerical transport algorithms will know that there are some issues with conservation of mass to be concerned about.
The semi-Lagrangian transport (SLT) method, is essentially a Lagrangian method that tracks a different set of parcels at each time step. Since the set of parcels differ at each time step, and they subsample the continuous field differently at each time step. Therefore, they can sample a different amount of mass at each time step. This leads to an apparent lack of conservation in SLT schemes. In our earlier work we "fixed" this conservation problem by adjusting the mass in the total number of parcels to be constant. I think the latest form of the algorithm is described in the paper...
19. Rasch, P.J., B. A. Boville, and G. P. Brasseur, 1995: A Three-Dimensional General Circulation Model with Coupled Chemistry for the Middle Atmosphere, \jgr, 100, 9041-9071.
The new advection algorithm named SPITFIRE (for SPlit Implementation of Transport using Flux Integral REpresentations) is a flux form algorithm that updates the tracer field in terms of densities within a grid volume. Because SPITFIRE monitors the amount of mass leaving one volume, and entering another, it is inherently a mass conserving scheme.
There is however an "apparent" mass source/sink that appears in MATCH which can cause some problems. The issue is this.....
At each time step, SPITFIRE accepts a tracer mixing ratio, and a statement of the amount of mass in a grid volume (expressed in terms of a pressure change in the volume). SPITFIRE then calculates a mass of tracer expressed as a tracer density (the product of the air mass times the tracer mixing ratio). It transports the tracer mass to give a new tracer density at the end of a time step. At the same time, it calculates a new air density by an identical algorithm. Then it calculates a new tracer mixing ratio by dividing the tracer density by the air density. It returns to the rest of the program an updated tracer density.
The problem comes in when people try to calculate the mass of tracer in the model. They do this by multiplying the tracer mixing ratio times a density, times a volume. This is equivalent to multiplying the mixing ratio by the pressure change across the layer, times the later area of the cell.The pressures involved in this calculation is the pressure interpolated from the input meteorological fields. The change in pressure across the cell over a time step is usually taken to be the pressure change interpolated from the input met fields.
This pressure change within each cell should reflect the mass entering or leaving each cell. But it doesn't. There is an inconsistency associated with the mass of air entering or leaving a cell, as calculated by the SPITFIRE algorithm, and the mass change as calculated from a time interpolation of the met. fields. So if you calculate the mass of tracer using the air mass from the met fields you will get a different answer from if you calculate the mass of tracer from the air mass predicted by SPITFIRE. This leads to an apparent source or sink of tracer mass in the model. But it is NOT due to the numerical algorithm SPITFIRE. It IS due to the met fields.
The inconsistency arises from a number of reasons:
- The model that calculated the pressure change in the met field almost certainly used another numerical algorithm (e.g. not SPITFIRE) to update the pressure change. For example, in the CCM we would calculate the pressure changes using a spectral method, and actually predict the log of surface pressure (which in turn can be transformed into a statement of the pressure in each layer). The CCM also uses a semi-implicit time stepping algorithm, which does not use the same discretization in time.
- The surface pressure may come from an objective analysis, rather than from a model calculation. In this case there is usually no explicit constraint that the mass field and the wind field need be consistent.
The bottom line is that without an adjustment, there is typically no reason to expect that there is a consistency in the mass and wind fields of a meteorological dataset, at least in terms of the control volume used by MATCH. To get such a consistency, one must adjust the mass field, or the wind field, or both, to explicitly enforce such a constraint. It is possible to adjust these fields to enforce the constraint. I have codes to do such an adjustment, but they are still in development.