Beyond the approaches described so far, there are other possible parameterisation methods that may be advantageous for addressing two deficiencies of current climate and NWP model parameterisations. The first of these deficiencies relates to the synchronisation and ordering of the application of different parameterisation schemes. In current models, with some exceptions [2009jff:cite?], each physics parameterisation is executed in turn at each grid point, with no interaction between the processes represented by the different parameterisations. For instance, cloud microphysics and radiation schemes are completely separate. This raises two questions. First, given two parameterisations A and B, is the result of computing the effect of process A followed by process B the same as that of computing B followed by A? In general, the answer is no. Second, which state should be used for computing the tendencies generated by each parameterisation scheme? Should we use model state x to calculate process tendencies for A and B as (dx/dt)A and (dx/dt)B and then use these tendencies to update the model state according to the combined tendency (dx/dt)A + (dx/dt)B, or should we calculate tendencies for one process, A say, and update the model state using the (dx/dt)A tendencies to give a new model state x', from which we then calculate tendencies due to B, (dx'/dt)B which we use to calculate a final model state x''. It is not clear which of these approaches is correct, and the interaction between the different schemes may interact with the model dynamics in undesireable ways.
The second deficiency of existing physics parameterisation approaches is that physics tendencies are calculated pointwise on the model grid. The problem with this is that we expect processes at the model grid scale to be among those represented worst in our models, due to the presence of numerical diffusion and hyperdiffusion at the grid scale. It would appear, intuitively, that we might be better served by performing physics computations on a coarser grid than that used for the model dynamics. In fact, some experiments [2009jff:cite] indicate that better results are achieved by performing physics computations on a finer grid than the dynamical grid. This somewhat counter-intuitive result appears to arise because of the sub-sampling of the model state provided by the interpolation needed to extract model state information for driving the fine grid physics. It is clear that existing models have some issues here, and alternative approaches may help either to improve the situation or at least to elucidate the problems that exist.
One very new idea for addressing both of these deficiencies is to implement non-local parameterisations. Instead of parameterising physical processes based on the model state at a single grid point, we would aim to supplement the dynamical equations of the model with further (partial differential) evolution equations representing the processes being parameterised. These additional equations are integrated in parallel with the model dynamical equations. The result is that questions of ordering of parameterisations do not arise (assuming correct numerical implementation of the integration of the parameterisation equations), that the parameterisations do not need to rely on the smallest grid scale information, and we may be able to develop parameterisations consistent with the model dynamics. One example where this approach may be particularly successful is in the parameterisation of tropical convection, where convective processes are closely linked to the initiation of convectively coupled waves, for which it is relatively straightforward to derive propagation equations that may be integrated in parallel with the model dynamics [2009jff:cite?].