On a fundamental problem in implementing flux-form advection schemes for tracer transport in 3-dimensional general circulation and chemistry transport models

A fundamental difficulty in 3-D models is addressed which can arise due to inconsistencies between advection schemes and winds. It is shown that a model will fail to meet certain basic criteria, desired of 3-D transport models, when the density field computed by the advection scheme from the winds differs from the implied density field based on the surface pressure and the sigma (or hybrid) coordinates. To allow a rigorous mathematical formulation, the focus is on the example of a mass flux advection scheme in a model where the winds and surface pressure are derived from different advection schemes (e.g. a spectral scheme in a climate model or a weather centre model): however, in principal the discussion applies to nearly any situation in which the pressure levels change in a model. To illustrate the potential severity of such problems, a mass conserving grid-to-grid transformation scheme is constructed which only uses the current tracer mass mixing-ratio distribution. It is shown that only one solution exists that is comprehensively valid for any arbitrary tracer distribution, and that this type of correction introduces an additional undesired artificial vertical diffusion component into the model transport that increases with increasing tracer mass mixing-ratio gradients and may exceed the physical vertical transport itself. It is demonstrated that the results of any supplementary fix, either mass fixer of grid-to-grid transformation, are generally unacceptable for global modelling applications. From this, it is concluded that the only alternative which can produce reliable results for any arbitrary tracer is to maintain a consistent grid throughout the entire model time step, where all changes in pressure levels due to modelled advection exactly match the changes implied by the surface pressure at the next time step. Although this is already done in some models, this would require significant changes in the structure of the advection scheme or its input wind fields in several other comtemporary general circulation and chemistry transport models.