Evaluating advection/transport schemes using interrelated tracers, scatter plots and numerical mixing diagnostics

Atmospheric tracers are often observed to be functionally related, and these relations can be physically or chemically significant. It is therefore highly desirable that the transport schemes used in chemistry and chemistry-climate models should not disrupt such functional relations in unphysical ways through numerical mixing or, indeed, unmixing. Here, diagnostics are proposed that quantify numerical mixing by a transport scheme for a single tracer, two tracers that are nonlinearly related, and three (or more) tracers that add up to a constant. For the two-tracer test, the question of how physically reasonable the numerical mixing is can be addressed by using scatter/correlation plots. Truncation errors will, in general, result in scatter points deviating from the preexisting functional curve and thereby introduce numerical mixing between the tracers. The proposed diagnostics quantify the mixing in terms of the normalized distances between the pre-existing functional curve and scatter points, and divide it into three categories: real mixing and two types of spurious numerical unmixing. For the three-tracer test, we quantify, in terms of standard error norms, how nearly a transport scheme can preserve the sum by transporting the individual tracers. The mixing diagnostics do not require the knowledge of the analytical solution to the transport problem for the individual tracers. However, using an idealized flow field and spatial distributions facilitates the use of the mixing diagnostics in transport scheme development. Hence we propose to exercise the new mixing diagnostics using an idealized but highly deformational analytical flow field. Example results using the CSLAM (Conservative Semi-LAgrangian Multi-tracer) scheme are presented.