Numerical advection schemes, cross‐isentropic random walks, and correlations between chemical species

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Abstract
The advection schemes used in numerical models of chemistry and transport at fixed resolution must unavoidably cause the models to misrepresent the transport in some way. This can include failure to establish or preserve the functional relations between long‐lived chemical tracers that are often observed in the atmosphere. We show that linear functional relations will be preserved exactly by purely linear advection schemes and also, less obviously, by certain “semi‐linear” flux‐limited schemes despite the unavoidable nonlinearity introduced by the flux limiter. In practice, semi‐linear flux‐limited schemes will also preserve nonlinear functional relations better than linear centered difference or spectral schemes that suffer from dispersion errors. The reason is that the dispersion errors lead to spurious oscillations of the mixing ratio field in physical space, artificially expanding the range of mixing ratios in any neighborhood, and hence to a spurious scatter in the relation between any two mixing ratio fields that are nonlinearly related to begin with. Examples of correlations not only preserved, but established, by real and model transport are discussed in this light, including the case of stratospheric transport on timescales of years, for which we discuss and extend earlier results on the ways in which tracer functional relations can arise, for sufficiently long‐lived tracers, purely from transport. The stratospheric results are shown not to depend on the quasi‐horizontal Fickian eddy diffusivity assumption used in the earlier work. The reason is that, whenever the quasi‐horizontal (isentropic) mixing is fast enough—even if it is non‐Fickian as expected in real stratospheric surf zones—the chaotic part of the quasi‐vertical, cross‐isentropic transport has the nature of a random walk with small vertical steps.