The Advection–Diffusion Problem for Stratospheric Flow. Part II: Probability Distribution Function of Tracer Gradients

Abstract

This paper is a continuation of the study of the advection?diffusion problem for stratospheric flow, and deals with the probability distribution function (PDF) of gradients of a freely decaying passive tracer. Theoretical arguments are reviewed and extended showing that mixing of a weakly diffused tracer by random large-scale flows produces a tracer gradient field whose probability distribution function has ?stretched exponential? tails P(|??|) ? exp(?b|??|?) with ? < 1. This contrasts with the lognormal distribution expected for advective mixing in the absence of diffusion. The non-Gaussian distribution of tracer gradients can be derived in terms of the statistics of strain rates of the random driving flow. It is shown that the tails of the gradient PDF provide information about the dissipation scale, the scale selectivity of the dissipation law, and the fluctuations of short-term strain. The gradient PDF is shown to contain information about tracer variability that is not present at all in the power spectrum of the tracer field. To show that the predictions remain valid for the gradient statistics of passive tracers driven by the well-organized lower-stratospheric flow with mixing barriers, a series of advection?diffusion simulations of a decaying passive tracer are presented. The mixing is driven by ECMWF winds on the 420-K isentropic surface using the high-resolution finite-volume model employed in Part I of this paper. It is found that the probability distribution function of the simulated tracer gradients is indeed stretched exponential, with the stretching parameter ? ≈ 0.55. The largest gradients are not found in the regions of highest Lyapunov exponents, but rather in the surf-zone regions adjacent to the reservoirs of high tracer fluctuation amplitude.