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The Advection–Diffusion Problem for Stratospheric Flow. Part I: Concentration Probability Distribution Function

Y. HuDepartment of the Geophysical Sciences, The University of Chicago, Chicago, Illinois

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R. T. PierrehumbertDepartment of the Geophysical Sciences, The University of Chicago, Chicago, Illinois

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Abstract

The mixing of a passive tracer by realistic time-dependent stratospheric flow (European Centre for Medium-Range Weather Forecasts winds) on an isentropic surface (420 K) is studied. Simulations of the advection–diffusion problem for an initially large-scale tracer field are carried out in the limit of weak diffusivity. Owing to chaotic advection, tracer variance is cascaded to small scales, where it can be dissipated despite the weak diffusivity. The tracer fluctuations are characterized in terms of their probability distribution function (PDF). The PDFs are characterized by a Gaussian core and “fat tails,” which fall more slowly than a Gaussian, and indicate anomalously high probability of extreme concentration fluctuations. Given the nonlinearity of many chemical reactions of interest, the anomalous prevalence of extreme fluctuations could have a profound effect on reactive tracers.

Zonal variations of tracer are homogenized globally leading to a unimodal PDF. Initially meridional variations are strongly influenced by the presence of mixing barriers. Meridional gradients homogenize only within mixing zones of limited latitudinal extent, bounded by permeable mixing barriers. The PDF becomes multimodal, with distinct populations of air caused by blending of the concentration values within each mixing zone. The Tropics is a zone of weak mixing, and serves as an important repository of stratospheric tracers, which are episodically ejected into surf zones in the form of filaments bearing extreme concentration values.

The shapes of the PDFs are discussed in terms of theoretical methods developed in the context of highly idealized mixing models. It is shown that such methods retain their utility when applied to realistic stratospheric mixing. The role of the probability distribution of finite-time Lyapunov exponents for the underlying trajectory problem is highlighted. The use of conditional averages of diffusion and dissipation is also illustrated. PDFs yielded by the idealized advection–diffusion problem are found to resemble those appearing in a GCM simulation of N2O.

The theoretical arguments and numerical results imply that under the assumption that the diffusivity is set so that the dissipation scale is comparable to model resolution, the concentration PDF eventually reaches a universal shape independent of model resolution after an initial transient stage. However, the width of the distribution, or equivalently the variance of the tracer fluctuation, increases algebraically as model resolution is refined.

Corresponding author address: Dr. Yongyun Hu, Dept. of Applied Mathematics, University of Washington, P.O. Box 352420, Seattle, WA 98195-2420. Email: yongyun@amath.washington.edu

Abstract

The mixing of a passive tracer by realistic time-dependent stratospheric flow (European Centre for Medium-Range Weather Forecasts winds) on an isentropic surface (420 K) is studied. Simulations of the advection–diffusion problem for an initially large-scale tracer field are carried out in the limit of weak diffusivity. Owing to chaotic advection, tracer variance is cascaded to small scales, where it can be dissipated despite the weak diffusivity. The tracer fluctuations are characterized in terms of their probability distribution function (PDF). The PDFs are characterized by a Gaussian core and “fat tails,” which fall more slowly than a Gaussian, and indicate anomalously high probability of extreme concentration fluctuations. Given the nonlinearity of many chemical reactions of interest, the anomalous prevalence of extreme fluctuations could have a profound effect on reactive tracers.

Zonal variations of tracer are homogenized globally leading to a unimodal PDF. Initially meridional variations are strongly influenced by the presence of mixing barriers. Meridional gradients homogenize only within mixing zones of limited latitudinal extent, bounded by permeable mixing barriers. The PDF becomes multimodal, with distinct populations of air caused by blending of the concentration values within each mixing zone. The Tropics is a zone of weak mixing, and serves as an important repository of stratospheric tracers, which are episodically ejected into surf zones in the form of filaments bearing extreme concentration values.

The shapes of the PDFs are discussed in terms of theoretical methods developed in the context of highly idealized mixing models. It is shown that such methods retain their utility when applied to realistic stratospheric mixing. The role of the probability distribution of finite-time Lyapunov exponents for the underlying trajectory problem is highlighted. The use of conditional averages of diffusion and dissipation is also illustrated. PDFs yielded by the idealized advection–diffusion problem are found to resemble those appearing in a GCM simulation of N2O.

The theoretical arguments and numerical results imply that under the assumption that the diffusivity is set so that the dissipation scale is comparable to model resolution, the concentration PDF eventually reaches a universal shape independent of model resolution after an initial transient stage. However, the width of the distribution, or equivalently the variance of the tracer fluctuation, increases algebraically as model resolution is refined.

Corresponding author address: Dr. Yongyun Hu, Dept. of Applied Mathematics, University of Washington, P.O. Box 352420, Seattle, WA 98195-2420. Email: yongyun@amath.washington.edu

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