FJ~L~/IER Physica A 263 (1999) 95-103 PHYSICA Turbulent Mixing of a Passive Scalar Alain Pumir Institut Non Lindaire de Nice, 1361 route des Lucioles, F06560, Valbonne, France Boris I. Shraiman Bell Labs, Lucent Technologies, 700 Mountain Ave, Murray Hill, NJ 07974, USA Eric D. Siggia Dept. of Physics, Cornell University, Ithaca, NY1~853, USA Abstract We review theory and experiment for the mixing of a passive scalar by a turbulent flow. If the scalar fluctuations are maintained steady by a uniform large scale gradient, the one-point distribution function of the scalar has exponential tails; a property readily explained in terms of the Lagrangian Green's function in path integral form. For higher order correlations and separations within the scaling regime of the turbulence itself, new anomalous exponents have been derived from the Hopf equation, expressing the stationarity of the correlation functions. We concentrate on the 3-point correlator whose scaling exponent is very different from Kolmogorov or mean field theory, and for which laboratory experiments can be compared with numerical solutions of the Hopf equation, and analytic theory based on representations of the group SL(2). Turbulence, someone remarked, is too important to ignore, and too hard to solve. Engineers concern themselves with the first part of this dichotomy, and professional "tur- bulators" with the second. The success of Komogorov scaling (K-41) seemed to guarantee the acceptance, of its less well founded successor (K-62), at least in so far as it defined the problem. Theory defined what was to be measured and progressively more refined and quantitative one-point measurements were given primacy over more qualitative visu- alizations. One of the firmest results of 40 years of research in pure turbulence, is that pure contemplation is not apt to yield a solution. Engineers, who were not concerned with constructing general theory, have amassed considerable information and understanding of real flows; these phenomena are the primary material from which new model problems should be extracted. This talk reviews recent successes by several groups [1 3] in calculating the properties of turbulent mixing at a level of rigor comparable to contemporary statistical mechan- ics. This is unprecedented in statistical fluid dynamics at high Reynolds' number and phenomenological similarities between scalar mixing and incompressible fluid turbulence 0378-4371/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0378-4371(98)00532-9