Self-similar decay in the Kraichnan model of a passive scalar

Gregory L Eyink Jack Xin

Fluid Dynamics and Shock Waves mathscidoc:1912.43851

Journal of Statistical Physics, 100, 679-741, 2000.8
We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0&lt;<i></i>&lt;2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent <i>a</i>(<i>d</i>/<i></i>)+1 are statistically realizable, where <i>d</i> is space dimension and <i></i>=2<i></i>. An infinite sequence of invariants <i>J</i> <sub>p</sub>, <i>p</i>=0,1,2,..., is pointed out, where <i>J</i> <sub>0</sub> is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants <i>J</i> <sub>0</sub> or <i>J</i> <sub>1</sub> must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariantthe first being <i>J</i> <sub>p</sub>converges at long times to a scaling
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  title={Self-similar decay in the Kraichnan model of a passive scalar},
  author={Gregory L Eyink, and Jack Xin},
  booktitle={Journal of Statistical Physics},
Gregory L Eyink, and Jack Xin. Self-similar decay in the Kraichnan model of a passive scalar. 2000. Vol. 100. In Journal of Statistical Physics. pp.679-741.
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