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We study the two-point correlation function of a freely decaying scalar in Kraichnans model of advection by a Gaussian random velocity field, stationary and white-noise in time but fractional Brownian in space with roughness exponent $0<zeta<2$, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions, by transforming the scaling equation to Kummers equation. It is shown that only those scaling solutions with scalar energy decay exponent $aleq (d/gamma)+1$ are statistically realizable, where $d$ is space dimension and $gamma =2-zeta$. An infinite sequence of invariants $J_ell, ell=0,1,2,...$ is pointed out, where $J_0$ is Corrsins integral invariant but the higher invariants appear to be new. We show that at least one of the first two invariants, $J_0$ or $J_1$, must be nonzero for realizable initial data. We classify initial data in long-time domains of attraction of the self-similar solutions, based upon these new invariants. Our results support a picture of ``two-scale decay with breakdown of self-similarity for a range of exponents $(d+gamma)/gamma < a < (d+2)/gamma,$ analogous to what has recently been found in decay of Burgers turbulence.
We study transport of a weakly diffusive pollutant (a passive scalar) by thermoconvective flow in a fluid-saturated horizontal porous layer heated from below under frozen parametric disorder. In the presence of disorder (random frozen inhomogeneities
There are many materials whose dielectric properties are described by a stretched exponential, the so-called Kohlrausch-Williams-Watts (KWW) relaxation function. Its physical origin and statistical-mechanical foundation have been a matter of debate i
The asymptotic decay of passive scalar fields is solved analytically for the Kraichnan model, where the velocity has a short correlation time. At long times, two universality classes are found, both characterized by a distribution of the scalar -- ge
The advection and mixing of a scalar quantity by fluid flow is an important problem in engineering and natural sciences. If the fluid is turbulent, the statistics of the passive scalar exhibit complex behavior. This paper is concerned with two Lagran
A broad range of membrane proteins display anomalous diffusion on the cell surface. Different methods provide evidence for obstructed subdiffusion and diffusion on a fractal space, but the underlying structure inducing anomalous diffusion has never b