Statistics of a Passive Scalar Advected by a Large-Scale 2D Velocity Field: Analytic Solution
Steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale. That corresponds to the so-called Batchelor regime where the velocity is replaced by its large-scale gradient. The probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate. The description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant. We have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics. The proof is valid for any finite correlation time of the velocity field. Whatever the statistics of the velocity field, the statistics of the passive scalar (averaged over time locally in space) is shown to approach gaussianity with increase in the Peclet number $Pe$ (the pumping-to-diffusion scale ratio). The first $n<\ln Pe$ simultaneous correlation functions are expressed via the flux of the square of the scalar and only one factor depending on the velocity field: the mean stretching rate, which can be calculated analytically in limiting cases. Non-Gaussian tails of the probability distributions at finite $Pe$ are found to be exponential.