THEORY OF RANDOM ADVECTION IN TWO DIMENSIONS
The steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described at scales less than the correlation length of the flow and larger than the diffusion scale. The probability distribution of the scalar is expressed via the probability distribution of the line stretching rate. The description of the line stretching can be reduced to the classical problem of studying the product of many matrices with a unit determinant. We found a change of variables which allows one to map the matrix problem into a scalar one and to prove thus a central limit theorem for the statistics of the stretching rate. The proof is valid for any finite correlation time of the velocity field. Whatever be the statistics of the velocity field, the statistics of the passive scalar in the inertial interval of scales is shown to approach Gaussianity as one increases the Peclet number Pe (the ratio of the pumping scale to the diffusion one). The first n < ln (Pe) simultaneous correlation functions are expressed via the flux of the squared scalar and only one unknown factor depending on the velocity field: the mean stretching rate. That factor can be calculated analytically for the limiting cases. The non-Gaussian tails of the probability distributions at finite Pe are found to be exponential.