Passive advection in nonlinear medium

Abstract

Forced advection of passive tracer, $\theta $, in nonlinear relaxational medium by large scale (Batchelor problem) incompressible velocity field at scales less than the correlation length of the flow and larger than the diffusion scale is considered. Effective theory explaining small scale scalar fluctuations is proven to be linear, asymptotic free (downscales from the scale of the pumping) and universal. Only three parameters are required to decribe exhaustively the small scale statistics of scalar difference: two velocity-dependent ones, average and dispersion ($\bar{\lambda}$ and $\Delta $ respectively) of the exponential stretching rate of a trial line element, and $\alpha $, standing for average rate of linear damping of small scale scalar fluctuations. $\alpha $ is an explicit functional of potential chracterized medium nonlinearity and amplitude of $\theta ^{2}$ flux pumped into the system. Structure functions show an extremely anomalous, intermittent behavior: $<|\delta \theta_{r}|^{q}> \sim r^{\xi_{q}}, \xi_{q} = \min {q,\sqrt{[ \frac{\bar{\lambda}}{\Delta}] ^{2} + \frac{2\alpha q}{\Delta}} - \frac{\bar{\lambda}}{\Delta}}$. No dissipative anomaly is found in the problem.