Passive advection in nonlinear medium

Turbulence is a very nonequilibrium state of nature, which becomes stationary if energy is supplied permanently at large scales. To construct a theory of turbulence means to describe temporal and spatial distributions of velocity and variety of different thermodynamic characteristics of the fluid, i.e., density if turbulence is compressible, temperature if thermo-advection is applied, relative concentration of components in the case of multicomponent ~color! flow, magnetic field distribution in a conducting fluid, etc. Dynamics of different fields describing a real turbulent flow is both nonlocal and nonlinear. We call the general situation active to emphasize the reciprocal character of interaction between velocity field and thermodynamic characteristic~s!. However, sometimes the effect of a thermodynamic field on the velocity distribution is suppressed. It takes place, for example, if scales are separated: a typical spatio-temporal scale of velocity is much larger than one of a thermodynamic quantity. The case when it is theoretically justified to neglect the effect of back reaction of thermodynamic field on velocity, in comparison with those of advection and nonlinearity, is called passive. The passiveness does not necessarily mean linearity. Moreover, our objective is to study the passive yet nonlinear situation. We consider a thermodynamic quantity u governed by