The Transport Equation for the Dispersal of Passive Tracers in a Nonuniform Turbulent Fluid: Numerical Simulations

The random advection of passive additives in a turbulent fluid plays an important role in solar physics, astrophysics, and atmospheric sciences. We concern ourselves here with the case where the fluctuations are not statistically homogeneous in space, and, hence, where the transport coefficients vary with position. Using a numerical model in which the fluid turbulence is defined kinematically, we show that the evolution of the distribution of passive tracers in the fluid is not always governed by the ordinary diffusion equation. We find it is governed by a more general transport equation whose form depends on the nature of the turbulence, particularly on its compressibility, or divergence. The more general transport equation resembles the ordinary diffusion equation, but the transport coefficient appears in two places and is raised to a power that depends on the divergence of the fluid velocity. If the flow has zero divergence, the case for incompressible turbulence, the resulting transport equation is found to be the regular diffusion equation.