Summary: A hyperbolic equation, analogous to the telegrapher’s equation in one dimension, is introduced to describe turbulent diffusion of a passive additive in a turbulent flow. The predictions of this equation, and those of the usual advection-diffusion equation, are compared with data on smoke plumes in the atmosphere and on heat flow in a wind tunnel. The predictions of the hyperbolic equation fit the data at all distances from the source, whereas those of the advection-diffusion equation fit only at large distances.
This hyperbolic equation is derived from an integro-differential equation for mean concentration which allows it to vary rapidly. If the mean concentration varies sufficiently slowly compared with the correlation time of the turbulence, the hyperbolic equation reduces to the advection-diffusion equation. However, if the mean concentration varies very rapidly, the hyperbolic equation should be replaced by the integro-differential equation.