Flatness of tracer density profile produced by a point source in turbulence

The average concentration of tracers advected from a point source by a multivariate normal velocity field is shown to deviate from a Gaussian profile. The flatness (kurtosis) is calculated using an asymptotic series expansion valid for velocity fields with short correlation times or weak space dependence. An explicit formula for the excess flatness at first order demonstrates maximum deviation from a Gaussian profile at time t of the order of five times the velocity correlation time, with a t^-1 decay to the Gaussian value at large times. Monotonically decaying forms of the velocity time correlation function are shown to yield negative values for the first order excess flatness, but positive values can result when the correlation function has an oscillatory tail.