A study of the connection between exit-time statistics and relative dispersion using a simple Lagrangian stochastic model

Abstract

We use a simple stochastic model of relative dispersion to explore the effect of departures from diffusive behaviour on the connection between exit-time statistics and the Richardson constant g for relative dispersion. We present our results as a function of the Lagrangian velocity structure function constant C ₀, which is a persistence parameter representing the strength of memory effects. For C ₀ → ∞, the model reduces to a diffusive process, for which we have analytical relationships between g and the exit-time statistics, while according to the best estimates from direct numerical simulation C ₀∼ 6–7 in three-dimensional turbulence. We calculate the Richardson constant g and the corresponding constants appearing in the inertial sub-range forms for moments of the exit time and of the inverse exit time, for both forward and backward relative dispersion. For C ₀ ∼ 6–7, and using model values for the first and third moments of the exit time, we find that the value for g estimated, assuming the diffusive relationships hold, is within about 20% of the ‘true’ value for forward relative dispersion but can be in error by up to a factor of 5 for backward dispersion. The errors are much larger if the corresponding moments of the inverse exit time are used.