Particle pair diffusion and persistent streamline topology in two-dimensional turbulence

Abstract

From observations of direct numerical simulations (DNS) of two-dimensional turbulence with inverse energy cascade, two physical pictures of particle pair diffusion are proposed based on persistent streamline topology associated with stagnation points. One picture describes the step-by-step separation process of individual pairs in a local frame moving with them, whereas the other serves as a statistical description of particle pair diffusion in a global frame which we define. These two pictures lead to the same characteristic time scale for particle pair diffusion. Based on this time scale, a new model of particle pair diffusion is proposed which predicts the temporal evolutions of the mean square separation, and of the probability density function (PDF) of separations. Our PDF equation turns out to be a generalization of Richardson's diffusion equation (Richardson L F 1926 Proc. R. Soc. A 110 709). DNS verifications support all the predictions of our model. A generalization of our approach to d-dimensional turbulence with energy spectrum proportional to k-p is given for the purpose of demonstrating that the PDF equation and the exponent of mean square separation are directly related with the fractal dimension of the spatial distribution of stagnation points.