Fluid-particle separation in the Batchelor regime with telegraph model of noise

We study the statistics of the relative separation between two fluid particles in a random flow. We confine ourselves to the Batchelor regime, i.e. we only examine the evolution of distances smaller than the smallest active scale of the flow, where the latter is spatially smooth. The Lagrangian strain is assumed as given in its statistics and is modelled by a telegraph noise. This is a stationary random Markov process, which can only take two values with known transition probabilities. The presence of two independent parameters (intensity of velocity gradient and flow correlation time) allows the definition of their ratio as the Kubo number, whose infinitesimal and infinite limits describe the delta-correlated and quasi-deterministic cases, respectively. However, the simplicity of the model enables us to write closed equations for the interparticle distance in the presence of a finite-correlated, i.e. coloured, noise. In 1D, the flow is locally compressible in every single realization, but the average ‘volume’ must keep finite. This provides us with a mathematical constraint, which physically reflects in the fact that, in the Lagrangian frame, particles spend longer time in contracting regions than in expanding ones. Imposing this condition consistently, we are able to find analytically the long-time growth rate of the interparticle-distance moments and, consequently, the senior Lyapunov exponent, which coherently turns out to be negative. Analysing the large-deviation form of the joint probability distribution, we also show the exact expression of the Cramér function, which happens to satisfy the well-known fluctuation relation despite the time irreversibility of the strain statistics. The 2D incompressible case is also studied. After showing a simple generalization of the 1D situation, we concentrate ourselves on the general isotropic case: the evolution of the linear and quadratic components is analysed thoroughly, while for higher moments, due to high computational cost, we focus on a restricted, though exact, dynamics. As a result, we obtain the moment asymptotic growth rates and the Lyapunov exponent (positive) in the two above-mentioned limits, together with the leading corrections. The quasi-deterministic limit turns out to be singular, while a perfect agreement is found with the already-known delta-correlated case.