Lyapunov exponents for particles advected in compressible random velocity fields at small and large Kubo numbers

TL;DR

This paper investigates how particles cluster in random velocity fields by calculating Lyapunov exponents across different Kubo numbers, revealing conditions for particle coalescence and the influence of flow compressibility.

Contribution

It provides new analytical and numerical methods for computing Lyapunov exponents in compressible random flows at various Kubo numbers, including resummation techniques and potential-based approaches.

Findings

Lyapunov exponents are always negative in 1D at finite Kubo numbers.

The sign of the maximal Lyapunov exponent in 2D depends on flow compressibility and Kubo number.

Transition point for particle coalescence is estimated up to Ku ≈ 0.5.

Abstract

We calculate the Lyapunov exponents describing spatial clustering of particles advected in one- and two-dimensional random velocity fields at finite Kubo numbers Ku (a dimensionless parameter characterising the correlation time of the velocity field). In one dimension we obtain accurate results up to Ku ~ 1 by resummation of a perturbation expansion in Ku. At large Kubo numbers we compute the Lyapunov exponent by taking into account the fact that the particles follow the minima of the potential function corresponding to the velocity field. The Lyapunov exponent is always negative. In two spatial dimensions the sign of the maximal Lyapunov exponent \lambda_1 may change, depending upon the degree of compressibility of the flow and the Kubo number. For small Kubo numbers we compute the first four non-vanishing terms in the small-Ku expansion of the Lyapunov exponents. By resumming these expansions we obtain a precise estimate of the location of the path-coalescence transition (where \lambda_1 changes sign) for Kubo numbers up to approximately Ku = 0.5. For large Kubo numbers we estimate the Lyapunov exponents for a partially compressible velocity field by assuming that the particles sample those stagnation points of the velocity field that have a negative real part of the maximal eigenvalue of the matrix of flow-velocity gradients.