Extension of the Oseledets theorem and preferential concentration of particles that sediment in turbulence

Abstract We reconsider the Oseledets theorem on the Jacobi matrix of a flow from the viewpoint of the singular value decomposition. We demonstrate that even though the infinite time limit of the orthogonal matrix in the decomposition depends on the trajectory, the rate of relaxation to it does not, and is given by differences of the Lyapunov exponents. We apply the results to the problem of infinitesimally close particles that rapidly sediment in turbulence. Here, in the particle frame, the flow changes fast and an effective white noise matrix process drives the separation. We erroneously stated in [1] that this matrix reduced to a $2\times 2$ one. Here we complete the theory for the observed elongation of particle structures in the vertical direction, while simultaneously correcting the previously made error. The key observation is that both orthogonal matrices in the singular value decomposition are degenerate. Their structure demonstrates that infinitesimal volumes of particles are compressed only horizontally. This implies that backward in time separation of close trajectories degenerates, resulting in the anisotropy of the radial distribution function (RDF).