Abstract
Stochastic flows mimicking 2D turbulence in compressible media are considered. Particles driven by such flows can collide and we study the collision (caustic) frequency. Caustics occur when the Jacobian of a flow vanishes. First, a system of nonlinear stochastic differential equations involving the Jacobian is derived and reduced to a smaller number of unknowns. Then, for special cases of the stochastic forcing, upper and lower bounds are found for the mean number of caustics as a function of Stokes number. The bounds yield an exact asymptotic for small Stokes numbers. The efficiency of the bounds is verified numerically. As auxiliary results we give rigorous proofs of the well known expressions for the caustic frequency and Lyapunov exponent in the one-dimensional model. Our findings may also be used for estimating the mean time when a 2D Riemann type partial differential equation with a stochastic forcing loses uniqueness of solutions.
Highlights
We address a stochastic flow [1] covered by the following Ito stochastic differential equations dr = vdt, dv = −(v/τ )dt + dw(t, r), (1)
To introduce the problem we focus on, let us supply (1) with initial conditions r|t=0 = a, v|t=0 = u0 (a) where u0 (a) is a given deterministic function, i.e., each particle starts from a certain position with a certain velocity completely determined by its position, and introduce the Jacobian
As for two dimensions, we found no analytical results whatsoever concerning with the dependence of ν on s, while some asymptotics for Lyapunov exponent (LE) were obtained in [2,6,7]
Summary
Let the initial position a be a random variable independent of the flow with probability density π a (x), for the conditional density πr(t,a) (x) conditioned on the flow is given by πr(t,a) (x) = π a (x)/J (t, x) It is well known [3] that for incompressible flows J (t, a) ≡ 1 , the density of the particles does not change at any point in the fluid. That results may be used for estimating the mean time when the corresponding SPDE for the Eulerian velocity field loses its uniqueness. Worth noting that if the dissipation term is replaced by a classical friction k∆u, we arrive at the well known stochastic Burger’s equation which has a unique solution for all t thereby no caustics may occur.
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