Material surfaces in stochastic flows: Integrals of motion and intermittency.

We consider the line, surface, and volume elements of fluid in stationary isotropic incompressible stochastic flow in d-dimensional space and investigate the long-time evolution of their statistic properties. We report the discovery of a family of d!-1 stochastical integrals of motion that are universal in the sense that their explicit form does not depend on the statistics of velocity. Only one of them has been discussed previously.

Long-term properties of finite-correlation-time isotropic stochastic systems.

We consider finite-dimensional systems of linear stochastic differential equations ∂_{t}x_{k}(t)=A_{kp}(t)x_{p}(t), A(t) being a stationary continuous statistically isotropic stochastic process with values in real d×d matrices. We suppose that the laws of A(t) satisfy the large-deviation principle. For these systems, we find exact expressions for the Lyapunov and generalized Lyapunov exponents and show that they are determined in a precise way only by the rate function of the diagonal elements of A.

Stationary scaling in small-scale turbulent dynamo problem.

We consider forced small-scale magnetic field advected by an isotropic turbulent flow. The random driving force is assumed to be distributed in a finite region with a scale smaller than the viscous scale of the flow. The two-point correlator is shown to have a stationary limit for any reasonable velocity statistics. Its spatial dependence is found to be a power law. The scaling exponent is found to be close to 3.

Turbulent transport in reaction-diffusion systems.

The impact of turbulent advection in reaction-diffusion systems is investigated for the viscous range of scales. We show that the population size can increase exponentially even in systems with density saturation, at the expense of exponential propagation of the reaction front. Exact expressions for scaling exponents of the density and population size are calculated in different intermediate asymptotics of the process. The system appears to demonstrate high intermittency.

Passive scalar transport by a non-Gaussian turbulent flow in the Batchelor regime.

We analyze passive scalar advection by a turbulent flow in the Batchelor regime. No restrictions on the velocity statistics of the flow are assumed. The properties of the scalar are derived from the statistical properties of velocity; analytic expressions for the moments of scalar density are obtained. We show that the scalar statistics can differ significantly from that obtained in the frames of the Kraichnan model.