Inertial particles driven by a telegraph noise

We present a model for the Lagrangian dynamics of inertial particles in a compressible flow, where fluid velocity gradients are modelled by a telegraph noise. The model allows for an analytic investigation of the role of time correlation of the flow in the aggregation-disorder transition of inertial particle. The dependence on Stokes and Kubo numbers of the Lyapunov exponent of particle trajectories reveals the presence of a region in parameter space (St, Ku) where the leading Lyapunov exponent changes sign, thus signaling the transition. The asymptotics of short and long-correlated flows are discussed, as well as the fluid-tracer limit.Comment: 8 pages, 6 figure

On the Aggregation of Inertial Particles in Random Flows

We describe a criterion for particles suspended in a randomly moving fluid to aggregate. Aggregation occurs when the expectation value of a random variable is negative. This random variable evolves under a stochastic differential equation. We analyse this equation in detail in the limit where the correlation time of the velocity field of the fluid is very short, such that the stochastic differential equation is a Langevin equation.Comment: 16 pages, 2 figure

The tail of the maximum of smooth Gaussian fields on fractal sets

We study the probability distribution of the maximum $M_S$ of a smooth stationary Gaussian field defined on a fractal subset $S$ of $\R^n$. Our main result is the equivalent of the asymptotic behavior of the tail of the distribution $\P(M_S>u)$ as $u\rightarrow +\infty.$ The basic tool is Rice formula for the moments of the number of local maxima of a random field

Strong Approximation of Empirical Copula Processes by Gaussian Processes

We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm

Ergodic properties of a model for turbulent dispersion of inertial particles

We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schroedinger equation in a random delta-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory

Quasi-Two-Dimensional Dynamics of Plasmas and Fluids

In the lowest order of approximation quasi-twa-dimensional dynamics of planetary atmospheres and of plasmas in a magnetic field can be described by a common convective vortex equation, the Charney and Hasegawa-Mirna (CHM) equation. In contrast to the two-dimensional Navier-Stokes equation, the CHM equation admits "shielded vortex solutions" in a homogeneous limit and linear waves ("Rossby waves" in the planetary atmosphere and "drift waves" in plasmas) in the presence of inhomogeneity. Because of these properties, the nonlinear dynamics described by the CHM equation provide rich solutions which involve turbulent, coherent and wave behaviors. Bringing in non ideal effects such as resistivity makes the plasma equation significantly different from the atmospheric equation with such new effects as instability of the drift wave driven by the resistivity and density gradient. The model equation deviates from the CHM equation and becomes coupled with Maxwell equations. This article reviews the linear and nonlinear dynamics of the quasi-two-dimensional aspect of plasmas and planetary atmosphere starting from the introduction of the ideal model equation (CHM equation) and extending into the most recent progress in plasma turbulence.U. S. Department of Energy DE-FG05-80ET-53088Ministry of Education, Science and Culture of JapanFusion Research Cente