A Solvable Model for Nonlinear Mean Field Dynamo

We formulate a solvable model that describes generation and saturation of mean magnetic field in a dynamo with kinetic helicity, in the limit of large magnetic Prandtl number. This model is based on the assumption that the stochastic part of the velocity field is Gaussian and white in time (the Kazantsev-Kraichnan ensemble), while the regular part describing the back reaction of the magnetic field is chosen from balancing the viscous and Lorentz stresses in the MHD Navier-Stokes equation. The model provides an analytical explanation for previously obtained numerical results.Comment: 6 page

The Simulation Of Slat Noise Applying Stochastic Sound Sources Based On Solenoidal Digital Filters (SDF)

A new fast and cheap stochastic approach is introduced to model the unsteady turbulent sound sources in the slat-cove of a high-lift airfoil. It is based on the spatial filtering of a random white-noise field and incoporates information about the integral length scale and the turbulent kinetic energy from a steady RANS computation. The stochastic method yields a solenoidal velocity field that is capable to reproduce exactly the well known analytical solution of the second-order two-point correlation tensor in case of homogeneous isotropic turbulence. Results for the sound generation at the slat are given for the underlying RANS mean-flow field being based on a Menter SST turbulence model with Kato-Launder modification. The results for the modeled turbulent flow-field and the radiated acoustic field exhibit physical meaningful characteristics

Analysis of a stochastic chemical system close to a sniper bifurcation of its mean field model

A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs for example in the modelling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) is studied. Our approach is based on the chemical Fokker Planck equation. To get some insights into advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, before the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size

From individual to collective behaviour of coupled velocity jump processes: a locust example

A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed. This modelling approach incorporates recent experimental findings on behaviour of locusts. It exhibits nontrivial dynamics with a “phase change” behaviour and recovers the observed group directional switching. Estimates of the expected switching times, in terms of number of individuals and values of the model coefficients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations with nonlocal and nonlinear right hand side is derived and analyzed. The existence of its solutions is proven and the system’s long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the effect of shrinking the interaction radius in the individual-based model when the number of individuals grows