Evolution of the vorticity-area density during the formation of coherent structures in two-dimensional flows

It is shown: 1) that in two-dimensional, incompressible, viscous flows the vorticity-area distribution evolves according to an advection-diffusion equation with a negative, time dependent diffusion coefficient and 2) how to use the vorticity-streamfunction relations, i.e., the so-called scatter-plots, of the quasi-stationary coherent structures in order to quantify the experimentally observed changes of the vorticity distribution moments leading to the formation of these structures.Comment: LaTeX, 15 pp., 2 eps figures. Some sections have been rewritten; referees' Comments have been include

Cyclic Markov chains with an application to an intermediate ENSO model

We develop the theory of cyclic Markov chains and apply it to the El Niño-Southern Oscillation (ENSO) predictability problem. At the core of Markov chain modelling is a partition of the state space such that the transition rates between different state space cells can be computed and used most efficiently. We apply a partition technique, which divides the state space into multidimensional cells containing an equal number of data points. This partition leads to mathematical properties of the transition matrices which can be exploited further such as to establish connections with the dynamical theory of unstable periodic orbits. We introduce the concept of most and least predictable states. The data basis of our analysis consists of a multicentury-long data set obtained from an intermediate coupled atmosphere-ocean model of the tropical Pacific. This cyclostationary Markov chain approach captures the spring barrier in ENSO predictability and gives insight also into the dependence of ENSO predictability on the climatic state

Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot

We introduce a new set of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy until a maximum entropy state is reached. The concept of generalized entropies is rigorously justified for continuous Hamiltonian systems undergoing violent relaxation. Tsallis entropies are just a special case of this generalized thermodynamics. Application of these results to stellar dynamics, vortex dynamics and Jupiter's great red spot are proposed. Our prime result is a novel relaxation equation that should offer an easily implementable parametrization of geophysical turbulence. This relaxation equation depends on a single key parameter related to the skewness of the fine-grained vorticity distribution. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations may have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in classes of equivalence and provide an aesthetic connexion between topics (vortices, stars, bacteries,...) which were previously disconnected.Comment: Submitted to Phys. Rev.