Motivated by the experimental study of Huang and Driscoll [Phys. Rev. Lett. 72, 2187 (1994)], we carry a systematic comparison between maximum entropy and restricted minimum enstrophy vortices. We find that, when the angular momentum constraint is reconsidered, the restricted minimum enstrophy principle provides a good and useful approximation of the statistical theory, except for large energies. In the case investigated by Huang and Driscoll, this approximation even yields a better agreement with experiments than the full statistical theory because the relaxation of the system toward the maximum entropy state is not complete. However, we show that this agreement cannot hold in more general cases, due to physical inconsistencies of minimum enstrophy principles. We also discuss the relevance of generalized thermodynamics (Tsallis entropy) to predict the equilibrium state.
This paper considers, in contraposition to the most probable states of quasi-inviscid theories, the status of the vorticity(co)-streamfunction(1/I) relations satisfied by the long-lived vortices observed in some numerical simulations of decaying two-dimensional turbulence and in experiments in stratified fluids. For the case co= -a1/l=sinh 1/1, the circular solutions can be expressed in terms of the IIIrd Painleve transcendent and dipolar solutions can be constructed by means of a Backlund transform.
We compare the quasi-stationary state obtained from a numerical integration of the two-dimensional Navier-Stokes equation, performed by Matthaeus et al. [Phys. Rev. Lett. 66, 2731 (1991)] with predictions of a mean field theory based on inviscid dynamics. We find that over a relatively short initial period, the theory does not apply, whereas at later stages its prediction for the quasi-stationary state is very good. We relate the failure and success of the inviscid theory to the relevance of viscous effects in the dynamics.
Abstract.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.
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