Mathematics – Logic
A thermodynamic approach to the construction of a phenomenological macroscopic model of developed turbulence in a compressible fluid is considered with regard for the formation of space-time dissipative structures. A set of random variables were introduced into the model as internal parameters of the turbulent-chaos subsystem. This allowed us to obtain, by methods of nonequilibrium thermodynamics, the kinetic Fokker-Planck equation in the configuration space. This equation serves to determine the temporary evolution of the conditional probability distribution function of structural parameters pertaining to the cascade process of fragmentation of large-scale eddies and temperature inhomogeneities and to analyze Markovian stochastic processes of transition from one nonequilibrium stationary turbulent-motion state to another as a result of successive loss of stability caused by a change in the governing parameters. An alternative method for investigating the mechanisms of such transitions, based on the stochastic Langevin-type equation intimately related to the derived kinetic equation, is also considered. Some postulates and physical and mathematical assumptions used in the thermodynamic model of structurized turbulence are discussed in detail. In particular, we considered, using the deterministic transport equation for conditional means, the cardinal problem of the developed approach-the possibility of the existence of asymptotically stable stationary states of the turbulent-chaos subsystem. Also proposed is the nonequilibrium thermodynamic potential for internal coordinates, which extends the well-known Boltzmann-Planck relationship for equilibrium states to the nonequilibrium stationary states of the representing ensemble. This potential is shown to be the Lyapunov function for such states. The relation is also explored between the internal intermittence in the inertial interval of scales and the fluctuations of the energy of dissipation. This study is aimed at constructing representative models of natural space environments. It develops a synergetic approach to modeling the structurized turbulence of astrophysical and geophysical systems, which was proposed by the author in previous papers (Kolesnichenko, 2002, 2003).
Kolesnichenko Aleksander V.
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