A parametrization of two-dimensional turbulence based on a maximum entropy production principle with a local conservation of energy

In the context of two-dimensional (2D) turbulence, we apply the maximum entropy production principle (MEPP) by enforcing a local conservation of energy. This leads to an equation for the vorticity distribution that conserves all the Casimirs, the energy, and that increases monotonically the mixing entropy ($H$-theorem). Furthermore, the equation for the coarse-grained vorticity dissipates monotonically all the generalized enstrophies. These equations may provide a parametrization of 2D turbulence. They do not generally relax towards the maximum entropy state. The vorticity current vanishes for any steady state of the 2D Euler equation. Interestingly, the equation for the coarse-grained vorticity obtained from the MEPP turns out to coincide, after some algebraic manipulations, with the one obtained with the anticipated vorticity method. This shows a connection between these two approaches when the conservation of energy is treated locally. Furthermore, the newly derived equation, which incorporates a diffusion term and a drift term, has a nice physical interpretation in terms of a selective decay principle. This gives a new light to both the MEPP and the anticipated vorticity method.Comment: To appear in the special IUTAM issue of Fluid Dynamics Research on Vortex Dynamic

Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a small-scale parametrization of 2D turbulence or serve as numerical algorithms to compute maximum entropy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows.Comment: 21 pages, 9 figure

Moment free energies for polydisperse systems

A polydisperse system contains particles with at least one attribute $\sigma$ (such as particle size in colloids or chain length in polymers) which takes values in a continuous range. It therefore has an infinite number of conserved densities, described by a density {\em distribution} $\rho(\sigma)$. The free energy depends on all details of $\rho(\sigma)$, making the analysis of phase equilibria in such systems intractable. However, in many (especially mean-field) models the {\em excess} free energy only depends on a finite number of (generalized) moments of $\rho(\sigma)$; we call these models truncatable. We show, for these models, how to derive approximate expressions for the {\em total} free energy which only depend on such moment densities. Our treatment unifies and explores in detail two recent separate proposals by the authors for the construction of such moment free energies. We show that even though the moment free energy only depends on a finite number of density variables, it gives the same spinodals and critical points as the original free energy and also correctly locates the onset of phase coexistence. Results from the moment free energy for the coexistence of two or more phases occupying comparable volumes are only approximate, but can be refined arbitrarily by retaining additional moment densities. Applications to Flory-Huggins theory for length-polydisperse homopolymers, and for chemically polydisperse copolymers, show that the moment free energy approach is computationally robust and gives new geometrical insights into the thermodynamics of polydispersity.Comment: RevTeX, 43 pages including figure

Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

We complete the literature on the statistical mechanics of point vortices in two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the multi-species Boltzmann-Poisson equation and establish a form of virial theorem. Using a maximum entropy production principle (MEPP), we derive a set of relaxation equations towards statistical equilibrium. These relaxation equations can be used as a numerical algorithm to compute the maximum entropy state. We mention the analogies with the Fokker-Planck equations derived by Debye and H\"uckel for electrolytes. We then consider the limit of strong mixing (or low energy). To leading order, the relationship between the vorticity and the stream function at equilibrium is linear and the maximization of the entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar to the Debye-H\"uckel approximation for electrolytes, except that the temperature is negative instead of positive so that the effective interaction between like-sign vortices is attractive instead of repulsive. This leads to an organization at large scales presenting geometry-induced phase transitions, instead of Debye shielding. We compare the results obtained with point vortices to those obtained in the context of the statistical mechanics of continuous vorticity fields described by the Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results but differences appear at the next order. In particular, the MRS theory predicts a transition between sinh and tanh-like \omega-\psi relationships depending on the sign of Ku-3 (where Ku is the Kurtosis) while there is no such transition for point vortices which always show a sinh-like \omega-\psi relationship. We derive the form of the relaxation equations in the strong mixing limit and show that the enstrophy plays the role of a Lyapunov functional