A complete theory of low-energy phase diagrams for two-dimensional turbulence steady states and equilibria

For the 2D Euler equations and related models of geophysical flows, minima of energy--Casimir variational problems are stable steady states of the equations (Arnol'd theorems). The same variational problems also describe sets of statistical equilibria of the equations. In this paper, we make use of Lyapunov--Schmidt reduction in order to study the bifurcation diagrams for these variational problems, in the limit of small energy or, equivalently, of small departure from quadratic Casimir functionals. We show a generic occurrence of phase transitions, either continuous or discontinuous. We derive the type of phase transitions for any domain geometry and any model analogous to the 2D Euler equations. The bifurcations depend crucially on a_4, the quartic coefficient in the Taylor expansion of the Casimir functional around its minima. Note that a_4 can be related to the fourth moment of the vorticity in the statistical mechanics framework. A tricritical point (bifurcation from a continuous to a discontinuous phase transition) often occurs when a_4 changes sign. The bifurcations depend also on possible constraints on the variational problems (circulation, energy). These results show that the analytical results obtained with quadratic Casimir functionals by several authors are non-generic (not robust to a small change in the parameters)

The Equivalence of the Lagrangian-Averaged Navier-Stokes-{\alpha} Model and the Rational LES model in Two Dimensions

In the Large Eddy Simulation (LES) framework for modeling a turbulent flow, when the large scale velocity field is defined by low-pass filtering the full velocity field, a Taylor series expansion of the full velocity field in terms of the large scale velocity field leads (at the leading order) to the nonlinear gradient model for the subfilter stresses. Motivated by the fact that while the nonlinear gradient model shows excellent a priori agreement in resolved simulations, the use of this model by itself is problematic, we consider two models that are related, but better behaved: The Rational LES model that uses a sub-diagonal Pade approximation instead of a Taylor series expansion and the Lagrangian Averaged Navier-Stokes-{\alpha} model that uses a regulariza- tion approach to modeling turbulence. In this article, we show that these two latter models are identical in two dimensions

Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations

We discuss the dynamics of zonal (or unidirectional) jets for barotropic flows forced by Gaussian stochastic fields with white in time correlation functions. This problem contains the stochastic dynamics of 2D Navier-Stokes equation as a special case. We consider the limit of weak forces and dissipation, when there is a time scale separation between the inertial time scale (fast) and the spin-up or spin-down time (large) needed to reach an average energy balance. In this limit, we show that an adiabatic reduction (or stochastic averaging) of the dynamics can be performed. We then obtain a kinetic equation that describes the slow evolution of zonal jets over a very long time scale, where the effect of non-zonal turbulence has been integrated out. The main theoretical difficulty, achieved in this work, is to analyze the stationary distribution of a Lyapunov equation that describes quasi-Gaussian fluctuations around each zonal jet, in the inertial limit. This is necessary to prove that there is no ultraviolet divergence at leading order in such a way that the asymptotic expansion is self-consistent. We obtain at leading order a Fokker--Planck equation, associated to a stochastic kinetic equation, that describes the slow jet dynamics. Its deterministic part is related to well known phenomenological theories (for instance Stochastic Structural Stability Theory) and to quasi-linear approximations, whereas the stochastic part allows to go beyond the computation of the most probable zonal jet. We argue that the effect of the stochastic part may be of huge importance when, as for instance in the proximity of phase transitions, more than one attractor of the dynamics is present

Controversy about the applicability of Tsallis statistics to the HMF model

Comment to "Nonextensive Thermodynamics and Glassy Behaviour in Hamiltonian Systems" by A. Rapisarda and A. Pluchino, Europhysics News 36, 202 (2005)

Sampling microcanonical measures of the 2D Euler equations through Creutz's algorithm: a phase transition from disorder to order when energy is increased

The 2D Euler equations is the basic example of fluid models for which a microcanical measure can be constructed from first principles. This measure is defined through finite-dimensional approximations and a limiting procedure. Creutz's algorithm is a microcanonical generalization of the Metropolis-Hasting algorithm (to sample Gibbs measures, in the canonical ensemble). We prove that Creutz's algorithm can sample finite-dimensional approximations of the 2D Euler microcanonical measures (incorporating fixed energy and other invariants). This is essential as microcanonical and canonical measures are known to be inequivalent at some values of energy and vorticity distribution. Creutz's algorithm is used to check predictions from the mean-field statistical mechanics theory of the 2D Euler equations (the Robert-Sommeria-Miller theory). We found full agreement with theory. Three different ways to compute the temperature give consistent results. Using Creutz's algorithm, a first-order phase transition never observed previously, and a situation of statistical ensemble inequivalence are found and studied. Strikingly, and contrasting usual statistical mechanics interpretations, this phase transition appears from a disordered phase to an ordered phase (with less symmetries) when energy is increased. We explain this paradox.Comment: 27 pages, 12 figure

Computing transition rates for the 1-D stochastic Ginzburg--Landau--Allen--Cahn equation for finite-amplitude noise with a rare event algorithm

In this paper we compute and analyse the transition rates and duration of reactive trajectories of the stochastic 1-D Allen-Cahn equations for both the Freidlin-Wentzell regime (weak noise or temperature limit) and finite-amplitude white noise, as well as for small and large domain. We demonstrate that extremely rare reactive trajectories corresponding to direct transitions between two metastable states are efficiently computed using an algorithm called adaptive multilevel splitting. This algorithm is dedicated to the computation of rare events and is able to provide ensembles of reactive trajectories in a very efficient way. In the small noise limit, our numerical results are in agreement with large-deviation predictions such as instanton-like solutions, mean first passages and escape probabilities. We show that the duration of reactive trajectories follows a Gumbel distribution like for one degree of freedom systems. Moreover, the mean duration growths logarithmically with the inverse temperature. The prefactor given by the potential curvature grows exponentially with size. The main novelty of our work is that we also perform an analysis of reactive trajectories for large noises and large domains. In this case, we show that the position of the reactive front is essentially a random walk. This time, the mean duration grows linearly with the inverse temperature and quadratically with the size. Using a phenomenological description of the system, we are able to calculate the transition rate, although the dynamics is described by neither Freidlin--Wentzell or Eyring--Kramers type of results. Numerical results confirm our analysis

Perturbative calculation of quasi-potential in non-equilibrium diffusions: a mean-field example

In stochastic systems with weak noise, the logarithm of the stationary distribution becomes proportional to a large deviation rate function called the quasi-potential. The quasi-potential, and its characterization through a variational problem, lies at the core of the Freidlin-Wentzell large deviations theory%.~\cite{freidlin1984}.In many interacting particle systems, the particle density is described by fluctuating hydrodynamics governed by Macroscopic Fluctuation Theory%, ~\cite{bertini2014},which formally fits within Freidlin-Wentzell's framework with a weak noise proportional to $1/\sqrt{N}$, where $N$ is the number of particles. The quasi-potential then appears as a natural generalization of the equilibrium free energy to non-equilibrium particle systems. A key physical and practical issue is to actually compute quasi-potentials from their variational characterization for non-equilibrium systems for which detailed balance does not hold. We discuss how to perform such a computation perturbatively in an external parameter $\lambda$, starting from a known quasi-potential for $\lambda=0$. In a general setup, explicit iterative formulae for all terms of the power-series expansion of the quasi-potential are given for the first time. The key point is a proof of solvability conditions that assure the existence of the perturbation expansion to all orders. We apply the perturbative approach to diffusive particles interacting through a mean-field potential. For such systems, the variational characterization of the quasi-potential was proven by Dawson and Gartner%. ~\cite{dawson1987,dawson1987b}. Our perturbative analysis provides new explicit results about the quasi-potential and about fluctuations of one-particle observables in a simple example of mean field diffusions: the Shinomoto-Kuramoto model of coupled rotators%. ~\cite{shinomoto1986}. This is one of few systems for which non-equilibrium free energies can be computed and analyzed in an effective way, at least perturbatively

Equilibrium statistical mechanics and energy partition for the shallow water model

The aim of this paper is to use large deviation theory in order to compute the entropy of macrostates for the microcanonical measure of the shallow water system. The main prediction of this full statistical mechanics computation is the energy partition between a large scale vortical flow and small scale fluctuations related to inertia-gravity waves. We introduce for that purpose a discretized model of the continuous shallow water system, and compute the corresponding statistical equilibria. We argue that microcanonical equilibrium states of the discretized model in the continuous limit are equilibrium states of the actual shallow water system. We show that the presence of small scale fluctuations selects a subclass of equilibria among the states that were previously computed by phenomenological approaches that were neglecting such fluctuations. In the limit of weak height fluctuations, the equilibrium state can be interpreted as two subsystems in thermal contact: one subsystem corresponds to the large scale vortical flow, the other subsystem corresponds to small scale height and velocity fluctuations. It is shown that either a non-zero circulation or rotation and bottom topography are required to sustain a non-zero large scale flow at equilibrium. Explicit computation of the equilibria and their energy partition is presented in the quasi-geostrophic limit for the energy-enstrophy ensemble. The possible role of small scale dissipation and shocks is discussed. A geophysical application to the Zapiola anticyclone is presented.Comment: Journal of Statistical Physics, Springer Verlag, 201