Fluctuations of large-scale jets in the stochastic 2D Euler equation

Two-dimensional turbulence in a rectangular domain self-organises into large-scale unidirectional jets. While several results are present to characterize the mean jets velocity profile, much less is known about the fluctuations. We study jets dynamics in the stochastically forced two-dimensional Euler equations. In the limit where the average jets velocity profile evolves slowly with respect to turbulent fluctuations, we employ a multi-scale (kinetic theory) approach, which relates jet dynamics to the statistics of Reynolds stresses. We study analytically the Gaussian fluctuations of Reynolds stresses and predict the spatial structure of the jets velocity covariance. Our results agree qualitatively well with direct numerical simulations, clearly showing that the jets velocity profile are enhanced away from the stationary points of the average velocity profile. A numerical test of our predictions at quantitative level seems out of reach at the present day

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

Efficiency at maximum power of a discrete feedback ratchet

Efficiency at maximum power is found to be of the same order for a feedback ratchet and for its open-loop counterpart. However, feedback increases the output power up to a factor of five. This increase in output power is due to the increase in energy input and the effective entropy reduction obtained as a consequence of feedback. Optimal efficiency at maximum power is reached for time intervals between feedback actions two orders of magnitude smaller than the characteristic time of diffusion over a ratchet period length. The efficiency is computed consistently taking into account the correlation between the control actions. We consider a feedback control protocol for a discrete feedback flashing ratchet, which works against an external load. We maximize the power output optimizing the parameters of the ratchet, the controller, and the external load. The maximum power output is found to be upper bounded, so the attainable extracted power is limited. After, we compute an upper bound for the efficiency of this isothermal feedback ratchet at maximum power output. We make this computation applying recent developments of the thermodynamics of feedback-controlled systems, which give an equation to compute the entropy reduction due to information. However, this equation requires the computation of the probability of each of the possible sequences of the controller's actions. This computation becomes involved when the sequence of the controller's actions is non-Markovian, as is the case in most feedback ratchets. We here introduce an alternative procedure to set strong bounds to the entropy reduction in order to compute its value. In this procedure the bounds are evaluated in a quasi-Markovian limit, which emerge when there are big differences between the stationary probabilities of the system states. These big differences are an effect of the potential strength, which minimizes the departures from the Markovianicity of the sequence of control actions, allowing also to minimize the departures from the optimal performance of the system. This procedure can be applied to other feedback ratchets and, more in general, to other control systems

Large Deviations in Fast-Slow Systems

The incidence of rare events in fast-slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior – such fluctuations are rare on short timescales but become ubiquitous eventually. This LDP involves an Hamilton-Jacobi equation whose Hamiltonian is related to the leading eigenvalue of the generator of the fast process, and is typically non-quadratic in the momenta – in other words, the LDP for the slow variables in fast-slow systems is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone. It is shown here that the eigenvalue problem for the Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables, and the fast variables enter quadratically the equation for the slow variables. These results are illustrated via examples, inspired by kinetic theories of turbulent flows and plasma, in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an SDE

Non-equilibrium statistical mechanics of the stochastic Navier–Stokes equations and geostrophic turbulence

International audienceTwo-dimensional and geophysical turbulent flows have the property to self organize and create large scale coherent jets and vortices. This is for instance one of the major processes for the dynamics of Earth's atmosphere. Following On-sager initial insight, based on conjugated works by mathematicians and physicists, this fundamental physical process has found some explanations in the framework of statistical mechanics. An important step, initiated twenty years ago, has been the study of the equilibrium statistical mechanics for the 2D Euler and the related quasi-geostrophic models (the Miller-Robert-Sommeria theory). Real geophysical and experimental flows are however dissipative and maintained by external forces. These lectures focus on recent theoretical development of the statistical mechanics of those non-equilibrium situations. Those progresses have been achieved using tools from field theory (path integrals and instantons), non-equilibrium statistical mechanics (large deviations, stochastic averaging). The aim of these lectures is to briefly introduce the theoretical aspects of this program in the simplest context: the 2D stochastic Euler or Navier-Stokes equations and the quasi-geostrophic equations. We review path integral representations of stochastic processes, large deviations for transition probabilities, action minimization, instanton theory, for general mechanical systems forced by random forces. We will apply this framework in order to predict equilibrium and non-equilibrium phase transitions for the 2D Euler, Navier-Stokes, and quasi-geostrophic dynamics, and to predict the rates of rare transitions between two attractors in situations of first order phase transitions. Kinetic theory of systems with long range interactions, both with and without stochastic external forces, are explained. Based on this kinetic theory, we predict non-equilibrium phase transitions, and discuss their recent experimental observations and numerical simulations. Even if the model we have considered so far are too simple academic models, the expected relevance of those approaches in the future for Earth atmosphere and climate dynamics is briefly discussed