Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states
aa r X i v : . [ phy s i c s . f l u - dyn ] A p r Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states
A. Naso, P.H. Chavanis and B. Dubrulle Laboratoire de Physique,Ecole Normale Sup´erieure de Lyon and CNRS (UMR 5672),46 all´ee d’Italie, 69007 Lyon, France Laboratoire de Physique Th´eorique (IRSAMC),CNRS and UPS, Universit´e de Toulouse,F-31062 Toulouse, France SPEC/IRAMIS/CEA Saclay, and CNRS (URA 2464),91191 Gif-sur-Yvette Cedex, France (Dated: To be included later)A simplified thermodynamic approach of the incompressible 2D Euler equation is consideredbased on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibriumstates are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these soleconstraints. We assume that these constraints are selected by properties of forcing and dissipation.We find that the vorticity fluctuations are Gaussian while the mean flow is characterized by alinear ω − ψ relationship. Furthermore, we prove that the maximization of entropy at fixed energy,circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophyat fixed energy and circulation. This provides a justification of the minimum enstrophy principlefrom statistical mechanics when only the microscopic enstrophy is conserved among the infinite classof Casimir constraints. Relaxation equations towards the statistical equilibrium state are derived.These equations can serve as numerical algorithms to determine maximum entropy or minimumenstrophy states. We use these relaxation equations to study geometry induced phase transitionsin rectangular domains. In particular, we illustrate with the relaxation equations the transitionbetween monopoles and dipoles predicted by Chavanis & Sommeria [J. Fluid. Mech. , 267(1996)]. We take into account stable as well as metastable states and show that metastable statesare robust and have negative specific heats. This is the first evidence of negative specific heats inthat context. We also argue that saddle points of entropy can be long-lived and play a role in thedynamics because the system may not spontaneously generate the perturbations that destabilizethem. PACS numbers: 05.20.-y Classical statistical mechanics - 05.45.-a Nonlinear dynamics and chaos - 05.90.+mOther topics in statistical physics, thermodynamics, and nonlinear dynamical systems - 47.10.-g Generaltheory in fluid dynamics - 47.15.ki Inviscid flows with vorticity - 47.20.-k Flow instabilities - 47.32.-y Vortexdynamics; rotating fluids
I. INTRODUCTION
Two-dimensional turbulence has the striking prop-erty of organizing spontaneously into large-scale coherentstructures. These coherent structures correspond to jetsand vortices in geophysical and astrophysical flows [1, 2].They can be reproduced in numerical simulations [3] andlaboratory experiments [4, 5] in either forced or unforcedsituations. These coherent structures involve the pres-ence of a mean flow and fluctuations around it. Themean flow turns out to be a steady state of the pure 2DEuler equations (without forcing and dissipation) at somecoarse-grained scale. In several cases, the steady state ischaracterized by a linear relationship between vorticity ω (or potential vorticity q = ω + h in the presence ofa topography h ) and stream function ψ . For example,the case of a linear q − ψ relationship was consideredearly by Fofonoff (1954) [6] as a simple model of oceaniccirculation. These “Fofonoff flows” were found to emergenaturally from random initial conditions in numerical ex-periments of forced and unforced 2D turbulence [7–11].However, these results are not expected to be general. There exists many other cases in 2D turbulence wherethe q − ψ relationship is not linear. The understandingand prediction of these quasi stationary states (QSS), inforced and unforced situations, is still a challenging prob-lem. Different approaches have been proposed to describethese QSSs.In the case of forced flows (for example oceanic flowsexperiencing a forcing by the wind and a dissipation), Ni-iler (1966) [12] and Marshall & Nurser (1986) [13] haveproposed that forcing and dissipation could equilibrateeach other in average and determine a QSS that is asteady state of the unforced and inviscid 2D Euler equa-tion. This QSS is characterized by a functional relation-ship q = f ( ψ ) between potential vorticity and streamfunction where the function f is selected by the proper-ties of forcing and dissipation. In particular, these au-thors discussed the properties that forcing must possessto generate Fofonoff flows.In the case of unforced flows, a phenomenological ap-proach, called the minimum enstrophy principle, was pro-posed by Bretherton & Haidvogel (1976) [14]. It is basedon the inverse cascade[55] process of Batchelor (1969)[15]. It is argued that, in the presence of a small vis-cosity, the (potential) enstrophy decays while the energyis approximately conserved. This is what Matthaeus &Montgomery (1980) [16] have called selective decay. Inthat case, we can expect that the system will relax to-wards a state that minimizes (potential) enstrophy atfixed energy. This is mainly a postulate. This principleleads to a steady state of the 2D Euler equation charac-terized by a linear q − ψ relationship. When applied togeophysical flows, this principle can give an alternativejustification of Fofonoff flows. The minimum enstrophyprinciple has been generalized by Leith (1984) [17] so asto take into account the conservation of angular momen-tum in order to describe isolated vortices. However, thisprinciple is purely phenomenological and it is difficult toestablish its domain of validity.A statistical mechanics approach of 2D turbulence hasbeen developed by Kraichnan (1967,1975) [18, 19] basedon the truncated Euler equations. In that case, the dy-namics conserves only the energy and the enstrophy (theother constraints of the Euler equation are lost by thetruncation). Since this system is Liouvillian, we can ap-ply the methods of statistical mechanics. This is theso-called energy-enstrophy statistical theory. In the ab-sence of topography, the truncated statistical mechanicspredicts a homogeneous flow with an equilibrium energyspectrum of the form E ( k ) = k/ ( a + bk ) correspond-ing to an equipartition distribution. In the presence ofa topography (or β -effect), this statistical mechanics ap-proach was generalized by Salmon, Holloway & Hender-shott (1976) [20]. It leads to a mean flow where the aver-age potential vorticity q and the average stream function ψ are related to each other by a linear relationship. Whenapplied to geophysical flows, this approach provides ajustification of Fofonoff flows from statistical mechanics.However, the fact that the truncated statistical mechan-ics breaks the conservation of some integrals of the 2DEuler equations may be considered as a limitation of thisapproach.Another statistical mechanics of 2D turbulence, basedon a point vortex approximation, was initiated by On-sager (1949) [21] and further developed by Joyce & Mont-gomery (1973) [22] and Lundgren & Pointin (1977) [23] ina mean field approximation. The statistical equilibriumstate maximizes the usual Boltzmann entropy (adaptedto point vortices) while conserving the energy and thenumber of vortices of each species. This statistical me-chanics predicts an equilibrium state where the relation-ship between vorticity and stream function is given bya Boltzmann distribution, or a superposition of Boltz-mann distributions (on the different species). However,the point vortex approximation is a crude approximationof real turbulent flows where the vorticity field is contin-uous.A statistical theory of 2D turbulence valid for contin-uous vorticity fields has been developed by Miller (1990)[24] and Robert & Sommeria (1991) [25]. This theorytakes into account all the constraints of the 2D Euler equation. The statistical equilibrium state maximizes amixing entropy while conserving energy, circulation andall the Casimirs. This leads to equilibrium states withmore general mean flows than in the previous approaches.In particular, the ω − ψ relationship and the fluctuationsaround it are determined by the initial conditions andcan take various shapes. However, some connections withthe earlier works can be found. For example, Robert &Sommeria (1991) [25] note that the results of the pointvortex approach can be recovered in the dilute limit oftheir statistical theory. On the other hand, Miller (1990)[24] notes that for specific initial conditions leading to aGaussian vorticity distribution at statistical equilibrium,the mean flow has a linear ω − ψ relationship similar tothat obtained from the minimum enstrophy principle. Fi-nally, Chavanis & Sommeria (1996) [26] consider a limitof strong mixing (or low energy) of the MRS theory inwhich βσψ ≪ c.g. = R ω d r atfixed energy and circulation. This justifies a form of invis-cid minimum enstrophy principle in a well-defined limitof the statistical theory[56]. This strong mixing limitalso makes a hierarchy among the Casimir constraints.To lowest order in the expansion βσψ ≪
1, leading toa linear ω − ψ relationship, only the circulation Γ andthe microscopic enstrophy Γ f.g. = R ω d r are important.To next orders, leading to nonlinear ω − ψ relationships,higher and higher moments Γ f.g.n = R ω n d r become rele-vant.Recently, an alternative statistical theory has been pro-posed by Ellis, Haven & Turkington (2002) [29] and fur-ther discussed by Chavanis (2005,2008) [30, 31] and Cha-vanis et al. (2010) [32]. These authors argue that, in realsituations where the flows are forced and dissipated atsmall scales, the conservation of all the constraints ofthe 2D Euler equation is abusive. They propose to keeponly the robust constraints (energy and circulation) andtreat the fragile constraints canonically. This amountsto prescribing a prior vorticity distribution instead of theCasimirs. This can be viewed as a grand microcanonicalversion of the Miller-Robert-Sommeria (MRS) theory. Inthe Ellis-Haven-Turkington (EHT) approach, the statis-tical equilibrium state maximizes a relative entropy (de-termined by the prior) while conserving energy and circu-lation. The mean flow turns out to maximize a general-ized entropy determined by the prior at fixed circulationand energy. For a Gaussian prior, the generalized entropyis proportional to minus the macroscopic enstrophy. Thisjustifies a minimum enstrophy principle from statisticalmechanics when the constraints are treated canonicallyand the prior is Gaussian. Furthermore, this approachallows to go beyond the minimum enstrophy principle byconsidering more complicated priors.We can give another interpretation of the EHT ap-proach. Indeed, Bouchet (2008) [33] notes that the EHTapproach provides a sufficient condition of MRS stabil-ity. Indeed, it is well-known in statistical mechanics andoptimization theory [34] that a solution of a maximiza-tion problem is always a solution of a more constraineddual maximization problem. Thus, grand microcanonicalstability (EHT) implies microcanonical stability (MRS).Therefore, if the mean flow maximizes a generalized en-tropy at fixed circulation and energy, then the corre-sponding Gibbs state is a MRS equilibrium state. How-ever, the reciprocal is wrong in case of ensemble inequiva-lence (between microcanonical and grand microcanonicalensembles) that is generic for systems with long-range in-teractions [34]. The mean flow associated to a MRS equi-librium state does not necessarily maximize a generalizedentropy at fixed circulation and energy. In this sense, theminimization of enstrophy at fixed circulation and energyprovides a sufficient, but not necessary, condition of MRSstability. A review of the connections between these dif-ferent variational principles has been given recently byChavanis (2009) [35].In this paper, we shall complement these different ap-proaches by considering a sort of intermediate situationbetween all these theories. We argue that, in most physi-cal situations, the system is forced and dissipated at smallscales. In some situations, forcing and dissipation equili-brate each other so that the system becomes, in average,statistically equivalent to the 2D Euler equation whereforcing and dissipation are “switched-off”. In particu-lar, a quasi stationary state (QSS) can form on a rela-tively short timescale. This QSS involves a mean flowand fluctuations arount it. We propose to describe thisstate in terms of the statistical mechanics of the 2D Eu-ler equation. However, we indirectly take into accountthe effects of forcing and dissipation in the choice of theconstraints. The energy and the circulation must be ob-viously conserved. By contrast, the conservation of allthe Casimir invariants is abusive and it is likely thatsome Casimir invariants will be destroyed by the forc-ing and the dissipation. We argue that some Casimirinvariants are more relevant than others and that theywill be selected by the properties of forcing and dissi-pation. In that case, the statistical equilibrium state isexpected to maximize the MRS entropy with these soleconstraints. It is not our goal here to determine howthese constraints are selected by the properties of forc-ing and dissipation. This is clearly a complicated prob-lem that has to be tackled by other methods. We shalljust use a heuristic approach and consider the situationwhere these relevant constraints are the circulation, theenergy and the microscopic enstrophy. In other words,among all the Casimir constraints, we only consider thequadratic one. We do not claim that this is the mostgeneral situation but simply that it is a case of physi-cal interest. We therefore consider the maximization ofthe MRS entropy at fixed circulation, energy and micro-scopic enstrophy and provide a detailed study of this (nontrivial) variational principle. Our maximization principleis similar in spirit to the approach of Kraichnan, which only considers the conservation of energy and enstrophy.However, we remain in physical space and work with theMRS entropy for the distribution of vorticity levels whileKraichnan works in Fourier space. Furthermor, we takeinto account the conservation of circulation.It may be noted that our approach is partly motivatedby results of experiments carried out in von Karman flows[36, 37]. In fact, the present paper, valid for the 2D Eu-ler equations, prepares the ground before considering themore complicated (but similar) case of 3D axisymmet-ric turbulence treated in [38]. In that later case, we showthe equivalence between maximum entropy states at fixedhelicity, angular momentum and microscopic energy andminimum macroscopic energy states at fixed helicity andangular momentum. Therefore, our two approaches areclosely related and provide a statistical basis for justify-ing the phenomenological minimum enstrophy (2D) andminimum energy (3D axisymmetric) principles [16].The paper is organized as follows.In Sec. II, we discuss some general properties of thesteady states of the 2D Euler equations. In particular, wemention the refined criterion of nonlinear dynamical sta-bility given by Ellis et al. [29] based on the maximizationof a pseudo entropy at fixed circulation and energy.In Sec. III, we recall the phenomenological minimumenstrophy principle based on selective viscous decay.In Sec. IV, we discuss the connection between maxi-mum entropy states and minimum enstrophy states. InSec. IV A, we consider the maximization of the MRS en-tropy at fixed energy, circulation and microscopic enstro-phy (energy-enstrophy-circulation statistical mechanics).This maximization problem leads to a statistical equilib-rium state with Gaussian fluctuations and a mean flowcharacterized by a linear ω − ψ relationship. In Sec.IV B, we introduce an equivalent, but simpler, maximiza-tion problem based on the maximization of an entropicfunctional of the coarse-grained vorticity at fixed circu-lation, energy and microscopic enstrophy. This gener-alized entropy has some similarities with the Renyi en-tropy [39, 40]. In Sec. IV C, we show the equivalence be-tween the maximization of entropy at fixed energy, circu-lation and microscopic enstrophy and the minimization ofmacroscopic enstrophy at fixed energy and circulation [57].Therefore, our simplified thermodynamic approach pro-vides a justification of the minimum enstrophy principle(and Fofonoff flows) from statistical mechanics when onlythe microscopic enstrophy is conserved among the infiniteclass of Casimir constraints.We also derive relaxation equations associated with thevarious maximization problems mentioned above (for thereader’s convenience, the formal derivation of these relax-ation equations is postponed to Appendix D). Associatedwith the basic variational problem, we derive a relax-ation equation for the vorticity distribution that increasesthe MRS entropy at fixed energy, circulation and micro-scopic enstrophy. In that case, the vorticity distribution ρ ( r , σ, t ) progressively becomes Gaussian and the meanflow ω ( r , t ) relaxes towards a steady state characterizedby a linear ω − ψ relationship. Associated with the simpli-fied variational problem, we derive a relaxation equationfor the mean flow that increases a generalized entropy(Renyi-like) while conserving energy, circulation and mi-croscopic enstrophy. In that case, the vorticity distribu-tion is always Gaussian, even in the out-of-equilibriumregime, with a uniform centered variance monotonicallyincreasing with time. Associated with the minimum en-strophy principle, we derive a relaxation equation forthe mean flow that dissipates the macroscopic enstrophywhile conserving energy and circulation. These relax-ation equations can serve as numerical algorithms to de-termine maximum entropy or minimum enstrophy stateswith appropriate constraints.In Sec. V, we study minimum enstrophy states atfixed energy and circulation in rectangular domains. Thisproblem was first considered by Chavanis & Sommeria(1996) [26] who report interesting phase transitions be-tween monopoles and dipoles depending on the geometryof the domain (e.g., the aspect ratio τ of a rectangu-lar domain) and on the value of the control parameterΓ /E . In particular, for Γ = 0, they report a transitionfrom a monopole to a dipole when the aspect ratio be-comes larger than τ c = 1 .
12. For Γ = 0 and τ < τ c ,the maximum entropy state is always a monopole andfor Γ = 0 and τ > τ c , the maximum entropy stateis a dipole for small Γ /E and a monopole for largeΓ /E . They also studied the metastability of the solu-tions (local entropy maxima) and the possible transitionbetween a direct and a reversed monopole. This work hasbeen followed recently in different directions. Venaille &Bouchet (2009) [41] have investigated in detail the natureof these phase transitions from the viewpoint of statisti-cal mechanics. In particular, they showed that the point(Γ = 0 , τ c = 1 .
12) corresponds to a bicritical point sepa-rating a microcanonical first order transition line to twosecond order transition lines. Keetels, Clercx & van Hei-jst (2009) [42] have studied minimum enstrophy statesin rectangular or circular domains with boundary condi-tions taking into account the effect of viscosity. Finally,Taylor, Borchardt & Helander (2009) [43] have shownthat the two types of solutions appearing in the studyof Chavanis & Sommeria [26] could explain the processof “spin-up” discovered by Clercx, Maassen & van Heijst(1998) [44]. In Sec. V, we recall and complete the mainresults of the approach of Chavanis & Sommeria [26] inorder to facilitate the discussion of the last section.In Sec. VI, we use the relaxation equations derived inAppendix D to illustrate the phase transitions describedin Sec. V. On the basis of the relaxation equations,we observe a persistence of unstable states that are sad-dle points of entropy. Therefore, we argue that unstablesaddle points of entropy may play a role in the dynamicsif the system does not spontaneously generate the per-turbations that can destabilize them. We also follow anhysteretic cycle as a function of the circulation wherethe hysteresis is due to the robustness of metastablestates (local entropy maxima). Finally, we briefly de- scribe the possibility of transitions between direct andreversed monopoles in the presence of stochastic forcing.Throughout this paper, we consider the simple caseof incompressible 2D flows without topography. How-ever, the main formalism of the theory can be general-ized straightforwardly to account for a topography (or a β -effect) by simply replacing the vorticity ω by the po-tential vorticity q = ω + h . Some applications will beconsidered in a companion paper [45]. We also assumethroughout the paper that the domain is of unit area A = 1. II. DYNAMICAL STABILITY OF STEADYSTATES OF THE 2D EULER EQUATION
We consider a two-dimensional incompressible and in-viscid flow described by the 2D Euler equations ∂ω∂t + u · ∇ ω = 0 , − ∆ ψ = ω, (1)where ω z = ∇ × u is the vorticity, ψ the stream functionand u the velocity field ( z is a unit vector normal to theflow). The 2D Euler equation admits an infinite numberof steady states of the form ω = f ( ψ ) , (2)where f is an arbitrary function. They are obtained bysolving the differential equation − ∆ ψ = f ( ψ ) , (3)with ψ = 0 on the domain boundary.To determine the dynamical stability of such flows, wecan make use of the conservation laws of the 2D Eulerequations. The 2D Euler equations conserve an infinitenumber of integral constraints that are the energy E = Z u d r = 12 Z ωψ d r , (4)and the Casimirs I h = Z h ( ω ) d r , (5)where h is an arbitrary function. In particular, all themoments of the vorticity Γ n = R ω n d r are conserved.The first moment Γ = R ω d r is the circulation and thesecond moment Γ = R ω d r is the enstrophy. Let usconsider a special class of Casimirs of the form S = − Z C ( ω ) d r , (6)where C is a convex function (i.e. C ′′ ≥ et al. [29]have shown that the maximization problemmax ω { S [ ω ] | E [ ω ] = E, Γ[ ω ] = Γ } , (7)determines a steady state of the 2D Euler equation that isnonlinearly dynamically stable. This provides a refinedcriterion of nonlinear dynamical stability. The criticalpoints of (7) are given by the variational principle δS − βδE − αδ Γ = 0 , (8)where β and α are Lagrange multipliers. This gives C ′ ( ω ) = − βψ − α ⇒ ω = F ( βψ + α ) , (9)where F ( x ) = ( C ′ ) − ( − x ). We note that ω ′ ( ψ ) = − β/C ′′ ( ω ) so that ω ( ψ ) is a monotonic function increas-ing for β < β >
0. This critical pointis a steady state of the 2D Euler equation. On the otherhand, it is a (local) maximum of the pseudo entropy atfixed energy and circulation iff − Z C ′′ ( ω )( δω ) d r − β Z δωδψ d r < , (10)for all perturbations δω that conserve energy and circu-lation at first order. In that case, it is formally non-linearly dynamically stable with respect to the 2D Eulerequations. This criterion is stronger than the well-knownArnol’d theorems that only provide sufficient conditionsof stability. We note, however, that the refined criterion(7) provides itself just a sufficient condition of nonlineardynamical stability. An even more refined criterion ofdynamical stability is given by the Kelvin-Arnol’d prin-ciple. A review of the connections between these differentstability criteria has been recently given by Chavanis [35]. III. THE MINIMUM ENSTROPHY PRINCIPLE
Let us consider the minimization of the enstrophy Γ = R ω d r at fixed circulation and energymin ω { Γ [ ω ] | E [ ω ] = E, Γ[ ω ] = Γ } . (11)The critical points are given by the variational principle δ Γ + 2 βδE + 2 αδ Γ = 0 , (12)where 2 β and 2 α are Lagrange multipliers (the factor 2has been introduced for compatibility with the results ofSec. IV C). This yields ω = − ∆ ψ = − βψ − α. (13)This is a steady state of the 2D Euler equation charac-terized by a linear ω − ψ relationship. On the other handit is a (local) minimum of enstrophy at fixed energy andcirculation iff Z ( δω ) d r + β Z δωδψ d r > , (14)for all perturbations δω that conserve energy and circu-lation at first order. There are several interpretations of the minimizationprinciple (11)[58]:(i) The minimum enstrophy principle was introducedin a phenomenological manner from a selective decayprinciple [14, 16, 17]. Due to a small viscosity, or othersource of dissipation, the enstrophy (fragile integral) isdissipated while the energy and the circulation (robustintegrals) are relatively well conserved[59]. It is then ar-gued that the system should reach a minimum enstrophystate at fixed circulation and energy. Note that there isno real justification for this last assumption as discussedin [27]. The enstrophy could decay without reaching itsminimum. Furthermore, the minimum potential enstro-phy principle is difficult to justify in terms of viscouseffects for the QG equations (see the Appendix of [45]).(ii) If we view Γ as a Casimir of the form (6), the mini-mization principle (11) is equivalent to the maximizationprinciple (7) for the pseudo entropy S = − Γ . In thiscontext, it determines a particular steady state of the2D Euler equation that is nonlinearly dynamically stableaccording to the refined stability criterion of Ellis et al. [29]. This provides another justification of the minimiza-tion problem (11) in relation to the inviscid 2D Eulerequation.(iii) For inviscid flows, the microscopic enstrophyΓ f.g. = R ω d r is conserved by the 2D Euler equationbut the macroscopic enstrophy Γ c.g. = R ω d r calculatedwith the coarse-grained vorticity decreases as enstrophyis lost in the fluctuations. Indeed, by Schwartz inequal-ity: Γ c.g. = R ω d r ≤ R ω d r = Γ f.g. . By contrast, theenergy E = R ωψ d r and the circulation Γ = R ω d r calculated with the coarse-grained vorticity are approx-imately conserved. This suggests an inviscid minimumenstrophy principle based on the minimization of macro-scopic enstrophy at fixed energy and circulation [26]. Inthis case, selective decay is due to the operation of coarse-graining, not viscosity.In the following, we shall discuss some connections be-tween the minimum enstrophy principle and the maxi-mum entropy principle. IV. CONNECTION BETWEEN MAXIMUMENTROPY STATES AND MINIMUMENSTROPHY STATESA. Energy-enstrophy-circulation statistical theory
Starting from a generically unstable or unsteady initialcondition, the 2D Euler equations are known to develop acomplicated mixing process leading ultimately to a quasistationary state (QSS), a vortex or a jet, on the coarse-grained scale. In order to describe this QSS and thefluctuations around it, we must introduce a probabilis-tic description. Let us introduce the density probability ρ ( r , σ ) of finding the vorticity level ω = σ at position r .Then, the local moments of vorticity are ω n = R ρσ n d r .In the statistical mechanics approach of Miller-Robert-Sommeria [24, 25], assuming that the system is strictlydescribed by the 2D Euler equation (no forcing and nodissipation), the statistical equilibrium state is expectedto maximize the mixing entropy S [ ρ ] = − Z ρ ln ρ d r dσ, (15)while conserving all the invariants (energy and Casimirs)of the 2D Euler equation. This forms the standard MRStheory.In the case of flows that are forced and dissipated atsmall scales, one may argue that forcing and dissipationwill compensate each other in average so that the systemwill again achieve a QSS that is a stationary solution ofthe 2D Euler equation. This QSS will be selected byforcing and dissipation. This fact is vindicated both inexperiments [36, 37] and numerical simulations [46]. Inorder to describe the fluctuations around this state, oneneeds to go one step further and obtain the vorticity dis-tribution. An idea is to keep the framework of the sta-tistical theory but argue that forcing and dissipation willalter the constraints. More precisely, forcing and dis-sipation will select some particular relevant constraintsamong all the invariants of the ideal 2D Euler equation.These constraints will determine the mean flow and thefluctuations around it. For example, we argue that thereexists physical situations in which only the conservationof energy, circulation and microscopic enstrophy are rele-vant (and of course the normalization condition). We donot claim that such situations are universal but simplythat they happen in some cases of physical interest. Thisseems to be the case for example in some oceanic situ-ations [6–20] (see also [47] for recent studies). We shalltherefore consider the maximization problemmax ρ { S [ ρ ] | E, Γ , Γ f.g. , Z ρdσ = 1 } , (16)where E = 12 Z ωψ d r = 12 Z ψρσ d r dσ, (17)Γ = Z ω d r = Z ρσ d r dσ, (18)Γ f.g. = Z ω d r = Z ρσ d r dσ. (19)The last constraint (19) will be called the microscopic (orfine-grained) enstrophy because it takes into account thefluctuations of the vorticity ω . It is different from themacroscopic (or coarse-grained) enstrophyΓ c.g. = Z ω d r . (20)which ignores these fluctuations. We haveΓ f.g. = Z ω d r + Γ c.g. , (21) where ω ≡ ω − ω is the local centered variance ofthe vorticity. The fluctuations of enstrophy are Γ fluct = R ω d r . In our terminology, the enstrophy will be called a fragile constraint because it cannot be expressed in termsof the coarse-grained field since ω = ω . While themicroscopic enstrophy Γ f.g. is conserved, the macroscopicenstrophy Γ c.g. is not conserved and decays. By contrast,the energy (17) and the circulation (18) will be called robust constraints because they can be expressed in termsof the coarse-grained fields. We shall come back to thisimportant distinction in Sec. IV C.The maximization problem (16) is what we shall call“the energy-enstrophy-circulation statistical theory”, orsimply, “the statistical theory” in this paper. We notethat a solution of (16) is always a MRS statistical equi-librium state, but the reciprocal is wrong in case of en-semble inequivalence because we have relaxed some con-straints (the other Casimirs). Here, we keep the robustconstraints E and Γ and only one fragile constraint Γ f.g. ,the quadratic one. We assume that these constraints areselected by the properties of forcing and dissipation. Aswe shall see, this assumption leads to Gaussian fluctua-tions and a mean flow characterized by a linear ω − ψ relationship. In principle, we can obtain more complexfluctuations and more complex mean flows (characterizedby nonlinear ω − ψ relationships) by keeping more andmore fine-grained moments Γ f.g.n> among the constraints.This can be a practical way to go beyond the Gaussianapproximation. However, the Gaussian approximation,leading to a linear ω − ψ relationship, is already an inter-esting problem presenting rich bifurcations (because theenergy constraint is nonlinear) [26], so that we shall stickto that situation.The critical points of (16) are solution of the varia-tional principle δS − βδE − αδ Γ − α δ Γ f.g. − Z ζ ( r ) δ (cid:18)Z ρdσ (cid:19) d r = 0 , (22)where β , α , α and ζ ( r ) are Lagrange multipliers. Thisyields the Gibbs state ρ ( r , σ ) = 1 Z ( r ) e − α σ e − ( βψ + α ) σ , (23)where the “partition function” is determined via the nor-malization condition Z ( r ) = Z e − α σ e − ( βψ + α ) σ dσ. (24)Therefore, in this approach, the distribution ρ ( r , σ ) ofthe fluctuations of vorticity is Gaussian and the centeredvariance of the vorticity ω ( r ) is uniform ω ≡ ω − ω = 12 α ≡ Ω . (25)On the other hand, the mean flow is given by ω = − Ω ( βψ + α ) . (26)This is a steady state of the 2D Euler equation charac-terized by a linear ω − ψ relationship. Then, the Gibbsstate can be rewritten ρ ( r , σ ) = 1 √ π Ω e − ( σ − ω )22Ω2 . (27)Since ω = Ω is uniform at statistical equilibrium, weget Γ f.g. = Ω + Γ c.g. . (28)Finally, a critical point of (16) is an entropy maximum at fixed E , Γ and Γ f.g. iff δ J ≡ − Z ( δρ ) ρ d r dσ − β Z δωδψ d r < δρ that conserve energy, circulation,microscopic enstrophy and normalization at first order(the proof is similar to the one given in [35] for relatedmaximization problems). Remark:
From Eqs. (15) and (27), we easily find that S = ln Ω . Then, using Eq. (28), we conclude that, atequilibrium, the entropy is given by S = 12 ln (cid:16) Γ f.g. − Γ c.g. (cid:17) . (30)Therefore, if there exists several local entropy maxima(metastable states) for the same values of the constraints E , Γ and Γ f.g. , the maximum entropy state is the onewith the smallest enstrophy Γ c.g. . This is a first re-sult showing connections between maximum entropy andminimum enstrophy principles. However, this equilib-rium result does not prove that the maximization of theentropy functional S [ ρ ] at fixed E , Γ and Γ f.g. is equiv-alent to the minimization of the enstrophy functionalΓ c.g. [ ω ] at fixed E and Γ (e.g. an entropy maximumcould be a saddle point of enstrophy). This equivalencewill be shown in Sec. IV C. Remark: the maximization problem (16) also arises inthe study of Kazantsev et al. [11] when the relaxationequations associated with the MRS statistical theory areclosed by using a Gaussian approximation (see [32] fordetails).
B. An equivalent but simpler variational principle
The maximization problem (16) is difficult to solve,especially regarding the stability condition (29), becausewe have to deal with a distribution ρ ( r , σ ). We shallintroduce here an equivalent but simpler maximizationproblem by “projecting” the distribution on a smallersubspace. To solve the maximization problem (16), wecan proceed in two steps[60]: (i) First step: We first maximize S at fixed E , Γ, Γ f.g. , R ρ dσ = 1 and a given vorticity profile ω ( r ) = R ρσ dσ . Since the specification of ω ( r ) determines E and Γ, thisis equivalent to maximizing S at fixed Γ f.g. , R ρ dσ = 1and ω ( r ) = R ρσ dσ . Writing the variational problem as δS − α δ Γ f.g. − Z λ ( r ) δ (cid:18)Z ρσ dσ (cid:19) d r − Z ζ ( r ) δ (cid:18)Z ρ dσ (cid:19) d r = 0 , (31)we obtain ρ ( r , σ ) = 1 √ π Ω e − ( σ − ω )22Ω2 , (32)with ω ≡ ω − ω = 12 α ≡ Ω . (33)Note that the centered variance of the vorticity ω ( r ) =Ω is uniform . Equation (33) also implies that α must be positive. We check that ρ is a global entropymaximum with the previous constraints since δ S = − R ( δρ ) ρ d r dσ < ρ so theirsecond variations vanish). Using the optimal distribution(32), we can express the entropy (15) in terms of ω writ-ing S [ ω ] ≡ S [ ρ ]. After straightforward calculations weobtain S = 12 ln Ω , (34)up to some unimportant constant terms. Note that Ω isdetermined by the constraint on the microscopic enstro-phy which leads toΩ = Γ f.g. − Z ω d r . (35)The second term is the macroscopic enstrophy associatedwith the coarse-grained flow: Γ c.g. = R ω d r . The rela-tion (35) can be used to express the entropy (34) in termsof ω alone. (ii) Second step: we now have to solve the maximiza-tion problem max ω { S [ ω ] | E, Γ , Γ f.g. } , (36)with S = 12 ln (cid:18) Γ f.g. − Z ω d r (cid:19) , (37) E = 12 Z ωψ d r , (38)Γ = Z ω d r . (39)The functional (37) might be called a generalized entropy.Interestingly, it resembles the Renyi entropy [39, 40]. (iii) Conclusion: finally, the solution of (16) is given byEq. (32) where ω is determined by (36). Therefore, (16)and (36) are equivalent but (36) is easier to solve becauseit is expressed in terms of ω ( r ) while (16) is expressed interms of ρ ( r , σ ).Up to second order, the variations of entropy (37) aregiven by∆ S = − (cid:18)Z ωδω d r + 12 Z ( δω ) d r (cid:19) − ) (cid:18)Z ωδω d r (cid:19) , (40)where Ω is given by Eq. (35). Considering the firstorder variations of the entropy, the critical points of (36)are determined by the variational problem δS − βδE − αδ Γ = 0 . (41)This yields ω = − Ω ( βψ + α ) . (42)We recover Eq. (26) for the mean flow. Combined withEq. (32), we recover the Gibbs state (23). Consideringnow the second order variations of the entropy (40), wefind that a critical point of (36) is a maximum of entropyat fixed energy, circulation and microscopic enstrophy iff − Z ( δω ) d r − β Z δωδψ d r − ) (cid:18)Z ωδω d r (cid:19) < , (43)for all perturbations δω that conserve circulation and en-ergy at first order (the conservation of microscopic en-strophy is automatically taken into account in our for-mulation). This stability condition is equivalent to Eq.(29) but much simpler because it depends only on theperturbation δω instead of the perturbation of the fulldistribution δρ . In fact, the stability condition (43) canbe simplified further. Indeed, using Eq. (42), we findthat the term in parenthesis can be written Z ωδω d r = − Ω Z ( βψ + α ) δω d r , (44)and it vanishes since the energy and the circulation areconserved at first order so that δE = R ψδω d r = 0 and δ Γ = R δω d r = 0. Therefore, a critical point of (36) isa maximum of entropy at fixed energy, circulation andmicroscopic enstrophy iff − Z ( δω ) d r − β Z δωδψ d r < , (45)for all perturbations δω that conserve circulation and en-ergy at first order. In fact, this stability condition canbe obtained more rapidly if we remark that the maxi-mization problem (36) is equivalent to the minimizationof the macroscopic enstrophy at fixed energy and circu-lation (see Sec. IV C). C. Equivalence with the minimum enstrophyprinciple
Since ln( x ) is a monotonically increasing function, itis clear that the maximization problem (36) is equivalentto max ω { S [ ω ] | E, Γ } , (46)with S = −
12 Γ c.g. , (47)Γ c.g. = Z ω d r , (48) E = 12 Z ωψ d r , (49)Γ = Z ω d r . (50)The functional S of the coarse-grained vorticity ω iscalled a “generalized entropy”. It is proportional to theopposite of the coarse-grained enstrophy. We have theequivalences (46) ⇔ (36) ⇔ (16) . (51)Therefore, the maximization of MRS entropy at fixed en-ergy, circulation and microscopic enstrophy is equivalentto the minimization of macroscopic enstrophy at fixedenergy and circulation. The solution of (16) is given byEq. (32) where ω is determined by (46) and Ω by Eq.(35). Therefore, (16) and (46) are equivalent but (46) iseasier to solve because it is expressed in terms of ω while(16) is expressed in terms of ρ . This provides a justifica-tion of the coarse-grained minimum enstrophy principlein terms of statistical mechanics when only the micro-scopic enstrophy is conserved among the Casimirs. Notethat, according to (7), the principle (46) also assures thatthe mean flow associated with the statistical equilibriumstate (16) is nonlinearly dynamically stable with respectto the 2D Euler equation.The critical points of (46) are given by the variationalproblem δS − βδE − αδ Γ = 0 . (52)This yields ω = − βψ − α. (53)This returns Eq. (26) for the mean flow (up to a trivialredefinition of β and α ). Together with Eq. (32), thisreturns the Gibbs state (23). On the other hand, thisstate is a maximum of S at fixed E and Γ iff − Z ( δω ) d r − β Z δωδψ d r < , (54)for all perturbations δω that conserve circulation and en-ergy at first order. This is equivalent to the criterion (45)as it should.We have thus shown the equivalence between the maxi-mization of MRS entropy at fixed energy, circulation andfine-grained enstrophy with the minimization of coarse-grained enstrophy at fixed energy and circulation. Thisequivalence has been shown here for global maximiza-tion. In Appendix A, we prove the equivalence for localmaximization by showing that the stability criteria (29)and (54) are equivalent. D. Equivalence with a grand microcanonicalensemble
In the basic maximization problem (16), the fine-grained enstrophy is treated as a constraint. Let us in-troduce a grand microcanonical ensemble by making aLegendre transform of the entropy with respect to thisfragile constraint Γ f.g. [35]. We thus introduce the func-tional S g = S − α Γ f.g. and the maximization problemmax ρ { S g [ ρ ] | E, Γ , Z ρdσ = 1 } . (55)A solution of (55) is always a solution of the more con-strained dual problem (16) but the reciprocal is wrong incase of “ensemble inequivalence” [35]. In the present case,however, we shall show that the microcanonical ensem-ble (16) and the grand microcanonical ensemble (55) areequivalent. This is because only a quadratic constraint(enstrophy) is involved.To solve the maximization problem (55) we can pro-ceed in two steps. We first maximize S g at fixed E , Γ, R ρ dσ = 1 and ω ( r ) = R ρσ dσ . This is equivalent tomaximizing S g at fixed R ρ dσ = 1 and ω ( r ) = R ρσ dσ ,and this leads to the optimal distribution (32) whereΩ = 1 / (2 α ) is now fixed. This is clearly the globalmaximum of S g with the previous constraints. Usingthis optimal distribution, we can now express the func-tional S g in terms of ω by writing S [ ω ] = S g [ ρ ]. Afterstraightforward calculations, we obtain S g = − Γ c.g. , (56)up to some constant terms (recall that Ω is a fixed pa-rameter in the present situation). In the second step, wehave to solve the maximization problemmax ω { S [ ω ] | E, Γ } . (57)Finally, the solution of (55) is given by Eq. (32) where ω is determined by (57). Therefore, the variational prin-ciple (55) is equivalent to (57). That this is true alsofor local maximization is shown in Appendix B of [32](in a more general situation). On the other hand, sinceΩ >
0, the maximization problem (57) is equivalent to (46). Since we have proven previously that (46) is equiv-alent to the microcanonical variational principle (16), weconclude that (16) and (55) are equivalent.
Remark: the grand microcanonical ensemble (55) cor-responds to the EHT approach with a Gaussian prior[29, 31, 32].
E. Connection between different variationalprinciples
Let us finally discuss the relationship between our ap-proach, Naso-Chavanis-Dubrulle (NCD), and the onesproposed by Miller-Robert-Sommeria (MRS) and Ellis-Haven-Turkington (EHT). To that purpose, we shallmake the connection between the variational principles[35]:(MRS) : max ρ { S [ ρ ] | E, Γ , Γ f.g.n> , Z ρdσ = 1 } , (58)(EHT) : max ρ { S χ [ ρ ] | E, Γ , Z ρdσ = 1 } , (59)(NCD) : max ρ { S [ ρ ] | E, Γ , Γ f.g. , Z ρdσ = 1 } . (60)(MaxS) : max ω { S [ ω ] | E, Γ } . (61)(MinΓ ) : min ω { Γ c.g. [ ω ] | E, Γ } , (62)where the functionals are S [ ρ ] = − Z ρ ( r , σ ) ln ρ ( r , σ ) d r dσ, (63) S χ [ ρ ] = − Z ρ ( r , σ ) ln (cid:20) ρ ( r , σ ) χ ( σ ) (cid:21) d r dσ, (64) S [ ω ] = − Z C ( ω ) d r , (65)Γ c.g. [ ω ] = Z ω d r , (66)with χ ( σ ) ≡ exp( − P n> α n σ n ) and C ( ω ) = − R ω [(ln ˆ χ ) ′ ] − ( − x ) dx where ˆ χ (Φ) ≡ R χ ( σ ) e − σ Φ dσ .In the framework of the MRS approach where all theCasimirs are conserved, the maximization of a “gener-alized entropy” S [ ω ] at fixed energy and circulation pro-vides a sufficient condition of MRS thermodynamical sta-bility [32, 33, 35]. However, the reciprocal is wrong incase of “ensemble inequivalence” between microcanonical0and grand microcanonical ensembles. Indeed, the coarse-grained vorticity field ω ( r ) associated with a MRS ther-modynamical equilibrium (i.e. a maximum of entropy S [ ρ ] at fixed energy, circulation and Casimirs) is not nec-essarily a maximum of generalized entropy S [ ω ] at fixedenergy and circulation (it can be a saddle point of gen-eralized entropy at fixed energy and circulation).In the framework of the EHT approach where the con-servation of the Casimirs is replaced by the specifica-tion of a prior vorticity distribution (i.e. the Casimirsare treated canonically), the maximization of a general-ized entropy S [ ω ] at fixed energy and circulation providesa necessary and sufficient condition of EHT thermody-namical stability [29, 31, 32]. Indeed, a vorticity dis-tribution ρ ( r , σ ) is a EHT thermodynamical equilibrium(i.e. a maximum of relative entropy S χ [ ρ ] at fixed energyand circulation) if and only if the corresponding coarse-grained vorticity field ω ( r ) is a maximum of generalizedentropy S [ ω ] at fixed energy and circulation.Thus, we symbolically have(MRS) ⇐ (EHT) ⇔ (MaxS) (67)Let us now specialize on the case of Gaussian distribu-tions.In the framework of the MRS approach where all theCasimirs are conserved, the minimization of macroscopicenstrophy Γ c.g. [ ω ] at fixed energy and circulation pro-vides a sufficient condition of MRS thermodynamical sta-bility for initial conditions leading to a Gaussian vorticitydistribution at equilibrium [32, 33, 35]. However, the re-ciprocal is wrong in case of “ensemble inequivalence”, i.e.the coarse-grained vorticity field ω ( r ) associated with aMRS thermodynamical equilibrium with Gaussian vor-ticity distribution is not necessarily a minimum enstro-phy state (it can be a saddle point of macroscopic enstro-phy at fixed energy and circulation).In the framework of the EHT approach where the con-servation of the Casimirs is replaced by the specifica-tion of a prior vorticity distribution, the minimizationof macroscopic enstrophy Γ c.g. [ ω ] at fixed energy andcirculation provides a necessary and sufficient conditionof EHT thermodynamical stability for a Gaussian prior[29, 31, 32], i.e. a vorticity distribution ρ ( r , σ ) is a EHTthermodynamical equilibrium with a Gaussian prior ifand only if the corresponding coarse-grained vorticityfield ω ( r ) is a minimum of macroscopic enstrophy Γ c.g. [ ω ]at fixed energy and circulation.In the framework of the NCD approach where onlythe microscopic enstrophy Γ f.g. [ ρ ] is conserved among theCasimir constraints, the minimization of macroscopic en-strophy Γ c.g. [ ω ] at fixed energy and circulation provides a necessary and sufficient condition of NCD thermodynam-ical stability. Indeed, a vorticity distribution ρ ( r , σ ) is aNCD thermodynamical equilibrium (i.e. a maximum ofentropy S [ ρ ] at fixed energy, circulation and microscopicenstrophy) if and only if the corresponding coarse-grainedvorticity field ω ( r ) is a minimum of macroscopic enstro-phy Γ c.g. [ ω ] at fixed energy and circulation. Thus, we symbolically have(MRS) ⇐ (EHT) ⇔ (MinΓ ) ⇔ (NCD) (68) Remark 1: the EHT and NCD approaches provide suf-ficient conditions of MRS stability. They are valuablein that respect as they are simpler to solve. They mayalso have a deeper physical meaning as discussed in theintroduction.
Remark 2: the equivalence between (EHT) for a Gaus-sian prior and (NCD) is essentially coincidental becausethese variational problems are physically very different.In particular, this agreement is only valid for a Gaussianprior and does not extend to more general cases.Finally, for completeness, we mention similar resultsobtained in [38] for axisymmetric flows. In the frame-work of the Naso-Monchaux-Chavanis-Dubrulle (NMCD)approach where only the microscopic energy E f.g. [ ρ ] isconserved among the Casimir constraints, the minimiza-tion of macroscopic energy E c.g. [ σ ] at fixed helicity andangular momentum provides a necessary and sufficient condition of NMCD thermodynamical stability, i.e. adistribution of angular momentum ρ ( r , η ) is a maximumof entropy S [ ρ ] at fixed helicity, angular momentum andmicroscopic energy E f.g. [ ρ ] if and only if the correspond-ing coarse-grained angular momentum distribution σ ( r )is a minimum of macroscopic energy E c.g. [ σ ] at fixed he-licity and angular momentum.Thus, we symbolically have(NMCD) ⇔ (Min E ) (69) V. PHASE TRANSITIONS IN 2D EULERFLOWSA. Minimum enstrophy states
In this section we study the maximization problemmax ω { S [ ω ] | E, Γ } , (70)where S = − R ω d r is the neg-enstrophy (the oppo-site of the enstrophy), E = R ωψ d r the energy andΓ = R ω d r the circulation. The maximization problem(70) can be interpreted as: (i) a criterion of nonlineardynamical stability with respect to the 2D Euler equa-tion (Sec. II), (ii) a phenomenological minimum enstro-phy principle (Sec. III), (iii) a sufficient condition ofMRS thermodynamical stability [33, 35], (iv) a neces-sary and sufficient condition of EHT thermodynamicalstability for a Gaussian prior [31, 32], (v) a necessaryand sufficient condition of thermodynamical stability inthe energy-enstrophy-circulation statistical theory whereonly the microscopic enstrophy is conserved among theCasimirs (Sec. IV). For simplicity and convenience, weshall call S the entropy.1We write the variational principle for the first ordervariations as δS − βδE − αδ Γ = 0 , (71)where β and α are Lagrange multipliers. This yields alinear ω − ψ relationship ω = − ∆ ψ = − βψ − α. (72)As before, we assume that the area of the domain is unityand we set h X i = R X d r . Taking the space average ofEq. (72), we obtain Γ = − β h ψ i − α so that the foregoingequation can be rewritten − ∆ ψ + βψ = Γ + β h ψ i , (73)with ψ = 0 on the domain boundary[61]. This is thefundamental differential equation of the problem. Theenergy and the entropy can then be expressed as E = − β (cid:0) h ψ i − h ψ i (cid:1) + 12 Γ h ψ i , (74) S = − β (cid:0) h ψ i − h ψ i (cid:1) −
12 Γ . (75)We shall study the maximization problem (70) by adapt-ing the approach of Chavanis & Sommeria [26] to thisspecific situation (these authors studied a related butnot exactly equivalent problem). We will see that thestructure of the problem depends on a unique controlparameter [26]: Λ = Γ √ E . (76)We note that the maximization problem (70) has beenstudied recently by Venaille & Bouchet [41] by using adifferent theoretical treatment. They performed a de-tailed analysis of the phase transitions associated with(70) in the context of statistical mechanics, emphasiz-ing in particular the notion of ensemble inequivalence.However, their approach is very abstract. Our study ismore direct and can offer a complementary discussion ofthe problem. The maximization problem (70) has alsobeen studied recently by Keetels et al. [42] with differentboundary conditions adapted to viscous flows.
B. The bifurcation diagram
In this section, we apply the methodology developedby Chavanis & Sommeria [26]. This methodology is rela-tively general: it is valid for an arbitrary domain and foran arbitrary linear operator. However, for illustration,we shall consider the Laplacian operator and a rectangu-lar domain.
1. The eigenmodes
We first assume thatΓ + β h ψ i = 0 , (77)corresponding to α = 0. In that case, the differentialequation (73) becomes − ∆ ψ + βψ = 0 , (78)with ψ = 0 on the domain boundary. Using the resultsof Appendix B, Eq. (78) has solutions only for β = β mn (eigenvalues) and the corresponding solutions (eigenfunc-tions) are ψ = (cid:18) E − β mn (cid:19) / ψ mn , (79)where we have used the energy constraint (74) to deter-mine the normalization constant. Substituting this resultin Eq. (77), we find that these solutions exist only forΛ = Λ mn with Λ mn = − β mn h ψ mn i . (80)For the eigenmodes h ψ mn i = 0 ( m or n even), we findΛ = 0 and for the eigenmodes h ψ mn i 6 = 0 ( m and n odd),we find Λ = Λ ′′ mn ≡ − β mn h ψ mn i = 0.
2. The solutions of the continuum
We now assume that Γ + β h ψ i 6 = 0 and we define φ = ψ Γ + β h ψ i . (81)In that case, the differential equation (73) becomes − ∆ φ + βφ = 1 , (82)with φ = 0 on the domain boundary. We also assumethat β = β mn . In that case, Eq. (82) has a uniquesolution that can be obtained by expanding φ on theeigenmodes. We get φ = X mn h ψ mn i β − β mn ψ mn , (83)where only the modes with h ψ mn i 6 = 0 are “excited”.For Γ = 0, taking the average of Eq. (81) and solvingfor h ψ i , we obtain h ψ i = Γ h φ i / (1 − β h φ i ). Therefore, thesolution of Eq. (73) is ψ = Γ φ − β h φ i . (84)Substituting this solution in the energy constraint (74),we obtain the “equation of state”:(1 − β h φ i ) = Λ ( h φ i − β h φ i ) . (85)2This equation determines β as a function of Λ. In par-ticular, it determines the caloric curve β ( E ) for a givenvalue of Γ = 0. Note that the equation of state involvesthe important function [26]: F ( β ) ≡ β h φ i − . (86)For Γ = 0, the solution of Eq. (73) is ψ = β h ψ i φ. (87)Taking the space average of this relation, we find thatthis solution exists only for a discrete set of temperatures β = β ( k ) ∗ satisfying F ( β ( k ) ∗ ) = 0. We shall note β ∗ ≡ β (1) ∗ the largest of these solutions. Substituting Eq. (87) inthe energy constraint (74), we find that the amplitude h ψ i is determined by E = − β h ψ i ( h φ i − h φ i ) . (88)Of course, the case Λ = 0 is also a limit case of theequation of state (85).
3. The mixed solutions
For β → β mn with m and n odd ( h ψ mn i 6 = 0), we findfrom Eq. (83) that φ ∼ h ψ mn i ψ mn / ( β − β mn ) → + ∞ leading to Λ → Λ ′′ mn and ψ → (2 E/β mn ) / ψ mn . There-fore, we recover the eigenfunction ψ mn as a limit case.The eigenfunctions with non vanishing average value aretherefore contained on the continuum branch.For β = β mn with m or n even ( h ψ mn i = 0), thesolution of Eq. (82) is not unique. It corresponds to themixed solutions φ = X m ′ n ′ h ψ m ′ n ′ i β mn − β m ′ n ′ ψ m ′ n ′ + χ mn ψ mn , (89)where χ mn is determined by the energy constraint (moreprecisely, it can be related to Λ by substituting Eq. (89)in Eq. (85) where now β = β mn ). These solutions form aplateau at fixed temperature β = β mn . For χ mn → + ∞ ,we recover the pure eigenmode ψ mn that exists at Λ = 0and for χ mn = 0, we connect the branch of continuumsolutions at Λ = Λ ′ mn .The general bifurcation diagram showing the eigen-modes, the solutions of the continuum and the mixedsolutions is represented in Fig. 2 of [26] (see also Figs. 1and 2 below). C. The geometry induced monopole/dipoletransition
For a given value of the control parameter Λ, we canhave an infinite number of solutions to Eq. (73) [26]. Wecan now use the entropy (75) to select the most probablestate (maximum entropy state) among all these solutions. For the eigenmodes, the entropy takes the simple form
S/E = β mn . In particular, for the eigenmodes ψ mn with m or n even ( h ψ mn i = 0) that exist only for Λ = 0, wehave SE (Λ = 0) = β mn . (90)For a rectangular domain elongated in the x direction,the eigenmode with the highest entropy at Γ = 0 is thedipole ( m, n ) = (2 ,
1) with temperature β ( τ ). There-fore, the maximum entropy state (or the minimum en-strophy state) corresponds to the mode with the largestscale. The modes with smaller scales ( m, n large) havelower entropy (higher enstrophy). Therefore, the max-imum entropy and minimum enstrophy principles se-lect the large-scale structures among the infinite classof steady states of the 2D Euler equation. This is a man-ifestation of the inverse cascade process.For the solutions of the continuum, the entropy can bewritten S/E = β (cid:18) h φ i β h φ i − (cid:19) − Λ . (91)For Λ = 0, this expression reduces to SE (Λ = 0) = β ( k ) ∗ . (92)The solution with highest entropy is the monopole withtemperature β ∗ ( τ ).For Λ = 0, we have a competition between themonopole β ∗ ( τ ) (continuum branch) and the dipole β ( τ ) (eigenmode)[62]. We must therefore compare theirentropy (or equivalently their inverse temperature) to se-lect the maximum entropy state. As shown in Chavanis& Sommeria [26], this selection depends on the geome-try of the domain. In a rectangular domain, it is foundthat the monopole has the highest entropy ( β ∗ > β )for τ < τ c = 1 .
12 while the dipole dominates ( β > β ∗ )for τ > τ c = 1 .
12. More generally, it can be shown thatthe entropy is a monotonically increasing function of theinverse temperature (for a fixed value of Λ). Therefore,at any Λ, the maximum entropy state is the one withthe highest inverse temperature [26]. The series of equi-libria β (Λ) is represented in Figs. 1 and 2 for a squaredomain and for a rectangular domain of aspect ratio 2,respectively. For τ < τ c , the maximum entropy state isthe direct monopole for any value of Λ. For τ > τ c , themaximum entropy state is the dipole for Λ < (Λ ′ ) andthe direct monopole for Λ > (Λ ′ ) . D. Stability analysis and ensemble inequivalence
For given values of E and Γ, there can exist differentcritical points of entropy S (canceling its first order vari-ations). They are solutions of the differential equation(73). For sufficiently small Λ, there exists an infinity of3 FIG. 1: Series of equilibria in a square domain ( τ = 1 < τ c ).In that case max { β ∗ , β } = β ∗ . The maximum entropy stateis the direct monopole (for Γ > < . For Λ < (Λ ′ ) ,the reversed monopole is metastable (local entropy maximum)as discussed in Sec. V D. Note that the metastable stateshave negative specific heats C = ∂E∂T = β E ∂ (1 /E ) ∂β < β ( E ) does not presentany phase transition (see Sec. V F). The vorticity profilesare plotted for Γ ≥
0. The red colour corresponds to positivevalues of the vorticity and the blue colour to negative values.FIG. 2: Series of equilibria in a rectangular domain with as-pect ratio τ = 2 > τ c . In that case max { β ∗ , β } = β . Themaximum entropy state is the dipole for Λ < (Λ ′ ) andthe direct monopole for Λ > (Λ ′ ) (the reversed monopolesare unstable). For Γ = 0, the caloric curve β ( E ) presents asecond order phase transition marked by the discontinuity of ∂β∂E ( E ) at E = E ′ as discussed in Sec. V F. solutions [26]. In the last section, we have compared thevalue of the entropy of these different solutions in order toselect the maximum entropy state. However, a more pre-cise study should determine which solutions correspondto global entropy maxima, local entropy maxima and sad-dle points. Saddle points of entropy are unstable andshould be rejected in principle (see, however Sec. VI B). By contrast, local entropy maxima (metastable states)can be long-lived for systems with long-range interac-tions. In practice, they are as much relevant as globalentropy maxima (stable states). In the following, usingan approach very close to the one followed by Chavanis& Sommeria [26] (but not exactly equivalent since thevariational problems differ), we determine sufficient con-ditions of instability . This will eliminate a large class ofsolutions that are unstable saddle points of entropy andgive the form of the perturbations that destabilize them.The remaining solutions are either stable or metastable.A critical point of entropy at fixed energy and circula-tion is a (local) maximum iff δ J = − Z ( δω ) d r − β Z δωδψ d r < , (93)for all perturbations that conserve energy and circulationat first order: δE = δ Γ = 0.(i) We first show that all the solutions with β < β are unstable (saddle points). To that purpose, we con-sider a perturbation of the form δω = ψ ( r ). The corre-sponding stream function is δψ = − β ψ ( r ). For thisperturbation, it is clear that δ Γ = R δω d r = 0 since h ψ i = 0. Furthermore, δE = R ψδω d r = 0 since ψ isorthogonal to the other eigenmodes and to the solutionsof the continuum (as they involve a summation (83) onthe eigenmodes with non zero average that are orthogo-nal to ψ ). Finally, a simple calculation shows that δ J = 12 (cid:18) ββ − (cid:19) > . (94)We have thus found a particular perturbation that in-creases the entropy at fixed energy and circulation.Therefore, the states with β < β are unstable (saddlepoints).(ii) We now show that if β < β ∗ the mode ψ exist-ing at Λ = 0 is unstable (saddle point). To that purpose,we consider a perturbation of the form δω = 1 − β ∗ φ ∗ (where φ ∗ is the solution of Eq. (82) corresponding to β = β ∗ ). The corresponding stream function is δψ = φ ∗ .For this perturbation, it is clear that δ Γ = R δω d r = 0since 1 − β ∗ h φ ∗ i = 0. Furthermore, δE = R ψ δω d r = 0since h ψ i = 0 and ψ is perpendicular to φ ∗ as ex-plained previously. Finally, after some simple algebrausing 1 − β ∗ h φ ∗ i = 0, we get δ J = 12 ( β ∗ − β )( h φ ∗ i − β ∗ h φ ∗ i ) > , (95)(the last term in parenthesis is positive as shown in Ap-pendix B). We have thus found a particular perturbationthat increases the entropy at fixed energy and circulation.Therefore, if β < β ∗ the mode ψ existing at Λ = 0is unstable. By continuity, the mixed solutions forminga plateau at β = β are also unstable if β < β ∗ sincethe two ends of the plateau are unstable.The maximization problem (70) corresponds to a con-dition of microcanonical stability which is relevant to our4problem since the circulation and the energy are con-served by the 2D Euler equation. However, it can beconvenient to establish criteria of canonical and grandcanonical stability. Indeed, the solution of a maximiza-tion problem is always solution of a more constraineddual maximization problem, but the reciprocal is wrongin case of ensemble inequivalence that is generic for sys-tems with long-range interactions [34]. Therefore, con-ditions of canonical and grand canonical stability pro-vide only sufficient conditions of microcanonical stabil-ity: grand canonical stability implies canonical stabil-ity which itself implies microcanonical stability (see, e.g.,[35]). This problem of ensemble inequivalence has beenstudied in detail by Venaille & Bouchet [41] and we brieflydiscuss it again bringing some complements regarding themetastable states (that are not considered in [41]).Considering the grand canonical ensemble, we have tomaximize the grand potential G = S − βE − α Γ (noconstraint problem). The condition of grand canonicalstability corresponds to inequality (93) for all variations δω . By decomposing the perturbation of the eigenmodesof the Laplacian, it is easy to show that the system isa maximum of grand potential iff β > β (where β isthe largest eigenvalue of the Laplacian). This is closelyrelated to the Arnol’d theorem (indeed, the grand po-tential is equivalent to the Arnol’d energy-Casimir func-tional [35]; furthermore, for a linear ω − ψ relationship,the Arnol’d theorem, which usually provides only a suffi-cient condition of grand canonical stability, now providesa necessary and sufficient condition of grand canonicalstability). Since grand canonical stability implies micro-canonical stability (but not the converse) we concludethat, if β > β , the system is a maximum of entropy atfixed circulation and energy.Considering the canonical ensemble, we have to max-imize the free energy J = S − βE at fixed circulation(one constraint problem). The condition of canonicalstability corresponds to inequality (93) for all variations δω that conserve circulation. By carefully taking intoaccount the constraint on the circulation, Venaille &Bouchet [41] show that the system is a maximum of freeenergy iff β > max { β , β ∗ } . In particular, the stateswith max { β , β ∗ } < β < β are stable in the canonicalensemble but unstable in the grand canonical ensemble.Thus, canonical and grand canonical ensembles are in-equivalent [41]. On the other hand, since canonical sta-bility implies microcanonical stability (but not the con-verse) we conclude that, if β > max { β , β ∗ } , the systemis a maximum of entropy at fixed circulation and energy.In particular, the states with E > Γ / (2Λ ) ≡ E (Γ)are stable in the canonical ensemble but unstable in thegrand canonical ensemble [41]. Note that the states with β < β ∗ are unstable in the canonical ensemble (they aresaddle points of free energy at fixed circulation). Thisresult can be obtained directly by considering a pertur-bation of the form δω = 1 − β ∗ φ ∗ like in (ii). For thisperturbation, δ Γ = 0. On the other hand, in the canon-ical ensemble, we do not need to impose δE = 0 so that this perturbation can be applied to any state leading to δ J = 12 ( β ∗ − β )( h φ ∗ i − β ∗ h φ ∗ i ) > , (96)which proves the result. By contrast, this argument doesnot work in the microcanonical ensemble since the cho-sen perturbation does not satisfy δE = 0 for all states.Therefore, when β ∗ > β , the states with β < β < β ∗ are unstable in the canonical ensemble (they are saddlepoints of free energy at fixed circulation) while they aremetastable in the microcanonical ensemble (they are lo-cal maxima of entropy at fixed circulation and energy).This is an interesting notion of ensemble inequivalencewhich affects metastable states (Venaille & Bouchet [41]show that the microcanonical and canonical ensemblesare equivalent for the fully stable states but the caseof metastable states is not considered in their study).In particular, we note that the metastable states with β < β < β ∗ have negative specific heats (see Fig. 1).This is allowed in the microcanonical ensemble but notin the canonical ensemble. Interestingly, this is the firstobservation of negative specific heats in that context.Combining all these results, we conclude that in themicrocanonical ensemble:(a) If β < β ∗ : the states are stable for β ≥ β ∗ ,unstable for β ≤ β and metastable for β < β < β ∗ , asshown in Fig. 1. Therefore, the direct monopole is stablefor any Λ and the reversed monopole is metastable forΛ < (Λ ′ ) .(b) If β > β ∗ : the states are stable for β ≥ β andunstable for β < β , as shown in Fig. 2. Therefore, thedipole is stable for Λ < (Λ ′ ) and the direct monopoleis stable for Λ > (Λ ′ ) . There is no metastable statein that case. E. The chemical potential
In Sec. V C, we have represented the inverse temper-ature β as a function of Λ. We shall now study how thechemical potential α depends on Λ. The chemical poten-tial is given by α = − β h ψ i − Γ. For the eigenmodes, α = 0 . (97)For the solutions of the continuum, assuming Γ = 0, andusing Eq. (84), we get α = Γ / ( β h φ i −
1) = Γ /F ( β ).Therefore, α √ E = Λ F ( β ) . (98)For Γ = 0, using Eq. (87), we obtain α √ E = ± p − β ∗ ( h φ ∗ i − h φ ∗ i ) , (99)which is a limit case of Eq. (98). The normalized chem-ical potential α/ √ E is plotted as a function of Λ in5Figs. 3 and 4, for a square domain and for a rectangulardomain of aspect ratio 2, respectively. To plot this curve,we have used Eqs. (85) and (98). For a given value of β , we can determine Λ by Eq. (85) and α/ √ E by Eq.(98). Therefore, we can obtain α/ √ E as a function ofΛ parameterized by β for the solutions of the continuum.For the mixed solutions, β = β mn is fixed and α/ √ E isa linear function of Λ given by Eq. (98). −6 −4 −2 0 2 4 6−10−8−6−4−20246810 Λ α / ( E ) / β * β β β * β β MPMPMN DD MN
FIG. 3: Relationship between α/ √ E and Λ in a square do-main (case β < β ∗ ). The solid lines correspond to the sta-ble ( β ≥ β ∗ ) and metastable ( β < β < β ∗ ) states. Un-stable states ( β ≤ β ) are represented by the dashed lines.The straight lines represent the mixed solutions with constanttemperature: β = β = β , β = β , β = β . −6 −4 −2 0 2 4 6−8−6−4−202468 Λ α / ( E ) / β β * β β β * β MPMN D
FIG. 4: Relationship between α/ √ E and Λ in a rectangu-lar domain of aspect ratio 2 (case β > β ∗ ). The solid linescorrespond to the stable states ( β ≥ β ). Unstable states( β < β ) are represented by the dashed lines. The straightlines represent the mixed solutions with constant tempera-ture: β = β , β = β , β = β . In Figs. 3 and 4, we have represented the series of equi-libria containing all the critical points of entropy. If wecontinue the series of equilibria to more and more un-stable states, the curve rolls up several times around theorigin (not shown). As indicated above, the series of equi- libria is parameterized by β . The branches correspondingto β > β are stable in the grand canonical, canonicaland microcanonical ensembles and the branches corre-sponding to β > max { β , β ∗ } are stable in the canonicaland microcanonical ensembles. The part of the branchescorresponding to max { β , β ∗ } < β < β are stable inthe canonical and microcanonical ensembles but not inthe grand canonical ensemble. For τ < τ c ( β < β ∗ ),the part of the branches corresponding to β < β < β ∗ are metastable in the microcanonical ensemble and un-stable in the other ensembles. Remark: in the grand canonical ensemble, the controlparameter is the chemical potential α and the conjugatedvariable is the circulation Γ. We must therefore considerΓ( α ) by rotating the curves of Figs. 3 and 4 by 90 o .Only the part of the curve with β > β (NW and SEquadrants) are stable in the grand canonical ensemble.There is a first order grand canonical phase transition at α = 0 marked by the discontinuity of the circulation Γ( α )between Γ = ± Λ ′′ √ E . Note that there is no metastablestates in the grand canonical ensemble because the stateswith β < β are all unstable. F. Description of phase transitions
We briefly discuss the nature of phase transitions asso-ciated with the maximization problem (70) and confirmthe results that Venaille & Bouchet [41] obtained by adifferent method. We also give a special attention to themetastable states that are not considered in [41].We shall describe successively the caloric curve β ( E )for a fixed Γ and the chemical potential curve α (Γ) for afixed E . As first observed by Chavanis & Sommeria [26],the nature of the phase transitions depends on the valueof max { β ∗ , β } . In a rectangular domain, this quantityis determined by the value of the aspect ratio τ . Wemust therefore consider two cases successively: τ < τ c and τ > τ c .
1. Caloric curve
The caloric curve corresponds to the stable part of theseries of equilibria β ( E ) containing global (stable) andlocal (metastable) maximum entropy states at fixed E and Γ. • Let us first consider τ < τ c corresponding tomax { β ∗ , β } = β ∗ as in Fig. 1. For Γ = 0, themaximum entropy state is the monopole and the caloriccurve is simply a straight line β ( E, Γ = 0) = β ∗ . Foreach value of the energy, we have two solutions with thesame inverse temperature β ∗ but different values of thechemical potential α (Γ = 0 , E ) = ± α (see Fig. 3).One solution is a monopole with positive vorticity atthe center (MP) and the other solution is a monopolewith negative vorticity at the center (MN). For Γ = 0,these solutions have the same entropy. Thus, the branch6 β ( E, Γ = 0) = β ∗ is degenerate. For Γ = 0, thecaloric curve β ( E, Γ = 0) can be deduced easily fromFig. 1[63]. The global maximum entropy state is thedirect monopole (for Γ > < E . For E > E ′ (Γ) ≡ Γ ′ ) ,the reversed monopole is metastable (local entropy max-imum). Note that the metastable states have negativespecific heats C ≡ dE/d (1 /β ) <
0. The caloric curve β ( E ) does not present any phase transition. • Let us now consider τ > τ c corresponding tomax { β ∗ , β } = β as in Fig. 2. For Γ = 0, themaximum entropy state is the dipole and the caloriccurve is simply a straight line β ( E, Γ = 0) = β . Foreach value of the energy, we have two solutions with thesame inverse temperature and the same chemical poten-tial α (Γ = 0 , E ) = 0 (see Fig. 4). One solution is adipole (+ , − ) with positive vorticity on the left and theother solution is a dipole ( − , +) with negative vortic-ity on the left (in Fig. 2, we have only represented thedipole ( − , +)). For Γ = 0, these solutions have the sameentropy. Thus, the branch β ( E, Γ = 0) = β is degen-erate. For Γ = 0, the caloric curve β ( E, Γ = 0) canbe deduced easily from Fig. 2. The maximum entropystate is the asymmetric (mixed) dipole (+ , − ) or ( − , +)for E > E ′ (Γ) ≡ Γ ′ ) and the direct monopole for E < E ′ (Γ) ≡ Γ ′ ) (the reversed monopoles are un-stable). The caloric curve β ( E ) presents a second orderphase transition marked by the discontinuity of ∂β∂E ( E )at E = E ′ (Γ).
2. Chemical potential curve
The chemical potential curve corresponds to the sta-ble part of the series of equilibria α (Γ) containing global(stable) and local (metastable) maximum entropy statesat fixed E and Γ. • Let us first consider τ < τ c corresponding tomax { β ∗ , β } = β ∗ as in Fig. 3. The global maximumentropy state is the monopole for any value of Γ. Con-sidering only fully stable states (global entropy maxima),there is a first order phase transition at Γ = 0 marked bythe discontinuity of α (Γ) while the entropy is continuous.When we pass from positive Γ to negative Γ, we pass dis-continuously (in terms of α but not in terms of β or S )from the monopole (MP) to the monopole (MN). In fact,due to the presence of long-lived metastable states (seeSec. VI C), we remain in practice on the monopole (MP)until the metastable branch disappears. Then we jumpon the monopole (MN) with discontinuity of α (and β and S ). This corresponds to a zeroth order phase tran-sition. • Let us now consider τ > τ c corresponding tomax { β ∗ , β } = β as in Fig. 4. The global max-imum entropy state is the asymmetric (mixed) dipole for | Γ | < Γ ′ ( E ) ≡ √ E Λ ′ and the direct monopolefor | Γ | > Γ ′ ( E ). There are two second order phasetransitions marked by the discontinuity of ∂α∂ Γ (Γ) at Γ = ± Γ ′ ( E ).
3. Phase diagram
The phase diagram in the ( τ, Λ) plane, including themetastable states, is plotted in Fig. 5. Depending onthe values of Λ and τ (and depending on the historyof the system in the zone of metastability), the maxi-mum entropy state is a dipole (D), a monopole (MP) ora monopole (MN). If we fix the circulation Γ, we obtainthe phase diagram in the ( τ, E ) plane. For Γ = 0, itshows the appearance of a second order phase transitionin β ( E ) for τ > τ c (for Γ = 0 there is no phase transition).If we fix the energy E , we obtain the phase diagram inthe ( τ, Γ) plane. As noted by Venaille & Bouchet [41],the point (Γ = 0 , τ = τ c ) is a bicritical point markingthe change from a first order to two second order phasetransitions in α (Γ). τ Λ MP(metastable) MN(metastable)
Monopole (MN)Monopole (MP)
Dipole(D) Λ ’ − Λ ’ FIG. 5: Phase diagram in the ( τ, Λ) plane showing the domainof stability of the direct monopoles and dipoles. We haveindicated by a dashed line the domain of metastability of thereversed monopoles.
Remark: for illustration, we have described the phasetransitions in the case of a rectangular domain and for theLaplacian operator. The generalization to an arbitrarydomain and a linear operator L is straightforward. Inthat case β is replaced by β ′ (the first eigenvalue of L with zero mean) and β is replaced by β ′′ (the firsteigenvalue of L with non zero mean). VI. RELAXATION TOWARDS MINIMUMENSTROPHY STATES
We shall now illustrate numerically the phase transi-tions discussed previously using the relaxation equationsintroduced in Appendix D. These relaxation equations7can serve as numerical algorithms to compute maximumentropy states or minimum enstrophy states with rele-vant constraints. Their study is also interesting in itsown right since these equations constitute non trivial dy-namical systems leading to rich bifurcations. Althoughthese relaxation equations do not provide a parametriza-tion of 2D turbulence (we have no rigorous argument forthat), they may however give an idea of the true evolutionof the flow towards equilibrium. In that respect, it wouldbe interesting to compare these relaxation equations withlarge eddy simulations (LES) of 2D turbulence. This will,however, not be attempted in the present paper.
A. Relaxation equations
We shall numerically solve the relaxation equation ofSec. D 3. For simplicity, we shall ignore the advectiveterm since we are just interested in describing the bifur-cations between the different equilibrium states. Then,by a proper rescaling of time, we can take D = 1 with-out loss of generality. The relaxation equation (D43)becomes ∂ω∂t = − ( ω + β ( t ) ψ + α ( t )) , (100)with the boundary condition ω = − α ( t ) on the edgeof the domain. The Lagrange multipliers β ( t ) and α ( t )evolve in time according to Eqs. (D44) and (D45) in or-der to conserve the circulation and the energy. This leadsto β ( t ) = Γ h ψ i − E h ψ i − h ψ i , (101) α ( t ) = − Γ h ψ i − E h ψ ih ψ i − h ψ i . (102)The rate of increase of entropy (neg-enstrophy) is˙ S = Z ( ω + βψ + α ) d r ≥ . (103)Therefore, the relaxation equation (100) with the con-straints (101) and (102) relaxes towards the maximumentropy state at fixed circulation and energy. Saddlepoints of entropy are linearly unstable to some pertur-bations (in particular those described in Sec. V D). B. Geometry induced phase transitions andpersistence of saddle points
We first consider the case of a square domain ( τ = 1 <τ c ) and take Γ = 0. For these values of parameters, therelaxation equation (100) admits an infinite number ofsteady states that are the solutions of Eq. (73). How-ever, the only stable solution is the monopole with inverse temperature β ∗ . It is the maximum entropy state at fixedcirculation and energy. In fact, for Γ = 0, this solutionis degenerate since the monopoles (MP) and (MN) havethe same entropy.Let us confront these theoretical results to a direct nu-merical simulation of Eq. (100). Starting from a genericinitial condition (made of Gaussian peaks with positiveand negative vorticity symmetrically distributed in thedomain to assure Γ = 0), we numerically find that thesystem spontaneously relaxes towards the dipole and re-mains in that state for a long time (see Fig. 6) althoughthis state is predicted to be unstable (see Sec. V C). Thissimple numerical experiment shows that unstable statescan be long-lived. In fact, the dipole is a saddle point ofentropy so that it is unstable only for very specific pertur-bations. If these perturbations are not generated spon-taneously during the relaxation process, the system canremain frozen in a saddle point for a long time. Anotherreason why the dipole has a long lifetime is due to the factthat the entropies of the monopole (stable) and dipole(unstable) are very close for Γ = 0 since β ∗ ≈ − . β ≈ − .
3. To check that the dipole is reallyunstable, we have introduced by hands (see the arrow inFig. 6) an optimal perturbation of the form δω = 1 − β ∗ φ ∗ (see Sec. V D). In that case, the dipole is immediatelydestabilized and the system quickly relaxes towards themonopole which is the maximum entropy state in thatcase. In the case shown in Fig. 6, we obtain a monopole(MP). If we introduce an optimal perturbation with theopposite sign, we get the monopole (MN). If we do notintroduce any perturbation by hand and just let the sys-tem evolve with the numerical noise, the dipole finallydestabilizes but this takes a long time (not shown) of theorder t ∼ FIG. 6: Starting from a generic initial condition with Γ = 0in a square domain, the system relaxes towards a dipole (firstplateau) although this solution is unstable (saddle point). At t = 20 (see arrow), an optimal perturbation is applied to thedipole which quickly destabilizes in a stable monopole (secondplateau). In the absence of optimal perturbation, the systemcan remain frozen in the dipole for a long time. We now consider a rectangular domain with aspect ra-8tio τ = 2 > τ c and again take Γ = 0. In that case, themaximum entropy state at fixed circulation and energy isthe dipole and the monopole is unstable (saddle point).Starting from a generic initial condition, the systemspontaneously relaxes towards the dipole and remains inthis state even if very large perturbations are applied (notshown). By contrast, if we start from the monopole, wenumerically observe that the system remains in that statefor a very long time although this state is unstable (seeSec. V C). If we apply by hands (see the arrow in Fig. 7)an optimal perturbation of the form δω = ψ (see Sec.V D), the monopole is immediately destabilized and thesystem quickly evolves towards the dipole (Fig. 7) whichis the maximum entropy state in that case. In the ab-sence of applied perturbation, we have not observed thedestabilization of the monopole on the timescale achievedin the numerical experiment (however, if we add the ad-vection term, the dipole is formed on a time of the order t ∼ FIG. 7: Starting from a monopole with Γ = 0 in a rectangu-lar domain ( τ > τ c ), the system remains in that state for along time (first plateau) although this state is unstable (sad-dle point). At t = 10 (see arrow), an optimal perturbationis applied to the monopole which quickly relaxes towards adipole (second plateau). In the absence of optimal perturba-tion, the system can remain frozen in the monopole state fora long time. In conclusion, this numerical study reveals that evenunstable states (saddle points of entropy) can be natu-rally selected by the system and persist for a long time.Indeed, these states are destabilized by a very partic-ular type of perturbations (that we call optimal) andsuch perturbations may not be necessarily generated bythe internal dynamics of the system. This suggests thatthe system can be frozen for a long time in a quasi sta-tionary state (QSS) that is not necessarily a stable ormetastable steady state of the 2D Euler equation. Itcan even be an unstable saddle point! This observationhas been made on the basis of the relaxation equationsthat are constructed so as to relax towards a maximumentropy state. However, the same phenomenon couldappear for real flows described by the Euler or Navier- Stokes equations in numerical simulations and laboratoryexperiments. This could be interesting to study in moredetail.
C. Metastability and hysteresis
We shall now describe the hysteretic cycle predictedby statistical mechanics (based on the neg-enstrophy) ina domain with aspect ratio τ < τ c . In Fig. 8, we plotthe entropy S/E as a function of the control parameterΛ. We shall assume that the energy E is fixed so thatΛ basically represents the circulation Γ. The hysteresisis due to the presence of metastable states (local entropymaxima) when − Λ ′ < Λ < Λ ′ . For 0 < Λ < Λ ′ , theglobal maximum entropy state is the direct monopole(MP) while the reversed monopole (MN) is metastable.For − Λ ′ < Λ <
0, the global maximum entropy state isthe direct monopole (MN) while the reversed monopole(MP) is metastable. Depending on how it has been pre-pared initially, the system can be found in the stable ormetastable state.
FIG. 8:
S/E ratio as a function of Λ in a square domain.
We start from a state with large Λ corresponding topositive temperature ( β > β < >
0, the (global) maxi-mum entropy state is the monopole (MP). For Λ = 0, weexpect a first order phase transition from the monopole(MP) to the monopole (MN) (see Sec. V C) marked bythe discontinuity of the chemical potential α (while β and S are still continuous). In fact, for − Λ ′ < Λ ≤
0, themonopole (MP) is metastable and robust so that the sys-tem remains on this branch. Therefore, in practice, thefirst order phase transition does not take place. How-ever, for Λ < − Λ ′ , the branch of monopoles (MP) be-comes unstable and the system jumps to the branch ofdirect monopoles (MN) which correspond to global en-9tropy maxima. This is marked by a discontinuity of en-tropy (zeroth order phase transition). If we decrease Λsufficiently, we enter in the region of positive tempera-ture states ( β > < < Λ < Λ ′ . Again, the first order phasetransition at Γ = 0 does not take place. For Λ > Λ ′ ,the branch of monopoles (MN) becomes unstable and thesystem jumps to the branch of direct monopoles (MP)which correspond to global entropy maxima. We havethus followed an hysteretic cycle as illustrated in Figs. 8and 9. FIG. 9: Hysteretic cycle in a square domain, obtained by nu-merical integration of Eq. (100). We have represented Λ(black) and β (red) as a function of time. Starting froma stable state with Λ ∈ [0; Λ ′ ] (MP), the system is regu-larly perturbed: at t = 10 , , , , , . ψ , and let the system re-lax. The effect of the Gaussian peak is to decrease Λ,while the eigenmode destabilizes the unstable states. For0 < Λ < Λ ′ we follow the stable branch (MP) of Fig. 8 andfor − Λ ′ < Λ <
0, we follow the metastable branch (MP).For Λ < − Λ ′ , the metastable solutions (MP) no longer ex-ist, and the system jumps to the upper branch (MN) of Fig.8. At t = 1450 , , , , . ψ . The value of Λ is then increased, andwe follow the stable branch (MN) for Λ ∈ [ − Λ ′ ; 0] and themetastable branch (MN) for Λ ∈ [0; Λ ′ ]. When Λ > Λ ′ , themetastable solutions (MN) no longer exists, and the systemjumps to the upper branch (MP) of Fig. 8. D. Bifurcations in the presence of a noise
For Γ = 0 in a square domain, the monopoles (MP)and (MN) are stable and have the same entropy but re-main quite distinct states (with opposite velocity). Thiscorresponds to a parity breaking for the final organiza-tion of the system [26]. In the presence of forcing, we expect to observe random transitions between these twosolutions[64] similar to those observed experimentally bySommeria [49] for 2D turbulence forced at small scale ina square box. Indeed, we are in a situation similar to thecase of a bistable system. To observe such transitions,one possibility is to introduce a stochastic noise in therelaxation equation (100). Unfortunately, for a simplewhite noise, we did not observe any transition and wehave not been able to find the properties of forcing thatallow such transitions to appear. This may be due to thehigh entropic barriere created by the unstable (dipole)solution. Therefore, in order to illustrate the main idea,we shall introduce a simple effective model.The relevant order parameter is the chemical potential α which takes the values ± α for the (stable) monopoles(MP) and (MN) and the value α = 0 for the (unsta-ble) dipole (see Fig. 3). We shall now introduce anentropic function S ( α ) modeled by a symmetric func-tion with three bumps (two maxima and one minimum).Since we know the entropy (by unit of energy) of themonopoles S monopoles = β ∗ and the entropy of the dipole S dipole = β , we find that S ( α ) = ( β − β ∗ ) " − (cid:18) αα (cid:19) + β ∗ . (104)When a forcing is present, we can propose that α becomesa stochastic variable described by a Langevin equation ofthe form dαdt = µS ′ ( α ) + √ Dη ( t ) , (105)where η ( t ) is a white noise. In the absence of forcing,Eq. (105) relaxes towards one maximum of S ( α ), themonopole (MP) or the monopole (MN), and stay therepermanently. In the presence of forcing, Eq. (105) de-scribes random transitions between these two states (seeFig. 10). This is the classical bistable system that hasbeen studied at length in statistical mechanics and Brow-nian theory [50].Random transitions have been observed in variousphysical systems in fluid mechanics (see, e.g., [46, 49, 51]and references therein). In the present study, we haveconsidered random transitions between a monopole (MP)and a reversed monopole (MN). They are associated withthe first order phase transition that takes place in asquare domain when the QSS has a linear ω − ψ rela-tionship. It would be interesting to see if they can be ob-tained directly from the forced Navier-Stokes equationsin situations where the ω − ψ relationship is close to lin-ear. Random transitions between a unidirectional flowand a dipole have been obtained recently by Bouchet &Simonnet [46] by solving numerically the forced Navier-Stokes equations in periodic domain. However, the situ-ation is different (and more complex) because these twostates are characterized by different ω − ψ relationships.Indeed, the forcing can change the shape of ω ( ψ ). In thesituation that we consider, the shape of ω ( ψ ) remains the0 FIG. 10: Solution of the stochastic equation (105) for µ =1 . D = 1 .
25 showing random transitions between themonopoles (MP) and (MN). The dipole is always unstable. same (linear) but the equation ∆ ψ = − ω ( ψ ) determiningthe QSS can admit two stable solutions (MP) and (MN).This situations is closer to that of a bistable system andwould be interesting to study numerically. VII. CONCLUSION
In this paper, we have studied the maximization ofthe Miller-Robert-Sommeria entropy S MRS at fixed en-ergy E , circulation Γ and microscopic enstrophy Γ f.g. and proved the equivalence with the minimization of themacroscopic enstrophy Γ c.g. at fixed energy E and cir-culation Γ. This provides a justification of the minimumenstrophy principle from statistical theory when only themicroscopic enstrophy is conserved among all the Casimirinvariants. We have suggested that relevant constraints(such as the microscopic enstrophy) are selected by theproperties of forcing and dissipation. Our simplified ther-modynamic approach leads to a mean flow characterizedby a linear ω − ψ relationship and Gaussian fluctuationsaround it. Such states can be relevant to describe certainoceanic flows [6–20]. More general flows with nonlinear ω − ψ relationships (and more general fluctuations) canbe constructed in principle by keeping other Casimir con-straints in addition to the microscopic enstrophy.We have studied the minimization of enstrophy at fixedenergy and circulation and analyzed the correspondingphase transitions with the approach of Chavanis & Som-meria [26]. We have discussed the link with the approachof Venaille & Bouchet [41]. We have proposed relaxationequations to solve this minimization problem (see [35]for generalizations) and used them to illustrate the phasetransitions.One interesting result of the simulations is the observa-tion that saddle points of entropy can be relevant in thedynamics. Indeed, these states are unstable only for par- ticular perturbations that are not necessarily generatedspontaneously by the system. As a result, they can belong-lived and robust. This observation may have inter-esting application in the case of von K´arm´an flows sinceit is found that Beltrami states are saddle points of en-ergy at fixed helicity, not energy minima [38]. Still, it isobserved experimentally [36, 37] that they are long-livedand robust.We have also discussed in detail the metastable statesthat were not considered in the study of Venaille &Bouchet [41]. For long-range interactions, metastablestates (local entropy maxima) are long-lived and theyare as much important as fully stable states (global en-tropy maxima). Interestingly, these metastable stateshave negative specific heats leading to a form of ensem-ble inequivalence between microcanonical and canonicalensembles (while these ensembles are equivalent at thelevel of fully stable states [41]). These metastable statescan lead to an hysteresis and to random transitions be-tween direct monopoles and reversed monopoles. Suchtransitions can also arise in more realistic fluid systemsand can have some importance in oceanography and me-teorology [46, 49, 51].A last remark may be in order. The MRS statisticaltheory of the 2D Euler equation, which is the most ba-sic and the most rigorous, takes into account an infinitenumber of constraints. When applied to real flows, thisis clearly unphysical and this leads to practical difficul-ties. It has been a subject of intense debate in the last 20years to find a practical way to deal with the constraints.Different approaches have been proposed: some considera point vortex approximation where only the energy andthe number of vortices in each species matter [22], someconsider since the start only a finite number of inviscidconstraints [18–20], some consider a strong mixing (orlow energy) limit of the MRS statistical theory whichmakes a hierarchy among the Casimir constraints [26],and some model the vorticity fluctuations by a prior dis-tribution [29–31]. In our recent works [32, 35], includingthe present one, we have not tried to determine whichapproach, if any, is the “best”. For the moment, wejust present different ways to deal with the constraintsand systematically study the corresponding variationalprinciples. We have also extended these variational prin-ciples to 3D axisymmetric flows [38]. These variationalprinciples have a long history in 2D turbulence and MHDand one virtue of our papers is to put several variationalprinciples in correspondance. The determination of the“best” approach is still a matter of debate and research. Appendix A: Equivalence between (29) and (54)
In Sec. IV, we have shown the equivalence of (16) and(46) for global maximization. In this Appendix, we showthe equivalence of (16) and (46) for local maximization,i.e. ρ ( r , σ ) is a (local) maximum of S [ ρ ] at fixed E , Γ,Γ f.g. and normalization if, and only if, the correspond-1ing coarse-grained vorticity ω ( r ) is a (local) minimumof Γ c.g. [ ω ] at fixed E and Γ. To that purpose, we showthe equivalence between the stability criteria (29) and(54). We use a general method similar to the one usedin [32, 52, 53] in related problems.We shall determine the optimal perturbation δρ ∗ ( r , σ )that maximizes δ J [ δρ ] given by Eq. (29) with the con-straints δω = R δρσ dσ , δ Γ f.g. = R δρσ dσd r = 0 and R δρ dσ = 0, where δω ( r ) is prescribed (it is only as-cribed to conserve circulation and energy at first order).Since the specification of δω determines δψ , hence thesecond integral in Eq. (29), we can write the variationalproblem in the form δ (cid:18) − Z ( δρ ) ρ d r dσ (cid:19) − Z λ ( r ) δ (cid:18)Z δρσ dσ (cid:19) d r − µδ (cid:18)Z δρσ dσd r (cid:19) − Z ζ ( r ) δ (cid:18)Z δρ dσ (cid:19) d r = 0 , (A1)where λ ( r ), µ and ζ ( r ) are Lagrange multipliers. Thisgives δρ ∗ ( r , σ ) = − ρ ( r , σ )( µσ + λ ( r ) σ + ζ ( r )) , (A2)and it is a global maximum of δ J [ δρ ] with the previ-ous constraints since δ ( δ J ) = − R ( δ ( δρ )) ρ d r dσ < δρ so their second varia-tions vanish). The Lagrange multipliers are determinedfrom the above-mentioned constraints. The constraints R δρ dσ = 0 and δω = R δρσ dσ lead to ζ ( r ) + λ ( r ) ω ( r ) + µω ( r ) = 0 , (A3) ζ ( r ) ω ( r ) + λ ( r ) ω ( r ) + µω ( r ) = − δω ( r ) . (A4)Now, the state ρ ( r , σ ) corresponds to the Gaussian distri-bution (27). Therefore, we have the well-known relations ω ( r ) = ω ( r ) + ω and ω ( r ) = ω ( r ) + 3 ω ( r ) ω where ω = Ω is uniform. Substituting these relations in Eqs.(A3) and (A4), and solving for λ ( r ) and ζ ( r ), we obtain λ ( r ) = − δω ( r ) ω − µω ( r ) , (A5) ζ ( r ) = ω ( r ) ω δω ( r ) + µω ( r ) − µω . (A6)Therefore, the optimal perturbation (A2) can be rewrit-ten δρ ∗ = − ρ (cid:20) − δωω ( σ − ω ) + µ (cid:8) ( σ − ω ) − ω (cid:9)(cid:21) . (A7)The Lagrange multiplier µ is determined by substitutingthis expression in the constraint R δρσ d r dσ = 0. Usingthe well-known identity ω ( r ) = ω ( r ) + 6 ω ω ( r ) + 3 ω valid for a Gaussian distribution, we obtain after somesimplifications µ = R ωδω d r ω . (A8)Therefore, the optimal perturbation (A2) is given by Eq.(A7) with Eq. (A8). Since this perturbations maximizes δ J [ δρ ] with the above-mentioned constraints, we have δ J [ δρ ] ≤ δ J [ δρ ∗ ]. Explicating δ J [ δρ ∗ ] using Eqs. (A7)and (A8), we obtain after simple calculations δ J [ δρ ] ≤ − ω Z ( δω ) d r − ω (cid:18)Z ωδω d r (cid:19) − β Z δωδψ d r . (A9)The r.h.s. returns the functional appearing in Eq. (43).We have already explained in Sec. IV B that for theclass of perturbations that we consider ( δ Γ = δE = 0)the second integral vanishes. Therefore, the foregoinginequality can be rewritten δ J [ δρ ] ≤ − ω Z ( δω ) d r − β Z δωδψ d r , (A10)where the r.h.s. is precisely the functional appearing inEq. (54). Furthermore, there is equality in Eq. (A10) iff δρ = δρ ∗ . This proves that the stability criteria (29) and(54) are equivalent. Indeed: (i) if inequality (54) is ful-filled for all perturbations δω that conserves circulationand energy at first order, then according to Eq. (A10),we know that inequality (29) is fulfilled for all pertur-bation δρ that conserves circulation, energy, fine-grainedenstrophy and normalization at first order; (ii) if thereexists a perturbation δω ∗ that makes δ J [ δω ] >
0, thenthe perturbation δρ ∗ given by Eq. (A7) with Eq. (A8)and δω = δω ∗ makes δ J [ δρ ] >
0. In conclusion, thestability criteria (29) and (54) are equivalent.
Appendix B: Eigenvalues and eigenfunctions of theLaplacian in a rectangular domain
We define the eigenfunctions and eigenvalues of theLaplacian by ∆ ψ n = β n ψ n , (B1)with ψ n = 0 on the domain boundary. These eigenfunc-tions are orthogonal and normalized so that h ψ n ψ m i = δ nm . Since − R ( ∇ ψ n ) d r = β n R ψ n d r , we note that β n <
0. Following Chavanis & Sommeria [26], we dis-tinguish two types of eigenmodes: the odd eigenmodes ψ ′ n such that h ψ ′ n i = 0 and the even eigenmodes ψ ′′ n suchthat h ψ ′′ n i 6 = 0. We note β ′ n and β ′′ n the correspondingeigenvalues.In a rectangular domain of unit area whose sides aredenoted a = √ τ and b = 1 / √ τ (where τ = a/b is theaspect ratio), the eigenmodes and eigenvalues are ψ mn = 2 sin( mπx/ √ τ ) sin( nπ √ τ y ) , (B2)2 β mn = − π (cid:18) m τ + τ n (cid:19) , (B3)where the origin of the Cartesian frame is taken at thelower left corner of the domain. The integer m ≥ x -axis and n ≥ y -axis. We have h ψ mn i = 0if m or n is even and h ψ mn i 6 = 0 if m and n are odd.The differential equation (82) can be solved analyt-ically by decomposing the field φ on the eigenmodesas φ = P mn c mn ψ mn and using the identity 1 = P mn h ψ mn i ψ mn . This yields Eq. (83) from which weobtain h φ i = X mn h ψ mn i β − β mn , (B4) h φ i = X mn h ψ mn i ( β − β mn ) = − d h φ i dβ . (B5)We note in particular that h φ i − β h φ i = − X mn β mn h ψ mn i ( β − β mn ) > . (B6) Appendix C: Temporal evolution of the differentmodes
The relaxation equation (100) can be solved analyti-cally by decomposing the vorticity and the stream func-tion on the eigenmodes of the Laplacian. Using thePoisson equation, we get ω ( r , t ) = P n a n ( t ) ψ n ( r ) and ψ ( r , t ) = P n b n ( t ) ψ n ( r ) with b n ( t ) = − a n ( t ) /β n . Sub-stituting these expressions in Eq. (100) and using theidentity 1 = P n h ψ n i ψ n , we obtain the ordinary differen-tial equations da n dt + (cid:18) − β ( t ) β n (cid:19) a n = − α ( t ) h ψ n i , (C1)for all n . The evolution of the Lagrange multipliers isgiven by Eqs. (101) and (102) with h ψ i = P n b n ( t ) h ψ n i and h ψ i = P n b n ( t ). The modes are coupled throughthe Lagrange multipliers in order to assure the conserva-tion of energy and circulation.In the grand canonical description in which β and α are constants, the foregoing differential equation can beintegrated straightforwardly, yielding a n ( t ) = (cid:18) a n (0) + α h ψ n i − β/β n (cid:19) e − (1 − β/β n ) t − α h ψ n i − β/β n . (C2)In that case, a steady state of the relaxation equationis stable iff β > β ′ where β ′ is the largest eigenvalue ofthe Laplacian. The condition β > β ′ is a necessary andsufficient condition for the steady state to be a global maximum of the grand potential G = S − βE − α Γ.That functional is related to the Arnol’d energy-Casimirfunctional used to settle the nonlinear dynamical stabilityof a steady state of the 2D Euler equation [35].
Appendix D: Relaxation equations1. Relaxation equations associated with themaximization problem (16)
In this Appendix, we construct relaxation equations as-sociated with the maximization problem (16) correspond-ing to the energy-enstrophy-circulation statistical theory.These relaxation equations can serve as a numerical al-gorithm to solve this constrained maximization problem.In the past, Robert & Sommeria [54] have proposed re-laxation equations that conserve all the Casimirs and in-crease the entropy. Here, we use a different approachbecause we want to conserve only the microscopic en-strophy (not all the Casimirs). Thus, the form of therelaxation equations will be different. In particular, theywill involve a current in the space of vorticity levels σ [32, 35] instead of a current in the space of positions [54].We construct a set of relaxation equations that increase S [ ρ ] while conserving E , Γ and Γ f.g. using a MaximumEntropy Production Principle. The dynamical equationthat we consider can be written as ∂ρ∂t + u · ∇ ρ = − ∂J∂σ , (D1)where J is an unknown current to be chosen so as toincrease S [ ρ ] while conserving the constraints. The localnormalization R ρdσ = 1 is satisfied provided that J → σ → ±∞ . Multiplying Eq. (D1) by σ and integratingover the levels, we get ∂ω∂t + u · ∇ ω = Z Jdσ ≡ X. (D2)Next, multiplying Eq. (D1) by σ and integrating overthe levels, we obtain ∂ω ∂t + u · ∇ ω = 2 Z Jσdσ. (D3)From Eqs. (D2) and (D3), we find that ∂ω ∂t + u · ∇ ω = 2 Z J ( σ − ω ) dσ. (D4)Using Eq. (D1), the time variations of S [ ρ ] are given by˙ S = − Z Jρ ∂ρ∂σ d r dσ, (D5)and the time variations of E, Γ , Γ f.g. are given by˙ E = Z Jψ d r dσ = 0 , (D6)3˙Γ = Z J d r dσ = 0 , (D7)˙Γ f.g. = 2 Z Jσ d r dσ = 0 . (D8)Following the Maximum Entropy Production Principle,we maximize ˙ S with ˙ E = ˙Γ = ˙Γ f.g. = 0 and the addi-tional constraint Z J ρ dσ ≤ C ( r , t ) , (D9)putting some physical bound on the diffusion current.The variational principle can be written in the form δ ˙ S − β ( t ) δ ˙ E − α ( t ) δ ˙Γ − α ( t ) δ ˙Γ f.g. − Z D ( r , t ) δ (cid:18)Z J ρ dσ (cid:19) d r = 0 , (D10)where β ( t ), α ( t ), α ( t ) and D ( r , t ) are time dependentLagrange multipliers associated with the constraints.This leads to the following optimal current J = − D (cid:20) ∂ρ∂σ + ρ ( β ( t ) ψ + α ( t ) + 2 α ( t ) σ ) (cid:21) . (D11)Therefore, the relaxation equation for the vorticity dis-tribution is ∂ρ∂t + u · ∇ ρ = ∂∂σ (cid:26) D (cid:20) ∂ρ∂σ + ρ ( β ( t ) ψ + α ( t ) + 2 α ( t ) σ ) (cid:21)(cid:27) . (D12)Integrating Eq. (D11) over σ , we obtain X = − D ( β ( t ) ψ + α ( t ) + 2 α ( t ) ω ) . (D13)Inserting Eq. (D13) into Eq. (D2) leads to the followingrelaxation equation for the mean flow ∂ω∂t + u · ∇ ω = − D ( β ( t ) ψ + α ( t ) + 2 α ( t ) ω ) . (D14)For the boundary condition, we shall take β ( t ) ψ + α ( t ) +2 α ( t ) ω = 0 on the domain boundary so as to be consis-tent with the equilibrium state where this quantity van-ishes in the whole domain. Since ψ = 0 on the boundary,we finally get ω = − α ( t ) / (2 α ( t )) on the domain bound-ary. A relaxation equation can also be written for thecentered variance ω . Using Eqs. (D11) and (D4), weobtain ∂ω ∂t + u · ∇ ω = 2 D (1 − α ( t ) ω ) . (D15)Finally, in Eqs. (D12), (D14) and (D15), the Lagrangemultipliers evolve so as to satisfy the constraints. Sub-stituting Eq. (D11) in Eqs. (D6), (D7) and (D8), weobtain the algebraic equations h ψ i β ( t ) + h ψ i α ( t ) + 4 Eα ( t ) = 0 , (D16) h ψ i β ( t ) + α ( t ) + 2Γ α ( t ) = 0 , (D17)2 Eβ ( t ) + Γ α ( t ) + 2Γ f.g. α ( t ) = 1 . (D18)where h X i = R X d r . Substituting ∂ρ/∂σ taken from Eq.(D11) in Eq. (D5) and using the constraints (D6)-(D8),we easily obtain ˙ S = Z J Dρ d r dσ, (D19)so that ˙ S ≥ D is positive. On the otherhand ˙ S = 0 iff J = 0 leading to the Gibbs state (23).From Lyapunov’s direct method, we conclude that theserelaxation equations tend to a maximum of entropy atfixed energy, circulation and microscopic enstrophy. Notethat during the relaxation process, the distribution ofvorticity is not Gaussian but changes with time accordingto Eq. (D12). The vorticity distribution is Gaussian onlyat equilibrium. Therefore, these relaxation equations de-scribe not only the evolution of the mean flow accordingto Eq. (D14) but also the evolution of the full vorticitydistribution according to Eq. (D12). We stress, however,that these equations are purely phenomenological andthat there is no compelling reason why they should givean accurate description of the real dynamics. However,they can be used at least as a numerical algorithm tocompute the statistical equilibrium state. Indeed, theseequations can only relax towards an entropy maximum atfixed energy, circulation and microscopic enstrophy, nottowards a minimum or a saddle point that are linearlyunstable with respect to these equations[65].
2. Relaxation equations associated with themaximization problem (36)
We shall now introduce a set of relaxation equations as-sociated with the maximization problem (36). We writethe dynamical equation as ∂ω∂t + u · ∇ ω = X, (D20)where X is an unknown quantity to be chosen so as toincrease S [ ω ] while conserving E , Γ and Γ f.g. . The timevariations of S are given by˙ S = − ( t ) Z ωX d r , (D21)where Ω ( t ) is determined by the constraint on micro-scopic enstrophy leading toΩ ( t ) = Γ f.g. − Z ω d r , (D22)at each time. On the other hand, the time variations of E and Γ are ˙ E = Z Xψ d r = 0 , (D23)˙Γ = Z X d r = 0 . (D24)4Following the Maximum Entropy Production Princi-ple, we maximize ˙ S with ˙ E = ˙Γ = 0 (the conservationof microscopic enstrophy has been taken into account inEq. (D22)) and the additional constraint X ≤ C ( r , t ) . (D25)The variational principle can be written in the form δ ˙ S − β ( t ) δ ˙ E − α ( t ) δ ˙Γ − Z D ( r , t ) δ (cid:18) X (cid:19) d r = 0 , (D26)and it leads to the optimal quantity X = − D (cid:18) β ( t ) ψ + α ( t ) + 1Ω ( t ) ω (cid:19) . (D27)Inserting Eq. (D27) in Eq. (D20), we obtain ∂ω∂t + u · ∇ ω = − D (cid:18) β ( t ) ψ + α ( t ) + 1Ω ( t ) ω (cid:19) , (D28)with ω = − α ( t )Ω ( t ) on the domain boundary. The La-grange multipliers evolve so as to satisfy the constraints.Substituting Eq. (D27) in Eqs. (D23)-(D24) and recall-ing Eq. (D22), we obtain the algebraic equationsΩ ( t ) = Γ f.g. − Z ω d r , (D29) h ψ i β ( t ) + h ψ i α ( t ) = − E Ω ( t ) , (D30) h ψ i β ( t ) + α ( t ) = − ΓΩ ( t ) . (D31)Substituting ω taken from Eq. (D27) in Eq. (D21) andusing the constraints (D23)-(D24), we easily obtain˙ S = Z X D d r , (D32)so that ˙ S ≥ D is positive. On the otherhand ˙ S = 0 iff X = 0 leading to the condition of equilib-rium (42). From Lyapunov’s direct method, we concludethat these relaxation equations tend to a maximum ofentropy at fixed energy, circulation and microscopic en-strophy.The relaxation equation (D28) is similar to Eq. (D14)but the constraints determining the evolution of the La-grange multipliers are different. More precisely, Eqs.(D30) and (D31) are equivalent to Eqs. (D17) and (D18)but Eq. (D16) has been replaced by Eq. (D29). In-deed, in the present approach, the vorticity distributionis always Gaussian during the dynamical evolution. It isgiven by Eq. (32) at any time, i.e. ρ ( r , σ, t ) = 1 p π Ω ( t ) e − ( σ − ω ( r ,t ))22Ω2( t ) . (D33) By contrast, in the approach of Sec. D 1, the vorticitydistribution changes with time. Therefore, the dynamicalevolution is different. However, in the two approaches,the equilibrium state is the same, i.e. it solves the maxi-mization problem (16). This is sufficient if we use theserelaxation equations as numerical algorithms to computethe maximum entropy state. Remark:
Using Eqs. (D20)-(D21), it is easy to showthat ˙Γ c.g. = − ( t ) ˙ S so that ˙Γ c.g. ≤ ( t ) ≥ Alternative relaxation equation: writing the r.h.s. ofEq. (D20) in the form of the divergence of a current inorder to conserve the circulation, and using a MEPP, weobtain a relaxation of the form [35]: ∂ω∂t + u · ∇ ω = ∇ · (cid:20) D (cid:18) ( t ) ∇ ω + β ( t ) ∇ ψ (cid:19)(cid:21) , (D34)where Ω ( t ) is given by Eq. (D29) and β ( t ) by β ( t ) = − R D ∇ ω · ∇ ψ d r Ω ( t ) R D ( ∇ ψ ) d r . (D35)The boundary conditions are ( ( t ) ∇ ω + β ( t ) ∇ ψ ) · n = 0on the domain boundary. This relaxation equation satis-fies the same general properties as Eq. (D28).
3. Relaxation equations associated with themaximization problem (46)
We shall introduce a set of relaxation equations associ-ated with the maximization problem (46). We write thedynamical equation as ∂ω∂t + u · ∇ ω = X, (D36)where X is an unknown quantity to be chosen so as toincrease S [ ω ] while conserving E and Γ. The time varia-tions of S are given by˙ S = − Z ωX d r . (D37)On the other hand, the time variations of E and Γ are˙ E = Z Xψ d r = 0 , (D38)˙Γ = Z X d r = 0 . (D39)Following the Maximum Entropy Production Princi-ple, we maximize ˙ S with ˙ E = ˙Γ = 0 and the additionalconstraint X ≤ C ( r , t ) . (D40)5The variational principle can be written in the form δ ˙ S − β ( t ) δ ˙ E − α ( t ) δ ˙Γ − Z D ( r , t ) δ (cid:18) X (cid:19) d r = 0 , (D41)and we obtain X = − D ( β ( t ) ψ + α ( t ) + ω ) . (D42)Substituting Eq. (D42) in Eq. (D36), we obtain ∂ω∂t + u · ∇ ω = − D ( β ( t ) ψ + α ( t ) + ω ) , (D43)with ω = − α ( t ) on the domain boundary. The Lagrangemultipliers β ( t ) and α ( t ) evolve so as to satisfy the con-straints. Substituting Eq. (D42) in Eqs. (D38) and(D39), we obtain the algebraic equations h ψ i β ( t ) + h ψ i α ( t ) = − E, (D44) h ψ i β ( t ) + α ( t ) = − Γ . (D45)Substituting ω taken from Eq. (D42) in Eq. (D37) andusing the constraints (D38)-(D39), we easily obtain˙ S = Z X D d r , (D46)so that ˙ S ≥ D is positive. On the otherhand ˙ S = 0 iff X = 0 leading to the condition of equilib-rium (53). From Lyapunov’s direct method, we concludethat these relaxation equations tend to a maximum of en-tropy (or a minimum of enstrophy) at fixed energy andcirculation. Alternative relaxation equation: writing the r.h.s. ofEq. (D36) in the form of the divergence of a current inorder to conserve the circulation, and using a MEPP, weobtain a relaxation of the form [35]: ∂ω∂t + u · ∇ ω = ∇ · [ D ( ∇ ω + β ( t ) ∇ ψ )] , (D47) β ( t ) = − R D ∇ ω · ∇ ψ d r R D ( ∇ ψ ) d r . (D48)The boundary conditions are ( ∇ ω + β ( t ) ∇ ψ ) · n = 0 on thedomain boundary. This relaxation equation satisfies thesame general properties as Eq. (D43). If we assume that D is constant, the foregoing equation can be rewritten ∂ω∂t + u · ∇ ω = D (∆ ω − β ( t ) ω ) , (D49) β ( t ) = − R ω d r E = S ( t ) E , (D50)where we have used an integration by parts to obtain thesecond term of Eq. (D50).
Remark 1 : Since the relaxation equations derived inthis section solve (11), they can also be used as a numeri-cal algorithm to construct nonlinearly dynamically stablestationary solutions of the 2D Euler equations character-ized by a linear ω − ψ relationship (see Secs. II and III)independently of the statistical mechanics interpretation. Remark 2 : Since the EHT thermodynamical equilib-rium with a Gaussian prior is equivalent to (46), the re-laxation equations derived in this section coincide with aparticular case of the relaxation equations derived in [32](the ones corresponding to a Gaussian prior).
Remark 3 : Since the optimization problems (16), (36)and (46) are equivalent, the corresponding relaxationequations derived in Appendices D 1, D 2 and D 3 havethe same equilibrium states. However, the dynamicsleading to these equilibrium states is different in eachcase because the constraints are different. [1] G.R. Flierl, Annu. Rev. Fluid Mech. , 493 (1987)[2] P.S. Marcus, Annu. Rev. Astron. Astrophys. , 523(1993)[3] J.C. McWilliams, J. Fluid Mech. , 21 (1984)[4] P. Tabeling, Phys. Rep. , 1 (2002)[5] H.J.H. Clercx, G.J.F. van Heijst, App. Mech. Rev. ,020802 (2009)[6] N.P. Fofonoff, J. Mar. Res. , 254 (1954)[7] G. Veronis, Deep-Sea Res. , 31 (1966)[8] A. Griffa, R. Salmon, J. Mar. Res. , 53 (1989)[9] P.F. Cummins, J. Mar. Res. , 545 (1992)[10] J. Wang, G.K. Vallis, J. Mar. Res. , 83 (1994)[11] E. Kazantsev, J. Sommeria, J. Verron, J. Phys. Oceano. , 1017 (1998)[12] P.P. Niiler, Deep-Sea Res. , 597 (1966)[13] J. Marshall, G. Nurser, J. Phys. Oceano. , 1799 (1986)[14] F.P. Bretherton, D.B. Haidvogel, J. Fluid. Mech. , 129(1976)[15] G.K. Batchelor, Phys. Fluid. Suppl. , 233 (1969)[16] W. Matthaeus, D. Montgomery, Ann. N.Y. Acad. Sci. , 203 (1980)[17] C.E. Leith, Phys. Fluid. , 1388 (1984)[18] R. Kraichnan, Phys. Fluid. , 1417 (1967)[19] R. Kraichnan, J. Fluid. Mech. , 155 (1975)[20] R. Salmon, G. Holloway, M.C. Hendershott, J. Fluid.Mech. , 691 (1976) [21] L. Onsager, Nuovo Cimento Suppl. , 279 (1949)[22] G. Joyce, D. Montgomery, J. Plasma Phys. , 107(1973)[23] T.S. Lundgren, Y.B. Pointin, J. Stat. Phys. , 323(1977)[24] J. Miller, Phys. Rev. Lett. , 2137 (1990)[25] R. Robert, J. Sommeria, J. Fluid. Mech. , 291 (1991)[26] P.H. Chavanis, J. Sommeria, J. Fluid. Mech. , 267(1996)[27] H. Brands, P.H. Chavanis, R. Pasmanter, J. Sommeria,Phys. Fluids , 3465 (1999)[28] P.H. Chavanis, J. Sommeria, J. Fluid. Mech. , 259(1998)[29] R. Ellis, K. Haven, B. Turkington, Nonlin. , 239 (2002)[30] P.H. Chavanis, Physica D , 257 (2005)[31] P.H. Chavanis, Physica D , 1998 (2008)[32] P.H. Chavanis, A. Naso, B. Dubrulle, [arXiv:0912.5096].[33] F. Bouchet, Physica D , 1978 (2008)[34] R. Ellis, K. Haven, B. Turkington, J. Stat. Phys. ,999 (2000)[35] P.H. Chavanis, Eur. Phys. J. B , 73 (2009).[36] R. Monchaux, F. Ravelet, B. Dubrulle, A. Chiffaudel, F.Daviaud, Phys. Rev. Lett. , 124502 (2006)[37] R. Monchaux, P.P. Cortet, P.H. Chavanis, A. Chiffaudel,F. Daviaud, P. Diribarne, B. Dubrulle, Phys. Rev. Lett. , 174502 (2008)[38] A. Naso, R. Monchaux, P.H. Chavanis, B. Dubrulle,[arXiv:0912.5102].[39] A. Renyi, Probability theory.
North-Holland Publ. Com-pany, Amsterdam (1970)[40] T.D. Frank,
Nonlinear Fokker-Planck Equations: Funda-mentals and Applications (Springer-Verlag, 2005)[41] A. Venaille, F. Bouchet, Phys. Rev. Lett. , 104501(2009)[42] G.H. Keetels, H.J.H. Clercx, G.J.F. van Heijst, PhysicaD , 1129 (2009)[43] J.B. Taylor, M. Borchardt, P. Helander, Phys. Rev. Lett. , 124505 (2009)[44] H.J.H. Clercx, S.R. Maassen, G.J.F. van Heijst, Phys.Rev. Lett. , 5129 (1998)[45] A. Naso, P.H. Chavanis, B. Dubrulle, in preparation.[46] F. Bouchet, E. Simonnet, Phys. Rev. Lett. , 094504(2009)[47] S. Dubinkina, J. Frank, J. Comput. Phys. , 1286(2007)[48] P.H. Chavanis, AIP Conf. Proc. , 39 (2008)[49] J. Sommeria, J. Fluid. Mech. , 139 (1986)[50] H. Risken, The Fokker-Planck equation (Springer, 1989)[51] R. Benzi, Phys. Rev. Lett. , 024502 (2005)[52] P.H. Chavanis, [arXiv:1002.0291][53] A. Campa, P.H. Chavanis, [arXiv:1003.2378][54] R. Robert, J. Sommeria, PRL (1992) 2776.[55] In 3D turbulence, energy cascades towards smaller andsmaller scales where it is dissipated by viscosity. In 2Dturbulence, the energy is transfered to larger and largerdistances in an inverse cascade (explaining its conserva-tion) whereas this is the enstrophy that is transfered toshorter and shorter scales, and dissipated.[56] In a different situation, Brands et al. (1999) [27] showthat, in case of incomplete relaxation, the statistical equi-librium state restricted to a “maximum entropy bubble”because of lack of ergodicity [28] can resemble a mini-mum enstrophy state. However, this agreement is ratherfortuitous and is not expected to be general. [57] This equivalence is not trivial. For example, in the MRSapproach where all the Casimirs are conserved, there ex-ists particular initial conditions leading to a GaussianGibbs state at equilibrium associated with a linear ω − ψ relationship [24]. However, we cannot conclude that thesestates are minimum enstrophy states (they could be justsaddle points of enstrophy at fixed circulation and en-ergy). We can only prove that a minimum of enstrophy(more generally a maximum of a “generalized entropy”)at fixed circulation and energy is MRS thermodynami-cally stable [33, 35] but this is not reciprocal: the coarse-grained vorticity field associated with a MRS equilibriumstate is not necessarily a minimum of enstrophy (or moregenerally a maximum of a “generalized entropy”) at fixedcirculation and energy (it could be a saddle point). Bycontrast, if we keep only the microscopic enstrophy as aconstraint among all the Casimirs, we shall prove herethat the Gibbs state is a maximum entropy state at fixedcirculation, microscopic enstrophy and energy iff the cor-responding coarse-grained vorticity field is a minimumof macroscopic enstrophy state at fixed circulation andenergy.[58] These interpretations can be generalized to any func-tional of the form S = − R C ( ω ) d r where C is convex[35].[59] If ν denotes the viscosity, we find from the Navier-Stokes equations that ˙Γ = − ν R ( ∇ ω ) d r . When ν → ∇ ω ) → + ∞ , as the flow develops small scales,the product ν R ( ∇ ω ) d r tends to a strictly positive fi-nite value. Therefore, the enstrophy decays. By contrast,˙ E = − ν Γ tends to zero when ν → et al. [42]and references therein.[62] As noted by Taylor et al. [43], the solution with h ψ i = 0(dipole) has zero angular momentum while the solutionwith h ψ i 6 = 0 (monopole) has nonzero angular momen-tum, even though the circulation is zero. This explainsthe “spin-up” phenomenon discovered in [44].[63] In fact, it is more convenient to plot β as a function of theinverse of the energy 1 /E as in Fig. 1 since the interestingbifurcations occurs for large values of the energy.[64] This idea was initially proposed in [26].[65] In fact, it is shown in Sec. VI that the system can remainblocked in an unstable state (saddle point of entropy) ifthe dynamics does not spontaneously develop the “dan-gerous” perturbations that make it unstable. This is be-cause the system is unstable for some perturbations butnot for all perturbations. Therefore, we must keep in7