ANALYSIS OF STATISTICAL EQUILIBRIUM MODELS OF GEOSTROPHIC TURBULENCE

Transcription

1 Journal of pplied Mathematics and Stochastic nalysis, 15:4 (2002), NLYSIS OF STTISTICL EQUILIBRIUM MODELS OF GEOSTROPHIC TURBULENCE RICHRD S. ELLIS 1and BRUCE TURKINGTON2 University of Massachusetts Department of Mathematics and Statistics mherst, M US rsellismath.umass.edu, turk@math.umass.edu KYLE HVEN Ne York University Courant Institute of Mathematical Sciences Ne York, NY US haven@cims.nyu.edu (Received January, 2002; Revised ugust, 2002) Statistical equilibrium lattice models of coherent structures in geostrophic turbulence, formulated by discretizing the governing Hamiltonian continuum dynamics, are analyzed. The first set of results concern large deviation principles (LDP's) for a spatially coarse-grained process ith respect to either the canonical and/or the microcanonical formulation of the model. These principles are derived from a basic LDP for the coarse-grained process ith respect to product measure, hich in turn depends on Cramer's Theorem. The rate functions for the LDP's give rise to variational principles that determine the equilibrium solutions of the Hamiltonian equations. The second set of results addresses the equivalence or nonequivalence of the microcanonical and canonical ensembles. In particular, necessary and sufficient conditions for a correspondence beteen microcanonical equilibria and canonical equilibria are established in terms of the concavity of the microcanonical entropy. complete characterization of equivalence of ensembles is deduced by elementary methods of convex analysis. The mathematical results proved in this paper complement the physical reasoning and numerical computations given in a companion paper, here it is argued that the statistical equilibrium model defined by a prior distribution on potential vorticity fluctuations and microcanonical conditions on total energy and circulation is natural from the perspective of geophysical applications. Key ords: Large Deviation Principle, Cramer's Theorem, Coarse Graining, Statistical Models of Turbulence, Hamiltonian Systems. MS subject classifications: 6F55, 60F10. 1 This research as supported by a grant from the Department of Energy (DE-FG02-99ER2536) and by a grant from the National Science Foundation (NSF-DMS-90052). 2 This research as supported by a grant from the Department of Energy (DE-FG02-99ER2536) and by a grant from the National Science Foundation (NSF-DMS ). Printed in the U.S by North tlantic Science Publishing Company 32

2 32 R. ELLIS, K. HVEN and B. TURKINGTON 1. Introduction The problem of developing a statistical equilibrium theory of coherent structures in todimensional turbulence or quasi-geostrophic turbulence has been the focus of numerous mathematics and physics papers during the past fifty years. In a landmark paper Onsager [23] studied a microcanonical ensemble of point vortices and argued that they cluster into a coherent large-scale vortex in the negative temperature regime corresponding to sufficiently large kinetic energies. The quantitative aspects of the point vortex model ere elaborated by Joyce-Montgomery [14], and subsequently these predictions ere compared ith direct numerical simulations of the end state of freely decaying turbulence [21]. nother approach due to Kraichnan [15], called the energy-enstrophy model, as based on spectral truncation of the underlying fluid dynamics. This Gaussian model as extensively applied to geophysical fluid flos [4, 2, 2]; for instance, to the organization of barotropic, quasigeostrophic flos over bottom topography [11]. The next major development as the Miller- Robert model [20, 26], hich enforced all the invariants of ideal motion and included the previous models as special or limiting cases. Recently, this general model has been scrutinized from the theoretical standpoint [30] and from the perspective of practical applications [6, 1]. These investigations have clarified the relation beteen various formulations of the general model and the simpler models, and they have revealed a formulation that has firm theoretical justification and rich physical applications. This formulation of the statistical equilibrium theory of coherent structures is described in our companion paper [10], and a particularly striking application of this model is given in another paper [31], here it is implemented to predict the permanent jets and spots in the active eather layer of Jupiter. The purpose of the current paper is to prove the theorems that legitimize the theoretical model developed in [10]. We therefore follo the line of development in that paper, in hich the nonlinear partial differential equation governing the dynamics of the fluid an infinitedimensional Hamiltonian system is used to motivate an equilibrium statistical lattice model, both in its microcanonical and canonical formulations. Our first set of main results pertains to the continuum limit of this model. These results are large deviation principles (LDP's) that is, exponential-order refinements of the la of large numbers for a spatially coarse-grained process hich justify the definition of the equilibrium states for the model. We base our direct proofs of these LDP's on an elementary large deviation analysis of the coarse-grained process ith respect to the product measure that underlies the lattice model. Our second set of main results concerns the relationship beteen the microcanonical and canonical equilibrium states; in particular, the equivalence or nonequivalence of the microcanonical and canonical ensembles ith respect to energy and circulation. The most interesting result is that microcanonical equilibria are often not realized as canonical equilibria. Significant examples of this remarkable phenomenon are given in [10, 31], here its impact on hydrodynamic stability is also discussed. The outline of the paper is as follos. In Section 2 e formulate the statistical equilibrium theory described in [10], and e define the coarse-grained process used in the analysis of the continuum limit. We state and prove the basic LDP for this process in Section 3, and then in Section 4 e present the LDP's for the microcanonical and canonical ensembles, relying on general theorems established in [9]. Then in Section 5 e turn to the issue of equivalence of those ensembles at the level of their equilibrium states, giving necessary and sufficient conditions for equivalence in terms of the concavity of the microcanonical entropy. In the concluding Section 6, e point to some of the physical implications of our results.

3 nalysis of Statistical Equilibrium Models Statement of Problem The continuum dynamics that underlies our statistical equilibrium lattice model can be vieed as a noncanonical Hamiltonian system ith infinitely many degrees of freedom. Namely, it can be ritten in the form `U `> ] $ L $ U œ ß ÐÞÑ here $ LÎ$ U denotes the variational derivative, and ] is an operator ith the property that ÖJ ß K» ' $ J $ K ].- $ U $ U is a Poisson bracket [22]. Here is a bounded spatial domain, and.- represents normalized Lebesgue measure on, normalized so that -ÐÑ œ. For the sake of definiteness, the flo domain is chosen to be the channel œ ÖB œ ÐB ß B ÑÀ ± B ± j Îß ± B ± j Î à ÐÞÑ thus.- œ.bîj j. The scalar function UœUÐBß>Ñrepresents either the vorticity in to-dimensional flo or the potential vorticity in quasi-geostrophic flo. U evolves according to `U `0 `1 `0 `1 `>, here `B `B `B `B ÒUß Ó œ! Ò0ß 1Ó» Þ ÐÞ$Ñ In this formula œ ÐBß>Ñ is the streamfunction defined by U œ P,, here P is a uniformly elliptic operator on and,œ,ðbñis a given continuous real-valued function on ; satisfies appropriate boundary conditions on the boundary of. Consequently, can be recovered from U via a Green's function 1ÐBßBÑ : ÐBÑ œ ' 1ÐBß B ÑÐUÐB Ñ,ÐB ÑÑ-Ð.B ÑÞ Hence (2.3) is an equation in U alone. This equation takes the Hamiltonian form (2.1) hen Nœ ÒUß Ó and LÐUÑ œ ' ÐBÑÐUÐBÑ,ÐBÑÑ-Ð.BÑ œ ' ÐUÐBÑ,ÐBÑÑ1ÐBß B ÑÐUÐB Ñ,ÐB ÑÑ-Ð.BÑ-Ð.B ÑÞ ÐÞ%Ñ Thus L is a quadratic form in U in hich interactions beteen UÐBÑ and UÐB Ñ are governed by the Green's function 1ÐBß B Ñ. This Hamiltonian formulation of to-dimensional or quasigeostrophic flo is discussed in [12, 22, 2, 29]. The Hamiltonian dynamics (2.1) conserves the total energy L and the total circulation

4 330 R. ELLIS, K. HVEN and B. TURKINGTON GÐUÑ œ ' ÐUÐBÑ,ÐBÑÑ-Ð.BÑÞ ÐÞ&Ñ In addition, there is an infinite family of Casimir invariants parametrized by continuous realvalued functions + and defined as EÐUÑ» ' +ÐUÐBÑÑ-Ð.BÑÞ ÐÞ'Ñ For the particular choice of a channel domain, the linear impulse ' B ÐUÐBÑ,ÐBÑÑ-Ð.BÑ is also conserved. For simplicity, e shall ignore it throughout this paper since it plays a role identical to G in the theory. When Pœ?» ` Î`B ` Î`B and,œ!, (2.3) is the vorticity transport equation, hich is an equivalent formulation of the Euler equations governing an incompressible, inviscid flo in to dimensions. When Pœ? and,œ B, (2.3) is referred to as the quasi-geostrophic equations; the Coriolis parameter! and the Rossby deformation radius Ð!ß Ó, and the effective bottom topography œ ÐB Ñ is a given real-valued function. With this choice of P and,, (2.3) governs the motion of a shallo, rotating layer of a homogeneous, incompressible, inviscid fluid in the limit of small Rossby number [2]. Throughout the rest of the paper, e refer to U, as in this quasi-geostrophic case, as the potential vorticity. 2.1 Statistical Model Since the governing dynamics (2.1) typically results in an intricate mixing of U on a range of scales, e consider ensembles of solutions, instead of individual deterministic solutions. This statistical mechanics approach is standard for a Hamiltonian system ith finitely many degrees of freedom in canonical form [2, 3, 24]. In order to carry over this methodology to the infinite dimensional Hamiltonian system (2.1), it is necessary to formulate an appropriate sequence of lattice models obtained by discretizing the system. Next e define these probabilistic lattice models. For each, the flo domain is uniformly partitioned into + microcells. We let denote a uniform lattice of + sites indexed by =, here each = corresponds to a unique microcell QÐ=Ñ of ; each QÐ=Ñ satisfies -ÖQÐ=Ñ œ Î+. The lattice discretizations of the Hamiltonian L and circulation G are given by L ÐÖUÐ=ÑÀ = Ñ» ÐUÐ=Ñ, Ð=ÑÑ1 Ð=ß = ÑÐUÐ= Ñ, Ð= ÑÑ ÐÞ(Ñ + = = and G ÐÖUÐ=ÑÀ = Ñ» ÐUÐ=Ñ, Ð=ÑÑß ÐÞ)Ñ + = here 1Ð=ß = Ñ is the average over QÐ=Ñ QÐ= Ñ of the Green's function 1ÐBß B Ñ, and, Ð=Ñ is the average over QÐ=Ñ of,:

5 nalysis of Statistical Equilibrium Models Ð=ß = Ñ» + ' ' 1ÐBß B Ñ-Ð.BÑ-Ð.B Ñ and, Ð=Ñ» + ',ÐBÑ-Ð.BÑÞ ÐÞ*Ñ QÐ=ÑQÐ= Ñ QÐ=Ñ The Casimirs (2.6) can be discretized similarly as functions EÐÖUÐ=ÑÀ = Ñ. We shall not attempt to prescribe a lattice dynamics under hich the functions L, G and E are exact invariants. In fact, a lattice dynamics conserving the nonlinear enstrophies E is not knon. The subtleties associated ith the rigorous formulation of this statistical model are fully discussed elsehere [10, 30]. Rather than pursue this direction any further here, e shall simply impose a joint probability distribution C on the lattice potential vorticities UÐ=Ñ, = à C captures the invariance of the nonlinear enstrophies E and the phase volume.u. Let 3 be a probability distribution on and let l be the support of 3. In order to simplify our presentation throughout the paper, e assume that l is compact. The extension to the case of noncompact support can be carried out via appropriate modifications of the + techniques that e use. We define the phase space H to be l, denote by U H the vector of lattice potential vorticities ÖUÐ=Ñß =, and let C be the product measure on H ith one-dimensional marginals 3. That is, for any Borel subset F of H, e define C ÖF œ ' C Ð.UÑ» ' 3Ð.UÐ=ÑÑÞ ÐÞ!Ñ F F = We refer to C as the prior distribution of the lattice potential vorticities. s is explained in [10], C can be interpreted as a canonical ensemble ith respect to lattice enstrophy E, in hich the distribution 3 is determined by the function +. lternatively, in the perspective adopted in the Miller-Robert model [19, 25], 3 is determined from the continuum initial data by 3 Ð.CÑ œ ' $ Ð.CÑ-Ð.BÑÞ UÐBß!Ñ While this choice of prior distribution is conserved by the continuum equations, it is not preserved under passage to a discretized dynamics. In practice, therefore, it is better to vie 3 as a free parameter in the model. For discussions on the choice of the prior distribution in specific applications, see [10, 6.1] and [31, page 1234]. 2.2 Definition of Ensembles In terms of the prior distribution C, e can form to different statistical equilibrium models by using either the microcanonical ensemble or the canonical ensemble ith respect to the invariants L and G. Given,!, for any Borel subset F of H the microcanonical ensemble is defined by Tß, ÖF» C ÖF ± L ÒI, ß I, Óß G Ò>, ß >, Ó Þ ÐÞÑ This is the conditional probability distribution obtained by imposing the constraints that L and G lie ithin the (small) closed intervals around I and >, respectively. Given real numbers and, the canonical ensemble is defined by T ÖF» ' exp Ð + L ÐUÑ + G ÐUÑÑC Ð.UÑß ÐÞÑ ß ß ^Ð ßÑ F

6 332 R. ELLIS, K. HVEN and B. TURKINGTON here ^Ð ß Ñ» ' expð + LÐUÑ + GÐUÑÑCÐ.UÑÞ H The real parameters and correspond to the inverse temperature and the chemical potential. 2.3 Continuum Limit The first set of main results of this paper concerns the continuum limits of the joint probability distributions (2.11) and (2.12). In this limit e obtain the continuum description of the potential vorticity that captures the large-scale behavior of the statistical equilibrium state. The key ingredient in this analysis is a coarse-graining of the potential vorticity field, defined by a local averaging of the random microstate U over an intermediate spatial scale. This process is constructed as follos. For such that, the flo domain is uniformly partitioned into macrocells denoted by Hß, á ß Hß. We assume that each macrocell satisfies -ÖH ß5 œ Î and contains +Î microcells QÐ=Ñand that lim max diamðhß5ñ œ!þ ÐÞ$Ñ Ä 5 Öß á ß For instance, can be dyadically refined into + œ microcells, and these can be collected into a partition of œ macrocells Hß5 satisfying (2.13). With respect to the collection of macrocells H ß5, e introduce a doubly-indexed process [ ß œ [ ß ÐUßBÑ, defined for U H and B by ß ßß5 H 5œ ß5 [ ÐUß BÑ» W ÐBÑß ÐÞ%Ñ here W œ W ÐUÑ» UÐ=ÑÞ ÐÞ&Ñ ßß5 ßß5 +Î = H ß5 For U H the pieceise-constant process [ ß ÐUß Ñ takes values in P ÐÑ and can be studied ith respect to either the microcanonical ensemble (2.11) or the canonical ensemble (2.12). We refer to [ ß as the coarse-grained process. With respect to each ensemble, our goal is to determine the set of P functions on hich [ ß concentrates in the continuum limit defined by first sending Ä and then Ä. More precisely, e seek to find the smallest subsets of PÐÑ, X and X, such that and Tß, Ö[ ß X Ä as Ä, Ä, and, Ä! ÐÞ'Ñ T Ö[ X Ä as Ä and Ä Þ ÐÞ(Ñ ß ß ß ß The subsets X and X ß consist of the most probable states for their respective ensembles and are referred to as the sets of equilibrium macrostates. In Section 4, each of these subsets is shon to arise by solving a variational principle. The variational principles are derived ß

7 nalysis of Statistical Equilibrium Models 333 from the LDP's for [ ß ith respect to the to ensembles. In order to prove these to LDP's, e first prove a basic LDP for [ ß ith respect to the product measure C defined in (2.10). This is done in the next section. 3. Basic LDP for the Coarse-Grained Process Before stating the basic LDP for [ ß ith respect to C, e first discuss the form of the rate function M. Throughout the paper the term rate function is used to describe any loer semicontinuous function mapping a complete, separable metric space into Ò!ß Ó. For α e define the cumulant generating function αc -ÐαÑ» log' / 3Ð.CÑß hich is finite because 3 has compact support, and for C e introduce the loer semicontinuous, conjugate convex function 3ÐCÑ» sup Ö C -Ð Ñ Þ α α α The normalized sums W ßß5 defined in (2.15) are sample means of the + Î i.i.d. random variables UÐ=Ñ, = H ß5. Therefore, Cramer's theorem [5] implies that, for each 5œßáß, the sequence ÖWßß5ß satisfies an LDP ith respect to C ith scaling constants +Î and rate function 3. In other ords, for any closed subset Jof and for any open subset K of lim sup +Î logcöwßß5 J Ÿ inf 3ÐCÑß Ä C J lim inf log ÖW K inf 3ÐCÑÞ Ä +Î C ßß5 C K The rate function for the LDP of [ ith respect to C is defined for ; P ÐÑ to be ß MÐ;Ñ» ' 3Ð;ÐBÑÑ-Ð.BÑÞ Ð$ÞÑ Since 3 is nonnegative and convex, it follos that M is ell-defined, nonnegative, and convex. We claim that M is loer semicontinuous ith respect to the strong topology on P ÐÑ. ssume that ; Ä ; in P ÐÑ. Then there exists a subsequence ; such 5 that lim Ä MÐ; Ñ œ lim inf Ä MÐ; Ñ and ; Ä ; a.s. Fatou's lemma and the loer semicontinuity of 3 on yield the desired conclusion: lim inf MÐ; Ñ œ lim inf ' 3Ð; ÐBÑÑ-Ð.BÑ 5 Ä Ä 5 ' lim inf 3Ð; ÐBÑÑ Ð.BÑ ' 3Ð;ÐBÑÑ Ð.BÑ œ MÐ;ÑÞ Ä

8 334 R. ELLIS, K. HVEN and B. TURKINGTON We no state the to-parameter LDP for [ ß ith respect to C. For any subset E of PÐ ÑMÐEÑ, denotes the infimum of Mover E. Theorem 3.1: Consider PÐÑ ith the strong topology. The sequence [ ß satisfies the LDP on PÐ Ñ, in the double limit Ä and Ä, ith rate function M. That is, for any strongly closed subset J of P Ð Ñ lim sup lim sup Ä Ä and for any strongly open subset K of lim inf lim inf Ä Ä + ß logc Ö[ J Ÿ MÐJ Ñß P ÐÑ + ß logc Ö[ K MÐKÑÞ We no prove the upper and loer large deviation bounds for [ ß in separate, but elementary steps. The folloing lemma is needed in both steps; it is an immediate consequence of Lemmas in [1]. Lemma 3.2: For each the sequence ÖÐWßß ßáßWßß Ñß satisfies the LDP on ith scaling constants +Î and the rate function Ð/ ßáß/ Ñ È 3Ð/ ÑÞ 5 5œ Proof of the Large Deviation Upper Bound: We define P to be the set of ; PÐ Ñ of the form ;ÐBÑœ 5œ / 5 H ÐBÑfor some / ßáß/. Let Jbe a ß5 strongly closed subset of PÐ Ñ. For is a closed subset of lim sup Ä + ß 5 H 5œ. By Lemma 3.2, it follos that logc Ö[ J J œ ÖÐ/ ßáß/ Ñ À / ÐBÑ J ß5 œ lim sup + logcöðwßß ßáßWßß Ñ J Ä Ÿ inf œ 3Ð/ 5 ÑÀÐ/ ßáß/ Ñ J 5œ œ inf ' 3Ð;ÐBÑÑÐ.BÑÀ; J P - Ÿ Ÿ inf ' 3Ð;ÐBÑÑÐ.BÑÀ; J - Ÿ œ MÐJÑ.

9 nalysis of Statistical Equilibrium Models 335 Sending Ä gives the desired large deviation upper bound. Proof of the Large Deviation Loer Bound: The folloing lemma is needed both here and later in the paper. We denote by ² ² the norm on PÐ Ñ. Lemma 3.3: Let ; be any function in P ÐÑ. For define ; ÐBÑ» ; Ð5Ñ ÐBÑ, here ; Ð5Ñ» ' ;ÐBÑ-Ð.BÑÞ H ß5 5œ H ß5 Then as Ä, ²; ; ² Ä!. Proof: For any given %! there exists a bounded Lipschitz function : mapping into and satisfying ²; : ² Ÿ& []. Since the operator mapping 0 P ÐÑÈ0 is an orthogonal projection, ²; : ² œ ²Ð; : Ñ ² Ÿ ²; : ² & Þ Ð$ÞÑ Hence it suffices to estimate ² : : ², here : is a Lipschitz function ith constant Q. straightforard calculation coupled ith (2.13) gives ² : : ² Ÿ Q Ö ÐH Ñ Ä! Ä Þ Ð$Þ$Ñ 5 Öß max ß5 á ß diam as This completes the proof. Given a strongly open subset K of P Ð Ñ, let ; be any function in K and choose &! so that FÐ;ß & Ñ, the ball in P Ð Ñ ith center ; and radius &, is a subset of K. Since ²; ; ² Ä! as Ä (Lemma 3.3), e can also choose R such that for all R FÐ; ß & ÎÑ FÐ;ß & Ñ. Since K ß &» œð/ ßáß/ Ñ À / 5 H ÐBÑ FÐ;ß % ÎÑ ß5 5œ is an open subset of, it follos from Lemma 3.2 that for all R lim inf log Ö[ K Ä + C ß lim inf logc Ö[ FÐ; ß & ÎÑ Ä + ß œ lim inf log ÖÐW ßáßW Ñ K Ä + C ßß ßß ß & inf œ 3Ð/ 5 ÑÀÐ/ ßáß/ Ñ K ß& 5œ 3Ð;Ð5ÑÑœ 3 ' ;ÐBÑÐ.BÑ - 5œ 5œ H ß5

10 336 R. ELLIS, K. HVEN and B. TURKINGTON ' 3Ð;ÐBÑÑ-Ð.BÑ œ ' 3Ð;ÐBÑÑ-Ð.BÑ œ MÐ;ÑÞ 5œH ß5 The last inequality in this display follos from Jensen's inequality. We have proved that for arbitrary ; K lim inf lim inf Ä Ä + ß logc Ö[ K MÐ;ÑÞ Taking the supremum of MÐ;Ñ over ; Kyields the desired large deviation loer bound. This completes the proof of Theorem LDP's ith Respect to the To Ensembles In this section e present LDP's for [ ß ith respect to the microcanonical and canonical ensembles defined in (2.11) and (2.12). The theorems follo from a general theory presented in [9]. The proofs of the LDP's depend in part on properties of various functionals given in the next subsection. 4.1 Properties of L, Gß L, and G We recall the definitions of the lattice energy L in (2.), the circulation G in (2.), and the functionals L and G in (2.4) and (2.5). The proofs of the LDP's ith respect to the to ensembles rely on boundedness and continuity properties of LÐ;Ñ and GÐ;Ñ for ; P ÐÑ and on the fact that uniformly over H, L and G are asymptotic to LÐ[ ß Ñ and GÐ[ ß Ñ; the latter to quantities are defined by replacing UÐBÑ in (2.4) and (2.5) by the PÐÑ -valued process [ ß ÐUßBÑ. The proof of this approximation for the circulation is immediate since it holds ith equality. For L the proof depends on the fact that the vortex interactions are long-range, being determined by the Green's function 1ÐBß B Ñ. For this reason, L is ell approximated by a function of the coarsegrained process [ ß. In physical parlance, the turbulence model under consideration is an asymptotically exact, local mean-field theory [20]. The next lemma proves the approximations as ell as the boundedness and continuity properties of L and G needed for the LDP's. Lemma 4.1: Ð+Ñ The sequence satisfies and [ ß lim lim sup ± LÐUÑ LÐ[ ß ÐUÑÑ ± œ! Ð%ÞÑ Ä Ä U H lim lim sup ± GÐUÑ GÐ[ ß ÐUÑÑ ± œ!þ Ð%ÞÑ Ä Ä U H Ð,Ñ L and G are bounded on the union, over Ÿ, of the supports of the C -distributions of [ ß. Ð-Ñ L and G are continuous ith respect to the eak topology on P ÐÑ and thus ith respect to the strong topology on PÐ Ñ. Proof: ( + ) short calculation shos that GÐUÑ œ GÐ[ ß ÐUÑÑ for all U H, thus proving (4.2). Carrying out the multiplication in the summand appearing in the definition of

11 nalysis of Statistical Equilibrium Models 33 % Ð3Ñ ÐÑ LÐUÑ, e rite LÐUÑ œ 3œ L ÐUÑ, here L has the summands UÐ=Ñ1Ð=ß = ÑUÐ= Ñ, ÐÑ Ð$Ñ L has the summands UÐ=Ñ1Ð=ß = Ñ, Ð= Ñ, L has the summands, Ð=Ñ1Ð=ß = ÑUÐ= Ñ, and Ð%Ñ L has the summands, Ð=Ñ1Ð=ß = Ñ, Ð= Ñ. Similarly, carrying out the multiplication in the integrand appearing in the definition of e rite LÐ[ ÐUÑÑ» ' Ð[ ÐUß BÑ,ÐBÑÑ1ÐBß B ÑÐ[ ÐUß B Ñ,ÐB ÑÑ-Ð.BÑ-ÐB ÑÞ ß ß ß LÐ[ ÐUÑÑ œ Ð3Ñ L Ð[ ÐUÑÑß ß 3œ ÐÑ ÐÑ % here L Ð[ ß ÐUÑÑ has the integrand [ ß ÐUß BÑ1ÐBß B Ñ[ ß ÐUß B Ñß L Ð[ ß ÐUÑÑ has Ð$Ñ the integrand [ ß ÐUß BÑ1ÐBß B Ñ,ÐB Ñ, L Ð[ ß ÐUÑÑ has the integrand Ð%Ñ,ÐBÑ1ÐBß B Ñ[ ß ÐUß B Ñ, and L Ð[ ß ÐUÑÑ has the integrand,ðbñ1ðbß B Ñ,ÐB Ñ. The limit (4.1) follos if e prove for 3 œ ß ß $ß % that lim lim sup ± L ÐUÑ L Ð[ ß ÐUÑÑ ± œ!þ Ð%Þ$Ñ Ä Ä U H Ð3Ñ The details of the proof are given for 3œ ; the cases 3œß$ß% are handled similarly. The verification of (4.3) for 3œ relies on the fact that since 3 has compact support l, there exists G such that ±UÐ=ѱ ŸGfor any, U H, and =. For Ÿ5, 5 Ÿ and Band B in e define Hß5 Hß5 5ß5œ Ð3Ñ 1 ÐBß B Ñ» 1 Ð5ß 5 Ñ ÐBÑ ÐB Ñß Ð%Þ%Ñ here 1 Ð5ß 5 Ñ» ' ' 1ÐBß B Ñ-Ð.BÑ-Ð.B ÑÞ Ð%Þ&Ñ Hß5 Hß5 Substituting the definition (2.14) (2.15) of [ ß ÐUßBÑ, e obtain ÐÑ L Ð[ ÐUÑÑ œ UÐ=ÑUÐ= Ñ' ' 1ÐBß B Ñ-Ð.BÑ-Ð.B Ñ ß + 5ß5œ = H = H ß5 H H ß5 ß5 ß5 œ UÐ=ÑUÐ= Ñ1 Ð5ß5 Ñ + 5ß5 œ = Hß5 = H ß5. Thus for any Ÿ,

12 33 R. ELLIS, K. HVEN and B. TURKINGTON sup U H œ sup U H ÐÑ ÐÑ ±L ÐUÑ L Ð[ ÐUÑѱ ß â 1Ð=ß=Ñ 1Ð5ß5Ñ UÐ=ÑUÐ= Ñ ß5 â = H â â + 5ß5 œ = H ß5 Ÿ G ±1 Ð=ß= Ñ 1 Ð5ß5 ѱ + 5ß5 œ = Hß5 = H ß5 œ G» ' ' Ò1ÐBß B Ñ 1 ÐBß B ÑÓ-Ð.BÑ-Ð.B Ñ» 5ß5œ = H = H ß5 ß5 QÐ=ÑQÐ= Ñ G Ÿ '' ± 1ÐBß B Ñ 1 ÐBß B Ñ ± -Ð.BÑ-Ð.B Ñ G G PÐ Ñ PÐ Ñ œ ² 1 1 ² Ÿ ² 1 1 ² Þ The Green's function 1ÐBß B Ñ is square-integrable on. Hence for the set ith the partition H ß5 Hß5, the function 1 defined in (4.4) (4.5) is the analogue of ; defined in Lemma 3.3. By an analogous proof, one shos that ²1 1 ² PÐ ÑÄ!. This limit and the preceding display yield the desired limit (4.3) in the case 3œ. Ð,Ñ For any Ÿ, U H, and =, e have ±UÐ=ѱ ŸG and thus sup B ±[ ß ÐUßBѱ ŸG. Hence the range of the support of the C-distribution of [ ß is a subset of Ö; P ÐÑÀ ² ; ² Ÿ G. Since, is bounded on and 1ÐBß B Ñ P Ð Ñ, it follos that L and G are bounded on the union of the supports of the C -distributions of [ ß. Ð-Ñ The eak continuity of GÐ;Ñ œ ' Ð;,Ñ.- is obvious. If ; Ä ; eakly in P ÐÑ, then by a property of the Green's function ' 1Ð ß B ÑÐ; ÐB Ñ,ÐB ÑÑ-Ð.B Ñ Ä ' 1Ð ß B ÑÐ;ÐB Ñ,ÐB ÑÑ-Ð.B Ñ strongly in P ÐÑÞ It follos that LÐ; Ñ Ä LÐ;Ñ. The proof of the lemma is complete.

13 nalysis of Statistical Equilibrium Models Microcanonical Model The LDP for [ ß ith respect to the microcanonical ensemble is stated in Theorem 4.2. We say that a constraint pair ÐIß > Ñ is admissible if ÐIß > Ñ œ ÐLÐ;Ñß GÐ;ÑÑ for some ; P ÐÑ ith MÐ;Ñ, and e let T denote the largest open subset of consisting of admissible constraint pairs. We call this domain T the admissible set for the microcanonical model. For a fixed constraint pair ÐIß > Ñ T, the rate function for the LDP is defined for ; P ÐÑ to be MÐ;Ñ WÐIß > Ñ if LÐ;Ñ œ I, GÐ;Ñ œ > ß M Ð;Ñ» œ Ð%Þ'Ñ otheriseß here WÐ Ñ is defined by WÐ Ñ» inf ÖMÐ; ÑÀLÐ; ÑœIßGÐ; Ñœ Þ Ð%Þ(Ñ ;µ µ µ µ > WÐ Ñ is called the microcanonical entropy. Theorem 4.2: Let ÐIß > Ñ T and consider P ÐÑ ith the strong topology. With respect Iß > to T ß,, [ ß satisfies the LDP on P Ð Ñ, in the triple limit Ä, Ä and, Ä!, ith Iß rate function M defined in Ð%Þ'Ñ. That is, for any strongly closed subset J of P Ð Ñ > lim lim sup lim sup, Ä! Ä Ä and for any strongly open subset + K of P ÐÑ lim lim inf lim inf, Ä! Ä Ä + ß, ß logt Ö[ J Ÿ M ÐJÑß ß, ß logt Ö[ K M ÐKÑÞ Proof: This LDP is a three-parameter analogue, involving the triple limit Ä, Ä, and, Ä!, of the to-parameter LDP given in Theorem 3.2 of [9]. The proof of that theorem is easily modified to the present three-parameter setting using the basic LDP in Theorem 3.1 and the properties of L, G, L, and G given in Lemma 4.1. For any admissible choice of I and > e define the set X of microcanonical equilibrium macrostates to be the set of ; PÐ Ñsuch that M Ð;Ñœ!. It follos from the definition of M that elements of X are exactly those functions that solve the folloing constrained minimization problem: minimize MÐ; µ Ñ over all µ ; P ÐÑ subject to the constraints LÐ;ÑœI µ and GÐ;Ñœ µ >. Ð%Þ)Ñ The optimum value in this variational principle determines the microcanonical entropy W defined in (4.). We point out an important consequence of the large deviation upper bound in Theorem Iß 4.2. Let E be a Borel subset of P ÐÑ hose closure E satisfies E X > œ g. Then M ÐEÑ!, and for all sufficiently small,! and all sufficiently large and Tß, Ö[ ß E Ÿ expð + M ÐE ÑÎÑÞ Ð%Þ*Ñ

14 340 R. ELLIS, K. HVEN and B. TURKINGTON Since the right-hand side of (4.9) converges to 0 as Ä, macrostates Unot lying in X have an exponentially small probability of being observed as a coarse-grained state in the continuum limit. The macrostates in X are therefore the overhelmingly most probable among all possible macrostates of the turbulent system. 4.3 Canonical Model The LDP for [ ß ith respect to the canonical ensemble is stated in Theorem 4.3. For any real values of and, the rate function for the LDP is defined for ; PÐÑto be here F Ð ß Ñ is defined by M ß Ð;Ñ» MÐ;Ñ LÐ;Ñ GÐ;Ñ F Ð ß Ñß Ð%Þ!Ñ F Ð ß Ñ œ inf ÖMÐ; Ñ LÐ; Ñ GÐ; Ñ Þ Ð%ÞÑ ;µ µ µ µ F Ð ß Ñ is called the canonical free energy. Theorem 4.3: Let Ð ß Ñ and consider P ÐÑ ith the strong topology. With res- pect to T ß ß, [ ß satisfies the LDP on P ÐÑ, in the double limit Ä and Ä, ith rate function M defined in (4.10). That is, for any strongly closed subset J of P ÐÑ ß lim sup lim sup Ä Ä and for any strongly open subset K of lim inf lim inf Ä Ä + ß ß ß ß logt Ö[ J Ÿ M ÐJÑß P ÐÑ + ß ß ß ß logt Ö[ K M ÐKÑß Proof: This LDP is a to-parameter analogue, involving the double limit Ä and Ä, of the one-parameter LDP given in Theorem 2.4 of [9]. The proof of that theorem relies on an application of Laplace's principle. The proof is easily modified to the present to-parameter setting using the basic LDP in Theorem 3.1 and the properties of L, G, L, and G given in Lemma 4.1. s is shon in Theorem of [], the boundedness properties of L and G given in Lemma 4.1 Ð,Ñ allo Laplace's principle to be applied in this setting. For a particular choice of and e define the set X ß of canonical equilibrium macro- states to be the set of ; PÐÑsuch that M ß Ð;Ñœ!. That is, the elements of X ß are exactly those functions that solve the folloing unconstrained minimization problem: minimize MÐ; µ Ñ LÐ; µ Ñ GÐ; µ Ñ over all µ ; P ÐÑÞ Ð%ÞÑ The optimum value in this variational principle determines the canonical free energy F Ð ß Ñ defined in (4.11). Comparing this minimization problem ith the corresponding constrained minimization problem (4.) arising in the microcanonical ensemble, e see that the parameters and in the former are Lagrange multipliers dual to the to constraints in the latter. s in the microcanonical case (see (4.9) and the associated discussion), the large deviation upper bound in Theorem 4.3 implies that the macrostates in X ß are overhelmingly the most probable among all possible macrostates of the turbulent system. 5. Equivalence of Ensembles

15 nalysis of Statistical Equilibrium Models 341 In this section e classify the possible relationships beteen the sets of equilibrium macrostates X and X ß. We begin by establishing that each of these equilibrium sets is nonempty. Then e turn to our second set of main results, hich concern the equivalence or lack of equivalence beteen the microcanonical and canonical ensembles. In many statistical mechanical models it is common for the microcanonical and canonical ensembles to be equivalent, in the sense that there is a one-to-one correspondence beteen their equilibrium macrostates. Hoever, in models of coherent structures in turbulence there can be microcanonical equilibria that are not realized as canonical equilibria. s is shon in our companion paper [10], and in a real physical application in [31], nonequivalence occurs often in the turbulence models and produces some of the most interesting coherent mean flos. 5.1 Existence of Equilibrium States We continue to assume that the support l of 3 is compact. It follos from this assumption that the MÐ;Ñ gros faster than quadratically as ² ; ² Ä ; the straightforard proof is left to the reader. Consequently, for any Q the sets O Q»Ö; P ÐÑÀMÐ;ÑŸQ are precompact ith respect to the eak topology of PÐ Ñ; this follos from the fact that each OQ is a subset of a closed ball in P ÐÑ. Moreover, MÐ;Ñ is loer semicontinous ith respect to the eak topology on PÐ Ñ. This property is an immediate consequence of the relationship MÐ;Ñ œ MÐ; Ñß Ð&ÞÑ sup here ; is the pieceise constant approximating function defined in Lemma 3.3. Indeed, since each function ;ÈMÐ;Ñ is eakly loer semicontinuous, the eak loer semicontinuity of MÐ;Ñ follos. Thus the sets O Q are closed ith respect to the eak topology. This property, in combination ith the eak precompactness of these sets, implies that the sets O Q are compact ith respect to the eak topology. Concerning the proof of (5.1), Jensen's inequality guarantees that MÐ; Ñ Ÿ MÐ;Ñ, hile Lemma 3.3 and the strong loer semicontinuity of M proved right after its definition in (3.1) yield lim inf Ä MÐ; Ñ MÐ;Ñ; (5.1) is thus proved. The existence of microcanonical and canonical equilibrium states no follos immediately from the direct methods of the calculus of variations. In the microcanonical case, for any admissible ÐIß > Ñ T a minimizing sequence exists that converges eakly to a minimizer ; X, by virtue of the properties of the objective functional Mjust derived and the continuity, ith respect to the eak topology, of the constraint functionals L and G (Lemma 4.1 Ð-ÑÑ. In the canonical case, since M gros superquadratically as ² ; ² Ä hile L gros quadratically and G linearly, it follos that for any Ð ß Ñ a minimizing sequence exists that converges eakly to a minimizer ; X ß. The details of this routine analysis are omitted. Typically, the solutions of the variational problems (4.) and (4.12) for the microcanonical and canonical ensembles, respectively, are expected to be unique. Indeed, numerical solutions of these problems, such as those carried out in [10], demonstrate that apart from degeneracies and bifurcations the equilibrium sets X and X ß are singleton sets. In the next subsection, ithout any special assumptions concerning uniqueness of equilibrium solutions for either model, e give complete and general results about the correspondence beteen these sets. In addition, the results given in the next subsection require no continuity or smoothness assumptions of the equilibrium solutions ith respect to the model parameters ÐIß > Ñ or Ð ß Ñ.

16 342 R. ELLIS, K. HVEN and B. TURKINGTON 5.2 Dual Thermodynamic Functions In the analysis to follo, e sho ho the properties of the thermodynamic functions for the microcanonical and canonical ensembles determine the correspondence, or lack of correspondence, beteen equilibria for these to ensembles. For the microcanonical ensemble the thermodynamic function is the microcanonical entropy WÐ Ñ defined in the constrained variational principle (4.), the solutions of hich constitute the equilibrium set X. For the canonical ensemble the thermodynamic function is the free energy F Ð ß Ñ defined in the unconstrained variational principle (4.11), the solutions of hich constitute the equilibrium set X ß. The basis for the equivalence of ensembles is the conjugacy beteen W and F; that is, F is the Legendre-Fenchel transform of W [13, 32]. This basic property is easily verified as follos: F Ð ß Ñ œ minömð;ñ LÐ;Ñ GÐ;ÑÑ Ð&ÞÑ ; œ inf minömð;ñ LÐ;Ñ GÐ;ÑÀ LÐ;Ñ œ Iß GÐ;Ñ œ > Iß > ; œ inf Ö I WÐIß Ñ > > œw Ð ß ÑÞ Consequently, F is a concave function of Ð ß Ñ, hich runs over. By contrast, W itself is not necessarily concave. The concave hull of W is furnished by the conjugate function of F, namely, F œw, hich satisfies the inequality WÐIß > Ñ Ÿ inf Ö I > F Ð ß Ñ œ W ÐIß > ÑÞ Ð&Þ$Ñ ß The relationship beteen microcanonical equilibria and canonical equilibria depends crucially on the concavity properties of the microcanonical entropy W. To this end, e introduce the subset V T on hich the concave hull W coincides ith W; that is, ÐIß > Ñ V if and only if W ÐIß > Ñ œ WÐIß > Ñ. Equivalently, V consists of those points ÐIß > Ñ T for hich there exists some Ð ß Ñ such that WÐI ß> Ñ Ÿ WÐ Ñ ÐI IÑ Ð> > Ñ Ð&Þ%Ñ for all ÐI ß > Ñ. This condition means that W has a supporting plane, ith normal determined by Ð ß Ñ, at the point Ð Ñ. Such points Ð Ñ are precisely those points of T at hich W has a nonempty superdifferential; this set consists of all Ð ß Ñ for hich (5.4) holds [13, 32]. s e ill see in the next subsection, the concavity set V plays a pivotal role in the criteria for equivalence of ensembles. 5.3 Correspondence beteen Equilibrium Sets The folloing theorem ensures that for constraint pairs in V the microcanonical equilibria are contained in a corresponding canonical equilibrium set, hile for constraint pairs in TÏV the microcanonical equilibria are not contained in any canonical equilibrium set. Theorem 5.1: Ð+Ñ If ÐIß > Ñ T belongs to V, then X X ß for some Ð ß Ñ. Ð,Ñ If ÐIß > Ñ T does not belong to V, then X X œ g for all Ð ß Ñ. ß

17 nalysis of Statistical Equilibrium Models 343 Proof: Ð+Ñ If ÐIß > Ñ belongs to V, then there exists Ð ß Ñ such that (5.4) holds for all ÐI ß > Ñ. Thus for all ÐI ß > Ñ I > WÐIß > Ñ Ÿ I > WÐI ß > ÑÞ By (5.2) and (4.11) this inequality implies that I > WÐ Ñ Ÿ inf Ö I > WÐ Ñ ÐI ß > Ñ œ F Ð ß Ñ œ minömð;ñ LÐ;Ñ GÐ;Ñ Þ ; To sho the claimed set containment, take any ; X and note that LÐ; Ñ œ I, GÐ; Ñ œ >, and MÐ; Ñ œ WÐIß > Ñ. Substituting these expressions into the preceding display yields MÐ;Ñ LÐ;Ñ GÐ;ÑŸ minömð;ñ LÐ;Ñ GÐ;Ñ Þ ; Since X ß consists of the minimizers of M L G, it follos that ; X ß. This completes the proof of Ð+Ñ. Ð,Ñ By (5.2) the hypothesis implies that for all Ð ß Ñ Then any ; X satisfies WÐ Ñ I > F Ð ß ÑÞ MÐ;Ñ LÐ;Ñ GÐ;Ñ œ WÐ Ñ I > F Ð ß Ñ œ minömð;ñ LÐ;Ñ GÐ;Ñ Þ ; Thus ; does not minimize M L G and hence does not belong to X ß. Since Ð ß Ñ is arbitrary, this completes the proof of Ð,Ñ. ccording to this theorem, henever V Á T, there are microcanonical equilibria that are not realized by any canonical equilibria. On the other hand, all canonical equilibria are contained in some microcanonical equilibrium set, and V is exhausted by the constraint pairs realized by all canonical equilibria. These further results are given in the next theorem. Theorem 5.2: Ð+Ñ The concavity set V consists of all constraint pairs realized by the canonical equilibria; that is, - ß ß V œ ÖLÐX Ñß GÐX ÑÑÀ Ð ß Ñ Þ Ð&Þ&Ñ Ð,Ñ Each canonical equilibrium set X ß consists of all microcanonical equilibria hose constraint pairs are realized by X ; that is, for any Ð ß Ñ ß X œ-öx ÀÐ Ñ ÐLÐX ÑßGÐX ÑÑ Þ Ð&Þ'Ñ ß ß ß Proof: Ð+Ñ The containment of V in the union is immediate from Theorem 5.1 Ð+Ñ. To sho the opposite containment e argue by contradiction, supposing that for some Ð ß Ñ and some ; X ß ÐIß > Ñ œ ÐLÐ; Ñß GÐ; ÑÑ TÏV. Then by (5.3) e find that ß

18 344 R. ELLIS, K. HVEN and B. TURKINGTON WÐ Ñ I > F Ð ß Ñ œ MÐ; Ñ Ÿ WÐ ÑÞ We thus obtain the desired contradiction. This completes the proof of part (a). Ð,Ñ The containment of X in the union is straightforard. Let ß ; X ß and set I œ LÐ; Ñ and > œ GÐ; Ñ. Then MÐ; Ñ I > Ÿ MÐ;Ñ LÐ;Ñ GÐ;Ñ for all ;. For those ; that satisfy the constraints LÐ;Ñ œ I, GÐ;Ñ œ >, e therefore find that MÐ; Ñ Ÿ MÐ;Ñ. Hence ; X. The opposite containment is also straightforard. If IœLÐ;Ñ µ and > œgð;ñ µ for some µ ; X, then for any ; X e have MÐ; Ñ Ÿ MÐ; µ Ñ. Since µ ; X, e obtain ß ß min ; ÖMÐ;Ñ LÐ;Ñ GÐ;Ñ œ MÐ; µ Ñ I > MÐ; Ñ LÐ; Ñ GÐ; Ñ. Hence ; X ß. This completes the proof of part Ð,Ñ. Theorems 5.1 and 5.2 allo us to classify the microcanonical constraint parameters ÐIß > Ñ according to hether or not equivalence of ensembles holds for those parameters. In fact, the admissible set T can be decomposed into three disjoint sets, here (1) there is a one-to-one correspondence beteen microcanonical and canonical equilibria, (2) there is a many-to-one correspondence from microcanonical equilibria to canonical equilibria, and (3) there is no correspondence. In order to simplify the precise statement of this classification, let us assume that the microcanonical entropy WÐ Ñ is differentiable on its domain T. Then, for each microcanonical parameter ÐIß > Ñ T there is a corresponding canonical parameter Ð ß Ñ determined locally by the familiar thermodynamic relations `W `W `I, `> œ œ Þ Under this assumption, e have the folloing classification, hich is a consequence of Theorems 5.1 and 5.2. ÐÑ Full equivalence. If ÐIß > Ñ belongs to V and there is a unique point of contact beteen W and its supporting plane at Ð Ñ, then X coincides ith I ß. ÐÑ Partial equivalence. If ÐIß > Ñ belongs to V but there is more than one point of contact beteen W and its supporting plane at Ð Ñ, then X is a strict subset of X ß. Moreover, X ß contains all those X for hich ÐI ß > Ñ is also a point of contact. Ð$Ñ Nonequivalence. If ÐIß > Ñ does not belonging to V, then X is disjoint from X ß. In fact, X is disjoint from all canonical equilibrium sets. For a complete discussion of results of this kind, e refer the reader to our paper [9], here e state and prove the corresponding results in a much more general setting and ithout the simplifying assumption that W is differentiable. 6. Concluding Discussion The theorems given in this paper support the theory developed in our companion paper [10], here e argue in favor of a statistical equilibrium model for geostrophic turbulence that is defined by a prior distribution on potential vorticity and microcanonical constraints on energy

19 nalysis of Statistical Equilibrium Models 345 and circulation. The first set of main results in the present paper furnishes an especially simple methodology for deriving the equilibrium equations and associated LDP's for that model. The second set of main results shos that the microcanonical ensemble has richer families of equilibrium solutions than the corresponding canonical ensemble, hich omits those microcanonical equilibrium macrostates corresponding to parameters ÐIß > Ñ not lying in the concavity set of the microcanonical entropy. The computations included in [10] for coherent flos in a channel ith zonal topography demonstrate that these omitted states constitute a substantial portion of the parameter range of the physical model. s this discussion shos, the equivalence-of-ensemble issue is a fundamental one in the context of these local mean-field theories of coherent structure henever the phenomenon of selforganization into coherent states is modeled. In recent ork [31], the statistical theory presented in [10] and analyzed in the present paper is applied to the active eather layer of the atmosphere of Jupiter. When appropriately fit to a Î-layer quasi-geostrophic model, the theory correctly produces large-scale features that agree qualitatively and quantitatively ith features observed by the Voyager and Galileo missions. For instance, for the channel domain in the southern hemisphere containing the Great Red Spot and White Ovals [31, Figure 2], the theory produces both the alternating east-est zonal shear flo and to anticyclonic vortices embedded in it. In particular, the size and position of the vortices closely match the size and position of the GRS and White Ovals, hile the zonally-averaged velocity profile is extremely close to the profile deduced by Limaye [16]. Similarly, for a northern hemisphere channel that contains no permanent vortices, the theory correctly predicts a zonal shear flo ith no embedded vortices. This agreement beteen theory and observation demonstrates the practical utility of the model analyzed in the preceding sections of the present paper. In Section 5.1 of [10], the nonlinear stability of the equilibrium states for the model is shon by applying a Lyapunov argument. The required Lyapunov functional is constructed from the rate function M for the coarse-grained process [ ß and the constraint functionals L and G. In the canonical case, this construction is identical ith a ell-knon method due to rnold [1], no called the energy-casimir method [12]. On the other hand, in the microcanonical case hen nonequivalence prevails, this method breaks don and hence it is necessary to give a more refined stability analysis. The required refinement, hich makes use of the concept of the augmented Lagrangian from constrained optimization theory, is presented in [10]. For example, the Jovian flos realized in [31] fall in this regime. Thus the variational principles derived from the large deviation analysis in Section 4 and examined in Section 5 also have important implications for hydrodynamic stability criteria. References [1] rnold, V.I., Mathematical Methods of Classical Mechanics, Springer-Verlag, Ne York 19. [2] Balescu, R., Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley & Sons, Ne York 195. [3] Balian, R., From Microphysics to Macrophysics, vol. I, (trans. by D. ter Haar and J.G. Gregg), Springer-Verlag, Berlin [4] Carnevale, G.F. and Frederiksen, J.S, Nonlinear stability and statistical mechanics of flo over topography, J. Fluid Mech. 15 (19), [5] Dembo,. and Zeitouni, O., Large Deviations Techniques and pplications, 2nd ed., Springer-Verlag, Ne York 199. [6] DiBattista, M.T., Majda,.J., and Grote, M.J., Meta-stability of equilibrium statistical structures for prototype geophysical flos ith damping and driving, Phys. D 151 (2001),

20 346 R. ELLIS, K. HVEN and B. TURKINGTON [] Dudley, R.M., Real nalysis and Probability, Chapman & Hall, Ne York 199. [] Ellis, R.S. and Dupuis, P., Weak Convergence pproach to the Theory of Large Deviations, John Wiley & Sons, Ne York 199. [9] Ellis, R.S., Haven, K., and Turkington, B., Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles, J. Stat. Phys. 101 (2000), [10] Ellis, R.S., Haven, K., and Turkington, B., Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flos, Nonlinearity 15 (2002), [11] Holloay, G., Eddies, aves, circulation, and mixing: statistical geofluid mechanics, nn. Rev. Fluid Mech. 1 (196), [12] Holm, D., Marsden, J., Ratiu, T., and Weinstein,., Nonlinear stability of fluid and plasma equilibria, Rev. Mod. Phys. 123 (195), [13] Ioffe,.D., and Tihomirov, V.M., Theory of External Problems, Elsevier North- Holland, Ne York 199. [14] Joyce, G. and Montgomery, D., Negative temperature states for the to-dimensional guiding center plasma, J. Plasma Phys. 10 (193), [15] Kraichnan, R., Statistical dynamics of to-dimensional flo, J. Fluid Mech. 6 (195), [16] Limaye, S.S., Jupiter: ne estimates of the mean zonal flo at the cloud level, Icarus 65 (196), [1] Lynch, J. and Sethuraman, J., Large deviations for processes ith independent increments, nn. Prob. 15 (19), [1] Majda,.J. and Holen, M., Dissipation, topography and statistical theories of large scale coherent structure, Commun. Pure ppl. Math. 50 (199), [19] Miller, J., Statistical mechanics of Euler equations in to dimensions, Phys. Rev. Lett. 65 (1990), [20] Miller, J., Weichman, P., and Cross, M.C., Statistical mechanics, Euler's equations, and Jupiter's red spot, Phys. Rev. 45 (1992), [21] Montgomery, D., Matthaeus, W.H., Stribling, W.T., Martinez, D., and Oughton, S., Relaxation in to dimensions and the sinh-poisson equation, Phys. Fluids 4 (1992), 3-6. [22] Morrison, P.J., Hamiltonian description of the ideal fluid, Rev. Mod. Phys. 0 (199), [23] Onsager, L., Statistical hydrodynamics, Nuovo Cimento 6 (1949), [24] Parisi, G., Statistical Field Theory, ddison-wesley, California 19. [25] Robert, R., maximum-entropy principle for to-dimensional perfect fluid dynamics, J. Stat. Phys. 65 (1991), [26] Robert, R. and Sommeria, J., Statistical equilibrium states for to-dimensional flos, J. Fluid Mech. 229 (1991), [2] Salmon, R., Lectures on Geophysical Fluid Dynamics, Oxford Univ. Press, Ne York 199. [2] Salmon, R., Holloay, G., and Hendershott, M.C., The equilibrium statistical mechanics of simple quasi-geostrophic models, J. Fluid Mech. 5 (196), [29] Shepherd, T.G., Symmetries, conservation las, and Hamiltonian structures in geophysical fluid dynamics, dv. Geophys. 32 (1990), [30] Turkington, B., Statistical equilibrium measures and coherent states in to-dimensional turbulence, Commun. Pure ppl Math. 52 (1999), [31] Turkington, B., Majda,., Haven, K., and DiBattista, M., Statistical equilibrium predictions of jets and spots on Jupiter, Proc. Nat. cad. Sci. US 9 (2001),

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