ANALYSIS OF STATISTICAL EQUILIBRIUM MODELS OF GEOSTROPHIC...

Journal of Applied Mathematics and Stochastic Analysis, :4 (2002), 341-361.15

ANALYSIS OF STATISTICAL EQUILIBRIUM MODELSOF GEOSTROPHIC TURBULENCE

RICHARD S. ELLIS and BRUCE TURKINGTON1 2

University of MassachusettsDepartment of Mathematics and Statistics

Amherst, MA 01003 USAE-mail: rsellismath.umass.edu, [email protected]

KYLE HAVENNew York University

Courant Institute of Mathematical SciencesNew York, NY 10012 USA

E-mail: [email protected]

(Received January, 2002; Revised August, 2002)

Statistical equilibrium lattice models of coherent structures in geostrophic tur-bulence, formulated by discretizing the governing Hamiltonian continuumdynamics, are analyzed. The first set of results concern large deviation principles(LDP's) for a spatially coarse-grained process with respect to either the canonicaland/or the microcanonical formulation of the model. These principles are derivedfrom a basic LDP for the coarse-grained process with respect to product measure,which 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 theHamiltonian equations. The second set of results addresses the equivalence ornonequivalence of the microcanonical and canonical ensembles. In particular, ne-cessary and sufficient conditions for a correspondence between microcanonicalequilibria and canonical equilibria are established in terms of the concavity of themicrocanonical entropy. A complete characterization of equivalence of ensemblesis deduced by elementary methods of convex analysis.

1This research was supported by a grant from the Department of Energy (DE-FG02-99ER25376) and by a grant from the National Science Foundation (NSF-DMS-9700852).

2This research was supported by a grant from the Department of Energy (DE-FG02-99ER25376) and by a grant from the National Science Foundation (NSF-DMS-9971204).

Printed in the U.S.A. ©2002 by North Atlantic Science Publishing Company 341

342 R. ELLIS, K. HAVEN and B. TURKINGTON

The mathematical results proved in this paper complement the physical reasoningand numerical computations given in a companion paper, where it is argued thatthe statistical equilibrium model defined by a prior distribution on potentialvorticity fluctuations and microcanonical conditions on total energy andcirculation is natural from the perspective of geophysical applications. Large Deviation Principle, Cramer's Theorem, Coarse Graining,´Key words:Statistical Models of Turbulence, Hamiltonian Systems. 76F55, 60F10.AMS subject classifications:

1. Introduction

The problem of developing a statistical equilibrium theory of coherent structures in two-dimensional turbulence or quasi-geostrophic turbulence has been the focus of numerousmathematics 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 acoherent large-scale vortex in the negative temperature regime corresponding to sufficientlylarge kinetic energies. The quantitative aspects of the point vortex model were elaborated byJoyce-Montgomery [14], and subsequently these predictions were compared with directnumerical simulations of the end state of freely decaying turbulence [21]. Another approachdue to Kraichnan [15], called the energy-enstrophy model, was based on spectral truncation ofthe underlying fluid dynamics. This Gaussian model was extensively applied to geophysicalfluid flows [4, 27, 28]; for instance, to the organization of barotropic, quasi-geostrophic flowsover bottom topography [11]. The next major development was the Miller-Robert model [20,26], which enforced all the invariants of ideal motion and included the previous models asspecial or limiting cases. Recently, this general model has been scrutinized from thetheoretical standpoint [30] and from the perspective of practical applications [6, 18]. Theseinvestigations have clarified the relation between various formulations of the general modeland the simpler models, and they have revealed a formulation that has firm theoreticaljustification and rich physical applications. This formulation of the statistical equilibriumtheory of coherent structures is described in our companion paper [10], and a particularlystriking application of this model is given in another paper [31], where it is implemented topredict the permanent jets and spots in the active weather layer of Jupiter. The purpose of the current paper is to prove the theorems that legitimize the theoreticalmodel developed in [10]. We therefore follow the line of development in that paper, in whichthe nonlinear partial differential equation governing the dynamics of the fluid an infinite-�dimensional 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 tothe continuum limit of this model. These results are large deviation principles (LDP's) that�is, exponential-order refinements of the law of large numbers for a spatially coarse-grained�process which justify the definition of the equilibrium states for the model. We base our directproofs of these LDP's on an elementary large deviation analysis of the coarse-grained processwith respect to the product measure that underlies the lattice model. Our second set of main results concerns the relationship between the microcanonical andcanonical equilibrium states; in particular, the equivalence or nonequivalence

Analysis of Statistical Equilibrium Models 343

of the microcanonical and canonical ensembles with respect to energy and circulation. Themost interesting result is that microcanonical equilibria are often not realized as canonicalequilibria. Significant examples of this remarkable phenomenon are given in [10, 31], whereits impact on hydrodynamic stability is also discussed. The outline of the paper is as follows. In Section 2 we formulate the statistical equilibriumtheory described in [10], and we define the coarse-grained process used in the analysis of thecontinuum limit. We state and prove the basic LDP for this process in Section 3, and then inSection 4 we present the LDP's for the microcanonical and canonical ensembles, relying ongeneral theorems established in [9]. Then in Section 5 we turn to the issue of equivalence ofthose ensembles at the level of their equilibrium states, giving necessary and sufficientconditions for equivalence in terms of the concavity of the microcanonical entropy. In theconcluding Section 6, we point to some of the physical implications of our results.

2. Statement of Problem

The continuum dynamics that underlies our statistical equilibrium lattice model can be viewedas a noncanonical Hamiltonian system with infinitely many degrees of freedom. Namely, itcan be written in the form

��

��

where denotes the variational derivative, and is an operator with the property that� � ��

��

� ��

is a Poisson bracket [22]. Here is a bounded spatial domain, and represents� � ��

normalized Lebesgue measure on , normalized so that . For the sake of� � �� definiteness, the flow domain is chosen to be the channel�

� � ��

thus .� � ��

The scalar function represents either the vorticity in two-dimensional flow� � �� or the potential vorticity in quasi-geostrophic flow. evolves according to�

��

� �� !�� , where � �

In this formula is the streamfunction defined by , where is a� � �� " � # "uniformly elliptic operator on and is a given continuous real-valued function on� # � #�� ; satisfies appropriate boundary conditions on the boundary of . Consequently, canbe recovered from via a Green's function :� �� $

� �� #��

$ $ $ $

Hence (2.3) is an equation in alone. This equation takes the Hamiltonian form (2.1) when�% � � �� &� and

�� #��

� �

�'�

� �� #�� #��

(

$ $ $ $� � �

� �

Thus is a quadratic form in in which interactions between and are� � �� $

governed by the Green's function . This Hamiltonian formulation of two-dimensional�� $

or quasi-geostrophic flow is discussed in [12, 22, 27, 29]. The Hamiltonian dynamics (2.1) conserves the total energy and the total circulation�

)�� #�� *��

�

In addition, there is an infinite family of Casimir invariants parametrized by continuous real-valued functions and defined as+

, �� +�� -��

�

For the particular choice of a channel domain , the linear impulse�

��

� �� #��

is also conserved. For simplicity, we shall ignore it throughout this paper since it plays a roleidentical to in the theory.) When and , (2.3) is the vorticity transport" � � � � � � � � � � � # � �

�

equation, which is an equivalent formulation of the Euler equations governing an incom-pressible, inviscid flow in two dimensions. When and ," � � � . # � � ��

(2.3) is referred to as the quasi-geostrophic equations; the Coriolis parameter and the� / Rossby deformation radius , and the effective bottom topography is. 0 � � 1 � � ��

a given real-valued function. With this choice of and , (2.3) governs the motion of a" #shallow, rotating layer of a homogeneous, incompressible, inviscid fluid in the limit of smallRossby number [27]. Throughout the rest of the paper, we refer to , 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 on a range of�scales, we consider ensembles of solutions, instead of individual deterministic solutions. Thisstatistical mechanics approach is standard for a Hamiltonian system with finitely many degreesof freedom in canonical form [2, 3, 24]. In order to carry over this methodology to the infinitedimensional Hamiltonian system (2.1), it is necessary to formulate an appropriate sequence of

lattice models obtained by discretizing the system. Next we define these probabilistic latticemodels. For each , the flow domain is uniformly partitioned into microcells. We let2 0 + � 2

� �2 2 2 denote a uniform lattice of sites indexed by , where each corresponds to a+ 3 3 0unique microcell of ; each satisfies . The lattice4�3� 4�3� �4�3�� +� � 2

discretizations of the Hamiltonian and circulation are given by� )

� ��3��3 0 �� 3�� # �3�� 3� 3 ��3 �� # �3 ��5�2 2 2 2 2�

+30 3 0

$ $ $� 2

2 2$

� ��

and

) ��3��3 0 �� 3�� # �3�� 6�2 2 2�+

30

�2

2

��

where is the average over of the Green's function , and� �3� 3 � 4�3�(4�3 � �� 2$ $ $

# �3� 4�3� #2 is the average over of :

and � �3� 3 � � + �� # �3� � + #�� 7�2 2 2$ $ $

24�3�4�3 � 4�3�

� � �$

� � �

The Casimirs (2.6) can be discretized similarly as functions ., ��3��3 0 ��2 2� We shall not attempt to prescribe a lattice dynamics under which the functions , and� )2 2

, ,2 2 are exact invariants. In fact, a lattice dynamics conserving the nonlinear enstrophies isnot known. The subtleties associated with the rigorous formulation of this statistical modelare fully discussed elsewhere [10, 30]. Rather than pursue this direction any further here, weshall simply impose a joint probability distribution on the lattice potential vorticities2

��3� 3 0 � ,, captures the invariance of the nonlinear enstrophies and the phase� 2 2

volume .�� Let be a probability distribution on and let be the support of . In order to simplify� � � �our presentation throughout the paper, we assume that is compact. The extension to the�case of noncompact support can be carried out via appropriate modifications of the techniquesthat we use. We define the phase space to be , denote by the vector of � 2 2

+2 � 0lattice potential vorticities , and let be the product measure on with��3�� 3 0 �� 2 2 2

one-dimensional marginals . That is, for any Borel subset of , we define� 8 2

�2 28 8 30

�8� � �� 3�� 2

We refer to as the prior distribution of the lattice potential vorticities.2

As is explained in [10], can be interpreted as a canonical ensemble with respect to2

lattice enstrophy , in which the distribution is determined by the function . Alter-, +2 �natively, in the perspective adopted in the Miller-Robert model [19, 25], is determined from�the continuum initial data by

� � ��9� � ��9� ��

��

While this choice of prior distribution is conserved by the continuum equations, it is notpreserved under passage to a discretized dynamics. In practice, therefore, it is better to view �

as a free parameter in the model. For discussions on the choice of the prior distribution inspecific applications, see [10, §6.1] and [31, page 12347].

2.2 Definition of Ensembles

In terms of the prior distribution , we can form two different statistical equilibrium models2

by using either the microcanonical ensemble or the canonical ensemble with respect to theinvariants and . Given , for any Borel subset of the microcanonical� ) : 82 2 2� ensemble is defined by

; �8� � �8 �� 0 �< � �< � ��) 0 � � � � �� <�2� 2 2 2

��

This is the conditional probability distribution obtained by imposing the constraints that �2

and lie within the (small) closed intervals around and , respectively. Given real) <2 �numbers and , the canonical ensemble is defined by� �

; �8� � �� + � �� + ) �� 2� � 2 2 2 2 2�

= � � �8

� � � �2� exp � �

where

= � � � � �� + � �� + ) �� 2 2 2 2 2 2� � � � � 2

exp

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 probabilitydistributions (2.11) and (2.12). In this limit we obtain the continuum description of thepotential 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, definedby a local averaging of the random microstate over an intermediate spatial scale. This�process is constructed as follows. For such that , the flow domain is uniformly partitioned into . 0 . � 2 > � .

macrocells denoted by , . We assume that each macrocell satisfies? @ �?.�� .�>.

��? � � � > + > 4�3�.�A . 2 . and contains microcells and that

lim max diam. B 1A 0 �� @ �> �

�? �� !�.

.�A

For instance, can be dyadically refined into microcells, and these can be� + � 22

collected into a partition of macrocells satisfying (2.13). With respect to the> � ?. .�A.

collection of macrocells , we introduce a doubly-indexed process ,? C � C �� .�A 2�. 2�.

defined for and by� 0 � 0 �2

C �� D � �� '�2�. 2�.�A ?

>

A��

�.

.�A

whereD � D �� 3� ��*�2�.�A 2�.�A

�+ >

30?2 .

.�A

�

For the piecewise-constant process takes values in and can be� 0 C �� &� " � � �2 2�.

studied with respect to either the microcanonical ensemble (2.11) or the canonical ensemble(2.12). We refer to as the coarse-grained process.C2�.

With respect to each ensemble, our goal is to determine the set of functions on which"

C 2 B 1 . B 12�. concentrates in the continuum limit defined by first sending and then .More precisely, we seek to find the smallest subsets of , and , such that" � � <�

��

; �C 0 � B � 2 B 1 . B 1 B ��-�<� <�2� 2�.

� �� as , , and

and

; �C 0 � B � 2 B 1 . B 1 ��5�2� � 2�. �� as and

The subsets and 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 subsetsis shown to arise by solving a variational principle. The variational principles are derivedfrom the LDP's for with respect to the two ensembles. In order to prove these twoC2�.

LDP's, we first prove a basic LDP for with respect to the product measure definedC2�. 2

in (2.10). This is done in the next section.

3. Basic LDP for the Coarse-Grained Process

Before stating the basic LDP for with respect to , we first discuss the form of the rateC2�. 2

function . Throughout the paper the term rate function is used to describe any lowerEsemicontinuous function mapping a complete, separable metric space into .� � 1 � For we define the cumulant generating function� �0

F� � � G ��9�� log��

�9

which is finite because has compact support, and for we introduce the lower semi-� �9 0continuous, conjugate convex function

H�9� � � 9 � F� ��0

sup� �

� �

The normalized sums defined in (2.15) are sample means of the i.i.d. randomD + >2�.�A 2 .

variables , . Therefore, Cramer's theorem [5] implies that, for each´��3� 3 0 ?.�A

A � �� @ �> �D �2 0 �. 2�.�A 2, the sequence satisfies an LDP with respect to with

scaling constants and rate function . In other words, for any closed subset of + > H �2 . �

lim sup log inf2 B 1

�D 0 � � I � H�9��9 0 �

�+ > 2 2�.�A2 .

and for any open subset of � �

lim inf log inf2 B 1

�D 0 �� / � H�9�9 0 �

�+ > 2 2�.�A2 .

The rate function for the LDP of with respect to is defined for to beC J 0 " � �2�. 2 �

E�J� � H�J�� !��

�

Since is nonnegative and convex, it follows that is well-defined, nonnegative, and convex.H EWe claim that is lower semicontinuous with respect to the strong topology on .E " � � �Assume that in . Then there exists a subsequence suchJ B J " � � J2 2

�A

that lim lim inf and a.s. Fatou's lemma and the lower2 B 1 2 2B 1 2 2A A AE�J � � E�J � J B J

semicontinuity of on yield the desired conclusion:H �

lim inf lim inf2 B 1 2 B 1

E�J � � H�J �� 2 2A

��

A �

/ H�J �� / H�J�� E�J�2 B 1� ��

lim infA

2A � �

We now state the two-parameter LDP for with respect to . For any subset ofC ,2�. 2

" � � E�, � E , � , denotes the infimum of over . Theorem 3.1: Consider with the strong topology. The sequence satisfies the" � � C

2�.�

LDP on , in the double limit and , with rate function . That is, for" � � 2 B 1 . B 1 E �any strongly closed subset of � " � � �

lim sup lim sup log. B 1 2 B 1

�C 0 � � I � E�� + 2 2�.2

and for any strongly open subset of � " � � �

lim inf lim inf log. B 1 2 B 1

�C 0 �� / � E��+ 2 2�.2

We now prove the upper and lower large deviation bounds for in separate, butC2�.

elementary steps. The following lemma is needed in both steps; it is an immediateconsequence of Lemmas 2.5 2.8 in [17].� Lemma 3.2: For each the sequence satisfies the LDP. ��D � @ � D �� 2 0 �2�.�� 2�.�>.

on with scaling constants and the rate function�>2 .

. + >

� � @ � � K H� �� > A

>

A��

�.

We define to be the set ofProof of the Large Deviation Upper Bound: ".

J 0 " � � J�� @ � 0 � >A�� A ? � >� � � � � of the form for some . Let be a� .

.�A .

strongly closed subset of . For " � � . 0 �

� � �� @ � � 0 � � �� 0 � �. � > A ?>

>

A��

� � � �. .�A.

.�is a closed subset of . By Lemma 3.2, it follows that�>.

lim sup log2 B 1

�C 0 � ��+ 2 2�.2

lim sup log� ��D � @ � D � 0 � �2 B 1

�> +

>2 2�.�� 2�.�> .

. 2

..

inf I � H� �� @ � � 0 ��>

>

A��A � > .

.

.

.� ��

inf� � H�J�� J 0 � L "� ��

� .

infI � H�J�� J 0 ��

�

.� � E��

Sending gives the desired large deviation upper bound. . B 1 �

The following lemma is needed both hereProof of the Large Deviation Lower Bound:and later in the paper. We denote by the norm on .M & M " � � � Lemma 3.3: Let be any function in . For defineJ " � � . 0 �

J �� J �A�� J �A� � > J�� . . ? . .

>

A�� ?

� �.

.�A

.�A

, where �

Then as , .. B 1 M J � J M B .

Proof: For any given there exists a bounded Lipschitz function mapping into� � �: � � � � and satisfying [7]. Since the operator mapping is anM J � M I � 0 " � � K �

.

orthogonal projection,

M J � M � M �J � � M I M J � M � �!�. . .� � � �

Hence it suffices to estimate , where is a Lipschitz function with constantM � M� � �.

4 � 1 . A straightforward calculation coupled with (2.13) gives

M � M I 4 & � �? �� B . B 1 �!!�A 0 �� @ �> �

� � . .�A.

max diam as � This completes the proof. �

Given a strongly open subset of , let be any function in and choose so� " � � J � : � �that , the ball in with center and radius , is a subset of . Since8�J� � " � � J ��

M J � J M B . B 1 N 0. as (Lemma 3.3), we can also choose such that for all. 0 N 8�J � � � 8�J� � . Since. � �

� � � � @ � � 0 � � �� 0 8�J � �.� � > A ? .>

>

A�� . .�A

..

is an open subset of , it follows from Lemma 3.2 that for all �>. . / N lim inf log

2 B 1�C 0 ��

+ 2 2�.2

lim inf log/ �C 0 8�J � ��2 B 1

�+ 2 2�. .2

�

lim inf log� ��D � @ � D � 0 � �2 B 1

�> +

>2 2�.�� 2�.�> .�

. 2

..

�

inf / � H� �� @ � � 0 ��>

>

A��A � > .�

.

.

.� ��

/ � H�J �A�� H > J�� > >

> >

A�� A��. .

?. .

. .

.�A

� � � � �

/ � H�J�� H�J�� E�J�� >

A��?

.

.�A

� ��

The last inequality in this display follows from Jensen's inequality. We have proved that forarbitrary J 0 �


�C 0 �� / � E�J��+ 2 2�.2

Taking the supremum of over yields the desired large deviation lower bound. � E�J� J 0 � �

This completes the proof of Theorem 3.1.

4. LDP's with Respect to the Two Ensembles

In this section we present LDP's for with respect to the microcanonical and canonicalC2�.

ensembles defined in (2.11) and (2.12). The theorems follow from a general theory presentedin [9]. The proofs of the LDP's depend in part on properties of various functionals given inthe next subsection.

4.1 Properties of , , and � ) � � )2 2

We recall the definitions of the lattice energy in (2.7), the circulation in (2.8), and the� )2 2

functionals and in (2.4) and (2.5). The proofs of the LDP's with respect to the two� )ensembles rely on boundedness and continuity properties of and for ��J� )�J� J 0 " � � �and on the fact that uniformly over , and are asympto- 2 2 2� )tic to and ; the latter two quantities are defined by replacing in��C � )�C � �� 2�. 2�.

(2.4) and (2.5) by the -valued process . The proof of this approxi-" � � C �� 2�.�

mation for the circulation is immediate since it holds with equality. For the proof depends�2

on the fact that the vortex interactions are long-range, being determined by the Green'sfunction . For this reason, is well approximated by a function of the coarse-�� $

2

grained process . In physical parlance, the turbulence model under consideration is anC2�.

asymptotically exact, local mean-field theory [20]. The next lemma proves theapproximations as well as the boundedness and continuity properties of and needed for� )the LDP's. Lemma 4.1: �+� The sequence satisfiesC2�.

lim lim sup. B 1 2 B 1 � 0

�� C �� '�� 2

2 2�.

and

lim lim sup. B 12 B 1� 0

�) ��)�C �� '� 2

2 2�.

�#� and are bounded on the union, over , of the supports of the -dis-� ) � I . � 2 2

tributions of .C2�.

and are continuous with respect to the weak topology on and thus with�F� � ) " � � �respect to the strong topology on ." � � � Proof: ( ) A short calculation shows that for all , thus+ ) �� )�C �� 02 2�. 2

proving (4.2). Carrying out the multiplication in the summand appearing in the definition of� �� 2 2

'H��

�H� ��2 2, we write , where has the summands�

��3�� 3� 3 ��3 � � ��3�� 3� 3 �# �3 � �2 2 2$ $ �� $ $ �!�

2 2, has the summands , has the sum-mands , and has the summands . Similarly,# �3�� 3� 3 ��3 � � # �3�� 3� 3 �# �3 �2 2 2 2 2

$ $ �'� $ $2

carrying out the multiplication in the integrand appearing in the definition of

��C �� C �� #�� C �� #�� 2�. 2�. 2�.�(

$ $ $ $� � �

� �

we write

��C �� C ��2�. 2.

'

H��

�H��where has the integrand � �C �� C �� C �� $ $

2�. 2�. 2�.

� �C �� C �� #�� C �� $ $ �!�2�. 2�. 2�. has the integrand , has the

integrand , and has the integrand#�� C �� C ��$ $ �'�2�. 2�.

#�� #�� $ $ .The limit (4.1) follows if we prove for thatH � �� !� '

lim lim sup. B 1 2 B 1� 0

�� C �� '!� 2

�H� �H�2 2�.

The details of the proof are given for ; the cases are handled similarly. TheH � � H � � !� 'verification of (4.3) for relies on the fact that since has compact support , thereH � � � �exists such that for any , , and .) � 1 ��3� � I ) 2 0 � 0 3 0 �2 2

For , and and in we define� I A A I > � �$ $. �

� �� A� A �� ''�. . ? ?$ $ $

>

A�A ��

�.

$.�A .�A$

where

� �A� A � � > �� '*�.$ $ $

.? ?

� �.�A .�A$

� �

Substituting the definition (2.14) (2.15) of , we obtain� C �� 2�.

� �C �� 3��3 � �� $ $ $2�.

� +

>

A�A �� 30?

>

3 0?

? ?

� � � �.

$

.2

.�A

$.�A$

.�A .�A$

� �

.� � � +

>

A�A �� 30?

3 0?

��3��3 �� A�A �� .

$.�A

$.�A$

$ $.

2

Thus for any ,� I . � 2

sup� 0

�� C �� 2

�� 2 2�.

sup � ��3��3 �� 0 2

� +

>

A�A �� 30?

3 0?

$ � �3�3 �� A�A �

� � � �.

$.�A

$.�A$

2 .$ $

2

I ) +

>

A�A �� 30?

3 0?

�� 3�3 �� A�A �� .

$.�A

$.�A$

2 .$ $

2

� �

� �� )

>

A�A �� 30?

3 0?4�3�4�3 �

$ $ $.

.

$.�A

$.�A$

$

� � � � � �

I �� )

$ $ $.

� ��

� �

� M � � � M I M � � � M ) ) . ." � ( � " � ( �

� � � � �

The Green's function is square-integrable on . Hence for the set with�� ( ($ � � � �the partition , the function defined in (4.4) (4.5) is the analogue of ? (? � � J.�A .�A . .$

defined in Lemma 3.3. By an analogous proof, one shows thatM � � � M B . " � ( � � � . This limit and the preceding display yield the desired limit

(4.3) in the case .H � � For any , , and , we have and thus�#� � I . � 2 � 0 3 0 ��3� � I ) �2 2

sup . Hence the range of the support of the -distribution of�0 2�. 2� �C �� I )

C �J 0 " � �� M J M I )� # �� 02�. 1 $ is a subset of . Since is bounded on and � �

" � ( � � )2� � , it follows that and are bounded on the union of the supports of the -

distributions of .C2�.

The weak continuity of is obvious. If weakly in�F� )�J� � �J � #�� J B J�� 2

" � � � , then by a property of the Green's function

� ��

�� &� � ��J �� #�� B �� &� � ��J�� #�� " � �$ 2 $ $ $ $ $ $ $ � � � strongly in

It follows that . The proof of the lemma is complete.��J � B ��J�2�

4.2 Microcanonical Model

The LDP for with respect to the microcanonical ensemble is stated in Theorem 4.2. WeC2�.

say that a constraint pair is admissible if for some�<� � �<� � � ��J��)�J�� J 0 " � � E�J� � 1 � � � with , and we let denote the largest open subset of consistingof admissible constraint pairs. We call this domain the admissible set for the�microcanonical model. For a fixed constraint pair , the rate function for the LDP�<� � 0� �is defined for to beJ 0 " � � �

E �J� � �'-�E�J� � D�<� � ��J� � < )�J� � �

1 �<�� if ,

otherwise

where is defined byD�<� ��

D�<� � � � �E�J ��J � � <�)�J � � � �'5�JO

O O O� �inf

D�<� �� is called the microcanonical entropy. Theorem 4.2: Let and consider with the strong topology. With res-�<� � 0 " � ��

pect to , satisfies the LDP on , in the triple limit , and; C " � � 2 B 1 . B 1<� 2� 2�.

��

� B E �'-� �, with rate function defined in . That is, for any strongly closed subset <��

of " � � �

lim lim sup lim sup log� B . B 1 2 B 1

; �C 0 � � I � E �� +

<� <�2� 2�.

2

� ��

and for any strongly open subset of� " � � �

lim lim inf lim inf log� B . B 1 2 B 1

; �C 0 �� / � E ��+

<� <�2� 2�.

2

� ��

This LDP is a three-parameter analogue, involving the triple limit ,Proof: 2 B 1. B 1 B , and , of the two-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 inTheorem 3.1 and the properties of , , , and given in Lemma 4.1.� ) � )2 2 �

For any admissible choice of and we define the set of microcanonical< � �<��

equilibrium macrostates to be the set of such that . It follows fromJ 0 " � � E �J� � <��

the definition of that elements of are exactly those functions that solve theE<� <�� following constrained minimization problem:

minimize over all subject to the constraints E�J � J 0 " � �O O ��'6�

��J � � < )�J � �O O and .�

The optimum value in this variational principle determines the microcanonical entropy Ddefined in (4.7). We point out an important consequence of the large deviation upper bound in Theorem 4.2.Let be a Borel subset of whose closure satisfies . Then, " � � , , L � P

� � <��

E �, � : : . 2�<�� , and for all sufficiently small and all sufficiently large and �

; �C 0 , � I �� + E �, � � �'7��<� <�

2� 2�. 2� �� exp

Since the right-hand side of (4.9) converges to 0 as , macrostates not lying in 2 B 1 � �<��

have an exponentially small probability of being observed as a coarse-grained state in thecontinuum limit. The macrostates in are therefore the overwhelmingly most probable�<��

among all possible macrostates of the turbulent system.

4.3 Canonical Model

The LDP for with respect to the canonical ensemble is stated in Theorem 4.3. For anyC2�.

real values of and , the rate function for the LDP is defined for to be� � �J 0 " � �

E �J� � E�J� � ��J� � )�J�� '� ��

where is defined by� � ��

� � � � �� E�J � � ��J � � )�J �� '��JO

O O Oinf

� � �� is called the canonical free energy. Theorem 4.3: Let and consider with the strong topology. With res-� � � 0 " � ��

pect to , satisfies the LDP on , in the double limit and ,; C " � � 2 B 1 . B 12� � 2�.

� � �

with rate function defined in That is, for any strongly closed subset of E � " � �� (4.10). �

lim sup lim sup log. B 1 2 B 1

; �C 0 � � I � E �� + 2� � 2�. �2

� � � �

and for any strongly open subset of � " � � �


; �C 0 �� / � E ��+ 2� � 2�. �2

� � � �

This LDP is a two-parameter analogue, involving the double limit andProof: 2 B 1. B 1 , of the one-parameter LDP given in Theorem 2.4 of [9]. The proof of that theoremrelies on an application of Laplace's principle. The proof is easily modified to the present

two-parameter setting using the basic LDP in Theorem 3.1 and the properties of , , ,� ) �2 2

and given in Lemma 4.1. As is shown in Theorem 1.3.4 of [8], the boundedness properties)of and given in Lemma 4.1 allow Laplace's principle to be applied in this setting. � ) �#� �

For a particular choice of and we define the set of canonical equilibrium macro-� � ��

states to be the set of such that . That is, the elements of areJ 0 " � � E �J� � � ��

exactly those functions that solve the following unconstrained minimization problem:

minimize over all E�J � � ��J � � )�J � J 0 " � � �'��O O O O� � �

The optimum value in this variational principle determines the canonical free energy � � �� defined in (4.11). Comparing this minimization problem with the corresponding constrainedminimization problem (4.8) arising in the microcanonical ensemble, we see that theparameters and in the former are Lagrange multipliers dual to the two constraints in the� �latter. As in the microcanonical case (see (4.9) and the associated discussion), the large deviationupper bound in Theorem 4.3 implies that the macrostates in are overwhelmingly the most��

probable among all possible macrostates of the turbulent system.

5. Equivalence of Ensembles

In this section we classify the possible relationships between the sets of equilibriummacrostates and . We begin by establishing that each of these equilibrium sets is� �<�

��

� �

nonempty. Then we turn to our second set of main results, which concern the equivalence orlack of equivalence between the microcanonical and canonical ensembles. In many statisticalmechanical models it is common for the microcanonical and canonical ensembles to beequivalent, in the sense that there is a one-to-one correspondence between their equilibriummacrostates. However, in models of coherent structures in turbulence there can bemicrocanonical equilibria that are not realized as canonical equilibria. As is shown in ourcompanion paper [10], and in a real physical application in [31], nonequivalence occurs oftenin the turbulence models and produces some of the most interesting coherent mean flows.

5.1 Existence of Equilibrium States

We continue to assume that the support of is compact. It follows from this assumption� �that the grows faster than quadratically as ; the straightforward proof is leftE�J� M J M B 1to the reader. Consequently, for any the sets 4 � 1 Q � �J 0 " � ��E�J� I 4�4

�are precompact with respect to the weak topology of ; this follows from the fact that" � � �each is a subset of a closed ball in . Moreover, is lower semicontinous withQ " � � E�J�4

�respect to the weak topology on . This property is an immediate consequence of the" � � �relationship

E�J� � E�J �� *��. 0

sup

.

where is the piecewise constant approximating function defined in Lemma 3.3. Indeed,J.since each function is weakly lower semicontinuous, the weak lower semi-J K E�J �.continuity of follows. Thus the sets are closed with respect to the weak topology.E�J� Q4

This property, in combination with the weak precompactness of these sets, implies that the setsQ4 are compact with respect to the weak topology. Concerning the proof of (5.1), Jensen'sinequality guarantees that , while Lemma 3.3 and the strong lowerE�J � I E�J�.

semicontinuity of proved right after its definition in (3.1) yield lim inf ;E E�J � / E�J�.B 1 .

(5.1) is thus proved. The existence of microcanonical and canonical equilibrium states now follows immediatelyfrom the direct methods of the calculus of variations. In the microcanonical case, for anyadmissible a minimizing sequence exists that converges weakly to a minimizer�<� � 0� �J 0 E�<��, by virtue of the properties of the objective functional just derived and thecontinuity, with respect to the weak topology, of the constraint functionals and (Lemma� )4.1 . In the canonical case, since grows superquadratically as while �F�� E M J M B 1 �grows quadratically and linearly, it follows that for any a minimizing) � � � 0� � �

sequence exists that converges weakly to a minimizer . The details of this routineJ 0 ��

analysis are omitted. Typically, the solutions of the variational problems (4.8) and (4.12) for the microcanonicaland canonical ensembles, respectively, are expected to be unique. Indeed, numerical solutionsof these problems, such as those carried out in [10], demonstrate that apart from degeneraciesand bifurcations the equilibrium sets and are singleton sets. In the next subsection,� �<�

��

� �

without any special assumptions concerning uniqueness of equilibrium solutions for eithermodel, we give complete and general results about the correspondence between these sets. Inaddition, the results given in the next subsection require no continuity or smoothnessassumptions of the equilibrium solutions with respect to the model parameters or�<� �� .

5.2 Dual Thermodynamic Functions

In the analysis to follow, we show how the properties of the thermodynamic functions for themicrocanonical and canonical ensembles determine the correspondence, or lack ofcorrespondence, between equilibria for these two ensembles. For the microcanonicalensemble the thermodynamic function is the microcanonical entropy defined in theD�<� ��constrained variational principle (4.7), the solutions of which constitute the equilibrium set� � � �<��. For the canonical ensemble the thermodynamic function is the free energy � � �defined in the unconstrained variational principle (4.11), the solutions of which constitute theequilibrium set .��

The basis for the equivalence of ensembles is the conjugacy between and ; that is, isD � �the Legendre-Fenchel transform of [13, 32]. This basic property is easily verified asDfollows:

min� � � � �� E�J� � ��J� � )�J�� *�J

inf min� �E�J� � ��J� � )�J��J� � <�)�J� � �<� J�

� � �

inf� � < � �D�<� ��<��

� ��

� D � � �R � �

Consequently, is a concave function of , which runs over . By contrast, itself is� � � �� D

not necessarily concave. The concave hull of is furnished by the conjugate function of ,D �namely, , which satisfies the inequality�R RR� D

D�<� � I � < � � � � �� D �<� � �*!��

� � �� inf RR

The relationship between microcanonical equilibria and canonical equilibria dependscrucially on the concavity properties of the microcanonical entropy . To this end, weDintroduce the subset on which the concave hull coincides with ; that is,� �S D DRR

�<� � 0 D �<� � � D�<� �� if and only if . Equivalently, consists of those pointsRR

�<� � 0 � � � 0� � � � � for which there exists some such that

D�< � � I D�<� � � �< �<�� *'�$ $ $ $� � � � � �

for all . This condition means that has a supporting plane, with normal�< � � 0 D$ $ � �determined by , at the point . Such points are precisely those points of� � � �<� � �<� �� at which has a nonempty superdifferential; this set consists of all for which (5.4)D � � �holds [13, 32]. As we will see in the next subsection, the concavity set plays a pivotal role in�the criteria for equivalence of ensembles.

5.3 Correspondence between Equilibrium Sets

The following theorem ensures that for constraint pairs in the microcanonical equilibria are�contained in a corresponding canonical equilibrium set, while for constraint pairs in the� �Tmicrocanonical equilibria are not contained in any canonical equilibrium set. Theorem 5.1: If belongs to , then for some .�+� �<� � 0 S � � �� <�

��

� �

If does not belong to , then for all .�#� �<� � 0 L � P � � �� <��

��

If belongs to , then there exists such that (5.4) holdsProof: �+� �<� � � � � 0� � � � �

for all . Thus for all �< � � 0 �< � �$ $ $ $� � �

� �� < � �D�<� � I < � �D�< � �$ $ $ $

By (5.2) and (4.11) this inequality implies that

inf� ��

< � �D�<� � I � < � �D�< � ��< � �$ $

$ $ $ $

min� � � � � �E�J� � ��J� � )�J��J

� � � � �

To show the claimed set containment, take any and note that ,J 0 ��J � � <� ��<��

)�J � � E�J � � �D�<� �� , and . Substituting these expressions into the precedingdisplay yields

E�J � � ��J � � )�J � I �E�J� � ��J� � )�J�� J

� � � �min

Since consists of the minimizers of , it follows that . This� � � �� E � � � ) J 0�

completes the proof of .�+� By (5.2) the hypothesis implies that for all �#� � � � 0� � �

D�<� � � < � � � � ��

Then any satisfiesJ 0� �<��

E�J � � ��J � � )�J �� D�<� � � < ��

min: � � � � �E�J� � ��J� � )�J��J

� � � � �

Thus does not minimize and hence does not belong to . Since J E � � � ) � � ��

is arbitrary, this completes the proof of . �#� �

According to this theorem, whenever , there are microcanonical equilibria that are� �Unot realized by any canonical equilibria. On the other hand, all canonical equilibria arecontained in some microcanonical equilibrium set, and is exhausted by the constraint pairs�realized by all canonical equilibria. These further results are given in the next theorem. Theorem 5.2: The concavity set consists of all constraint pairs realized by the canonical�+� �

equilibria; that is,

� � � � � �� )� �� 0 � �**��

�#� Each canonical equilibrium set consists of all microcanonical equilibria��

whose constraint pairs are realized by ; that is, for any � � ��

� � � � ��

� � �<�� <� � 0 �� )� �� *-��

The containment of in the union is immediate from Theorem 5.1 . ToProof: �+� �+��show the opposite containment we argue by contradiction, supposing that for some and� � �� some . Then by (5.3) we find thatJ 0 � �<� � � ��J ��)�J �� 0 T� � ��

D�<� � � < � � � � � � � E�J � I D�<� ��

We thus obtain the desired contradiction. This completes the proof of part (a). The containment of in the union is straightforward. Let and set�#� J 0��

< � ��J � � )�J � E�J � � < � I E�J� � ��J� � )�J�� and . Then for� � �� all . For those that satisfy the constraints , , we therefore find thatJ J ��J� � < )�J� � �E�J � I E�J� J 0� �. Hence .�<��

The opposite containment is also straightforward. If and for< � ��J � � )�J �O O�some , then for any we have . Since , weJ 0 J 0 E�J � I E�J � J 0O O O� ��

��

<�

obtain

minJ

�E�J� � ��J� � )�J�� E�J � � < �� O� � � ��

./ E�J � � ��J � � )�J ��

Hence . This completes the proof of part .J 0 �#��

Theorems 5.1 and 5.2 allow us to classify the microcanonical constraint parameters �<� ��according to whether or not equivalence of ensembles holds for those parameters. In fact, theadmissible set can be decomposed into three disjoint sets, where� (1) there is a one-to-one correspondence between 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 micro-canonical entropy is differentiable on its domain . Then, for each microcanonicalD�<� �� parameter there is a corresponding canonical parameter determined�<� � 0 � � �� locally by the familiar thermodynamic relations

� �� D �D�< �,

�

Under this assumption, we have the following classification, which is a consequence ofTheorems 5.1 and 5.2. If belongs to and there is a unique point of�� <� �Full equivalence. � �

contact between and its supporting plane at , then coincides withD �<� �� <��

<� �� .�� <� � If belongs to but there is more than one point ofPartial equivalence. � �

contact between and its supporting plane at , then is a strict subsetD �<� �� <��

of . Moreover, contains all those for which is also a point� � � ��

� �< � $ $$ $

�< � �of contact.

�!� �<� � If does not belonging to , then is disjoint fromNonequivalence. � � �<��

� ��

�<�. In fact, is disjoint from all canonical equilibrium sets.

For a complete discussion of results of this kind, we refer the reader to our paper [9], wherewe state and prove the corresponding results in a much more general setting and without thesimplifying assumption that is differentiable.D

6. Concluding Discussion

The theorems given in this paper support the theory developed in our companion paper [10],where we argue in favor of a statistical equilibrium model for geostrophic turbulence that isdefined by a prior distribution on potential vorticity and microcanonical constraints on energyand circulation. The first set of main results in the present paper furnishes an especially simplemethodology for deriving the equilibrium equations and associated LDP's for that model. Thesecond set of main results shows that the microcanonical ensemble has richer families ofequilibrium solutions than the corresponding canonical ensemble, which omits thosemicrocanonical equilibrium macrostates corresponding to parameters not lying in the�<� ��concavity set of the microcanonical entropy. The computations included in [10] for coherentflows in a channel with zonal topography demonstrate that these omitted states constitute asubstantial portion of the parameter range of the physical model. As this discussion shows, theequivalence-of-ensemble issue is a fundamental one in the context of these local mean-fieldtheories of coherent structure whenever the phenomenon of self-organization into coherentstates is modeled. In recent work [31], the statistical theory presented in [10] and analyzed in the presentpaper is applied to the active weather layer of the atmosphere of Jupiter. When appropriatelyfit to a -layer quasi-geostrophic model, the theory correctly produces large-scale� � � features that agree qualitatively and quantitatively with features observed by the Voyager andGalileo missions. For instance, for the channel domain in the southern hemisphere containingthe Great Red Spot and White Ovals [31, Figure 2], the theory produces both the alternatingeast-west zonal shear flow and two anticyclonic vortices embedded in it. In particular, the sizeand position of the vortices closely match the size and position of the GRS and White Ovals,while the zonally-averaged velocity profile is extremely close to the profile deduced byLimaye [16]. Similarly, for a northern hemisphere channel that contains no permanentvortices, the theory correctly predicts a zonal shear flow with no embedded vortices. Thisagreement between theory and observation demonstrates the practical utility of the modelanalyzed in the preceding sections of the present paper. In Section 5.1 of [10], the nonlinear stability of the equilibrium states for the model isshown by applying a Lyapunov argument. The required Lyapunov functional is constructedfrom the rate function for the coarse-grained process and the constraint functionals E C �2�.

and . In the canonical case, this construction is identical with a well-known method due to)Arnold [1], now called the energy-Casimir method [12]. On the other hand, in the

microcanonical case when nonequivalence prevails, this method breaks down and hence it isnecessary to give a more refined stability analysis. The required refinement, which makes useof the concept of the augmented Lagrangian from constrained optimization theory, ispresented in [10]. For example, the Jovian flows realized in [31] fall in this regime. Thus thevariational principles derived from the large deviation analysis in Section 4 and examined inSection 5 also have important implications for hydrodynamic stability criteria.

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