ANALYSIS OF STATISTICAL EQUILIBRIUM MODELS OF GEOSTROPHIC TURBULENCE

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 with 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 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 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 between microcanonical equilibria and canonical equilibria are established in terms of the concavity of the microcanonical entropy. A complete characterization of equivalence of ensembles is deduced by elementary methods of convex analysis.


Introduction
The problem of developing a statistical equilibrium theory of coherent structures in twodimensional 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 were elaborated by Joyce-Montgomery [14], and subsequently these predictions were compared with direct numerical simulations of the end state of freely decaying turbulence [21].Another approach due to Kraichnan [15], called the energy-enstrophy model, was based on spectral truncation of the underlying fluid dynamics.This Gaussian model was extensively applied to geophysical fluid flows [4,27,28]; for instance, to the organization of barotropic, quasi-geostrophic flows over 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 as special or limiting cases.Recently, this general model has been scrutinized from the theoretical standpoint [30] and from the perspective of practical applications [6,18].These investigations have clarified the relation between 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], where it is implemented to predict 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 theoretical model developed in [10].We therefore follow the line of development in that paper, in which 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 law of large numbers for a spatially coarse-grained process which 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 with respect to the product measure that underlies the lattice model.
Our second set of main results concerns the relationship between the microcanonical and canonical equilibrium states; in particular, the equivalence or nonequivalence of the microcanonical and canonical ensembles with 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], where its impact on hydrodynamic stability is also discussed.
The outline of the paper is as follows.In Section 2 we formulate the statistical equilibrium theory described in [10], and we 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 we present the LDP's for the microcanonical and canonical ensembles, relying on general theorems established in [9].Then in Section 5 we 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, we point to some of the physical implications of our results.

Statement of Problem
The continuum dynamics that underlies our statistical equilibrium lattice model can be viewed as a noncanonical Hamiltonian system with infinitely many degrees of freedom.Namely, it can 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 .# shallow, rotating layer of a homogeneous, incompressible, inviscid fluid in the limit of small Rossby number [27].Throughout the rest of the paper, we refer to , as in this quasigeostrophic case, as the potential vorticity.

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.This statistical mechanics approach is standard for a Hamiltonian system with 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 we define these probabilistic lattice models.
For each , the flow domain is uniformly partitioned into microcells.We let We shall not attempt to prescribe a lattice dynamics under which the functions , and ) are exact invariants.In fact, a lattice dynamics conserving the nonlinear enstrophies is not known.The subtleties associated with the rigorous formulation of this statistical model are fully discussed elsewhere [10,30].Rather than pursue this direction any further here, we shall simply impose a joint probability distribution on the lattice potential vorticities 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 techniques that we use.We define the phase space to be , denote by the vector of 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 to 2 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 not preserved 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 in specific applications, see [10, §6.1] and [31, page 12347].

Definition of Ensembles
In terms of the prior distribution , we can form two different statistical equilibrium models 2 by using either the microcanonical ensemble or the canonical ensemble with respect to the invariants and .Given , for any Borel subset of the microcanonical ensemble is defined by 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 The real parameters and correspond to the inverse temperature and the chemical potential.

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 we 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 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 , .
. studied with respect to either the microcanonical ensemble (2.11) or the canonical ensemble (2.12).We refer to as the coarse-grained process.C 2 .With respect to each ensemble, our goal is to determine the set of functions on which " C 2 B 1 .B 1 2 .concentrates in the continuum limit defined by first sending and then .More precisely, we seek to find the smallest subsets of , and , such that as , , and and 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 subsets is shown to arise by solving a variational principle.The variational principles are derived from the LDP's for with respect to the two ensembles.In order to prove these two C 2 .LDP's, we first prove a basic LDP for with respect to the product measure defined C 2 . 2 in (2.10).This is done in the next section. .

Basic LDP for the Coarse-Grained Process
We define to be the set of Proof of the Large Deviation Upper Bound: of the form for some .Let be a The last inequality in this display follows from Jensen's inequality.We have proved that for arbitrary J 0 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.

LDP's with Respect to the Two Ensembles
In this section we present LDP's for with respect to the microcanonical and canonical C 2 .ensembles defined in (2.11) and (2.12).The theorems follow 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.

Properties of
We recall the definitions of the lattice energy in (2.7), the circulation in (2.8), and the 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- ( , we write , where has the summands , has the summands , has the summands , and has the summands .Similarly, carrying out the multiplication in the integrand appearing in the definition of , then by a property of the Green's function $ strongly in It follows that .The proof of the lemma is complete.J B J 2

Microcanonical Model
The LDP for with respect to the microcanonical ensemble is stated in Theorem 4.2.We C 2 .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 consisting of 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 be J 0 " where is defined by 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 in Theorem 3.1 and the properties of , , , and given in Lemma 4.1.) ) , and for all sufficiently small and all sufficiently large and 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 the continuum limit.The macrostates in are therefore the overwhelmingly most probable < among all possible macrostates of the turbulent system.

Canonical Model
The LDP for with respect to the canonical ensemble is stated in Theorem 4.3.For any C 2 .real values of and , the rate function for the LDP is defined for to be J 0 " where is defined by This LDP is a two-parameter analogue, involving the double limit and Proof: 2 B 1 .B 1 , 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 two-parameter setting using the basic LDP in Theorem 3.1 and the properties of , , , 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 macrostates to be the set of such that .That is, the elements of are J 0 " E J exactly those functions that solve the following unconstrained minimization problem: The optimum value in this variational principle determines the canonical free energy defined in (4.11).Comparing this minimization problem with the corresponding constrained minimization problem (4.8) arising in the microcanonical ensemble, we see that the parameters 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 deviation upper bound in Theorem 4.3 implies that the macrostates in are overwhelmingly the most probable among all possible macrostates of the turbulent system.

Equivalence of Ensembles
In this section we classify the possible relationships between the sets of equilibrium macrostates 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 or lack of equivalence between 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 between their equilibrium macrostates.However, in models of coherent structures in turbulence there can be microcanonical equilibria that are not realized as canonical equilibria.As is shown 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 flows.

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 left E J M J M B 1 to the reader.Consequently, for any the sets 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 with Q " E J 4 respect to the weak topology on .This property is an immediate consequence of the " relationship 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 Q 4 This property, in combination with the weak precompactness of these sets, implies that the sets Q 4 are compact with respect to the weak topology.Concerning the proof of (5.1), Jensen's inequality guarantees that , while Lemma 3.3 and the strong lower E 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 immediately from the direct methods of the calculus of variations.In the microcanonical case, for any admissible 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 the continuity, 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 routine J 0 analysis are omitted.
Typically, the solutions of the variational problems (4.8) 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 and are singleton sets.In the next subsection, < without any special assumptions concerning uniqueness of equilibrium solutions for either model, we give complete and general results about the correspondence between these sets.In addition, the results given in the next subsection require no continuity or smoothness assumptions of the equilibrium solutions with respect to the model parameters or < .

Dual Thermodynamic Functions
In the analysis to follow, we show how the properties of the thermodynamic functions for the microcanonical and canonical ensembles determine the correspondence, or lack of correspondence, between equilibria for these two ensembles.For the microcanonical ensemble the thermodynamic function is the microcanonical entropy defined in the D < 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 the equilibrium set .The basis for the equivalence of ensembles is the conjugacy between and ; that is, is D the Legendre-Fenchel transform of [13,32].This basic property is easily verified as D follows: 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.

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 T microcanonical equilibria are not contained in any canonical equilibrium set. Theorem The containment of in the union is immediate from Theorem 5.1 .To Proof: + + show the opposite containment we argue by contradiction, supposing that for some and some .Then by (5.3) we find that J 0 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 that J 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 , we J 0 J 0 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, the admissible set can be decomposed into three disjoint sets, where 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 there is no correspondence.In order to simplify the precise statement of this classification, let us assume that the microcanonical entropy is differentiable on its domain .Then, for each microcanonical D < parameter there is a corresponding canonical parameter determined < 0 locally by the familiar thermodynamic relations .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], where we state and prove the corresponding results in a much more general setting and without the simplifying assumption that is differentiable.D

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 is defined by a prior distribution on potential vorticity and microcanonical constraints on energy 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 shows that the microcanonical ensemble has richer families of equilibrium solutions than the corresponding canonical ensemble, which omits those microcanonical equilibrium macrostates corresponding to parameters not lying in the < concavity set of the microcanonical entropy.The computations included in [10] for coherent flows in a channel with zonal topography demonstrate that these omitted states constitute a substantial portion of the parameter range of the physical model.As this discussion shows, the equivalence-of-ensemble issue is a fundamental one in the context of these local mean-field theories of coherent structure whenever the phenomenon of self-organization into coherent states is modeled.
In recent work [31], the statistical theory presented in [10] and analyzed in the present paper is applied to the active weather layer of the atmosphere of Jupiter.When appropriately fit to a -layer quasi-geostrophic model, the theory correctly produces large-scale features that agree qualitatively and quantitatively with 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-west zonal shear flow and two 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, while 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 flow with no embedded vortices.This agreement between 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 shown by applying a Lyapunov argument.The required Lyapunov functional is constructed from 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 is necessary to give a more refined stability analysis.The required refinement, which makes use of the concept of the augmented Lagrangian from constrained optimization theory, is presented in [10].For example, the Jovian flows 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.
the microcanonical entropy.
On the other hand, all canonical equilibria are contained 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.