Bruce Turkington

Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles

We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions of an unconstrained minimization problem in which the constraint in the first problem is replaced by a Lagrange multiplier. The analysis of equivalence and nonequivalence of ensembles reduces to the following question in global optimization. What are the relationships between the set of solutions of the constrained minimization problem that characterizes microcanonical equilibrium macrostates and the set of solutions of the unconstrained minimization problem that characterizes canonical equilibrium macrostates? In general terms, our main result is that a necessary and sufficient condition for equivalence of ensembles to hold at the level of equilibrium macrostates is that it holds at the level of thermodynamic functions, which is the case if and only if the microcanonical entropy is concave. The necessity of this condition is new and has the following striking formulation. If the microcanonical entropy is not concave at some value of its argument, then the ensembles are nonequivalent in the sense that the corresponding set of microcanonical equilibrium macrostates is disjoint from any set of canonical equilibrium macrostates. We point out a number of models of physical interest in which nonconcave microcanonical entropies arise. We also introduce a new class of ensembles called mixed ensembles, obtained by treating a subset of the dynamical invariants canonically and the complementary set microcanonically. Such ensembles arise naturally in applications where there are several independent dynamical invariants, including models of dispersive waves for the nonlinear Schrödinger equation. Complete equivalence and nonequivalence results are presented at the level of equilibrium macrostates for the pure canonical, the pure microcanonical, and the mixed ensembles.

Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows

Statistical equilibrium models of coherent structures in two-dimensional and barotropic quasi-geostrophic turbulence are formulated using canonical and microcanonical ensembles, and the equivalence or nonequivalence of ensembles is investigated for these models. The main results show that models in which the energy and circulation invariants are treated microcanonically give richer families of equilibria than models in which they are treated canonically. For each model, a variational principle that characterizes its equilibrium states is derived by large deviation techniques. An analysis of the two different variational principles resulting from the canonical and microcanonical ensembles reveals that their equilibrium states coincide if and only if the microcanonical entropy function is concave. Numerical computations implemented for geostrophic turbulence over topography in a zonal channel demonstrate that nonequivalence of ensembles occurs over a wide range of the model parameters and that physically interesting equilibria are often omitted by the canonical model. The nonlinear stability of the steady mean flows corresponding to microcanonical equilibria is established by a new Lyapunov argument. These stability theorems refine the well-known Arnold stability theorems, which do not apply when the microcanonical and canonical ensembles are not equivalent.

An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles

This short paper presents a nontechnical introduction to the problem of nonequivalent microcanonical and canonical ensembles. Both the thermodynamic and the macrostate levels of definition of nonequivalent ensembles are introduced. The many relationships that exist between these two levels are also explained in simple physical terms.

Vortex rings: existence and asymptotic estimates

The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter X; as A —» oo the vortex ring tends to a torus whose cross-section is an infinitesimal disc. 0. Introduction. The study of steady vortex rings in an ideal fluid has been the subject of many investigations (see, for example, [3], [19] and the references given there). The classical examples are Helmholtzs rings of small cross-section [17] and Hills spherical vortex [18]. A general existence theorem for vortex rings was first established by Fraenkel and Berger [13] (see also the very recent work [5], [20] with a similar approach); this paper also contains an excellent survey of the subject. The approach in [13] is based on a variational principle for the stream function. More recently Benjamin [4] developed a new approach based on a variational principle for the vorticity. This approach is more natural since (i) the vorticity has compact support (whereas the stream function does not) and (ii) the quantities involved in the variational principle have direct physical significance. In this paper we establish the existence of vortex rings by a new method. As in [4] we formulate the problem in a variational form for the kinetic energy as a functional of the vorticity. We take the admissible functions to vary in the set S^ of functions f(x) satisfying: f (x) = f (r, z) = f (r, z) where x = (r, 0, z), (0.1) i , , j r2

STATISTICAL EQUILIBRIUM MEASURES AND COHERENT STATES IN TWO-DIMENSIONAL TURBULENCE

(x) dx = I, j

Statistical Equilibrium Computations of Coherent Structures in Turbulent Shear Layers

(x)dx<l, 0 <?(*)< A, i.e., an axisymmetric flow with prescribed impulse (= 1), circulation (< 1) and vortex strength (< X); in [4] f is taken to vary over all rearrangements of a given function f0(r, z). Our approach seems technically simpler; it has the further advantage that it leads to vortex rings with, essentially, any given vorticity function, such as (0.2) fit) = cl{l>0] (c > 0), (0.3) fit) = c(t + )p (c>0,B>0). The method of solving our variational problem is in some sense an adaptation of the method of Auchmuty [1] and Auchmuty and Beals [2] (see also [14]-[16]) who Received by the editors May 22, 1980. AMS (MOS) subject classifications (1970). Primary 35J20, 76G05; Secondary 31A15, 35J05. 1 The first author is partially supported by National Science Foundation Grant MCS-781 7204.

Statistical equilibrium predictions of jets and spots on Jupiter.

The equilibrium statistics of the Euler equations in two dimensions are studied, and a new continuum model of coherent, or organized, states is proposed. This model is defined by a maximum entropy principle similar to that governing the Miller-Robert model except that the family of global vorticity invariants is relaxed to a family of inequalities on all convex enstrophy integrals. This relaxation is justified by constructing the continuum model from a sequence of lattice models defined by Gibbs measures whose invariants are derived from the exact vorticity dynamics, not a spectral truncation or spatial discretization of it. The key idea is that the enstrophy integrals can be partially lost to vorticity fluctuations on a range of scales smaller than the lattice microscale, while energy is retained in the larger scales. A consequence of this relaxation is that many of the convex enstrophy constraints can be inactive in equilibrium, leading to a simplification of the mean-field equation for the coherent state. Specific examples of these simplified theories are established for vortex patch dynamics. In particular, a universal relation between mean vorticity and stream function is obtained in the dilute limit of the vortex patch theory, which is different from the sinh relation predicted by the Montgomery-Joyce theory of point vortices.

Derivation of maximum entropy principles in two-dimensional turbulence via large deviations

A numerical method is developed to treat the statistical equilibrium model of coherent structures in two-dimensional turbulence. In this model the vorticity, which fluctuates on a microscopic scale, is described macroscopically by a local probability distribution. A coherent vortex is identified with a most probable macrostate, which maximizes entropy subject to the constraints dictated by the complete family of conserved quantities for incompressible, inviscid flow. Attention is focused on the special case corresponding to vortex patches, and a simple, robust, and efficient algorithm is proposed in this case. The form of the iterative algorithm and its convergence properties are derived from the variational structure of the statistical equilibrium problem. Solution branches are computed for the shear layer configuration, and the results are interpreted in terms of the dynamical phenomena of rollup and coalescence.

The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

An equilibrium statistical theory of coherent structures is applied to midlatitude bands in the northern and southern hemispheres of Jupiter. The theory imposes energy and circulation constraints on the large-scale motion and uses a prior distribution on potential vorticity fluctuations to parameterize the small-scale turbulent eddies. Nonlinearly stable coherent structures are computed by solving the constrained maximum entropy principle governing the equilibrium states of the statistical theory. The theoretical predictions are consistent with the observed large-scale features of the weather layer if and only if the prior distribution has anticyclonic skewness, meaning that intense anticyclones predominate at small scales. Then the computations show that anticyclonic vortices emerge at the latitudes of the Great Red Spot and the White Ovals in the southern band, whereas in the northern band no vortices form within the zonal jets. Recent observational data from the Galileo mission support the occurrence of intense small-scale anticyclonic forcing. The results suggest the possibility of using equilibrium statistical theory for inverse modeling of the small-scale characteristics of the Jovian atmosphere from observed features.

Generalized canonical ensembles and ensemble equivalence

The continuum limit of lattice models arising in two-dimensional turbulence is analyzed by means of the theory of large deviations. In particular, the Miller–Robert continuum model of equilibrium states in an ideal fluid and a modification of that model due to Turkington are examined in a unified framework, and the maximum entropy principles that govern these models are rigorously derived by a new method. In this method, a doubly indexed, measure-valued random process is introduced to represent the coarse-grained vorticity field. The natural large deviation principle for this process is established and is then used to derive the equilibrium conditions satisfied by the most probable macrostates in the continuum models. The physical implications of these results are discussed, and some modeling issues of importance to the theory of long-lived, large-scale coherent vortices in turbulent flows are clarified.