Statistical mechanics of quasi-geostrophic flows on a rotating sphere
Statistical mechanics provides an elegant explanation to the appearance of coherent structures in two-dimensional inviscid turbulence: while the fine-grained vorticity field, described by the Euler equation, becomes more and more filamented through time, its dynamical evolution is constrained by some global conservation laws (energy, Casimir invariants). As a consequence, the coarse-grained vorticity field can be predicted through standard statistical mechanics arguments (relying on the Hamiltonian structure of the two-dimensional Euler flow), for any given set of the integral constraints.
It has been suggested that the theory applies equally well to geophysical turbulence; specifically in the case of the quasi-geostrophic equations, with potential vorticity playing the role of the advected quantity. In this study, we demonstrate analytically that the Miller-Robert-Sommeria theory leads to non-trivial statistical equilibria for quasi-geostrophic flows on a rotating sphere, with or without bottom topography. We first consider flows without bottom topography and with an infinite Rossby deformation radius, with and without conservation of angular momentum. When the conservation of angular momentum is taken into account, we report a case of second order phase transition associated with spontaneous symmetry breaking. In a second step, we treat the general case of a flow with an arbitrary bottom topography and a finite Rossby deformation radius. Previous studies were restricted to flows in a planar domain with fixed or periodic boundary conditions with a beta-effect.
In these different cases, we are able to classify the statistical equilibria for the large-scale flow through their sole macroscopic features. We build the phase diagrams of the system and discuss the relations of the various statistical ensembles.
Keywords: Classical phase transitions (Theory), Phase diagrams (Theory), Metastable states, Turbulence
- 1 Introduction
- 2 Statistical Mechanics of the quasi-geostrophic equations
3 The equilibrium mean flow in the barotropic case () without bottom topography ()
- 3.1 Fixed energy and circulation
- 3.2 Fixed energy, circulation and angular momentum
- 4 General case: quasi-geostrophic flow over a topography.
- 5 Discussion
- 6 Conclusion
- A Solid-body rotations
- B Minimum energy for a flow with given angular momentum
An important characteristic of two-dimensional turbulent fluid flows is the emergence of coherent structures: in the 80s, numerical simulations [1, 2] showed that a turbulent flow tends to organize itself spontaneously into large-scale coherent vortices for a wide range of initial conditions and parameters. Laboratory experiments reported similar observations [3, 4, 5, 6]. Large-scale coherent structures are also ubiquitous in planetary atmospheres and in oceanography. Due to the long-lived nature of these structures, it is very appealing to try to understand the reasons for their appearance and maintenance through a statistical theory.
This endeavour is supported by theoretical arguments: as first noticed
by Kirchhoff, the equations for a perfect fluid flow can be recast in
a Hamiltonian form, which makes them a priori suitable for
standard statistical mechanics treatments, as a Liouville theorem
automatically holds. The first attempt along these lines was Onsager’s
statistical theory of point vortices . One peculiar
outcome of Onsager’s theory is the appearance of negative temperature
states at which large-scale vortices
Subsequent attempts essentially considered truncations of the equations of motion in the spectral space. Lee  obtained a Liouville theorem in the spectral phase space and constructed a statistical theory taking into account only the conservation of energy. Kraichnan built a theory on the basis of the conservation of the quadratic invariants: energy and enstrophy [19, 20]. The theory mainly predicts an equilibrium energy spectrum corresponding to an equipartition distribution. This spectrum has been extensively confronted with experiments and numerical simulations (e.g. [21, 22]) but the discussion remains open .
More recently, Miller [24, 25] and Robert & Sommeria [26, 27] independently developed a theory for the continuous vorticity fields, taking into account all the invariants of motion. Due to the infinite number of these invariants, the rigorous mathematical justification is more elaborate than previous approaches and relies on convergence theorems for Young measures [28, 29]. Miller  provides two alternative derivations, perhaps more heuristic, the first one being based on phase space-counting ideas similar to Boltzmann’s classical equilibrium statistical mechanics (see also Lynden-Bell ), while the second one uses a Kac-Hubbard-Stratonovich transformation. The MRS theory was checked against laboratory experiments  and numerical studies [32, 33] in a wide variety of publications .
One of the main interest of applying statistical mechanical theories to inviscid fluid flows is that it provides a very powerful tool to investigate directly the structure of the final state of the flow, regardless of the temporal evolution that leads to this final state. From a practical point of view, such a tool would of course be of great value as it is well-known that turbulence simulations are very greedy in terms of computational resources. In some rare cases, computations can be carried out analytically and it is even possible to elucidate the final organization of the flow directly from the mean field equations obtained from statistical mechanics. In any case, the interest is also theoretical since equilibrium statistical mechanics of inviscid fluid flows can be seen as a specific example of long-range interacting systems , whose statistical mechanics is known to yield peculiar behaviors, in phase transitions and ensemble inequivalence [36, 37, 38, 39, 40, 41]. As an example, statistical mechanics provided valuable insight in the understanding of a von Karman experiment, in particular in transitions between different flow regimes [42, 43], fluctuation-dissipation relations  and Beltramization [45, 46, 31, 47].
One particular area where avoiding long numerical turbulence simulations would be highly beneficial is geophysics. Jupiter’s great red spot provides a prototypical example of application of statistical mechanics to geophysical fluid dynamics [25, 48, 49, 50, 51], in which valuable insight is gained from the statistical theory. Even before, the Kraichnan energy-enstrophy theory was extensively used to discuss energy and enstrophy spectra in the atmosphere [52, 53] and topographic turbulence [54, 55]. However, only one study  considers the global equilibrium flow in a spherical geometry, with encouraging results, but this study does not investigate the structure of the flow in a systematic way. Statistical mechanics of the continuous vorticity field conserving all the invariants has also been applied to the Earth’s oceans, focusing either on small-scale parameterizations [57, 58, 59, 60, 61] or on meso-scale structures  (and in particular the Fofonoff flow [39, 40, 63]).
In this study, we investigate analytically the statistical equilibria of the large-scale general circulation of the Earth’s atmosphere, modelled by the quasi-geostrophic equations, taking into account the spherical geometry (with possible bottom topography) and the full effect of rotation, in the framework of the MRS theory. More precisely, we show that in the absence of a bottom topography, due to the spherical geometry, the solution to the statistical mechanics problem can be derived in a very simple way. The result is, however, highly non trivial because, when the conservation of angular momentum is properly accounted for, it leads to a second order phase transition associated with a spontaneous symmetry breaking. Since all the previous studies used a -effect instead of the full Coriolis parameter and focused on rectangular bounded regions rather than on the full sphere, this simple solution was not noticed before. We draw the phase diagrams of the system in both microcanonical and grand-canonical ensembles. The relations between the two statistical ensembles is described in detail and we present a refined notion of marginal equivalence of ensemble (see also ). In the presence of a bottom topography, we obtain semi-persistent equilibria reminiscent of the structures observed in the atmosphere. They correspond to saddle points of entropy. Strictly speaking, they are unstable since they can be destabilized by certain infinitesimal perturbations belonging to particular subspaces of the dynamical space: for these saddle points of the entropy surface, there is at least one direction along which the entropy increases while the constraints remain satisfied. However, it may take a long time before the system spontaneously generates these perturbations. Therefore, these states may persist for a long time before finally being destabilized . In the atmospheric context, these semi-persistent equilibrium states could account for situations of atmospheric blocking where a large scale structure can form for a few days before finally disappearing.
In section 2, we present the general statistical mechanics of the quasi-geostrophic equations. In section 3 we obtain the structure of the equilibrium mean flow in the particular case of a sphere without bottom topography in the limit of infinite Rossby deformation radius, with and without conservation of angular momentum. In section 4, we examine the effect of the bottom topography and of the Rossby deformation radius. Section 5 presents a discussion of the obtained results and a comparison with previously published results, while conclusions are presented in section 6.
2 Statistical Mechanics of the quasi-geostrophic equations
2.1 Definitions and notations
We consider here an incompressible, inviscid, fluid on the two dimensional sphere (denoted to keep notations simple). The coordinates are where is the polar angle (the latitude is thus ) and the azimuthal angle. In the following, for any quantity , we note its average value over the whole domain:
We introduce the eigenvectors of the Laplacian on the sphere. These are the spherical harmonics with eigenvalues :
where are the associated Legendre polynomials and . The scalar product on the vector space of complex-valued functions on the sphere is defined as usual as
where the bar denotes complex conjugaison, so that the spherical harmonics form an orthonormal basis of the Hilbert space :
Note that .
For applications to the Earth, we shall take the inverse of the Earth’s rotation rate as the time unit and we set in the radial direction so that the Earth mean radius is the length unit. Hence all the analytical calculations are carried out on the unit sphere , while we retain the dependance in the calculations to stress the effect of rotation in the formulae, even though for numerical applications, we will always take .
2.2 The quasi-geostrophic equations
We consider here the simplest model for geophysical flows: the one-layer quasi-geostrophic equations, also called the equivalent barotropic vorticity equations. We assume that the velocity field satisfies the incompressibility condition , so that we can introduce a stream function such that , and define the potential vorticity as
where is the topography and the Rossby deformation radius . The evolution of the potential vorticity is given by the quasi-geostrophic equation
In other words, the flow conserves potential vorticity. Together with the fact that the flow is incompressible, this implies the conservation of the integral of any function of potential vorticity , called Casimir invariants ( being an arbitrary function). In particular, any moment of the potential vorticity is conserved. will be called here the circulation and the potential enstrophy. The energy, given by
is also conserved. Finally, due to the spherical symmetry, one may also consider a supplementary invariant: the integral over the domain of the vertical component of angular momentum
where is the zonal component of velocity. In A, we show that, for a solid-body rotation, the dynamical invariants and are not independent: they obey a relation of the form , with . We also show (B) that, for any flow, .
For fluid motion on a rotating sphere, the term includes the Coriolis parameter . Following , the general form we will consider here is with the bottom topography and the average height of the fluid. The relative vorticity is and the absolute vorticity . In the limit of infinite Rossby deformation radius () and no topography (), we recover the 2D Euler equations. Introducing the Poisson brackets on the sphere
the quasi-geostrophic equation (7) reads
It is well known in the case of the Euler (or quasi-geostrophic in a planar domain) equations that the Poisson bracket form implies that the steady states of the equations correspond to with an arbitrary function. In fact, due to the particular geometry considered here, the form of the steady-states must be slightly refined. Let us consider solutions of the quasi-geostrophic equations of the form . Substituting this relation into equation (11), we obtain
so that , with an arbitrary function. This is the general form of the solutions of the quasi-geostrophic equations which are stationary in a frame rotating with angular velocity with respect to the initial reference frame (which rotates with angular velocity ). When , we recover the previous relationship. However, due to the spherical symmetry, there is no reason to select the reference frame a priori.
In the next section, we show that statistical mechanics allows to select a particular function on the grounds that it is the most probable equilibrium state respecting the constraints.
2.3 Maximum entropy states
If we were to inject a droplet of dye in a turbulent two-dimensional flow, we would observe a complex mixing where the originally regular patch of dye turns into finer and finer filaments as time goes on. After a while, the filaments are so intertwined that the dye seems homogeneously distributed over the fluid to a human eye: the coarse-grained dye concentration is homogeneous. In the quasi-geostrophic equations, it is potential vorticity that is mixed by the flow (equation (7)). The crucial difference is that the advected quantity is no longer a passive tracer but plays an active role in the dynamics. Due to the conservation constraints associated to the quasi-geostrophic equations, the potential vorticity mixing will not lead to an homogeneous coarse-grained distribution. In particular, the energy constaint prevents complete mixing. We wish to determine what this final coarse-grained state will be, regardless of the details of the fine-grained structure of the potential vorticity field. Analogously to classical statistical mechanics [68, 69, 70], after identifying the correct description for microstates (exact fine-grained vorticity field) and macrostates (the coarse-grained vorticity field, mathematically represented as a Young measure), one selects the macrostate that maximizes the statistical entropy subject to the relevant macroscopic constraints (conserved quantities), as developed by Miller and Robert [24, 25, 26, 27]. The underlying fondamental property is that an overwhelming majority of microstates lie in the vicinity of the equilibrium macrostate. The implicit separation of scales between microstates and macrostates implies that the contributions of the small-scale fluctuations of vorticity are discarded in the macroscopic quantities (strictly speaking, this is true for the energy but not for the Casimirs: computing the moments of the vorticity distribution using the fine-grained distribution or the coarse-grained distribution yields different results). As a consequence, the Miller-Robert-Sommeria (MRS) theory is a mean-field theory [71, 25]. Note also that albeit all the dynamically conserved quantities of the equations are imposed as constraints in the statistical mechanics, the topological constraints are not conserved: a connected vorticity domain should remain connected through time, while in the statistical mechanics the only thing that is conserved is the area of this domain.
At the microscopic level, the potential vorticity is fully determined by the initial conditions and the evolution equation (7). At the macroscopic level, we consider the coarse-grained potential vorticity as a random variable with probability distribution : the probability that the potential vorticity has the value with an error at point is . The potential vorticity distribution characterizes the macroscopic state. The potential vorticity distribution must satisfy the normalization condition at each point of the domain, and the mean value of the potential vorticity is given by . We introduce a stream function corresponding to the ensemble-mean potential vorticity through . The statistical entropy of the probability distribution is
We are looking for the probability distribution that maximizes the statistical entropy functional subject to the constraints mentioned in section 2.2: global conservation of energy and Casimir functionals. The conservation of all the Casimirs is equivalent to the conservation of the area of each potential vorticity level . Hence the statistical equilibria must satisfy
where and are respectively the Lagrange multipliers associated with the conservation of energy, potential vorticity levels, angular momentum, and normalization.
The resulting potential vorticity probability density is the Gibbs state
where and . Due to the normalization condition, the partition function is also given by
and the ensemble-mean potential vorticity satisfies the usual relation
The right-hand side of this equation is a certain function of the relative stream function . Hence, for given values of the Lagrange multipliers and , the statistical entropy maximization procedure selects a functional relationship between potential vorticity and relative stream function at steady-state: . This describes a flow rotating with angular velocity with respect to the terrestrial frame. Therefore, statistical mechanics selects steady states of the QG equations in a rotating frame. The resulting mean field equation is simply
This is the general mean field equation for equilibrium states of the quasi-geostrophic equations. In the limit , one recovers the well-known mean field equation for the Euler equation. Note that in the case of only two potential vorticity levels and , the partition function is simply , so that after straightforward computations, we find the relationship with , , . With and , we recover the relationship :
To determine the statistical equilibrium state, we have to solve the mean field equation (17), relate the Lagrange multipliers to the constraints and study the stability of the solutions (whether they are entropy maxima or saddle points). If several entropy maxima are found for the same parameters (conserved quantities), we must distinguish metastable states (local entropy maxima) from fully stable states (global entropy maxima).
2.4 The linear relationship
In practice, the mean field equation (17) is difficult to solve because the function is in general nonlinear due to the conservation of all the moments of the fine-grained potential vorticity. Besides, it is generally difficult to relate the Lagrange multipliers to the conserved quantitites. The two levels system of  provides an example of a case where it is possible to write down explicitly the relationship but the meanfield equation (18) is not analytically solvable without further approximations. Nevertheless, efficient numerical methods do exist, like for instance the algorithm of Turkington and Whitaker  or the method of relaxation equations [73, 74].
To go further with analytical methods, a common solution is to linearize the relationship. Several justifications of this procedure can be given, which can be grossly classified in two types of approaches. In the first approach, one simply discards the effect of the high-order fine-grained potential vorticity moments (it would be possible to include them one by one hierarchically), while in the second approach, their effect is prescribed through a gaussian prior distribution for small-scale potential vorticity (in the general theory, one can specify a non-gaussian prior, which would lead to a nonlinear relationship).
In the limit of strong mixing , the argument of the exponential in the partition function is small and a power expansion of can be carried out. The rigorous computation is presented in  and yields a linear relationship. This power series expansion can be done at virtually any order. At first-order, equation (17) becomes identical to the mean field equation obtained by minimizing the coarse-grained enstrophy with fixed energy, circulation and angular momentum. The value of the fine-grained enstrophy is fixed by the initial condition and we always have . The strong mixing limit thus corresponds to cases where the energy, circulation, angular momentum (called robust invariants because they are expressed in terms of the coarse-grained potential vorticity) and fine-grained enstrophy are the only important invariants and the higher-order moments of the fine-grained enstrophy (called fragile invariants) do not play any role. This can be seen as a form of justification in the framework of statistical mechanics of inviscid fluids of the early phenomenological minimum enstrophy principle suggested by ,  and  on the basis of the inverse cascade of Batchelor  for finite viscosities. The connection between the inviscid statistical theory and the phenomenological selective decay approach is discussed at length in ,  and .
For any given energy, one can find a vorticity level distribution such that the function is linear. This corresponds to a gaussian (see ). Indeed, if is a Gaussian with mean value and standard deviation , the analytical computation of the partition function is straightforward
and the mean flow satisfies the equation
Furthermore, if the flow maximizes at fixed energy and circulation, then it is granted to be thermodynamically stable in the MRS sense [82, 39] (see also ). In the approach of Ellis, Haven and Turkington , is interpreted as a prior distribution for the high-order moments of potential vorticity (fragile constraints): arguing that real flows are subjected to forcing and dissipation at small scales, Ellis et al.  objected that conservation of the fragile constraints (which depend on the fine-grained field) is probably irrelevant. They suggested to treat these constraints canonically by fixing the Lagrange multiplier instead of itself. Chavanis [83, 84] showed that this is equivalent to maximizing a relative entropy with a prescribed prior distribution for the small-scale potential vorticity. The ensemble-mean coarse-grained potential vorticity then is a maximum of a generalized entropy functional with fixed values of the robust invariants (energy, circulation and angular momentum), where is a convex function determined by the prior . The linear relation (20) corresponds to a gaussian prior and, in this case, the generalized entropy is minus the coarse-grained enstrophy, i.e. .
As an intermediate case, Naso et al.  take up the argument that the conservation of some Casimirs is broken by small-scale forcing and dissipation, but instead of prescribing a prior small-scale vorticity distribution, they suggest that the relevant invariants to keep are determined directly by forcing and dissipation (which, on average, equilibrate so that the system reaches a quasi-stationary state). They show that maximizing the Miller-Robert-Sommeria entropy with fixed energy, circulation, angular momentum and fine-grained enstrophy is equivalent (for what concerns the macroscopic flow) to minimizing the coarse-grained enstrophy at fixed energy, circulation and angular momentum. Furthermore, the fluctuations around this macroscopic flow are gaussian.
Note that one does not necessarily need to justify physically the linear relationship: we may argue that we are just studying a subset of the huge and notoriously difficult to compute class of MRS statistical equilibria.
In the following sections, we shall study the mean equilibrium flow
for the quasi-geostrophic equations on the sphere based on these
equivalent formulations of the variational problem: we use the
generalized entropy where is the area of the
2.5 Statistical ensembles and variational problems
In this study, we shall consider the maximization of the generalized entropy , as explained in the previous section, with either fixed energy and circulation, or fixed energy, circulation, and angular momentum. The corresponding variational problems can be written as
These constrained variational problems correspond to the microcanonical ensemble and the function is called the entropy. We shall also consider the dual variational problems with relaxed constraints
In both cases, the corresponding statistical ensemble will be termed
Clearly, the critical points of the microcanonical and grand-canonical variational problems are the same, due to the Lagrange multiplier theorem. However the nature of these critical points (maximum, minimum, saddle point) can differ. Nevertheless, it is straightforward to convince oneself that a maximum of the relaxed variational problem is also a maximum of the constrained variational problem (grand-canonical stability implies microcanonical stability) and that a saddle point of the constrained variational problem is necessarily a saddle point of the relaxed variational problem (microcanonical instability implies grand-canonical instability). More detailed relationships between the constrained and relaxed variational problems can be found in [36, 37, 82, 74].
It is thus possible that a maximum in the constrained variational problem (that is an equilibrium state in the microcanonical ensemble) may not be reached in the grand-canonical ensemble. Such a situation happens when no grand-canonical equilibrium has the prescribed energy, circulation, and angular momentum if relevant. In this case, we speak of ensemble inequivalence. More precisely, we have just described ensemble inequivalence at the macrostate level [36, 86]. Another characterization of ensemble inequivalence, referred to as ensemble inequivalence at the thermodynamic level, is linked to the concavity of the entropy [36, 37, 86]. Indeed, the functions and are linked by Legendre-Fenchel transformations. Using convex analysis , it can be proved that the transformation is invertible only when is a concave function. A particular indicator of ensemble inequivalence is the microcanonical specific heat. It is easily proved that when computed with the grand-canonical probability distribution, the specific heat is always positive. On the other hand, no such result holds in the microcanonical ensemble. Thus, a negative microcanonical specific heat indicates ensemble inequivalence [88, 89, 90, 41].
In practice, the microcanonical ensemble is the natural one to treat problems where the system is large enough to be considered isolated, as in astrophysics or geophysical flows. Yet, it is always more convenient mathematically to deal with a relaxed variational problem than directly with the constrained variational problem. For this reason, it is customary to consider canonical or grand-canonical ensembles even in these cases. Meanwhile, one must keep in mind that the physical interpretation of the canonical or grand-canonical ensemble may not be straightforward. In our case, it is not easy to see what physical object would play the role of a reservoir of energy, circulation or angular momentum. In other words, it is not clear how the Lagrange multipliers and are fixed, and which physical quantity they represent. Strictly speaking, the canonical and grand-canonical ensembles are physically relevant only when they are equivalent with the microcanonical ensemble (or when there is a good physical reason to be otherwise). On the contrary, the microcanonical ensemble is always physically relevant.
Remark: We have justified the variational principles (21)-(24) as conditions of thermodynamical stability. We note that the very same variational principles can also be interpreted as sufficient conditions of nonlinear dynamical stability with respect to the QG equations [37, 74]. Therefore, the stable states that we shall determine are both dynamically and thermodynamically stable.
3 The equilibrium mean flow in the barotropic case () without bottom topography ()
3.1 Fixed energy and circulation
In this section, we consider the following variational problems
are respectively the (normalized) entropy, energy and circulation. The critical points satisfy
where and are Lagrange multipliers associated with the conservation of circulation and energy, respectively. This leads to the linear relationship
Averaging over space, we obtain , but since the integral of is the circulation of the velocity on the domain boundary, hence vanishing in the case of the sphere. Thus, and the relationship reads
If we let such that , the mean field equation reduces to the simple Helmholtz equation
Here, we note that is an eigenvector of the Laplacian on the sphere (with eigenvalue ). Therefore if we introduce the operator , satisfies . To solve equation (31), we have to distinguish two cases, depending whether the inverse temperature is an eigenvalue of the Laplacian or not .
Note that the Lagrange multiplier is only related to the mean value of the stream function on the domain, , which has no effect on the structure of the flow (the stream function is defined up to an unimportant additive constant). This is not surprising as in the case considered in this section (), we must always have . Therefore, neither nor will intervene in the following discussion. To keep the notations simple in the sequel, we shall make the gauge choice and identify and . This also implies .
Case : the continuum solution
In this case, equation (31) reduces to , but is invertible, hence
The relation between the energy and the Lagrange multiplier can be solved:
(see the caloric curve on figure 1), so that the thermodynamic potentials (entropy) and (free energy) are
Knowing the equilibrium streamfunction , we can compute the zonal and meridional components of the velocity field, respectively and :
where is the Earth’s mean radius. The equilibrium motion of the fluid is thus a simple solid body rotation with angular velocity
In particular, the equilibrium velocity distribution is purely zonal, vanishing at the poles with a maximum at the equator. At low statistical temperatures (), the rotation of the fluid has the same sign as the solid body rotation of the Earth, while high statistical temperatures () correspond to counter-rotating flows. Examples of such zonal wind profiles are drawn on figure 2 for various statistical temperatures . Note that for any given value of the energy, the two types of solutions coexist, due to the symmetry , as is clear from the caloric curve shown in figure 1.
Remark: when (corresponding to ), we find that so that there is no rotation in the inertial frame. In a sense, the Earth rotation is canceled.
Let us now suppose that with . The solutions of equation (31) form a dimensional affine space: if is such that then the space of solutions is simply . Specifically, the general solution reads
where are arbitrary coefficients, constrained only by the fixed value of the energy. Clearly, the expressions for the energy and the entropy become
so that the thermodynamic potentials are given by
For a fixed value of , equation (42) means that the energy can have any value greater than , depending on the coefficients . This degeneracy is apparent in figure 1: each time the Lagrange multiplier reaches an eigenvalue of the Laplacian, we have a plateau of the caloric curve. The degeneracy is in fact multiple: for each point of the plateau, characterized by , we have a whole dimensional sphere of solutions, with radius .
In the grand-canonical ensemble, the grand-potential has the same value for all the states on the plateau .
Strictly speaking, is a forbidden value since the solution space is then empty. Nevertheless, one can consider that as , the streamfunction diverges proportionally to :
Similarly, the energy diverges as .
Nature and stability of the critical points
So far, we have only found the critical points of the variational problems (25) and (26). It remains to determine their nature: minimum, maximum or saddle points of the entropy functional. To that purpose, we introduce the grand-potential functional . A critical point of entropy at fixed energy and circulation is a local maximum if, and only, if
for all perturbations that conserve energy and circulation at first order. This is the stability condition in the microcanonical ensemble. In the grand-canonical ensemble, the stability condition becomes for all perturbations .
Clearly if , and the point is a maximum of with respect to perturbations conserving the energy and circulation. Actually, this remains true as long as (see ). In fact, for all perturbations , even those which break the conservation of the constraints: the flow is grand-canonically stable (which implies microcanonical stability). This is related to the Arnold sufficient condition of nonlinear dynamical stability .
Conversely, for , let us show that the statistical equilibria computed in this section are in fact saddle points of the entropy. It suffices to consider perturbations proportional to eigenvectors of the Laplacian. Let such that . This perturbation conserves the circulation, and the variation of the energy at first order is where is the stream function of the basic mean flow (hence a linear combination of and , and possibly , , in the degenerate case). By the orthogonality property of the spherical harmonics, if () or and . For this particular perturbation , we have
so that if . Hence, perturbations proportional to eigenvectors of the Laplacian of order with suffice to destabilize the flow. The particular perturbation for instance destabilizes the equilibrium mean flow as soon as . In particular, no degenerate mode is stable. We have proven here microcanonical instability, which implies grand-canonical instability. As a consequence, for any given energy, there is only one stable equilibrium state (solid line on figure 1 which corresponds to ). The dashed lines on figure 1 correspond to unstable saddle points.
Summary of the results
In the microcanonical ensemble, there is only one equilibrium state (global entropy maximum) for each energy. It corresponds to the solid line on figure 1. The associated equilibrium flow is a counter-rotating solid-body rotation. The other states are unstable saddle points. In the canonical ensemble, there is an equilibrium state only for .The other states are unstable saddle points. The ensembles are equivalent. The statistical temperature is given by
It is negative when . The second derivative of the entropy is negative:
which means that is a concave function, in accordance with our findings of ensemble equivalence. Furthermore, the heat capacity is positive and can be computed explicitly (see figure 3):
3.2 Fixed energy, circulation and angular momentum
Due to the axial symmetry, there is another relevant conserved quantity that can be taken into account in the variational problem: the angular momentum
The critical points of the variational problems
which leads to the relationship . According to section 2.2, the solutions of this equation correspond to states that are steady in a frame rotating with angular velocity : indeed, this relation is of the form with . Imposing , that is neglecting conservation of the angular momentum, amounts to considering only the solutions which are stationary in the reference frame rotating with the angular velocity of the Earth. These solutions were described in the previous section. As in the previous section, spatial averaging yields and the relationship becomes
Making again the gauge choice , leading to , and setting , we find that is given by the Helmholtz equation
We now discuss the resolution of the Helmholtz equation (57) as in the previous section.
Case : the continuum solution
In this case, is invertible and is proportional to the first eigenmode of the Laplacian, so that
The equilibrium flow is a solid-body rotation with angular velocity
The potential vorticity is . We can compute the energy, angular momentum, and entropy:
The thermodynamic potentials and are given by
As always true for solid-body rotations (see B), the energy and angular momentum are linked by , with . This relation is independent of . Hence in the microcanonical ensemble, the continuum solution exists only on the curve . For there is no such solution.
where and are arbitrary coefficients. The requirement for the stream function to be real-valued imposes . The corresponding energy, angular momentum and entropy are given by
The Lagrange multiplier , associated with the conservation of angular momentum, is determined by the relation which can be inverted to yield
The entropy and grand-potential are given by
We shall see that these solutions are unstable saddle points in both ensembles. In the microcanonical ensemble, when , this degenerate solution reduces to the continuum solution.
In this case, equation (57) admits solutions only if the right-hand side vanishes, i.e. when . Then, the equilibrium flow has the general form
where is a real coefficient and a complex coefficient, linked by the energy and angular momentum requirements. Setting , the energy, angular momentum and entropy read
so that is in fact fixed by the angular momentum while and depend on both and :
Introducing the angle such that and , the stream function reads
When , this solution coincides with the continuum solution: it is a solid-body rotation. When , the flow has wave-number one in the longitudinal direction; it is a dipole with the angle playing the role of a phase. The phase is arbitrary (it is not determined by the constraints). The stream function can be re-written as