Kinetic theory of Onsager’s vortices in two-dimensional hydrodynamics

Kinetic theory of Onsager’s vortices in two-dimensional hydrodynamics

Pierre-Henri Chavanis Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de Toulouse, F-31062 Toulouse, France
Abstract

Starting from the Liouville equation, and using a BBGKY-like hierarchy, we derive a kinetic equation for the point vortex gas in two-dimensional (2D) hydrodynamics, taking two-body correlations and collective effects into account. This equation is valid at the order where is the number of point vortices in the system (we assume that their individual circulation scales like ). It gives the first correction, due to graininess and correlation effects, to the 2D Euler equation that is obtained for . For axisymmetric distributions, this kinetic equation does not relax towards the Boltzmann distribution of statistical equilibrium. This implies either that (i) the “collisional” (correlational) relaxation time is larger than , where is the dynamical time, so that three-body, four-body… correlations must be taken into account in the kinetic theory, or (ii) that the point vortex gas is non-ergodic (or does not mix well) and will never attain statistical equilibrium. Non-axisymmetric distributions may relax towards the Boltzmann distribution on a timescale of the order due to the existence of additional resonances, but this is hard to prove from the kinetic theory. On the other hand, 2D Euler unstable vortex distributions can experience a process of “collisionless” (correlationless) violent relaxation towards a non-Boltzmannian quasistationary state (QSS) on a very short timescale of the order of a few dynamical times. This QSS is possibly described by the Miller-Robert-Sommeria (MRS) statistical theory which is the counterpart, in the context of two-dimensional hydrodynamics, of the Lynden-Bell statistical theory of violent relaxation in stellar dynamics.

I Introduction

In 1949, Onsager onsager () published a seminal paper in which he laid down the foundations of the statistical mechanics of vortices in two-dimensional hydrodynamics 111His paper contains two additional major results: (i) the quantization of the circulation of vortices in superfluids (this is mentioned as a footnote in his paper); (ii) the inviscid dissipation of energy in 3D turbulence for singular Euler solutions.. He considered the point vortex gas as an idealization of more realistic vorticity fields and discovered that negative temperature states are possible for this system 222His argument for the existence of negative temperatures was given two years before Purcell & Pound purcell () reported the presence of negative “spin temperatures” in an experiment on nuclear spin systems.. At negative temperatures, corresponding to high energies, like-sign vortices have the tendency to cluster into “supervortices” similar to the large-scale vortices (e.g. Jupiter’s great red spot) observed in the atmosphere of giant planets. If the point vortices all have the same sign, one gets a monopole. If they have different signs, one gets a dipole made of two clusters of opposite sign, or possibly a tripole made of a central vortex of a given sign surrounded by two vortices of opposite sign. Therefore, statistical mechanics may explain the ubiquity of large-scale vortices observed in geophysical and astrophysical flows.

The qualitative arguments of Onsager were developed more quantitatively in a mean field approximation by Joyce & Montgomery jm (); mj (), Kida kida () and Pointin & Lundgren pl (); lp (), and by Onsager himself in unpublished notes esree (). The statistical theory predicts that the point vortex gas should relax towards an equilibrium state described by the Boltzmann distribution. Specifically, the equilibrium stream function is solution of a Boltzmann-Poisson equation. Several mathematical works caglioti (); k93 (); es (); ca2 (); kl () have shown how a proper thermodynamic limit could be rigorously defined for the point vortex gas (in the Onsager picture). It is shown that the mean field approximation becomes exact in the limit with (where is the number of point vortices and is the circulation of a point vortex).

A practical limitation of Onsager’s theory resides in the approximation of a continuous flow by a discrete collection of point vortices. This is clearly an idealization that he was aware of: “The distributions of vorticity which occur in the actual flow of normal liquids are continuous” onsager (). This approximation also leads to some ambiguity in the construction of a statistical model of realistic flows: “What set of discrete vortices will best approximate a continuous distribution of vorticity?” onsager ().

The statistical mechanics of continuous vorticity fields was developed later by Miller miller () and Robert & Sommeria rs (). The same theory appears in a paper (in russian) by Kuzmin kuzmin () eight years before but his contribution is less well-known. The Miller-Robert-Sommeria (MRS) statistical theory is based on the 2D Euler equation which describes incompressible and inviscid flows. It predicts that the 2D Euler equation can reach a statistical equilibrium state (or metaequilibrium state) on a coarse-grained scale as a result of a mixing process. Recently, the MRS theory has been applied to geophysical and astrophysical flows, notably to oceanic circulation kazantsev (), jovian vortices (Jupiter’s great red spot) turkington (); bs (); sw (), Fofonoff flows vb (); vblong (); naso () and oceanic rings and jets vr ().

The MRS theory shares many analogies with the theory of violent relaxation developed by Lynden-Bell lb () for collisionless stellar systems described by the Vlasov equation. The analogy between two-dimensional vortices and self-gravitating systems was mentioned by Onsager in a letter to Lin: “At negative temperatures, the appropriate statistical methods have analogues not in the theory of electrolytes, but in the statistics of stars…” esree (). This analogy has been further developed in houchesPH ().

At that stage, we would like to make some observations that will be important in the following. First of all, point vortices do not exist in nature 333An exception concerns the case of non-neutral plasmas under a strong magnetic field jm (); mj (). In the so-called “guiding-center approximation”, this system consists in long filaments of charge parallel to the magnetic field and moving under their mutual electric field with the velocity. The mathematical description of this system is equivalent to a collection of point vortices in which the charge of a filament corresponds to the circulation of a vortex.. Fundamentally, the physical system to consider is the continuous vorticity field that is solution of the 2D Euler equation. The point vortex model is a particular solution of the 2D Euler equation of the form . However, there are rigorous results mp () which show that any smooth solution of the 2D Euler equation may be approximated arbitrarily well over a finite time interval by a collection of point vortices with as . The physical situation is completely different in the case of stellar systems. Basically, a stellar system is a discrete collection of point mass stars. For in a proper thermodynamic limit, this discrete system can be approximated by a continuous distribution function that is solution of the Vlasov equation. This is the reverse situation with respect to 2D hydrodynamics! It is nevertheless interesting to further develop the analogy between a system of stars in astrophysics and a system of point vortices in 2D hydrodynamics.

It is well-known in astrophysics that a stellar system undergoes two successive types of relaxation henon (). In the “collisionless regime”, one can ignore correlations between stars due to finite effects and make a mean field approximation. In that case, the evolution of the system is governed by the Vlasov equation. The Vlasov-Poisson system can undergo a violent collisionless relaxation towards a quasi stationary state (QSS) on a few dynamical times. This corresponds to the mixing of the distribution function by the mean field. This mixing process is responsible for the apparent regularity of galaxies. This is precisely what the Lynden-Bell lb () statistical theory tries to describe. Unfortunatelly, in astrophysics, the Lynden-Bell theory does not give a good prediction of the structure of the whole galaxy because of the problem of incomplete relaxation in the outer part of the system where mixing is inefficient. On a longer timescale, in the so-called “collisional regime”, one must take into account correlations between stars due to finite effects (graininess) 444Actually, the evolution of the distribution function is not due to real collisions between stars (like in the theory of gases) but to weak deflexions due to binary encounters with relatively large impact parameter bt (). For simplicity, we shall however call them “collisions”.. These correlations are expected to drive the system towards the ordinary statistical equilibrium state described by the Boltzmann distribution for . This mixing process is due to discrete effects and takes place on a very long timescale. Indeed, the “collisional” relaxation time scales like and it diverges for bt (). For self-gravitating systems, the Boltzmann distribution is not reached in practice because of evaporation spitzer () and the gravothermal catastrophe antonov (); lbw (). Even if the statistical mechanics of self-gravitating systems is complicated because of the peculiar nature of the interaction, the concepts of violent collisionless relaxation towards a QSS and of slow collisional relaxation towards the Boltzmann distribution are fundamental and can be transposed to other systems with long-range interactions. For example, they have been clearly illustrated for the HMF model cdr ().

The same distinction between collisionless and collisional relaxation applies to the system of point vortices. In the “collisionless regime”, one can ignore correlations between point vortices due to finite effects and make a mean field approximation. In that case, the evolution of the system is governed by the 2D Euler equation. The 2D Euler-Poisson system can undergo a process of violent collisionless relaxation towards a quasi stationary state (QSS) on a few dynamical times. This corresponds to the mixing of the vorticity by the mean field. This mixing process leads to the formation of large scale vortices similar to those seen in planetary atmospheres. This is precisely what the MRS statistical theory tries to describe. On a longer timescale, called the “collisional regime”, one must take into account correlations between point vortices due to finite effects (graininess) 555Here again, the evolution of the vorticity field is not due to real collisions between point vortices but to distant binary encounters. For simplicity, we shall however call them “collisions”.. These correlations are expected to drive the system towards the ordinary statistical equilibrium state, described by the Boltzmann distribution, for . This mixing process is due to discrete effects and takes place on a very long timescale. As we shall see, the scaling with of the “collisional” relaxation time of point vortices is not firmly established.

It is of paramount importance not to confuse the Lynden-Bell (or MRS) statistical theory and the Boltzmann statistical theory which apply to very different timescales in the evolution of a Hamiltonian system with long-range interactions. This is related to the non-commutation of the limits and . Mathematically speaking, the QSS is reached when the limit is taken before the limit while the Boltzmann distribution is reached when the limit is taken before the limit. The Lynden-Bell (or MRS) theory makes the statistical mechanics of a continuous field evolving according to the Vlasov (or 2D Euler) equation while the Boltzmann theory makes the statistical mechanics of a discrete system of particles (stars or point vortices) evolving according to Hamiltonian equations. The first describes the QSS that is formed after a few dynamical times and the second describes the statistical equilibrium state that is reached for . The distribution predicted by Lynden-Bell (or MRS) is different from the Boltzmann distribution due to additional conservation laws in the Vlasov (or Euler) equation. In fact, the Lynden-Bell (or MRS) statistical mechanics and the Boltzmann statistical mechanics describe two types of mixing occurring at different scales: in the process of violent “collisionless” relaxation, the smooth distribution function (or vorticity field ) that evolves according to the Vlasov (or Euler) equation is mixed by the mean field and the coarse-grained distribution function (or vorticity field ) reaches a statistical equilibrium state. In the process of “collisional” relaxation, the discrete distribution of particles or (stars or point vortices) that evolves according to the Klimontovich equation (equivalent to the Hamilton equations) is mixed by discrete effects and the smooth distribution of particles or reaches a statistical equilibrium state.

Onsager onsager () assumed ergodicity of the point vortex gas and determined the ordinary statistical equilibrium state by evaluating the density of states and the entropy of the Hamiltonian system. However, he did not mention the source of mixing leading to the statistical equilibrium state. As we have explained, there are two sources of mixing in the point vortex gas: the fluctuations of the mean field during the phase of violent relaxation and the fluctuations due to discrete effects during the phase of collisional relaxation. The first one is very rapid and the second one very slow. It is likely that Onsager had in mind the mixing due to the fluctuations of the mean field. Indeed, he was basically interested in the 2D Euler equation, not by finite effects (“collisions”) in the point vortex gas. As we have explained, only continuous vorticity fields make sense in 2D hydrodynamics. The fact that he used the point vortex model was just a question of commodity. At that time, nobody knew how to make the statistical mechanics of a continuous vorticity flow. However, the statistical equilibrium state of the 2D Euler equation should be described by the MRS distribution, not by the Boltzmann distribution. The Boltzmann distribution describes the statistical equilibrium state of the point vortex gas that is reached for due to the development of correlations (finite effects) between point vortices. It is not clear whether Onsager had anticipated these two very different regimes related to the subtle non-commutation of the and limits.

In the present paper, we shall be essentially interested in the slow “collisional” relaxation of the point vortex gas towards the Boltzmann distribution due to finite effects. Its fast “collisionless” relaxation towards a QSS for will be only briefly discussed. Now, the relevance of the Boltzmann distribution for point vortices when is not clearly established because it relies on an assumption of ergodicity (or sufficient mixing) that may not be realized 666In fact, it has been proven that the point vortex gas is non-ergodic khanin (). However, statistical mechanics does not require strict ergodicity. An efficient mixing over the energy surface will suffice to justify the use of the microcanonical ensemble and of the Boltzmann distribution for . However, our concern here is that the point vortex gas may not mix well enough due to the effect of “collisions” (correlations, finite effects) to justify the applicability of statistical mechanics and of the Boltzmann distribution.. If we want to prove that the point vortex gas truly relaxes towards the Boltzmann distribution for , and if we want to determine the relaxation time (in particular its scaling with ), we must develop a kinetic theory. The MRS statistical theory is also based on an assumption of ergodicity, or efficient mixing, and the process of violent relaxation of the 2D Euler equation towards a QSS should also be described in terms of a kinetic theory (for the coarse-grained vorticity field). This kinetic theory is more complicated to develop and will not be considered here. We only refer to rsmepp (); rr (); csr (); quasi () for some investigations in this direction.

In previous papers preR (); pre (); bbgky (); kindetail (), we have developed a kinetic theory of 2D point vortices and we have derived a kinetic equation for the evolution of the smooth vorticity field taking two-body correlations into account. This equation is valid at the order and provides therefore the first order correction to the Euler equation obtained for . This kinetic equation was derived by using various methods such as the projection operator formalism, the quasilinear theory and the BBGKY hierarchy. In these works, we focused on (distant) two-body collisions and neglected collective effects. This leads to a kinetic equation similar to the Landau landau () equation in plasma physics 777Contrary to the Landau equation, our derived kinetic equation does not exhibit any divergence, suggesting that collective effects are not crucial for point vortices contrary to the case of plasma physics where they account for Debye shielding and regularize the large-scale divergence appearing in the Landau equation.. A kinetic theory of the point vortex gas had been previously developed by Dubin & O’Neil dubin () in the context of non-neutral plasmas under a strong magnetic field. They used the Klimontovich formalism and took collective effects into account. This leads to a kinetic equation similar to the Lenard-Balescu lenard (); balescu () equation in plasma physics. Their work was pursued in sdprl (); sd2 (); dubinjin (); dubin2 () in different directions. In a recent paper klim (), we used the Klimontovich formalism to derive a Fokker-Planck equation describing the evolution of a test vortex in a bath of field vortices, taking collective effects into account.

The purpose of the present paper is two-fold. One, physical, motivation is to discuss the implications of the kinetic theory of two-dimensional point vortices. An important conclusion of our analysis is that the relevance of the Boltzmann distribution is not established by the present-day kinetic theory. More precisely, we show that, if the point vortex gas ever relaxes towards the statistical equilibrium state, this takes place on a very long time, larger than (for axisymmetric distributions). The collisional relaxation of point vortices is therefore a very slow process and requires to take into account three-body, four-body… correlations, of order , …, that are ignored so far in the kinetic theory. It is also possible that mixing by “collisions” is not sufficient to drive the system towards the Boltzmann distribution. We emphasize, however, that the point vortex gas can rapidly reach a QSS as a result of a “collisionless” violent relaxation. However, this QSS is described by the Miller-Robert-Sommeria, or Lynden-Bell, distribution, not by the Boltzmann distribution. Another, more technical, motivation of our paper is to derive the Lenard-Balescu-type kinetic equation for point vortices from the BBGKY hierarchy. This derivation (which constitutes the main result of the paper) is new and completes our previous derivation bbgky () where collective effects were neglected. Our approach closely follows the method of Ichimaru ichimaru () for deriving the Lenard-Balescu equation in plasma physics. We also consider the relaxation of a test vortex in a bath of field vortices described by a Fokker-Planck equation. Finally, we mention open problems and future directions of research.

We note that other kinetic theories of point vortices exist but they apply to situations different from the one we will consider here. For example, Nazarenko & Zakharov zakharov () obtained a kinetic equation for point vortices with different intensities moving on the background of a shear flow and experiencing “hard” collisions. Marmanis marmanis () and Newton & Mezic newtonmezic () derived kinetic equations for a vortex gas viewed as a coupling, via the Liouville equation, between monopoles, dipoles and tripoles. Finally, Sire & Chavanis renormalization () developed a kinetic theory of three-body collisions (dipoles hitting monopoles) with application to the context of 2D decaying turbulence. General results about point vortices can be found in the book of Newton newton ().

Ii Two-dimensional point vortices: Evolution of the system as a whole

ii.1 The Boltzmann distribution

We consider an isolated system of point vortices with identical circulation on an infinite plane. Their dynamics is fully described by the Kirchhoff-Hamilton equations kirchhoff (); newton ():

(1)

where the coordinates of the point vortices are canonically conjugate. We shall denote the potential of interaction by . This Hamiltonian system conserves the energy , the circulation , the angular momentum and the impulse . We take the origin of the system of coordinates at the center of vorticity so that (we shall ignore this constraint in the following). The angular momentum fixes the vortex size . For , we expect this system to reach a statistical equilibrium state described by the microcanonical distribution onsager (); lp (); bbgky ():

(2)

where is the density of states. The microcanonical entropy of the system is defined by . The microcanonical temperature and the angular velocity are then given by and . As first realized by Onsager onsager (), the temperature of the point vortex gas can be negative. At negative temperatures , corresponding to high energy states, point vortices of the same sign group themselves in “supervortices” similar to large-scale vortices in planetary atmospheres. We define the thermodynamic limit as in such a way that the normalized energy , the normalized temperature , the normalized angular momentum and the normalized angular velocity are of order unity (these scalings result from simple dimensional analysis). We can renormalize the parameters so that the circulation of the vortices scales like and the vortex size like . Then, the inverse temperature scales like , the energy like , the angular momentum like and the angular velocity like . On the other hand, the total circulation and the dynamical time are of order unity.

In the thermodynamic limit defined previously, it can be rigorously proven caglioti (); k93 (); es (); kl (); ca2 () that the -body distribution function factorizes in a product of one-body distribution functions

(3)

Therefore, the mean field approximation becomes exact in the thermodynamic limit . Furthermore, the one-body distribution function , or equivalently the smooth density of point vortices or the smooth vorticity field , is the solution of a maximum entropy principle mj (); houchesPH ():

(4)

where

(5)

are the Boltzmann entropy, the mean field energy, the circulation and the angular momentum. Here, is the mean field stream function produced by the smooth distribution of vortices according to the Poisson equation

(6)

The mean velocity of a point vortex is then where is a unit vector normal to the flow. Fundamentally, the Boltzmann entropy is defined by where is the number of microstates (complexions), specified by the precise position of each point vortex, corresponding to a given macrostate, specified by the smooth vorticity field giving the average number of point vortices in macrocells of size . Using the Stirling formula for , we obtain the expression (5) of the Boltzmann entropy. Introducing Lagrange multipliers and writing the variational principle in the form , we obtain the mean field Boltzmann distribution

(7)

where is a positive constant. Substituting this relation in the Poisson equation (6), we obtain the Boltzmann-Poisson equation

(8)

like in the theory of electrolytes (for ) or in the statistics of stars (for ). The statistical equilibrium state is then obtained by solving this equation and relating the Lagrange multipliers , (or ) and to the constraints , and lp (). Then, we have to make sure that the resulting distribution is a maximum of at fixed circulation, energy and angular momentum (most probable state), not a minimum or a saddle point.

ii.2 BBGKY-like hierarchy and expansion

In order to establish whether the point vortex gas will reach the Boltzmann distribution (7) predicted by statistical mechanics and determine the timescale of the relaxation, in particular its scaling with , we need to develop a kinetic theory of point vortices. Basically, the evolution of the -body distribution of the point vortex gas is governed by the Liouville equation

(9)

where

(10)

is the velocity of point vortex due to its interaction with the other vortices. Here, denotes the exact stream function in produced by the discrete distribution of point vortices and denotes the exact velocity induced by point vortex on point vortex . The Liouville equation (9), which is equivalent to the Hamiltonian system (1), contains too much information to be of practical use. In general, we are only interested in the one-body or two-body distributions 888Nevertheless, a conclusion of our study will be that higher order distributions should also be considered.. From the Liouville equation (9) we can construct a complete BBGKY-like hierarchy for the reduced distributions . It reads bbgky ():

(11)

This hierarchy of equations is not closed since the equation for the one-body distribution involves the two-body distribution , the equation for the two-body distribution involves the three-body distribution , and so on… The idea is to close the hierarchy by using a systematic expansion of the solutions in powers of in the thermodynamic limit .

The first two equations of the hierarchy are

(12)
(13)

We decompose the two- and three-body distributions in the suggestive form

(14)
(15)

This is similar to the Mayer mayer () decomposition in plasma physics. Substituting Eqs. (14) and (II.2) in Eqs. (12) and (13) and simplifying some terms, we obtain

(16)
(17)

Equations (16) and (17) are exact for all but they are not closed. We shall close these equations by expanding the solutions in powers of for . In this limit, the correlation functions scale like . In particular, and . On the other hand, and . We are aiming at obtaining a kinetic equation that is valid at the order . Let us consider the terms in Eq. (17) one by one. The first and second terms are of order . They represent the transport of the two-body correlation function by the mean flow. The third term represents the effect of “soft” binary collisions between vortices; it is of order . If we consider only these first three terms (as done in our previous paper bbgky ()), we obtain a kinetic equation that is the counterpart of the Landau equation in plasma physics. The fourth term represents the effect of “hard” binary collisions between vortices. This is the term which, together with the previous ones, gives rise to the Boltzmann equation in the theory of gases. It is of order but it may become large at small scales so its effect is not entirely negligible. For example, in plasma physics, hard collisions must be taken into account in order to regularize the logarithmic divergence that appears at small scales in the Landau and Lenard-Balescu equations. In the case of point vortices, there is no divergence at small scales in the kinetic equation that we shall obtain. Therefore, in this paper, we shall ignore the contribution of this term (but we note that it would be interesting to study it specifically). The fifth term is of order and it corresponds to collective effects. In plasma physics, this term leads to the Lenard-Balescu equation. It takes into account dynamical Debye screening and regularizes the divergence at large scales that appears in the Landau equation. The main contribution of this work will be to take this term into account in the kinetic theory of point vortices in order to obtain a Lenard-Balescu-type kinetic equation from the BBGKY hierarchy. The last two terms are of the order and they will be neglected. In particular, at the order , we can neglect the three-body correlation function. In this way, the hierarchy of equations is closed.

If we introduce the smooth vorticity field and the two-body correlation function , we get at the order :

(18)
(19)

We have introduced the mean velocity in created by all the vortices

(20)

and the fluctuating velocity created by point vortex on point vortex :

(21)

We also recall that the exact velocity created by point vortex on point vortex can be written

(22)

where is the binary potential of interaction between point vortices. Equations (18) and (II.2) are exact at the order . They form the right basis to develop a kinetic theory of point vortices at this order of approximation.

ii.3 The limit : the 2D Euler equation (collisionless regime)

In the limit for a fixed time , the correlations between point vortices can be neglected and the -body distribution factorizes in a product of one-body distributions:

(23)

Therefore, for long-range interactions, the mean field approximation is exact at the thermodynamic limit . Substituting the factorization (23) in the Liouville equation (9), and integrating on , , …, we find that the smooth vorticity field of the point vortex gas is solution of the 2D Euler equation

(24)

This equation also results from Eq. (18) if we neglect the correlation function in the right-hand side. The 2D Euler equation describes the collisionless evolution of the point vortex gas for times smaller than . In practice, is large so that the domain of validity of the 2D Euler equation is huge. The 2D Euler equation is the counterpart of the Vlasov equation in plasma physics and stellar dynamics. It can undergo a process of mixing and violent relaxation towards a quasistationary state (QSS) on a very short timescale, of the order of a few dynamical times . This QSS has the form of a large-scale vortex. Miller miller () and Robert & Sommeria rs () have developed a statistical mechanics of the 2D Euler equation to predict these QSSs. The MRS theory is the counterpart of the Lynden-Bell theory for collisionless stellar systems lb (). The analogy between two-dimensional vortices and stellar systems is developed in houchesPH ().

ii.4 The order : an exact kinetic equation (collisional regime)

If we want to describe the collisional evolution of the point vortex gas, we need to consider finite effects. Equations (18) and (II.2) are valid at the order so they describe the evolution of the system on a timescale of order . The equation for the evolution of the smooth vorticity field is of the form

(25)

where is a “collision” term analogous to the one arising in the Boltzmann equation in the theory of gases. In the present context, there are no real collisions between point vortices. The term on the right-hand side of Eq. (18) is due to the development of correlations between vortices. It is induced by the two-body correlation function which is determined in terms of the vorticity by Eq. (II.2). Our aim is to derive a kinetic equation that is valid at the order and that gives the first correction to the Euler equation.

The formal solution to Eq. (II.2) is

(26)

where the propagator satisfies the equation

(27)

with the initial condition . Equation (27) can be viewed as a linearized version of the 2D Euler equation (see Appendix A). Indeed, if we make the replacement in Eq. (83) and linearize it with respect to [see Eq. (84)], we obtain Eq. (27) in which plays the role of . Consequently, the propagator obeys the linearized Euler equation.

Substituting Eq. (II.4) in Eq. (18), we obtain a kinetic equation

(28)

that is exact at the order . If we neglect collective effects, we recover the generalized Landau equation (115) that was derived in our previous articles pre (); bbgky (); kindetail (). Equation (II.4) is a complicated non-Markovian integrodifferential equation. It is furthermore coupled to Eq. (27) which determines the evolution of the propagator. In order to resolve this coupling, it is necessary to consider the timescales involved in the dynamics. We shall argue that, for a given vorticity profile , the two-body correlation function relaxes to its asymptotic form on a timescale short compared with that on which changes appreciably. This is the equivalent of the Bogoliubov ansatz in plasma physics. It is expected to be a very good approximation for since the two-body correlation function relaxes on a few dynamical times while the vorticity field changes on a collisional relaxation time of the order or larger. Therefore, it is possible to neglect the time variation of in the calculation of the collision term and extend the time integration to . This amounts to replacing the two-body correlation function in Eq. (18) by its asymptotic value for a given vorticity profile . After the correlation function has been obtained as a functional of , the time dependence of can be reinserted. With this Bogoliubov ansatz (or adiabatic hypothesis), the kinetic equation (II.4) can be rewritten

(29)

Similarly, the equation for the propagator takes the form

(30)

with the initial condition . The two equations (II.4) and (30) are now completely decoupled. For a given vorticity profile at time , one can solve Eq. (30) to obtain and determine the collision term in the right-hand side of Eq. (II.4). Then, the vorticity profile evolves with time on a slow timescale according to Eq. (II.4). Interestingly, the structure of this kinetic equation bears a clear physical meaning in terms of generalized Kubo relations bbgky (). This equation is valid at the order and, for , it reduces to the (smooth) 2D Euler equation (24) which describes the collisionless evolution of the point vortex gas.

The kinetic equation (II.4) is, of course, equivalent to the pair of equations

(31)
(32)

which correspond to the first two equations of the BBGKY-like hierarchy at the order within the Bogoliubov ansatz. These equations, supplemented by Eq. (30) for the propagator, provide the formal solution of the problem in the general case. In order to obtain more explicit expressions, we have to consider particular types of flow.

Iii Explicit kinetic equation for axisymmetric flows

iii.1 Laplace-Fourier transforms

We consider an axisymmetric distribution of point vortices that is a stable steady state of the 2D Euler equation. Therefore, the vorticity field evolves in time only because of the development of correlations between point vortices due to finite effects (graininess). In that case, an explicit form of the kinetic equation can be derived.

For an axisymmetric flow, introducing a system of polar coordinates, the vorticity field and the two-body correlation function can be written as and , and the mean velocity as . On the other hand, according to Eq. (22), the radial velocity (in the direction of ) created by point vortex on point vortex , is

(33)

where is symmetric in and and even in (see Appendix B). In that case, Eqs. (31) and (II.4) take the form

(34)
(35)

For convenience, we have not written the time in the vorticity field appearing in the correlation function. As we have previously explained, the vorticity profile is assumed “frozen” on the short timescale that we consider to compute the asymptotic expression of the correlation function and the collision term (Bogoliubov ansatz). The time will be restored at the end in the kinetic equation.

We now expand the potential of interaction in Fourier series

(36)

and perform similar expansions for and . In terms of these Fourier transforms, Eqs. (34) and (III.1) can be rewritten

(37)
(38)

Introducing the Laplace transform of (see Appendix A for the definition of Laplace transforms) and integrating on time , we get

(39)

where is the Laplace contour in the complex -plane. The integration over can be performed by closing the contour by an infinite semicircle in the upper half-plane. Since vanishes for , the only contribution of the integral comes from the pole at . Using the residue theorem, we obtain