Vortices andTwo-Dimensional FluidMotionC. Eugene Wayne

The study of fluid motions is of obviousimportance for a host of applicationsranging in scale from the microscopicto the atmospheric. Since we live in athree-dimensional world, it may be less

obvious why the understanding of two-dimensionalfluid flows is of interest. However, in many appli-cations, such as the atmosphere or the ocean, thefluid domain is much smaller in one direction thanin the other two—and also smaller than the typicalsize of features of interest in the fluid. For example,in the case of the atmosphere, the thickness is afew tens of kilometers, while the lateral extent istens of thousands of kilometers and the diameterof a feature such as a hurricane can be severalhundreds of kilometers. Furthermore, in both theatmosphere and the ocean, the applicability of atwo-dimensional approximation is enhanced bytwo additional effects: the stratification of themedium (which reduces the effective thickness ofthe domain) and the rotation of the earth, whichtends to reduce variations in the vorticity field withheight and means that in any cross-sectional plane,the flow is effectively two-dimensional. In suchcircumstances a two-dimensional approximation tothe fluid motion can provide very accurate insightsinto the behavior of the physical system.

Even more interesting is the fact that two- andthree-dimensional fluids behave in qualitativelydifferent fashions. In three-dimensional flowsenergy typically flows from large-scale features tosmall ones until it is dissipated by the viscosityof the fluid. In two dimensions the phenomenontends to reverse itself, and the energy concentratesitself in a few large vortex-like structures. This

C. Eugene Wayne is professor of mathematics at BostonUniversity. His email address is cew@math.bu.edu.

phenomenon, known as the “inverse cascade”,manifests itself in a striking visual way through thecoalescence of many small vortices into a smallernumber of larger vortices.

Figure 1. Atmospheric vortices formed by windflowing past the Aleutian Islands, captured by

Landsat 7 [17]. Note that in this image, thevorticity field cannot be directly visualized.

Instead, one views passive tracer particles (i.e.,clouds!) that are carried along by the

background flow and which are believed toaccurately mimic the vorticity field.

A beautiful visualization of this effect wascreated by Maarten Rutgers in turbulent soapfilms (see Figure 2). The patterns make visibledifferences in the vorticity of the fluid. Thevorticity will be defined more precisely below

10 Notices of the AMS Volume 58, Number 1

but basically represents the rotational speed of thefluid either clockwise or counterclockwise. In thisfigure, the flow begins above the top of the pictureand falls under the influence of gravity toward thebottom of the picture, so going down in the pictureindicates a later stage in the evolution of the flow.The tendency of the vorticity to organize itself intolarger and larger structures is clearly visible.

Figure 2. Two-dimensional turbulent flowvisualized in a soap film by MaartenRutgers—for an even more striking illustrationof this phenomenon, see the video clip underthe “Turbulence” section ofhttp://maartenrutgers.org/ [20].

The growth in the size of vortices and thereduction in their numbers is also visible innumerical experiments, such as those displayedin Figure 3, that are the result of research bythe vortex dynamics group at the TechnischeUniversiteit Eindhoven (TUE), Netherlands. One ofthe main goals in the study of two-dimensionalturbulent flows has been to understand and explainthis inverse cascade and in particular to explain thetendency of the vorticity to coalesce into a smallerand smaller number of larger and larger vortices.In this article I will explain how by exploiting ideasfrom the kinetic theory of gases one can show that

almost all two-dimensional viscous flows eventuallyapproach a single, large vortex.

Figure 3. A numerical simulation of atwo-dimensional turbulent flow. The figuresdisplay the vorticity field (with blue and redrepresenting fluid swirling in oppositedirections) at successively later and later timesand clearly indicate the tendency of regions ofvorticity of like sign to coalesce into a smallerand smaller number of larger vortices [21].

The typical way of describing the motion of afluid is through its velocity field, v(z, t); that is,through a vector field that at each point in spaceand time gives the velocity of the fluid at thatpoint. However, more than 150 years ago Helmholtzrealized that in addition to the velocity, the vorticityof the fluid carries important information aboutthe nature of the flow. As mentioned above, thevorticity roughly measures the swirl in a fluid. Moreprecisely, the vorticity is defined as the curl of thefluid velocity,

ω(z, t) = ∇× v(z, t).

Note that we see already an important differencebetween two and three dimensions—in two di-mensions, only one component of the vorticityis nonzero, and thus we can treat the vorticityessentially as a scalar field.

For an incompressible fluid, the fluid velocitysatisfies the Navier-Stokes equations, the systemof coupled nonlinear partial differential equations

∂tv+ v · ∇v = ν∆v− 1ρ∇p ,(1)

∇ · v = 0 ,(2)

whereν is the kinematic viscosity of the fluid (whichwe will assume is constant), ρ is the fluid density(which is constant due to the incompressibilitycondition), and p = p(z, t) is the pressure inthe fluid. The first of these equations is justNewton’s law for the fluid, with the left-handside representing the acceleration of the fluidand the right-hand side the forces acting on it.We will assume that the only forces present arethe internal viscous forces, modeled by the firstterm on the right-hand side, and the pressureforces, represented by the second. External forcesacting on the fluid could be incorporated byadding additional terms to the right-hand side ofthe equation. We will also ignore the effects of

January 2011 Notices of the AMS 11

boundaries on the fluid by assuming that the fluidoccupies all of Rd with d = 2,3.

In order to determine how the vorticity evolves,one can take the curl of both sides of the first of theNavier-Stokes equations. The dynamics representedby the two- and three-dimensional equations arestrikingly different despite the close relationshipbetween the equations. In three dimensions onehas the system of equations(3)∂tω(z, t)−ω ·∇v(z, t)+v ·∇ω(z, t) = ν∆ω(z, t),while in two dimensions one has only the single,scalar equation

(4) ∂tω(z, t)+ v · ∇ω(z, t) = ν∆ω(z, t).(We will use a boldface ω to denote the vorticityvector and an ω to denote the single, nonzerocomponent of the vorticity in two dimensions.)The presence of the “vortex stretching term”,−ω ·∇v(z, t) in (3), is a critical physical differencebetween these two equations. For certain specialfluid configurations, this term can lead to a sort offeedback mechanism in which the vorticity beginsto grow. While it is not known if this growthcan continue without bound, there is no obviousmechanism to stop its growth, and this is thephysical source of the uncertainty as to whetheror not smooth solutions of the Navier-Stokesequations exist for all time in three dimensions.The fact that it is still unknown whether or not thepartial differential equations that are believed todescribe such a basic system as fluid motion haveunique, smooth solutions makes this obviously anextremely important question, and the successfulresolution of this question (or the discovery ofan example demonstrating the formation of asingularity in the solution in finite time) waschosen by the Clay Mathematics Institute as one ofthe one-million-dollar Millennium Prize Problems.The role of the vortex-stretching term and itsrelationship both to the possible formation ofsingularities and the analysis of the Navier-Stokesequation is discussed in detail in [5]. In contrast,in two dimensions the absence of this term allowsone to construct global solutions of the two-dimensional vorticity equation, even for initial datawith very little regularity [8].

One respect in which the vorticity formulationof the fluid equations is less convenient than thevelocity-pressure formulation is that the velocitystill appears in equations (3) and (4) for theevolution of the vorticity. However, if we rememberthat the vorticity is the curl of a divergence-freevector field (i.e., the velocity), then we can recoverthe vorticity field from the velocity field via theBiot-Savart law, which inverts the curl, and whichin the two-dimensional case on which we will focusfrom now on takes the form

(5) v(z, t) = 14π

∫R2

(z− z)⊥

|z− z|2 ·ω(z, t)dz .

Here, if z = (x, y) ∈ R2, then z⊥ = (−y, x). Thusthe velocity can be regarded as a linear, butnonlocal, function of the vorticity. With this pointof view the two-dimensional vorticity equation(4) can be regarded as a nonlinear heat equationin which the nonlinear term is quadratic andnonlocal. As we will see later, this relationship withthe heat equation will play an important role inunderstanding solutions of (4).

Let’s now look more closely at equation (4) andtry to understand the influence of various terms inthe equation. First consider the case in which ν = 0(the inviscid case). In this case, if we pretend for themoment that the velocity field is given to us, ratherthan being determined by the vorticity, then theequation becomes simply a transport equation inwhich the vorticity is carried along by the velocityfield. In reality, the situation is more complicated,because as the vorticity is advected by the velocityfield, the velocity field itself changes in responseto the changing vorticity; and in order to obtainan accurate model of the evolution of the vorticity,one must incorporate the “feedback” of the vortexmotion on the velocity field.

Helmholtz, and then later and more system-atically Kirchhoff, made the assumption that thevorticity could be written as a finite sum of pointvortices (i.e., delta functions) whose positionsmoved in response to the velocity field they cre-ated. Note that the velocity field of a point vortexcan be computed explicitly from the Biot-Savart law,and using this, Helmholtz and Kirchhoff could trackthe dynamical evolution of the velocity field in theirmodel and account for the feedback the vortex mo-tion creates. Thus, if one assumes that the vorticityfield can be written asω(z, t) =

∑Nk=1 Γkδ(z−zj(t)),

where Γj is the strength of the j th vortex and zj(t)is its position and substitutes this into the ν = 0case of (4) (and interprets the solution in anappropriate weak sense—see [13] for details), thenone finds explicit ordinary differential equationsfor the locations of the centers of the vortices. Ifone sets zj(t) = (xj(t), yj(t)), then one finds that

(6)

xj(t) = −1

2π

∑k 6=j

Γk yj − yk|zj − zk|2,

yj(t) =1

2π

∑k 6=j

Γk xj − xk|zj − zk|2.

These equations have explicit solutions for anumber of simple arrangements of small num-bers of vortices. For instance, one can easilycheck that two vortices of equal strength willmove on a circle about the point midway be-tween them, while two vortices of oppositestrength will move on parallel lines, in a di-rection perpendicular to the line joining them (seeFigure 4).

The set of equations (6) turns out to have anumber of remarkable properties [18]. For instance,

12 Notices of the AMS Volume 58, Number 1

Vortices of opposite strengthVortices of equal strength

Figure 4. The dynamics of the centers of a pairof point vortices of equal strength and ofopposite strength.

it is a Hamiltonian system in which the Hamiltonianis proportional to the sum of the logarithm of thedistance between pairs of vortices; furthermore,if the number of vortices is three or less, itis a completely integrable Hamiltonian system.However, if one has four or more vortices, onetypically finds chaotic solutions.

In spite of the special properties of the point-vortex equations, analytic solution for generalinitial data becomes quickly impossible for morethan a small number of vortices. (An exceptionto this general rule are the equilibrium andrelative equilibrium solutions, for which there areinteresting connections with analogous solutionsin the N-body problem in celestial mechanics [1].)However, given the Hamiltonian nature of theequations of motion and the chaotic nature of theirsolutions for large numbers of vortices, it is natural(at least in retrospect) to attempt to understand thebehavior of large collections of vortices with theaid of statistical mechanics. Thus, if one considersthe initial vorticity distribution in Figure 3, onecan imagine it as a “gas” of point vortices thatinteract with the other vortices through a potentialenergy function in which the energy of a vortex pairis proportional to the logarithm of the distancebetween them. Lars Onsager may have been the firstperson to adopt this point of view, and it led him toa remarkable conclusion [7]. Onsager found that thestatistical mechanical description of a collection ofpoint vortices moving according to the equationsof motion (6) could support states in which aparameter analogous to the absolute temperaturein a traditional statistical mechanical system wasactually negative. Furthermore, Onsager realizedthat a consequence of these negative temperaturestates was that vortices of like sign would coalesceand that this could explain the tendency observedin Figure 3 for large vortices of each sign to formas the fluid evolves. As Onsager himself put it ([19],quoted in [7])

It stands to reason that the largecompound vortices formed in thismanner will remain as the only con-spicuous features of the motion.

Thus Onsager had provided a means of explaininghow large vortices could form from randomcollections of small vortices, provided the effectsof viscosity are ignored and assuming that thehypotheses that underlie the theory of statisticalmechanics are satisfied.

It is not only in the theoretical understandingof fluid flows that the Helmholtz-Kirchhoff pointvortex model has played an important role. Eventhough one cannot solve the equations (6) analyti-cally for more than a few vortices, they are perfectlyamenable to numerical solution, and this idea hasformed the basis of “vortex methods” or “meshlessmethods” in computational fluid mechanics. Inthis approach, one first approximates the initialdistribution of vorticity by a collection of point vor-tices (or, more frequently in numerical approaches,smoothed vortices with finite size cores). The keyquantities are the location and strength of eachof the vortices. The vortex strength is typicallyconserved, and thus the evolution of the fluid canbe tracked by following the locations of the centersof the vortices via a system of ordinary differentialequations such as (6). It can be shown that vor-tex methods give convergent approximations (asthe number of points used in the approximationtends to infinity) to inviscid fluid flows (thoughsometimes with relatively slow convergence rate).However, one problem that can arise is related toOnsager’s observation that in a large collectionof vortices, those of like sign will tend to clumptogether. Thus, after some time, large parts ofthe computational domain may have only a veryfew vortices, which leads to a loss of informationabout the flow in these regions. For a furtherdiscussion of the advantages and disadvantages ofusing vortex methods to numerically approximatetwo-dimensional flows see the recent monographof Majda and Bertozzi [12], while [2] containsa survey of recent improvements in the vortexmethod, with a particular focus on how one canincorporate viscous effects in the method.

Thus far, we have mainly discussed the limitingcase of the Navier-Stokes equation in which theviscosity is zero. In realistic fluids (with thespectacular exception of super fluids) the viscositymay be small but is never zero, so we next considerits effects on the preceding scenarios. One can showthat for finite (sometimes short) times, the solutionof the weakly viscous Navier-Stokes equation withappropriate initial data is well approximated bysolutions of the point-vortex model [13]. However,these short-time results cannot provide insight intothe long-time phenomena that occur in the “inversecascade”. Indeed, we’ll see that even within thelong-time regime there are two distinct time scales,one on which the inviscid phenomena predictedby Onsager appear and a second, typically longer,time scale over which viscous effects manifest

January 2011 Notices of the AMS 13

Figure 5. The vorticity field around aswimming “fish”, computed with a modern

vortex method [6]

themselves and which we show below lead to theformation of a single large vortex in the system.

If we ignore the nonlinear term and focus onthe linear terms in the equation, we just find thetwo-dimensional heat equation

(7) ∂tω = ν∆ω .In this equation we know that the effects of theLaplacian term are to spread things out—or diffusethem—at a rate proportional to ν. Indeed, if wewere to take a point vortex, i.e., a Dirac δ-mass,as an initial condition, we know that the effect ofthe equation is just to “smear” this point out intoa Gaussian. More precisely, if we take an initialcondition ω(z,0) = αδ(z) for (4), and we ignorethe nonlinear term in the equation, then we findimmediately that the resulting solution is

(8) ω(z, t) = α4πνt

e−|z|2/(4νt) .

Remarkably, this explicit Gaussian turns out tobe an exact solution not only of the linear heatequation but also of the nonlinear two-dimensionalvorticity equation, known as the Oseen-Lamb vortex.To see the reason for this, recall that since the fluidis incompressible, its velocity field is divergencefree. In this case the Helmholtz decompositionimplies the existence of a stream function ψ(z)such that v(z) = (∂yψ,−∂xψ). Thus the vorticityis related to the stream function via the equation

(9) ω(z, t) = ∂xv2 − ∂yv1 = −∆ψ(z, t).In the present example of the Oseen-Lamb vortex,in which the vorticity is a purely radial functiondepending only on |z|, solving Poisson’s equationwill give a stream function that is also purely radialand that in turn gives rise to a purely tangentialvelocity field. Since the nonlinear term in the

vorticity equation consists of the dot product ofthe velocity field with the gradient of the vorticity,we will have the dot product of a tangential with aradial vector; i.e., the value of the nonlinear term iszero when evaluated on the velocity and vorticityfields of the Oseen-Lamb vortex. In this case thevorticity equation reduces to the heat equation,which, as we have already remarked, is solved bythe Oseen-Lamb vortex.

Note that in the expression for the vorticity fieldof the Oseen vortex the space and time variablesare linked in a special fashion. This suggests that itmay be convenient to study the vorticity equation innew variables, so-called scaling variables. With thisin mind we define new dependent and independentvariables through the change of variables(10)

ω(x, y, t)= 11+ t w

(x√

1+ t ,y√

1+ t , log(1+ t)).

If we define ξ = x√1+t , η =

y√1+t , ζ = (ξ, η), and

τ = log(1+ t), then in terms of these new variablesthe two-dimensional vorticity equation takes theform(11)

∂tw = ν∆w + 12∇· (ζw)−u ·∇w ≡ Lνw −u ·∇w,

where u is just the velocity field, rewritten in termsof the scaling variables, and the derivatives inthe Laplacian and divergence are now taken withrespect to ζ instead of z.

Note that in terms of these variables, the familyof Oseen vortices are all fixed points of this equation,i.e., the functions

ω(ξ,η) = Ωα(ξ, η) = α4πe−|ζ|2/4ν

= α4πe−(ξ2+η2)/4ν

(12)

are all stationary solutions for equation (11). Notethat we have normalized the Gaussian so thatthe parameter α of the Oseen vortex gives thetotal vorticity (i.e., the integral of the vorticity) ofthe solution. Note further that both the vorticityequation, (4), and the rescaled vorticity equation,(11), conserve the total vorticity. The presence ofthis family of fixed points suggests that we mightbe able to use ideas from dynamical systems theoryto study the stability of these fixed points andto try to determine whether they can explain theasymptotic behavior of some or all of the solutionsof this partial differential equation.

There are (at least) two different dynamicalsystems approaches that can be applied to studythe stability and influence on the asymptotics ofthe fixed points (12); namely:

• Linearization about the fixed point and theconstruction of invariant manifolds in thephase space corresponding to the variousspectral subspaces of the linearization, or

• Lyapunov functionals.

14 Notices of the AMS Volume 58, Number 1

In this article I’ll focus on the Lyapunov func-tional approach because of its relationship to thestatistical mechanics point of view that we havediscussed above. However, one can also use theinvariant manifold approach to analyze the asymp-totic behavior of solutions of (11). In contrast to theLaplacian, whose spectrum is the entire negativereal axis, the linear operator, Lν , when consideredon spaces of functions that go to zero rapidly atinfinity, has a spectrum that consists of a set ofdiscrete eigenvalues, plus a half-plane of essentialspectrum. One can construct finite-dimensional,invariant manifolds corresponding to the spanof the eigenfunctions of these eigenvalues thatdescribe very precisely the asymptotics of smallsolutions of (11) [9]. Moreover, the eigenfunctionscorresponding to these discrete eigenvalues forma convenient basis with respect to which onecan systematically extend the Helmholtz-Kirchhoffpoint-vortex model described above to include theeffects of viscosity and finite core size [16].

Recall that a Lyapunov functional for a dynamicalsystem is a continuous function, bounded belowon the phase space of the dynamical system, whichwhen evaluated on an orbit of the dynamicalsystem is monotonic nonincreasing and boundedbelow. (Very colloquially, it is a function whosevalue decreases with time when evaluated alongsolutions of the dynamical system.) Recall thatour goal is to understand how the vorticity formslarge structures after very long times. One wayof characterizing the long-time asymptotics of adynamical system is through theω-limit set, whichis the set of points in the phase space which atrajectory approaches arbitrarily closely to as timetends to infinity. If the trajectory approaches astable fixed point, then the ω-limit set will consistjust of this fixed point. However, the ω-limitset can also be a periodic orbit or even somechaotic attractor. Note that it is not immediatelyapparent that every trajectory will have anω-limit.This follows if the solutions of the dynamicalsystem satisfy certain compactness conditions.For the remainder of the article we will assumethat the solutions of the vorticity equation satisfythese conditions, though proving this takes somework—the details are explained in [10].

A key tool in locating the ω-limit set is theLaSalle Invariance Principle, which says that givena Lyapunov functional for a dynamical system, theω-limit set of a trajectory must lie in the set onwhich the Lyapunov function is constant (whenevaluated along an orbit). More precisely, if thepoints in the phase space of the dynamical systemare denoted by w , if the flow or semi-flow definedby the dynamical system is denoted by Φt , and ifthe Lyapunov functional is denoted by H(w) (andit is differentiable), then theω-limit set must lie in

the set of points

(13) E = w | ddtH(Φt(w))|t=0 = 0.

Let’s now return to rescaled vorticity equation(11) and make use of one more analogy. So far, wehave considered the equation in which we ignoredthe dissipative term and retained only the timederivative and the nonlinearity, and we have alsoignored the nonlinear term and retained only thedissipative term. Let’s finally retain all the termsin (11) but ignore for the moment the fact that thevelocity field is linked to the vorticity and pretendthat it is just some given, divergence-free vectorfield. In this case if we use the fact that ∇ · u = 0,we can write (11) as

(14) ∂τw = ν∆w −∇ · (w∇U) ,

where ∇U(ζ, τ) = u(ζ, τ) − 12ζ. This equation is

just the Fokker-Planck equation, which describesthe evolution of the probability distribution ofthe location of a particle in a gas confined by thepotential U and with diffusive effects modeled bythe term ν∆w . Equation (14) has been studiedextensively by physicists and mathematicians formore than a century, and, in particular, motivatedby Boltzmann’s theory that the entropy of sucha system should never decrease, Lyapunov func-tionals have been developed that are based onthe entropy. (See [14] for a nice discussion ofthe interplay between physics and analysis in thisproblem.)

The classical entropy function for solutions of(11) would be S[w](t) =

∫R2 w(ζ) lnw(ζ)dζ, but

as explained in [14] it is often more useful to studythe relative entropy, that is, the entropy relative tothe expected asymptotic state of the system. In thiscase our candidate for the asymptotic state of thesystem is one of the Oseen vortices Ωα defined in(12), which results in a relative entropy functional

(15) H[w](τ) =∫R2w(ζ, τ) ln

(w(ζ, τ)Ωα(ζ)

)dζ.

Of course, so far, there is no proof that this isa Lyapunov functional for the (rescaled) vorticityequation. It is a candidate, suggested by the analogybetween (11) and the Fokker-Planck equation, butin the actual vorticity equation the fluid velocity isnot independent of the vorticity (and in particularthe vorticity equation, unlike the Fokker-Planckequation, is a nonlinear equation). Furthermore,a second problem is apparent from formula (15).Since solutions of the Fokker-Planck equationrepresent probability densities, it is natural toassume that they are nonnegative, and consequentlythe logarithm in (15) is well defined. However, itis quite unnatural to assume that solutions ofthe vorticity equation are all of one sign—typicalsolutions intermingle regions of positive vorticitywith regions of negative vorticity, and for such

January 2011 Notices of the AMS 15

solutions it becomes impossible to define therelative entropy functional.

We’ll return to the second of these problems ina moment, but first consider whether or not (15) atleast defines a Lyapunov functional for solutionsof the vorticity equation which are everywherepositive. As we’ll explain below, if the solution ofthe vorticity equation is positive at some instant oftime, it will remain so for all later times. Assumingthat the solution is positive for some time t andthat the vorticity tends to zero sufficiently rapidlyas |ζ| → ∞ so that the integral in (15) converges,we can compute the derivative of the entropy, andwe find:(16)ddτH[w](τ) =

∫R2

(1+ ln

(w(ζ, τ)Ωα(ζ)

))∂w∂τdζ.

If we now use (11) to rewrite the time derivative ofthe vorticity and then integrate by parts (and usethe relationship between vorticity and velocity), wefind

(17)ddτH[w](τ) = −

∫R2w(ζ)

∣∣∣∣∇( wΩα)∣∣∣∣2

dζ.

Since w is positive, (17) shows that for such solu-tions the relative entropy function is a Lyapunovfunction. One can also show H is bounded belowfor w in an appropriate function space (and thatthe compactness properties mentioned above alsohold), and hence the LaSalle Invariance Principleholds. This means that the ω-limit set for anypositive solution of the vorticity equation must liein the set where d

dτH[w](τ) = 0. Since w(ζ, t) > 0,(17) implies that the only way the time derivativeof the entropy can vanish is if

(18)(wΩα)= constant;

i.e., if the solution is an Oseen vortex. This meansthat for positive solutions of the vorticity equationthe only possible asymptotic states are the Oseenvortices.

As we remarked above, the assumption thatsolutions are positive is a priori a very unnaturalone for the vorticity equation, and thus we nowturn to the question of how to treat solutionsthat change sign. To deal with such solutions,we need another Lyapunov functional. The clueto finding this second Lyapunov function is theobservation we made earlier about the similarityof the two-dimensional vorticity equation (4) to anonlinear heat equation. Closer inspection showsthat, just like the heat equation, solutions of (4)satisfy a maximum principle. In particular:

• A solution that is positive (for all z) forsome time t0 will remain positive for anylater time t > t0, and

• If the initial condition for the vorticityequation satisfies ω(z,0) ≥ 0 for all z,then the solution will be strictly positivefor all times t > 0.

Note that these remarks also hold for solutionsof the rescaled vorticity equation (11). As a conse-quence of these two observations, we find a second,surprisingly simple, Lyapunov functional, namelythe L1(R2)-norm of the solution. Define

(19) K[w](τ) =∫R2|w(ζ, τ)|dζ.

To see that this is a Lyapunov functional, split thesolution of (11) into its positive and negative parts,i.e., write w(ξ,η, τ) = w+(ξ, η, τ) − w−(ξ, η, τ),where w± are both nonnegative and satisfy theequations:

(20) ∂τw± = Lνw± − u · ∇w±.Here, u is the total velocity field—i.e., the veloc-ity field associated with w(ζ, τ) rather than thatassociated with either w+ or w−. We note thatone can show (by undoing the change to scalingvariables) that solutions of (20) also satisfy twoproperties listed above. If we choose initial con-ditions for w± so that w+(ζ,0) = sup(w(ζ,0),0)and w−(ζ,0) = − inf(w(ζ,0),0), then w±|τ=0 areboth nonnegative and have disjoint support. By themaximum principle, ifω±|τ=0 6≡ 0, both w± will bestrictly positive for all positive times. From this wefind that

K[w](τ) =∫R2|w+(ζ, τ)−w−(ζ, τ)|dζ

≤∫R2w+(ζ, τ)dζ +

∫R2w−(ζ, τ)dζ

=∫R2w+(ζ,0)dζ +

∫R2w−(ξ, η,0)dζ(21)

=∫R2|w+(ζ,0)dζ −w+(ζ,0)|dζ

= K[w](0) ,where the equality in the middle of (21) followsfrom the fact that solutions of (20) conserve “mass”(i.e., the integral of the solution), as do solutions ofthe vorticity equation. From (21) we see that K is aLyapunov functional for solutions of the vorticityequation. Furthermore, recalling that the maximumprinciple implies that both w± are strictly positive,we see that the first inequality in (21) will be a strictinequality unless either w+ or w− is identicallyzero. Thus the Lyapunov functional K is strictlydecreasing except on the set of functions that iseither strictly positive or strictly negative, and,appealing again to the LaSalle invariance principle,we see that the ω-limit set of solutions must lie inthe set where K is not strictly decreasing—i.e., inthe set of either everywhere positive or everywherenegative solutions.

If we now put together our two Lyapunov func-tionals, we have the following conclusion, namely,

16 Notices of the AMS Volume 58, Number 1

for general solutions, the Lyapunov functional Kimplies that the ω-limit set must lie in the setof solutions of one sign. However, for solutionsof one sign, the relative entropy functional, H,implies that the ω limit set must be an Oseenvortex. So far, we have been somewhat vague aboutthe function space on which we work, but in factthese results hold for any solution whose initialvorticity is absolutely integrable (for this and othertechnical details we refer the reader to [10]) andthus we have

Theorem 1.1. Any solution of the two-dimensionalvorticity equation whose initial vorticity is in L1(R2)and whose total vorticity

∫R2 ω(z,0)dz 6= 0 will tend,

as time tends to infinity, to the Oseen vortex withparameter α =

∫R2 ω(z,0)dz.

Extensions and ConclusionsTheorem 1.1 implies that with even the slightestamount of viscosity present, two-dimensional fluidflows will eventually approach a single large vortex.However, if the viscosity is small, this convergencemay take a very long time. Furthermore, Onsager’soriginal calculations of vortex coalescence were foran inviscid fluid model, which suggests that somesort of coalescence should occur independentof the viscosity—and, in particular, on a timescale that does not depend on the viscosity. Thus,while Theorem 1.1 says that eventually all two-dimensional viscous flows will approach an Oseenvortex, there should be a variety of interesting andimportant behaviors that manifest themselves inthe fluid prior to reaching the asymptotic statedescribed in the theorem.

One of the most important physical effects,and one of the hardest to understand from amathematical point of view, concerns the mergerof two or more vortices. Clearly such mergersmust take place in order for the multitude ofsmall vortices present in an initially turbulentflow to coalesce into the small number of largevortices predicted by Onsager. Furthermore, asdiscussed in [15], this process plays a key rolein many nonturbulent flows such as the wingtipvortices that form behind an airplane wing, asillustrated in Figure 6. As the authors of [15]explain, although the phenomenon is obviouslya three-dimensional one, and three-dimensionaleffects undoubtedly influence the details of theflow, the two-dimensional dynamics “contain all theingredients necessary to explore and understandthe physics involved in vortex merging”. Whilethe Oseen vortex that characterizes the long-time asymptotics of the flow has the propertythat the effects of the nonlinear terms in thevorticity equation vanish, both numerical andexperimental studies show that the merger processis highly nonlinear and involves the filamentationand interpenetration of the two vortices into one

co-rotating vorticesmerging

counter-rotatingvortices

Figure 6. The merger of wingtip vorticesbehind an airplane (from [15]). Although theflow is obviously three-dimensional, much ofthe process of vortex merger can beunderstood by considering cross-sections ofthe flow as if they were two-dimensionalvortices.

Figure 7. An experimental dye visualization ofthe merger of two-dimensional vortices, from[15]

another, as shown in Figure 7. While physicallybased criteria exist to predict when merger willoccur, a rigorous mathematical understandingof this phenomenon is so far almost completelyabsent.

A second interesting phenomenon that is par-ticularly noticeable in the numerical simulationsof two-dimensional flows on bounded domains isthe creation and persistence of metastable struc-tures. For square domains with periodic boundaryconditions, the total vorticity is forced to be zero,and as a consequence the asymptotic state is thezero state. Nonetheless, a number of different,very long-lived, metastable states are observed[22]. The origin and properties of these states inthe two-dimensional Navier-Stokes equation is stillnot understood, but statistical mechanical ideashave again been used to propose an explanationassociated with the different time scales on whichenergy and entropy are dissipated [4]. Similarmetastable phenomena also occur in the weaklyviscous Burgers’ equation, which is often usedas a simplified testing ground for understandingthe Navier-Stokes equations. There, the long-timeasymptotics are again governed by a family ofself-similar states, analogous to the Oseen vorticesin the two-dimensional Navier-Stokes equations.However, for very long times (exponentially long inthe reciprocal of the viscosity!), one observes not

January 2011 Notices of the AMS 17

the self-similar state but rather a special family ofsolutions known as “diffusive N-waves” [11], [3]. Asin the expected behavior of the two-dimensionalNavier-Stokes equation, the metastable states inBurgers’ equation are closely related to theN-wavesof the inviscid equation, while the self-similar as-ymptotic state depends crucially on the presenceof dissipation in the systems. Because of thesimpler nature of Burgers’ equation, one can showthat the metastable states form a one-dimensionalattractive invariant manifold in the phase spaceof the equation, and one can speculate that asimilar dynamical systems explanation might ac-count for the metastable behavior observed in thetwo-dimensional Navier-Stokes equation, as it hasfor the long-time asymptotics of solutions.

In summary, two-dimensional fluid motionspresent interesting differences with three-dimensional fluids from both the mathematicaland physical points of view. In spite of the factthat we live in a three-dimensional world, inmany situations a two-dimensional fluid model isappropriate. One important situation in which thisis the case and for which a two-dimensional fluidmodel is often used is the earth’s atmosphere. Intwo dimensions, it is particularly convenient tostudy the evolution of the vorticity, rather thanwork directly with the velocity field of the fluid.Ever since Helmholtz and Kirchhoff developedan ordinary differential equation model to de-scribe the evolution of point vortices, dynamicalsystems ideas have played an important role inunderstanding the evolution of the vorticity intwo-dimensional flows, a theme that continues topay dividends to the present day.

A distinctive feature of two-dimensional flows isthe “inverse cascade” of energy from small scales tolarge ones. Lars Onsager first sought to explain thisphenomenon by studying the statistical mechanicsof large collections of inviscid point vortices. WhileOnsager’s observation about inviscid flows remainsunexplained, dynamical systems ideas—in this caseLyapunov functionals inspired by kinetic theory—have been used to show that in the presence of anarbitrarily small amount of viscosity, essentiallyany two-dimensional flow whose initial vorticityfield is absolutely integrable will evolve as time goesto infinity toward a single, explicitly computablevortex.

AcknowledgmentsThe research of the author is supported in partby the NSF through grant DMS-0908093. It is apleasure to acknowledge many discussions of thesubject matter of this article with my collaborators,Margaret Beck, Thierry Gallay, Ray Nagem, GuidoSandri, and David Uminsky. I also wish to thankPaul Newton and two anonymous referees fornumerous suggestions that improved this article.

Note: Graphic images in this article are courtesyof: (Figure 1) NASA Earth Observatory; (Figure 2)Prof. X. L. Wu, Prof. W. I. Goldberg, Universityof Pittsburgh Physics Department, and MaartenRutgers, Ph.D.; (Figure 3) H. J. H. Clercx and G. J. F.van Heijst, TU Eindhoven, NL, Eindhoven Universityof Technology (copyright F. van Heijst/HJH Clercx);(Figure 4) author; (Figure 5) Jeff D. Eldredge andJournal of Computational Physics, Elsevier; (Figures6 and 7) Patrice Meunier, Stéphane Le Dizès,Thomas Leweke and Comptes Rondus Physique,Elsevier Masson.

References[1] H. Aref, P.K. Newton, M.A. Stremler, T. Tokieda,

and D.L. Vainchtein, Vortex crystals, Advances inApplied Mechanics, pages 1–79, 2002.

[2] Lorena A. Barba, Vortex Method for ComputingHigh-Reynolds Number Flows: Increased Accuracywith a Fully Mesh-less Formulation, Ph.D. thesis,California Institute of Technology, 2004.

[3] Margaret Beck and C. Eugene Wayne, Using globalinvariant manifolds to understand metastability inthe Burgers equation with small viscosity, SIAM Jour-nal on Applied Dynamical Systems 8, no. 3 (2009),1043–1065.

[4] E. Caglioti, M. Pulvirenti, and F. Rousset, The 2Dconstrained Navier-Stokes equation and intermedi-ate asymptotics, J. Phys. A 41, no. 34:344001 (2008),9 pp.

[5] Charles R. Doering and J. D. Gibbon, Applied Anal-ysis of the Navier-Stokes Equations, Cambridge Textsin Applied Mathematics, Cambridge University Press,Cambridge, 1995.

[6] J. Eldredge, Numerical simulation of the fluid dy-namics of 2d rigid body motion with the vortexparticle method, Journal of Computational Physics221, no. 2 (2007), 626–648.

[7] Gregory L. Eyink and Katepalli R. Sreenivasan,Onsager and the theory of hydrodynamic turbulence,Rev. Modern Phys. 78, no. 1 (2006), 87–135.

[8] Isabelle Gallagher and Thierry Gallay, Unique-ness for the two-dimensional Navier-Stokes equationwith a measure as initial vorticity, Math. Ann. 332,no. 2 (2005), 287–327.

[9] Thierry Gallay and C. Eugene Wayne, Invariantmanifolds and the long-time asymptotics of theNavier-Stokes and vorticity equations on R2, Arch.Ration. Mech. Anal. 163, no. 3 (2002), 209–258.

[10] , Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math.Phys. 255, no. 1 (2005), 97–129.

[11] Y.-J. Kim and A. E. Tzavaras, Diffusive N-waves andmetastability in the Burgers’ equation, SIAM J. Math.Anal., 33, no. 3 (2001), 607–633 (electronic).

[12] Andrew J. Majda and Andrea L. Bertozzi, Vorticityand Incompressible Flow, volume 27 of CambridgeTexts in Applied Mathematics, Cambridge UniversityPress, Cambridge, 2002.

[13] Carlo Marchioro, On the inviscid limit for a fluidwith a concentrated vorticity, Comm. Math. Phys.196, no. 1 (1998), 53–65.

[14] P. A. Markowich and C. Villani, On the trend toequilibrium for the Fokker-Planck equation: An inter-

18 Notices of the AMS Volume 58, Number 1

play between physics and functional analysis, Mat.Contemp. 19 (2000), 1–29. VI Workshop on PartialDifferential Equations, Part II (Rio de Janeiro, 1999).

[15] Patrice Meunier, Stéphane Le Dizès, and ThomasLeweke, Physics of vortex merging, Comptes RendusPhysique 6, no. 4–5 (2005), 431–450.

[16] Raymond Nagem, Guido Sandri, David Uminsky,and C. Eugene Wayne, Generalized Helmholtz-Kirchhoff model for two-dimensional distributedvortex motion, SIAM J. Appl. Dyn. Syst. 8, no. 1 (2009),160–179.

[17] NASA, Image of the Day, August 8, 2004,http://http://earthobservatory.nasa.gov/IOTD/view.php?id=4718.

[18] Paul K. Newton, The N-vortex Problem, volume 145of Applied Mathematical Sciences, Springer-Verlag,New York, 2001.

[19] Lars Onsager, Statistical hydrodynamics, Il NuovoCimento 6 (1949), 279–287.

[20] Maarten Rutgers, http://http://www.maartenrutgers.org/science/turbulence/gallery.html.

[21] Eindhoven Two-dimensional TurbulenceGroup at the Technishe Universiteit,http://http://web.phys.tue.nl/nl/de_faculteit/capaciteitsgroepen/transportfysica/fluid_dynamics_lab/turbulence_vortex_dynamics/2d_turbulence/.

[22] Z. Yin, D. C. Montgomery, and H. J. H. Clercx,Alternative statistical-mechanical descriptions ofdecaying two-dimensional turbulence in terms of“patches” and “points”, Physics of Fluids 15, no. 7(2003), 1937–1953.

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