C. R. Acad. Sci. Paris, t. 329, Série II b, p. 41–46, 2001Mécanique des fluides/Fluid Mechanics

Formation of ‘clusters’ of vortices on a sphereVadim PAVLOV a, Daniel BUISINE a, Viktor GONCHAROV b

a DMF, UFR de Mathématiques Pures et Appliquées, université de Lille-1, 59655 Villeneuve d’Ascq, Franceb Institut de la Physique Atmosphèrique, Academie des Sciences de la Russie, 109017 Moscou, Russie

(Reçu le 9 mai 2000, accepté après révision le 23 octobre 2000)

Abstract. This paper applies the Hamiltonian approach to the two-dimensional motion of anincompressible fluid on a curvilinear surface. The method has been used to formulategoverning equations of motion, and to interpret the evolution of a system consisting ofN ∼ 102–103 localized two-dimensional vortices on a sphere. The analysis shows that theinstability appears immediately, forming initial disorganized structures which develop andfinally ‘burst’. The system evolves to a few separated vortex ‘spots’ which are quasi-stable. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

localized vortices / interactions / fragmentation / clusters

Formation de « clusters » de tourbillons sur une sphère

Résumé. L’approche hamiltonienne est utilisée pour formuler les équations gouvernant le mouve-ment de l’ensemble desN ∼ 102–103, tourbillons localisés décrivant l’écoulement d’unfluide incompressible sur une surface sphèrique. Cette approche est mise en oeuvre dansl’étude de la stabilité et de l’évolution temporelle de deux configurations initiales simples :une suite de plots répartis sur l’équateur d’une part et un jet équatorial d’autre part. Lesrésultats numériques montrent le développement des instabilités et les appariements quiconduisent á la formation des grandes structures tourbillonnaires quasi-stables. 2001Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

tourbillons localisés / interactions / fragmentation / clusters

Version française abrégée

De nombreuses études ont été consacrées à la compréhension du comportement des tourbillonsocéaniques et atmosphériques observés dans la nature. Le mouvement de ces tourbillons et leurs interactionsdépendent fortement des contraintes physiques imposées, telles que par exemple la courbure du globeterrestre, sa rotation, la stratification et autres particularités de l’atmosphère et de l’océan. Une étudethéorique incluant simultanément toutes ces facteurs est un problème très complexe. C’est pourquoi le rôlede ces facteurs a été le plus souvent étudié soit séparement, soit dans le cadre d’hypothèses simplificatrices.Dans les conditions géophysiques, l’intensité des structures tourbillonnaires océaniques et atmosphériquesdépasse très largement la vorticité moyenne ambiante. Ces structures peuvent être étudiées dans le cadre duconcept des tourbillons localisés, supposés non visqueux (en effet, le nombre de Reynolds caractéristiqueRe∼ 108 1, les dimensions des tourbillonsD R, oùR est le rayon de la Terre). Ceci nous conduità décrire les déplacements et l’interaction de tourbillons localisés à partir du modèle mathématique desingularités tourbillonnaires, considérées comme solutions bidimensionnelles des équations du mouvement

Note présentée par René MOREAU.

S1620-7742(00)01287-3/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 41

V. Pavlov et al.

non visqueux. Une telle approche est largement utilisée en hydrodynamique geo- et astrophysique (voir[2,3] et la bibliographie présentée).

Nous présentons succintement l’approche hamiltonienne [4] de l’étude de l’écoulement bidimensionneld’un fluide incompressible sur une surface. Nous nous intéressons à la dynamique des tourbillons surune surface sphèrique, ainsi qu’à la mise en oeuvre de l’AH dans l’étude de la stabilité et de l’évolutiontemporelle de deux configurations initiales simples : une suite de plots répartis sur l’équateur d’une part etun jet équatorial d’autre part.

La dynamique des écoulements incompressibles qui est régie par l’équation (1), peut être reformuléesous la forme (2) :∂tΩα = Ωα,H =

∫dx′ Ωα,Ω′β(δH/δΩ′β), où les crochets de Poisson sont dé-

taillés en [4]. En supposant que l’écoulement est bidimensionnel, introduisons la composante contrava-riante de la vorticitéΩ3 ≡Ω, normale à la surface. Posons que le champ de vorticitéΩ est décrit complète-ment à partir deN singularités d’intensitéγi placées aux points~ζi(t) de coordonnées(ζ1

i (t), ζ2i (t)) soit :

Ω =∑i γiδ(

~ζ−~ζi(t)). La composante contravariante de la vorticitéΩ est donc une fonctionnelle des coor-donnéesζαi (t). L’hamiltonien du système (l’énergie cinétique) s’écrit :H =− 1

2

∫∫d~ζ d~ζ′Ω Ω′G(~ζ, ~ζ′) =

− 12

∑Ni6=j γiγjG(~ζi, ~ζj). La fonction de GreenG(~ζ, ~ζ′) est solution de l’équation∆G(~ζ, ~ζ′) = δ(~ζ, ~ζ′)−

V −1, oùV est l’aire de la surface. En calculant les crochets de Poisson pourζαi (t) (3), on peut établir les

équations∂tζαi = ζαi ,H =∫

dx′ ζαi , ζ′βj (δH/δζ′

βj ) = εαβg

−1/2i γ−1

i (δH/δζβj ) (4). Sur la sphère derayon unité, prenons pour coordonnées généraliséesζαi (t) la longitudeζ1

i = θi et la latitudeζ2i = φi. On

peut montrer que, pour une sphère, la fonction de GreenG s’écrit :G(cosβij) = (4π)−1 ln(1 − cosβij)(5), oùβij est l’angle séparant deux singularités. Les équations (4) et (5) permettent d’obtenir le système(6) qui a été résolu numériquement par un schéma de Runge–Kutta du quatrième ordre. Un des résultats decalcul est présenté sur lafigure 1.

Le résultat important de ce travail est de montrer que l’approche hamiltonienne mise en oeuvre pour unensemble de tourbillons en mouvement sur une sphère, débouche sur un système dynamique suffisammentriche pour rendre compte des instabilités, des interactions et des appariements qui conduisent à la formationdes clusters.

1. Introduction. Basic equations

A number of theoretical and experimental studies have been devoted to the understanding of the dynamicsof atmospheric and oceanic vortices which are frequently observed in nature. The interest in the problem isalso driven by the anxiety of civilization’s impact on the environment (see, for example, [1], for a reviewon the ozone hole, or work on the stratospheric polar vortex). Compared to traditional fluid dynamics,atmospheric and oceanic vortex dynamics includes a number of physical restrictions (spherical surface,stratification, etc.) which strongly affect the motion and interaction of vortex fields. A theoretical studycombining all these factors is a very complicated problem, and therefore they have been frequentlyinvestigated separately, or with simplified assumptions. The vorticity in real atmospheric and oceanic eddies(rings or typhoons) often largely exceeds the background vorticity: these structures have a relatively largelifetime too. On the other hand, in the approximation ofRe−1 ∼ 10−8 1 when a fluid is considered asinviscid, the governing hydrodynamical equations admit the singular vortices as its solution. Thus, singularvortices as a mathematical model can be used as basic elements in the following approach. In spite ofsome difficulties in the interpretation of physical results, the concept of localized (singular) vortices islargely used in problems of geo- and astrophysical hydrodynamics (see, for example, [2,3] and referencestherein). In many cases the study of the dynamics of these vortices and their interaction is simpler than inanalogous problems for continuous vortex distribution – an arbitrary initial hydrodynamical field can berepresented in the form of superposition of fields generated by vortices. Thus, the resulting vortex field can

42

Formation of ‘clusters’ of vortices on a sphere

be presented as a result of the interaction of the localized vortices, and the averaged vorticity is defined viatheir superposition.

The dynamics of incompressible flows is governed by equations:

∂tvα + vβ∂βvα = ∂α(ρ−1p+ χ), ∂βvβ = 0 (1)

wherevα are velocity components (α,β = 1,2,3) in the Cartesian system of coordinates,∂t is the partialderivative of a field variable with respect to time,p is pressure,ρ is density (furtherρ= 1).

The system (1) may be (see [4] and references therein) presented on the phase space of the vorticity fieldΩα = εαµβ∂µvβ (εαµβ is the Levi-Civita tensor) in Hamiltonian form as:

∂tΩα = Ωα,H=

∫dx′

Ωα,Ω

′β

(δH/δΩ′β

)(2)

where the Hamiltonian,H, of the system is the quantity functionally dependent on the fields,Ωα. TheHamiltonian structure of hydrodynamical models consists of the Hamiltonian,H, given by the total energyexpressed in terms of field variables,Ωα, and of the functional Poisson bracket,. Conservation of energyfollows from the given formulation, since∂tH = H,H= 0. Here and throughout this work, the primedenotes that the field variables depend on the space coordinatex′, dx= dx1 dx2 dx3. The skew-symmetricfunctional Poisson bracket in equation (2) is defined for the given model by the expressionΩα,Ω′β =

εασγεγλνεβνµ∂σΩl∂µδ(x−x′). For contravariant components,Ωα of the vorticity when using coordinatesx = (x1, x2, x3) is not Cartesian, we obtainΩα,Ω′β = g−1/2εασγεγλνε

βνµ∂σΩλ∂µg−1/2δ(x − x′).

Heregαβ is the metric tensor andg is its determinant.For incompressible flows when the fluid particle moves along non intersecting stationary fluid surfaces,

we introduce the orthogonal curvilinear coordinatesx1, x2, x3: coordinate linesx3 coincide with vortexones while coordinate linesx1 and x2 lie on the fluid surfaces. In this case,v = v1, v2,0 and thevorticity Ω = 0,0,Ω. Here Ω = g−1/2(∂1v2 − ∂2v1). The Poisson bracket reduces [4] toΩ,Ω′ =εαβg−1/2(∂Ω/∂xα)(∂/∂xβ)δ(x−x′)g−1/2 (α,β = 1,2). The contravariant vorticityΩ obeys theequation∂tΩ + vα∂αΩ = 0 (α= 1,2), and differs from the usual ‘physical’ vorticityΩg1/2(g11g22)−1/2.Both definitions coincide only in cases wheng = g11g22.

Let us assume thatx1 coincides with the streamlines of the unperturbed stationary problem. For thiswide class of layer models geometric properties of the space associated with such coordinate systems arecharacterized only by componentsg11, g22, g33 of the metric tensor and by its determinantg which aredeemed independent ofx1, just as is the velocity profile of the unperturbed flow. The kinetic energy of thefluid may be written in the formH=− 1

2

∫∫d~ζ d~ζ′ΩΩ′G(~ζ, ~ζ′). Green’s functionG(~ζ, ~ζ′) is a solution of

the equation∆G(~ζ, ~ζ′) = δ(2)(~ζ, ~ζ′)−V −1, whereV is the ‘volume’ of a domain where the delta-functionis defined (for a spherical surface, for example, the ‘volume’ isV = 4πR2 – the surface has a ‘volume’, buthas no boundaries). IfV →∞, one has the traditional equation.

Let us consider an evolution of a system consisting ofN singular vortices:

Ω =∑i

γiδ(2)(~ζ − ~ζi(t)

)= g−1/2

∑i

γiδ(1)(ζ1 − ζ1

i (t))δ(1)(ζ2 − ζ2

i (t))

The full vorticity is given by∫

d~ζ′Ω(~ζ′) =∑i γi. Here,γi is independent of time intensities of the vortices,

~ζi = (ζ1i , ζ

2i ) are their coordinates dependent on time,δ(2)(~ζ − ~ζi) is the 2D-function of Dirac,~ζi = ~ζi(t).

Calculating the Poisson bracketsζαi , ζβj , we findζαi , ζ

βj = γ−1

i g−1/2i δijε

αβ . Heregi is calculated atthe point where the vortex is localized. Thus, in terms of variablesζαi the dynamics of a system of singular

43

V. Pavlov et al.

vortices is described by the functional equations:

∂tζαi =

ζαi ,H

=

∫dx′ζαi , ζ

β′j

(δH/δζβ′j

)= εαβγ−1

i g−1/2(∂H/∂ζβi

)(3)

The expression forH = − 12

∑Ni,j γiγjG(~ζi, ~ζj) has a shortcoming by having an uncertainty which arises

from the energy of the interaction becoming infinite when wheni = j. But the diagonal terms in theHamiltonian can be excluded, because they are independent of the space coordinates as a result of theindependence ofGii. Spherical coordinates are an example of curvilinear coordinates which satisfy thisrequirement.

On a sphere the location of vortices is given by longitudeθ and latitudeφ. Coordinatesζαi are(θi, φi),whereα = 1,2, tensorεαβ has componentsε12 = −ε21 = 1, ε11 = −ε22 = 0. We have, thus,ζ1 = θ,ζ2 = ϕ, ζ3 = r, g11 = r2, g22 = r2 sin2 θ, g33 = 1, andg = r4 sin2 θ > 0. For a spherical surface thebasic system of dimensionless equations becomes∂tθi = (γi sin θi)

−1∂φiH, ∂tφi = −(γi sin θi)−1∂θiH.

Here,∂t is the dimensionless derivative-operator with respect to time, applied to a spherical coordinateof a vortex, time is measured in units ofτ = r2/γ0, γ0 = max(|γi|), γi = γi/γ0, |γi| 6 1, i = 1, . . . ,N,∂φi = ∂/∂φi, ∂θi = ∂/∂θi, 0 6 θ 6 π, 0 6 φ − 2πk 6 2π, k = 0,±1,±2, . . . . If the regular current isabsent, the Hamiltonian is given by the expression where the diagonal terms are absent in the sum

∑′j,k.

For the spherical surface, Green’s function is given by:

G(cosβij) =−(4π)−1∞∑l=1

[2l+ 1][l(l+ 1)

]−1Pl(cosβij)≡ (4π)−1 ln(1− cosβij)

Here,Pl(cosβ) are Legendre polynomials. Finally, we obtain from (3) the following equations:

∂tθi =−1

2(γi sinθi)

−1∑j 6=i

γj∂φiG, ∂tφi =1

2(γi sinθi)

−1∑j 6=i

γj∂θiG (4)

2. Results of calculations and discussion

Equation (4) has been treated by means of a 4-order Runge–Kutta scheme. The time step is restrictedby the condition∆t < η∆ψ/ sup(|∂tθi|, |∂tφi|) where∆ψ is the characteristic angle between two pointvortices. To avoid the polar singularity due to the metrics, a polar cap of0,01 radian is excluded from thedomain. The energy conservation has been evaluated during the calculations. The HamiltonianH has beentried with small variations:−0.01924016 H 6 −0.0192344, during the process of iterating fromt = 0to t= 400, with a root-mean-square error< 3× 10−4. We initialize the system ofm equidistant clustersof vortices centered at the latitudeθc = π/2, and composed ofn2 point vortices. The parameters of theproblem are: (a) the numberm of clusters, (b) the fractional areaµ occupied by the clusters supposed tobe sufficiently small – this parameter is defined as the total area enclosed by the vortex patches dividedby 4π: µ 'mδϕδθ sin θc/4π. The trajectories and the distribution density of the clouds of point vorticesare given here in the framework of the so-called ‘sinus’ – representation where the coordinates are definedby x = ϕ sin θ andy = θ. The evolution of 5 equatorial patches each containing 64 point vortices whenµ = 0.157 is analyzed initially. The patches, which are initially at sufficiently short distances, exchangeimmediately with each other numerous vortices and merge rapidly forming a perturbed strip which can bequalified as turbulent. Originating from this unstable strip, 4 structures are formed without ‘pushing out’ ofvortices att= 130. Later, appearance of 2 or 4 structures is accompanied by the pushing out of vortices.Finally 3 structures are formed with the vortex density comparable to one of initial clusters. We also analyzefragmentation of an equatorial, initially homogeneous, jet (figure 1). The jet consists ofN = 100 point

44

Formation of ‘clusters’ of vortices on a sphere

Figure 1. Destruction of the strip and formation of 3 or 4 clusters att < 305.

Figure 1. Fragmentation d’un jet équatorial et formation de 3 ou 4 clusters pourt < 305.

vortices. The distribution corresponds to an equatorial jet with the following parameters: maximum of thevelocity on the axis and transversal variations of the velocity∆Vφ,i = ±0.1. The sliding near the edgesof the jet is the cause of the appearance of instability which is displayed for the first time att = 158 andt= 193. Figure 1 illustrates fluctuations on edges of the strip in a form of small quasi-symmetrical rings.These structures develop betweent = 226 and t = 246. Very rapidly appearing oscillations lead to thedestruction of the strip and to the formation of 3 or 4 clusters att= 305. These structures develop into twolarge clusters of positive vorticity and three weak zones of negative vorticity (t > 374). The process areaccompanied by isolated vortices ejected during the process of transition.

The work was carried out for several reasons: (i) the questions regarding what the Hamiltonian looks likeand what the structure of the canonical equations is in concrete situations, are not as trivial as they mayappear at first glance, (ii) the development of the Hamiltonian approach [4] would remain incomplete if

45

V. Pavlov et al.

no practical application of the theoretical analysis were given. As an application of the method, we haveexamined two of the simplest configurations of flows on the surface of a sphere: a system ofN point vorticesinitially regularly distributed, and an equatorial jet. We have found that the vortex dynamics contains achange of vortices (vortex pairs) among vortex patches, the appearance of fragmentation of the structures.We have analyzed also the stability and the fragmentation of the initially homogeneous equatorial jet. Themost important result of the direct simulation (forN ∼ 102) is that the system forms cluster structures (seealso predictions from the statistical mechanics of point vortices [5–7] whenN →∞).

This work was partially supported by the Russian Fundamental Research Foundation under grant No.00-05-64019.

References

[1] McIntyre, M.I., Atmospheric dynamics: some fundamentals with observational implications, in: J.C. Gille,G. Visconti (Eds.), Proc. Int. School of Physics “Enrico Fermi”, 1991.

[2] Aref, H., Integrable, chaotic, and turbulent vortex motion in two-dimensional flows, Ann. Rev. Fluid Mech. 15(1983) 345–389.

[3] Polvani, L.M., Dritschel, D.G., Wave and vortex dynamics on the surface of a sphere, J. Fluid Mech. 255 (1993)35–64.

[4] Goncharov, V.P., Pavlov, V.I., Some remarks on the physical foundation of the Hamiltonian description of fluidmotions, European J. Mech. B/Fluids 16 (4) (1997) 509–555.

[5] Onsager, L., Statistical hydrodynamics, Nuovo Cimento 6 (supp) (1949) 279–287.[6] Brands, H., Chavanis, P.H., Pasmanter, R., Sommeria, J., Maximum entropy versus minimum enstrophy, Phys.

Fluids 11 (11) (1999) 3465–3477.[7] Berdichevsky, V. L., Statistical mechanics of point vortices, Phys. Rev. E 51 (5) (1995) 4432–4452.

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