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Formation of `clusters' of vortices on a sphere
C. R. Acad. Sci. Paris, t. 329, Série II b, p. 41–46, 2001 Mécanique des fluides/Fluid Mechanics
Formation of ‘clusters’ of vortices on a sphere Vadim PAVLOV a , Daniel BUISINE a , Viktor GONCHAROV b a b
DMF, UFR de Mathématiques Pures et Appliquées, université de Lille-1, 59655 Villeneuve d’Ascq, France 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 an incompressible fluid on a curvilinear surface. The method has been used to formulate governing equations of motion, and to interpret the evolution of a system consisting of N ∼ 102 –103 localized two-dimensional vortices on a sphere. The analysis shows that the instability appears immediately, forming initial disorganized structures which develop and finally ‘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 mouvement de l’ensemble des N ∼ 102 –103 , tourbillons localisés décrivant l’écoulement d’un fluide incompressible sur une surface sphèrique. Cette approche est mise en oeuvre dans l’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. Les résultats numériques montrent le développement des instabilités et les appariements qui conduisent á la formation des grandes structures tourbillonnaires quasi-stables. 2001 Académie 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 tourbillons océaniques et atmosphériques observés dans la nature. Le mouvement de ces tourbillons et leurs interactions dépendent fortement des contraintes physiques imposées, telles que par exemple la courbure du globe terrestre, sa rotation, la stratification et autres particularités de l’atmosphère et de l’océan. Une étude théorique incluant simultanément toutes ces facteurs est un problème très complexe. C’est pourquoi le rôle de 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 atmosphériques dépasse très largement la vorticité moyenne ambiante. Ces structures peuvent être étudiées dans le cadre du concept des tourbillons localisés, supposés non visqueux (en effet, le nombre de Reynolds caractéristique Re ∼ 108 1, les dimensions des tourbillons D R, où R est le rayon de la Terre). Ceci nous conduit à décrire les déplacements et l’interaction de tourbillons localisés à partir du modèle mathématique de singularités tourbillonnaires, considérées comme solutions bidimensionnelles des équations du mouvement Note présentée par René M OREAU. S1620-7742(00)01287-3/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
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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 bidimensionnel d’un fluide incompressible sur une surface. Nous nous intéressons à la dynamique des tourbillons sur une surface sphèrique, ainsi qu’à la mise en oeuvre de l’AH dans l’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. La dynamique des écoulements incompressibles qui est régie par l’équation (1), peut être reformulée R β β sous la forme (2) : ∂t Ωα = {Ωα , H} = dx0 {Ωα , Ω0 }(δH/δΩ0 ), où les crochets de Poisson sont détaillés en [4]. En supposant que l’écoulement est bidimensionnel, introduisons la composante contravariante de la vorticité Ω3 ≡ Ω, normale à la surface. Posons que le champ de vorticité Ω est décrit complètement à partir de N singularités d’intensité γi placées aux points ζ~i (t) de coordonnées (ζi1 (t), ζi2 (t)) soit : P Ω = i γi δ(ζ~ − ζ~i (t)). La composante contravariante de la vorticité Ω est donc une fonctionnelle des coorRR ~ ζ~0 ) = dζ~ dζ~0 Ω Ω0 G(ζ, données ζiα (t). L’hamiltonien du système (l’énergie cinétique) s’écrit : H = − 12 P N ~ ζ~0 ) est solution de l’équation ∆ G(ζ, ~ ζ~0 ) = δ(ζ, ~ ζ~0 ) − − 21 i6=j γi γj G(ζ~i , ζ~j ). La fonction de Green G(ζ, −1 α V , où V est l’aire de la surface. En calculant les crochets de Poisson pour ζi (t) (3), on peut établir les R −1/2 −1 β β γi (δH/δζjβ ) (4). Sur la sphère de équations ∂t ζiα = {ζiα , H} = dx0 {ζiα , ζ 0 j }(δH/δζ 0 j ) = εαβ gi α rayon unité, prenons pour coordonnées généralisées ζi (t) la longitude ζi1 = θi et la latitude ζi2 = φi . On peut montrer que, pour une sphère, la fonction de Green G s’écrit : G(cos βij ) = (4π)−1 ln(1 − cos βij ) (5), où βij 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 de calcul est présenté sur la figure 1. Le résultat important de ce travail est de montrer que l’approche hamiltonienne mise en oeuvre pour un ensemble de tourbillons en mouvement sur une sphère, débouche sur un système dynamique suffisamment riche pour rendre compte des instabilités, des interactions et des appariements qui conduisent à la formation des clusters.
1. Introduction. Basic equations A number of theoretical and experimental studies have been devoted to the understanding of the dynamics of atmospheric and oceanic vortices which are frequently observed in nature. The interest in the problem is also driven by the anxiety of civilization’s impact on the environment (see, for example, [1], for a review on 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 study combining all these factors is a very complicated problem, and therefore they have been frequently investigated 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 large lifetime too. On the other hand, in the approximation of Re −1 ∼ 10−8 1 when a fluid is considered as inviscid, the governing hydrodynamical equations admit the singular vortices as its solution. Thus, singular vortices as a mathematical model can be used as basic elements in the following approach. In spite of some difficulties in the interpretation of physical results, the concept of localized (singular) vortices is largely used in problems of geo- and astrophysical hydrodynamics (see, for example, [2,3] and references therein). In many cases the study of the dynamics of these vortices and their interaction is simpler than in analogous problems for continuous vortex distribution – an arbitrary initial hydrodynamical field can be represented 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 via their superposition. The dynamics of incompressible flows is governed by equations: ∂t v α + vβ ∂β vα = ∂α (ρ−1 p + χ),
∂β vβ = 0
(1)
where vα are velocity components (α, β = 1, 2, 3) in the Cartesian system of coordinates, ∂t is the partial derivative of a field variable with respect to time, p is pressure, ρ is density (further ρ = 1). The system (1) may be (see [4] and references therein) presented on the phase space of the vorticity field Ωα = εαµβ ∂µ vβ (εαµβ is the Levi-Civita tensor) in Hamiltonian form as: Z ∂t Ωα = {Ωα , H} =
dx0 Ωα , Ω0β δH/δΩ0β
(2)
where the Hamiltonian, H, of the system is the quantity functionally dependent on the fields, Ωα . The Hamiltonian structure of hydrodynamical models consists of the Hamiltonian, H, given by the total energy expressed in terms of field variables, Ωα , and of the functional Poisson bracket {, }. Conservation of energy follows from the given formulation, since ∂t H = {H, H} = 0. Here and throughout this work, the prime denotes that the field variables depend on the space coordinate x0 , dx = dx1 dx2 dx3 . The skew-symmetric functional Poisson bracket in equation (2) is defined for the given model by the expression {Ωα , Ω0β } = εασγ εγλν εβνµ ∂σ Ωl ∂µ δ(x − x0 ). For contravariant components, Ωα of the vorticity when using coordinates 0 x = (x1 , x2 , x3 ) is not Cartesian, we obtain {Ωα , Ω β } = g −1/2 εασγ εγλν εβνµ ∂σ Ωλ ∂µ g −1/2 δ(x − x0 ). αβ Here g is the metric tensor and g is its determinant. For incompressible flows when the fluid particle moves along non intersecting stationary fluid surfaces, we introduce the orthogonal curvilinear coordinates x1 , x2 , x3 : coordinate lines x3 coincide with vortex ones while coordinate lines x1 and x2 lie on the fluid surfaces. In this case, v = {v1 , v2 , 0} and the vorticity Ω = {0, 0, Ω}. Here Ω = g −1/2 (∂1 v2 − ∂2 v1 ). The Poisson bracket reduces [4] to {Ω, Ω0 } = εαβ g −1/2 (∂Ω/∂xα )(∂/∂xβ )δ(x − x0 )g −1/2 (α, β = 1, 2). The contravariant vorticity Ω obeys the equation ∂t Ω + v α ∂α Ω = 0 (α = 1, 2), and differs from the usual ‘physical’ vorticity Ωg 1/2 (g11 g22 )−1/2 . Both definitions coincide only in cases when g = g11 g22 . Let us assume that x1 coincides with the streamlines of the unperturbed stationary problem. For this wide class of layer models geometric properties of the space associated with such coordinate systems are characterized only by components g11 , g22 , g33 of the metric tensor and by its determinant g which are profile of the unperturbed flow. The kinetic energy of the deemed independent of x1 , just as is the velocity RR ~ ζ~0 ). Green’s function G(ζ, ~ ζ~0 ) is a solution of dζ~ dζ~0 ΩΩ0 G(ζ, fluid may be written in the form H = − 21 0 (2) 0 −1 ~ ζ~ ) − V , where V is the ‘volume’ of a domain where the delta-function ~ ζ~ ) = δ (ζ, the equation ∆G(ζ, is defined (for a spherical surface, for example, the ‘volume’ is V = 4πR2 – the surface has a ‘volume’, but has no boundaries). If V → ∞, one has the traditional equation. Let us consider an evolution of a system consisting of N singular vortices: Ω=
X i
X γi δ (2) ζ~ − ζ~i (t) = g −1/2 γi δ (1) ζ 1 − ζi1 (t) δ (1) ζ 2 − ζi2 (t) i
R P The full vorticity is given by dζ~0 Ω(ζ~0 ) = i γi . Here, γi is independent of time intensities of the vortices, ζ~i = (ζi1 , ζi2 ) are their coordinates dependent on time, δ (2) (ζ~ − ζ~i ) is the 2D-function of Dirac, ζ~i = ζ~i (t). −1/2 δij εαβ . Here gi is calculated at Calculating the Poisson brackets {ζiα , ζjβ }, we find {ζiα , ζjβ } = γi−1 gi α the point where the vortex is localized. Thus, in terms of variables ζi the dynamics of a system of singular
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V. Pavlov et al.
vortices is described by the functional equations: Z ∂t ζiα = ζiα , H = dx0 ζiα , ζjβ0 δH/δζjβ0 = εαβ γi−1 g −1/2 ∂H/∂ζiβ
(3)
PN The expression for H = − 21 i,j γi γj G(ζ~i , ζ~j ) has a shortcoming by having an uncertainty which arises from the energy of the interaction becoming infinite when when i = j. But the diagonal terms in the Hamiltonian can be excluded, because they are independent of the space coordinates as a result of the independence of Gii . Spherical coordinates are an example of curvilinear coordinates which satisfy this requirement. On a sphere the location of vortices is given by longitude θ and latitude φ. Coordinates ζiα are (θi , φi ), where α = 1, 2, tensor εαβ has components ε12 = −ε21 = 1, ε11 = −ε22 = 0. We have, thus, ζ 1 = θ, ζ 2 = ϕ, ζ 3 = r, g11 = r2 , g22 = r2 sin2 θ, g33 = 1, and g = r4 sin2 θ > 0. For a spherical surface the basic system of dimensionless equations becomes ∂t θi = (γi sin θi )−1 ∂φi H, ∂t φi = −(γi sin θi )−1 ∂θi H. Here, ∂t is the dimensionless derivative-operator with respect to time, applied to a spherical coordinate of a vortex, time is measured in units of τ = 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 P0 is absent, 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(cos βij ) = −(4π)−1
∞ X
−1 [2l + 1] l(l + 1) Pl (cos βij ) ≡ (4π)−1 ln(1 − cos βij )
l=1
Here, Pl (cos β) are Legendre polynomials. Finally, we obtain from (3) the following equations: X 1 γj ∂φi G, ∂t θi = − (γi sin θi )−1 2 j6=i
X 1 ∂t φi = (γi sin θi )−1 γj ∂θi G 2
(4)
j6=i
2. Results of calculations and discussion Equation (4) has been treated by means of a 4-order Runge–Kutta scheme. The time step is restricted by the condition ∆t < η ∆ψ/ sup(|∂t θi |, |∂t φi |) where ∆ψ is the characteristic angle between two point vortices. To avoid the polar singularity due to the metrics, a polar cap of 0, 01 radian is excluded from the domain. The energy conservation has been evaluated during the calculations. The Hamiltonian H has been tried with small variations: −0.0192401 6 H 6 −0.0192344, during the process of iterating from t = 0 to t = 400, with a root-mean-square error < 3 × 10−4 . We initialize the system of m equidistant clusters of vortices centered at the latitude θc = π/2, and composed of n2 point vortices. The parameters of the problem are: (a) the number m of clusters, (b) the fractional area µ occupied by the clusters supposed to be sufficiently small – this parameter is defined as the total area enclosed by the vortex patches divided by 4π: µ ' mδϕδθ sin θc /4π. The trajectories and the distribution density of the clouds of point vortices are given here in the framework of the so-called ‘sinus’ – representation where the coordinates are defined by x = ϕ sin θ and y = θ. The evolution of 5 equatorial patches each containing 64 point vortices when µ = 0.157 is analyzed initially. The patches, which are initially at sufficiently short distances, exchange immediately with each other numerous vortices and merge rapidly forming a perturbed strip which can be qualified as turbulent. Originating from this unstable strip, 4 structures are formed without ‘pushing out’ of vortices at t = 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 analyze fragmentation of an equatorial, initially homogeneous, jet (figure 1). The jet consists of N = 100 point
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Formation of ‘clusters’ of vortices on a sphere
Figure 1. Destruction of the strip and formation of 3 or 4 clusters at t < 305. Figure 1. Fragmentation d’un jet équatorial et formation de 3 ou 4 clusters pour t < 305.
vortices. The distribution corresponds to an equatorial jet with the following parameters: maximum of the velocity on the axis and transversal variations of the velocity ∆Vφ,i = ±0.1. The sliding near the edges of the jet is the cause of the appearance of instability which is displayed for the first time at t = 158 and t = 193. Figure 1 illustrates fluctuations on edges of the strip in a form of small quasi-symmetrical rings. These structures develop between t = 226 and t = 246. Very rapidly appearing oscillations lead to the destruction of the strip and to the formation of 3 or 4 clusters at t = 305. These structures develop into two large clusters of positive vorticity and three weak zones of negative vorticity (t > 374). The process are accompanied 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 like and what the structure of the canonical equations is in concrete situations, are not as trivial as they may appear at first glance, (ii) the development of the Hamiltonian approach [4] would remain incomplete if
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no practical application of the theoretical analysis were given. As an application of the method, we have examined two of the simplest configurations of flows on the surface of a sphere: a system of N point vortices initially regularly distributed, and an equatorial jet. We have found that the vortex dynamics contains a change 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. The most important result of the direct simulation (for N ∼ 102 ) is that the system forms cluster structures (see also predictions from the statistical mechanics of point vortices [5–7] when N → ∞). 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 fluid motions, 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.
46
Formation of ‘clusters’ of vortices on a sphere Vadim PAVLOV a , Daniel BUISINE a , Viktor GONCHAROV b a b
DMF, UFR de Mathématiques Pures et Appliquées, université de Lille-1, 59655 Villeneuve d’Ascq, France 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 an incompressible fluid on a curvilinear surface. The method has been used to formulate governing equations of motion, and to interpret the evolution of a system consisting of N ∼ 102 –103 localized two-dimensional vortices on a sphere. The analysis shows that the instability appears immediately, forming initial disorganized structures which develop and finally ‘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 mouvement de l’ensemble des N ∼ 102 –103 , tourbillons localisés décrivant l’écoulement d’un fluide incompressible sur une surface sphèrique. Cette approche est mise en oeuvre dans l’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. Les résultats numériques montrent le développement des instabilités et les appariements qui conduisent á la formation des grandes structures tourbillonnaires quasi-stables. 2001 Académie 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 tourbillons océaniques et atmosphériques observés dans la nature. Le mouvement de ces tourbillons et leurs interactions dépendent fortement des contraintes physiques imposées, telles que par exemple la courbure du globe terrestre, sa rotation, la stratification et autres particularités de l’atmosphère et de l’océan. Une étude théorique incluant simultanément toutes ces facteurs est un problème très complexe. C’est pourquoi le rôle de 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 atmosphériques dépasse très largement la vorticité moyenne ambiante. Ces structures peuvent être étudiées dans le cadre du concept des tourbillons localisés, supposés non visqueux (en effet, le nombre de Reynolds caractéristique Re ∼ 108 1, les dimensions des tourbillons D R, où R est le rayon de la Terre). Ceci nous conduit à décrire les déplacements et l’interaction de tourbillons localisés à partir du modèle mathématique de singularités tourbillonnaires, considérées comme solutions bidimensionnelles des équations du mouvement Note présentée par René M OREAU. 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 bidimensionnel d’un fluide incompressible sur une surface. Nous nous intéressons à la dynamique des tourbillons sur une surface sphèrique, ainsi qu’à la mise en oeuvre de l’AH dans l’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. La dynamique des écoulements incompressibles qui est régie par l’équation (1), peut être reformulée R β β sous la forme (2) : ∂t Ωα = {Ωα , H} = dx0 {Ωα , Ω0 }(δH/δΩ0 ), où les crochets de Poisson sont détaillés en [4]. En supposant que l’écoulement est bidimensionnel, introduisons la composante contravariante de la vorticité Ω3 ≡ Ω, normale à la surface. Posons que le champ de vorticité Ω est décrit complètement à partir de N singularités d’intensité γi placées aux points ζ~i (t) de coordonnées (ζi1 (t), ζi2 (t)) soit : P Ω = i γi δ(ζ~ − ζ~i (t)). La composante contravariante de la vorticité Ω est donc une fonctionnelle des coorRR ~ ζ~0 ) = dζ~ dζ~0 Ω Ω0 G(ζ, données ζiα (t). L’hamiltonien du système (l’énergie cinétique) s’écrit : H = − 12 P N ~ ζ~0 ) est solution de l’équation ∆ G(ζ, ~ ζ~0 ) = δ(ζ, ~ ζ~0 ) − − 21 i6=j γi γj G(ζ~i , ζ~j ). La fonction de Green G(ζ, −1 α V , où V est l’aire de la surface. En calculant les crochets de Poisson pour ζi (t) (3), on peut établir les R −1/2 −1 β β γi (δH/δζjβ ) (4). Sur la sphère de équations ∂t ζiα = {ζiα , H} = dx0 {ζiα , ζ 0 j }(δH/δζ 0 j ) = εαβ gi α rayon unité, prenons pour coordonnées généralisées ζi (t) la longitude ζi1 = θi et la latitude ζi2 = φi . On peut montrer que, pour une sphère, la fonction de Green G s’écrit : G(cos βij ) = (4π)−1 ln(1 − cos βij ) (5), où βij 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 de calcul est présenté sur la figure 1. Le résultat important de ce travail est de montrer que l’approche hamiltonienne mise en oeuvre pour un ensemble de tourbillons en mouvement sur une sphère, débouche sur un système dynamique suffisamment riche pour rendre compte des instabilités, des interactions et des appariements qui conduisent à la formation des clusters.
1. Introduction. Basic equations A number of theoretical and experimental studies have been devoted to the understanding of the dynamics of atmospheric and oceanic vortices which are frequently observed in nature. The interest in the problem is also driven by the anxiety of civilization’s impact on the environment (see, for example, [1], for a review on 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 study combining all these factors is a very complicated problem, and therefore they have been frequently investigated 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 large lifetime too. On the other hand, in the approximation of Re −1 ∼ 10−8 1 when a fluid is considered as inviscid, the governing hydrodynamical equations admit the singular vortices as its solution. Thus, singular vortices as a mathematical model can be used as basic elements in the following approach. In spite of some difficulties in the interpretation of physical results, the concept of localized (singular) vortices is largely used in problems of geo- and astrophysical hydrodynamics (see, for example, [2,3] and references therein). In many cases the study of the dynamics of these vortices and their interaction is simpler than in analogous problems for continuous vortex distribution – an arbitrary initial hydrodynamical field can be represented 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 via their superposition. The dynamics of incompressible flows is governed by equations: ∂t v α + vβ ∂β vα = ∂α (ρ−1 p + χ),
∂β vβ = 0
(1)
where vα are velocity components (α, β = 1, 2, 3) in the Cartesian system of coordinates, ∂t is the partial derivative of a field variable with respect to time, p is pressure, ρ is density (further ρ = 1). The system (1) may be (see [4] and references therein) presented on the phase space of the vorticity field Ωα = εαµβ ∂µ vβ (εαµβ is the Levi-Civita tensor) in Hamiltonian form as: Z ∂t Ωα = {Ωα , H} =
dx0 Ωα , Ω0β δH/δΩ0β
(2)
where the Hamiltonian, H, of the system is the quantity functionally dependent on the fields, Ωα . The Hamiltonian structure of hydrodynamical models consists of the Hamiltonian, H, given by the total energy expressed in terms of field variables, Ωα , and of the functional Poisson bracket {, }. Conservation of energy follows from the given formulation, since ∂t H = {H, H} = 0. Here and throughout this work, the prime denotes that the field variables depend on the space coordinate x0 , dx = dx1 dx2 dx3 . The skew-symmetric functional Poisson bracket in equation (2) is defined for the given model by the expression {Ωα , Ω0β } = εασγ εγλν εβνµ ∂σ Ωl ∂µ δ(x − x0 ). For contravariant components, Ωα of the vorticity when using coordinates 0 x = (x1 , x2 , x3 ) is not Cartesian, we obtain {Ωα , Ω β } = g −1/2 εασγ εγλν εβνµ ∂σ Ωλ ∂µ g −1/2 δ(x − x0 ). αβ Here g is the metric tensor and g is its determinant. For incompressible flows when the fluid particle moves along non intersecting stationary fluid surfaces, we introduce the orthogonal curvilinear coordinates x1 , x2 , x3 : coordinate lines x3 coincide with vortex ones while coordinate lines x1 and x2 lie on the fluid surfaces. In this case, v = {v1 , v2 , 0} and the vorticity Ω = {0, 0, Ω}. Here Ω = g −1/2 (∂1 v2 − ∂2 v1 ). The Poisson bracket reduces [4] to {Ω, Ω0 } = εαβ g −1/2 (∂Ω/∂xα )(∂/∂xβ )δ(x − x0 )g −1/2 (α, β = 1, 2). The contravariant vorticity Ω obeys the equation ∂t Ω + v α ∂α Ω = 0 (α = 1, 2), and differs from the usual ‘physical’ vorticity Ωg 1/2 (g11 g22 )−1/2 . Both definitions coincide only in cases when g = g11 g22 . Let us assume that x1 coincides with the streamlines of the unperturbed stationary problem. For this wide class of layer models geometric properties of the space associated with such coordinate systems are characterized only by components g11 , g22 , g33 of the metric tensor and by its determinant g which are profile of the unperturbed flow. The kinetic energy of the deemed independent of x1 , just as is the velocity RR ~ ζ~0 ). Green’s function G(ζ, ~ ζ~0 ) is a solution of dζ~ dζ~0 ΩΩ0 G(ζ, fluid may be written in the form H = − 21 0 (2) 0 −1 ~ ζ~ ) − V , where V is the ‘volume’ of a domain where the delta-function ~ ζ~ ) = δ (ζ, the equation ∆G(ζ, is defined (for a spherical surface, for example, the ‘volume’ is V = 4πR2 – the surface has a ‘volume’, but has no boundaries). If V → ∞, one has the traditional equation. Let us consider an evolution of a system consisting of N singular vortices: Ω=
X i
X γi δ (2) ζ~ − ζ~i (t) = g −1/2 γi δ (1) ζ 1 − ζi1 (t) δ (1) ζ 2 − ζi2 (t) i
R P The full vorticity is given by dζ~0 Ω(ζ~0 ) = i γi . Here, γi is independent of time intensities of the vortices, ζ~i = (ζi1 , ζi2 ) are their coordinates dependent on time, δ (2) (ζ~ − ζ~i ) is the 2D-function of Dirac, ζ~i = ζ~i (t). −1/2 δij εαβ . Here gi is calculated at Calculating the Poisson brackets {ζiα , ζjβ }, we find {ζiα , ζjβ } = γi−1 gi α the point where the vortex is localized. Thus, in terms of variables ζi the dynamics of a system of singular
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vortices is described by the functional equations: Z ∂t ζiα = ζiα , H = dx0 ζiα , ζjβ0 δH/δζjβ0 = εαβ γi−1 g −1/2 ∂H/∂ζiβ
(3)
PN The expression for H = − 21 i,j γi γj G(ζ~i , ζ~j ) has a shortcoming by having an uncertainty which arises from the energy of the interaction becoming infinite when when i = j. But the diagonal terms in the Hamiltonian can be excluded, because they are independent of the space coordinates as a result of the independence of Gii . Spherical coordinates are an example of curvilinear coordinates which satisfy this requirement. On a sphere the location of vortices is given by longitude θ and latitude φ. Coordinates ζiα are (θi , φi ), where α = 1, 2, tensor εαβ has components ε12 = −ε21 = 1, ε11 = −ε22 = 0. We have, thus, ζ 1 = θ, ζ 2 = ϕ, ζ 3 = r, g11 = r2 , g22 = r2 sin2 θ, g33 = 1, and g = r4 sin2 θ > 0. For a spherical surface the basic system of dimensionless equations becomes ∂t θi = (γi sin θi )−1 ∂φi H, ∂t φi = −(γi sin θi )−1 ∂θi H. Here, ∂t is the dimensionless derivative-operator with respect to time, applied to a spherical coordinate of a vortex, time is measured in units of τ = 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 P0 is absent, 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(cos βij ) = −(4π)−1
∞ X
−1 [2l + 1] l(l + 1) Pl (cos βij ) ≡ (4π)−1 ln(1 − cos βij )
l=1
Here, Pl (cos β) are Legendre polynomials. Finally, we obtain from (3) the following equations: X 1 γj ∂φi G, ∂t θi = − (γi sin θi )−1 2 j6=i
X 1 ∂t φi = (γi sin θi )−1 γj ∂θi G 2
(4)
j6=i
2. Results of calculations and discussion Equation (4) has been treated by means of a 4-order Runge–Kutta scheme. The time step is restricted by the condition ∆t < η ∆ψ/ sup(|∂t θi |, |∂t φi |) where ∆ψ is the characteristic angle between two point vortices. To avoid the polar singularity due to the metrics, a polar cap of 0, 01 radian is excluded from the domain. The energy conservation has been evaluated during the calculations. The Hamiltonian H has been tried with small variations: −0.0192401 6 H 6 −0.0192344, during the process of iterating from t = 0 to t = 400, with a root-mean-square error < 3 × 10−4 . We initialize the system of m equidistant clusters of vortices centered at the latitude θc = π/2, and composed of n2 point vortices. The parameters of the problem are: (a) the number m of clusters, (b) the fractional area µ occupied by the clusters supposed to be sufficiently small – this parameter is defined as the total area enclosed by the vortex patches divided by 4π: µ ' mδϕδθ sin θc /4π. The trajectories and the distribution density of the clouds of point vortices are given here in the framework of the so-called ‘sinus’ – representation where the coordinates are defined by x = ϕ sin θ and y = θ. The evolution of 5 equatorial patches each containing 64 point vortices when µ = 0.157 is analyzed initially. The patches, which are initially at sufficiently short distances, exchange immediately with each other numerous vortices and merge rapidly forming a perturbed strip which can be qualified as turbulent. Originating from this unstable strip, 4 structures are formed without ‘pushing out’ of vortices at t = 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 analyze fragmentation of an equatorial, initially homogeneous, jet (figure 1). The jet consists of N = 100 point
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Formation of ‘clusters’ of vortices on a sphere
Figure 1. Destruction of the strip and formation of 3 or 4 clusters at t < 305. Figure 1. Fragmentation d’un jet équatorial et formation de 3 ou 4 clusters pour t < 305.
vortices. The distribution corresponds to an equatorial jet with the following parameters: maximum of the velocity on the axis and transversal variations of the velocity ∆Vφ,i = ±0.1. The sliding near the edges of the jet is the cause of the appearance of instability which is displayed for the first time at t = 158 and t = 193. Figure 1 illustrates fluctuations on edges of the strip in a form of small quasi-symmetrical rings. These structures develop between t = 226 and t = 246. Very rapidly appearing oscillations lead to the destruction of the strip and to the formation of 3 or 4 clusters at t = 305. These structures develop into two large clusters of positive vorticity and three weak zones of negative vorticity (t > 374). The process are accompanied 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 like and what the structure of the canonical equations is in concrete situations, are not as trivial as they may appear at first glance, (ii) the development of the Hamiltonian approach [4] would remain incomplete if
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no practical application of the theoretical analysis were given. As an application of the method, we have examined two of the simplest configurations of flows on the surface of a sphere: a system of N point vortices initially regularly distributed, and an equatorial jet. We have found that the vortex dynamics contains a change 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. The most important result of the direct simulation (for N ∼ 102 ) is that the system forms cluster structures (see also predictions from the statistical mechanics of point vortices [5–7] when N → ∞). 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 fluid motions, 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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