Vortex polygons

Vortex polygons: dynamics and associated particle advectionA. C. Espinosa Ramírez1, a) and Oscar Velasco Fuentes1

Departamento de Oceanografía Física, CICESE, Ensenada, Baja California, México

(Dated: 22 April 2021)

This paper presents a numerical study of the advection of passive and active particles by three and four equal vorticeslocated on the vertices of a polygon. The vortices, which have either singular or uniform vorticity distribution, areimmersed in an incompressible, unbounded, and inviscid fluid. Under these conditions, a regular polygon rotatessteadily if it consists of point vortices, and unsteadily if it consists of Rankine vortices. When the point-vortex polygonis perturbed by making it slightly irregular, the flow becomes time periodic. In this case, dynamical-systems methods,such as lobe dynamics and Poincaré maps, serve to compute the fluid exchanged between different regions and thearea of the chaotic sea. Both quantities are found to grow with the amplitude of the perturbation: the former does itin a weakly nonlinear way, the latter in a piecewise linear manner. The Rankine-vortex polygons always produce atime-aperiodic flow which depends on their relative size. Small vortices deform slightly and produce particle advectionwhich is analogous to the perturbed point-vortex case; large vortices deform strongly and merge to form a single one.The critical distance for merger is found to be δ/a≈ 3.6 and δ/a≈ 3.2 for three and four vortices, respectively, wherea is the vortex radius, and δ is the side length of the polygon. In both cases the vortices expel the largest amount ofvortical fluid at their critical distance, thus producing the least efficient merger.

I. INTRODUCTION

Polygonal arrangements of vortices have been observed innature and the laboratory at various scales of time and space:in superfluid helium (Yarmchuk, Gordon, and Packard, 1979),around the center of hurricanes (Kossin and Schubert, 2004),in the atmosphere of Jupiter (Grassi et al., 2018), and in arotating water tank (Di Labbio et al., 2020).

Long before these observations, the dynamics of vortexpolygons had received considerable attention. It is well knownthat a regular polygon of N point vortices of equal strengthΓ rotates uniformly without change of shape and is stable ifN < 7 (Thomson, 1883; Havelock, 1931). Thus, when con-sidering only the relative positions of the point vortices, thesystem is steady. Under the same consideration, an irregularpolygon of point vortices moves periodically if N = 3,4 (Aref,1979; Aref and Pomphrey, 1982). There exist also steadypolygons of finite-area vortices: they have uniform vorticitywithin specific contours that have only been computed nu-merically, and their stability depends on both the number ofvortices and their size (Dritschel, 1985; Dhanak, 1992). Thenonlinear evolution of large vortices has revealed the mergerof three and four vortices (Dritschel, 1986), a phenomenonalso observed in polygons of vortices with smooth vorticitydistribution (Swaminathan et al., 2016).

The study of the motion of fluid particles in the velocityfield of interacting vortices has occupied a prominent place inthe literature on chaotic advection since the inception of theterm (Aref, 2002). The periodicity of perturbed point-vortextrigons and tetragons makes them ideal models for the study ofparticle advection with the classic tools of dynamical-systemstheory: Poincaré maps, invariant manifolds, Melnikov func-tion, etc. (see, e.g. Holmes, 1990). The case of three pointvortices has been studied by Kuznetsov and Zaslavsky (1998,

a)Electronic mail: aespinosa@cicese.edu.mx

2000), the case of four point vortices by Boatto and Pierre-humbert (1999).

Since a finite-vortex polygon moves steadily only when thevortices have a very specific shape, substituting these with cir-cular vortices (of the same area and circulation) amounts toperturbing the steady state. The behavior of these Rankine-vortex polygons, and the associated particle advection, can bequalitatively deduced from previous studies of pairs of finite-area vortices (Velasco Fuentes, 2001, 2005).

Small vortices rotate with an almost constant angular ve-locity while undergoing little deformation. Consequently, thevelocity field can always be decomposed into a steady meanflow plus a weak, time-aperiodic perturbation. The methodsfor the study of particle advection in these type of flows arestraightforward extensions of those used in the time-periodiccase (see Velasco Fuentes, 2001, 2005, for examples, and ref-erences therein for the theory). We therefore expect particletrajectories to be regular in most of the flow domain; and tobe chaotic in a multiply-connected region, formed by the thinto moderately thick stripes that replace the separatrices of theunperturbed flow.

In contrast, sufficiently large vortices undergo merger,which is characterized by a phase of large deformation wheneven the angular velocity of the system is not well defined.Consequently, it is not possible to find a decomposition(steady flow plus time dependent perturbation) that is validfor all time. The methods alluded to in the previous paragraphmay still be applied, but they present some practical difficul-ties. Methods that do not require a dominant mean flow toexist, or to be known, like the finite time Lyapunov exponent(FTLE), may then be applied (see, e.g., Haller, 2001; Shad-den, Lekien, and Marsden, 2005).

The purpose of this study is to find the regions of the param-eter space where the behaviors described above occur (that isto say, the critical distance for the merger of Rankine-vortextrigons and tetragons) and to quantify the advection of passiveand active (vortical) particles in the different regimes.

Vortex polygons 2

II. POINT-VORTEX POLYGONS

The evolution of M point vortices is governed by 2N ordi-nary differential equations (Kirchhoff, 1883):

dxi

dt=− 1

2π

N

∑j=1j 6=i

γ jyi− y j

r2i j

(1)

dyi

dt=

12π

N

∑j=1j 6=i

γ jxi− x j

r2i j

(2)

where ri j is the distance between vortices i and j; and, forthe purposes of this study, all point vortices have the samecirculation: γ j = Γ for all j. We integrated these equationsnumerically with a fourth-order Runge-Kutta scheme.

A set of N point vortices located on the vertices of aregular polygon rotates with constant angular velocity (Ω =Γ(N− 1)/4πR2, where R is the radius of the polygon) whilepreserving their relative positions. In a reference system thatrotates with the polygon the flow is steady and its topology ischaracterized by the following elements (see figure 1): an el-liptic stagnation point at the center of the polygon; one N-gonof pseudo-elliptic points corresponding to the point-vortices(they are called pseudo-elliptic because the velocity is infiniteat the location of the vortices but these singularities are sur-rounded by closed streamlines); one N-gon of elliptic stagna-tion points, sometimes called ghost vortices, corresponding torecirculation zones; and two N-gons of hyperbolic points withtheir corresponding separatrices. These curves are streamlinesthat start at one hyperbolic point and end at the same or a dif-ferent hyperbolic point; in the language of dynamical systems,they are the superposition of a stable and an unstable manifold(see, e.g., Rom-Kedar, Leonard, and Wiggins, 1990). The sep-aratrices divide the flow domain into 2N +3 zones: N regionsassociated to the point vortices, all labeled 1 in Fig. 1; region 2surrounds the center of the polygon; N regions correspondingto the ghost vortices, all labeled 3; region 4 separates type-1regions from type-2 and type-3 regions; and region 5 is theexterior flow, extending from the vicinity of the polygon toinfinity.

Important differences in the flow geometry arise as thenumber of vortices changes. For N = 3,4 the boundaries ofall type-1 and type-3 regions are homoclinic separatrices: eachone ends at the hyperbolic stagnation point where it starts. ForN > 4 all separatrices are heteroclinic: they start and end atdifferent points. As a consequence, when N = 3,4 the type-1regions form a disjoint set, whereas for N > 4 each region 1touches two regions of the same type at one point. We willlater invoke this topological difference to explain the behaviorof finite area vortices.

We perturb the point-vortex system by distorting the poly-gon. So, in the initial condition, the trigon becomes an isosce-les triangle instead of an equilateral one, and the tetragon be-comes a rectangle instead of a square, as illustrated in Fig.2. In both cases the ensuing motion of the point vortices istime-periodic (Aref, 1979; Aref and Pomphrey, 1982), with

FIG. 1. Flow topology of point-vortex polygons: (a) trigon, (b)tetragon, (c) pentagon. Filled black dots represent the point vor-tices, empty black dots represent elliptic points, and crosses repre-sent hyperbolic points. The numbers identify different flow regions,as explained in the text.

a period that grows with the perturbation ε = d/R, where dmeasures the displacement of the vortices from their equilib-rium positions.

The periodicity of the vortex motion enables us to use aPoincaré map —the map of the particle position [x(t0),y(t0)]to its position one period later [x(t0 + T ),y(t0 + T )], wheret0 is an arbitrary time and T is the period— for the study ofparticle motion in the velocity field of perturbed point-vortextrigons and tetragons. We constructed the map by samplingthe position of a particle (relative to the polygon) every timethe point vortices return to their initial configuration (in gen-eral, the vortices do not return to their initial positions but thisis of no consequence: only relative positions play a role in the

Vortex polygons 3

d

d/2

RR

d/2

FIG. 2. Point vortices located at the vertices of polygons of ra-dius R. Filled dots represent the steady configuration (regular poly-gons), empty dots represent the perturbed configuration (irregularpolygons), the distance d measures the deformation of the polygon.

dynamics).If the perturbation is small, the hyperbolic stagnation points

of the steady case survive in the form of hyperbolic fixedpoints of the Poincaré map; similarly, the elliptic and pseudo-elliptic points survive as elliptic fixed points of the Poincarémap. The separatrices, on the other hand, break-up: the sta-ble and unstable manifolds no longer coincide with each otherbut they transversally intersect at an infinite number of points.We determined this entangled curves as follows. To obtain theunstable manifold, we placed a short line crossing the stagna-tion point of the instantaneous flow along the stretching di-rection and numerically computed its evolution until time nT ,where n is a small integer and T is the period of the vortexmotion. Since the flow is time periodic, this curve representsthe manifold at any time t = mT , where m is an integer. Thestable manifold can be obtained in a similar way, but the ini-tial line must be placed along the squeezing direction and theintegration must be done backwards in time until time −nT .We, however, did not compute the stable manifold directly:because of space and time symmetries of the flow the stablemanifold is a mirror image of the unstable manifold.

These intersecting manifolds constitute a (numerical)demonstration that particle trajectories are chaotic in some re-gions of the plane, and can be used to study mass transport be-tween different regions through lobe dynamics (Rom-Kedar,Leonard, and Wiggins, 1990). A lobe is the area boundedby segments of the stable and unstable manifolds which arein turn limited by two successive primary intersections ofthose curves (see Samelson and Wiggins, 2006, for a com-plete description of the method). Once a lobe is identified,we compute its area by numerically evaluating the integralsA =

∫xdy = −

∫ydx, where (x,y) are the coordinates of par-

ticles located along the lobe’s boundary. We applied lobedynamics to quantify the exchange of mass between a vortex(type-1 region) and its surroundings (region 4) as a functionof the perturbation ε = d/R. Figure 3 shows the lobes for apoint-vortex trigon with perturbation ε = 0.07. These repre-sent the fluid exchanged between the vortex and its surround-ings in one iteration of the Poincaré map, as follows: the fluidlocated in the dark red region moves to the light red region,thus escaping the vortex; the fluid located in the dark blue re-gion moves to the light blue region, thus entering the vortex.

Figure 4 shows the lobe area as a function of the pertur-bation for both trigons and tetragons. The dimensionlesslobe area is well approximated by second order polynomi-als: A = 2.15ε2 + 0.45ε and A = ε2 + 0.13ε , for trigons andtetragons, respectively, where A = AL/Ain with AL the lobe

FIG. 3. Lobes produced by a point-vortex trigon with a perturbationof ε = 0.07. Fluid in the dark red lobe moves out (to the light redlobe), fluid in the dark blue lobe moves in (to the light blue lobe),

area and Ain the area of region 1 in the unperturbed case (seeFig. 1). Note, however, that for the range of values of ε usedhere, the dominant term is the linear one. We did not com-puted lobe areas for larger perturbations because the mani-folds soon become very complex, making it challenging toselect the correct curves that define one lobe. For example,figure 5 shows, with a black line, the manifolds of a perturbedpoint-vortex tetragon for ε = 0.28.

In order to determine the area occupied by the chaotic seawe constructed Poincaré sections by computing long orbits(v 1000T ) of a small number of particles (v 100). Figure 6ashows an example of such a section, where the color of everyorbit depends on its position along the line of initial conditionsvisible on the upper left quadrant. It is easy to identify parti-cles with different qualitative behaviors: blue particles are inthe exterior flow, red particles are trapped by one of the vor-tices, green particles belong to a chain of stability islands notpresent in the steady flow, and the yellow particles occupy thechaotic sea. Note that the green particles, which were initiallylocated inside the elongated island on the top, appear to jumpto the island on the bottom, where no particles were initiallylocated; the red particles, on the other hand, are confined tothe region surrounding one of the vortices. The reason is thatthe former rotate around a periodic orbit of the Poincaré map,whereas the latter rotate around a fixed point of the map. Eachof the five white areas in figure 6a surrounds a fixed point ofthe map; like the red area, they are separated from the chaoticsea by a permanent, impenetrable boundary (a KAM curve,see, e.g., Rom-Kedar, Leonard, and Wiggins, 1990). Takinginto account the ergodic property of chaotic orbits, the factthat we are only interested in the largest chaotic sea, and theflow geometry of vortex polygons, we identify a chaotic orbitby the distribution of its radial coordinate r. Indeed, figure 6ashows that r is distributed within narrow ranges for blue, greenand red particles, while it is in the range 0–2R for yellow parti-

Vortex polygons 4

0 0.05 0.1 0.150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

AL/A

in

FIG. 4. Lobe areas for point-vortex trigons and tetragons as a func-tion of the perturbation ε . AL is the area of the lobe, and Ain is thearea of the unperturbed region associated with a point vortex (thetype-1 regions of Fig. 1). The markers show direct calculations us-ing the chaotic tangle (black circles for trigons; gray triangles fortetragons); the lines represent quadratic regressions obtained withthe condition that they must pass through the origin.

cles. Once we identified these chaotic orbits, we plotted themin order to generate a two-color image, like the white-gray fig-ure 6b, and computed the area of the chaotic sea as the fractionof colored pixels in the flow region represented in the image.

Fig. 7 shows the area of the chaotic sea as a function of theperturbation of point-vortex trigons and tetragons. We foundthat it grows linearly with the perturbation until a jump occurs,when the largest sea merges with a smaller sea associated withthe outer stability islands (compare frames b and c in figure6); after the jump the linear growth continues at a differentrate. This behavior will occur wherever the boundaries of twodifferent chaotic seas meet, for example in point-vortex pen-tagons and hexagons. We do not expect to see a similar phe-nomenon for larger N because of the instability of the vortexmotion.

III. RANKINE-VORTEX POLYGONS

A. Dynamics

The equation of motion for the two-dimensional, unforcedflow of an incompressible, inviscid fluid can be written in thefollowing form:

∂ω

∂ t+~u ·∇ω = 0, (3)

where ~u = (u,v) is the velocity on the (x,y) plane and ω =∂v/∂x−∂u/∂y is the component of the vorticity in the direc-

FIG. 5. Poincaré sections of particle trajectories and manifolds fora point-vortex tetragon with perturbation ε = 0.28. The chaotic tan-gle is formed by one stable and one unstable manifolds (each rep-resented by a black line) and is contained in the chaotic sea (repre-sented mostly by green dots).

tion perpendicular to this plane. In the problems we study herethe vorticity is zero everywhere, except for a finite number ofpatches of uniform vorticity; therefore, the contour dynam-ics method is ideally suited to compute the flow evolution.The equation of motion then becomes (Zabusky, Hughes, andRoberts, 1979):

dxk

dt=− 1

2π∑

j∆ω j

∮log |xk−x j|

∂x j

∂ s jds j, (4)

where x j and xk are the vector positions along the j-th and k-th contours, respectively; ∆ω j is the vorticity jump across thej-th contour; and s j is the arc length along the j-th contour.

To evaluate the integral in equation (4), we represented thecontours with a finite number of nodes; to solve the differ-ential equation we used a fourth-order Runge-Kutta schemewith fixed time step: dt = Te/1000, where Te is the eddyturnover time (in what follows times will be given in unitsof Te). The vortex contours were initially represented with100− 120 nodes, but this number increased with the lengthof the contour as the flow evolved. Since the fluid is inviscidand incompressible, the vorticity and area inside a contour areconstant even though some initial conditions lead to strong de-formation of the vortex patch; so we used the area to monitorthe quality of the integration.

Rankine-vortex polygons are not a steady solution of theEuler equations: they continuously change their shape asthey rotate, with a slightly varying angular velocity, aroundtheir common center. Furthermore, the vortices never recovertheir original circular shape, so the flow is not periodic ei-ther. Hence, a Rankine-vortex polygon may be viewed as anaperiodic perturbation of a point-vortex polygon —the per-turbation parameter being a/R, where a is the radius of thevortices and R is the radius of the polygon. We, however,

Vortex polygons 5

FIG. 6. Measuring the area of the chaotic sea produced by perturbedpoint-vortex trigons. (a) Positions of 60 passive particles at each of1000 iterations of the Poincaré map for ε = 0.14; the color of thedots indicates the particle’s initial position along the line on the topleft quadrant; the gray dots represent the point vortices, (b) particlesin the chaotic sea for ε = 0.14, (c) particles in the chaotic sea forε = 0.1.

found it more convenient to report our results as a functionof the size of the polygon relative to the size of the vortices:α = δ/a, where δ = 2Rsin(π/N) is the length of the poly-gon side. Note that δ is also the distance between the centersof two adjacent vortices; thus α is directly comparable to theintercentroid distance used in the well-known case of two vor-tices (Velasco Fuentes, 2005).

Our analysis centers on the evolution of regular Rankine-vortex trigons and tetragons as a function of α . To validateand compare the new results, we also analyzed the case oftwo vortices (digon) and, in order to verify one hypothesis,we did some computations for pentagons.

It is well known that two Rankine vortices display three

0 0.1 0.2 0.3 0.40

2

4

6

8

10

Acs

/Ain

FIG. 7. Area of the chaotic sea for point-vortex trigons and tetragonsas a function of the perturbation ε . Acs is the area of the chaoticsea, and Ain is the area of the unperturbed region associated with apoint vortex (the type-1 regions of Fig. 1). The markers show directcalculations using the Poincaré map (black circles for trigons; graytriangles for tetragons); the lines represent linear regressions.

types of interaction: merger, exchange and elastic (see, e.g.,Velasco Fuentes, 2005). We found that Rankine-vortex trigonsand tetragons either merge or interact elastically, but no initialcondition leads to a mere exchange of mass. The critical dis-tance that separates the merger and elastic interaction regimesis α ≈ 3.6 for trigons and α ≈ 3.2 for tetragons.

We claim that there is no exchange regime for vortex N-gons with N = 3,4 because the flow topology of these is qual-itatively different from all other cases (N = 2,5,6, . . . ). Fig-ure 1, which shows the steady flow geometry of point-vortexpolygons, also represents the initial flow geometry of Rank-ine vortices of small and intermediate sizes. When N = 3,4(Fig. 1a,b) there are no contact points between adjacent type-1 regions, there is thus no mechanism for the exchange ofmass between these regions. When N has any other value(Fig. 1c shows the case N = 5) adjacent type-1 regions toucheach other at a hyperbolic stagnation point; thus, as the flowevolves, these regions can exchange fluid.

In order to test this hypothesis, we computed the evolutionof Rankine-vortex pentagons: we found that for α > 2.8 thevortices interact elastically, for α < 2.5 they merge into a sin-gle vortex, and between these two values each vortex retainsits identity while exchanging some mass with its neighbors.

B. Advection in the elastic-interaction regime

The results obtained with point-vortex polygons suggestthat passive particles in the velocity field of Rankine-vortexpolygons are subject to chaotic advection too.

Vortex polygons 6

In the elastic-interaction regime (large α , relatively smallvortices) the Eulerian velocity field varies moderately aroundits initial condition, evolving smoothly and slowly; thereforeits character must manifest in the Lagrangian dynamics (see,e.g., Velasco Fuentes, 2005, and references therein). Thismeans that each of the 2N stagnation points of hyperbolic typesurvives in the form of a hyperbolic trajectory. This is simplythe path of a fluid particle that attracts one set of particles: thestable manifold; and repels another set of particles: the unsta-ble manifold (where attraction and repulsion merely describethe motion, without implying causation).

We numerically computed the manifolds at time t as fol-lows (see Velasco Fuentes, 2001, for a detailed description ofthe method). To obtain the unstable one, we placed a shortline crossing the stagnation point of the instantaneous flow,observed at t−δ t, along the repelling direction and computedits evolution until time t. To obtain the stable one, we placed ashort line crossing the stagnation point of the instantaneousflow, observed at t + δ t, along the attracting direction andcomputed its evolution, backward in time, until time t.

Figure 8 shows the chaotic tangle for Rankine-vortex poly-gons just above the critical distance for merger. Note the qual-itative similarity of frame b with Fig. 5, which shows a point-vortex tetragon. The most important feature is the presenceof multiple intersections of the manifolds, which implies theexistence of chaotic particle trajectories in the velocity field ofRankine-vortex trigons and tetragons. The area of the chaoticsea grows with the radius of the vortices (i.e. with decreas-ing α), although we did not measure it, as we did for point-vortices, because the aperiodic nature of the Rankine-vortexevolution makes it necessary to use long time integrations oflarge number of particles.

In the mass-exchange regime (observed for vortices of in-termediate size when N = 2,5,6 . . . ) the geometry of thechaotic tangle is similar to the elastic regime described above,with one important difference: some active fluid particles (thevortex filaments) enter the chaotic sea.

C. Advection in the merger regime

The cause of vortex merger has been the subject of intenseinvestigation, several physical mechanisms have been pro-posed: the filaments, the skew-symmetric part of the vorticityfield, the strain field, etc. (see, e.g. Melander, Zabusky, andMcwilliams, 1988; Meunier, Le Dizès, and Leweke, 2005;Brandt and Nomura, 2006). They all illuminate some aspectsof the phenomenon but they all know exceptions: vorticeswith particular vorticity distributions which do not merge evenwhen they satisfy the required conditions, or the other wayaround (Velasco Fuentes, 2005). Therefore, here we charac-terize and quantify the phenomenon, and leave its cause for afuture study.

In the merger regime (large vortices, i.e. α below the crit-ical value) the condition of slowly evolving velocity field isnot satisfied. Consequently, most hyperbolic stagnation pointsobserved in the initial velocity field (figure 1) fail to appear inthe Lagrangian dynamics. Generally, the inner N-gon of hy-

FIG. 8. Chaotic tangles for Rankine-vortex polygons computed att = 0. (a) Trigon of relative size α = 3.6; (b) tetragon of relative sizeα = 3.2. The shaded areas represent the vortices; the blue and orangelines represent the unstable and stable manifolds, respectively.

perbolic stagnation points (see Fig. 1) is the first to disappear,in what could be considered the merger event proper. Theouter N-gon of hyperbolic stagnation points disappears at alater stage, as these points gradually lose hyperbolicity whilethe system acquires circular symmetry, see Haller (2001) forthe theory and Velasco Fuentes (2001) for an application tothe merger of two vortices.

Figure 9 shows the unstable manifolds for vortex trigons intheir two regimes. Frame (a) shows an elastic interaction (α =3.6) at time t = 4.6: the vortices are clearly oval but they showno filamentation and the manifolds are well separated fromthe boundaries of the vortices (note that Fig. 8a shows thiscase too, but at time t = 0). Frame (b) shows larger vortices(α = 3.2) at time t = 2.5, after the merger has taken placebut before the axisymmetrization has been completed. A triadof particle trajectories still show hyperbolic behavior at thisstage.

Thus, the merger process leads to strong stirring for a shorttime only, a phenomenon usually referred to as transient chaos(Tél, 2015). This contrasts with the intense, continuous stir-ring produced by polygons of point vortices or relatively smallRankine vortices.

The outcome of a merger event depends critically on thenumber of vortices (N). When two vortices merge the result

Vortex polygons 7

FIG. 9. Unstable manifolds for Rankine-vortex trigons in the elasticinteraction regime, (a) α = 3.6, t = 4.6; and the merger regime, (b)α = 3.2, t = 2.5. Shaded areas represent the vortices, red dots markapproximate intersections of hyperbolic trajectories with the time-slice, and blue lines represent the unstable manifolds (in each frameone manifold is drawn with a thicker line).

is a more or less elongated vortex which can be considered asa single patch (in the inviscid case, a filament of irrotationalfluid always survives between the merged vortices, but it isso thin that it may be safely neglected). The merger of morethan two vortices results in a ring of vortical fluid surroundinga core of irrotational fluid. The core and the annulus are tri-angular if they are the result of the merger of a vortex trigon(Fig. 10), square if their origin is a tetragon, and so on. Theirrotational core is what survives of the region 2 of the initialcondition (see figure 1); its area increases with N: in trigonsit is about 25 % of the area of one of the initially circular vor-tices, in tetragons it is about 80 %. The vortical annular regionalso captures irrotational fluid from the vicinity of the vortices(region 4 in the initial condition); this appears to occur alwaysin small amounts, equivalent to about 1.5 % of the area of oneof the vortices.

The efficiency of the merger process is defined as the frac-tion of the vortices masses (i.e., areas, in a divergenceless,two-dimensional flow) that ends up in the final vortex. Todetermine it, we manually identified the expelled filaments,as illustrated in Fig. 10, and calculated their area at differ-ent stages in the flow evolution. Figure 11 shows averages ofthese calculations.

Rankine-vortex trigons and tetragons eject the largestamount of fluid, as measured by the area of the filaments, just

FIG. 10. A Rankine-vortex trigon of relative size α = 2.57 aftermerger. (a) One of the expelled filaments is represented in red attime t=1.43; the start of the filament, denoted by the black line, wasmanually chosen. (b) At time t=3.06 the filaments are significantlylonger, but their area is the same.

below the critical distance for merger. This means that the ef-ficiency of the merger process is lowest at the critical distanceand increases as α decreases. Merger efficiency goes from≈ 30 % to ≈ 80 % for N = 3, and from ≈ 60 % to ≈ 90 %for N = 4. This is in contrast with the case of two vortices,where efficiency is almost 100 % at the critical distance andhas a minimum, ≈ 75 %, at some smaller α (Velasco Fuentes,2005).

Several methods have been developed to study Lagrangiantransport by arbitrary unsteady velocity fields: see Had-jighasem et al. (2017) for a critical comparison of a dozenof them, and Xu (2019) for a recent application to the study oftransport by a dipolar vortex. The weakly unsteady flows in-duced by vortex polygons of point vortices and small Rankinevortices are best studied with the robust methods of classicaldynamical systems (see, e.g. Holmes, 1990); the merger ofvortex polygons, being strongly unsteady at some stages, is abetter candidate for the application of the newer methods.

The finite-time Lyapunov exponent (FTLE) measures therate at which particle trajectories separate over a finite time in-

Vortex polygons 8

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7A

f / A

v

2 vortices3 vortices4 vorticescritical value for 2 vorticescritical value for 3 vorticescritical value for 4 vortices

FIG. 11. Area of filaments expelled by Rankine-vortex polygonsas a function of relative size α . A f is the area of a filament, Av isthe area of a Rankine vortex. The vertical, dashed lines indicate thecorresponding critical distance for merger.

terval. Thus, an FTLE field reveals the Lagrangian dynamicsof the flow for a given time interval: the maxima (ridges) ap-proximate the stable manifolds when the exponents are com-puted forward in time and the unstable manifolds when theexponents are computed backward in time (see, e.g., Shad-den, Lekien, and Marsden, 2005). Figure 12 shows the FTLEfield, computed as described in Haller (2001), for a trigon ofrelative size α = 2.5. A regular mesh of 500× 500 particleswere advected from t = 0 to t = 2.1; consequently, the ridgesof the FTLE field (dark streaks) expose the stable manifolds:note the lobes pointing in the trigon’s anticlockwise rotationand compare them to the stable manifolds of the trigon in theelastic interaction regime (the orange curves in Fig 8). Thereis, however, a fundamental difference: the presence of stablemanifolds inside the vortices (indicated by the ridges). This isa necessary condition for the occurrence of filamentation and,for Rankine-vortex trigons and tetragons, of merger.

IV. CONCLUSIONS

Irregular point-vortex polygons and regular Rankine-vortexpolygons exhibit unsteady motion that leads to chaotic advec-tion in the vicinity of the vortices.

In point-vortex trigons and tetragons the exchange of fluidbetween the cores and their surroundings grows weakly non-linearly with the perturbation. The chaotic sea, i.e. the regionof the flow domain that is subjected to intense stirring, growspiecewise linearly. These two metrics also show that, undera comparable perturbation, the trigons stir the fluid more effi-ciently than the tetragons.

Regular polygons of three, four and five Rankine vorticesmerge when the initial dimensionless distance between ad-jacent vortices (i.e., the polygon side length divided by thevortex radius) is smaller than α ≈ 3.6,3.2,2.5, respectively.

FIG. 12. FTLE field in the initial condition (integration time: t = 0–2.1) for a Rankine-vortex trigon of relative size α = 2.5. Darkercolors denote higher values of FTLE, the white circle indicates theboundary of one of the vortices.

Hence, the critical distance for merger decreases with N inthis range of values. Preliminary results indicate that the crit-ical distance increases for N > 5, seemingly approaching amaximum value (α > 3). At this point, it might be of interestto mention that for large N the polygon is analogous to an in-finite row of equally spaced Rankine vortices, whose criticaldistance for merger is α ≈ 3.8.

During merger the vortices always expel some filaments sothat only a fraction of their mass ends up in the final vortex. InRankine-vortex trigons and tetragons this fraction is smallestjust below the critical distance for merger: about 30 % fortrigons and 60 % for tetragons. This is in sharp contrast withpairs of Rankine vortices which, at the critical distance, mergewith an almost 100 % efficacy.

Natural extensions of the present work (to better approx-imate the atmospheric flows mentioned in the introduction)would use vortices with a smooth vorticity distribution andinclude the effects of the rotation and curvature of the planet.

Acknowledgements

We are grateful to the anonymous reviewers for commentsand criticisms on an earlier version of this paper. This researchwas supported by CONACyT (México) through a postgradu-ate scholarship to A.C.E.R.

Data availability

The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.

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