Stability of relative equilibria of three vortices

Transcription

1 Stability of relative equilibria of three vortices Hassan Aref Citation: Physics of Fluids (994-present), 0940 (009); doi: 006/606 View online: View Table of Contents: Published by the AIP Publishing On: Wed, 0 Nov 0 0:08:5

2 PHYSICS OF FLUIDS, Stability of relative equilibria of three vortices Hassan Aref Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia 406, USA and Center for Fluid Dynamics, Technical University of Denmark, Lyngby DK-800, Denmark Received 9 May 009; accepted August 009; published online 5 September 009 Three point vortices on the unbounded plane have relative equilibria wherein the vortices either form an equilateral triangle or are collinear While the stability analysis of the equilateral triangle configurations is straightforward, that of the collinear relative equilibria is considerably more involved The only comprehensive analysis available in the literature, by Tavantzis and Ting Phys Fluids, 9 988, is not easy to follow nor is it very physically intuitive The symmetry between the three vortices is lost in this analysis A different analysis is given based on explicit formulas for the three eigenvalues determining the stability, including a new formula for the angular velocity of rotation of a collinear relative equilibrium A graphical representation of the space of vortex circulations is introduced, and the resultants between various polynomials that enter the problem are used This approach adds considerable transparency to the solution of the stability problem and provides more physical understanding The main results are summarized in a diagram that gives both the stability and instability of the various collinear relative equilibria and their sense of rotation 009 American Institute of Physics doi:006/606 I INTRODUCTION A relative equilibrium of point vortices is a configuration that moves without change in shape or size, as if the vortices formed a rigid body All configurations of two point vortices on the unbounded plane are relative equilibria Three point vortices have relative equilibria of two kinds First, three vortices of any circulations placed at the vertices of an equilateral triangle are a relative equilibrium Second, for almost all vortex triples it is possible to find one or more collinear configurations that are relative equilibria In this paper we determine all relative equilibria of three vortices as a function of the vortex circulations and analyze their linear stability As would be expected for this classical problem, numerous partial results are scattered about in the literature going back, at least, to Thomson For a given triple of vortex circulations whether a relative equilibrium is stable or unstable is usually easy to discover from the geometrical analyses of three-vortex motion pioneered in 877 by Gröbli and refined over the course of a century by Synge and, independently, by Aref 4 See also Ref 5 for a recent account of Synge s approach However, the only systematic analysis of the linear stability problem for the collinear relative equilibria, of which the author is aware, is the 988 paper by Tavantzis and Ting 6 The present study arose from an attempt to make this analysis more transparent in terms of physics In particular, the author felt that the symmetry between the three vortices inherent in the problem, which is so useful in much of the theory of three-vortex motion, was lost in the analysis in Ref 6 So far as the author can tell, all the results obtained here are consistent with those of Ref 6 Additional details and insights have emerged in the present analysis, in particular as regards the sense of rotation of the collinear equilibrium configurations The analysis given here also produces explicit formulas for the angular frequency of rotation and for the oscillation frequency or growth rate of infinitesimal perturbations in terms of the physical variables characterizing the relative equilibrium This information should also be obtainable from the analysis in Ref 6, but it seems quite laborious to do so in general The phase diagram in vortex circulation space, which we introduce in Fig, is very helpful in getting one s bearings in this problem It was, to the best of the author s knowledge, first used in a 978 Saclay report by Conte and de Seze 7 These authors also used the notion of resultants between two polynomials to track how various relationships depend on the circulations that enter the coefficients of those polynomials They had access to and used a symbolic mathematics package called AMP, 8 something that is routine today but was quite novel 0 years ago The stability analysis in the report 7 is incomplete it takes up only about a page Unfortunately, a full account was never published in the archival journal literature According to Conte 9 private communication, 004 a version of the report was included in his Thèse d État defended on June 4, 979 The ideas introduced in this work prove very useful, as we shall see The stability problem for the equilateral triangle configurations is straightforward by comparison with the case of collinear relative equilibria and is included mostly for completeness II GENERAL ANALYSIS OF RELATIVE EQUILIBRIA We start with some general considerations regarding relative equilibria of an arbitrary number of point vortices N This section initially follows the analysis in the review paper by Aref et al 0 and establishes our notation The point vortex equations on the unbounded plane, taken as the complex plane, without background flow are /009/9/0940//$500, American Institute of Physics On: Wed, 0 Nov 0 0:08:5

3 0940- Hassan Aref Phys Fluids, N dz = dt i =, z z N = + = 0 where the line over a symbol means complex conjugation, and the prime on the summation means The timedependent vortex positions are denoted as z t, =,,N The constant circulations of the vortices are These equations may be found in many places in the literature, eg, Ref or Ref We now assume that the configuration of vortices is instantaneously moving as a rigid body, ie, that the velocity of every vortex is made up of a translation and a rotation, dz = V + iz, dt Equations 4 and 7 allow us to classify relative equilibria of point vortices The form of these equations is simple: two linear equations in two unknowns, V and The condition for a unique solution is that the determinant of the coefficient matrix on the left hand side be nonzero, ie, that L = I X + Y 0 For L0, then, we find unique solutions for V and, V = il i 0 ix + iy ii = X + iy i L, a where V is complex and is real Both are the same for all vortices Substituting the Ansatz into Eq, we obtain in place of the ordinary differential equations ODEs a set of algebraic equations V iz = i = z z N To establish necessary conditions for a solution to exist, we multiply Eq by and z, in turn, and sum over Thus, multiplying by we obtain V + ix + iy =0, where is the first symmetric function of the vortex circulations, N =, = and the real quantities X and Y are the components of linear impulse, N X + iy = z = Next, multiplying Eq by z, and summing yields, after a brief calculation, VX iy + ii = i, where is the second symmetric function of the vortex circulations, = N, and I is the angular impulse, N I = z = We note for future reference that = il 0 X iy i = L b The right hand sides of these equations depend only on combinations of the first and second symmetric functions of the vortex circulations, and, and on the integrals of motion X, Y, and I or L Thus, if the Ansatz is valid at some instant, it will be valid for all time We may now reason as follows maintaining the assumption L0 For 0, we may assume X=Y =0, since an inconsequential shift in the origin of coordinates will otherwise assure this This amounts to choosing the center of vorticity z cv = = N z N = X + iy = as the origin of coordinates Then, using the new coordinates, V=0 from Eq 4, and L= I c, where the subscript reminds us that I is being calculated about the center of vorticity The vortices rotate as a rigid body about the center of vorticity with an angular velocity given by Eq 7 or I c = 4 Equation 4 includes the possibility for =0 that the vortex configuration is stationary For =0 and X +Y = L0, we have =0 by Eq b and the motion consists of pure translation with a velocity given by Eq a, V = = N X + iy 4i X + Y, 5 where we have used Eq 0 with =0 The case L=0 remains to be considered Equations 4 and 7 are no longer independent For 0 we may again arrange X=Y =0 by a shift in the origin of coordinates When the origin is so chosen, L=0 implies I c =0 It also follows from Eq 4 that V=0 and from Eq 7 that we must have =0 The configuration can either rotate or be stationary The angular velocity must be found by returning to the basic equations in their algebraic form, Eq It is unknown at On: Wed, 0 Nov 0 0:08:5

4 0940- Stability of relative equilibria of three vortices Phys Fluids, present whether stationary configurations with L = 0 exist We shall, however, see momentarily that a stationary configuration with L=0 requires at least four vortices For =0, L=0 implies X=Y =0 Equation 4 is then satisfied identically and Eq 7 becomes I = Since 0 for =0, we must have I0, and the configuration rotates with an angular frequency given by this equation However, the center of rotation needs to be determined as part of the analysis since the center of vorticity is at infinity For this determination we must return to the point vortex equations in their algebraic form, Eq We now specialize the analysis to three-vortex motion Among three circulations two must be of the same sign, and we can assume these two circulations to be positive without loss of generality The case of two negative vortices then follows by reflection in the plane of flow or by time reversal Since the numbering of the vortices is also at our disposal, we may adopt the convention that the three circulations satisfy in every case For three vortices every relative equilibrium is known to be either an equilateral triangle or a collinear configuration 4 The equilateral triangle configuration is a relative equilibrium for every choice of vortex circulations Given a triple of vortex circulations,,, there are, in principle, two equilateral triangle relative equilibria, one for clockwise orientation of vortices,,, and one for counterclockwise orientation The motion and stability of these two configurations parallel one another modulo the obvious adjustments, and so we shall treat only one orientation Since the geometry of these relative equilibria is independent of the vortex circulations, the stability analysis for the equilateral triangle case is rather straightforward On the other hand, for the collinear relative equilibria, of which there can be three, two, one or even none depending on the circulations, the analysis is considerably more complicated For these the configuration geometry albeit just three points along a line depends on the vortex circulations, and this coupling between configuration and vortex circulations is, as we shall see, the main source of complexity both in determining the configurations and in the stability calculations For an equilateral triangle of side s we have L= s Hence, L0 is equivalent to 0 For 0 our present analysis tells us that L= I c = s and the vortices must rotate about the center of vorticity with an angular frequency given by Eq 4 or = /s As we shall see from the explicit solution for the equilateral triangle relative equilibria, this formula also gives the angular velocity of rotation in the case =0 when Eq 4 becomes indeterminate For =0 the triangle translates with the velocity of translation found previously See also the considerations leading to Eq 5 below The special case of the three-vortex problem with =0 can be solved in full detail as shown by Rott and Aref, 4 including for the case L=0 One finds that the vortices form (a) (b) (c) (d) FIG Four collinear relative equilibria singled out by the analysis a,, =,,, unstable; b,,, sum of circulations is zero, stable; c,/,, marginally stable; d,, 5, unstable Configurations a c rotate clockwise; d is stationary Linear scale differs from configuration to configuration Solid dots correspond to positive, open dots to negative circulation The radii of the dots scale with the magnitude of the circulation a collinear relative equilibrium in this case The positions are given by z t = e it, z t = e it, z t = e it, where is a scale factor, and =, with =, 6a 6b 7 the third symmetric function of the three vortex circulations We note that 0 since both and are negative when =0, ie, the configuration rotates clockwise An example of such a configuration is shown in Fig The relative equilibrium Eq 6a appears already in Gröbli s thesis, although the general analysis of the three-vortex problem with total circulation zero was not given in detail in that work Another important special case arises by considering a configuration of three vortices that is stationary: V=0, =0 Then Eq 4 is identically satisfied and, according to Eq 7, =0 Since =0 implies 0, we may assume or arrange by a shift in coordinates that z + z + z =0 8a For a stationary configuration Eqs, for N=, take the form On: Wed, 0 Nov 0 0:08:5

5 Hassan Aref Phys Fluids, z z + z z =0, z z + z z =0 z z + z z =0, These are clearly equivalent to z z = z z z z, = z z, and then, recalling =0, to z + z + z =0 z z = z z, 8b Equations 8a and 8b are two linear relations between the coordinates z, z, and z and thus show that the vortices must be collinear We take this common line to be the x-axis The solution has the form x =, x =, x =, 9 where is a real scale factor An example of such a configuration is shown in Fig This solution also appears in Ref In fact, the solutions 6a, 6b, and 9 are closely related, as we shall see Recalling that =0, one finds easily for this solution that I c = For three vortices, then, all stationary configurations have L= I c = 0 Hence, stationary configurations with L=0 or, equivalently, I c =0 must involve at least four vortices In fact, Hampton et al 5 showed that all stationary four-vortex configurations also have L0 If stationary N-vortex configurations with L = 0 exist, they must involve at least five vortices dz dt = i s z z + z z i = s z X + iy, where s is the common length of the three sides of the triangle For 0 we introduce the center of vorticity z cv from Eq, z cv = X + iy = z + z + z + + Equation may then be written as dz = iz z cv, dt =,,, a where = s b Since z cv is a fixed point in the plane, these equations show that the vortices rotate about the center of vorticity with angular frequency This derivation includes the case =0 Solving Eq 0 along with Eq we obtain the solution z t = z cv + se it, z t = z cv + se it, 4 z t = z cv + se it, III EQUILATERAL TRIANGLE RELATIVE EQUILIBRIA Let us now give the complete solution for the equilateral triangle relative equilibria If vortices,, and appear counterclockwise, we have z z = z z, z z = z z = z z, where =e i/ From the basic equation we then have dz dt = i s z z + z z = i s z X + iy, dz dt = i s z z + z z = i s z X + iy, 0 with given by Eq b The solution in Eq 4 is the counterpart for three vortices of the well-known solution to the two-vortex problem There is a free scale parameter, s, the side length of the equilateral triangle The form of Eq 4 makes the side parallel to the x-axis of coordinates at t=0: z 0 z 0=s It is not difficult to check that the solution 4 satisfies Eq with N= The reader may find it instructive to write the formulas corresponding to Eq 4 when vortices,, and appear clockwise For =0 the right hand sides of Eq may all be written as the same velocity V = X + iy is, which is constant throughout the motion Hence, the vortex triangle translates like a rigid body Now, when =0, L= X +Y = s for an equilateral triangle Using Eq 0 we get X + Y = + + s The speed of translation V is then On: Wed, 0 Nov 0 0:08:5

6 Stability of relative equilibria of three vortices Phys Fluids, V = s The formula for the speed of translation of a vortex pair with constituent vortices of circulation = can be written in the somewhat contrived form V = + s to bring out the analogy with Eq 5 This is also the limiting form of Eq 5 when one of the three circulations tends to 0 IV COLLINEAR EQUILIBRIA OF THREE VORTICES For a collinear relative equilibrium we take the instantaneous line through the vortices to be the x-axis, and we then have the conditions for a relative equilibrium in the form x x = x x + x x +x x x x, so we also have x x + x x = x x x x, which shows that x x x x 0 Hence, there can be no solution of Eq 6 for b=0, a0 For a neutral vortex triple there are no translating collinear relative equilibria This result also follows, of course, from the full analysis of the case =0,4 If b0, we may assume a=0, since we can always shift the origin of coordinates by a/b and this will eliminate any nonzero a from Eq 6 We now have x + x + x =0 8a Multiplying the first of Eq 6 by x x, the second by x x, the third by x x, and adding, we obtain an equation that does not contain the circulations, a + bx = a + bx = a + bx = +, x x x x +, x x x x +, x x x x 6 where a=iv and b= are related to the velocity of translation and the angular frequency of rotation, respectively Since b is real, and the right hand sides in Eq 6 are real, it follows that a must also be real, ie, V is pure imaginary In other words, if the line of vortices is to translate, it must do so in a direction perpendicular to the line We have already dealt with the special case a=b=0 Solutions for this case exist only if =0 and are given by Eq 9 The reader may wish to verify that, indeed, Eq 9 satisfies Eq 6 with 0 on the left hand side If b=0 but a0, solutions will only exist for =0 This follows from the general theory, but is not hard to verify independently by multiplying the first of Eq 6 by, the second by, the third by, and adding Now, if we subtract the second of Eq 6 from the first, we find x x + or, using =0, x x + x x x x + x x x x =0 x x =0, 7 This equation, which does not contain the circulations, has no solutions, however To see this note that it may be rewritten as x x x x x x =, x x x x = x x x x This shows that 0x x x x But x x x + x x x + x x x =0 8b This elegant result was given by Gröbli For vortex triples with = =, and any value of, we have the solution x =0, x = x of Eqs 8a and 8b These include the stationary configuration for = /, a special case of Eq 9 In the restricted space of vortex triples with two identical vortices these configurations have invariant geometry as the strength of the third vortex is varied This suggests that the stability analysis will simplify for such configurations since geometry and circulations are decoupled This family of relative equilibria may not be the most mathematically exciting solutions of Eq 6, but we shall see that they play an important role in the theory They are also important physically, and they are certainly among the most celebrated relative equilibria of three vortices Three vortices on a line became a much studied topic in geophysical fluid dynamics with the discovery of the vortex tripole by van Heijst and Kloosterziel, 6 Kloosterziel and van Heijst, 7 and van Heijst et al 8 The vortex triple,, with the same geometry is shown in Fig An asymmetrical rotating configuration with the sum of the circulations equal to 0 is also shown For a general discussion of Eqs 8a and 8b consider the variables = x x, = x x, = x x, 9 which are not independent since + + =0 Equation 8b may then be written as x + x + x =0 0 From Eq 8a and any two of Eq 9 we get the transformation formulas inverse to Eq 9 when 0, On: Wed, 0 Nov 0 0:08:5

7 Hassan Aref Phys Fluids, x =, x =, x = For =0 the condition 8a tells us that =, which with Eq 0 gives = = It follows that L is proportional to and so vanishes Hence, we return to the solution 6a and 6b We note in passing that the general solution of the system x + x + x =0, is x = x + x + x =0, x, = x, =, 4a 4b where is an arbitrary parameter For =0, Eq combined with Eq yields the second equation 4a Equation 4b then gives the solution 6a For =0, Eqs 8a and 8b are equivalent to Eq 4a We get the solution 9 by rewriting Eq 4b using =0 This common algebraic origin of the two special solutions 6a and 9 seems to have gone unnoticed Continuing with the general case 0, Eq 8a is now identically satisfied, and Gröbli s relation, Eq, becomes + + =0 5 Choose one of the ratios between the s as a new independent variable, eg, z = = x x, x x and note that we then have = + = z; Thus Eq 5 becomes z + +z + z z + which is a cubic equation for determining z, 6a = = z +z 6b z +z ++ z =0, pz = + z + + z + z + =0 7 This equation with a relabeling of vortices and appears as Eq 5 of Ref 6 A related equation appears in Ref 9 The cubic equation 7 will always have at least one real root According to Eq 6a, in order for a root of pz to be physically acceptable, it must be 0, Since p0 = + and p = +, the cases + =0 and + =0 require special consideration If we imagine vortices and placed on the real axis, with vortex to the left of vortex, x x, we see that vortex to the right of vortex, x x, implies 0z When vortex approaches vortex from the right, z tends to the disallowed value z=0 For vortex between vortices and, x x x, we have z0 The disallowed value z= corresponds to vortex coinciding with vortex Finally, for vortex to the left of vortex, x x, we have z Three special values of z deserve highlighting For z= vortex is at the midpoint of For pz to have the root z= we must have =, and so the three vortex circulations are identical For z= vortex is at the midpoint of ; p =0 implies = For z= vortex is at the midpoint of ; p =0 implies = If vortex is to the right of vortex, we will return to the situation explored above after half a turn of the configuration Hence, this analysis covers all possible cases The real roots of the cubic polynomial pz, Eq 7, except for z=0,, give the possible collinear relative equilibria The explicit formulas for the roots of pz as functions of,, and are known from what is usually called Cardano s formula; see Ref 0 for an elegant exposition of the theory of the algebraic solution of the cubic If we set Z = + z + +, Eq 7 is transformed to Z +HZ + G =0, where z = Z + +, 8a 8b H = , 8c G = Remarkably, one finds that the discriminant D of this cubic is quite simple this result was found using the symbolic algebra program MATHEMATICA, D = 4 G + H = , 8d where the symmetric functions of the vortex circulations,,, and, have been introduced previously A Special cases When =0, we see that D0 Hence, there is only one real root of pz For the solution 6a we have : : = : :, cf Eq Thus, from Eq 6a, we find the root / and, indeed, direct substitution shows that p / =0 when =0 When =0, we have 0 but 0 Thus D0 and there are three real roots of pz For the stationary solution 9 we have : : = : :, see the equations lead On: Wed, 0 Nov 0 0:08:5

8 Stability of relative equilibria of three vortices Phys Fluids, ing up to Eq 8b and Eq 65 later Thus, from Eq 6a, we find the root / and, indeed, direct substitution shows that p / =0 when =0 Let us also consider the special cases + =0 and + =0 where a disallowed root of pz, ie, z=0 or z=, needs to be factored out since it leads to two of the vortices coinciding, a physically unacceptable situation For = the polynomial pz in Eq 7 has the factor z, and pz = z + z + + z + 9a If =4 0, ie, if /, this equation has three real solutions: the physically unacceptable z = 0 and the two roots of the quadratic in square brackets, z = Our convention on relative magnitudes is so the inequality / is always satisfied Of these two solutions the other unacceptable value, z=, arises if, and only if, = In this case there is only the one collinear relative equilibrium for the vortex circulation triple,, : z= or x = x +x =0 This configuration is shown in Fig For = the polynomial pz in Eq 7 has the factor z+, pz = z + + z + z + 9b If 4 + =4 0, ie, if /, this equation has three real solutions: the physically unacceptable z= and the two roots of the quadratic in square brackets, z = In this case the roots of Eq 8b are given as 0 n + Z n = H cos, n = 0,,, cos = G, H 40 where H and G are as in Eq 8c, and H is negative The corresponding values of z, obtained through Eq 8a, will be designated z p 0, z p, and z p The superscripts are to avoid confusion with the vortex positions z and z We have z p z p p z 0 with equality at points or curves in vortex circulation space where two or more of the roots merge When the discriminant vanishes, two of the roots Z 0, Z, and Z coincide The cubic has three coincident roots when G=0 and H=0 This can only occur for the circulation triple,, 5 4 When the discriminant is positive, the cubic has just one real root This root can be written as 0 Z = G + D/ H G + D /, 4 where the quantities H, G, and D were given in Eqs 8c and 8d This Z-value must be converted to a value of z through Eq 8a The formulas developed here clearly imply a rather complicated dependence of the positions of the vortices in a collinear relative equilibrium on the circulations Given a root of the cubic equation 7, z, and imposing the condition that x + x + x =0, we find for 0 that the positions of the three vortices along the line are, cf Eq and Eqs 6a and 6b, x = z + +, x = z, x = + z +, 4 Our convention on relative magnitudes restricts us to consider the interval / Of the two solutions the other unacceptable value z=0 arises if, and only if, = For this case there is, again, only the collinear equilibrium z= or x = x +x =0 For = /, ie, for the vortex circulations,, =,,, there is also only one collinear relative equilibrium corresponding to z= Up to a scale factor, the vortex positions in this configuration may be given as x =, x =, and x = This configuration is shown in Fig For 0 /, ie, for vortex triples,, =,r, with 0r/ there are no collinear relative equilibria We now return to the main analysis B The general case In the general case, the number of real roots of the cubic equation 7 will depend on the sign of the discriminant Eq 8d There will be three real solutions to Eq 7 when the discriminant, Eq 8d, is negative, ie, when where is a scale factor, =x x =, that may be chosen freely For =0 the positions were given by Eq 6a The angular frequency of rotation of a collinear relative equilibrium is given by the general relation 4 However, for collinear relative equilibria we shall find an expression for derived directly from the determining equations to be more readily applicable We have x =, x =, x = From these by subtraction, and using Eq 0, = + + = By permutation of indices we obtain the following set of equations: On: Wed, 0 Nov 0 0:08:5

9 Hassan Aref Phys Fluids, = + + = + + = + + From these one obtains, again using Eq 0 and assuming 0, = For the special case =0, L= I c =0, the frequency of rotation requires a different manipulation of Eqs 4, aswe shall see in Sec VII C We pause to consider the special case of two identical vortices, which is of interest in itself but also will play an important role in the general development of the theory C The case of two identical vortices For,, =,,r, we have G=0, H= 5+4r There are three solutions of the cubic when 5 4 r For these we have =/ and Z 0 = 5+4r, Z = 5+4r, Z =0 The corresponding values of z are, from Eq 8a, z 0 p = + 5+4r, z p = 5+4r, z p = These could have been obtained directly from Eq 7, which in this case takes the form z +z +rz +r=0 The cubic is seen to have the root z= Factoring out a linear term corresponding to this root gives a quadratic equation from which the remaining two solutions are obtained Equation 4 with the roots z 0 p, z p, and z p leads to the following configurations up to a scale factor : i: x = + r 5+4r +r, x = + r 5+4r +r, x = 5+4r +r, ii: x = r 5+4r +r, x = r 5+4r +r, x = 5+4r +r, 45 iii: x = x =, x =0, respectively Configuration i is identical to configuration ii if we rotate it half a turn and relabel the identical vortices and For r= we get the three collinear arrangements of three identical vortices, labeled,,, on a line, from left to right: i--; ii --; iii -- We reiterate that the solution iii exists for all r, including the important case r= when the total circulation is zero Solutions i and ii exist only for 5 4 r At r= the roots z p 0 and z p attain the unacceptable values 0 and, respectively In i we see that x and x coincide for r= In ii x and x coincide For this value of r, then, there is again just the one collinear relative equilibrium iii In the ranges 5 4 r and r there are three collinear relative equilibria Turning to the angular frequency of rotation, we see from Eq 44 that for solution iii, = +8r+r =+r +r For r= this configuration is stationary Indeed, except for a scale factor, it is consistent with the general form 9 For r= 5 4, Z=0 is a triple root of the cubic This configuration rotates clockwise with angular frequency = / For even smaller r we have only the one real root Z=0Itis this solution that persists into the range r 5 4 For r= we have =0 and = /, twice the clockwise angular frequency of the r= 5 4 configuration This is a special case of the solution given in Eqs 6a and 6b Since solutions i and ii for r= are not of the form 9, in particular, x 0, they must correspond to rotating configurations and must, therefore, have I c =0 In general, we get the rotation frequency from Eq 44 with z=z p 0, Note that Thus, z p 0, z p 0, + = +r, z p 0, + z p 0, + =+r 0, = +r+r +r + r+r+r = +r +r This has the finite, positive value =/ for r= For r= these solutions disappear They do so by vortex coinciding either with vortex solution i or with vortex solution ii As this occurs parametrically, through a sequence of collinear relative equilibria, the expression for the On: Wed, 0 Nov 0 0:08:5

10 Stability of relative equilibria of three vortices Phys Fluids, TABLE I Different regimes of three-vortex motion in terms of and from Ref 4 Case Example 0 0 0,, 0 0 0,, 0 0 =0,, ,, 5 0 =0 0,, ,,4 angular frequency of rotation of these solutions must diverge D Main analysis continued With these results in hand we return, once again, to the main analysis In the general case and we exclude the singular cases = and =, which we dealt with separately the number of real solutions one or three is determined by the sign of the discriminant Eq 8d There will be three real solutions to Eq 7 when Let us consider the various cases in Table I, which is the basis for the geometrical classification of three-vortex motion in Refs and 4, in light of this condition The examples are merely intended to provide a concrete set of circulations that satisfy the conditions stated for each particular regime They do not necessarily capture all possibilities in that regime We shall need some well known estimates First, by Cauchy s inequality = =, ie, This inequality is valid regardless of the signs of the vortex circulations Next, the inequality between the arithmetic mean / and the harmonic mean / implies 9 This inequality is valid when all vortex circulations are positive Thus, in case in Table I we have =, so Eq 46 is satisfied, and there must always be three collinear equilibria In case the negative circulation is larger than / + and thus, a fortiori, larger than We have 0 but and are both 0, so +6, and there must again always be three collinear equilibria In case the inequality 46 degenerates to 0, which is, of course, correct for that case There are, thus, three collinear equilibria in case as well One of these is stationary and we found it explicitly in Eq 9 Case 4 is the most complicated: we have the possibility that Eq 7 reduces to a quadratic polynomial, and there may then be two collinear equilibria, or one, or none In general, for given and, we can consider the two sides of the inequality 46 as functions of in the range + / + The right hand side of 46 is a cubic polynomial in which vanishes at the two ends of this interval and is negative everywhere in between At the lower limit of the interval it has a double root The left hand side is a quadratic polynomial, which is positive equal to at the lower limit of the interval, but negative equal to 6 at the upper limit Closer examination eg, calculating the discriminant for the cubic in given by Eq 46 with an equal sign, again most simply done by using a symbolic algebra program shows that there is a unique value of within the interval in question, let us call it,at which the two sides of Eq 46 are equal Thus, for between the lower limit + and, the inequality 46 will be violated, and there is just one collinear equilibrium except for the special case + =0 discussed previously For between and / + the inequality 46 is satisfied and there are three collinear equilibria Unfortunately, the general expression for in terms of and is rather complicated and does not seem to offer further insight For = =, however, we have the simple result = 5/4 This algebraic discussion is clarified in Sec IV E when we visualize the space of circulation triples In case 5 we have, trivially, that Eq 46 is never satisfied and there is just one collinear equilibrium, which we have already determined, Eqs 6a and 6b Similarly, in case 6, the right hand side of Eq 46 is negative, whereas the left hand side is positive There is again just one collinear equilibrium In summary, as we progress through the cases listed in Table I, we go, roughly speaking, from having three collinear equilibria to having just one The crossover occurs within case 4, which precipitates a more detailed analysis These results are in accord with the geometrically based conclusions from the phase diagrams of Synge and Aref 4 The more precise quantitative criteria given here and the expression for the discriminant of the cubic in terms of symmetric functions appear to be new E Graphical representation To obtain an overview of solutions it helps to represent things graphically In the main case where the sum of the three vortex circulations is nonzero, let = /, =,, Consider a trilinear plot with coordinates,, and, as shown in Fig The equilateral triangle has height The central triangle corresponds to,, and being all positive This is the region that we called case in Table I If we allow to become negative, still keeping 0, we move into the wedge-shaped region to the upper left in the diagram in Fig This region encompasses case, case, and case 4 from Table I although we restrict ourselves to half the region 0 Consider the curve that in trilinear coordinates is given by On: Wed, 0 Nov 0 0:08:5

11 Hassan Aref Phys Fluids, c 4b 4e 4d 4a κ κ κ 6 FIG The trilinear,, diagram used to classify regimes of three-vortex motion The various curves dividing up circulation space are explained in the text The circle circumscribed about the basic equilateral triangle is given by Eq 47 The dashed lines are + =0 and + =0, respectively The cusped curve is given by Eq 46 Labels,, etc, give the various regimes summarized in Table I The convention on the three circulations restricts interest to a subset of the diagram + + =0 47 This is the equation for a circle It passes through the three vertices of the equilateral triangle and, thus, is the circumscribed circle of that triangle In Fig we show this circle, and the regimes corresponding to cases 4 and case 6 from Table I Case consists of half of the arc of the circle separating the two regimes and 4 The further partitioning of regime 4 into several pieces is explained below The regime corresponding to case 5 is not available in the diagram in Fig since this representation relies on 0 Fortunately, detailed results for the case of vanishing total circulation are available independently,4 The regime corresponding to case 6 is half of the lower right wedge in the diagram Although we show the full diagram, and curves such as Eq 47 extended throughout, our convention restricts us in Fig to a subset of the full plane This very useful diagram appears in the work by Conte and de Seze 7 and Conte 9 We may now draw the curve corresponding to Eq 46 in the diagram in Fig A threefold symmetric curve with three cusps arises Points inside this curve correspond, in general, to three collinear relative equilibria Exceptions On: Wed, 0 Nov 0 0:08:5

12 0940- Stability of relative equilibria of three vortices Phys Fluids, arise on the lines + =0 and + =0 and their point of intersection Points outside the curve correspond, in general, to only one collinear relative equilibrium Along the line + =0 outside the curve there are no collinear relative equilibria We also need to analyze points on the curve We clearly see why case 4 is the most complicated regime It contains portions corresponding to three, two, one, and zero collinear relative equilibria The cubic curve and the lines + =0 and + =0 partition regime 4 into five pieces, labeled 4a through 4e and our interest is confined to the half wherein 0 On the lines + =0 and + =0 there are just two collinear relative equilibria, as we have seen, and at their point of intersection just one This is true only for the piece of these lines that is inside the cusped cubic curve As we cross the cubic, the significance of the lines + =0 and + =0 changes from implying two solutions or exceptionally just one to implying no solutions at all On the cusped curve itself the discriminant of the cubic pz vanishes, and we have, in general, just two solutions, and exceptionally just one Fixing and and varying in the range + + / corresponds in Fig to moving along a line : = : from where it intersects the circle =0 to infinity At some point this line intersects the cusped curve That point of intersection defines the intermediate value of that we designated earlier For =, ie, =, the line goes through the cusp itself V LINEAR STABILITY OF THE RELATIVE EQUILIBRIA OF THREE VORTICES: GENERAL THEORY We are now ready to investigate the linear stability of the equilateral triangle and collinear relative equilibria of three vortices Since the geometry of the equilateral triangle equilibria is decoupled from the vortex circulations, the analysis for this case will prove much simpler than for the case of the collinear relative equilibria The equations of motion are Eqs specialized to N=, dz dt i = + z z z z, dz dt i = + z z z z, i 48 dz dt = + z z z z Most of the relative equilibria correspond to solid body rotation z t=z 0 e it, =,,, where the angular frequency of rotation is determined as part of the solution We perturb such a solution by setting z t=z 0 + te it, where the t are infinitesimal perturbations with respect to which we shall linearize Inclusion of the factor e it in the perturbation is for convenience in calculation, as we shall see presently Expanding to first order in the small perturbations, we obtain the linearized perturbation equations d dt d dt d dt i = i z 0 z 0 + i = i z 0 z 0 + i = i z 0 z 0 + z 0 z 0, z 0 z 0, 49 z 0 z 0 Factors e it have been canceled in these equations The coefficient matrix on the right hand side is, thus, independent of time We may write these linearized perturbation equations in an easily understood vector-matrix notation, where = i + A, 50a A = z 0 z 0 + z 0 z 0 z 0 z 0 z 0 z 0 z 0 z 0 + z 0 z 0 z 0 z 0 z 0 z 0 z 0 z 0 z 0 z 0 z 0 z 0 + z 0 z 0 50b Differentiating Eq 50a once more, we get = + AA 5 = i + A = + A + ia i + A, or, taking the complex conjugate and effecting the cancellation, We may now proceed as follows First, we find two general eigenvectors of AA The most obvious of these is,,, which is an eigenvector of A with eigenvalue 0 since the rows in A sum to zero Hence,,, is also an eigenvector On: Wed, 0 Nov 0 0:08:5

13 0940- Hassan Aref Phys Fluids, of AA with eigenvalue 0 Only slightly less obvious is the eigenvector z 0 =z 0,z 0,z 0 Clearly, the first row of Az 0 is z 0 0 z z 0 z 0 + z 0 0 z z 0 z 0 = z 0 0 z + z 0 z 0 = z 0 The two other rows similarly produce z 0 and z 0, respectively Thus, Az 0 =z 0 and AAz 0 = z 0 The vector z 0 is, therefore, an eigenvector of AA with eigenvalue Next, since TrAA=TrAA=TrAA, the trace of the matrix AA is real, and since we know two of the eigenvalues to be real, the third eigenvalue must also be real If we expand the perturbation in eigenmodes of AA, wesee from Eq 5 that t, the amplitude of an eigenmode with eigenvalue, will evolve according to the ODE, d dt = + 5 If the eigenmode is stable For the solution for t is oscillatory For = we would have t = 0t+0 For the stability problem under investigation here we do not envision perturbations with a rate of change in the coordinates, only with a displacement at time t=0, ie, 0=0 The eigenmodes found so far, corresponding to eigenvalues 0 and, thus, are stable and neutrally stable, respectively, to linear order The issue is whether the third eigenvalue, which we now designate as, is larger or smaller than We have TrAA= + Hence, if we calculate this trace, we have determined the last eigenvalue, which, in turn, is decisive insofar as linear stability of the relative equilibrium is concerned Stability results aside, the perturbation,, changes the linear impulse of the configuration unless =0 The perturbation z 0 changes the angular impulse unless it happens to vanish for the relative equilibrium under consideration In a stability calculation we would typically avoid perturbations that change the integrals of motion At this point the details of the calculation depend on which relative equilibrium we are considering In fact, for the collinear relative equilibria we shall be able to work directly with matrix A rather than AA Thus, we treat the stability of the equilateral triangle relative equilibria and the collinear equilibria in separate sections VI LINEAR STABILITY OF THE EQUILATERAL TRIANGLE RELATIVE EQUILIBRIA With vortices,, and in counterclockwise order, we get z 0 z 0 =s, z 0 z 0 =s, and z 0 z 0 = s, where =e i/ Thus, A = + + s 5 + A straightforward, if slightly tedious, calculation gives AA = s , with and as the two symmetric functions of the vortex circulations introduced previously The trace of this matrix is TrAA = + s + + = s 55 + = s, or = s 56 We see that if 0, that = if =0, and that if 0 The equilateral triangle relative equilibrium, then, is linearly stable if = + + 0, marginally stable if =0, and unstable if 0 The results of the graphical analysis of the three-vortex problem,4 show that this conclusion from linearized stability holds for finite amplitude perturbations as well For 0 the triangle will pulsate around the equilibrium configuration when perturbed with angular frequency In the last step we have used the result that any polynomial that is symmetric in its arguments, here,,, can be expressed in terms of the symmetric polynomials,, and For this case = /s, Eq b Hence, the expression for the trace gives the third eigenvalue of AA, = s, 57 since from Eq 5 the eigenmode evolves as e t =e it For = = = this frequency is =/s = The observation that the pulsation frequency and the rotation frequency coincide for an equilateral triangle of identical vortices is due to Thomson In general, in the stable case, On: Wed, 0 Nov 0 0:08:5

14 0940- Stability of relative equilibria of three vortices Phys Fluids, = = + +, 58 so the frequency of oscillation is always smaller than the frequency of rotation with equality only in the case of identical vortices This elegant form of the result was given in Ref 7 The eigenvector corresponding to eigenvalue is found to be /,/,/ after a short calculation, orin general terms, z 0 0 z, z 0 0 z, z 0 0 z 59 If the sum of the circulations vanishes, the equilateral triangle translates z 0 t=z 0 +Vt with V given by Eq 5 We consider perturbations of the form z t=z 0 +Vt+ t The analysis follows that already given but with =0 Hence, eigenvalues 0 now a double root of the characteristic polynomial and /s are obtained From =0 and Eq 0, however, we see that is negative and the translating equilateral triangle of vortices is always unstable This is in accord with the detailed solution for this case,4 We now turn to the calculation that is the main motivation for this paper VII LINEAR STABILITY OF THE COLLINEAR RELATIVE EQUILIBRIA In this case A is a real matrix, = b Using this expression in the result for +, Eq 6a, we arrive at a similar expression for, = c As a complement to Eq 6a we have = d a result that will be put to good use shortly Although we shall focus exclusively on the eigenvalue, Eq 6c, in what follows, we note that the eigenvector corresponding to this eigenvalue is in all cases given by a similar expression to Eq 59 for the equilateral triangle relative equilibrium, viz,,, 6 A = A = The eigenvector z 0 =x,x,x of AA, which we shall call x in this case, is now real, so the general relation Az 0 =z 0 becomes Ax=x, and x is already an eigenvector of A with eigenvalue The vector,, is, of course, still an eigenvector of A with eigenvalue 0 If we let denote the third eigenvalue now of A not of AA we have from the coefficient of the quadratic term in the characteristic polynomial of A, or simply from setting the trace of the matrix to the sum of the eigenvalues, that This vector consists of displacements of the vortices along the line of the relative equilibrium Further, AA=A and our general criterion for linear instability is that A has an eigenvalue larger than The eigenvalues of A are 0,, and, where is given by Eq 6c We have linear stability so long as In practice, we shall first determine the sign of for a given collinear relative equilibrium If is positive, we then consider the condition for absence of linear instability in the form Similarly, if is negative, the condition to be checked is Let us use Eqs 6b and 6c to determine the stability of collinear relative equilibria in the special case treated in Sec IV C,,, =,,r For r=0 this corresponds to a passive particle in the field of two identical vortices With the circulations,, =,,r, the polynomial pz simplifies considerably, Tr A = = + 6a We now recall the previously derived expression for, Eq 44, pz =z +z +rz +r In several of the following formulas we have, for simplicity, omitted factors of or, equivalently, used units where = We have restored in formulas where physical units seem essential, such as angular frequency of rotation and the like As we saw in Sec IV C, pz has the roots On: Wed, 0 Nov 0 0:08:5

15 Hassan Aref Phys Fluids, z 0 p = + 5+4r, z p = 5+4r, z p = Consider first the solution corresponding to the root z p of pz which exists for all values of r Vortex is at the origin, x =0; vortices and at equal distances from it, x = x From Sec IV C or Eq 6b we have the angular frequency of rotation =+r s u Hence, for r we have counterclockwise rotation For r= we have a stationary equilibrium For r the configuration rotates clockwise In particular, for r =, when the sum of the circulations vanishes, 0 From Eq 6c we get =4+r For r because of our convention we are not interested in r, we have and both positive, the condition for linear stability is, and the configuration is unstable since For r=0 vortex degenerates to a passive particle The instability is clear in this case from considering the flow induced by the two vortices which has a saddle point at the origin For r= the equilibrium is stationary, =0Itisunstable since 0, whereas the stability criterion has degenerated to the requirement =0 For 5 4 r we have 0, the condition for linear stability is, and the configuration is unstable since We have marginal stability for r= 5 4 since = For r 5 4 we still have 0, the condition for linear stability is still, but the configuration is linearly stable since now satisfies the criterion One may think of the strong, negative central vortex as the stabilizing agent We see that a vortex of absolute strength 5 4 is sufficient to accomplish the task of stabilizing the configuration A fortiori we have linear stability for r= Consider next the solutions corresponding to z p 0, These are of interest only for 5 4 r For r=, z p 0 z p takes on the disallowed value 0 so vortices and coincide and the angular frequency must diverge For r= 5 4 these two roots coincide with z p, which becomes a triple root The angular frequency of rotation, n, n=0,, as determined in Sec IV C or from Eq 6b, is the same for both roots, 0, = +r +r From Eq 6c we see that for these configurations 0, is independent of r, 0, = The stability results follow For r, we have 0 0,, and the condition for stability is 0, 0, 0, The inequality 0, 0, is clearly satisfied To satisfy FIG Schematic of the n=0 collinear relative equilibrium solution for,, =,,r The positions along the common vertical line are shown as a function of r Solid dots correspond to positive vortices, open dots to negative vortices The radii of the dots scale with the magnitude of the circulation Clockwise and counterclockwise rotation are indicated with the dividing line at r= Stability and instability are indicated by s and u, respectively, with the dividing line at r= Note the singularity at r= two vortices occupy the same position and that vortex has become a passive particle at r=0 The vortex positions are plotted from Eq 45 0, 0,, we require r Thus, the collinear relative equilibria corresponding to the roots z p 0, are linearly unstable for r This includes the case r=0 where vortex has degenerated to a passively advected particle We have marginal stability for r=, and these relative equilibria are stable for r For r= we have the triple root and a singularity in 0, For 5 4 r, on the other side of the singularity, 0, 0, and the condition for stability is 0, 0, 0, The inequality 0, 0, is clearly satisfied The inequality 0, 0, again requires r Hence, the collinear relative equilibria corresponding to the roots z p 0, are linearly stable also for 5 4 r, ie, we have linear stability for the entire interval 5 4 r In summary, the solution for n= has a fixed geometry with vortex at the midpoint of the two positive, identical vortices and This configuration is unstable everywhere within the cusped curve, ie, for 5 4 r It reverses sense of rotation, from counterclockwise to clockwise, as it passes through the stationary equilibrium for r= This solution continues as the sole real root of pz outside the cusped curve, ie, for r 5 4 It is marginally stable for r= 5 4 and becomes stable as one crosses the cusped curve The configuration retains its clockwise sense of rotation for all r 5 4 The solutions for n=0,, on the other hand, change from unstable for r to stable for r but their counterclockwise sense of rotation is unchanged They become marginally stable for r= They become unphysical for r= When they are reborn for 5 4 r, they have reversed their sense of rotation but remain stable At r= 5 4 they merge smoothly with the solution for n= and disappear We summarize this analysis in Fig, which shows a schematic of the configurations for n=0 as a function of r The configurations for n= produce a similar diagram For On: Wed, 0 Nov 0 0:08:5 r