Self-Similar Collapse of n Point Vortices

J Nonlinear Sci DOI 10.1007/s00332-014-9207-8

Self-Similar Collapse of n Point Vortices Henryk Kudela

Received: 3 July 2013 / Accepted: 27 March 2014 © Springer Science+Business Media New York 2014

Abstract A system of a finite number of linear vortices in self-similar motion and under suitable conditions that are related to the initial positions and circulations of the vortices can collapse to the point with finite time. It is shown how the initial positions that lead to the collapse can be found numerically. An explicit solution for the selfsimilar collapse of the trajectories is derived. Examples of a collapsing system of 3, 7, and 30 vortices are given. A description is given of how to obtain the curves of collapsing positions of vortices if one particular system of vortices has already been found. Examples of such curves are given. Keywords

Point vortex · Collapse · Inviscid fluid

Mathematics Subject Classification

76B47 · 76B04 · 37B04

1 Introduction The study of the dynamics of point vortices on the plane has been a very active research field for a long time. The equations of motion for a point vortex system [Eq. (1) below] that we will use today appeared for the first time in the paper by Helmholtz (1867). In Kirchhoff (1876), it was shown that the motion of a system of vortices could

Communicated by Paul Newton. Electronic supplementary material The online version of this article (doi:10.1007/s00332-014-9207-8) contains supplementary material, which is available to authorized users. H. Kudela (B) Wrocaw University of Technology, Wrocław, Poland e-mail: [email protected]

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be expressed as a Hamiltonian system. In 1877, Gröbli in his dissertation studied analytically the problem of the motion of three vortices. He was able to show that three vortices could collapse to a point in finite time (for the historical background see Aref 1983, 1988; Aref et al. 1992). A three-vortex system possesses three independent integrals having commuting Poisson brackets. In such a case, based on Liouville’s theorem, a Hamiltonian system can be explicitly reduced to the quadratures (Kozlov 2003). But it must be said that reducing the Hamiltonian and computing analytically the motion of the three vortices is not trivial task (Aref 1979). Unfortunately, Gröbli’s work was completely forgotten. The analytical results of Gröbli were rediscovered by Novikov for three identical vortices (Novikov 1975; Novikov and Sedov Yu 1979) and generalized by Aref (1979). The collapse of vortices belongs to one of the most interesting problems related to the dynamics of vortices. The problem represents the great interest shown in the theoretical investigation of two-dimensional turbulence as a model for the aggregation of vorticity. Due to the fact that the distance between vortices changes during motion and brings the system to a different length scale, a collapse can be considered an elementary act in two-dimensional turbulence kinetics (Leoncini et al. 2000; Novikov 1980; Aref 1983). A collapsing system of vortices can be one of the scenarios related to the loss of the uniqueness of Euler equations in three dimensions. We must remember that point vortices are represented by a set of three-dimensional rectilinear vortex lines. The collapse solutions of vortices in two dimensions become collapse solutions in three-dimensional one, despite the fact that vorticity is extended to infinity. This paper describes how to find numerically the positions of n-vortices (n ≥ 3) that, being in self-similar motion, converge to a point in finite time. To the author’s best knowledge, this is the first time such a demonstration has been made.

2 Equations of Motion We start by recalling the equations of motion of a vortex system. Let us assume that there are n-point vortices on the plane with distinct positions z = (z 1 , z 2 , . . . , z n ) ∈ Cn , z k = xk + i yk and circulations (intensities) 1 , 2 , . . . , n , with each  j ∈ R \ 0. The differential equations that describe the motion of n-point vortices are (Newton 2001; Kochin et al. 1965) dz k (t) 1 i  = vk (z(t)) = j , dt 2π zk − z j n

(1)

j=1

where the prime on the summation indicates an omission of the term with j = k. The following fundamental identities are associated with system (1) (Kochin et al. 1965; O’Neil 1987): n  k=1

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k vk = 0,

(2)

J Nonlinear Sci n  k n 

k z k vk =

i  k  j , 2π

(3)

d  k  j ln (z k − z j ). dt 2πi

(4)

k> j

k vk v k =

k

k> j

Using these identities, it is not hard to prove the following integrals of motion (Kochin et al. 1965; Hernandez-Gaduno and Lacomba 2004, 2006; Newton 2001): n n 1   H =−  j k ln r jk = const., where r jk = |z j − z k |, 2π j k, k= j n 

M = Mx + i M y = V = S=

n  k n 

 k

n 

jxj + i

j=1

dyk dxk xk − yk dt dt

 j y j = const.,

(5)

(6)

j=1



=

1  k  j , = const., 2π

(7)

k> j

 j (x 2j + y 2j ) = const.

(8)

j=1

Invariant (5) is expressed through the mutual distance between the vortices and can be interpreted as an inner interaction energy of the vortices. From Kirchhoff (1876) we know that system (1) can be expressed, using (5), in the Hamiltonian formulation (Newton 2001; Borisov and Mamaev 2005) k

∂H dxk = , dt ∂ yk

k

∂H dyk =− . dt ∂ xk

(9)

Invariant (6) is called a linear impulse, and (8) is called an angular impulse (Aref 1979). It is possible to derive one more invariant as a linear combination of (6) and (7) (Newton 2001; Kimura 1987): L=



k  j rk2j = σ S − (Mx2 + M y2 ) = const.,

(10)

k> j

where σ =



k = const.

(11)

k

The invariant L plays a certain role in the study of the self-similar collapse of vortices (Aref 1979; Kimura 1987; Novikov and Sedov Yu 1979; Borisov and Mamaev 2005). The L is expressed in terms of the mutual separation of the vortices r jk . It is clear that when a system collapses, every ri j → 0 and the invariant L must be equal to

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zero, L = 0 . Our interest here is to find numerically the initial positions of n vortices that collapse in finite time. As was remarked in Aref (2010), due to the invariants of motion, it seems impossible that the distance between any two vortices become less than the smallest distance between any pair of vortices that was initially given. But as was explored for n = 3 in Aref (1979, 2010), Novikov and Sedov Yu (1979), and Kimura (1987), three vortices that are in a self-similar motion, under some additional conditions for circulation and the initial positions of the vortices, can be collided in the center of the vorticity in finite time. Here it will be numerically demonstrated that such a collapse is possible for any n > 3. 3 Self-Similar Motion of n Vortices on the Plane The dynamic of the n-vortices system is determined by the collection of n intensities k , k = 1, . . . , n and the initial position of the vortices z k (0). It is worth noting that the transformation z = az + b, inserted into Eq. (1), gives 1 i  1 dz k (t) = j . dt a 2π zk − z j n

a

(12)

j=1

We see that this results in dividing all intensities k by |a|2 = aa. Scaling all k does not change the shape of the trajectories but only scales the time t/|a|2 (Newton 2001; Synge 1949). The shapes of trajectories are invariant to the translations, rotations, and dilatations. One can call the given configuration [z] = (z 1 , z 2 , . . . , z n ) equivalent or similar to [z] ∼ [z] if for some a, b ∈ C the system [z  ] is obtained from [z] by the transformation z i = a(z i + b) (O’Neil 1987). Assuming that σ = 0, we may shift the center of vorticity to the zero of the coordinate system xc = Mx /σ = 0,

yc = M y /σ = 0.

(13)

This means that Mx = 0 and M y = 0. We adopt the following definition of the self-similar collapsing motion of n vortices (see O’Neil 1987). Definition 1 An n-vortices system is in self-similar collapsing motion if there exists the complex function λ(t) = λr (t) + iλi (t), with λr (t) < 0 and λi (t) = 0, such that for all k we have dz k = vk (z 1 , z 2 , . . . , z n ) = λ(t)z k , dt

k = 1, 2, . . . , n.

(14)

To find the solution of (14), we introduce the new variables (r (t), ϕ(t)) and assume the following form of the solution: z k = z k (0)r (t)eiϕ(t) ,

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r (0) = 1, ϕ(0) = 0, k = 1, 2, . . . , n.

(15)

J Nonlinear Sci

Inserting (15) into (1) one obtains 

dr dϕ +i dt dt



j i    −iϕ(t) 2π re z k (0) − z j (0) n

eiϕ(t) z k (0) =

j

λ(0) 1 z k (0). = vk (z(0)) = r r

(16)

Comparing the real and imaginary parts of both sides of Eq. (16) results in the following differential equation for r (t) and φ(t): dr 1 = λr (0), dt r

r

dϕ 1 = λi (0). dt r

(17)

λi (0) ln(2λr (0) + 1). 2λr (0)

(18)

Direct integration of (17) gives r=

 2λr (0)t + 1,

ϕ=

The solution of Eq. (15) has the form  z k (t) = 2λr (0)t + 1 e

 λ (0) i 2λir (0) ln(2λr (0)t+1)

1

λi (0)

z k (0) = (2λr (0)t + 1) 2 +i 2λr (0) , k = 1, 2, . . . , n.

(19)

Solution (19) represents a logarithmic spiral that depends on the initial position of the vortices z(0) = (z 1 (0), z 2 (0), . . . , z n (0)). The collision (collapse) time is given by Tc = −

1 . 2λr (0)

(20)

It is easy to check that when t → Tc , then z k (t) → 0, r (t) → 0, and ϕ(t) → +∞ when λi (0) > 0 or ϕ(t) → −∞ when λi (0) < 0. From (14), it results that for t = 0 we have vk (z(0)) = λ(0)z k (0),

k = 1, 2, . . . , n.

(21)

If the positions z k (0) are found, then λ(0) can be calculated, λ(0) =

vk (0) = λr (0) + iλi (0), z k (0)

for k = 1, 2 . . . , n,

(22)

as well as the collision time Tc (20). If λr (0) > 0, then the vortex system expands. To change the direction of motion, one should change the sign of the circulations of all vortices to the opposite one (Newton 2001; Synge 1949). When the real part of λ(t) equals zero, λr = 0, then the vortices are in relative equilibrium (O’Neil 1987; Palmore 1982; Kimura 1987; Barreiro et al. 2012). The whole system rotates as a solid body, and Tc = ∞.

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Applying the solution z k = z k (0)r (t)eϕ(t) to the Hamiltonian (5) we have ⎞ n  1  j k ⎠ ln |r (t)|. H (t) = H (0) − ⎝ 2π ⎛

(23)

k> j

Thus, the conservation of the Hamiltonian (5) during self-similar motion requires that V [invariant (7)] be zero (V = 0). 4 Algebraic Equation for Initial Position of Collapsing Vortices For the completeness and clarity of this paper we quote here two propositions given by O’Neil (1987). Proposition 1 If invariants M = 0 and V = 0, then also invariants S = 0 and L = 0. Proof From definition (14) one has vk − v j = λ(t)(z k − z j ). Using identity (2) one can write σ vk =

n 

 j (vk − v j ) =

n 

j=1

Next, using identity (3), vk M = vk

n

n  j=1

=

 j λ(z k − z j ) = λ(σ z k − M).

(24)

j=1

n 

j=1 v j  j z j

jz j −

n 

= 0, we have

vjjz j

j=1

 j z j (vk − v j ) =

j

n 

 j z j λ(z k − z j ) = z k M − S.

(25)

j=1

Since M = 0, then also S = 0. It is clear from (10) that L = 0 as well.



From the assumed definition (14) results v j z k = vk z j ,

k, i = 1, . . . , n k = j.

(26)

O’Neil (1987) proved that there are only n − 3 independent equations of the form (26). Proposition 2 Suppose that the invariants V = 0 and M = 0. If v1 z j − v j z 1 = 0 for n − 3 values of index j, then v1 z j − v j z 1 = 0 for every index j, j = 1, 2 . . . , n. Proof Let us assume that D j = v1 z j − v j z 1 = 0 for j = 4, 5, . . . , n. For j = 2, 3, using identity (2) we have 2 D 2 + 3 D 3 =

n  j=1

123

 j (v1 z j − v j z 1 ) = v1

n  j=1

 j z j − z1

n  j=1

 j v j = v1 M = 0.

(27)

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Next, using (3) one can write  2 z 2 D 2 + 3 z 3 D 3 =

n 

 j z j (v1 z j − v j z 1 )

j=1

= v1

n 

 j z j z j − z1

j=1

n 

 j z j v j = v1 S − z 1 i V = 0. (28)

j=1

We obtain the system of equations 

2  3 z 2 2 3 z 3



   D2 0 = . D3 0

(29)

The determinant of algebraic system (29) is different from zero because it is assumed that z i = z j for every i, j = 1, . . . , n. This means that D2 = 0 and D3 = 0. This completes the proof.

From what it was said above follows that the collapsing configuration of vortices must be common zeros of the real and the imaginary parts of the complex functions f k = v1 z k − vk z 1 , k = 1, n − 3 and Mx = 0, M y = 0, S = 0. The number of equations is equal to 2n − 3. The configuration space of n vortices needs 2n real numbers. To close the algebraic system we need three additional equations. To reduce the number of unknowns it was fixed one of the vortex position coordinates, z n = (xn , yn ) hoping that the rest of the vortices arrange themselves to fulfill conditions for a collapse.  Additionally we include in the system a real part of the n i vi ) = 0. This identity provides no information identity (2) f n−2 = Re( i=1 about the structure of the collapsing position of the vortices but it should still be true for any vortex system. The following nonlinear system of algebraic equations arises: f 1 = Re[v1 z 2 ] − Re[v2 z 1 ] = 0,

(30)

f 2 = I m[v1 z 2 ] − I m[v2 z 1 ] = 0, ..........................................

(31)

f n−6 = I m[v1 z n−3 ] − I m[vn−3 z 1 ] = 0, f 2n−5 = Mx = 0,

(32) (33)

f 2n−4 = M y = 0, f 2n−3 = S = 0,  n   f 2n−2 = Re i vi = 0.

(34) (35) (36)

i=1

Thus, the nonlinear system of 2n − 2 equations, with a fixed position of the last vortex z n = (xn , yn ), was solved using the Newton method. To obtain a numerical solution of an algebraic nonlinear system of equations, one should make a good guess

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regarding the initial solution (starting point). An unconstrained optimization procedure with the Levenberg–Marquart algorithm was used to find the minimum of the function

F(x1 , y1 , . . . , xn−1 , yn−1 ) =

2n−2 

f i2 .

(37)

i=1

After some trials with the distribution of x1 , y1 , . . . , xn−1 , yn−1 , one should be able to minimize the function F(x1 , y1 , . . . , xn−1 , yn−1 ) ∼ 10−20 and use these points as the starting points for the Newton procedure applied to the system (30)–(36). The convergence of the Newton procedure should be very fast because when the minimum of F is small, the points z j are close to the solution of the system (30)–(36). Next, one should check whether z i = z j for i = j and z j = 0, j = 1, 2, . . . , n. The λ(0) (22) should have the same value for each i = 1, . . . , n. This assures that collapsing time (20) will be the same for all vortices. The invariants Mx , M y , and S should equal zero with the assumed numerical precision. In seeking the particular positions of collapsing vortices it is natural to ask the following question: do other collapsing systems of vortices exist around these particular positions? To investigate the problem, we use the fact that the last equation, Eq. (36), in the algebraic system (30)–(36) was used only to close it. Using the solution of the system (30)–(36) we are able to calculate the Hamiltonian H0 , which depends on the distance between vortices, ri j = |z i − z j |, i > j. Now perturbing slightly the value of the Hamiltonian H (0) and replacing the last Eq. (36) by H = H0 + H = const., where H is a small number, and using as initial starting points the old positions z j, j = z j , j = 1, 2, . . . , n, it is possible to calculate the new collapsing positions  1, 2, . . . , n. Doing this repeatedly and enlarging and decreasing the Hamiltonian, one can obtain the sequence of the collapsing positions of vortices. Joining the positions of each vortex when the Hamiltonian is changed in some interval [H1 , H2 ] one obtains n different curves. Points with the same Hamiltonian form a system of collapsing vortices. It turns out that such a procedure sometimes makes it possible to determine the positions of vortices that are very close to relative equilibrium (the collapse time Tc takes a large value ∼ 104 ). All calculations are done using Mathematica v.9.01. Mathematica has the desired procedures: FindMinium to minimize function (37), FindRoot to solve the nonlinear system of Eqs. (30)–(36), and NDSolve to solve the system of differential Eqs. (1). It is very important to carry the calculations with high enough digital precision. In Mathematica the digital precision can be fixed using a parameter that is called a “WorkingPrecision”. Sending to the procedures the parameter “W orkig Pr ecision → w” cause that the numerical calculations are done using the numbers with w-digit precision (Ruskeepää 2009). All numerical values in this paper were written with precision of the six decimal digits.

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5 Numerical Examples 5.1 Collapse of Three Vortices To clarify our numerical procedure and to check the numerical algorithm, we start from the well-known problem concerning the collapse of three vortices (Aref 2010; Kimura 1987). If one of the vortex positions is fixed, then it is assumed that (x3 , y3 ) = (1, 0), and the collapsing system of three vortices is determined only by S = 0, Mx = 0, and M y = 0 and, additionally, by the real part of identity (2). The circulations of the vortices are  = (1 , 2 , 3 ) = (−2, −2, 1). The collapsing trajectories are presented in Fig. 1. They were obtained using formula (19). The trajectories were also found using the solution of differential Eq. (1) using NDSolve. The difference between them is less than 10−13 . The critical time is Tc = 2.739533, H (0) = −0.0381743, and WorkingPrecision → 32 is chosen. The minimum of the function F(x1 , y1 , x2 , y2 ) is ∼ 10−43 , the center of vorticity (xc , yc ) and angular impulse S are ∼ 10−33 . Now, with H (0) the last Eq. (36) in system (30)–(36) is replaced by H = H (0) ± H , where H = 0.0001 and the system of algebraic equations is solved again using as the starting solution the old collapsing positions. This made it possible to find the new set of collapsing vortices. Repeating this procedure, increasing and decreasing of Hamiltonian, and joining the points z j that are the solution of (30)–(36), curves we call collapsing position curves are obtained. The points on the curves have different Hamiltonian values. The set of points with the same Hamiltonian values make up the collapsing set of vortices. The numerical results are presented in Fig. 2. When the Hamiltonian H is decreased to Hc2 ≈ −0.220636, the vortices 1 and 2 are moved to location 1 and 2 . Vortices 1 , 2 , and 3 lie, with a precision of ∼ ±10−5 , on a straight line and are very close to relative equilibrium. Their collapsing time is ∼ 104 . An increase in Hamiltonian H (0) to H ≈ 10−8 causes the displacement of vortices 1 and 2 to positions 1 and 2 . With a numerical precision of ∼ ±10−4 , vortices 1 , 2 , and 3 form an equilateral triangle. One can check that Hamiltonian for an equilateral triangle when V = 0 is exactly equal to zero. The only relative equilibrium for the three-vortex

Fig. 1 Collapsing trajectories of three vortices: (1 , 2 , 3 ) = (−2, −2, 1), H (0) = −0.0381743, Tc = 2.73953. The fixed third vortex had coordinates (x3 , y3 ) = (1, 0)

2 0.2

3

0.0 0.2 0.4

1

0.6 0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

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J Nonlinear Sci Fig. 2 Curves of the collapsing positions; the continuous line 1 , 1 and 2 , 2 . Hamiltonian is changed from H1 = −0.2206356, that corresponds to the vortices position (1 , 2 , 3), to H2 ≈ 10−8 that relates to the points (1 , 2 , 3). Vortex 3, has a fixed position, (x3 , y3 ) = (1, 0)

Fig. 3 Dependence of collapse time on Hamiltonian values

0.6

2’’

0.4

2

0.2 0.0

1’

2’

3

0.2 0.4

1 1’’

0.6 0.5

0.0

0.5

1.0

TCr 1000 500 100 50 10 5 1

Fig. 4 Trajectories for shortest collapsing time; Tc = 1.570796, H = −0.145840

0.20

0.15

0.10

0.2

H 0.00

0.05

2

3

0.0 0.2

1

0.4 0.6 0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

system are a straight line (positions 1 , 2 , and 3) and the equilateral triangle (vortices 1 , 2 , 3) (Aref 2010; Newton 2001; Hernandez-Gaduno and Lacomba 2004). In Fig. 3 the dependence of the collapse time on the Hamiltonian value is presented. Close to relative equilibrium Tc (H ) is very steep, and approaching that point Hamiltonian increments of H ∼ 10−7 are used. The shortest time is Tcmin = 1.570796 and H (Tcmin ) = −0.145840, and trajectories for that minimal collapsing time Tcmin are presented in Fig. 4. Choosing the collapsing points near the relative equilibrium one obtains the trajectories with many turns around of the vortex center (Fig. 5).

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J Nonlinear Sci Fig. 5 Collapsing trajectories of three vortices for Hamiltonian value H = −0.0004413 and collapse time Tc = 25.827932

0.5

2 3

0.0

1

0.5

0.5

Fig. 6 Distance of vortices ri (t) = |z i (t)|, i = 1, 2, 3 from singular point (0, 0). The collapse time and Hamiltonian value are Tc = 25.827932 and H = −0.000441. The numbers 1, 2, 3 related to the numbers of the vortices presented in Fig. 5

0.0

0.5

1.0

r 1.0 0.8

3

0.6

2 0.4

1

0.2 0.0 0

5

10

15

20

25

t

To convince oneself that vortices approach the singular point (0, 0) the distance of vortices ri (t) = |z i (t)| to singular point (0, 0) can be plotted (Fig. 6). All vortices should reach the singular point in the same time and ri (t) should not cross each other. From Fig. 6 one can see that vortices accelerate approaching the singular point and the acceleration r¨ grows to the −∞. 5.2 Collapse of Seven Vortices Figure 7 presents the collapse trajectories for seven vortices. The collapse time is Tc = 2.710570 and the Hamiltonian has a value of H (0) = −0.544423. Replacing Eq. (36) by the perturbed Hamiltonian H = H (0) ± H and repeating the procedure described in Sect. 5.1 produces the collapsing position curves (Fig. 8). The Hamiltonian is changed from H1 = −0.4394600 (points with superscript  ) to H2 = −0.588194 (points with superscript  ). Both collections of points, those with the superscript  and those with  , are close to relative equilibrium. Crosses on the curves indicate the locations of the vortices used in Fig. 7. The curves are irregular and quickly change directions and slopes. Graph Tc (H ) is presented in Fig. 9. The cross on the curve indicates the Hamiltonian value of H and Tc for the solution presented in Fig. 7.

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J Nonlinear Sci Fig. 7 Collapse trajectories of seven vortices. The Hamiltonian and the collapse time have values of H = −0.544423 and Tc = 2.710570, respectively. The intensities of the vortices i are  = (1, 1, −2, −2, −2, 3/2, −2)

1.5

2

1.0

5

0.5

7 0.0

4

1

0.5

6 3

1.0 1.0

0.5

0.0

0.5

1.0

1.5

2’’ 1.0

5’’ 0.5

2’

4’’

5’ 2

6’’

1

7

4’ 3’’

2

0.5

1’

1.0

6’

1’’

1

3’ 1.5

Fig. 8 Curves of collapsing positions. Hamiltonian was changed from H1 = −0.439461 (points with superscript  ) to H2 = −0.588194 (points with  ). The crosses on the curves marked the collapsing position of vortices from

Fig. 9 Dependence of the collapse time on Hamiltonian value. The cross on the graph indicates (H, Tc ) for the solution presented on the Fig. 7

TCr 1000 500 100 50 10 5 1 0.60

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0.55

0.50

0.45

H

J Nonlinear Sci Fig. 10 Trajectories of collapse system of seven vortices with circulation opposite to that in Fig. 7,  = (−1, −1, 2, 2, 2, −3/2, 2). The Hamiltonian: H = −0.324734; collapse time: Tc = 2.460261

6 1

7 0.5

4 3 0 0.6

0.22

2

0 0.2

0.6

0.5

5 1

1 Fig. 11 Curves of the collapsing positions. The Hamiltonian values are changed from H1 = −0.27223407 (the points with the superscript  ) to H2 = −0.324634 (the points with the superscript  ). The collapse time for H1 = −0.27223407 is Tc = 30540 and for H2 = −0.324634 it is Tc = 2.460261

1.5

6’

6’’ 1.0

7 0.5

4’ 3’ 3’’ 1.5

1.0

4’’

0.5

0.5

2’’

1.0

1.5

2’ 0.5

5’’

5’

1.0

1’’ 1.5

1’

Sometimes one finds positions with a negative collapse time. This means that the vortices will expand, and λr > 0 in (14). In such cases, it is enough to change the sign of the intensities i to the opposite one to obtain the collapse of vortices. The solution for such cases is presented in Fig. 10. The curves of the collapsing vortices, when the Hamiltonian was changed from H1 = −0.272234 (the configurations with the superscript  ) to H2 = −0.324634 (the configurations with the superscript  ) were shown in Fig. 11. It may be noticed some order and symmetry in the positions of the vortices. The vortices 1 , 2 , 4 , 5 and 6 lie

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J Nonlinear Sci Fig. 12 Dependence of collapse time on Hamiltonian value for seven-vortex system and with a circulation of vortices as in Fig. 10

TC 104 1000

100

10

1 0.33

0.32

0.31

0.30

0.29

0.28

H 0.27

on the straight line, the points 3 and 7 had the same distance form that straight line. The fixed point 7 has the coordinates (0.6, 0.72). The points with superscript  are near to the relative equilibria (the collapse time there was Tc 30540.). For the points with superscript  the collapse time was equal to Tc = 2.460261. In this case the collapsing position curves did not end on the relative equilibrium, (see Fig. 12). It was impossible to continue the calculations because xc , yc and S starts to deviate from zero suddenly. 5.3 Collapse of 30 Vortices The last example relates to a system of 30 vortices. Five different values for the intensities i are used: 1 to 6 are equal to√−1 , 7 to 12 are equal to 588/625(−1+ √ 11),  to 18 are equal to 84/125(1 − √ 11), 19 to 24 are equal to 12/25(−1 + 13 √ 11), and  to  are equal to 1/5(6 − 11). In addition, σ = 2058/625(−1 + 30 25 √ 11) ≈ 7.62818, and WorkingPrecision→ 400 was chosen. The minimum of the function (37) was obtained, F ∼ = 10−740 . The values of Mx , M y , and S are equal to zero −400 . The solutions are presented in Fig. 13. The collapse time is Tc = 7.3839299, 10 the Hamiltonian has a value of H = −3.72977147, and the position of the fixed vortex used in the solution of system (30)–(36) is indicated in Fig. 13 by the dashed-line circle, x30 , y30 = (11/10, 11/10). Figure 14 presents a close-up graph of curves ri (t) = |z i (t)| in the time interval [7.38, T c], Tc = 7.383930. The graph clearly shows that the vortices are accelerating as they move to the singular point, and the acceleration reaches minus infinity at (0, 0). The curves of the collapsing positions of the vortices, parameterized by Hamiltonian values, are presented in Fig. 15. The Hamiltonian changes from H1 = −3.334771 (numbered thick points in Fig. 15) to H2 = −3.7377714 with steps H = 0.0005. The thick numbered points in Fig. 15 are close to the relative equilibrium positions (the numerically obtained collapse time is Tc 5.2 × 103 ), H1 = −3.334771. Along the curves the invariants Mx , M y , and S are ∼ 10−400 . Similar to the case of Fig. 11, it is impossible to continue the calculations for a Hamiltonian value lower than H2 because suddenly xc , yc , and S started to deviate from zero. The dependence of the collapse time on the Hamiltonian values Tc (H ) is shown in Fig. 16. The time value for the right side of the graph is Tc ∼ 5200 and decreased to time Tc = 7.383.

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2

1

0

1

2

3

2

1

0

1

2

3

4

Fig. 13 Trajectories of collapse system of 30 vortices. Vortex circulations: (1 to 6) = −1, (6 to 12) = √ √ √ 588/625(−1 + 11), (13 to 18) = 84/125(1 − 11), (19 to 24) = 12/25(−1 + 11), and (25 to 30) = √ 1/5(6 − 11), H = −3.72977147, Tc = 7.383930

r 0.05 0.04 0.03 0.02 0.01 0. 7.38

7.382

7.384

t

Fig. 14 Distance ri = |z i (t)|, i = 1, . . . , 30 of the vortices from the singular point (0, 0), in interval [7.38, Tc ]. The collapse time Tc = 7.3839299 is indicated by the vertical line on the right end of the graph

The instantaneous portraits of the trajectories of the 30 vortices after the times t = 5, t = 25, t = 100, and t = 200, at a collapse time of Tc = 5197.80 and H1 = −3.334771, are presented in Fig. 17. In this case, the collapse time is rather large, and the tightly packed trajectories make many turns around the center of the vortices before reaching the singular point (0.0). The instantaneous distribution of the streamlines generated by the collapsing vortices may be obtained by solving the Poisson equation for stream function ψ and vorticity ω. The vorticity distribution of the point vortex system is given by the sum of the delta Dirac measures (Newton 2001):

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4

3 9

18 2

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30 7 21 2 1 29 28 10 15

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12 14

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11 19 17

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4 Fig. 15 Curves of collapsing positions of vortices parameterized by Hamiltonian values in interval [−3.334771, −3.7377714]. Numbered thick points: vortices very close to relative equilibrium H1 = −3.334771, Tc ∼ = 5197.80. Fixed vortex indicated by circle, (x30 , y30 ) = (1.1, 1.1) Fig. 16 Dependence of collapse time on Hamiltonian values for data as in Fig. 13

TC

1000

100

10

1

ω(x, y) =

n 

3.7

3.6

 j δ(x − x j )δ(y − y j ).

3.5

3.4

H 3.3

(38)

j=1

The relations between the stream function ψ(x, y), vorticity ω, and velocities (u, v) are ψ(x, y) = −ω(x, y), ∂ψ , v = − ∂ψ u= ∂x , ∂y

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t

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Fig. 17 Instantaneous portraits of trajectories of 30 vortices after times t = 5, t = 25, t = 100, and t = 200. Collapse time – Tc ∼ = 5197.80. Thick points: initial positions of vortices; blue: negative circulation, red: positive circulation (Color figure online)

where = ∂∂x 2 + ∂∂y 2 . The stream function ψ, due to the linearity of Eq. (39), is equal to the sum of the stream functions ψ j of each vortex, 2

ψ(x, y) =

n  j=1

2

ψ j (x, y),

where ψ j (x, y) = −

j  ln (x − x j )2 + (y − y j )2 , 2π (41)

The instantaneous pictures of the streamlines for t = 0, t = 3, t = 6, and t = 7.38 are presented in Fig. 18. The red thick points indicate the positions of the positive vortices, the blue points the negative vortices. The square in the middle of the frames indicate the vorticity center (0, 0). To visualize the dynamics of the changes in the distribution of the streamlines, the streamline ψ = 0 in Fig. 18) is drawn as a thick line. The dark area around the vortices indicate the area where the negative gage pressure is smaller than −0.75. At the center of the vortex the gage pressure p goes to −∞.

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Fig. 18 Instantaneous pictures of streamlines of collapsing vortices for t = 0, t = 3, t = 6, and t = 7.38. Thick streamline line: streamline of zero value, ψ = 0. Collapse time – Tc = 7.38393; gray: negative gage pressure less than −0.75 (Color figure online)

The gage pressure was calculated using Bernoulli’s law:

p=−

u 2 + v2 . 2

(42)

As one can see, the smudges around the vortices and the line ψ = 0 are irregular, which is indicated by the strong interactions between the vortices. As the vortices converge to the singular point (0, 0), the smudges of the negative pressure join, but in the frame T = 6 one can still see the white area around the vorticity center that is surrounded by the vortices. At the frame T = 7.38 (collapse time: Tc = 7.3839299) the low-pressure smudge creates a regular disc. The streamlines become regular circles as if produced by a single vortex.

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Because the invariant V = 0 and σ = 0, one can write  k> j

⎛ k  j = ⎝

n  j=1

⎞2 j⎠ −

n  ( j )2 = 0

(43)

j=1

This means that the collapsed vortices do not annihilate their intensities in the center and the resulting vortex has an absolute value larger than that of any of the vortices n 2 that took part in the collapse  R = ± j=1  j , (Novikov and Sedov Yu 1979). 6 Concluding Remarks It was demonstrated numerically that the system of finite number of linear vortices being in the self-similar motion, under suitable initial conditions can collapse to the point with finite time. After shifting the center of vorticity to the zero of the coordinate system and fixing the vortex position (xn , yn ), the collapse took place when the circulation of vortices and their initial positions satisfied: V = 0, Mx = 0, M y = 0, S = 0 (four equations) and 2(n − 3) equations for the real and imaginary parts n v j z k = vk z j . i vi ) = 0. To close the system for 2n equations it was also included identity Re( i=1 The collapse of the vortices to the point in the finite time changes the topology of the flow field and may play an important role in topologically driven transitions where vortices are involved, Leoncini et al. (2000). The collapsing or expanding phenomena can be regeared as an aggregation of vortices into one vortex or as the splitting of one vortex into n vortices. It seems that the collapsing-vortex phenomenon may have relevance in the meteorological phenomenon of polar cyclone (arctic vortex) evolution (http://en.wikipedia. org/wiki/Cyclone). As described at http://en.wikipedia.org/wiki/Cyclone, a polar cyclone is a low-pressure weather system that usually has a horizontal length scale on the order of ∼1,000 km. Meteorological satellite imagery reveals many small-scale cloud vortices. A comparison of the spirals of collapsing trajectories of vortices with a photo of a polar cyclone taken by satellite (NASA/GSFC 2012) reveals an encouraging similarity. Figure 19 shows an enlarged portrait of trajectories of seven collapsing

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vortices presented in Fig. 10 that are putted on the satellite picture (NASA/GSFC 2012). The trajectories seem to fit well to the spiral arms of the cyclone. Of course, confirmation of the collapsing phenomenon in the dynamics of a cyclone requires more observation and verification. Spiral structures are natural candidates for the role of so-called generic structures of turbulent flow and have been used many times in the study of turbulence (see Moffat 1993; Catracis 2011 and references therein). The self-similar collapse of vortices that leads to the logarithmic spiral solution (19) may turn out to be useful tools for the study of turbulent flow. The collapsing-vortices problem relates to a widely discussed problem of loss uniqueness of Euler’s equation for inviscid flow (Wayne 2011). It is well-known that under the regularity of the initial data the existence of the global solutions of Euler equation can be proved (Gallagher and Gally 2005; Wolibner 1933). Using the initial conditions in the form of a point vortex distribution for a vorticity, those results are not applicable. On the the other hand, it is known that point vortex approximations converge to two-dimensional Euler equations (Goodman et al. 1990). In some case, the regular solution and the singular one may not be far from each other. It was proved that the two-dimensional point-vortex model is a very useful, though simple, approximation to many important physical systems (Wayne 2011; Aref 2007a,b). It also serves as a basis for a very efficient numerical method, the so-called vortex particle method, for the solution of the Navier–Stokes equations for two-dimensional flow (Kudela and Malecha 2009). Acknowledgments The author would like to thank Prof. Grzegorz Karch from the Institute of Mathematics of Wrocaw University for the initial review of the manuscript.

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