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Periodic, aperiodic and chaotic motion of drift wave vortices in a circular region
‘3uo.i. .Solrrons & Frarmls Vol. 2. No. 6. PP. 597-607. 1992 Printed in Great Britain
0960.0779/92$5.00 + .m 0 1992 Pergamon Press Ltd
Periodic, Aperiodic and Chaotic Motion of Drift Wave Vortices in a Circular Region HIDEAKI Interdisciplinary
Graduate
School
of Engineering
SHIBAHARA Sciences.
MITSUO Research
Institute
for Applied
Mechanics, (Received
Kyushu
University,
Kasuga,
Fukuoka
816, Japan
KONO
Kyushu 31 July
University,
Kasuga,
Fukuoka
816,
Japan
1992)
Abstract-The modulated point vortex model associated with the Hasegawa-Mima equation is used to study the dynamical behavior of drift wave vortices in a bounded region. When a circle is taken as the boundary of the region, the potential for the vortices can be constructed with the aid of the circle theorem which replaces the boundary by the mirror images of the vortices inside the circle and ensures that the potential for the vortices is zero on the boundary. Periodic, aperiodic and chaotic motions are realized depending on the number of vortices and the magnitude of the value of the diamagnetic drift which breaks the rotational symmetry and has a tendency to drive vortices into chaotic motion.
1. INTRODUCTION
Vortices are rather long-lived entities as long as the viscosity is negligible, which is the case for high temperature plasmas, and in fact can be responsible for long range transport in macroscopic systems. They have been the subject of intense study, especially in connection with anomalous transport in a magnetically confined plasma where drift wave vortices are supposed to play central roles. Studies on dynamics of drift wave vortices, however, have mainly been numerically performed based on the Hasegawa-Mima equation [l], which is not easy to solve analytically. The modulated point vortex model introduced by Zabusky and McWilliams [2] first and then formulated by Kono and Horton [3] has been shown to describe characteristic features of dynamics of drift wave vortices as far as the vortex dynamics is concerned. An advantage of introducing point vortices is to convert a nonlinear partial differential equation into a system of ordinary differential equations which are easier to solve. Notwithstanding a successful description of fluid vortices in terms of the point vortex, boundary effects have to be considered because real plasma does not extend to infinity. In the present paper, therefore, we study the dynamical behavior of drift wave vortices in a bounded region. Similar numerical studies on dynamics of point vortices in a bounded region are performed based on a two-dimensional Euler equation [4]. A crucial difference of the Hasegawa-Mima equation from Euler’s equation, is the existence of the drift term giving rise to dispersive waves. Because of the diamagnetic drift the vorticity attached to each point vortex is no longer constant but varies in space and time. In Section 2 we derive the equation of point vortices in a bounded region for which a circle is taken. The potential for the vortices is constructed replacing the boundary by the mirror images of the vortices 597
HIDEAICI
598
SHIBAHARA
and MITSUO
KONO
inside the region. In Section 3 the stability of the fixed points of the equations is analyzed for the cases of a single vortex and two vortices inside the circle. In Section 4 the solutions are numerically solved to give a variety of motions. i.e. periodic, aperiodic and chaotic motion depending on the number of the vortices and the magnitude of the diamagnetic drift. The last section is devoted to discussions.
2. EQUATION
OF MOTION
The equation of motion equation is given by:
FOR POINT
VORTICES
for point vortices [2,3,5]
IN A BOUNDED
REGION
associated with the Hasegawa-Mima
dz, -=dt where zn is a complex variable related to the coordinate (.rn, ~1,) of the center of a point vortex through Z, = X, + iy, and Z, is the complex conjugate to *Il. K,(I,-0- zal) is the modified Bessel function of the second kind. The dynamical behavior of the vortices under equation (1) are shown to recover the various characteristics revealed by numerically solving the Hasegawa-Mima equation. Since the vortices propagate in pairs parallel or antiparallel to the diamagnetic direction, however, more than two vortices are finally dissolved into isolated or paired vortices no matter how complicated the intermediate state of the interactions is. This situation suggests that the dynamics of the vortices confined in a bounded region is very different from that in a space extended to infinity. Although detailed dynamics must depend on the shape of the bounded regions. we may expect to extract characteristic features common to vortex dynamics in various types of the bounded regions. If, for simplicity, we take a circle with a radius R as a bounded region. the circle theorem [6] gives the potential for the vortices inside the circle as:
On the circle (Z = RI/z), V(Z) is zero, implying the boundary is a stream line or an equipotential line. In addition, equation (2) has no singularity at z = 0. Then the equation describing the dynamics of the vortices in the circle is given by:
(3)
3. FIXED
POINTS
OF THE DYNAMICAL
EQUATIONS
For the case of one vortex in the circle, equation equation (3) becomes:
AND THEIR
(3) is integrable.
STABILITY
If z is put as pit’,
-= dp 0’ dt d0 -=dt
-R=
2rp3 (K + U*P ~0s W%(R’/p
- P>,
(4)
Vortices
in a circular
region
599
whose solution is given by: 8 tan1
A tan [~(t - r,,)] - tan (t&/2) = -
1 + (l/A)tan[~(t
A = &:
- to)]tan(8,/2)
’
;;;j.
(0 = -$
(5)
K,(R?/p
- p)V[Kz
- (u*p)2].
Note that w and A should be replaced by io and iA for (u,P)~ > 2. The trajectory is a circle for (0,~)’ < 2 and is an arc connecting the initial position pe”u with one of the fixed points pe’lH* where 8, = cos-1 (-K/U&) when (u&)~ > 2 (Fig. 1). The dynamics of two or more vortices in the circle is sensitive to their initial configurations and the magnitude of the value of u,. First we consider the case of a vortex-pair with vorticities of the same magnitude and opposite sign (K, = -K~ = K > 0). In this case, the fixed points are obtained by putting z, = olle”, for the two cases of the configurations symmetric with respect to the y-axis (the diamagnetic direction) (z, = -zz; p, = p, = p and 13, + BZ = n) and the origin (zl = -z2; p1 = pz = p and 8, - & = rr). For the first case the fixed point is given by (pr, 0,) = (p*, 0) where p* is determined by: K,Gw
+ w/P*)2w*(R2/P* + P*>- WR’/P* - P*)l = 0.
(6)
For R = 1, p* is equal to 0.4629. The stability of the fixed point can be analyzed by linearizing equation (3) around the fixed point with respect to perturbations 6p, and 68, K elm’. The eigenvalue (LI is expressed as: (IJ = $
KK,(@,),
(7)
where N is a positive constant given by: *z =
2 R’ - pii
1 + R2 - pi P*
K&P,)
+ R2(RJ - P$ Ko(R2/p,
K1(2pd
&W’/P, K,(~P*)
24
- PJ
KI(~P,)
+ PA
(8)
11 ’
Equation (7) indicates that the fixed point is stable [Fig. 2(a)]. This result is applied to the case where the initial positions deviate largely from the fixed point as long as two vortices are initially located symmetrically with respect to the y-axis (Fig. 3). Each of them follows
Fig. 1. The
trajectory
of a vortex.
(a) (o,p)*
C K?; and (b) (u,p)?
Z K*.
HIDEAKI~HIBAHARA
600
and
MITWO
KONO
r
Fig. 2. The
Fig. 3. The
trajectories of a vortex-pair (K, = -IQ .vt = ~1 = O(t), = 0): and (b) XI = -x2
trajectories
of
a (s,
vortex-pair = -x2 = 0.2,
= 4.0) initially = 0.2. ~1 = -yz
near to the fixed point: (a) I, = -.v? = 0.3464 (0, = a/3) for II* = 0.1.
(K, = -q = 4.0) initially symmetric with .VI = ~2 = 0): (a) o* = 0: and (b) u* = 2.0.
respect
to
the
= 0.4.
y-axis
a closed trajectory centered at the fixed point p = p*, independent of the magnitude of LI*. The vortices propagate in a pair up to the boundary, and then change their partners with their own mirror images. New pairs move along the boundary of the circle to trace semicircles. Then they again exchange their partners with the original ones and repeat the previous trajectories. The periodic orbit suggests the existence of an integral for this case although we have not found it yet. On the other hand for the latter case the fixed points are homogeneously distributed on the circle of the radius p*, that is, 8, remains arbitrary. The stability analysis provides the same result as before. However, since the fixed points are distributed densely on the circle, the orbit with an initial position that slightly deviates from one of the fixed points drifts back and forth along the circle of p = p*, rotating around the fixed points [Fig. 2(b)]. This drift on the circle can be seen from the equation for 8,:
-df’t = dt
- 2
K1(2p,)a’6p
+ 2
K,(2p,)68
X
sinf3,,
(9)
where 6p and Sr3(= 6( 8, - 19~)) are given by the linearized equations: 6p = 6p,cos(wt), 68 = 2cudpasin (cot),
(10)
Vortices
in a circular
region
601
where 6p,, is an initial deviation from p.+. Differentiating equation (9) with respect to t and taking the time average of the resultant equation, we obtain: $
6 = -Q’sinE,
(11)
where $ = 2 (0,) + rr and Q = (0,/27r)K,(2p,)v( ~50~). In deriving equation (ll), we have used R/cc, = ~u,&,/K<< 1. Equation (11) is nothing but a pendulum equation and its solution is given by: sin:
= ksn(Qt
+ K(k),
where k = sin (&,/2) and K(k) is the complete maximum oscillation angle is given by: (Ol)mas = i[sin-’
k),
elliptic integral.
(12) From equation
(12), the
k - ~1 = 6,(t = 0),
(13)
which is confirmed by the numerical results shown in Fig. 2(b). The above result suggests that the orbit is stable as long as the condition Q/w << 1 is satisfied.
4. NUMERICAL
According to the initial configurations are numerically examined. 4.1.
RESULTS
and number
of the vortices, the following
cases
Two vortices symmetric with respect to the origin
When the initial position of the vortex-pair (opposite sign and same magnitude of vorticity) deviates from one of the fixed points, the trajectory is periodic for u* = 0, quasiperiodic for small u* and chaotic for finite u* (Fig. 4). For small values of u*, the trajectory is basically a semicircle and drifts back and forth to cover the region given by equation (13) [Fig. 4(b)]. A s u* increases, the length of the drift on the circle centered at p* becomes large and at the same time its magnitude changes discontinuously, cycle by cycle, leading to mixing which is an indication of onset of chaos [Fig. 4(d)]. The onset of chaos is also confirmed by a dramatic change in the frequency spectrum of the time sequence of the orbit, that is, the change from a line spectrum to a broad spectrum. By examining the frequency spectrum, we find that the critical value of u* for the onset of chaos is approximately 1.75 for the particular parameters used in Fig. 4. A vortex-pair with short distance shows a variety of motions depending on the magnitudes of u* as is illustrated in Fig. 5. A like-sign vortex-pair with the same magnitude of vorticity moves in such a way that the pair rotates together for u* = 0 and becomes quasiperiodic for u, # 0 (Fig. 6). When the magnitudes of the vorticity are different, the trajectories are periodic at paticular configration for u.+ = 0 and quasiperiodic otherwise. A finite value of u* leads to chaotic trajectories (Fig. 7). 4.2.
Two vortices on the same radius
When a vortex-pair starts from nearby positions on the same radius, the trajectory is periodic for LJ* = 0 and chaotic for finite LJ*. As the initial distance between the two vortices is increased, the trajectory becomes aperiodic for u* = 0, and chaotic with an
HIDEAKI SHIBAHARA and Mrrsuo
602
Fig. 4. The
trajectories
(x1 = -x2
KONO
of a vortex-pair (~1 = -KZ = 4.0) initially symmetric with respect to the origin y, = -y2 = 0.1): (a) v* = 0; (b) o* = 0.005; (c) o* = 0.5; and (d) u* = 2.5.
= V/3/10,
increase in u*. Further increase in u* leads to an almost periodic motion and then again to a random motion (Fig. 8). The trajectory of a like-sign vortex-pair with the same magnitude of vorticity is periodic or quasiperiodic, again depending on the initial positions for U* = 0 and is quasiperiodic and chaotic for u* # 0 (Fig. 9). 4.3.
Three and more than three vortices
When identical vortices are initially located at the vertices of an equilateral polygon centered at the origin, then they rotate together for u* = 0, while they follow quasiperiodic orbits for u* # 0. This belong to the same family of dynamics as that shown in Fig. 6. When plus and minus sign vortices are alternately located at the vertices of an equilateral polygon initially, the motion is periodic for u* - 0 and chaotic for u* # 0 (Fig. 10). A
Vortices
Fig. 5. The
trajectories yl = -yz
of a vortex-pair = -0.0125): (a)
in a circular
region
603
(~1 = -IQ = 0.5) with a short distance (XI = -x2 (b) II* = 0.5: (c) v* = 2.0; and (d) II+ = 50.0.
U* = 0.05;
(~1 = ~2 = 4.0) = -?‘r = d/2/5):
respect initially symmetric with (a) II* = 0; and (b) v* = 2.0.
= 0.0217,
to ihe origin
system with an odd number of vortices with at least one with opposite sign of vorticity initially located at the vertices of an equilateral polygon exhibits chaotic motion independent of the magnitude of U* (Fig. 11). 5. DISCUSSION
As is shown in the previous section, vortices in a bounded region exhibit complicated behavior similar to a billiard problem. It is the diamagnetic drift that has a tendency to drive the system into chaotic motion.
HIDEAKI
604
Fig. 7. The
Fig. 8. The
trajectories
trajectories
SHIBAHARA
and MITSJO
KONO
of two different vortices (~1 = 6.0. tq = 2.0) initially symmetric with (x1 = -x2 = 0.6018. y1 = -yz = 0): (a) I)+ = 0: and (b) v* = 5.0.
radius of a vortex-pair (ICI = -KZ = 4.0) initially on the same yI = y2 = 0): (a) o* = 0: (b) U* = 2.0: (c) o* = 4.0; and (d) U* = 5.0.
respect
to the origin
(x1 = .rl,k2 = -0.4,
Vortices
in a circular
region
a vortex-pair (~1 = ~2 = 4.0) initially on the same yI = yz = 0): (a) o+ = 0: and (b) 0% = 6.0.
605
Fig. 9. The
trajectories
of
Fig. 10. The equilateral
trajectories square (st
of two vortex-pairs (it = -KZ = K~ = -K~ = 4.0) initially located at the vertices of an = -x1 = 0.4, x1 = sJ = 0. yt = ~3 = 0, yz = -,va = 0.4): (a) II* = 0; and (b) II* = 1.0.
Fig. 11. The
trajectories
of
three vortices equilateral
IQ = ~3 = 4.0) initially (KI = -4.0. triangle: (a) o* = 0: and (b) U+ = 2.0.
radius
located
(xt = 0.175,
at
the
vertices
x2 = 0.8,
of
an
Similar numerical simulations [4] have been done based on two-dimensional Euler vortices whose interaction potential is logarithmic. When the bounded region is taken as a circle, a system with two vortices is completely integrable and does not exhibit chaotic motion [7]. One of the bounded regions for a vortex-pair to show chaotic motion for the Euler vortices is shown to be a semicircle [S]. The reason we have chaotic motion of a
606
HIDEAKI
SHIBAHAKA
and MITSUO
KONO
vortex-pair in a circle boundary is that our vorticity depends on the vortex position because of the diamagnetic drift which breaks the rotational symmetry and causes mixing. It is noted, however, that for a plasma with cylindrical symmetry the diamagnetic drift is azimuthal, and therefore the role of the diamagnetic drift in causing chaotic motion in the present work based on a slab geometry, may be substantially reduced in the purely cylindrical system. The equation in a cylindrical system is given by:
dz, dt
where we have assumed a Gaussian profile in a radial coordinate for the background density. The numerical solution of equation (14) with the same parameters used in Fig. 4 which is for the slab geometry is given in Fig. 12. In fact they are quite different from each other as I)* increases.
L Fig. 12. The
trajectories
of a vortex-pair
in a cylindrical
system
for the same
parameters
as those
in Fig. 4.
Vortices
in a circular
region
607
It is interesting to note that some of the numerical results obtained in the previous sections are similar to those found by El Naschie [9] who discussed the existence of strange nonchaotic attractors for the fluid contained in a cylindrical tank. According to Ref. [9], the equation describing the motion of a fluid particle is equivalent to the damped pendulum equation with two-frequency quasiperiodic forcing. Our two vortex system is four-dimensional and is characterized by two frequencies which are the rotation frequency around the fixed point and that of the drifting oscillation. The existence of these two frequencies may play a role similar to the two-frequency quasiperiodic forcing. This could be the reason for the similarity. However, our many vortex systems with higher dimensions exhibit the same sort of evolution, suggesting a possibility of different interpretations, though we have not confirmed this yet. One interesting problem is to examine whether a coherent structure (an aggregation of like-sign vortices) is formed when many point vortices are initially randomly distributed in a bounded region, this is now under way. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
A. N. M. J. D. L. Y. Y. M.
Hasegawa and K. Mima. Phys. ReL*. Lerr. 39, 205 (1977); Phys. Fluids 21, 87 (1978) J. Zabuskv and J. C. McWilliams. Phvs. Nuid~ 25, 2175 (1982). Kono and W. Horton, Phys. Nuids B3. 3255 (1991). C. Hardin and J. P. Mason. Phys. NuidJ 27, 1583 (1984). D. Hobson. Phys. Fluids A3, 3027 (1991). M. Mime-Thomson. Tlteorericnl Hydrodynunlics. p. 154. Macmillan. London (1964). Kimura. Y. Kusumoto and H. Hashimoto. J. Phys. Sot. Japm 53. 2988 (1984). Kimura and H. Hashimoto. J. Phys. Sot. Japm 55. 5 (1986). S. El Naschie. Compurer Marh. Applic. 23. 25 (1992); Nuovo Cimento 107B. 589 (1992).
0960.0779/92$5.00 + .m 0 1992 Pergamon Press Ltd
Periodic, Aperiodic and Chaotic Motion of Drift Wave Vortices in a Circular Region HIDEAKI Interdisciplinary
Graduate
School
of Engineering
SHIBAHARA Sciences.
MITSUO Research
Institute
for Applied
Mechanics, (Received
Kyushu
University,
Kasuga,
Fukuoka
816, Japan
KONO
Kyushu 31 July
University,
Kasuga,
Fukuoka
816,
Japan
1992)
Abstract-The modulated point vortex model associated with the Hasegawa-Mima equation is used to study the dynamical behavior of drift wave vortices in a bounded region. When a circle is taken as the boundary of the region, the potential for the vortices can be constructed with the aid of the circle theorem which replaces the boundary by the mirror images of the vortices inside the circle and ensures that the potential for the vortices is zero on the boundary. Periodic, aperiodic and chaotic motions are realized depending on the number of vortices and the magnitude of the value of the diamagnetic drift which breaks the rotational symmetry and has a tendency to drive vortices into chaotic motion.
1. INTRODUCTION
Vortices are rather long-lived entities as long as the viscosity is negligible, which is the case for high temperature plasmas, and in fact can be responsible for long range transport in macroscopic systems. They have been the subject of intense study, especially in connection with anomalous transport in a magnetically confined plasma where drift wave vortices are supposed to play central roles. Studies on dynamics of drift wave vortices, however, have mainly been numerically performed based on the Hasegawa-Mima equation [l], which is not easy to solve analytically. The modulated point vortex model introduced by Zabusky and McWilliams [2] first and then formulated by Kono and Horton [3] has been shown to describe characteristic features of dynamics of drift wave vortices as far as the vortex dynamics is concerned. An advantage of introducing point vortices is to convert a nonlinear partial differential equation into a system of ordinary differential equations which are easier to solve. Notwithstanding a successful description of fluid vortices in terms of the point vortex, boundary effects have to be considered because real plasma does not extend to infinity. In the present paper, therefore, we study the dynamical behavior of drift wave vortices in a bounded region. Similar numerical studies on dynamics of point vortices in a bounded region are performed based on a two-dimensional Euler equation [4]. A crucial difference of the Hasegawa-Mima equation from Euler’s equation, is the existence of the drift term giving rise to dispersive waves. Because of the diamagnetic drift the vorticity attached to each point vortex is no longer constant but varies in space and time. In Section 2 we derive the equation of point vortices in a bounded region for which a circle is taken. The potential for the vortices is constructed replacing the boundary by the mirror images of the vortices 597
HIDEAICI
598
SHIBAHARA
and MITSUO
KONO
inside the region. In Section 3 the stability of the fixed points of the equations is analyzed for the cases of a single vortex and two vortices inside the circle. In Section 4 the solutions are numerically solved to give a variety of motions. i.e. periodic, aperiodic and chaotic motion depending on the number of the vortices and the magnitude of the diamagnetic drift. The last section is devoted to discussions.
2. EQUATION
OF MOTION
The equation of motion equation is given by:
FOR POINT
VORTICES
for point vortices [2,3,5]
IN A BOUNDED
REGION
associated with the Hasegawa-Mima
dz, -=dt where zn is a complex variable related to the coordinate (.rn, ~1,) of the center of a point vortex through Z, = X, + iy, and Z, is the complex conjugate to *Il. K,(I,-0- zal) is the modified Bessel function of the second kind. The dynamical behavior of the vortices under equation (1) are shown to recover the various characteristics revealed by numerically solving the Hasegawa-Mima equation. Since the vortices propagate in pairs parallel or antiparallel to the diamagnetic direction, however, more than two vortices are finally dissolved into isolated or paired vortices no matter how complicated the intermediate state of the interactions is. This situation suggests that the dynamics of the vortices confined in a bounded region is very different from that in a space extended to infinity. Although detailed dynamics must depend on the shape of the bounded regions. we may expect to extract characteristic features common to vortex dynamics in various types of the bounded regions. If, for simplicity, we take a circle with a radius R as a bounded region. the circle theorem [6] gives the potential for the vortices inside the circle as:
On the circle (Z = RI/z), V(Z) is zero, implying the boundary is a stream line or an equipotential line. In addition, equation (2) has no singularity at z = 0. Then the equation describing the dynamics of the vortices in the circle is given by:
(3)
3. FIXED
POINTS
OF THE DYNAMICAL
EQUATIONS
For the case of one vortex in the circle, equation equation (3) becomes:
AND THEIR
(3) is integrable.
STABILITY
If z is put as pit’,
-= dp 0’ dt d0 -=dt
-R=
2rp3 (K + U*P ~0s W%(R’/p
- P>,
(4)
Vortices
in a circular
region
599
whose solution is given by: 8 tan1
A tan [~(t - r,,)] - tan (t&/2) = -
1 + (l/A)tan[~(t
A = &:
- to)]tan(8,/2)
’
;;;j.
(0 = -$
(5)
K,(R?/p
- p)V[Kz
- (u*p)2].
Note that w and A should be replaced by io and iA for (u,P)~ > 2. The trajectory is a circle for (0,~)’ < 2 and is an arc connecting the initial position pe”u with one of the fixed points pe’lH* where 8, = cos-1 (-K/U&) when (u&)~ > 2 (Fig. 1). The dynamics of two or more vortices in the circle is sensitive to their initial configurations and the magnitude of the value of u,. First we consider the case of a vortex-pair with vorticities of the same magnitude and opposite sign (K, = -K~ = K > 0). In this case, the fixed points are obtained by putting z, = olle”, for the two cases of the configurations symmetric with respect to the y-axis (the diamagnetic direction) (z, = -zz; p, = p, = p and 13, + BZ = n) and the origin (zl = -z2; p1 = pz = p and 8, - & = rr). For the first case the fixed point is given by (pr, 0,) = (p*, 0) where p* is determined by: K,Gw
+ w/P*)2w*(R2/P* + P*>- WR’/P* - P*)l = 0.
(6)
For R = 1, p* is equal to 0.4629. The stability of the fixed point can be analyzed by linearizing equation (3) around the fixed point with respect to perturbations 6p, and 68, K elm’. The eigenvalue (LI is expressed as: (IJ = $
KK,(@,),
(7)
where N is a positive constant given by: *z =
2 R’ - pii
1 + R2 - pi P*
K&P,)
+ R2(RJ - P$ Ko(R2/p,
K1(2pd
&W’/P, K,(~P*)
24
- PJ
KI(~P,)
+ PA
(8)
11 ’
Equation (7) indicates that the fixed point is stable [Fig. 2(a)]. This result is applied to the case where the initial positions deviate largely from the fixed point as long as two vortices are initially located symmetrically with respect to the y-axis (Fig. 3). Each of them follows
Fig. 1. The
trajectory
of a vortex.
(a) (o,p)*
C K?; and (b) (u,p)?
Z K*.
HIDEAKI~HIBAHARA
600
and
MITWO
KONO
r
Fig. 2. The
Fig. 3. The
trajectories of a vortex-pair (K, = -IQ .vt = ~1 = O(t), = 0): and (b) XI = -x2
trajectories
of
a (s,
vortex-pair = -x2 = 0.2,
= 4.0) initially = 0.2. ~1 = -yz
near to the fixed point: (a) I, = -.v? = 0.3464 (0, = a/3) for II* = 0.1.
(K, = -q = 4.0) initially symmetric with .VI = ~2 = 0): (a) o* = 0: and (b) u* = 2.0.
respect
to
the
= 0.4.
y-axis
a closed trajectory centered at the fixed point p = p*, independent of the magnitude of LI*. The vortices propagate in a pair up to the boundary, and then change their partners with their own mirror images. New pairs move along the boundary of the circle to trace semicircles. Then they again exchange their partners with the original ones and repeat the previous trajectories. The periodic orbit suggests the existence of an integral for this case although we have not found it yet. On the other hand for the latter case the fixed points are homogeneously distributed on the circle of the radius p*, that is, 8, remains arbitrary. The stability analysis provides the same result as before. However, since the fixed points are distributed densely on the circle, the orbit with an initial position that slightly deviates from one of the fixed points drifts back and forth along the circle of p = p*, rotating around the fixed points [Fig. 2(b)]. This drift on the circle can be seen from the equation for 8,:
-df’t = dt
- 2
K1(2p,)a’6p
+ 2
K,(2p,)68
X
sinf3,,
(9)
where 6p and Sr3(= 6( 8, - 19~)) are given by the linearized equations: 6p = 6p,cos(wt), 68 = 2cudpasin (cot),
(10)
Vortices
in a circular
region
601
where 6p,, is an initial deviation from p.+. Differentiating equation (9) with respect to t and taking the time average of the resultant equation, we obtain: $
6 = -Q’sinE,
(11)
where $ = 2 (0,) + rr and Q = (0,/27r)K,(2p,)v( ~50~). In deriving equation (ll), we have used R/cc, = ~u,&,/K<< 1. Equation (11) is nothing but a pendulum equation and its solution is given by: sin:
= ksn(Qt
+ K(k),
where k = sin (&,/2) and K(k) is the complete maximum oscillation angle is given by: (Ol)mas = i[sin-’
k),
elliptic integral.
(12) From equation
(12), the
k - ~1 = 6,(t = 0),
(13)
which is confirmed by the numerical results shown in Fig. 2(b). The above result suggests that the orbit is stable as long as the condition Q/w << 1 is satisfied.
4. NUMERICAL
According to the initial configurations are numerically examined. 4.1.
RESULTS
and number
of the vortices, the following
cases
Two vortices symmetric with respect to the origin
When the initial position of the vortex-pair (opposite sign and same magnitude of vorticity) deviates from one of the fixed points, the trajectory is periodic for u* = 0, quasiperiodic for small u* and chaotic for finite u* (Fig. 4). For small values of u*, the trajectory is basically a semicircle and drifts back and forth to cover the region given by equation (13) [Fig. 4(b)]. A s u* increases, the length of the drift on the circle centered at p* becomes large and at the same time its magnitude changes discontinuously, cycle by cycle, leading to mixing which is an indication of onset of chaos [Fig. 4(d)]. The onset of chaos is also confirmed by a dramatic change in the frequency spectrum of the time sequence of the orbit, that is, the change from a line spectrum to a broad spectrum. By examining the frequency spectrum, we find that the critical value of u* for the onset of chaos is approximately 1.75 for the particular parameters used in Fig. 4. A vortex-pair with short distance shows a variety of motions depending on the magnitudes of u* as is illustrated in Fig. 5. A like-sign vortex-pair with the same magnitude of vorticity moves in such a way that the pair rotates together for u* = 0 and becomes quasiperiodic for u, # 0 (Fig. 6). When the magnitudes of the vorticity are different, the trajectories are periodic at paticular configration for u.+ = 0 and quasiperiodic otherwise. A finite value of u* leads to chaotic trajectories (Fig. 7). 4.2.
Two vortices on the same radius
When a vortex-pair starts from nearby positions on the same radius, the trajectory is periodic for LJ* = 0 and chaotic for finite LJ*. As the initial distance between the two vortices is increased, the trajectory becomes aperiodic for u* = 0, and chaotic with an
HIDEAKI SHIBAHARA and Mrrsuo
602
Fig. 4. The
trajectories
(x1 = -x2
KONO
of a vortex-pair (~1 = -KZ = 4.0) initially symmetric with respect to the origin y, = -y2 = 0.1): (a) v* = 0; (b) o* = 0.005; (c) o* = 0.5; and (d) u* = 2.5.
= V/3/10,
increase in u*. Further increase in u* leads to an almost periodic motion and then again to a random motion (Fig. 8). The trajectory of a like-sign vortex-pair with the same magnitude of vorticity is periodic or quasiperiodic, again depending on the initial positions for U* = 0 and is quasiperiodic and chaotic for u* # 0 (Fig. 9). 4.3.
Three and more than three vortices
When identical vortices are initially located at the vertices of an equilateral polygon centered at the origin, then they rotate together for u* = 0, while they follow quasiperiodic orbits for u* # 0. This belong to the same family of dynamics as that shown in Fig. 6. When plus and minus sign vortices are alternately located at the vertices of an equilateral polygon initially, the motion is periodic for u* - 0 and chaotic for u* # 0 (Fig. 10). A
Vortices
Fig. 5. The
trajectories yl = -yz
of a vortex-pair = -0.0125): (a)
in a circular
region
603
(~1 = -IQ = 0.5) with a short distance (XI = -x2 (b) II* = 0.5: (c) v* = 2.0; and (d) II+ = 50.0.
U* = 0.05;
(~1 = ~2 = 4.0) = -?‘r = d/2/5):
respect initially symmetric with (a) II* = 0; and (b) v* = 2.0.
= 0.0217,
to ihe origin
system with an odd number of vortices with at least one with opposite sign of vorticity initially located at the vertices of an equilateral polygon exhibits chaotic motion independent of the magnitude of U* (Fig. 11). 5. DISCUSSION
As is shown in the previous section, vortices in a bounded region exhibit complicated behavior similar to a billiard problem. It is the diamagnetic drift that has a tendency to drive the system into chaotic motion.
HIDEAKI
604
Fig. 7. The
Fig. 8. The
trajectories
trajectories
SHIBAHARA
and MITSJO
KONO
of two different vortices (~1 = 6.0. tq = 2.0) initially symmetric with (x1 = -x2 = 0.6018. y1 = -yz = 0): (a) I)+ = 0: and (b) v* = 5.0.
radius of a vortex-pair (ICI = -KZ = 4.0) initially on the same yI = y2 = 0): (a) o* = 0: (b) U* = 2.0: (c) o* = 4.0; and (d) U* = 5.0.
respect
to the origin
(x1 = .rl,k2 = -0.4,
Vortices
in a circular
region
a vortex-pair (~1 = ~2 = 4.0) initially on the same yI = yz = 0): (a) o+ = 0: and (b) 0% = 6.0.
605
Fig. 9. The
trajectories
of
Fig. 10. The equilateral
trajectories square (st
of two vortex-pairs (it = -KZ = K~ = -K~ = 4.0) initially located at the vertices of an = -x1 = 0.4, x1 = sJ = 0. yt = ~3 = 0, yz = -,va = 0.4): (a) II* = 0; and (b) II* = 1.0.
Fig. 11. The
trajectories
of
three vortices equilateral
IQ = ~3 = 4.0) initially (KI = -4.0. triangle: (a) o* = 0: and (b) U+ = 2.0.
radius
located
(xt = 0.175,
at
the
vertices
x2 = 0.8,
of
an
Similar numerical simulations [4] have been done based on two-dimensional Euler vortices whose interaction potential is logarithmic. When the bounded region is taken as a circle, a system with two vortices is completely integrable and does not exhibit chaotic motion [7]. One of the bounded regions for a vortex-pair to show chaotic motion for the Euler vortices is shown to be a semicircle [S]. The reason we have chaotic motion of a
606
HIDEAKI
SHIBAHAKA
and MITSUO
KONO
vortex-pair in a circle boundary is that our vorticity depends on the vortex position because of the diamagnetic drift which breaks the rotational symmetry and causes mixing. It is noted, however, that for a plasma with cylindrical symmetry the diamagnetic drift is azimuthal, and therefore the role of the diamagnetic drift in causing chaotic motion in the present work based on a slab geometry, may be substantially reduced in the purely cylindrical system. The equation in a cylindrical system is given by:
dz, dt
where we have assumed a Gaussian profile in a radial coordinate for the background density. The numerical solution of equation (14) with the same parameters used in Fig. 4 which is for the slab geometry is given in Fig. 12. In fact they are quite different from each other as I)* increases.
L Fig. 12. The
trajectories
of a vortex-pair
in a cylindrical
system
for the same
parameters
as those
in Fig. 4.
Vortices
in a circular
region
607
It is interesting to note that some of the numerical results obtained in the previous sections are similar to those found by El Naschie [9] who discussed the existence of strange nonchaotic attractors for the fluid contained in a cylindrical tank. According to Ref. [9], the equation describing the motion of a fluid particle is equivalent to the damped pendulum equation with two-frequency quasiperiodic forcing. Our two vortex system is four-dimensional and is characterized by two frequencies which are the rotation frequency around the fixed point and that of the drifting oscillation. The existence of these two frequencies may play a role similar to the two-frequency quasiperiodic forcing. This could be the reason for the similarity. However, our many vortex systems with higher dimensions exhibit the same sort of evolution, suggesting a possibility of different interpretations, though we have not confirmed this yet. One interesting problem is to examine whether a coherent structure (an aggregation of like-sign vortices) is formed when many point vortices are initially randomly distributed in a bounded region, this is now under way. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
A. N. M. J. D. L. Y. Y. M.
Hasegawa and K. Mima. Phys. ReL*. Lerr. 39, 205 (1977); Phys. Fluids 21, 87 (1978) J. Zabuskv and J. C. McWilliams. Phvs. Nuid~ 25, 2175 (1982). Kono and W. Horton, Phys. Nuids B3. 3255 (1991). C. Hardin and J. P. Mason. Phys. NuidJ 27, 1583 (1984). D. Hobson. Phys. Fluids A3, 3027 (1991). M. Mime-Thomson. Tlteorericnl Hydrodynunlics. p. 154. Macmillan. London (1964). Kimura. Y. Kusumoto and H. Hashimoto. J. Phys. Sot. Japm 53. 2988 (1984). Kimura and H. Hashimoto. J. Phys. Sot. Japm 55. 5 (1986). S. El Naschie. Compurer Marh. Applic. 23. 25 (1992); Nuovo Cimento 107B. 589 (1992).