RAPID COMMUNICATIONS
PHYSICAL REVIEW A 74, 041603共R兲 共2006兲
Bending-wave instability of a vortex ring in a trapped Bose-Einstein condensate
1
T.-L. Horng,1 S.-C. Gou,2 and T.-C. Lin3
Department of Applied Mathematics, Feng Chia University, Taichung 40074, Taiwan
Department of Physics, National Changhua University of Education, Changhua 50058, Taiwan
3
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan
共Received 31 March 2006; published 18 October 2006兲
2
Using a velocity formula derived by matched asymptotic expansion, we study the dynamics of a vortex ring
in an axisymmetric Bose-Einstein condensate in the Thomas-Fermi limit. The trajectory for an axisymmetrically placed and oriented vortex ring shows that it generally precesses in a condensate. The linear instability
due to bending waves is investigated both numerically and analytically. General stability boundaries for various
perturbed wave numbers are computed. Our analysis suggests that a slightly oblate trap is needed to prevent the
vortex ring from becoming unstable.
DOI: 10.1103/PhysRevA.74.041603
PACS number共s兲: 03.75.Lm, 03.75.Kk, 67.40.Vs
Vortices are fundamental excitations in gases or liquids,
characterized by circulation of fluid around a core. Among
all vortical structures, vortex rings 共VRs兲 with closed-loop
cores are perhaps the most familiar to our daily experience;
their compact and persistent nature has fascinated many researchers for a long time 关1兴. They can be found in various
scales in nature, from the well-known smoke rings of cigarettes to the VRs observed in the wakes of aircraft. Remarkably, quantized VRs with cores of angstrom size have been
proved to exist when charged particles are accelerated
through superfluid helium 关2,3兴. The recent achievement of
quantized vortices in a trapped Bose-Einstein condensate
共BEC兲 关4–6兴 have suggested the possibility of producing
VRs in ultracold atoms. Several schemes for producing VRs
in atomic BECs have been put forward 关7–12兴. In particular,
Feder et al. 关10兴 have proposed using dynamical instabilities
in the condensate to make a dark soliton decay into VRs.
Based on this scheme, VRs in a trapped BEC were first realized experimentally by Anderson et al. 关13兴.
In fluids, a VR can move along its axis with a selfinduced velocity. Despite their solitary nature, VRs are susceptible to azimuthally wavy distortions and these so-called
bending waves may be amplified under certain circumstances. The bending-wave instability of classical VRs has
been extensively studied in the past few decades, and its
inviscid instability was first explored by Widnall et al. 关14兴;
it is a short-wave instability, characterized by k␰ ⬃ O共1兲,
where k is the wave number of the unstable wave and ␰ is the
size of vortex core. In this Rapid Communication, we examine the bending-wave instability of a VR in a trapped BEC.
The vortex dynamics in a superfluid generally resembles that
in a normal inviscid fluid even though the circulation is
quantized in the superfluid. However, some essential differences remain regarding the bending-wave instability. For example, the core size of a quantized vortex is extremely small
in a large BEC 关15兴, where we may assume ␰ → 0. Accordingly, k →⬁ under the condition k␰ ⬃ O共1兲, and therefore the
Widnall instability would never occur in this limiting case.
Yet the trapping potential causes vortex stretching, and the
bending-wave instability with wavelength comparable to the
ring radius may still occur.
To analyze the stability of the VR in a trapped BEC, we
1050-2947/2006/74共4兲/041603共4兲
use a scheme developed by Svidzinsky and Fetter, which
utilizes the quantum analog of Biot-Savart law to determine
the local velocity for each element of the vortex 关16兴. The
velocity formula is derived from the time dependent
Gross-Pitaevskii equation by the method of matched
asymptotic expansions in the Thomas-Fermi 共TF兲 limit.
To be specific, we shall consider a trapping potential
2 2
V共x兲 = m共␻⬜
r + ␻z2z2兲 / 2 in the cylindrical coordinates
共r , ␪ , z兲, with the aspect ratio defined by ␭ = ␻z / ␻⬜. The
density profile of the condensate is given by ␳共x兲
2
= ␳0共1 − r2 / R⬜
− z2 / Rz2兲 in the TF limit, where R⬜
2 1/2
= 共2␮ / m␻⬜兲 and Rz = 共2␮ / m␻z2兲1/2 are, respectively, the radial and axial TF radii of the trapped BEC; ␮ is the chemical
potential and ␳0 = ␮m / 4␲ប2a is the central particle density.
Thus, the velocity of a vortex line element at x in a nonrotating trap is given by 关16兴
冉
v共x兲 = ⌳共␰, ␬兲 ␬b̂ +
冊
t̂ ⫻ ⵜV共x兲
,
␮␳共x兲/␳0
共1兲
−2
+ ␬2 / 8兲; t̂ is the unit vecwhere ⌳共␰ , ␬兲 = 共− ប / 2m兲ln共␰冑R⬜
tor tangent to the vortex line at x, and b̂ is the associated
binormal unit vector; ␬ is the curvature of the vortex line at
x. Here the vortex is assumed to carry one quantum of circulation.
The first term in the large parentheses of Eq. 共1兲 is from
the local induction approximation 共LIA兲, when the vortex is
treated as an infinitely thin filament. It says that the motion is
self-induced by the local curvature and is heading in the
direction of b̂. In fact, the LIA alone ensures that the arc
length between any two points on the vortex line remains
invariant and thus the vortex is not stretched, which means
the bending-wave instability will never happen 关17兴. Nevertheless, the external potential brings in a stretching force,
t̂ ⫻ ⵱V共x兲, which makes the bending-wave instability possible. Suppose a circular VR of radius r is initially formed
axisymmetrically and remains so afterward, so that t̂ = e␪,
␬b̂ = 共1 / r兲ez. Since ␰ r⬍ R⬜, the logarithmic term in
⌳共␰ , ␬兲 is predominated by the numerical factor ln ␰, whose
magnitude is large as ␰ → 0. Hence ⌳共␰ , ␬兲 varies very
041603-1
©2006 The American Physical Society
RAPID COMMUNICATIONS
PHYSICAL REVIEW A 74, 041603共R兲 共2006兲
HORNG, GOU, AND LIN
(b)
(a)
10
0.4
5
0.2
v
0
8
0
λ
z
z
10
−0.2
4
−5
−0.4
2
−10
0
5
r
10
0
100
T
200
0
FIG. 1. 共a兲 Phase portrait for Eq. 共3兲 with ␭ = 1 and R⬜ = 10. The
dashed line indicates the TF limit. 共b兲 Axial velocity ż 共scaled in
trap units兲 as a function of T = ⌳t for various r共0兲. The solid,
dashed, dotted, and dash-dotted lines represent the cases of
r共0兲 / req = 0.5, 0.9, 1.1, and 1.5, respectively.
slowly with r and we may treat it as a constant. The motion
equations then becomes
1
2␭2z
ṙ =
,
⌳
G共r,z兲
␪˙ = 0,
1
− 2r
1
ż =
+ ,
⌳
G共r,z兲 r
共2兲
2
where G共r , z兲 = R⬜
− r2 − ␭2z2, and the length, time, and en−1
ergy are scaled by 共ប / m␻⬜兲1/2, ␻⬜
, and ប␻⬜, respectively.
˙
The equation ␪ = 0 is redundant. Numerical calculations of
the equations of r and z show that the VR moves up and
down along z axis cyclically with its radius expanding and
shrinking simultaneously. Dividing ṙ by ż, we obtain the differential equation relating r and z,
2␭2rz
dr
= 2
,
dz R⬜ − 3r2 − ␭2z2
共3兲
of which the solution is rG共r , z兲 = C, where C is a constant.
To explore this solution further, we obtain a family of contours with different C’s for a given ␭, indicating the cyclic
motion of the VR as shown in Fig. 1共a兲 and its associated
ring’s z component velocity varying with time for several C’s
in Fig. 1共b兲. Among all precession solutions, there is a
unique stationary solution with req = R⬜ / 冑3, zeq = 0, which is
independent of ␭ and agrees with the result in Ref. 关8兴. Letting rmin and rmax be the minimum and maximum values
2
2
− rmin
兲
of r for a certain C, we then have C = rmin共R⬜
2
2
= rmax共R⬜ − rmax兲. Integrating the radial equation in Eq. 共2兲
2
2
with r共0兲 = rmin, p = 共rmax
− rmin
兲1/2, and q = 关rmax共2rmin + rmax兲兴,
we find that r can be expressed in a closed form of t, i.e.,
r共t兲 =
rminrmax
.
rmaxcn2␶ + rminsn2␶
6
共4兲
Here sn ␶ = ␶ − 共1 + k2兲␶3 / 3 ! + 共1 + 14k2 + k4兲␶5 / 5 ! + ¯ and cn
␶ = 1 − ␶2 / 2 ! + 共1 + 4k2兲␶4 / 4 ! + ¯ are Jacobian elliptic functions 关18兴, with ␶ = 2⌳␭qt / C, k = p / q.
To study the bending-wave stability of the unperturbed
solution given above, let us initialize the coordinates as
r共t兲 = r0共t兲 + ␧r1共␪ , t兲, ␪共t兲 = ␪0共t兲 + ␧␪1共t兲, and z共t兲 = z0共t兲
+ ␧z1共␪ , t兲, where ␧r1, ␧␪1, and ␧z1 denote the small pertur-
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
r
0
FIG. 2. Stability boundary curves of 兩n 兩 = 1 to 10 共from bottom
to top兲 for R⬜ = 10. The solid line is the stability boundary of
兩n 兩 = 1; the circle of 兩n 兩 = 2; the square of 兩n 兩 = 3, and so forth. For
兩n 兩 ⫽ 1, the region above each boundary curve is unstable for the
corresponding mode and stable below. For 兩n 兩 = 1, the boundary
curve is simply ␭ = 1; above this it is stable and below it 共including
it兲 unstable. The cross on the horizontal axis denotes req.
bations with ␧ 1. According to the Seret-Frenet formula
␬b̂ = ␬t̂ ⫻ n̂ = 共⳵x / ⳵s兲 ⫻ 共⳵2x / ⳵s2兲, where s is the arc length.
Currently, it is more suitable to parametrize t̂ and b̂ in terms
of ␪ rather than s. Using the chain rule and the relation ds
= 兩⳵x / ⳵␪ 兩 d␪, it follows that t̂ = 共⳵x / ⳵␪兲 兩 ⳵x / ⳵␪兩−1 and ␬b̂
= 共⳵x / ⳵␪兲 ⫻ 共⳵2x / ⳵␪2兲 兩 ⳵x / ⳵␪兩−3, and thus, to the linear
−1
order of ␧, we obtain t̂ = ␧r1⬘r−1
0 er + e␪ + ␧z1⬘r0 ez and ␬b̂
−2
−2
−2
= ␧z1⬙r0 er − ␧z1⬘r0 e␪ + 共r0 − ␧r1 − ␧r1⬙兲r0 ez, where the primed
notations indicate the derivatives with respect to ␪. Substituting t̂ and ␬b̂ above into Eq. 共1兲 and expanding all terms in
power series of ␧, we see that 共r0 , ␪0 , z0兲 satisfy the zerothorder equations 共2兲. To the first order of ␧, we get a set of
linearized equations for r1 and z1. For further study of the
linear stability for bending waves, we express r1共␪ , t兲
= R1共t兲exp共in␪兲, z1共␪ , t兲 = Z1共t兲exp共in␪兲 where n is an integer.
Accordingly, we have
4␭2z0共r0R1 + ␭2z0Z1兲
2␭2Z1
Ṙ1 − n2Z1
=
+
,
+
⌳
G共r0,z0兲
G2共r0,z0兲
r20
共5兲
2R1
4r0共r0R1 + ␭2z0Z1兲
Ż1 共n2 − 1兲R1
−
=
−
.
⌳
G共r0,z0兲
G2共r0,z0兲
r20
共6兲
Here we have ignored the equation of ␪1, which is redundant.
In general, r0共t兲, z0共t兲, R1共t兲, and Z1共t兲 can only be solved
numerically. The linear stability will depend on ␭, 兩n兩, and
r0共0兲, by letting z0共0兲 = 0 without loss of generality, and the
stability boundaries are shown in Fig. 2. If we focus on the
linear stability of a stationary ring, i.e., with r0共t兲 = req,
z0共t兲 = zeq, and further assume R1共t兲 = ␣ exp共−i␻t兲, Z1共t兲
= ␤ exp共−i␻t兲, the condition for nontrivial ␣ and ␤ after substituting the expression above into Eqs. 共5兲 and 共6兲 is given
by
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RAPID COMMUNICATIONS
PHYSICAL REVIEW A 74, 041603共R兲 共2006兲
BENDING-WAVE INSTABILITY OF A VORTEX RING…
2
i␻R⬜
册
共7兲
冋
z共n兲
1 共␪,t兲
册冋册
⬀
1
␥n
ei共n␪−␻兩n兩t兲
z 0
−1
5
From Eq. 共7兲, we obtain the dispersion relation for the dis−2
关共n2 − ␭2兲共n2 − 3兲兴1/2. Obviously,
turbance, ␻兩n兩共␭兲 = ± 3⌳R⬜
␻兩n兩 is either purely real or purely imaginary. Real ␻兩n兩 corresponds to a traveling wave with fixed amplitude such that the
unperturbed solution is stable. If ␻兩n兩 is otherwise imaginary,
with a complex-conjugate pair, such that one of them implies
that the amplitude of the bending wave grows exponentially
in time, the unperturbed solution is unstable. It is easy to
show that when 共1兲 0ⱕ ␭ ⬍ 1, ␻兩n兩 is real for all 兩n 兩 ⫽ 1; 共2兲
1 ⱕ ␭ ⱕ 2, ␻兩n兩 is real for all n; 共3兲 ␭ ⬎ 2, ␻兩n兩 is real for
兩n 兩 ⱕ 1 or 兩n 兩 ⱖ ␭. Hence, we conclude that the ring is absolutely stable when 1 ⱕ ␭ ⱕ 2, for which the BEC is purely
spherical to slightly oblate.
The normal mode of the stable wave is given by
r共n兲
1 共␪,t兲
z 0
−1
共8兲
2
where ␥n = i␻兩n兩R⬜
/ 3⌳共n2 − ␭2兲 is the ratio between ␣ and ␤
for a given 兩n兩. The ␲ / 2 phase difference between ␣ and ␤
implies a helical wave along the circumference of the VR.
Since ␻兩n兩 always comes in pairs with the same magnitude
but opposite sign, there must be two identical helical waves
traveling in opposite directions, and the result is a standing
wave rotating around the unperturbed core axis, which is
named a bending wave. For n = 0, the rotation is precisely the
precession of a circular ring with r0共0兲 ⯝ req, and the frequency obtained from the inverse of Eq. 共4兲 by setting
rmin req rmax is exactly ␻0. The mode with 兩n 兩 = 1 is very
special. Since the ring is slightly shifted horizontally from its
original equilibrium position, it tilts due to the unbalanced
force and then wobbles around the xy plane. For 兩n 兩 ⱖ 2, the
bending wave looks like a petal.
The normal mode for the unstable wave is obtained by
letting ␻兩n兩 in Eq. 共8兲 be imaginary, when the bending wave,
unlike the stable one, is not propagating at all but just grows
exponentially in time. The growth of an unstable wave of
兩n 兩 = 8 is shown in Fig. 3, where the mode is initially excited
and continues to grow for ␭ = 冑65. As the disturbance continues to grow and the vortex ring continues to be stretched,
the petal shape becomes more pronounced as shown in Fig.
3共d兲. The vortex dynamics is catastrophically disrupted when
the circumference of the ring diverges. The growth of an
unstable wave of 兩n 兩 = 1 on a stationary ring is somewhat
different from that for other n’s and is shown in Fig. 4. In
this case, the excited VR wobbles at first as in the stable
mode, then gradually turns over, and is seriously distorted at
last. In most situations, however, the perturbation is generally random and only the unstable mode with the largest
growth rate will be selected to grow and dominate the instability. This most unstable mode is determined by solving
d␻兩n兩 / dn = 0, and we find that its wave number is closest to
共␭2 / 2 + 3 / 2兲1/2 for ␭ ⬎ 2. For 0 ⱕ ␭ ⬍ 1, the only unstable
−2
mode is 兩n 兩 = 1 with a growth rate 3⌳R⬜
共2 − 2␭2兲1/2 which
reaches its maximum at ␭ = 0.
5
5
0
y −5
−5 x
(c)
(d)
z 0
−1
5
0
y −5
−5 x
0
5
2
1.5
0
−5 x
z 0
−1
5
5
0
y −5
(f)
(e)
5
0
y −5
0
0
−5 x
0.4
|F(r)|
3⌳共n2 − 3兲
(b)
(a)
=0
0.2
0
10
1
0
5
T
10
n
5
T
0
30
20
10
FIG. 3. Growth of unstable waves of 兩n 兩 = 8 on a stationary
vortex ring at various T = ⌳t, where ␭ = 冑65, R⬜ = 10. T⫽共a兲 0, 共b兲
4.90, 共c兲 7.90, and 共d兲 11.82. 共e兲 The circumference L of the vortex
ring as a function of time. 共f兲 Development of the Fourier components of r. In the long-time limit, the 兩n 兩 = 8 mode is most amplified,
yet its higher harmonics 共兩n 兩 = 16, 24, etc.兲 are also amplified.
The stability criterion 1 ⱕ ␭ ⱕ 2, where ␭ = 1 , 2 represents,
respectively, the stability boundary for the mode 兩n 兩 = 1 , 2,
indicates that a stationary VR is unstable in both cigar- and
pancake-shaded BECs. This can be understood as follows.
When the BEC is in a cigar shape, as observed in 兩n 兩 = 1
mode shown in Fig. 4, the wobbling VR is easier to overturn
and gets strongly distorted afterward. When the BEC is in a
pancake shape, we can take the 兩n 兩 = 8 mode shown in Fig. 3
as an example. The reason that ␭ must be large enough so as
to destabilize this mode can be explained in the following
way. For a stationary ring, ␬b̂ and F are balanced vectors
lying on xy plane, where F = ␳0t̂ ⫻ ⵜV共x兲 / ␮␳共x兲 in Eq. 共1兲.
When the VR is slightly perturbed, the bending wave rotates
around the unperturbed core axis owing to the small imbalance between ␬b̂ and F which almost remains on the xy
plane. Take any tip of the petal in Fig. 3 as an example. As
(a)
(b)
0
−5
−10
0
z
−5
−10
z
5
0
y −5
5
0
y −5
5
−50 x
(c)
0
−5
−10
z
5
0
y −5
(e)
5
−50 x
(d)
0
z
−5
−10
5
−50x
5
0
y −5
(f)
5
−50
x
2
3
2
|F(r)|
3⌳共n2 − ␭2兲
L(T)/L(0)
2
− i␻R⬜
L(T)/L(0)
冋
det
1
0
200
T 100
100
200
T
300
1
0
0
2
4
n
6
8
FIG. 4. Growth of unstable waves of 兩n 兩 = 1 on a stationary
vortex ring at various T = ⌳t, with ␭ = 1 / 冑2, R⬜ = 10. T⫽共a兲 0, 共b兲
200.42, 共c兲 236.20, and 共d兲 256.94. 共e兲 The circumference L of the
vortex ring as a function of time. 共f兲 Development of the Fourier
components of r.
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HORNG, GOU, AND LIN
the tip rotates more up from the xy plane, the component of
F in −n̂ direction gets larger chiefly due to ␭2z, the z component of ⵱V. This −n̂ component acts as a stretching force
that will enhance the petal shape if it is large enough. Naturally an instability threshold exists for ␭, and ␭ ⬎ 8 is needed
here to destabilize 兩n 兩 = 8 mode. As 兩n兩 gets larger, so does the
curvature ␬. Hence, this instability threshold also becomes
larger, and this explains the instability criterion ␭2 ⬎ n2 when
兩n 兩 ⱖ 2. Unlike the stationary VR, the stability boundary for a
precessing VR can only be determined numerically, and the
result is shown in Fig. 2. We see that a precessing VR generally has a smaller instability threshold than the stationary
one. In addition, the larger the precession trajectory is, the
smaller the threshold becomes. This is because the instability
of a precessing VR always happens when it is expanding and
moving downward, where a larger trajectory would give a
smaller ␳共x兲, and thus increase the magnitude of F.
In summary, we have investigated the dynamics of a
single VR and its stability in an axisymmetric BEC in the TF
limit. Under axisymmetric initial conditions, the motion
equations for the VR are solved analytically. The bendingwave instability follows to be studied and the stability
boundaries are computed for various combinations of ␭, n,
and r0共0兲 both numerically and analytically. The main contribution of our studies suggests that the VR can be stable
only in a slightly oblate trap.
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