- Email: [email protected]
Stability of vortex rings in a model of superflow
Physica D 69 (1993) 163-171 North-Holland
SDI: 0167-2789(93)E0212-T
Stability of vortex rings in a model of superflow L.M. Pismen a'b and A . A . N e p o m n y a s h c h y a'c aCenter for Research in Nonlinear Phenomena, Technion - Israel Institute of Technology, 32000 Haifa, Israel bDepartment of Chemical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel CDepartment of Mathematics, Technion- Israel Institute of Technology, 32000 Haifa, Israel Received 26 January 1993 Revised manuscript received 29 April 1993 Accepted 24 May 1993 Communicated by F.H. Busse
Stability of an isolated vortex ring is studied in the framework of the Ginzburg-Landau model using the nonlocal equation of motion. It is shown that an instability which might have been caused by nonlocal effects in the long-scale theory falls into the range of wavelengths comparable with the healing length. A higher-order effect of acoustic emission is found to play a stabilizing role, since the dissipation of the energy of perturbations by isotropic emission is sufficiently strong to restore the circular shape with only a small loss of the momentum.
1. Introduction
Line vortices play a premier role both in conventional fluid mechanics and in superfluid dynamics. On the most basic level, the dynamics of vortices is identical in both systems, the only distinction being the quantization of circulation in superfluids. The classical Biot-Savart law governing the vortex motion can be derived from the Ginzburg-Landau model of superfluids as well as from the Euler equation of incompressible inviscid fluid dynamics. Many well-known studies of vortex motion use a still cruder local induction approximation (LIA) which can be formally justified by a large value of the logarithm of the ratio ~ of the transverse dimension of the line vortex (coinciding, in the case of superfluids, with the healing length) to the radius of curvature. The LIA leads to the beautiful result by Hasimoto [8] reducing the equation of motion of the line vortex, expressed through the geometric invariants of the curve (curvature and torsion), to the nonlinear Schr6dinger equation (NLS) for the complex filament function. The fundamental flaw of the LIA lies, however, in the absence of vortex stretching; thus, even small nonlocal corrections causing changes of the length of the vortex are apt to cause strong deviations from LIA results at sufficiently long times. The inadequacy of the LIA should be felt particularly strongly when the stability of line vortex configurations is studied. Due to the self-focussing nature of the NLS featuring in Hasimoto's theory, its plane wave solutions, corresponding to spiral configurations, must be unstable, as indeed was proven in a straightforward way by Newton and Keller [14]. The exceptional case was the circle which turned out to be neutrally stable. A neutrally stable situation, evidently, might be resolved when weaker effects are taken into account, and abandoning the LIA should be the first step. Working with the full Ginzburg-Landau model, Grant [6] found the spectrum of small-amplitude oscillations of an isolated vortex ring (with the wavelength far exceeding the core size) via a complicated Elsevier Science Publishers B.V.
164
L.M. Pismen, A.A. Nepomnyashchy / Stability of vortex rings in a model of superflow
asymptotic expansion procedure in the coordinate frame specifically suited to the distorted ring geometry. We shall reproduce his results in section 3 using the general nonlocal equation of motion derived (to the first-order in the ratio • of the core size to the ring radius) with the help of the method of matched asymptotic expansion in the core region [15], and valid for weakly curved vortex filaments of an arbitrary shape. We note that destabilization due to nonlocal effects is theoretically possible, although numerically the instability range falls outside the applicability region of the long-scale theory. As the neutral stability persists in the framework of the nonlocal Biot-Savart theory, effects of a higher order in • should be considered. In higher orders, similarities between conventional and superfluid dynamics are largely lost, since the dynamics becomes sensitive to dissipation mechanisms. Stability of vortices in an incompressible fluid depends on the weak viscous dissipation [20,17]. On the other hand, the expected dissipation mechanism in superfluids is the sound emission stemming from their weak compressibility. The study of stability of quantized vortex rings is of interest in view of the role played by quantized vortices in superfluid turbulence [18] and a conjectured role of vortex rings in superfluid A-transition [19]. The role of vortex-sound interactions is particularly important in this context. The importance of compressibility effects in the motion of vortex rings was indicated by recent computations by Frisch et al. [4,5] using the full Ginzburg-Landau model rather than the Biot-Savart law. Jones et al. [9] considered higher-order corrections to the axially symmetric vortex ring solution, and found that expansion or contraction of the ring is forbidden due to the integral energy and momentum conservation, and, consequently, vortex rings with sufficiently large radii are stable. The question that remained unanswered was whether ring oscillations can trigger a change of the radius by generating acoustic emission that would carry away the energy and the momentum of superflow. This is the principal question we shall address in this paper. Rather than trying to extend the equation of vortex motion to the second order in •, we shall use in section 4 the integral energy and momentum balances to account for losses due to the sound emission. The acoustic radiation due to the weak compressibility of the superfluid is determined using the method of asymptotic matching of solutions in the vortical and acoustic flow regions [10]. Our conclusion is that the acoustic dissipation of the energy of perturbations of the ring is sufficiently strong to restore the circular shape with only a small loss of the momentum.
2. A model of superfluid We consider the defocussing nonlinear Schr6dinger (Gross-Pitaevskii) equation -iu, = ½V2u + •-2(1 - [ u l 2 ) u .
(1)
The small parameter of the problem is the ratio • of the core size (healing length) to a characteristic macroscopic scale of the system. The standard form of eq. (1) is obtained by setting u = ~ e i~ and separating the real and imaginary parts:
1/1
~,=~7
V ¢_
iv l ) + e - 2 ( 1 _
- ¢, = ½¢V2~ + V¢ . V ~ .
~/2) ,
(2) (3)
L.M. Pismen, A . A . Nepomnyashchy / Stability of vortex rings in a model of superflow
165
Topologically nontrivial solutions of eq. (1) are vortex lines characterized by the circulation condition V~--2"rm, where the integration is carried out over an arbitrary enclosing contour. Stable vortices carry the unit topological charge n = --1. Eqs. (2), (3) can be rewritten, respectively, in the form of the Bernoulli equation
,+½1vl2+p=0,
(4)
and the continuity equation p, + V. (pv) = 0.
(5)
Here p = lul 2 and v = V ~ are identified, respectively, with the density and velocity fields, and p is the pressure introduced via the equation of state p = , - 2 ( p _ 1) - ~
1
V2X/~.
(6)
Far from the vortex core, the last term in the equation of state eq. (6) can be neglected, and it reduces to a linear equation of state of a weakly compressible fluid. Pressure can then be eliminated from the Bernoulli and continuity equations yielding a nonlinear wave equation for the potential ~ : • ,, - c2V2~ + Vff~.V~ t = 0.
(7)
The velocity of sound is c = X / d p / d p = ¢-1 >> 1. When compressibility is neglected, the velocity field is just a potential flow of an ideal fluid generated by a line vortex, and can be computed using the Biot-Savart law [1]. The Biot-Savart integral diverges on the line vortex itself, and the equation of vortex motion is obtained by matching with the solution of the full system (2), (3) in the core region [15]. When the velocity field is nonstationary, an acoustic wave is emitted. The acoustic field generated by a moving line vortex carries both momentum and energy, and is responsible for an effective dissipation leading to the decay of classical (incompressible) vortex configurations. An early example was given by Klyatskin [11] who showed that acoustic radiation causes the separation of a rotating pair of point vortices with the like charges in the plane. Other planar effects, including collapse of a pair of oppositely charge vortices of unequal strength, were predicted by Gryanik [7]. The general treatment of acoustic emission by vortex motions at low Mach numbers is outlined in section 4.
3. Oscillations of a vortex ring
Consider a weakly distorted unit circle r = a[1 + ~?(~b)], z = a~(~b). Retaining only terms linear in the distortion functions ~(~b), ~(~b), one can write the following expressions for the position, tangent, normal, and binormal vectors in the Cartesian frame with the origin at the center of the undistorted circle: r(~b) = a([1 + ~/(~b)] cos ~b, [1 + ~/(¢k)] sin ~b, ~(~b)), t(~b) = ( - s i n ¢k + ~/'(~b) cos ~b, cos ~b + ~'(~b) sin ~b, ~'(~b)), n(~b) = ( - c o s ~b - r/'(cb) sin ok, - s i n ~b + ~/'(~b) cos ~b, ¢"(~b)), b(~b) = (¢"(~b) cos ~b + ¢'(~b) sin ~b, ¢"(~b) sin ~b - ¢'(~b) cos ~b, 1).
(8)
166
L.M. Pismen, A.A. Nepomnyashchy / Stability of vortex rings in a model of superflow
The local curvature is r = a-'[1 - 7($) - n"(~b)] •
(9)
The Biot-Savart integral defining the velocity induced at ~b = 0 is computed [15] by separating the contour into the near region [~bl < 6 and the far region 8 < I~b[< ~r, where 6 is a small parameter that, however, should far exceed the size of the vortex core ~. The near integral is computed in the local Frenet-Serret frame (t(0), n(0), b(0)), and the divergency at [~bl-->0 is eliminated by matching with a solution in the core region. This yields the local part of the induced velocity v~ = ½K(0)b(0)In ab,[1 _+ ~(0)] E
1K(0)b(0)(ln a_~b~+ ~7(0)) ,
(10)
where b is an ~7(1) constant. A numerical computation [15] gives b = 2X/2 e -°'61s = 1.5246. The far integral is most conveniently computed in the Cartesian frame. Using the unnormalized tangent vector dl = r'(~b) d~b = a(1 + ~)t d~b yields, in the linear approximation, 2"n'-6
,,=_½
2-tr-8
f q(,)×a,(,) iq( )l 3
1 f sin(l F(,) )
- 8a
(11)
8
where q(~b)= r ( 0 ) - r(~b) is the vector from the reference point to a point on the distorted circle, and
F(~b) =
+ 2 sin2(½~b)
~'(~b)(1 - cos ~) + [~(0) - ~(~b] sin ~b . ½(1 + c°s ~) ~(~) - 1(3 - c°s ~b) ~(0) - ~'(~b) sin ~b
(12)
The 6--->0 limit of the integral (11) is - ~
1
In 6(~"(0), - ~ ' ( 0 ) , 1 - 7 ( 0 ) -
77"(0)) =
- ½ K(0) in 6 b(0).
The divergency can be eliminated by adding and extracting an integral 2~-8
--~K(0) b(0) J
f(~b) d~b,
8
where f(th)~2/~b at (h--->O. The most convenient choice of a compensating function is f(~b) = 1/sin(½ $ ) which gives the far integral for the undistorted unit circle
1
i
d~b 1 4 sin'S½q)) - ~ In ~ .
8
The total induced velocity at the reference point is expressed then as 2-e
1
O = l ) 1 +V2=½K(O) b(O)(ln--~---+2~7(O)+~"(O))+T6-a f G(~b) dq~ o sina(½~b) ' where
(13)
L.M. Pismen, A . A . Nepomnyashchy / Stability o f vortex rings in a model o f superflow
// if'(4,) sin 4, - ~"(0)(1 - cos 4,) + [~(0) - ~(4,)] cos 4, '~ G(4,) = | [U(4,) + ~'(0)1(1 - cos 4,) + [~'(0) - ~'(4,)] sln 4' 1. \½(1 + cos 4')[~7(4') -77(0)] + rt"(0)(1 - cos 4') - 7 / ( 4 ' ) sin 4 ' /
167
(14)
Presenting the small deviations as a Fourier series = Z ~7. ei"*,
~ = ~ ~. el"*,
(15)
we obtain from eqs. (13), (14) the equations of motion for the spectral components ~7., ~.:
it. = a-2f.~.,
• = a-2g.19..
(16)
The coefficients f., g. are computed by integrating by parts:
L = - ~ n1 -_ - ~ n1
2
In
4ab +~
- n sin n4, sin 4, + n2(1 - cos 4,) + (1 - cos n4,) cos 4, d4, sin3(½4,)
0
4ab + (n 2 - - ] ) I . - 1 2in
g. _-1~(n2 _ 1) In 4abe +2--1 81 i
2
-nsinn4,sin4,+n2(1-c°s4,)+½(1-c°sn4,)(l+cos4,)d4,sin3(½4,)
0
- 1~(n z - 1) In 4ab + 1 E
"2--
(n z-¼)l. +¼n z
(17)
where ~12
I.
=
sin4,
0
d4, =
k=l 2 k
1 "
The frequency ~0~ =a-2~F2-f,g, vanishes at n = 1, which corresponds to a displacement without deformation. The expressions (17) coincide with those found by Grant [6] when the difference in scaling is taken into account. It is notable that the action of local and nonlocal terms is oppositely directed. Consider, for example, a segment of a distorted circle that has advanced ahead of the rest of the curve and is bent forward. The binormal is inclined locally towards the axis of motion, and therefore the local component of the induced velocity is directed inward, as is seen, indeed, from eq. (17). On the other hand, the nonlocal component of the velocity induced at this location by the rest of the circle, that is trailing behind, is directed outward. Formally, the local component must be stronger due to the presence of the logarithm of a large quantity. The nonlocal component turns out, however, to be growing with n faster than the local one. This is seen in fig. 1 showing the values of n corresponding to the change of the sign o f f , and g, at different values of the logarithm of the ratio x = 4able. Instability is observed when the coefficients f, and g, are of the same sign, which happens within a certain band of wavenumbers that shifts towards larger n with decreasing ~. Note that a similar phenomenon of destabilization of a vortex ring at a certain wavelength dependent on the vorticity distribution within the core is known in conventional fluid mechanics [17]. The intervals of instability for lower modes are indicated by horizontal lines in fig. 1. We note that the theory breaks down when the perturbation wavelength becomes comparable with the core size, i.e., at n~ = if(l). Numerically,
168
L.M. Pismen, A . A . Nepomnyashchy / Stability of vortex rings in a model of superflow n 10
/
.....I 1.5
I
I
I
2.5
3
3.5
in x
Fig. 1. Values of n corresponding to the change of the sign of f. (left curve) and g. (fight curve) at different values of the logarithm of the ratio x = 4ab/~. The intervals of instability are indicated by horizontal lines.
nE = ~(1); thus, the instability range falls into the region where the long-scale theory is no longer applicable. The common asymptotic value of both nonlocal terms in eq. (17) at n---~o0 is ln2[ln(4n)+ C - ½], where C is the Euler constant. The instability interval lies asymptotically at ln(ab/~n) = C - ½ , and its width vanishes at n---~oo. Numerically, e n / a = b exP(½- C ) ~ 1.41, which is outside the applicability region of the long-scale theory. The leading term in the short-scale asymptotics of the frequency, ~o = (n2/2a 2) In n, is determined exclusively by the nonlocal term, and coincides with the long-scale asymptotics of the spectrum of core oscillations obtained long ago for a straight-line vortex by Pitaevskii [16]. At wavelengths of the order of the core radius, the vortex is expected to be stabilized due to the depletion of the superfluid density within the core, and, indeed, numerical calculations [2,3] indicated that no instabilities arose in this range.
4. Sound emission by an oscillating vortex ring Effects of the second order in ~ can modify the results based on the Biot-Savart law. The effect appearing in this approximation is the emission of sound. We shall base the following analysis of the acoustic effects on the method of matched asymptotic expansion developed by Kambe [10], that expl6its a wide separation between the characteristic scale a of a vortex flow region and the acoustic wavelength h. These scales are well separated since the motion of the vortex ring is strongly subsonic, the Mach number being of the same order of magnitude as the ratio ~/a of the core size to the radius of the ring. The inner and outer solutions are matched at distances s from the ring center satisfying the inequality a ,~ s ~ A. In the inner region s = a , the spatial derivatives are scaled as V - a -1, and the estimate for the characteristic frequency following from the Bernoulli equation is to ~ a -2. In this region, both the first
L.M. Pismen, A.A. Nepomnyashchy / Stability of vortex rings in a model of superflow
169
and the last terms in eq. (7) can be neglected; the flow is effectively incompressible, and the velocity field can be computed using the Biot-Savart integral. For the purpose of matching with the acoustic region, it is sufficient to compute the flow potential at distances large compared with the ring radius a, which is given [1,10] by the multipole expansion
(P = p . v I + O: VVI+ ".., S
(18)
s
where
P=¼~r×dl,
O=-~r®r×dl.
(19)
It is easy to see that, for a weakly distorted circle, the dipole term vanishes in the linear approximation. The tensor O in the quadrupole term is presented after integrating by parts as c
O(t) = - ¼a3 ~b
-cos24, sin 24, 0
sin 24, 0 cos 24, 0 0 0
~'(4,,t) d4,.
(20)
A nonvanishing contribution to O is given by the second harmonic nonplanar distortion ~(4,, t)oc e i(2¢'÷0''). Higher harmonics contribute to higher terms of the multipole expansion, which, as one can show, generate acoustic waves with the intensity proportional to higher powers of the Mach number. Since the quadrupole flow field v = V ~ decays ocs-3 at s--,o% the terms ~tt and V2~ in eq. (7) are both scaled as s -4 at large distances, whereas the nonlinear term is scaled as s -6, and can be neglected. The solution of the resulting linear wave equation in the far (acoustic) region can also be presented in the form of a multipole expansion with coefficients determined by matching with the inner region. The leading quadrupole term for a vortex ring with a stationary center is given [10] by
For a ring propagating along the z axis with the speed v o = (2a) -1 ln(4ab/e) ~ c = E -1, eq. (21) should be applied in the comoving frame, and the distance from the center of the ring s replaced by the distance ~ from the point where the center had been located at the moment the sound wave was emitted. The coordinate of this point in the comoving frame is z = -ovo/c. A short computation yields o- = s + zM + ~t(M2), where M = vo/c ~ 1 is the Mach number. Transforming to cylindrical coordinates and using the second harmonic nonplanar circle distortion with the amplitude ; A brings eq. (21) to the form
~=lAa3e2i6
0 [1 0 (lei(O,t_k~)) ]
where k = to/c is the acoustic wavenumber. The contribution to • that has the slowest rate of decay at s--~ oo is obtained by applying the derivatives in eq. (22) to the exponential term only: = -8-A\-~/( a ~3 (kr) a e i(°~t-k~r + 2 ' )
.
(23)
The evolution equations for the average ring radius and for the amplitude of oscillations can be
L.M. Pismen, A . A . Nepomnyashchy / Stability o f vortex rings in a model o f superflow
170
obtained with the help of integral momentum and energy balances. Both energy and momentum of the flow field due to the vortex ring are concentrated in the inner region, while the outward flux due to acoustic emission can be computed in the outer wave field. The total kinetic energy of the flow field generated by an undistorted circular vortex is [1] T = 2"rr~ v- (r x d/) = 2~r2a In 4a____bb~
(24)
The only nonvanishing component of the total momentum is P~ = "~ ¢ r x dl = 2'rr2a2
T
(25)
Vo
Corrections to both values are quadratic in distortions. The energy flux density in the acoustic wave is [12] j = pv. Using p = - ~ t , v = V~ with • given by eq. (23), and integrating over the surface of a large cylinder with the radius and half-height R centered at the center of the vortex ring, gives the total energy loss due to acoustic emission: J = Re ~ ~*n .V~ dS R
R
=4,rr n--,=limR e ( f ~*(r, R)~zz(r, R) r dr + R f ~*, (R, z) qbr(R, z) dz ) o
o 1
x 57/2 dx = ~67r AZa6ck6 f (1 1+ +X2)
~--"~5 'l'l" A2a6ck 6 .
(26)
Note that ak = aw/c ~ (ac) -1 ~ 1; thus, the acoustic effects are very weak. The acoustic emission by higher harmonics is of a still higher order in ~, or in the Mach number. The leading term in the far field acoustic potential generated by higher harmonics is expressed as
~A a
ein, 0" ei(,%t_~,~) ~ A Or
Or n
(kr) n e i(°"'-k-~+n*)
(27)
This yields T ~ A2c(ak) 2(n+l). The momentum flux density, or acoustic stress, is [12] tr = v ® v = Re(V~* ® Vq~). We compute it on the surface of the same large cylinder as above. A nonvanishing contribution to the flux of the z-component of the total momentum is obtained due to the asymmetry of Vz = Oq~ OZ =
+M
Oor
due to the motion of the ring. The term proportional to the Mach number is responsible for the braking action leading to a decrease of the average radius of the ~ing, whereas the isotropic part of the acoustic emission causes the damping of deviations from the circular shape. Integrating over either end of the cylinder gives the total loss of the z-momentum due to acoustic emission:
L.M. Pismen, A.A. Nepomnyashchy / Stability of vortex rings in a model of superflow R
= 8rrM R--,~limR
f s-~l~(r,
0 = 1 7 r A 2 M a 6 k 6 = OoC-2j.
171
1
R)12r dr = ~Tr A 2 M a 6 k 6
fxS
(1 + x2) 7/2
0
(28)
O n e can see that the acoustic loss o f the m o m e n t u m is relatively inefficient, as the ratio of 2 f / P z to J / T is M 2 ~ 1. This ratio is o f the same o r d e r o f the m a g n i t u d e also for w e a k e r radiation due to the higher harmonics. T h u s , the e n e r g y o f perturbations is dissipated by isotropic acoustic emission b e f o r e it can cause substantial loss of the m o m e n t u m that could lead to the collapse o f the vortex ring.
Acknowledgements
L . M . P . a c k n o w l e d g e s the s u p p o r t o f the F u n d for P r o m o t i o n o f R e s e a r c h at the T e c h n i o n . A . A . N . a c k n o w l e d g e s the s u p p o r t o f the B a r e c h a F o u n d a t i o n .
References
[1] G.K. Batchelor, Introduction to Fluid Mechanics (Cambridge Univ. Press, Cambridge, 1967). [2] A.L. Fetter, Phys. Rev. A 140 (1965) 452. [3] A.L. Fetter, Ann. Phys. 70 (1972) 67. [4] T. Frisch, Y. Pomeau and S. Rica, Phys. Rev. Lett. 69 (1992) 452. [5] T. Frisch and S. Rica, Physica D 61 (1992) 158. [6] J. Grant, J. Phys. A 4 (1971) 695. [7] V.M. Gryanik, Izv. AN SSSR Fiz. Atm. Ok. 19 #2 (1983) 203. [8] H. Hasimoto, J. Fluid Mech. 51 (1972) 477. [9] C.A. Jones, S.J. Putterman and P.H. Roberts, J. Phys. A 19 (1986) 2991. [10] T. Kambe, J. Fluid Mech. 173 (1986) 643. [11] V.I. Klyatskin, Izv. AN SSSR Mekh. Zh. Gaz. #6 (1966) 87. [12] L.D. Landau and E.M. Lifschitz, Fluid Mechanics (Pergamon, Oxford, 1980). [13] M.J. Lighthill, Proc. R. Soc. London A 211 (1952) 564. [14] P. Newton and J.B. Keller, SIAM J. Appl. Math. 47 (1987) 959. [15] L.M. Pismen and J. Rubinstein, Physica D 47 (1991) 353. [16] L.P. Pitaevskii, Sov. Phys. JETP 13 (1961) 451. [17] P.G. Saffman, J. Fluid Mech. 84 (1978) 625. [18] K.W. Schwarz, Phys. Rev. B 31 (1985) 5782. [19] G.A. Williams, J. Low Temp. Phys. 89 (1992) 91. [20] S.E. Widnall, Ann. Rev. Fluid Mech. 7 (1975) 141.
SDI: 0167-2789(93)E0212-T
Stability of vortex rings in a model of superflow L.M. Pismen a'b and A . A . N e p o m n y a s h c h y a'c aCenter for Research in Nonlinear Phenomena, Technion - Israel Institute of Technology, 32000 Haifa, Israel bDepartment of Chemical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel CDepartment of Mathematics, Technion- Israel Institute of Technology, 32000 Haifa, Israel Received 26 January 1993 Revised manuscript received 29 April 1993 Accepted 24 May 1993 Communicated by F.H. Busse
Stability of an isolated vortex ring is studied in the framework of the Ginzburg-Landau model using the nonlocal equation of motion. It is shown that an instability which might have been caused by nonlocal effects in the long-scale theory falls into the range of wavelengths comparable with the healing length. A higher-order effect of acoustic emission is found to play a stabilizing role, since the dissipation of the energy of perturbations by isotropic emission is sufficiently strong to restore the circular shape with only a small loss of the momentum.
1. Introduction
Line vortices play a premier role both in conventional fluid mechanics and in superfluid dynamics. On the most basic level, the dynamics of vortices is identical in both systems, the only distinction being the quantization of circulation in superfluids. The classical Biot-Savart law governing the vortex motion can be derived from the Ginzburg-Landau model of superfluids as well as from the Euler equation of incompressible inviscid fluid dynamics. Many well-known studies of vortex motion use a still cruder local induction approximation (LIA) which can be formally justified by a large value of the logarithm of the ratio ~ of the transverse dimension of the line vortex (coinciding, in the case of superfluids, with the healing length) to the radius of curvature. The LIA leads to the beautiful result by Hasimoto [8] reducing the equation of motion of the line vortex, expressed through the geometric invariants of the curve (curvature and torsion), to the nonlinear Schr6dinger equation (NLS) for the complex filament function. The fundamental flaw of the LIA lies, however, in the absence of vortex stretching; thus, even small nonlocal corrections causing changes of the length of the vortex are apt to cause strong deviations from LIA results at sufficiently long times. The inadequacy of the LIA should be felt particularly strongly when the stability of line vortex configurations is studied. Due to the self-focussing nature of the NLS featuring in Hasimoto's theory, its plane wave solutions, corresponding to spiral configurations, must be unstable, as indeed was proven in a straightforward way by Newton and Keller [14]. The exceptional case was the circle which turned out to be neutrally stable. A neutrally stable situation, evidently, might be resolved when weaker effects are taken into account, and abandoning the LIA should be the first step. Working with the full Ginzburg-Landau model, Grant [6] found the spectrum of small-amplitude oscillations of an isolated vortex ring (with the wavelength far exceeding the core size) via a complicated Elsevier Science Publishers B.V.
164
L.M. Pismen, A.A. Nepomnyashchy / Stability of vortex rings in a model of superflow
asymptotic expansion procedure in the coordinate frame specifically suited to the distorted ring geometry. We shall reproduce his results in section 3 using the general nonlocal equation of motion derived (to the first-order in the ratio • of the core size to the ring radius) with the help of the method of matched asymptotic expansion in the core region [15], and valid for weakly curved vortex filaments of an arbitrary shape. We note that destabilization due to nonlocal effects is theoretically possible, although numerically the instability range falls outside the applicability region of the long-scale theory. As the neutral stability persists in the framework of the nonlocal Biot-Savart theory, effects of a higher order in • should be considered. In higher orders, similarities between conventional and superfluid dynamics are largely lost, since the dynamics becomes sensitive to dissipation mechanisms. Stability of vortices in an incompressible fluid depends on the weak viscous dissipation [20,17]. On the other hand, the expected dissipation mechanism in superfluids is the sound emission stemming from their weak compressibility. The study of stability of quantized vortex rings is of interest in view of the role played by quantized vortices in superfluid turbulence [18] and a conjectured role of vortex rings in superfluid A-transition [19]. The role of vortex-sound interactions is particularly important in this context. The importance of compressibility effects in the motion of vortex rings was indicated by recent computations by Frisch et al. [4,5] using the full Ginzburg-Landau model rather than the Biot-Savart law. Jones et al. [9] considered higher-order corrections to the axially symmetric vortex ring solution, and found that expansion or contraction of the ring is forbidden due to the integral energy and momentum conservation, and, consequently, vortex rings with sufficiently large radii are stable. The question that remained unanswered was whether ring oscillations can trigger a change of the radius by generating acoustic emission that would carry away the energy and the momentum of superflow. This is the principal question we shall address in this paper. Rather than trying to extend the equation of vortex motion to the second order in •, we shall use in section 4 the integral energy and momentum balances to account for losses due to the sound emission. The acoustic radiation due to the weak compressibility of the superfluid is determined using the method of asymptotic matching of solutions in the vortical and acoustic flow regions [10]. Our conclusion is that the acoustic dissipation of the energy of perturbations of the ring is sufficiently strong to restore the circular shape with only a small loss of the momentum.
2. A model of superfluid We consider the defocussing nonlinear Schr6dinger (Gross-Pitaevskii) equation -iu, = ½V2u + •-2(1 - [ u l 2 ) u .
(1)
The small parameter of the problem is the ratio • of the core size (healing length) to a characteristic macroscopic scale of the system. The standard form of eq. (1) is obtained by setting u = ~ e i~ and separating the real and imaginary parts:
1/1
~,=~7
V ¢_
iv l ) + e - 2 ( 1 _
- ¢, = ½¢V2~ + V¢ . V ~ .
~/2) ,
(2) (3)
L.M. Pismen, A . A . Nepomnyashchy / Stability of vortex rings in a model of superflow
165
Topologically nontrivial solutions of eq. (1) are vortex lines characterized by the circulation condition V~--2"rm, where the integration is carried out over an arbitrary enclosing contour. Stable vortices carry the unit topological charge n = --1. Eqs. (2), (3) can be rewritten, respectively, in the form of the Bernoulli equation
,+½1vl2+p=0,
(4)
and the continuity equation p, + V. (pv) = 0.
(5)
Here p = lul 2 and v = V ~ are identified, respectively, with the density and velocity fields, and p is the pressure introduced via the equation of state p = , - 2 ( p _ 1) - ~
1
V2X/~.
(6)
Far from the vortex core, the last term in the equation of state eq. (6) can be neglected, and it reduces to a linear equation of state of a weakly compressible fluid. Pressure can then be eliminated from the Bernoulli and continuity equations yielding a nonlinear wave equation for the potential ~ : • ,, - c2V2~ + Vff~.V~ t = 0.
(7)
The velocity of sound is c = X / d p / d p = ¢-1 >> 1. When compressibility is neglected, the velocity field is just a potential flow of an ideal fluid generated by a line vortex, and can be computed using the Biot-Savart law [1]. The Biot-Savart integral diverges on the line vortex itself, and the equation of vortex motion is obtained by matching with the solution of the full system (2), (3) in the core region [15]. When the velocity field is nonstationary, an acoustic wave is emitted. The acoustic field generated by a moving line vortex carries both momentum and energy, and is responsible for an effective dissipation leading to the decay of classical (incompressible) vortex configurations. An early example was given by Klyatskin [11] who showed that acoustic radiation causes the separation of a rotating pair of point vortices with the like charges in the plane. Other planar effects, including collapse of a pair of oppositely charge vortices of unequal strength, were predicted by Gryanik [7]. The general treatment of acoustic emission by vortex motions at low Mach numbers is outlined in section 4.
3. Oscillations of a vortex ring
Consider a weakly distorted unit circle r = a[1 + ~?(~b)], z = a~(~b). Retaining only terms linear in the distortion functions ~(~b), ~(~b), one can write the following expressions for the position, tangent, normal, and binormal vectors in the Cartesian frame with the origin at the center of the undistorted circle: r(~b) = a([1 + ~/(~b)] cos ~b, [1 + ~/(¢k)] sin ~b, ~(~b)), t(~b) = ( - s i n ¢k + ~/'(~b) cos ~b, cos ~b + ~'(~b) sin ~b, ~'(~b)), n(~b) = ( - c o s ~b - r/'(cb) sin ok, - s i n ~b + ~/'(~b) cos ~b, ¢"(~b)), b(~b) = (¢"(~b) cos ~b + ¢'(~b) sin ~b, ¢"(~b) sin ~b - ¢'(~b) cos ~b, 1).
(8)
166
L.M. Pismen, A.A. Nepomnyashchy / Stability of vortex rings in a model of superflow
The local curvature is r = a-'[1 - 7($) - n"(~b)] •
(9)
The Biot-Savart integral defining the velocity induced at ~b = 0 is computed [15] by separating the contour into the near region [~bl < 6 and the far region 8 < I~b[< ~r, where 6 is a small parameter that, however, should far exceed the size of the vortex core ~. The near integral is computed in the local Frenet-Serret frame (t(0), n(0), b(0)), and the divergency at [~bl-->0 is eliminated by matching with a solution in the core region. This yields the local part of the induced velocity v~ = ½K(0)b(0)In ab,[1 _+ ~(0)] E
1K(0)b(0)(ln a_~b~+ ~7(0)) ,
(10)
where b is an ~7(1) constant. A numerical computation [15] gives b = 2X/2 e -°'61s = 1.5246. The far integral is most conveniently computed in the Cartesian frame. Using the unnormalized tangent vector dl = r'(~b) d~b = a(1 + ~)t d~b yields, in the linear approximation, 2"n'-6
,,=_½
2-tr-8
f q(,)×a,(,) iq( )l 3
1 f sin(l F(,) )
- 8a
(11)
8
where q(~b)= r ( 0 ) - r(~b) is the vector from the reference point to a point on the distorted circle, and
F(~b) =
+ 2 sin2(½~b)
~'(~b)(1 - cos ~) + [~(0) - ~(~b] sin ~b . ½(1 + c°s ~) ~(~) - 1(3 - c°s ~b) ~(0) - ~'(~b) sin ~b
(12)
The 6--->0 limit of the integral (11) is - ~
1
In 6(~"(0), - ~ ' ( 0 ) , 1 - 7 ( 0 ) -
77"(0)) =
- ½ K(0) in 6 b(0).
The divergency can be eliminated by adding and extracting an integral 2~-8
--~K(0) b(0) J
f(~b) d~b,
8
where f(th)~2/~b at (h--->O. The most convenient choice of a compensating function is f(~b) = 1/sin(½ $ ) which gives the far integral for the undistorted unit circle
1
i
d~b 1 4 sin'S½q)) - ~ In ~ .
8
The total induced velocity at the reference point is expressed then as 2-e
1
O = l ) 1 +V2=½K(O) b(O)(ln--~---+2~7(O)+~"(O))+T6-a f G(~b) dq~ o sina(½~b) ' where
(13)
L.M. Pismen, A . A . Nepomnyashchy / Stability o f vortex rings in a model o f superflow
// if'(4,) sin 4, - ~"(0)(1 - cos 4,) + [~(0) - ~(4,)] cos 4, '~ G(4,) = | [U(4,) + ~'(0)1(1 - cos 4,) + [~'(0) - ~'(4,)] sln 4' 1. \½(1 + cos 4')[~7(4') -77(0)] + rt"(0)(1 - cos 4') - 7 / ( 4 ' ) sin 4 ' /
167
(14)
Presenting the small deviations as a Fourier series = Z ~7. ei"*,
~ = ~ ~. el"*,
(15)
we obtain from eqs. (13), (14) the equations of motion for the spectral components ~7., ~.:
it. = a-2f.~.,
• = a-2g.19..
(16)
The coefficients f., g. are computed by integrating by parts:
L = - ~ n1 -_ - ~ n1
2
In
4ab +~
- n sin n4, sin 4, + n2(1 - cos 4,) + (1 - cos n4,) cos 4, d4, sin3(½4,)
0
4ab + (n 2 - - ] ) I . - 1 2in
g. _-1~(n2 _ 1) In 4abe +2--1 81 i
2
-nsinn4,sin4,+n2(1-c°s4,)+½(1-c°sn4,)(l+cos4,)d4,sin3(½4,)
0
- 1~(n z - 1) In 4ab + 1 E
"2--
(n z-¼)l. +¼n z
(17)
where ~12
I.
=
sin4,
0
d4, =
k=l 2 k
1 "
The frequency ~0~ =a-2~F2-f,g, vanishes at n = 1, which corresponds to a displacement without deformation. The expressions (17) coincide with those found by Grant [6] when the difference in scaling is taken into account. It is notable that the action of local and nonlocal terms is oppositely directed. Consider, for example, a segment of a distorted circle that has advanced ahead of the rest of the curve and is bent forward. The binormal is inclined locally towards the axis of motion, and therefore the local component of the induced velocity is directed inward, as is seen, indeed, from eq. (17). On the other hand, the nonlocal component of the velocity induced at this location by the rest of the circle, that is trailing behind, is directed outward. Formally, the local component must be stronger due to the presence of the logarithm of a large quantity. The nonlocal component turns out, however, to be growing with n faster than the local one. This is seen in fig. 1 showing the values of n corresponding to the change of the sign o f f , and g, at different values of the logarithm of the ratio x = 4able. Instability is observed when the coefficients f, and g, are of the same sign, which happens within a certain band of wavenumbers that shifts towards larger n with decreasing ~. Note that a similar phenomenon of destabilization of a vortex ring at a certain wavelength dependent on the vorticity distribution within the core is known in conventional fluid mechanics [17]. The intervals of instability for lower modes are indicated by horizontal lines in fig. 1. We note that the theory breaks down when the perturbation wavelength becomes comparable with the core size, i.e., at n~ = if(l). Numerically,
168
L.M. Pismen, A . A . Nepomnyashchy / Stability of vortex rings in a model of superflow n 10
/
.....I 1.5
I
I
I
2.5
3
3.5
in x
Fig. 1. Values of n corresponding to the change of the sign of f. (left curve) and g. (fight curve) at different values of the logarithm of the ratio x = 4ab/~. The intervals of instability are indicated by horizontal lines.
nE = ~(1); thus, the instability range falls into the region where the long-scale theory is no longer applicable. The common asymptotic value of both nonlocal terms in eq. (17) at n---~o0 is ln2[ln(4n)+ C - ½], where C is the Euler constant. The instability interval lies asymptotically at ln(ab/~n) = C - ½ , and its width vanishes at n---~oo. Numerically, e n / a = b exP(½- C ) ~ 1.41, which is outside the applicability region of the long-scale theory. The leading term in the short-scale asymptotics of the frequency, ~o = (n2/2a 2) In n, is determined exclusively by the nonlocal term, and coincides with the long-scale asymptotics of the spectrum of core oscillations obtained long ago for a straight-line vortex by Pitaevskii [16]. At wavelengths of the order of the core radius, the vortex is expected to be stabilized due to the depletion of the superfluid density within the core, and, indeed, numerical calculations [2,3] indicated that no instabilities arose in this range.
4. Sound emission by an oscillating vortex ring Effects of the second order in ~ can modify the results based on the Biot-Savart law. The effect appearing in this approximation is the emission of sound. We shall base the following analysis of the acoustic effects on the method of matched asymptotic expansion developed by Kambe [10], that expl6its a wide separation between the characteristic scale a of a vortex flow region and the acoustic wavelength h. These scales are well separated since the motion of the vortex ring is strongly subsonic, the Mach number being of the same order of magnitude as the ratio ~/a of the core size to the radius of the ring. The inner and outer solutions are matched at distances s from the ring center satisfying the inequality a ,~ s ~ A. In the inner region s = a , the spatial derivatives are scaled as V - a -1, and the estimate for the characteristic frequency following from the Bernoulli equation is to ~ a -2. In this region, both the first
L.M. Pismen, A.A. Nepomnyashchy / Stability of vortex rings in a model of superflow
169
and the last terms in eq. (7) can be neglected; the flow is effectively incompressible, and the velocity field can be computed using the Biot-Savart integral. For the purpose of matching with the acoustic region, it is sufficient to compute the flow potential at distances large compared with the ring radius a, which is given [1,10] by the multipole expansion
(P = p . v I + O: VVI+ ".., S
(18)
s
where
P=¼~r×dl,
O=-~r®r×dl.
(19)
It is easy to see that, for a weakly distorted circle, the dipole term vanishes in the linear approximation. The tensor O in the quadrupole term is presented after integrating by parts as c
O(t) = - ¼a3 ~b
-cos24, sin 24, 0
sin 24, 0 cos 24, 0 0 0
~'(4,,t) d4,.
(20)
A nonvanishing contribution to O is given by the second harmonic nonplanar distortion ~(4,, t)oc e i(2¢'÷0''). Higher harmonics contribute to higher terms of the multipole expansion, which, as one can show, generate acoustic waves with the intensity proportional to higher powers of the Mach number. Since the quadrupole flow field v = V ~ decays ocs-3 at s--,o% the terms ~tt and V2~ in eq. (7) are both scaled as s -4 at large distances, whereas the nonlinear term is scaled as s -6, and can be neglected. The solution of the resulting linear wave equation in the far (acoustic) region can also be presented in the form of a multipole expansion with coefficients determined by matching with the inner region. The leading quadrupole term for a vortex ring with a stationary center is given [10] by
For a ring propagating along the z axis with the speed v o = (2a) -1 ln(4ab/e) ~ c = E -1, eq. (21) should be applied in the comoving frame, and the distance from the center of the ring s replaced by the distance ~ from the point where the center had been located at the moment the sound wave was emitted. The coordinate of this point in the comoving frame is z = -ovo/c. A short computation yields o- = s + zM + ~t(M2), where M = vo/c ~ 1 is the Mach number. Transforming to cylindrical coordinates and using the second harmonic nonplanar circle distortion with the amplitude ; A brings eq. (21) to the form
~=lAa3e2i6
0 [1 0 (lei(O,t_k~)) ]
where k = to/c is the acoustic wavenumber. The contribution to • that has the slowest rate of decay at s--~ oo is obtained by applying the derivatives in eq. (22) to the exponential term only: = -8-A\-~/( a ~3 (kr) a e i(°~t-k~r + 2 ' )
.
(23)
The evolution equations for the average ring radius and for the amplitude of oscillations can be
L.M. Pismen, A . A . Nepomnyashchy / Stability o f vortex rings in a model o f superflow
170
obtained with the help of integral momentum and energy balances. Both energy and momentum of the flow field due to the vortex ring are concentrated in the inner region, while the outward flux due to acoustic emission can be computed in the outer wave field. The total kinetic energy of the flow field generated by an undistorted circular vortex is [1] T = 2"rr~ v- (r x d/) = 2~r2a In 4a____bb~
(24)
The only nonvanishing component of the total momentum is P~ = "~ ¢ r x dl = 2'rr2a2
T
(25)
Vo
Corrections to both values are quadratic in distortions. The energy flux density in the acoustic wave is [12] j = pv. Using p = - ~ t , v = V~ with • given by eq. (23), and integrating over the surface of a large cylinder with the radius and half-height R centered at the center of the vortex ring, gives the total energy loss due to acoustic emission: J = Re ~ ~*n .V~ dS R
R
=4,rr n--,=limR e ( f ~*(r, R)~zz(r, R) r dr + R f ~*, (R, z) qbr(R, z) dz ) o
o 1
x 57/2 dx = ~67r AZa6ck6 f (1 1+ +X2)
~--"~5 'l'l" A2a6ck 6 .
(26)
Note that ak = aw/c ~ (ac) -1 ~ 1; thus, the acoustic effects are very weak. The acoustic emission by higher harmonics is of a still higher order in ~, or in the Mach number. The leading term in the far field acoustic potential generated by higher harmonics is expressed as
~A a
ein, 0" ei(,%t_~,~) ~ A Or
Or n
(kr) n e i(°"'-k-~+n*)
(27)
This yields T ~ A2c(ak) 2(n+l). The momentum flux density, or acoustic stress, is [12] tr = v ® v = Re(V~* ® Vq~). We compute it on the surface of the same large cylinder as above. A nonvanishing contribution to the flux of the z-component of the total momentum is obtained due to the asymmetry of Vz = Oq~ OZ =
+M
Oor
due to the motion of the ring. The term proportional to the Mach number is responsible for the braking action leading to a decrease of the average radius of the ~ing, whereas the isotropic part of the acoustic emission causes the damping of deviations from the circular shape. Integrating over either end of the cylinder gives the total loss of the z-momentum due to acoustic emission:
L.M. Pismen, A.A. Nepomnyashchy / Stability of vortex rings in a model of superflow R
= 8rrM R--,~limR
f s-~l~(r,
0 = 1 7 r A 2 M a 6 k 6 = OoC-2j.
171
1
R)12r dr = ~Tr A 2 M a 6 k 6
fxS
(1 + x2) 7/2
0
(28)
O n e can see that the acoustic loss o f the m o m e n t u m is relatively inefficient, as the ratio of 2 f / P z to J / T is M 2 ~ 1. This ratio is o f the same o r d e r o f the m a g n i t u d e also for w e a k e r radiation due to the higher harmonics. T h u s , the e n e r g y o f perturbations is dissipated by isotropic acoustic emission b e f o r e it can cause substantial loss of the m o m e n t u m that could lead to the collapse o f the vortex ring.
Acknowledgements
L . M . P . a c k n o w l e d g e s the s u p p o r t o f the F u n d for P r o m o t i o n o f R e s e a r c h at the T e c h n i o n . A . A . N . a c k n o w l e d g e s the s u p p o r t o f the B a r e c h a F o u n d a t i o n .
References
[1] G.K. Batchelor, Introduction to Fluid Mechanics (Cambridge Univ. Press, Cambridge, 1967). [2] A.L. Fetter, Phys. Rev. A 140 (1965) 452. [3] A.L. Fetter, Ann. Phys. 70 (1972) 67. [4] T. Frisch, Y. Pomeau and S. Rica, Phys. Rev. Lett. 69 (1992) 452. [5] T. Frisch and S. Rica, Physica D 61 (1992) 158. [6] J. Grant, J. Phys. A 4 (1971) 695. [7] V.M. Gryanik, Izv. AN SSSR Fiz. Atm. Ok. 19 #2 (1983) 203. [8] H. Hasimoto, J. Fluid Mech. 51 (1972) 477. [9] C.A. Jones, S.J. Putterman and P.H. Roberts, J. Phys. A 19 (1986) 2991. [10] T. Kambe, J. Fluid Mech. 173 (1986) 643. [11] V.I. Klyatskin, Izv. AN SSSR Mekh. Zh. Gaz. #6 (1966) 87. [12] L.D. Landau and E.M. Lifschitz, Fluid Mechanics (Pergamon, Oxford, 1980). [13] M.J. Lighthill, Proc. R. Soc. London A 211 (1952) 564. [14] P. Newton and J.B. Keller, SIAM J. Appl. Math. 47 (1987) 959. [15] L.M. Pismen and J. Rubinstein, Physica D 47 (1991) 353. [16] L.P. Pitaevskii, Sov. Phys. JETP 13 (1961) 451. [17] P.G. Saffman, J. Fluid Mech. 84 (1978) 625. [18] K.W. Schwarz, Phys. Rev. B 31 (1985) 5782. [19] G.A. Williams, J. Low Temp. Phys. 89 (1992) 91. [20] S.E. Widnall, Ann. Rev. Fluid Mech. 7 (1975) 141.