Instability of a Vortex Ring due to Toroidal Normal Fluid Flow in Superfluid 4He
aa r X i v : . [ c ond - m a t . o t h e r] S e p Instability of a Vortex Ring due to ToroidalNormal Fluid Flow in Superfluid He Bhimsen K. ShivamoggiUniversity of Central FloridaOrlando, Fl 32816-1364
Abstract
Vortex rings self-propelling in superfluid He are shown to be driven unstableby a toroidal normal fluid flow. This instability has qualitative similarities withthe
Donnelly-Glaberson instability of Kelvin waves on a vortex filament driven bythe normal fluid flow along the vortex filament. The growth rate of the presentinstability is found to be independent of the radius of the vortex ring.
Following Landau [1], one considers the superfluid He below the
Lambda point asan inviscid, irrotational fluid with thermal excitations superposed on that fluid. Theseexcitations are modeled by a normal fluid whose interactions via mutual friction withthe superfluid are mediated by vortices. The mutual friction models the scattering ofthermal excitations by the vortices (Feynman [6]) and was confirmed experimentallyvia the second sound attenuation in liquid He (Hall and Vinen [7], [8]). Vinen [9] gavea phenomenological derivation of the mutual friction force. Vortices in superfluid Heare, as Onsager [10] suggested, linear topological defects with the superfluid densityvanishing at the vortex core and the circulation around a vortex line quantized, whichwas confirmed experimentally by Vinen [11]. Thanks to the circulation quantizationconstraint, as Feynman [6] suggested, the only possible turbulent motion in a superfluidis a disordered motion of tangled vortex lines.Vortex rings are self-propelled three-dimensional toroidal structures (Saffman [12]) ,which are believed to play a key role in the mechanism of superfluid turbulence (Svis-tunov [14], Vinen [15]). Superfluid turbulence has been suggested to be the kineticsof merging and splitting vortex rings rather than the kinetics of tangled vortex lines(Nemirovski [16], Walmsley and Golov [17]). The emission of vortex rings from re-connections between vortex filaments is believed to facilitate the energy transfer from The mutual friction is known (Schwarz [2], Shivamoggi [3], [4]) to play the dual roles of driving forceand drag force and hence to produce both growth and decay of the vortex line length. We do not have anadequate understanding of the underlying roton-vortex scattering process in superfluid He yet (Donnelly[5]). In his vortex theory of matter, Kelvin [13] proposed to explain the behavior of atoms by consideringthem as vortex rings and knots in ether.
Kelvin-wave cascade in the ultra-lowtemperature regime (Svistunov [14], Zhu et al. [18], Kursa et al. [19]). Walmsley etal. [20] generated superfluid turbulence experimentally via collisions in a beam of uni-directional vortex rings in superfluid He in the limit of zero temperature ( . K ).Rayfield and Reif [21] used ions coming from a radioactive cathode to produce vortexrings and gave direct experimental confirmation for the existence of quantized circu-lation. The vortex ring nucleation process sets in when the ions are accelerated byimposed electric fields and reach a critical velocity (Walmsley and Golov [17]) . Themotion of the vortex rings was controlled and detected by tagging each ring with atrapped ion and the applied electric field enabled tuning the ring radius r to partic-ular values. Vortex rings in superfluids, thanks to the topological robustness due tothe quantization condition on the circulation, tend to be very stable (in contrast to theircounterparts in hydrodynamics) especially at very low temperatures, where the dissipa-tive effects are very small. On the other hand, the decay of the vortex rings in superfluid He was used by Bewley and Sreenivasan [22] to demonstrate energy dissipation in su-perfluid He near the lambda point through the energy transfer from the superfluid tothe normal fluid via mutual friction. Direct observation of vortex cores in superfluid He was accomplished by Bewley et al. [23] and Fonda et al. [24] by using small solidhydrogen particles as traces in liquid He.It may be noted that the generation of vorticity in superfluid He signifies, on theother hand, the local destruction of superfluidity (Landau [1]). So, vortices in superfluid He essentially behave like classical vortex filaments, barring quantum mechanicalfeatures associated with their circulations and extremely thin cores and inclusion ofthe mutual friction force . This was very adequately confirmed by the numericalsimulations of Schwarz [2], [27].The extremely thin cores of vortex filaments in superfluid He lead to a singularityin the vortex self-advection velocity according to the
Biot-Savart law in hydrodynamicswhich is resolved by an asymptotic calculation (Da Rios [28], Arms and Hama [29])called the local induction approximation (LIA). Arms and Hama [29] used the LIAto investigate the evolution of a perturbed vortex ring in hydrodynamics. Kiknadzeand Mamaladze [30] extended this investigation to consider evolution of a perturbedvortex ring in superfluid He and found that the mutual friction causes a decay of theperturbation on the vortex ring. The purpose of this paper is to investigate the effect ofa toroidal normal fluid flow on a perturbed vortex ring in superfluid He. Low temperature conditions favor the vortex-ring generation by keeping the thermal excitations(phonons) sufficiently low and hence the energy loss small. At higher temperatures, rotons appear andcause large energy losses. This scenario is however violated in vortex reconnection processes between two neighboring vortexfilaments which involve sharp distortions of the vortex filaments (Paoletti el al. [25], Bewley et al. [26])and the concomitant generation of Kelvin waves associated with helical displacements of the vortex cores(Svistunov [14]). The Kelvin waves on a linear vortex filament are known to be driven unstable by the normal fluid flow along the vortex filament (Shivamoggi [3], [4]). Stability of a Vortex Ring in a Superfluid
Upon including the mutual friction force (Hall and Vinen [7]- [9], Bekarevich and Kha-latnikov [31]) exerted by the normal fluid on a vortex ring, the self-advection velocityof the vortex ring as per the LIA is given by(the HVBK model ): v = γκ ˆ t × ˆ n + α ˆ t × (cid:0) U − γκ ˆ t × ˆ n (cid:1) − α ′ ˆ t × (cid:2) ˆ t × (cid:0) U − γκ ˆ t × ˆ n (cid:1)(cid:3) (1)where U is the normal fluid velocity (taken to be constant in space and time and pre-scribed (Schwarz [2], [27]), κ is the average curvature, and ˆ t and ˆ n are unit tangent andunit normal vectors, respectively, to the vortex ring, and γ = Γ ln( c/κa ) , where Γ isthe quantum of circulation, c is a constant of order unity and a ≈ . × − cm isthe effective core radius of the filament. α and α ′ are the mutual friction coefficientswhich are small (except near the lambda point) so the short-term vortex ring evolutionis only weakly affected by the mutual friction. However, it provides for a mechanismto stretch the vortex ring (which is inextensional in the LIA). The mutual friction termassiciated with α plays the dual roles of driving force and drag force (Schwarz [2], [27],Shivamoggi [3], [4]). We drop here the mutual friction term associated with α ′ because, • α > α ′ (Vinen and Niemela [32]), • it does not produce physically significant effects in comparison with those pro-duced by the mutual friction term associated with α (Shivamoggi [3], [4]).Let us write in cylindrical coordinates (Arms and Hama [29]), r = ( r + ˆ r )ˆ i r + ( wt + ˆ z )ˆ i z (2)where r is the unperturbed radius of the vortex ring, ˆ r and ˆ z are the deviations fromthe circular vortex ring in the r and z directions, respectively, and w is the uniformtranslational self-propelling velocity of the circular ring, given by Kelvin’s formula , w = Γ2 πr ln (cid:18) r a − (cid:19) . (3)Next, noting (Shivamoggi [3], [4]) that the destabilizing effect of the normal fluidflow is produced by the toroidal normal-fluid flow velocity component along the vor-ticity vector (see Appendix) . We therefore take here, U = U θ ˆ i θ . (4) Strictly speaking, the normal fluid flow should be determined as part of the meso-scale solution byaccounting for the back reaction of the vortices on the normal fluid. However, the HVBK model is valid ifthe length scales characterizing the flow in question are much larger than the intervortical distance so thevortex lines can be considered to be organized into polarized bundles. According to LIA, an arbitrary vortex filament experiences a self-induced motion, which may be ap-proximated locally as that of an osculating vortex ring of radius same as the local radius of curvature of thevortex filament. The normal fluid flow along the axis of the vortex ring was found in numerical simulations to keep thevortex ring stable (Kivotides et al. [33]. ˆ r t = σ ˆ z θθ + ασ (ˆ r θθ + ˆ r ) − αr U θ ˆ z θ (5a) ˆ z t = − σ (ˆ r θθ + ˆ r ) + ασ ˆ z θθ + αr U θ ˆ r θ (5b)where σ ≡ γ/r .Looking for solutions of the form, ˆ q ( θ, t ) ∼ e i ( mθ − ωt ) (6)equations (5a), (5b) give, ω + iασω (2 m − − σ m ( m − − iασ mU θ r (2 m −
1) = 0 . (7)Noting that α is small, (7) gives ω ≈ − iασ " (2 m − ∓ U θ r (2 m − σm ( m − ± σm p m − . (8)(8) shows that the vortex ring develops an instability produced by the toroidal normalfluid flow velocity component U θ (as in the case of a linear vortex filament (Shivamoggi[3], [4]) (see Appendix for a qualitative picture of this instability)). This instabilityalso has qualitative similarities with the Donnelly-Glaberson instability (Cheng et al.[34], Glaberson et al. [35]) of Kelvin waves on a vortex filament driven by the normalfluid flow along the undisturbed vortex filament. If one takes m = kr (Kiknadzeand Mamaladze [30]), (8) further shows that the growth rate for the present instabilityis essentially independent of the radius r of the vortex ring. This instability wouldmaterialize if the time required for this instability to develop is smaller than the timecharacterizing viscous decay of the toroidal normal fluid flow (which may be taken tobe ∼ O ( r /ν ) , ν being the kinematic viscosity of the normal fluid), i.e., kν (cid:0) k r − (cid:1) αU θ (2 k r − < (9)which favors large toroidal normal fluid flows and large vortex rings.It may be noted that, if the vortex ring remains closed, m = 1 , , ... , so the motionis periodic around the periphery of the vortex; m = 1 corresponds to the trivial case ofa uniform displacement of a circular vortex ring.Note that (8) reduces, • in the limit U θ ⇒ , to the result of Kiknadze and Mamaladze [30] - the effectof mutual friction is now to cause only decay of the perturbation on the vortexring; • in the limit α ⇒ (the hydrodynamics limit), to the result of Arms and Hama[29]. 4 Discussion
The effect of mutual friction on a self-propelling vortex ring in superfluid He is to pro-duce a decay of a perturbation imposed on the ring. However, a toroidal normal fluidflow is found to drive a vortex ring unstable. This instability has qualitative similaritieswith the
Donnelly-Glaberson instability of Kelvin waves on a vortex filament drivenby the normal fluid flow along the vortex filament. The growth rate of the present in-stability is further found to be essentially independent of the radius of the vortex ring.
Appendix: Stability of a Vortex Filament in a Superfluid
Consider a vortex filament aligned essentially along the x-axis in a superfluid (Shiva-moggi [3], [4]). Writing in cartesian coordinates, r = x ˆ i x + ˆ y ( x, t )ˆ i y + ˆ z ( x, t )ˆ i z (A.1)taking U = U ˆ i x (A.2)and neglecting the nonlinear terms, we obtain from equation (1), ˆ y t = − σ ˆ z xx + ασ ˆ y xx + αU ˆ z x (A.3a) ˆ z t = σ ˆ y xx + ασ ˆ z xx − αU ˆ y x . (A.3b)Putting Φ ≡ ˆ y + i ˆ z (A.4)equations (A.3) give, i Φ t = − σ Φ xx + αU Φ x (A.5)which may be viewed as a Schr¨odinger type equation for a non-conservative system(Caldirola [36], Kanai [37]). If one puts, Φ( x, t ) = Ψ( x ) e − iωt (A.6)equation (A.5) leads to σ Ψ xx − αU Ψ x + ω Ψ = 0 (A.7)which represents a harmonic oscillator with negative damping.Looking for solutions of the form Φ( x, t ) ∼ e i ( kx − ωt ) (A.8)equation (A.5) leads to: 5 = iαkU + σk . (A.9)(A.9) shows the destabilization of the circularly-polarized Kelvin waves propagatingalong the vortex filament . Acknowledgments
This work was started when the author held a visiting research appointment at theEindhoven University of Technology supported by a grant from the Burgerscentrum.The author is thankful to Professor Gert Jan van Heijst for his hospitality and helpfuldiscussions. I wish to express my sincere thanks to Professors Carlo Barenghi, GrishaFalkovich, Andrei Golov, Ladik Skrbek, Katepalli Sreenivasan and William Vinen fortheir valuable remarks. I am also thankful to Dr. Demosthenes Kivotides and LeosPohl for helpful discussions.
References [1] L. D. Landau:
J. Phys. (USSR) , 71, (1941).[2] K. W. Schwarz: Phys. Rev. B , 5782, (1985).[3] B. K. Shivamoggi: Phys. Rev. B , 012506, (2011).[4] B. K. Shivamoggi: Euro. Phys. J. B , 275, (2013).[5] R. J. Donnelly: Quantized Vortices in Helium II
Cambridge Univ. Press, (1991).[6] R. P. Feynman:
Prog. Low Temp. Phys. , 17, (1955).[7] H. E. Hall and W. F. Vinen: Proc. Roy. Soc. (London)
A 238 , 204, (1956).[8] H. E. Hall and W. F. Vinen:
Proc. Roy. Soc. (London)
A 238 , 215, (1956).[9] W. F. Vinen: in
Quantum Fluids , Ed. D.F. Brewer, North-Holland, p.74, (1966).[10] L. Onsager:
Proc. Int. Conf. Theor. Phys. , p. 877, Science Council of Japan,(1953).[11] W. F. Vinen:
Proc. Roy. Soc. (London)
A 260 , 218, (1961).[12] P. G. Saffman:
Vortex Dynamics , Cambridge Univ. Press, (1992).[13] Lord Kelvin:
Phil. Mag. , 15, (1867).[14] B. V. Svistunov: Phys. Rev. B , 3647, (1995).[15] W. F. Vinen: Phys. Rev. B , , 1410, (2000). This result continues to hold in the nonlinear regime as well (Shivamoggi [3], [4]).
Phys. Rep. , 85, (2013).[17] P. W. Walmsley and A. I. Golov:
Phys. Rev. Lett. , 245301, (2008).[18] T. Zhu, M. L. Evans, R. A. Brown, P. M. Walmsley and A. I. Golov:
Phys. Rev.Fluids , 044502, (2016).[19] M. Kursa, K. Bajer and T. Lipniacki: Phys. Rev. B , , 014515, (2011).[20] P. M. Walmsley, P. A. Tompsett, D. E. Zmeev and A. I. Golov: Phys. Rev. Lett. , 125302, (2014).[21] G. W. Rayfield and F. Reif:
Phys. Rev. Lett. , 305, (1963).[22] G. P. Bewley and K. R. Sreenivasan: J. Low Temp. Phys. , 84, (2009).[23] G. P. Bewley, D. P. Lathrop and K. R. Sreenivasan:
Nature , 588, (2006).[24] E. Fonda, D. P. Meichle, N. T. Ouelette, S. Hormoz and D. P. Lathrop:
Proc. Nat.Acad. Sci. , , 4707, (2014).[25] M. S. Paoletti, M. E. Fisher and D. P. Lathrop: Physica D , 1367, (2010).[26] G. P. Bewley, M. S. Paoletti, K. R. Sreenivasan and D. P. Lathrop:
Proc. Nat.Acad. Sci. , 13707, (2008).[27] K. W. Schwarz:
Phys. Rev. B , 2398, (1988).[28] L. S. Da Rios: Rend. Circ. Mat. Palermo , 117, (1906).[29] R. J. Arms and F. R. Hama: Phys. Fluids , 553, (1965).[30] L. Kiknadze and Yu. Mamaladze: J. Low Temp. Phys. , 321, (2002).[31] I. L. Bekarevich and I. M. Khalatnikov:
Sov. Phys. JETP , 643, (1961).[32] W. F. Vinen and J. J. Niemela: J. Low Temp. Phys. , 167, (2002).[33] D. Kivotides, C. F. Barenghi and D. C. Samuels:
Science , 777, (2000).[34] D. K. Cheng, M. W. Cromar and R. J. Donnelly:
Phys. Rev. Lett. , 433, (1973).[35] W. I. Glaberson, W. W. Johnson and R. M. Ostermeier: Phys. Rev. Lett. , 1179,(1974).[36] P. Caldirola: Nuovo Cimento , 393, (1941).[37] E. Kanai: Prog. Theor. Phys.3