Amplitude of waves in the Kelvin-wave cascade
AAmplitude of waves in the Kelvin-wave cascade
V. B. Eltsov and V. S. L’vov Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland Department of Chemical and Biological Physics,Weizmann Institute of Science, Rehovot 76100, Israel
Development of experimental techniques to study superfluid dynamics, in particular, applicationof nanomechanical oscillators to drive vortex lines, enables potential observation of the Kelvin-wave cascade on quantized vortices. One of the first questions which then arises in analysis of theexperimental results is the relation between the energy flux in the cascade and the amplitude ofthe Kelvin waves. We provide such relation based on the L’vov-Nazarenko picture of the cascade.Remarkably, the amplitude of the waves depends on the energy flux extermely weakly, as power onetenth.
In quantum turbulence, velocity fluctuations and vor-tex reconnections drive oscillating motion of quantizedvortices – Kelvin waves [1]. Kelvin waves interact non-linearly and support a cascade of energy towards smallerlength scales and larger wave numbers. In the currentlyaccepted picture of quantum turbulence [2], a quasi-classical hydrodynamics energy cascade at length scaleslarger than the intervortex distance (cid:96) is followed, aftersome cross-over region, by the Kelvin-wave cascade atscales smaller than (cid:96) [3]. The mutual friction dampsKelvin waves very efficiently, and the cascade is expectedto start to develop when the mutual friction α is well be-low 10 − [4]. As temperature and α decreases, the cas-cade extends to progressively smaller length scales andeventually, at the lowest temperatures, it is damped byemission of bosonic [5] or fermionic [6, 7] quasiparticlesby the oscillating vortex cores.The theory of the Kelvin-wave cascade was the subjectof controversy [8–12], until finally the L’vov-Nazarenkomodel got supported by numerical simulations [13, 14].The theory is built for a straight vortex with uniform oc-cupation of Kelvin wave modes along the length. Suchsituation never occurs in a typical experiment on quan-tum turbulence. Recently, progress in experimental tech-niques [15–17] enables controllable excitation of waves onstraight or nearly straight vortices, see Fig. 1 for possi-ble setups. Such experiments have potential to observeKelvin-wave cascade directly and thus allow comparisonto the theory. One of the first questions which analysisof such experiments poses is the relation of the energyflux carried by the cascade (observed, e.g., as an increaseof the damping of a nanomechanical agitator) to the am-plitude of the excited Kelvin waves. We provide suchrelation in this work.We assume that the Kelvin-wave cascade on a vortex oflength L [cm] carries the energy flux ˜ (cid:15) [erg/s] and startsfrom the wave number k min [cm − ]. Our goal is to findthe amplitude A k [cm] of the Kelvin wave with the wavenumber k [cm − ]. We start by noting that in the localinduction approximation the energy of a vortex line E v is given by the product of its length L and the vortex tension ν s E v = ν s L , ν s = ρ s κ Λ4 π , Λ = ln (cid:16) (cid:96)a (cid:17) . (1)Here ρ s is the superfluid density, κ is the circulation quan-tum, a is the vortex core radius and (cid:96) is the mean in-tervortex spacing or the size of the enclosing volume, inthe case of a single vortex. For a spiral Kelvin wave ofthe radius A k and wavelength λ k = 2 π/k , the increase ofthe length compared to that of the straight vortex is L k = (cid:18)(cid:113) λ k + (2 πA k ) − λ k (cid:19) Lλ k ≈ L π A k λ k , (2)where we assumed that A k (cid:28) λ k . Thus the total energydue to Kelvin waves is E kw = ±∞ (cid:88) k = ± k min ν s L k = L ∞ (cid:88) k = k min ν s A k k = L ν s k min (cid:90) ∞ k min A k k dk . (3)Comparing this result to the expression of the energy viathe Kelvin-wave frequency ω k and the combined occupa-tion number N k for modes with ± k [18] E kw = ρ s L ∞ (cid:90) k min E k dk , E k = ω k N k , ω k = κ Λ4 π k , (4)we find A k = k min κ N k . (5)The L’vov-Nazarenko spectrum is [18] E k = C LN κ Λ (cid:15) / Ψ / k / , C LN ≈ . , (6a)Ψ = 8 π Λ κ (cid:90) ∞ k min E k dk . (6b)Here (cid:15) is the energy flux per unit length and per unitmass. It is related to the flux ˜ (cid:15) as (cid:15) = ˜ (cid:15)Lρ s , [ (cid:15) ] = cm s . (7) a r X i v : . [ c ond - m a t . o t h e r] M a r vorticesmechanicalagitators(a) (b) FIG. 1. Fig. 1. Example configurations of vortex lines, agi-tated to generate Kelvin waves. (a) A single vortex, attachedto an oscillating device. (b) An array of vortices, stretchedbetween parallel plates and agitated by shear or torsional os-cillations of the plates.
Solving Eq. (6) for Ψ we getΨ = (12 πC LN ) / (cid:15) / κ / k / (8)and from Eq. (5) finally A k = 2 (cid:18) π C LN (cid:19) / k / (cid:15) / κ / k / ≈ . k / κ / k / (cid:18) ˜ (cid:15)Lρ s (cid:19) / . (9)Checking dimensions we find correctly [ A k ] = cm . Notethat A k ∝ ˜ (cid:15) / . Thus determination of the amplitudefrom the energy flux should be relatively reliable, whilethe reverse procedure is bound to be very uncertain.The total increase of the vortex line length due toKelvin waves can be found from the energy as L kw = E kw /ν s , where E kw is given by Eqs. (4), (6a) and (8): L kw = E kw ν s = L / (3 πC LN ) / (cid:15) / κ / k / . (10)Thus for the relative increase we get a simple formula L kw L = E kw E v = Ψ2 . (11)In cases, where instead of a single vortex, one con-siders a vortex array with the total length L occupyingvolume V with the density L = L/V = (cid:96) − (Fig. 1b), itmight be more convenient to operate with the standard3-dimensional energy flux ε per unit mass and unit vol-ume, [ ε ] =cm s − . Having geometry of Fig. 1b in mind,it is easy to see that ε = (cid:15) L . Then for the increase L kw of the vortex-line density due to Kelvin waves, we findusing Eqs. (8) and (11) L kw L = Ψ2 = (cid:104) πC LN ) εb L κ (cid:105) / ≈ . (cid:16) εb L κ (cid:17) / , (12) where we introduced b = k min (cid:96) ∼ . (13)We note that the numerical value of the prefactor inEqs. (9) and (12) should be taken with caution. In thecalculations we assume that the total energy of Kelvinwaves can be found by the integral (4) limited from be-low by k min with the scale-invariant spectrum (6). Inreality this spectrum was derived for k (cid:29) k min whilethe main contribution to E kw is coming from the region k (cid:39) k min . Behavior of the Kelvin-wave spectrum in thislong-wavelengths region may be different and, in general,is not universal.In some applications, the tilt θ of a vortex carryingKelvin waves with respect to the direction of the straightvortex is of interest. The averaged tilt angle can be con-nected to the length increase L kw = (cid:90) L (cid:113) θ ( z ) d z − L (cid:39) (cid:90) L tan θ ( z ) d z = 12 (cid:104) tan θ ( z ) (cid:105) L . (14)Together with Eq. (11) this results in (cid:104) tan θ ( z ) (cid:105) (cid:39) L kw L = Ψ , (15)where Ψ is given by Eq. (8).For example, let us consider a vortex of length L =100 µ m in superfluid He ( κ = 9 . · − cm /s, Λ = 17, ρ s = 0 .
14 g/cm ). Vortex is agitated with the fre-quency f = 30 kHz which we assume to set the longestKelvin wave length k min = (cid:112) π f /κ Λ ≈ . · cm − , λ k min ≈ . µ m. If the energy flux over the Kelvin-wavecascade is ˜ (cid:15) = 10 − erg/s, then we find that the ampli-tude of the waves at the largest scale is A k min ≈ . µ m,increase of the vortex length L kw ≈ µ m and the av-eraged tilt angle (cid:104) θ (cid:105) ≈ ◦ . We see that even such amoderate flux, which corresponds to working against thefull vortex tension ν s over ˜ (cid:15)/ν s f ≈ . µ m per period ofthe drive, can bring the vortex on the edge of the regimewhere the turbulence of Kelvin waves may still be con-sidered as weak.To conclude, we have found the dependence of the am-plitude of the Kelvin waves, of the length increase of thevortex, and of the average vortex tilt on the energy fluxcarried by the Kelvin-wave cascade. The results are ap-plicable in the regime of weak turbulence of Kelvin waves,which is uniform along the vortex. We stress that the am-plitude of the Kelvin waves, generated when a vortex ismechanically agitated, does not necessary coincide withthe amplitude of the motion of the agitator. Solving theproblem of excitation of Kelvin waves in a realistic ex-perimental geometry remains a task for future research.The work has been supported by the European Re-search Council (ERC) under the European Union’s Hori-zon 2020 research and innovation programme (GrantAgreement No. 694248). [1] R.J. Donnelly, Quantized Vortices in Hellium II (Cam-bridge University Press, 1991).[2] W.F. Vinen,
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