E.V. Kozik and B.V. Svistunov (KS) paper "Symmetries and Interaction Coefficients of Kelvin waves", arXiv:1006.1789v1, [cond-mat.other] 9 Jun 2010, contains a comment on paper "Symmetries and Interaction coefficients of Kelvin waves", V. V. Lebedev and V. S. L'vov, arXiv:1005.4575, 25 May 2010. It relies mainly on the KS text "Geometric Symmetries in Superfluid Vortex Dynamics}", arXiv:1006.0506v1 [cond-mat.other] 2 Jun 2010. The main claim of KS is that a symmetry argument prevents linear in wavenumber infrared asymptotics of the interaction vertex and thereby implies locality of the Kelvin wave spectrum previously obtained by these authors. In the present note we reply to their arguments. We conclude that there is neither proof of locality nor any refutation of the possibility of linear asymptotic behavior of interaction vertices in the texts of KS.
The article considers implications of tilt symmetry (symmetry with respect to tilting of the coordinate axis with respect to which vortex motion is studied) in the non-linear dynamics of Kelvin waves. The conclusion is that although the spectrum of Kelvin wave is not tilt-invariant, this does not compromise tilt invariance of the Kelvin-wave cascade vividly argued now in the theory of superfluid turbulence. The article investigates the effect of strong kelvon interaction on the power-law exponent for the Kelvin-wave cascade and suggests a simple picture of the crossover from the classical Kolm…
I discuss the Comment by L'vov and Nazarenko (arXiv:1208.4593) aiming at refutation of my perviously published criticism of their mechanism of the Kelvin-wave cascade. It is important that in their Comment L'vov and Nazarenko admitted that the Hamiltonian, from which they derived their mechanism, is not tilt-invariant. This provides full ammunition to their critics, who believe that their mechanism is in conflict with tilt symmetry of the Kelvin-wave dynamics in an isotropic space.
A Kolmogorov-type cascade of Kelvin waves--the distortion waves on vortex lines--plays a key part in the relaxation of superfluid turbulence at low temperatures. We propose an efficient numeric scheme for simulating the Kelvin wave cascade on a single vortex line. The idea is likely to be generalizable for a full-scale simulation of different regimes of superfluid turbulence. With the new scheme, we are able to unambiguously resolve the cascade spectrum exponent, and thus to settle the controversy between recent simulations [1] and recently developed analytic theory [2]. [1] W.F. Vinen, M. Tsu…
As long as vorticity quantization remains irrelevant for the long-wave physics, superfluid turbulence supports a regime macroscopically identical to the Kolmogorov cascade of a normal liquid. At high enough wavenumbers, the energy flux in the wavelength space is carried by individual Kelvin-wave cascades on separate vortex lines. We analyze the transformation of the Kolmogorov cascade into the Kelvin-wave cascade, revealing a chain of three distinct intermediate cascades, supported by local-induction motion of the vortex lines, and distinguished by specific reconnection mechanisms. The most pr…
Theoretical considerations are made of superfluid turbulence in the Kelvin wave cascade regime at low temperatures (T < 1K) and length scales of the order or smaller than the intervortical distance. The energy spectrum is shown to be in accord with the Kolmogorov scaling. The vortex line decay equation is shown to have an underlying Hamiltonian framework. Effects of spatial intermittency (exhibited in laboratory experiments) on superfluid turbulence are incorporated via the fractal nature of the vortex lines, for length scales of the order or smaller than the intervortical distance. The spatia…
Kelvin waves propagating on quantum vortices play a crucial role in the phenomenology of energy dissipation of superfluid turbulence. Previous theoretical studies have consistently focused on the zero-temperature limit of the statistical physics of Kelvin-wave turbulence. In this letter, we go beyond this athermal limit by introducing a small but finite temperature in the form of non-zero mutual friction dissipative force; A situation regularly encountered in actual experiments of superfluid turbulence. In this case we show that there exists a new typical length-scale separating a quasi-inerti…
We performed numerical simulations of decaying quantum turbulence by using a generalized Gross-Pitaevskii equation, that includes a beyond mean field correction and a nonlocal interaction potential. The nonlocal potential is chosen in order to mimic He II by introducing a roton minimum in the excitation spectrum. We observe that at large scales the statistical behavior of the flow is independent of the interaction potential, but at scales smaller than the intervortex distance a Kelvin wave cascade is enhanced in the generalized model. In this range, the incompressible kinetic energy spectrum o…
Reconnections of quantum vortex filaments create sharp bends which degenerate into propagating Kelvin waves. These waves cascade their energy down-scale and their waveaction up-scale via weakly nonlinear interactions, and this is the main mechanism of turbulence at the scales less than the inter-vortex distance. In case of an idealised forcing concentrated around a single scale k0, the turbulence spectrum exponent has a pure direct cascade form -17/5 at scales k>k0 and a pure inverse cascade form -3 at k<k0. However, forcing produced by the reconnections contains a broad range of Fourier modes…
We study the statistical and dynamical behavior of turbulent Kelvin waves propagating on quantized vortices in superfluids, and address the controversy concerning the energy spectrum that is associated with these excitations. Finding the correct energy spectrum is important because Kelvin waves play a major role in the dissipation of energy in superfluid turbulence at near-zero temperatures. In this paper, we show analytically that the solution proposed in Ref. \cite{10LN} enjoys existence, uniqueness and regularity of the pre-factor. Furthermore, we present numerical results of the dynamical …
We present a comprehensive statistical study of free decay of the quantized vortex tangle in superfluid $^4$He at low and ultra-low temperatures, $0\leqslant T \leqslant 1.1\,$K. Using high resolution vortex filament simulations with full Biot-Savart vortex dynamics, we show that for ultra-low temperatures $T\lesssim 0.5 \,$K, when the mutual friction parameters $α\simeq α' < 10^{-5}$, the vortex reconnections excite Kelvin waves with wave lengths $λ$ of the order of the inter-vortex distance $\ell$. These excitations cascade down to the resolution scale $Δξ$ which in our simulations is of the…
We propose a different type of Landau instability in trapped Bose-Einstein condensates by a helically moving environment. In the presence of quantized vortices, the instability can cause spontaneous radiation and amplification of Kelvin waves. This study gives a microscopic understanding of the Donnelly-Glaberson instability which was known as a hydrodynamic instability in superfluid helium. The Donnelly-Glaberson instability can be a powerful tool for observing the dispersion relation of Kelvin waves, vortex reconnections, and quantum turbulence in atomic Bose-Einstein condensates.
On the basis of recently proposed scenario of the transformation of the Kolmogorov cascade into the Kelvin-wave cascade, we develop a theory of low-temperature cutoff. The theory predicts a specific behavior of the quantized vortex line density, $L$, controlled by the frictional coefficient, $α(T) \ll 1$, responsible for the cutoff. The curve $\ln L(\ln α)$ is found to directly reflect the structure of the cascade, revealing four qualitatively distinct wavenumber regions. Excellent agreement with recent experiment by Walmsley {\it et al.} [arXiv:0710.1033]--in which $L(T)$ has been measured do…
We comment on the claim by Lebedev and L'vov [arXiv:1005.4575] that the symmetry with respect to a tilt of a quantized vortex line does not yet prohibit coupling between Kelvin waves and the large-scale slope of the line. Ironically, the counterexample of an effective scattering vertex in the local induction approximation (LIA) attempted by Lebedev and L'vov invalidates their logic all by itself being a notoriously known example of how symmetries impose stringent constraints on kelvon kinetics---not only the coupling in question but the kinetics in general are absent within LIA. We further exp…
Kelvin waves (kelvons)--helical waves on quantized vortex lines--are the normal modes of vortices in a superfluid. At zero temperature, the only dissipative channel of vortex dynamics is phonon emission. Starting with the hydrodynamic action, we derive the Hamiltonian of vortex-phonon interaction, thereby reducing the problem of the interaction of Kelvin waves with sound to inelastic elementary excitation scattering. On the basis of this formalism, we calculate the rate of sound radiation by superfluid turbulence at zero temperature and estimate the value of short-wavelength cutoff of the turb…
By assuming a self-similar structure for Kelvin waves along vortex loops with successive smaller scale features, we model the fractal dimension of a superfluid vortex tangle in the zero temperature limit. Our model assumes that at each step the total energy of the vortices is conserved, but the total length can change. We obtain a relation between the fractal dimension and the exponent describing how the vortex energy per unit length changes with the length scale. This relation does not depend on the specific model, and shows that if smaller length scales make a decreasing relative contributio…
The energy transfer mechanism in zero temperature superfluid turbulence of helium-4 is still a widely debated topic. Currently, the main hypothesis is that weakly nonlinear interacting Kelvin waves transfer energy to sufficiently small scales such that energy is dissipated as heat via phonon excitations. Theoretically, there are at least two proposed theories for Kelvin-wave interactions. We perform the most comprehensive numerical simulation of weakly nonlinear interacting Kelvin-waves to date and show, using a specially designed numerical algorithm incorporating the full Biot-Savart equation…
We study the statistical properties of the Kelvin waves propagating along quantized superfluid vortices driven by the Gross-Pitaevskii equation. No artificial forcing or dissipation is added. Vortex positions are accurately tracked. This procedure directly allows us to obtain the Kevin-waves occupation-number spectrum. Numerical data obtained from long time integration and ensemble-average over initial conditions supports the spectrum proposed in [L'vov and Nazarenko, JETP Lett 91, 428 (2010)]. Kelvin wave modes in the inertial range are found to be Gaussian as expected by weak-turbulence pred…
Dynamics of quantized vortex lines in a superfluid feature symmetries associated with the geometric character of the complex-valued field, $w(z)=x(z)+iy(z)$, describing the instant shape of the line. Along with a natural set of Noether's constants of motion, which---apart from their rather specific expressions in terms of $w(z)$---are nothing but components of the total linear and angular momenta of the fluid, the geometric symmetry brings about crucial consequences for kinetics of distortion waves on the vortex lines---the Kelvin waves. It is the geometric symmetry that renders Kelvin-wave ca…
Rotating turbulence is ubiquitous in nature. Previous works suggest that such turbulence could be described as an ensemble of interacting inertial waves across a wide range of length scales. For turbulence in macroscopic quantum condensates, the nature of the transition between the quasiclassical dynamics at large scales and the corresponding dynamics at small scales, where the quantization of vorticity is essential, remains an outstanding unresolved question. Here we expand the paradigm of wave-driven turbulence to rotating quantum fluids where the spectrum of waves extends to microscopic sca…
We study the 3D forced-dissipated Gross-Pitaevskii equation. We force at relatively low wave numbers, expecting to observe a direct energy cascade and a consequent power-law spectrum of the form $k^{-α}$. Our numerical results show that the exponent $α$ strongly depends on how the inverse particle cascade is attenuated at $k$'s lower than the forcing wave number. If the inverse cascade is arrested by a friction at low $k$'s, we observe an exponent which is in good agreement with the weak wave turbulence prediction $k^{-1}$. For a hypo-viscosity, a $k^{-2}$ spectrum is observed which we explain…
