We study reconnections of quantum vortices by numerically solving the governing Gross-Pitaevskii equation. We find that the minimum distance between vortices scales differently with time before and after the vortex reconnection. We also compute vortex reconnections using the Biot-Savart law for vortex filaments of infinitesimal thickness, and find that, in this model, reconnection are time-symmetric. We argue that the likely cause of the difference between the Gross-Pitaevskii model and the Biot-Savart model is the intense rarefaction wave which is radiated away from a Gross-Pitaeveskii reconnection. Finally we compare our results to experimental observations in superfluid helium, and discuss the different length scales probed by the two models and by experiments.

Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations. PACS numbers: 67.25.dk, 47.32.C, 47.27.Gs Below a critical temperature, liquid helium becomes a two-fluid system in which an inviscid superfluid component coexists with a viscous normal fluid component. The flow of the superfluid is irrotational: superfluid vorticity is confined to vortex lines of atomic thickness around which the circulation takes a fixed value κ (the quantum of circulation). Superfluid turbulence [1, 2] is easily created by stirring either helium isotope ( 4 He or 3 He-B), and consists of a tangle of reconnecting vortex filaments which interact with each other and with the viscous normal fluid (which may be laminar or turbulent). The most important observable quantity is the vortex line density L (vortex length per unit volume), from which one infers the average distance between vortex lines, ≈ L −1/2 . Our interest is in the properties of superfluid turbulence and their similarities with ordinary turbulence.Experiments [3,4] have revealed that, if the superfluid turbulence is driven by grids or propellers, the distribution of the turbulent kinetic energy over length scales larger than obeys the celebrated k −5/3 Kolmogorov scaling observed in ordinary (classical) turbulence. Here k is the magnitude of the three-dimensional wavenumber (wavenumber and frequency are related by k = f /v, wherev is the mean flow). Numerical calculations performed using either the vortex filament model [5,6] or the Gross-Pitaevskii equation [7,8] confirm the Kolmogorov scaling. It is thought that the effect arises from the partial polarization of the vortex lines [1, 2, 9], but such effect has never been clearly identified. Another important experimental observation is that in both 4 He [10] and 3 He-B [11], the frequency spectrum of the fluctuations of L has a decreasing f −5/3 scaling typical of passive objects [6,12] advected by a turbulent flow. This latter result seems to contradict the interpretation of L as a measure of superfluid vorticity, ω = κL which is usually made in the literature [1,2,11,[13][14][15].In fact, from dimensional analysis, the vorticity spec- * andrew.baggaley@gla.ac.uk trum corresponding to the Kolmogorov law should increase with f (as f 1/3 ), not decrease. Since the vortex line density is a positive quantity, a better analogy is to the enstrophy spectrum: however in classical turbulence this spectrum is essentially flat [16,17], in disagreement with the helium experiments [10,11].The aim of this letter is to reconcile these two sets of observations (each separa...

To explain the observed decay of superfluid turbulence at very low temperature, it has been proposed that a cascade of Kelvin waves (analogous to the classical Kolmogorov cascade) transfers kinetic energy to length scales which are small enough that sound can be radiated away. We report results of numerical simulations of the interaction of quantized vortex filaments. We observe the development of the Kelvin-waves cascade, and compute the statistics of the curvature, the amplitude spectrum (which we compare with competing theories) and the fractal dimension. [16,17].If the temperature is relatively large (more than 1 K in 4 He), the turbulent kinetic energy contained in the superfluid vortices is transferred by the mutual friction [18] into the viscous gas of thermal excitations (the normal fluid) and then decays into heat; therefore a constant supply of energy (continuous stirring for example) is needed to maintain the intensity of the turbulence. If the temperature is relatively small (less than 1 K in 4 He), the normal fluid is negligible, but, despite the absence of viscous dissipation, the turbulence still decays [19,20]. The Kelvin-waves cascade [21][22][23] was proposed to explain this surprising effect.A Kelvin wave is a rotating sinusoidal or helical displacement of the core of vortex filaments away from its unperturbed position [24][25][26]. The dispersion relation of a Kelvin wave of angular frequency ω and wavenumber k along a straight vortex is [27]where a 0 is the vortex core radius and K n is the modified Bessel function of order n. In the long-wavelength approximation (ka 0 ≪ 1), the angular frequency reduces towhere γ = 0.5772 is Euler's constant; the negative sign in this expression means that the Kelvin wave propagates in the direction opposite to the orientation of the unperturbed vorticity. Eq. 1 was originally derived for a thin, hollowcored vortex in a perfect Euler fluid; a similar expression was obtained for a vortex in a Bose-Einstein condensate [28,29]. The simplified dispersion formula ω(k) = ck 2 , where c is a constant, is often used in the Kelvin-waves cascade literature.The Kelvin-waves cascade is the process in which the nonlinear interaction of Kelvin waves creates waves of shorter and shorter wavelength λ = 2π/k. At high enough temperature the mutual friction would quickly damp out the shorter Kelvin waves [27], but at low temperatures the cascade proceeds unhindered, until k is large enough that sound is efficiently radiated away (phonon emission) by rapidly rotating vortices [30][31][32][33]. There is thus an analogy * Electronic address: a.w.baggaley@ncl.ac.uk † Electronic address: c.f.barenghi@ncl.ac.uk