Finite-temperature effects in helical quantum turbulence

Physical Review

The theory of the He II thermal counterflow process in wide id)10 ' cm) channels is investigated on the assumption that both the normal and superQuid components make a transition from a laminar to a turbulent type of Qow. A critical heat current Wo is identified with the superQuid transition. The superQuid turbulent state is taken to be essentially that described by Vinen in terms of quantized vortex line and has an associated mutual friction. A second critical heat current lV, is identi6ed with the normal-Quid transition. It is argued that this transition is essentially of a classical turbulent type, with the added condition that the critical value of the Reynolds number must depend on the extent of mutual-friction coupling. This interpretation is shown to be consistent with experimentally observed critical heat currents, as well as with critical-velocity sects found in other types of Qow. The assumption of two crticial heat currents de6nes three distinct Qow regions. Xt is shown that these three regions are essentially the same as those found experimentally by Allen, Gri5.ths, and Osborne. On the basis of some simplifying assumptions regarding the normal-Quid turbulent state, the temperature and pressure gradients accompanying thermal counterQow are calculated. Comparison with experiment shows good qualitative and often quantitative agreement. It is also shown that the model developed can be successfully used to interpret experiments involving flows of a nonthermal counterQow type.

Physical Review B, 2021

We report an experimental study of oscillatory thermal counterflow of superfluid 4 He and its transition to quantum turbulence inspired by the work of Kotsubo and Swift [ Phys. Rev. Lett. 62, 2604 (1989)]. We use a pair of transversally oriented second-sound sensors to provide direct proof that upon exceeding a critical heat flux, quantized vorticity is generated in the antinodes of the longitudinal resonances of the oscillating counterflow. Building on modern understanding of oscillatory flows of superfluid 4 He [D. Schmoranzer et al., Phys. Rev. B 99, 054511 (2019)], we re-evaluate the original data together with ours and provide grounds for the previously unexplained temperature dependence of critical velocities. Our analysis incorporates a classical flow instability in the normal component described by the dimensionless Donnelly number, which is shown to trigger quantum turbulence at temperatures below ≈ 1.7 K. This contrasts with the original interpretation based on the dynamics of quantized vortices, and we show that for oscillatory counterflow, such an approach is valid only at temperatures above ≈ 1.8 K. Finally, we demonstrate that the instabilities occurring in oscillatory counterflow are governed by the same underlying physics as those in flow due to submerged oscillators and propose a unified description of high Stokes number coflow and counterflow experiments.

2010

In 2048 3 simulation of quantum turbulence within the Gross-Pitaevskii equation we demonstrate that the large scale motions have a classical Kolmogorov-1941 energy spectrum E(k) ∝ k −5/3 , followed by an energy accumulation with E(k) ≃ const at k about the reciprocal mean intervortex distance. This behavior was predicted by the L'vov-Nazarenko-Rudenko bottleneck model of gradual eddy-wave crossover [J. Low Temp. Phys., 153, 140-161 (2008)], further developed in the paper.

Physics Reports, 2013

The term ''quantum turbulence'' (QT) unifies the wide class of phenomena where the chaotic set of one dimensional quantized vortex filaments (vortex tangles) appear in quantum fluids and greatly influence various physical features. Quantum turbulence displays itself differently depending on the physical situation, and ranges from quasiclassical turbulence in flowing fluids to a near equilibrium set of loops in phase transition. The statistical configurations of the vortex tangles are certainly different in, say, the cases of counterflowing helium and a rotating bulk, but in all the physical situations very similar theoretical and numerical problems arise. Furthermore, quite similar situations appear in other fields of physics, where a chaotic set of one dimensional topological defects, such as cosmic strings, or linear defects in solids, or lines of darkness in nonlinear light fields, appear in the system. There is an interpenetration of ideas and methods between these scientific topics which are far apart in other respects. The main purpose of this review is to bring together some of the most commonly discussed results on quantum turbulence, focusing on analytic and numerical studies. We set out a series of results on the general theory of quantum turbulence which aim to describe the properties of the chaotic vortex configuration, starting from vortex dynamics. In addition we insert a series of particular questions which are important both for the whole theory and for the various applications. We complete the article with a discussion of the hot topic, which is undoubtedly mainstream in this field, and which deals with the quasi-classical properties of quantum turbulence. We discuss this problem from the point of view of the theoretical results stated in the previous sections. We also included section, which is devoted to the experimental and numerical suggestions based on the discussed theoretical models.

Acta Applicandae Mathematicae

We propose a mathematical interpolation between several regimes of energy cascade in quantum turbulence in He II. On the basis of a physical interpretation of such mathematical expression we discuss in which conditions it is expected to appear an intermediate $k^{2}$ k 2 regime (equipartition regime) in the transition region between the hydrodynamic regime and the Kelvin wave regime (namely, between the $k^{-5/3}$ k − 5 / 3 and $k^{-1}$ k − 1 regions in coflow situations and between the $k^{-3}$ k − 3 and $k^{-1}$ k − 1 regions in counterflow situations). It is seen that if the energy rate transfer from the hydrodynamic region to the Kelvin wave region is sufficiently slow, such equipartition region will be present, but for higher values of such energy rate transfer it will disappear. For high rates of the energy rate transfer, the transition regime between the hydrodynamic and the Kelvin wave regimes will be monotonous, characterized by a negative exponent of $k$ k between $-5/3$ −...

arXiv preprint arXiv: …, 2009

Quantum turbulence that exhibits vortex creation, annihilation and interactions is demonstrated as an exact solution of the time-dependent, free-particle Schroedinger equation evolved from a smooth random-phased initial condition. Relaxed quantum turbulence in 2D and 3D exhibits universal scaling in the steady-state energy spectrum as k −1 in small scales. Due to the lack to dissipation, no evidence of Kolmogorov-type forward energy cascade in 3D or inverse energy cascade in 2D is found, but the rotational and potential flow components do approach equi-partition in the scaling regime. In addition, the 3D vortex line-line correlation exhibits universal behavior, scaled as ∆r −2 , where ∆r is the separation between any two vortex line elements, in fully developed turbulence. We also show that quantum vortex is not frozen to the matter, nor is the vortex motion induced by other vortices via Biot-Savart's law. Thus, quantum vortex is actually a nonlinear wave, propagating at a speed very different from a classical vortex.

Turbulence in superfluid helium is unusual and presents a challenge to fluid dynamicists because it consists of two coupled, inter penetrating turbulent fluids: the first is inviscid with quantised vorticity, the second is viscous with continuous vorticity. Despite this double nature, the observed spectra of the superfluid turbulent velocity at sufficiently large length scales are similar to those of ordinary turbulence. We present experimental, numerical and theoretical results which explain these similarities, and illustrate the limits of our present understanding of superfluid turbulence at smaller scales.

2009

We present a numerical study of quantum turbulence within the three-dimensional Gross-Pitaevskii equation, concentrating on the direct energy cascade in the case of a forced-dissipated system.

Tangles of a quantized vortex line of initial density Lð0Þ ∼ 6 × 10 3 cm −2 and a variable amplitude of fluctuations of flow velocity Uð0Þ at the largest length scale are generated in superfluid 4 He at T ¼ 0.17 K, and their free decay LðtÞ is measured. If Uð0Þ is small, the excess random component of the vortex line length first decays as L ∝ t −1 until it becomes comparable with the structured component responsible for the classical velocity field, and the decay changes to L ∝ t −3=2. The latter regime always ultimately prevails, provided the classical description of U holds. A quantitative model of coexisting cascades of quantum and classical energies describes all regimes of the decay. Turbulent flow in classical fluids can be described [1] as a superposition of coherent vortices, possessing a large nonequilibrium energy and responsible for the cascade of this energy towards smaller length scales, and a random incoherent flow, which is in equilibrium and dissipates at the smallest scales by viscosity. The energy spectrum, i.e., contributions from velocity fluctuations at different length scales, adjusts self-consistently to maintain the continuity of the cascade's energy flux [2,3]. Vortices in superfluid 4 He are different [4] in that all of them have filamentary cores surrounded by an inviscid flow of identical velocity circulation κ ¼ h=m ¼ 0.997 × 10 −3 cm 2 s −1 (h and m being the Planck's constant and atomic mass of 4 He, respectively) [4]. Hence, turbulence in this system [quantum turbulence (QT)] is a tangle of vortex lines [5,6]. Yet, QT might be an analog of the classical scenario in that there are two coexisting structures [7]: One (flow around bundles of vortex lines) possesses all properties of the classical coherent vortices, while another incoherent component (flow around individual lines) is responsible for the transfer of energy towards the dissipa-tive processes at smaller scales and adjusts its own extent self-consistently. Importantly, the concept of the energy cascade is still potent [8]. Of special interest is the limit of zero temperature, T ¼ 0, at which QT is nondissipative down to length scales much smaller than the typical distance between vortices, l ¼ L −1=2 , where L is the length of the vortex line per unit volume. The nature and rate of the corresponding energy cascade and ultimate dissipative processes remain open questions of fundamental importance [9,10]. Two extreme cases of QT in the T ¼ 0 limit have been studied [11-13] and revealed different types of free decay LðtÞ. One ("ultraquantum" or "Vinen QT") is a random tangle of vortex lines with negligible velocity fluctuations at length scales r ≫ l. Such a tangle is fully described by L. Another limit ("quasiclassical" or "Kolmogorov QT") is that of partially polarized tangles with the dominant contribution to the energy coming from a flow around many vortex lines. Here a second parameter is required, the amplitude of velocity fluctuations U at the integral length scale L i-usually of the order of the container size D. Many questions remained. Which of these regimes is transient and which is the ultimate "equilibrium" type? Which parameters describe their interplay? What is the ratio of the contributions from the coherent and random components to the vortex length in the equilibrium state? Our experiment, in which vortex tangles were generated with a known value of U, answers these questions. The energy, per unit mass, of the turbulent state is the volume-averaged E ¼ 1 2 hv 2 i with velocity v given by the Biot-Savart integral over all vortex lines. We consider a developed bulk QT, for which L i ≫ l. Then there are two major contributions to the energy, E ¼ E q þ E c. In the near field r ≪ l, the quantum energy is dominated by the velocity of fluid circulating around individual lines: E q ≈ γκ 2 L; ð1Þ where γ ≈ lnðl=a 0 Þ=4π and the vortex core radius is a 0 ≈ 1.3 Å [4]. In our experiments, l is within the range 0.14-2 mm; hence, γ ≈ 1.2 AE 0.1 ≈ const. On the other hand, in the far field r ≫ l (classical length scales), the flow velocity arises from the contributions of many aligned vortex lines. If the forcing is at length scale D and Re s ≡ ðUD=κÞ ≫ 1 [14], then the coarse-grained velocity field should obey classical fluid dynamics with no dis-sipation. Hence, the Kolmogorov-like energy cascade [2] is expected, with the classical energy dominated by U: E c ≈ 1 2 U 2 : ð2Þ In the T ¼ 0 limit, the energy could be removed by either phonon emission due to short-wavelength Kelvin waves [15,16] or the diffusion of small vortex rings [9,10,

2013

Superfluid Turbulence is unusual and presents a challenge to fluid dynamicists because it consists of two coupled, inter penetrating turbulent fluids: the first is inviscid with quantised vorticity, the second is viscous with continuous vorticity. Despite this double nature, the observed spectra of the superfluid turbulent velocity at sufficiently large length scales are similar to those o ordinary turbulence. We present experimental, numerical and theoretical results which explain these similarities, and illustrate the limits of our present understanding of superfluid turbulence at smaller scales.