SPECIAL FEATURE: INTRODUCTION INTRODUCTION SPECIAL FEATURE: Introduction to quantum turbulence

Carlo F. Barenghia,1, Ladislav Skrbekb, and Katepalli R. Sreenivasanc aJoint Quantum Centre Durham-Newcastle and School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom; bFaculty of Mathematics and Physics, Charles University, 12116 Prague, Czech Republic; and cDepartment of Physics and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 The term quantum turbulence denotes the turbulent motion of quantum fluids, systems such as superfluid helium and atomic Bose–Einstein condensates, which are characterized by quantized vorticity, superfluidity, and, at finite temperatures, two-fluid behavior. This article intro- duces their basic properties, describes types and regimes of turbulence that have been observed, and highlights similarities and differences between quantum turbulence and classical turbulence in ordinary fluids. Our aim is also to link together the articles of this special issue and to provide a perspective of the future development of a subject that contains aspects of fluid mechanics, atomic physics, condensed matter, and low-temperature physics.

quantized vortices | absolute zero | Kolmogorov spectrum

Turbulence is a spatially and temporally com- Quantum Fluids resulting either in superconductivity in plex state of fluid motion. Five centuries ago, In this series of articles, we shall be concerned charged systems of electrons in crystal lattices Leonardo da Vinci noticed that water falling almost exclusively with superfluid 4He, the or in superfluidity in systems consisting of into a pond creates eddies of motion. Today, B-phase of superfluid 3He,and,toalesser neutral atoms. The outcome is surprisingly turbulence still provides physicists, applied extent, with ultracold atomic gases. These similar to what happens for bosons: Super- mathematicians, and engineers with a con- systems exist as fluids at temperatures on the fluidity arises from the formation of a co- tinuing challenge. Leonardo realized that the order of a Kelvin, milliKelvin, and micro- herent particle field that can be described ‡ motion of water shapes the landscape. Today’s Kelvin, respectively. Their constituents are using the formalism of the order parameter researchers appreciate that many physical either bosons (such as 4He atoms with zero or a condensate macroscopic wave function. processes, from the generation of the Galactic spin) or fermions (such as 3He atoms with Superfluidity of 4He was experimentally magnetic field to the efficiency of jet engines, spin 1/2). This difference is fundamental: discovered by Kapitza and by Allen and depend on turbulence. the former obey Bose–Einstein statistics and Misener in 1938 although it is now believed The articles in this collection are devoted the latter Fermi–Dirac quantum statistics. that Kamerlingh Onnes must have had su- to a special form of turbulence known as Letusconsideranideal(noninteracting) perfluid helium in his apparatus when he first quantum turbulence (1–3), which appears gas of bosons first. Under normal conditions liquefiedheliuminLeidenin1908.Heand in quantum fluids. Quantum fluids differ at room temperature, the de Broglie wave- other pioneers of low-temperature physics from ordinary fluids in three respects: (i) length λ of each atom is much smaller than soon discovered that, below a critical tem- they exhibit two-fluid behavior at nonzero the average separation d between the atoms; perature Tλ ≅ 2:17 K, liquid helium displays temperature or in the presence of impu- if the temperature T is lowered, λ increases unusual behavior. They therefore called it rities, (ii) they can flow freely, without the until, if T is sufficiently small, λ becomes helium I and helium II, respectively, above dissipative effect of viscous forces, and larger than d, and the quantum–mechanical and below this temperature. Still, in 1938, (iii) their local rotation is constrained to wave aspects become dominant. The result- London linked the properties of helium II to discrete vortex lines of known strength ing phase transition, called the Bose–Einstein Bose–Einstein condensation. Further prog- (unlike the eddies in ordinary fluids, which condensation (5), is characterized by a mac- ress in understanding superfluidity of 4He are continuous and can have arbitrary size, roscopic number of bosons occupying the was driven by the work of Landau based on shape, and strength). Superfluidity and quan- state of zero energy. Although the possibility different considerations. tized vorticity are extraordinary manifesta- of Bose–Einstein condensation was raised in The physical properties of normal liquid tions of quantum mechanics at macroscopic- 1924–1925, its direct experimental demon- 3He at milliKelvin temperatures are well de- length scales. stration in dilute ultracold atomic gases oc- scribed in the frame of the Fermi liquid theory Recent experiments have highlighted quan- curred only in 1995. To achieve superfluidity titative connections, as well as fundamental (flow without friction), another ingredient is Author contributions: C.F.B., L.S., and K.R.S. performed research differences, between turbulence in quantum necessary: The particles must interact with and wrote the paper. fluids and turbulence in ordinary fluids each other. The authors declare no conflict of interest. (classical turbulence). The relation between In fermionic systems, at temperatures 1To whom correspondence should be addressed. E-mail: c.f. thetwoformsofturbulenceisindeeda much lower than a characteristic Fermi tem- [email protected]. † common theme in the articles collected here. perature, particles occupy the interior of The term quantum turbulence was introduced into the literature Because different scientific communities (low- the Fermi sphere in the momentum space, in 1986 by R. J. Donnelly in a symposium dedicated to – the memory of G. I. Taylor (4). temperature physics, condensed-matter phys- with only relatively few particle hole pairs, ‡ What determines the need for a quantum mechanical description ics, fluid dynamics, and atomic physics) have called excitations. Attractive interaction be- is not the absolute value of temperature but whether it is contributed to progress in quantum turbu- tween fermions leads to an instability and lower than a certain characteristic temperature of the system (for † example, the Fermi temperature); instances of quantum fluids lence, the aim of this article is to introduce the formation of Cooper pairs, which are at high temperature are exciton-polariton condensates ( ≈ 300 K) the main ideas in a coherent way. bosons undergoing Bose condensation, and neutron stars (106 to 109 K).

www.pnas.org/cgi/doi/10.1073/pnas.1400033111 PNAS | March 25, 2014 | vol. 111 | suppl. 1 | 4647–4652 Downloaded by guest on September 29, 2021 of Landau. Note the striking difference in emission of quasiparticles called rotons). In mechanism when helium is cooled through kinematic viscosities of 4He (the lowest of a more general sense, the Landau criterion the superfluid second-order phase transition: all known fluids, two orders of magnitude less applies to any superfluid: On exceeding cer- The phase of ψ, unable to adjust everywhere than water’s) and of 3He near the superfluid tain critical velocity (which in fermionic at the same time, leaves vortex lines as defects. transition (comparable with that of air or superfluids is called the Landau pair-braking The possibility of quantum turbulence was olive oil). Superfluidity of 3He was theoreti- velocity), it becomes energetically favorable first raised by Feynman (6); soon afterward, cally predicted by Pitaevskii and experimen- to generate quasiparticles, which means the Vinen showed that a turbulent vortex tangle tally discovered by Osheroff, Richardson, and onset of dissipation. can be generated in the laboratory (7) by Lee in 1973. Cooper pairs consisting of two What makes superfluid hydrodynamics applying a heat flux to helium II. Two 3 He atoms (which themselves are fermions), particularly interesting is that the circulation properties of vortex lines are important for # v · dr rotating about their center of mass, are bosons integral C s is equal either to the quantum turbulence. The first is the mutual of total spin and orbital numbers equal to quantum of circulation κ = h=m or to zero, friction force (7, 8), which couples the su- one. This property allows three different depending on whether or not the integration perfluid and the normal fluid. It arises from projections on quantization axes in or- path C encloses a quantized vortex line; here, the scattering of thermal quasiparticles (con- h ’ m bital and spin spaces, and, as a conse- is Planck sconstantand the mass of the stituents of the normal fluid) by the velocity 4 quence, several different superfluid phases relevant boson (one atom in He, a Cooper field of the vortex lines. The second property of 3He exist. The classification of them and pair in 3He-B). This quantization condition, 3 comprises Kelvin waves, which are helical of the types of quantized vorticity in He is suggested by Onsager and experimentally displacements of the vortex core that rotate beyond the scope of this article (in partic- confirmed by Vinen, arises from the exis- with angular velocity ω ∼ κk2, where k is the ular, because quantum turbulence has been tence and the single-valuedness of a complex, one-dimensional wavenumber (shorter waves ψ studied only in the B phase). macroscopic superfluid wave function ,and rotate faster). Kelvin waves arise from the Finally, recently developed experimental the usual quantum mechanical prescription tension of the vortex lines (the kinetic energy methods of laser and evaporative cooling thatthevelocityisproportionaltothegra- ψ of a circulating superfluid about a unit length have opened up a new road to ultralow- dient of the phase of . As a consequence, the of line). Their direct observation is reported temperature physics: MicroKelvin clouds of superflow is not only inviscid (like the ideal ∇ × v = in the article by Fonda et al. (9). At finite dilute atoms were generated and nanoKelvin Euler fluid), but also potential ( s 0). temperatures, Kelvin waves are damped by temperatures achieved to explore quantum- Vortex lines can be viewed as holes with mutualfriction,but,below1K,theypropa- degenerate gases, providing additional work- circulation. Moving around the vortex axis, gate more freely and lead to acoustic emission ing fluids to study quantum turbulence. the phase of ψ changes by 2π (multiple values at large values of k. The transfer of energy to The two-fluid model, introduced in the of κ areunstableinheliumII),corresponding such large k by a Kelvin wave cascade context of 4He first by Tizsa and (based on to a persistent azimuthal superfluid velocity (analogous to the Kolmogorov cascade of different considerations) by Landau, is a of the form vs = κ=ð2πrÞ where r is the radial classical turbulence) explains the observed convenient level of description of quantum distance from the axis. An isolated vortex line 4 decay of turbulence at low temperatures, as turbulence. Below Tλ, He is described as a is thus a stable topological defect. On its axis, discussed in the articles by Barenghi et al. viscous normal fluid (a gas of thermal exci- real and imaginary parts of ψ vanish; the tations, called phonons and rotons, that narrow region where the density drops from (10) and by Walmsley et al. (11); in the carry the entire entropy content) coexist- its value at infinity to zero is proportional to weak-amplitude regime, Kelvin waves can ing with an inviscid superfluid (related but the healing (or coherence) length ξ, which be studied using wave-turbulence theory not equal to the condensate fraction). The depends on the strength of the interaction [see the article by Kolmakov et al. (12)]. density of helium II, essentially temperature- between the bosons. In 4He, ξ ≈ 10−10m; in The difference between the ideal fluid and independent, can be decomposed into ρ = 3He-B, ξ is about 100 times larger, and in the superfluid can be better appreciated by noticing the link between superfluidity and ρn + ρs,wherethenormalfluid’s and super- atomic condensates even larger (1/100–1/10 superconductivity, and the relation between fluid’s densities, ρn and ρs, respectively, are of the system size). strongly temperature-dependent: in the low- Quantized vortex lines are nucleated in- an ideal conductor and a superconductor; the temperature limit (T → 0), helium is entirely trinsically or extrinsically (that is, from al- latter behaves as an ideal diamagnetic sub- superfluid (ρs=ρ → 1, ρn=ρ → 0) whereas, in ready existing vortex lines, which, twisting stance that, below certain critical conditions, the high temperature limit (T → Tλ), super- under the influence of the superflow and then expels the externally applied magnetic field fluidity vanishes (ρs=ρ → 0, ρn=ρ → 1). At reconnecting, generate new vortex loops). from its interior (the Meissner effect). In temperatures below 1 K (where ρn=ρ = 0:07), Nucleation is opposed by a potential barrier, superfluidity, the corresponding feature is 3 4 in the absence of He impurities, He can be which, upon exceeding a critical velocity vc, that the superflow is always curl-free, or po- considered more or less a pure superfluid. canbeovercomeeitherthermallyorby tential, independently on whether rotating or Similar considerations apply for superfluid quantum tunneling. In helium II (except quiescent samples were cooled through the 3 He; its B phase can be considered a pure close to Tλ), intrinsic nucleation requires superfluid transition. Quantized vortices exist superfluid below about 200 μK. vc ≈ 10m=s, large enough to make it unlikely in superconductors and may reconnect [the The normal fluid and the superfluid sup- unless induced by a fast-moving ion. Experi- physical quantity that is quantized here is the − port two independent velocity fields vn and mentally reported values of vc ≈ 10 2m=sare magnetic flux, in units 2πZ=ð2eÞ,where2e is vs, respectively, and the superfluid compo- associated with extrinsic nucleation from the charge of two electrons constituting a nent flows without viscous dissipation. Based remnant vortices pinned to the walls of Cooper pair]. The motion of vortex lines and on the form of the dispersion relation, Lan- the container. In 3He-B, both intrinsic and flux tubes, however, is not the same: If dis- dau predicted that the superfluidity of 4He extrinsic nucleation are possible, and (as in placed, the former move (almost) along the − disappears at flow velocities exceeding a crit- atomic condensates) vc ≈ 10 3m=s. Vortex binormal and the latter (almost) along the icalvalueofabout60m/s(duetothe lines can also form by the Kibble–Zurek normal direction (13), which possibly explains

4648 | www.pnas.org/cgi/doi/10.1073/pnas.1400033111 Barenghi et al. Downloaded by guest on September 29, 2021 the absence of quantum turbulence in quantum turbulence created by ultrasonic The microscopic model is the Gross–Pit- INTRODUCTION superconductors. transducersatabout1:5K (24). An improved aevski equation for a Bose–Einstein conden- SPECIAL FEATURE: technique based on negative ions has been sate, obtained (after suitable approximations) Experimental Methods for Quantum recently introduced for measurements of from the Hamiltonian of a Bose gas under- Turbulence decaying quantum turbulence below 1 K (25, going two-bodies collisions. The Madelung Some of the experimental methods used to 26). Negative ions (electron bubbles) are transformation makes the hydrodynam- probe turbulence in ordinary viscous fluids injected by a sharp field-emission tip and ics interpretation of the wave function ψ have been used for quantum turbulence. manipulatedbyanappliedelectricfield.Bare apparent, yielding the classical continuity They include small Pitot tubes (14) to mea- ions dominate for T > 0:8 K whereas, for equation and a modified Euler equation. sure pressure-head fluctuations (giving access T < 0:7 K, they become self-trapped in the It differs from the classical Euler equation to velocity probability density distributions, core of quantized vortex rings of diameter because of the presence of the so-called structure functions, and energy spectra) and about 1 μm(theringsareintrinsicallynu- quantum pressure, which, differentiating a a plethora of small mechanical oscillators, cleated when the ions are accelerated past vc superflow from a perfect Euler flow, is re- such as spheres, wires, nanowires, grids, and by an imposed electric field). Short pulses of sponsible for vortex reconnections (31), for quartz tuning forks (for recent review, see ref. ions or rings are sent across the experimental sound pulses at reconnection events (32), 15), that can generate and detect quantum cell.Therelativereductionoftheamplitude and for the nucleation of vortices near a turbulence. of pulses of ions or rings detected at the boundary (e.g., an ion) (33) or a strong den- Direct visualization is invaluable in clas- collector on the opposite side of the helium sity variation (e.g., cavitation) (24, 34). sical turbulence: Methods such as particle cell after their interaction with quantum tur- The Gross–Pitaevskii equation has been image and particle-tracking velocimetry, bulence is converted to vortex-line density. used to study turbulence in atomic conden- applied to scientific and industrial prob- In 3He-B, information on quantum tur- sates (35, 36), but its application to superfluid lems, give quantitative data and qualitative bulence has been obtained using NMR (27). helium is only qualitative. Its dispersion re- information such as flow patterns. Although Another experimental technique in 3He-B is lation does not exhibit the roton minimum the application of these methods to cryo- the Andreev scattering of quasiparticles by and is valid only near T = 0. For generaliza- genic flows is difficult for reasons that are the velocity field of quantized vortices (28), tions to finite temperature, we refer the both technical (e.g., optical access to the described in the article by Fisher et al. (29). reader to the article of Berloff et al. (37). experimental volume) and fundamental Finally, in atomic Bose–Einstein conden- One approach worth mentioning is the (e.g., the presence of two velocity fields and sates, vortices are created by stirring or Zaremba–Nikuni–Griffin formalism (38), interaction (16) of quantum vortices with shaking the trap, phase imprinting, or mov- which couples the Gross–Pitaevskii equa- particles—in most cases, μm-sized frozen ing a laser “spoon” across the condensate; tion to a Boltzmann equation for the thermal flakes of solid hydrogen or deuterium), it images are taken after releasing the trap and cloud of noncondensed atoms, allowing has already led to important results on the expanding the condensate, as explained by atomic collisions within the thermal cloud direct observation of individual quantized White et al. (30). and between thermal cloud and condensate: vortices (17), their reconnections (18), La- The outcome of this self-consistent interac- grangian velocity (19), and acceleration (20) Theoretical Models of Quantum tion is the emergence (39) of dissipative statistics. Another visualization technique Turbulence effects on vortex motion (mutual friction). (21), based on fluorescence, employs neutral Unlike classical turbulence, studied on the A mesoscale approach that shuns effects at p – ξ He2 triplet molecules as tracers. solid ground of the Navier Stokes equation, the scale of is the vortex-filament model of An advantage of these conventional there is no single equation governing the Schwarz (40), which represents vortex lines methods is that they allow direct comparison motion for quantum turbulence, but rather as space curves of infinitesimal thickness and between classical turbulence above Tλ and a hierarchy of models at different length circulation κ.AtT = 0, a point on a vortex quantum turbulence below Tλ,but,inthe scales, each with its own limitations. It is as if line moves with the total superfluid velocity latter case, care must be taken to understand one is unable to describe the trees and the at that point—which is the self-induced ve- whether particles trace the motion of the forest in a unified way: We need one (mi- locity calculated using the Biot–Savart law, normal fluid, the superfluid, or the vortex croscopic) model that accounts for the close- plus any externally imposed superflow. For lines. The current status of the subject is up details of one tree or few trees, a second T > 0, the motion results from the balance of described in the article by Guo et al. (22). (mesoscopic) model that, from further away, Magnus and friction forces. Schwarz’sinsight Second-sound attenuation is the most does not resolve the details of the trees but was to recognize that, to describe quantum powerful (and historically the oldest) (7) still distinguishes individual trees as isolated turbulence, his equation of motion must be measurement tool in 4He, revealing the sticks, and a third (macroscopic) model that supplemented with an algorithmic procedure vortex-line density L—the total length of does not resolve trees at all but recognizes to reconnect vortex lines that approach each the quantized vortex line in a unit volume where the forest is sparser or denser. In this other by a distance less than the discretiza- (23). Because second sound is an antiphase spirit, we note that helium turbulence is tion distance along the lines (thus moving oscillation of normal and superfluid compo- characterized by a wide separation of length away from the realm of Euler dynamics). nents, this technique cannot be used below 1 scales ξ ℓ D,whereξ (already defined) The vortex-filament model is perhaps the K(becausethereislittlenormalfluid)orin is a measure of the vortex core, ℓ is the aver- most useful and flexible numerical tool for 3He at any temperature (second-sound waves age distance between vortex lines (usually quantum turbulence in 4He and 3He-B; the are overdamped by the large viscosity of the estimated as ℓ ≈ L−1=2), and D is the size state of the art is described by Baggaley and normal fluid). of the system: typically, ξ ≈ 10−10min4He Hänninen (41). It is therefore important to Helium ions have been successfully used to (10−8min3He-B), ℓ ≈ 10−5m, and D ≈ appreciate its limitations. The first is that detect quantum vortices in 4He—for example, 10−2m. In atomic condensates, these scales (unlike the Gross–Pitaevskii equation) it does to investigate the decay of inhomogeneous are not as widely separated: ξ < ℓ < D. not describe acoustic losses of energy by

Barenghi et al. PNAS | March 25, 2014 | vol. 111 | suppl. 1 | 4649 Downloaded by guest on September 29, 2021 rapidly rotating Kelvin waves at very low dependence of superfluid and normal fluid coupled turbulent systems, one in which the temperature. The second limitation is that components and the relatively high kine- vorticity is continuous and the other with the computational cost of the Biot–Savart law matic viscosity of 3He-B compared with 4He quantized vortex lines. The mutual friction grows rapidly as N2 (where N is the number suggest the following natural distinction: transfers energy from one fluid to the other of discretization points N along the vortex i) Pure quantum (superfluid) turbulence in so that it can act both as a source and a sink lines). To speed up his calculations, Schwarz low-temperature 4He and 3He-B. This is of energy for each fluid. Moreover, by replaced the Biot–Savart law with its local conceptually the simplest (but experimen- combining thermal and mechanical drives, induction approximation, which neglects any tally the most challenging) form of quantum special types of turbulence can be generated vortex interaction and requires an arbitrary turbulence: a single turbulent superfluid (the in which the mean superfluid and normal mixing step to achieve a statistically steady normal fluid being absent or negligible). This fluid velocities are not necessarily the same. stateofturbulence(42)—an approximation prototype of turbulence—a tangle of quan- For example, thermal counterflow is induced that is thought as unsatisfactory by todays’s tized vortex line—can be excited at small by applying a voltage to a resistor (heater) standards. This problem was recently solved length scales by injecting ions or vortex rings located at the closed end of a channel that (43, 44) by adapting to vortex dynamics the (26), or at larger length scales using vibrating is open to a superfluid 4He bath at the other N log N tree-algorithm created for computa- objects (15, 28), or by suddenly halting the end. The heat flux is carried away from the tional astrophysics (45). The third difficulty rotation and destabilizing an existing vortex heater by the normal fluid alone, and, by of Schwarz’s model is that (with notable ex- lattice (25). Besides the residual friction po- conservation of mass, a superfluid current ceptions) (46, 47) the normal fluid velocity is tentially caused by thermal excitations, dissi- occurs in the opposite direction. In this way imposed rather than computed self-consis- pation of kinetic energy is possible due to a relative (counterflow) velocity is created tently by solving the Navier–Stokes equation acoustic emission from short and rapidly along the channel that is proportional to the (suitably modified by a friction term): again, rotating Kelvin waves (55) and from vortex applied heat flux that is quickly accompanied the reason is the computational cost involved. reconnections (31). The length scale required by a tangle of vortex lines (7, 64). The The problem of self-consistency is solved, for efficient acoustic emission is much shorter normalfluidislikelytobelaminarforsmall at a more macroscopic level, by the Hall– than the typical curvature at the quantum heat fluxes and probably turbulent for large − = Vinen–Bekharevich–Khalatnikhov (HVBK) length scale ℓ ≈ L 1 2, but can be achieved heat fluxes. Another special case is pure equations, originally developed for rotating by a Kelvin wave cascade—which is the superflow (23), generated both mechanically helium. The HVBK equations describe the energy transfer to increasingly smaller scales and thermally in a channel whose one end or motion of fluid parcels containing a large arising from the nonlinear interaction of both ends are covered by superleaks (walls number of parallel vortex lines. Coarse Kelvinwaves(56,57);thismechanismis with holes so tiny that they are permeable graining allows treating superfluid vorticity discussed by Barenghi et al. (10)). In 3He-B, only to the superfluid component). as a continuous classical field, generalizing the larger vortex core limits the wavenumber A second possible classification of quan- the original two-fluid equations of Landau. range of the Kelvin wave cascade, but Caroli– tum turbulence is based on the form of the The HVBK equations successfully predict Matricon bound states in the vortex core energy spectrum EðkÞ—the distribution of the oscillation of a rotating vortex lattice, the provide an additional dissipation mecha- kinetic energy over the wavenumbers k (in- Glaberson instability, the instability of he- nism (58). verse length scales). Two limiting regimes lium Couette flow, and the transition to ii) Quantum turbulence with friction in have been tentatively identified: Taylor vortices (48, 49), flows for which the finite-temperature 3He-B. The main feature i) Vinen or ultra-quantum turbulence. This assumption is valid that the vortex lines are of this form is that the highly viscous normal is a random vortex tangle with a single locally aligned within each fluid parcel. Ap- fluid is effectively clamped to the walls. The dominant length scale, ℓ. It has long been plication of the HVBK equations to turbu- mutual friction force acts on all length scales argued (7) that steady counterflow tur- 4 lence is not justified for randomly oriented and affects the dynamics of quantized vorti- bulence at nonzero temperature in He is in vortex lines, as the net superfluid vorticity ces, damping the energy of Kelvin waves into this regime although the energy spectrum has in each fluid parcel would be zero, yielding the normal fluid. The role of friction increa- never been directly measured experimen- zero friction, despite the nonzero vortex- ses with rising temperature to the point that, tally. A recent numerical calculation (65) of line density. Modifications of the HVBK upon exceeding a critical temperature, tur- counterflow turbulence driven by a uniform equations have been developed, neglecting bulence can be suppressed altogether. Tem- normal flow has shown a broad energy the vortex tension and approximating the perature thus plays an analogous role to that spectrum around k ≈ 1=ℓ, confirming this mutual friction (50, 51). Such models prob- of the Reynolds number in classical turbu- state. At very low temperatures, ultraquan- 4 ably underestimate friction dissipation, but lence (27). This type of quantum turbulence tum turbulence has been produced in He by the coupled motion of both fluids is com- is discussed in the article by Eltsov et al. (59). ion injection (26). The main experimental 4 puted self-consistently. Shell models of tur- iii) Two coupled turbulent fluids in He. evidence (26, 66) is that, if the vortex tangle This type of turbulence is easily generated by is left to decay, the vortex line density de- bulence (52, 53) and Leith models (54) are − variants of the HVBK equations trading stirring helium II by mechanical means [e.g., creases as L ∼ t 1, in agreement with a phe- spatial information for that in the k-space. towed grids (60–62) and propellers (14)], by nomenological model of Vinen (7), which ultrasound (24), or by forcing it using grids indeed assumes a random, homogeneous, and Types and Regimes of Quantum and flows past bluff bodies in wind tunnels isotropic vortex configuration. The same Turbulence (63). Both superfluid and normal fluid com- L ∼ t−1 decay has been observed in numer- It is useful to classify the various types of ponent are turbulent. Because of its double ical simulations (67), which also confirmed turbulent flows that can be generated in nature, this is the most general and chal- that the energy spectrum remains broadly a quantum fluid, keeping in mind that more lenging type of quantum turbulence, generally concentrated near k ≈ 1=ℓ. classification schemes might well be pos- richer than classical turbulence in conven- ii) Kolmogorov or semiclassical turbu- sible. To start with, the strong temperature tional viscous fluids, presenting us with two lence. This regime is quite similar to classical

4650 | www.pnas.org/cgi/doi/10.1073/pnas.1400033111 Barenghi et al. Downloaded by guest on September 29, 2021 turbulence, as the energy spectrum contains flexibility than helium, as physical parameters challenging task is the development of INTRODUCTION an inertial range and closely displays the can be engineered and vortices can be in- miniature special probes capable of probing SPECIAL FEATURE: − = celebrated K41 scaling EðkÞ ∼ k 5 3 over dividually manipulated, and, thanks to the quantum turbulence, resolving all scales in- 1=D k =ℓ (thus, most of the energy weak interactions, are a testing ground for cluding quantum scales smaller than ℓ. resides at the largest length scales). Direct the theory. Two-components (75) and spinor On the other hand, the existence of the evidence of Kolmogorov scaling is provided condensates (76) are rich new systems for ultraquantum regime is a warning signal that by experiments at high and intermediate turbulence. Dipolar condensates (77) may not all quantum turbulent flows are related to temperatures (14, 63) and numerical sim- open the possibility of turbulence with un- classical flows. The various types and regimes ulations (65) in which the vortex tangle is usual vortex interaction. These opportunities ofquantumturbulencethatwehaveidenti- driven by a turbulent normal fluid. At very are discussed in the article by White et al. (30). fied provide a rich range of problems that we low temperatures, the experimental evi- should solve using hydrodynamics models Outlook dence (26) is based on the observed decay (the spirit is similar to how the known −3=2 Quantum turbulence is a relatively young L ∼ t ,which,ithasbeenargued(1,66, planetary atmospheres are explained by the field of research compared with conventional 68), is consistent with the Kolmogorov spec- same physical principles under different T = turbulence in viscous fluids, which has slowly trum. At 0, numerical simulations based parameters). – but steadily progressed over several centuries. on both the vortex-filament model (68 70) Problems that seem particularly challeng- – The early works on quantum turbulence (7) and the Gross Pitaevskii equation (71, 72) ing involve either two turbulent cascades were mainly concerned with counterflow as have produced spectra consistent with the taking place in the same fluid in different k−5=3 a problem of heat transfer unique of liquid scaling. Further numerical studies have regions of k-space (the Kolmogorov cascade revealed that the Kolmogorov energy spec- helium II. It was only after the seminal experiments of Donnelly, Tabeling, and co- and the Kelvin waves cascade) or two active trum is associated with the presence of turbulent superfluids affecting each other metastable bundles of polarized quantized workers (14, 61-62) that the attention shifted to concepts such as energy spectrum and [e.g., two-component cold gases (75) and, vortices (65, 70, 73). This insight opens when experimentally realized (78), 3He–4He the possibility of stretching such bundles vorticity decay, which are typical of the fluid 3 4 dynamics literature. These and other experi- mixtures with both He and He superfluid]. (stretching of individual quantum vortices For complexity and difficulty, the closest is not possible because of the quantization ments showed that, over length scales much larger than the mean intervortex distance ℓ, analog in classical physics is perhaps the condition). The polarization of (part of) the problem of coupled turbulent velocity and vortex tangle is also discussed in the article of quantum turbulence mimics (66, 68) the magnetic fields in astrophysical magneto- Vinen and Skrbek (74) on turbulence gen- properties of classical turbulence, hinting (in ’ hydrodynamics. erated by oscillating objects; its importance the spirit of Bohr s old quantum theory) that The temperature of the cosmic microwave lies in the fact that, in classical turbulence, many quanta of circulation yield classical background radiation is about 2:7K,andthe vortex stretching is thought to be responsible behavior. This result, together with the very 4 coldest place found in the Universe is the for the dissipationless transfer of energy from low kinematic viscosity of He, suggests that Boomerang Nebula ( ≈ 1K),5,000light- large to small scales (energy cascade). quantum turbulence can be used to study years away from Earth in the constella- An important issue is the normal fluid’s classical problems such as the temporal decay tion of Centaurus. Thus, further experimental profile in various types of channel and pipe of homogeneous, isotropic turbulence or the studies of quantum turbulence, probing flowsofheliumII.Forexample,innumerical long-standing puzzle of the Loitsianskii in- simulations of counterflow turbulence driven variant. In general, highly turbulent flows physical conditions not known to Nature by uniform normal fluid, the energy spectrum are needed to tackle these problems. It is at temperatures many orders of magnitude k ≈ =ℓ not difficult to generate such flows with lower, may uncover phenomena not yet broadly peaks at the mesoscales 1 (65) 4 (as in ultraquantum turbulence), but the liquid He, and the huge-capacity liquefiers known to physics. L ∼ t−3=2 of the European Organization for Nuclear experimentally observed decay is ACKNOWLEDGMENTS. We thank W. F. Vinen for stimu- (64) (typical of quasiclassical turbulence). Research (CERN) are already considered for lating discussions and constructive criticism. L.S. acknowl-  Thus, either large-scale structures already the purpose within the European project edges support from Czech Science Foundation Grant GACR European High-Performance Infrastructures 203/14/02005S and from the European Commission under exist in steady-state counterflow or are the 7th Framework Programme EuHIT. C.F.B. is grateful to generated by halting the normal fluid. in Turbulence (EuHIT), in the frame of the the Engineering and Physical Sciences Research Council for A third classification is suggested by the seventh European Union initiative. A more Grant EP/I019413/1. relative magnitude of ξ, ℓ,andD.Quantum turbulence in atomic Bose–Einstein conden- 1 Skrbek L, Sreenivasan KR (2012) Developed quantum turbulence 9 Fonda E, Meichle DP, Ouellette NT, Hormoz S, Lathrop DP (2014) sates lacks the wide separation of scales typ- and its decay. Phys Fluids 24:011301–011347. Direct observation of Kelvin waves excited by quantized vortex ical of liquid helium: The size of typical 2 Vinen WF, Niemela JJ (2002) Quantum turbulence. J Low Temp reconnection. 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