
==== Front
Proc Natl Acad Sci U S A
Proc Natl Acad Sci U S A
PNAS
Proceedings of the National Academy of Sciences of the United States of America
0027-8424
1091-6490
National Academy of Sciences

38457491
202302256
10.1073/pnas.2302256121
persPerspectiveapp-physApplied Physical Sciences405
447
Perspective
Physical Sciences
Applied Physical Sciences
Phenomenology of transition to quantum turbulence in flows of superfluid helium
Skrbek Ladislav ladislav.skrbek@mff.cuni.cz
a 1 2 https://orcid.org/0000-0002-6016-1194

Schmoranzer David a 1 https://orcid.org/0000-0001-9334-2453

Sreenivasan Katepalli R. katepalli.sreenivasan@nyu.edu
b 1 2 https://orcid.org/0000-0002-3943-6827

aFaculty of Mathematics and Physics, Charles University, Prague 121 16, Czech Republic
bDepartment of Physics, Courant Institute of Mathematical Sciences, Tandon School of Engineering, New York University, New York, NY 11201
2To whom correspondence may be addressed. Email: ladislav.skrbek@mff.cuni.cz or katepalli.sreenivasan@nyu.edu.
Edited by David Weitz, Harvard University, Cambridge, MA; received September 18, 2023; accepted January 17, 2024

1L.S., D.S., and K.R.S. contributed equally to this work.

8 3 2024
19 3 2024
8 9 2024
121 12 e230225612118 9 2023
17 1 2024
Copyright © 2024 the Author(s). Published by PNAS.
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

Transition from laminar to turbulent states of classical viscous fluids is complex and incompletely understood. Transition to quantum turbulence (QT), by which we mean the turbulent motion of quantum fluids such as helium II, whose physical properties depend on quantum physics in some crucial respects, is naturally more complex. This increased complexity arises from superfluidity, quantization of circulation, and, at finite temperatures below the critical, the two-fluid behavior. Transition to QT could involve, as an initial step, the transition of the classical component, or the intrinsic or extrinsic nucleation of quantized vortices in the superfluid component, or a simultaneous occurrence of both scenarios—and the subsequent interconnected evolution. In spite of the multiplicity of scenarios, aspects of transition to QT can be understood at a phenomenological level on the basis of some general principles, and compared meaningfully with transition in classical flows.

quantum turbulence
superfluidity
flow instability
transition to turbulence
Czech Science Foundation GA\v{C}R 20-00918S Ladislav SkrbekDavid Schmoranzer Czech Science Foundation GA\v{C}R 20-00918S Ladislav SkrbekDavid Schmoranzer
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pmc1. Scope of the Article

Transition to turbulence of classical viscous flows is an intriguing and complex problem of fluid dynamics (1–3). Transition to quantum turbulence (QT) (4), which is the term that denotes the turbulent motion of quantum fluids, is more tortuous. Quantum fluids, which are systems such as superfluid helium, neutron stars, or atomic Bose–Einstein condensates, are characterized by quantized vorticity, superfluidity and, at finite temperatures below the critical, by a two-fluid behavior. This added complexity limits our knowledge of the transition process in quantum flows. Yet, the time is ripe to summarize and critically examine the accumulated knowledge on the subject and compare its characteristic features with those in classical fluids. This Perspective is written in the same spirit as our recent article on the phenomenology of QT in helium superfluids (5). We briefly describe basic properties of quantum fluids and follow it up by a description of special features relevant to transition to QT, critically discussing what is known, and what open questions remain to be explored.

2. Basic Properties of Quantum Fluids

We shall be concerned mostly with superfluid 4He, the B-phase of superfluid 3He and, briefly, with ultracold atomic gases, which are quantum fluids existing at temperatures of the order of a Kelvin, milli-Kelvin, and micro-Kelvin, respectively, corresponding to the majority of relevant experiments. Constituents of quantum fluids are either bosons (such as 4He atoms with zero spin, obeying Bose–Einstein statistics) or fermions (such as 3He atoms with spin 1/2, obeying Fermi–Dirac statistics). This difference is fundamental. At room temperature, the de Broglie wavelength λdB of each atom is much smaller than the average separation between the atoms; if the temperature T is lowered, λdB increases and the quantum-mechanical wave aspects become dominant. In the case of bosons, phase transition called the Bose–Einstein condensation occurs, characterized by a macroscopic number of bosons occupying the state of zero energy. For superfluidity (flow without friction), bosons must interact collectively. In the case of fermions, at temperatures much lower than a characteristic Fermi temperature, particles occupy the Fermi sphere in the momentum space, except relatively few particle–hole pairs. Attractive interaction between fermions mediated by magnons (spin density waves) leads to the formation of Cooper pairs which, similar to bosons undergoing Bose condensation, results in superfluidity in systems consisting of neutral atoms such as 3He or superconductivity in charged systems of electrons in crystal lattices. Superfluidity arises from the formation of a coherent particle field that can be described using the condensate macroscopic wave function.

The two-fluid model (6), introduced by Tizsa and Landau, is conveniently used for phenomenological description of quantum fluids. Below the so called λ-temperature Tλ, 4He (known for historical reasons as He II while the normal liquid above Tλ is termed as He I) is described as a viscous normal fluid (a gas of thermal excitations in the form of phonons and rotons) carrying the entire entropy content, coexisting with the inviscid superfluid. The density of He II is ρ=ρn+ρs, where the densities of the normal fluid, ρn, and superfluid, ρs, are temperature-dependent: in the T→0 limit He II is entirely superfluid, whereas superfluidity vanishes as T→Tλ. Similar considerations apply for 3He displaying superfluidity below about 1 mK, and its B phase can be considered a pure superfluid below about 200 μK. Generally, a rich variety of two-fluid flows can be generated. Mechanical forcing usually results in coflows, the closest analogues to classical viscous flows in which normal fluid and superfluid move, on the average, with the same mean velocity in the same direction. The two components can also be made to flow with different average velocities (or against each other in the extreme), which is the situation referred to as counterflow.

For certain conditions (which we will discuss shortly), the normal fluid flow and the inviscid superflow are two independent velocity fields un and us. However, the picture changes when quantized vortices appear in the superfluids. For our purpose, these so-called vortex lines can be viewed as topological line defects with circulation that is quantized in units κ=h/m, where h is Planck’s constant and m the mass of the superfluid particle (4He atom in He II, a Cooper pair in 3He-B). This quantization condition was proposed by Onsager (7, 8) and experimentally confirmed by Vinen (9). It arises from the existence and the single-valuedness of a complex, macroscopic superfluid wave function Ψ and the quantum mechanical prescription that the velocity is proportional to the gradient of the phase of Ψ. As a consequence, the superflow is potential: ∇×us=0 everywhere, except inside the cores of quantized vortices of size ξ. In He II, ξ≈1 Å, whereas it is about 100 times larger in 3He-B, and larger still in atomic condensates (1/100 to 1/10 of the system size in current studies).

The creation of a vortex line is opposed by a potential barrier. In a uniform superflow parallel to a solid wall, a vortex at a distance x from the wall and normal to the flow experiences two Magnus forces: one of magnitude ρsκus away from the wall and another of magnitude ρsκ2/(4πx) toward the wall due to its image. The resulting potential energy has the approximate maximum of [ρsκ2/(4π)]ln(x/ξ) at x=κ/(4πus). In He II, except very close to the superfluid transition, this potential barrier cannot be overcome either thermally or by quantum tunneling.

Consistent with these considerations, the superflow remains frictionless up to an intrinsic critical velocity ucrint≈10 m/s that is much larger than that observed in experiments. Additionally, recent theoretical calculations (10) in the frame of generalized Gross–Pitaevskii model suggest that in He II a breakdown of superfluidity due to roton emission as well as the intrinsic critical velocity for vortex nucleation in the wake of very small moving objects depends on their size. In practice, however, frictionless macroscopic superflows of He II break down by an extrinsic process in which existing small lengths of vortex line, called seeds, elongate and reconnect under the influence of the superflow. There must still be some small effective barrier because there would otherwise be no frictionless flow. In macroscopic He II samples, seeds are almost always present because the remnant vortices might have been present from an earlier experiment: The walls of the vessel containing He II are rough in comparison with the vortex core size, providing pinning centers to anchor remnant vortices whose density in the bulk samples of He II can be estimated (11). In fact, special approaches are needed to create a macroscopic He II sample that is entirely free of remnant vortices, which is the reason why the critical velocities for vortex generation in He II are observed to be low in practice, of order of a few cm/s. Alternatively, vortices might be formed by the Kibble–Zurek mechanism (12) as helium is cooled through the second order superfluid transition: the phase of Ψ is unable to adjust everywhere and leaves vortex lines as topological defects. Vortex lines thus appear spontaneously while cooling helium through the critical temperature, though they decay and disappear. Due to the much bigger size of vortex core in 3He-B both intrinsic and extrinsic vortex nucleation are possible.

Vortex lines support Kelvin waves, which are the propagating helical motions on the vortex core, arising from the tension of the vortex line and the Magnus forces acting between its segments (6). A perturbation of wavelength λw=2π/k where k is the wavenumber, propagates along the vortex line in the form of a circularly polarized wave that has the dispersion relation ω≈[(κk2)/(4π)] ln [1/(kξ)], where ω is the angular frequency of the wave, as was first derived by Lord Kelvin (13) for waves on a classical thin-cored vortex filament in an inviscid fluid. The difference between classical Kelvin waves and those in superfluid helium is that the circulation κ is quantized in the latter.

Vortex lines are the reason why the normal fluid and superfluid components are coupled. The coupling is caused by the so-called mutual friction (6, 14), arising from the fact that phonons and rotons, which are the constituents of the normal fluid of He II, are scattered off the cores of quantized vortices. One consequence of this effect is that, at finite temperatures, Kelvin waves are damped by mutual friction. As an illustration, consider the motion of the vortex ring of circulation κ, radius R and core radius ξ≪R. If we increase (decrease) the radius R of the ring, the ring travels slower (faster) and has more (less) energy. A ring traveling in the positive x-direction in the presence of uniform superfluid and normal fluid velocities of us and un aligned in the same direction, R obeys the relation (15) dR/dt=γ/ρsκun−us−uR, where γ is the friction coefficient of the normal fluid. Since γ→0 as T→0, the ring’s radius and energy remain constant. If T>0 but un=us=0, dR/dt<0 and the ring shrinks, losing energy by friction to the background normal fluid at rest. If un−us≠0, the ring will shrink (grow) gaining (losing) energy from (to) the normal fluid depending on whether un−us−uR>0 or un−us−uR<0. In particular, if us=0, the ring expands when the normal fluid flows in the same direction as the moving ring. This feature is important for understanding the transition to QT in thermal counterflow in He II.

A second effect that is essential for the transition to QT is known as Donnelly–Glaberson instability, which was discovered experimentally (16) and explained (17, 18) for the configuration of a rotating bucket of He II. When rotating with angular velocity Ω, a lattice of rectilinear quantized vortices of areal density nv=2Ω/κ is formed, aligned in the direction of Ω, mimicking solid body rotation of viscous fluids. If heat is applied in the direction of rotation axis, the vortex lines become unstable to the exponential growth of Kelvin waves of initially infinitesimal amplitude. These unstable vortex lines reconnect and eventually create a tangle, an essential ingredient of QT.

Finally, a point on the nomenclature: by superfluid turbulence we mean the turbulence of the superfluid component; the obvious situation occurs in the T→0 limit, where there is no normal fluid and the turbulence is a tangle characterized by vortex lines of length L per unit volume, known as the vortex line density. The situation is more complex at finite temperatures. The superfluid turbulence—i.e., the vortex tangle in the superfluid component—interacts via the mutual friction force with the normal component that itself might or might not be turbulent. By QT, we inclusively mean turbulence of any quantum fluid displaying quantized vorticity and superfluidity. QT can be simpler than classical turbulence in the limit of T→0, but is generally more complex at finite temperatures.

3. Transition to Superfluid Turbulence as T→0

This limit is the simplest to consider, and unique without a classical counterpart. Indeed, QT in this limit occurs from inviscid (potential) flow and might take one of two well established forms, known as Vinen (or ultra-quantum) turbulence or Kolmogorov (or quasi-classical) turbulence, characterized by different forms of energy spectra and temporal decays, as discussed in refs. 6 and 5. For a quick reference, Vinen turbulence has only one characteristic length scale ℓ=L−1/2, where L decays, in the absence of sources, as t−1, with t denoting time; the energy also decays as t−1. The Kolmogorov turbulence is characterized by energy transfer across a multitude of scales, in which L decays as t−3/2 and the energy as t−3/2. It is more difficult to generate steady potential flow in the T→0 limit than to generate oscillatory flows, which has led, from the time of discovery of superfluidity, to a larger number of studies of flows created by oscillating wires, grids, spheres, cantilevers, torsional pendulums or tuning forks immersed in liquid helium; for reviews, see refs. 19 and 20 and references therein. These devices can be used at any temperature, including the T→0 limit in both He II and 3He-B; moreover, similar geometric configurations can be used to generate classical turbulence, which enables direct comparison between classical and quantum cases.

He II experiments by the Prague and Lancaster groups, which use vibrating wires (21), grids (22) and tuning forks (23, 24), report up to three critical velocities of hydrodynamic origin. Among others, these studies form the basis of plausible explanations for the behavior of oscillatory flows as T→0 (25).

The first critical velocity, connected to frequency shifts rather than changes in the drag on the oscillators, is associated with seeds of quantized vortex loops near the surface of the oscillator (22), possibly forming a thin layer. These seed vortices must be pinned on the surface, which can almost always be considered rough in He II. The response of a vortex loop of length b can be described via normal modes of excited Kelvin waves with wave numbers kn=nπ/b. This response will be adiabatic (i.e., the vortex at any instant will be at its equilibrium position) if the oscillation frequency is significantly lower than that of the fundamental resonant Kelvin mode. The superflow distorts the vortices. The impulse required to create a vortex loop is ρsκS, where S is the area of vortex loop, which is time-dependent in an oscillatory flow. For such a pinned vortex loop, we can write the rate of change of the momentum of the oscillating structure of the bare mass M0 as[1] Meffdudt=M0dudt+Menhdudt+ρsκdSprdt,

where Menh is the hydrodynamically enhanced mass of the structure and Spr is the change of the loop area projected in the direction perpendicular to superflow. If the vortex responds adiabatically, we can write dSpr/dt=(dSpr/du)(du/dt), so that Meff=M0+Menh+ρsκdSpr/du. Although the quantity dSpr/du must be found by appropriate simulations for a given configuration, this simple approach (26) illustrates the mechanism of how the effective mass of the oscillating structure increases, leading to the shift in the resonant frequency of the flow due to an oscillatory body, with no appreciable increase in dissipation. Because the finite viscosity of the normal fluid lowers the Q-value of the oscillatory flow, the first critical velocity is hard to observe in the two-fluid regime above 1 K.

An initial vortex loop of arbitrary orientation tends to twist in the oscillatory superflow and eventually self-reconnect upon exceeding a second critical velocity. These vortex loops carry energy and momentum and propagate into the bulk, eventually merging in a random turbulent tangle, creating the Vinen type of QT (5, 6). It is usually accompanied by hysteresis (detectable with amplitude sweeps) and an increase in the drag, though the measured drag coefficient is only of the order 10−1 or 10−2. Thus, this state differs from classical turbulent flow by the absence of large structures of a wake.

It is only upon exceeding the third critical velocity, observed to be of the order of 1 m/s and above (25), that the drag coefficient starts to grow toward unity even in the T→0 limit (in practice, of order 10 mK for He II). This happens when a turbulent wake is created at length scales exceeding the intervortex distance mimicking the classical turbulent wake, representing quasi-classical or Kolmogorov QT (5, 6). This assertion is experimentally impossible to check in 3He-B, as the cooling power of cryostats at submilli-Kelvin temperatures is insufficient to compensate for strong energy dissipation.

4. Transition to QT in 3He-B at Finite Temperature

The last remark brings us naturally to the case that is next in simplicity: transition to QT in superfluid 3He-B. Here above about 0.2Tc, where the critical temperature Tc is 1 to 2 mK (depending on the pressure), the normal fluid is rather viscous and remains essentially at rest in the frame of reference of the container. The terms QT and superfluid turbulence in fermionic 3He-B have the same meaning and represent the dynamics of the tangle of quantized vortex line coupled by the mutual friction to a thick “soup” of the normal fluid. This type of transition to turbulence does not have a classical counterpart. To understand this statement, consider the equation for the coarse-grained superfluid velocity,[2] ∂us∂t+∇μ=(1−α′)us×ω+αω^×(ω×us),

obtained by Sonin (27) from the Euler equation, after averaging vortex lines in the frame of reference of the normal fluid. Symbols α and α′ are the mutual friction coefficients of 3He-B (28), μ is the chemical potential, ω the coarse-grained vorticity, and ω^ the unit vector in the direction of ω. This equation describes the superfluid velocity field sufficiently well on scales larger than the averaging length. Since un=0, we deal only with us. Rescaling t→(1−α′)t leads to[3] ∂us∂t+∇μ=us×ω+qω^×(ω×us).

This equation is quite different from the Navier–Stokes equation (29), where the relative importance of the inertial and viscous terms is given by the Reynolds number, Re, which depends on the fluid density and viscosity, as well as the speed and the geometry of the flow. Here, the role of Re is played by the dimensionless parameter Reeff=1/q=(1−α′)/α. The beauty of Eq. 3 is the general result that the state of the flow depends only on a single temperature dependent parameter 1/q, regardless of the actual geometry of the flow: because of the high viscosity of the normal fluid in 3He-B, the temperature plays here the role of Re in classical viscous flows.

We now discuss this transition in 3He-B from the point of view of Donnelly–Glaberson instability, leading to the production of quantized vorticity via self-reconnections of seed vortex loops. Kopnin (30) developed a model for the onset of superfluid turbulence in terms of an initial avalanche-like multiplication of vortices to form a turbulent vortex tangle, applicable both for 3He-B and He II. The superflow can be characterized by the so-called superfluid Reynolds number Res=UnsH/κ, where H is the system size, the kinematic viscosity of the ordinary Reynolds number is replaced by the circulation quantum of the same dimension and Uns is the mean superfluid velocity with respect to the normal component (counterflow velocity); in 3He-B, Uns=Us. The condition Res≈1 corresponds to Feynman’s celebrated criterion that a vortex gets unpinned from a container wall when it is energetically favorable. However, as we already discussed, quantized vortices are not necessarily created even when Res>1 and the superfluid remains in a vortex-free state. We know this to be true in the rotating bucket of 3He-B: the normal fluid rotates with the container and the superfluid is at rest, and quantum vortices are generated only when much higher counterflow velocities that suggested by the Feynman criterion occur at the perimeter of the vessel.

QT can be initiated below about 0.6Tc if seeded intrinsically or an extrinsic injection takes place.* It develops when initial vortices start to multiply, reconnect and form a tangle (29). The model due to Kopnin assumes that vortex multiplication occurs first within a certain region in the fluid near the location of seed loops from where the vortex tangle is created before spreading into the rest of the fluid. The loops spread into the bulk if Uns>usi, where usi≈κ/ℓ is the self-induced velocity for a vortex loop of length ℓ. This process is accompanied by the dissipation of kinetic energy until the equilibrium density of loops is reached. The rate of variation of the vortex-loop density due to the viscous component becomes dL/dt=−Bα(Uns−usi) with the numerical factor B≈1. These considerations lead to an equation applicable to 3He-B, similar to the celebrated Vinen equation [derived for the thermal counterflow of He II (31–34)], given by[4] dLdt=βK(UnsL3/2−κL2).

Note that the coefficient βK in Eq. 4 can be positive or negative, depending on the mutual friction parameters. For βK<0, the rate of extraction of vortex loops exceeds the rate of multiplication; there is no time for vortices to multiply since all the seeds and the newly created vortices are immediately swept away into the bulk fluid. The number of vortices in the final state is essentially equal to the number of initial vortices, and the turbulent state is not formed. This corresponds to the case of 3He-B at temperatures that are high enough to prevent the formation of QT. For βK>0 the vortices multiply faster, the number of vortex loops created is large with each new vortex loop serving as a source for more. An avalanche-like multiplication occurs, leading to the formation of a turbulent vortex tangle. As the vortex loops grow in number, the self-induced velocity increases and, finally, the vortex line density saturates at Lsat, corresponding to the Vinen equilibrium density for counterflow turbulence.

5. Transition to QT in He II at Finite Temperature

In He II at T⪆1K, the mean free path of thermal excitations—rotons and phonons—is short enough to allow classical hydrodynamic description. When stirred, both the superfluid and normal components easily become turbulent, creating a unique double turbulent state, consisting of a continuum of normal fluid eddies and tangles of discrete vortex lines, interacting with each other. Under some conditions, remarkable similarities arise between this double turbulent state of He II and classical turbulence but they behave differently in other circumstances.

As already remarked, it is easier to study oscillatory flows of He II around vibrating objects, as steady and sustained flow is more difficult to achieve. Moreover, the theoretical analysis benefits from comparisons with oscillatory flows of classical fluids of low viscosity using the same experimental setup. So we start with a brief overview of oscillatory viscous flows in He I and He gas.

A. Transition to Turbulence in Classical Oscillatory Flows.

In oscillatory flows a second characteristic scale, L2, emerges in addition to the linear size of the obstacle, L1. The governing Navier–Stokes equation can then be written (35) in the dimensionless form[5] ωU∂u′∂t′+U2L1(u′·∇′u′+∇′p′)=νUL22Δ′u′,

where u′=u/U, t′=ωt and the position vector r′=r/Li. The characteristic length scales L1,2 are used together with the characteristic velocity U to estimate the maximum magnitude of the respective velocity derivatives, and ν is the kinematic viscosity of the fluid. An independent time scale emerges from the angular frequency of oscillation, ω. Generally, the choice of L1 and L2 depends on the body shape and flow parameters. Candidates may include the typical body size D, the surface roughness Rq, or the Stokes boundary layer thickness (viscous penetration depth), defined as δ=2η/(ρω), where η=νρ denotes the dynamic viscosity of the fluid. When δ≪D, we say that the body oscillates in the high-frequency regime so that the Stokes number St=D2/(πδ2) is large. Depending on body geometry (especially surface roughness and sharp corners), δ or D may take the role of L1 (related to the highest expected tangential velocity derivative) in the Navier–Stokes equation, but it is always δ that takes the place of L2 (creating the largest velocity derivative in any direction). When Rq≫δ, or when sharp corners are present, we can safely put L1=L2=δ, and the Navier–Stokes equation contains only one dimensionless parameter, the boundary layer-based Reynolds number Reδ≡(δρU)/η. Assuming that the instability in the laminar flow of this type occurs upon reaching a critical value Reδcr, it immediately follows that the corresponding critical velocity scales as νω, as has been experimentally verified, e.g., in the normal liquid He I and gaseous 4He for the flow due to oscillating quartz forks (36). This regime allows one to understand many features of oscillatory quantum flows.

B. Transition to Turbulence of Oscillatory Flows of He II at High Stokes Numbers.

Let us apply the above considerations to He II, assuming the normal component to be laminar and the superflow to be potential. We replace ρ by ρn, decompose the pressure into partial pressures of the normal and superfluid components, p=ps+pn, and replace δ by δn=2η/(ρnω), where η denotes the dynamic viscosity of He II. Again, if δn≪D, and Rq≫δn, one can set L1=L2=δn, and the Navier–Stokes equation depends on a single dimensionless parameter—the Donnelly number Dn≡(δnρnU)/η—as†[6] 2∂u′∂t′+Dn(u′·∇′u′+∇′pn′)=Δ′u′.

Note that Dn becomes equivalent to Reδ at the superfluid transition temperature Tλ, allowing direct comparison of He II with classical oscillatory flows. A detailed treatment is given in ref. 35, from which we state the principal result that the drag coefficient due to the laminar normal fluid is given as CDn=Φ/Dn, where the constant Φ is determined purely by the geometry of the oscillator and the mode shape. Thus, in the laminar case, the drag due to the normal component is fully described by laws of classical fluid dynamics. For turbulent flows of this type, where only the normal component contributes to the nonlinear drag, a unique function CDn(Dn) is expected, if Dn is the only dynamical parameter of importance. Departures from this function must then signify an instability of the superfluid component; if it becomes turbulent at some critical velocity us,cr, a marked increase is observed in the drag coefficient CDn above that measured in a classical fluid.

Some oscillators permit analytical solution of the Navier–Stokes equation for the drag coefficient in the normal fluid. Approximating the vibrating wire as an infinite smooth cylinder, one obtains CDn=4π/Dn (36); for a smooth disc CDn=2/Dn (37). For tuning fork oscillators (38) the numerical prefactor depends on the aspect ratio of the tines (37).

For comparing classical and quantum criteria for transition, it is necessary to define a superfluid critical velocity. Assuming that the superfluid Reynolds number Res=usLs/κ reaches some critical value at the transition, it is reasonable to assume that the characteristic length scale Ls is the oscillation amplitude us/ω, and therefore us,cr∝κω. Hanninen & Schoepe (39, 40) argued that the onset of superfluid turbulence in oscillatory flows is universal, and showed that[7] us,cr≈8κω/σ,

where the numerical factor σ depends weakly on the mutual friction parameters, implying a slow increase of us,cr with temperature. For a temperature change from 0.4 to 1.9 K, and for a sphere of 100 μm diameter oscillating at 236 Hz, the increase of us,cr is about 10% (39).

This description‡ of high-Stokes-number oscillatory flow of He II in the temperature range of two-fluid behavior has been verified experimentally (37) with vibrating wire resonators, tuning fork, double-paddle, and torsionally oscillating disc. Whether the classical hydrodynamic instability or Donnelly–Glaberson instability occurs first depends on the geometry and size of the oscillator, as well as the temperature. A crossover between the two could occur because of the steep temperature dependence of the kinematic viscosity of the normal fluid. Upon increasing oscillation amplitude, either instability can live on its own until eventually it serves as a trigger for the other, mediated by the mutual friction or by pressure forces.

At velocities sufficiently above the critical, where one observes a developed turbulent drag regime, the normal and superfluid components are coupled by mutual friction and contribute to the drag together. He II then behaves as a single quasi-classical fluid and the classical definition of the drag coefficient applies: CD=2F/(AρU2), where ρ is the total density, with the expected tendency toward a temperature-independent constant value of order unity (19).

C. Hysteresis and Intermittent Switching.

Interesting results have come from the pioneering work of Schoepe and coworkers with oscillating spheres (42–44). The relation between the driving force and the transverse oscillation velocity is linear at low drives, indicating a laminar flow, but is replaced at higher drives by a nearly quadratic dependence, indicating a turbulent regime. The transitional response is accompanied by hysteresis and, at the lowest temperature, also by an unstable region where the response switches intermittently between the two states. This intermittent switching has also been observed with various mechanical oscillators, e.g., tuning forks (45).

A variety of interesting experiments relevant to the transition to both classical and QT has been reported. In particular, the vibrating wire resonators (46), as well as spheres (42) and grids (47), clearly show the transition from hydrodynamic behavior at temperatures above about 1 K to ballistic behavior at milli-Kelvin temperatures. Vibrating wire resonators in 4He have been thoroughly studied, by the Lancaster group, reporting the history-dependent behavior (48) as well as two critical velocities (49), and by the Osaka group (50–53). Here illuminating results have been obtained with a rather smooth, thin wire of diameter 2.5 μm oscillating at a frequency of 610 Hz.§ If the wire is surrounded initially by He I, pumped through the λ-transition and cooled rapidly to 1.4 K, the initial response of the wire indicated very strong damping, even at low drives, with very strong hysteretic effects. Repeated scanning of the drive, however, eventually led to a steady and reproducible response with no hysteresis, exhibiting a transition from a laminar regime to a turbulent regime upon reaching a wire velocity of about 3cm/s. The Osaka group repeated the measurements later at 35 mK, directly filling the region around the wire with helium (53) through a small 0.1 mm pinhole at mK temperatures. In this case, the response remained laminar up to the largest velocity that could be imposed, above 1m/s.

These observations confirm that transition to QT in bulk 4He occurs primarily via extrinsic vortex nucleation. Indeed, a convincing experimental study revealing the role of remnant vortices has been performed by Yano et al. (54), using two vibrating wire resonators. Both were placed in a very low temperature fluid filled slowly through a tiny orifice. Sometimes, it was found that for one of the vibrating wires (but not for the other), the critical velocity was >1 m/s and yet the vortices could not be measured. That this high critical velocity was associated with the absence of any attached remnant vortex was confirmed by generating turbulence with the second wire, causing a large reduction in the first wire’s critical velocity. The reason is that the second wire, to which a few remnant vortices were attached, emits a beam of vortex rings (55), which attach to the first wire and produce turbulence at a lower critical velocity.

D. Second Sound as Oscillating Counterflow of He II.

QT in He II can be generated by high-amplitude second sound. We refer to this form of QT as ac counterflow turbulence, as second sound represents counterflow oscillations of the normal and superfluid components. Kotsubo & Swift (56, 57) were the first to generate this form of QT, using cylindrical second sound resonators. Longitudinal second sound was generated externally, via a superleak,¶ by magnetically driven bellows. The critical velocity was found to be independent of the resonator diameter, displaying a square root frequency dependence. The signature of QT was the flat top in the second sound resonance curve. This feature was later observed also when generating QT thermally (58, 59) by placing a heater at one side of the resonator and detecting the second sound amplitude by a sensitive thermometer on the opposite side.

At low velocities, only viscous drag is generated by the normal fluid, obeying a universal scaling law in terms of the Donnelly number. Upon exceeding the critical value, Dncr, corresponding to a velocity un,cr, the normal component undergoes the turbulent transition, possibly triggering the generation of quantized vortices in the superfluid as well. On the other hand, quantized vorticity may be generated directly via the Donnelly–Glaberson instability. Attempts have been made to analyze the turbulent instability in an ac counterflow in this manner (56, 57) without the consideration of classical flow instabilities, but the observed systematic temperature dependence of critical velocities could not be explained. In practice, which instability occurs first depends on both the geometry of the oscillator and the temperature that determines the viscosity of He II and the densities of both components. In experiments on flow due to mechanical resonators, the comparison of the two criteria for the transition is straightforward: In a coflow, the velocities of the normal and superfluid components are practically identical. For ac thermal counterflow, noting that un,cr=ρs/ρnus,cr, we can find a common dimensionless parameter. For example, the superfluid critical velocity us,cr may be converted to the effective critical Donnelly number[8] Dncr,eff=ρsρnδnus,crνn(T),

where νn(T)=η/ρn. Unlike the true Dncr describing the classical instability, the critical value of Dncr,eff is no longer expected to be constant. In fact, requiring a constant value of the correct critical parameter, us,cr, also requires Dncr,eff to be a function of temperature. However, Dncr,eff scales with the frequency in the same way as the true Dncr, as both un,cr and us,cr have the same frequency dependence ∝ω1/2.

It is remarkable that, within experimental accuracy, for 1.2K≤T≤1.7K, different experiments such as the mechanically driven second sound (56, 57) and thermally driven ac counterflow (58, 59) display the onset of the transition to QT at the same Dncr=16±3, suggesting that the transition is triggered by the instability in oscillatory flow of the normal component. Above ≈1.8 K, however, Dncr decreases with increasing temperature, in accordance with Eq. 8. This behavior is explained by the instability in the superfluid component by the Donnelly–Glaberson mechanism. Thus, a crossover of the mechanisms of turbulence generation in oscillatory counterflow is experimentally observed: one related to a classical instability of the normal fluid, dominating at lower temperatures in the two-fluid regime, and the other purely a consequence of quantized vortex dynamics in the superfluid component, which dominates at higher temperatures. This observation strongly suggests that transition to QT in oscillatory coflow and ac counterflow is governed by the same underlying physics, although the crossover of the instabilities occurs in opposite directions for the counterflows and coflows. We emphasize that this hydrodynamic approach is applicable only in the temperature range where He II displays the two-fluid behavior.

6. Transition to QT in Steady Flows of He II

Although the transition to turbulence of viscous flows through pipes (and channels) have been studied since the seminal work of Reynolds (60) in 1883, the problem is still one of active research (3). Classical fluid dynamics experiments on transition take great care on aspects such as flow management, entrance length and surface roughness. As a result, many experimental norms have been established for creating transitional flows, but comparable care and concern has not been evinced for He II flows presumably because the emphasis has been different.

A. Classification of Flows.

The pioneering quantum pipe flow experiment of Allen and Misener (61) in 1938 is directly relevant to the discovery of superfluidity. Since then, superfluid pipe/channel flows have continued to be extensively studied, addressing issues of stability and transition to QT. Using mechanical and/or thermal drives, a rich variety of two-fluid channel flows of He II can be generated, as illustrated in Fig. 1. Mechanical forcing almost always results in coflows, the closest analogues to classical viscous channel flows, in which normal fluid and superfluid move with the same mean velocity in the same direction. The two components of He II can also be made to flow with different mean velocities, which is the situation of counterflow. A particular case of counterflow is the thermal counterflow, with no net mass flow, and the ratio of normal and superfluid velocities is set by the temperature. Various sub-classifications are possible. The thermal counterflow confined in one direction is called the channel counterflow. If it is generated by point-like heaters, it is termed spherical counterflow. Another special case is the pure superflow, corresponding to a net superfluid flow, while the normal component remains stationary on the average. Pure superflows can be generated both mechanically (e.g., by compressing bellows, as for coflows) and thermally (as for counterflow), using suitable superleaks. If an additional heater is placed above the superleak, converting a part of the superfluid through-flow into normal fluid, any flow ratio of the two components can be achieved (62). We shall now discuss these cases separately.

Fig. 1. Illustration of three types of two-fluid channel flows of He II. Top: Thermal counterflow is easily generated by placing a heat source (red) at the closed end of a channel that is open to the helium bath at the other end. Various types of seed vortex loops are shown, serving as sources for extrinsic vortex nucleation via self-reconnection (a–c), producing free vortex loops (d), that further interact and eventually generate a tangle of QT. Middle: Coflow is any flow generated by classical means, for example pressure-driven steady flows or flows generated by towed grids, i.e., flows where the two components are forced by the same means. Bottom: Pure superflow is analogous to thermal counterflow but only the superfluid component flows through the channel, on average. The flow occurs upon forcing He II by pressure through superleaks (gray), which are impenetrable to the viscous normal component but allow a through-flow of the inviscid superfluid component. Internal normal fluid flow in the channel is, however, possible.

B. Coflow.

Because of considerable experimental difficulties, it is perhaps not surprising that a transition experiment similar to Reynolds’ basic case in classical fluids does not exist in He II, either at T=0 or at finite T. The scant experiments that do exist suffer from several shortcomings, and it is difficult to obtain quantitative information from them. For these reasons, we choose not to say anything further on this topic except emphasizing the need for new and careful studies.

C. Channel Counterflow.

This flow can be set up by applying a voltage to a resistor (heater) located at channel that is open to a helium bath. A steady counterflow velocity uns is established when a heater is switched on. If the applied heat flux q˙ is less than a critical value q˙c1, there are no vortex lines in the flow (except for the remnant ones), hence no mutual friction. The normal fluid flow, whose kinematic viscosity νn=η/ρn, can be characterized by the Donnelly number Dn=undρn/η. On the other hand, if q˙>q˙c1, the heat transport is affected by the emergence of a vortex tangle. The pressure drop along the channel increases with q˙, but only slightly, according to the so-called Allen–Reekie rule (63). Thermal counterflow in a wide channel was first systematically studied in an outstanding series of papers by Vinen (31). He introduced a phenomenological model based on the concept of a random vortex tangle characterized by a single variable, the (approximately homogeneous) vortex line density L, which was found to scale with uns as L=γ2(uns−unscr)2, where unscr is the critical velocity typically of the order of 1 mm/s and γ is a temperature dependent parameter. L also sets the only (quantum) length scale of this simple approach, namely the mean inter-vortex distance ℓ∼L−1/2.

A closer look at thermal counterflow shows, however, that this quantum flow is more complex. Historically, careful and extensive measurements of temperature differences ΔT along counterflow channels performed by Tough’s group (64, 65) clearly indicate that several steady counterflow turbulent states are possible, depending on the geometry. Above a first critical velocity unscr1 a vortex tangle is created in the superfluid while the normal fluid remains laminar [the so called T I state, where L1/2≈γ1uns (64, 65)]; its velocity profile is altered by mutual friction, probably acquiring a flattened-tail shape (66, 67), but the details are still under investigation. Numerical simulations of thermal counterflow between two plates with the distance dp apart, under the assumption of laminar parabolic normal fluid velocity profile, show that the local vortex line density L(x) across the flow is not constant but peaks around x/dp≈0.15 from the walls (68). The turbulence in the superfluid component is of the Vinen type, as the temporal decay of vortex line density originating from the T I state obeys the prediction of the Vinen equation L(t)∝1/t for late times (31), as was recently verified experimentally (69, 70).

Upon reaching a second critical velocity unscr2, the turbulence appears stronger [the so called T II state, where L1/2≈γ2uns with γ2>γ1 (64, 65, 71)]. As suggested theoretically (72) and confirmed by flow visualization (73), in T II state, the normal fluid is turbulent. The existence of large eddies in the normal fluid subsequently causes, via mutual friction, large superfluid eddies, and He II enters a remarkable double turbulent regime. The large-scale normal and superfluid eddies are partially coupled, but move on average in opposite directions. In the steady T-II state, there are two energy inputs to the superfluid component: At the quantum length scale ℓ creating a quantum peak of the energy density (74), and the classical energy input at the large scale of the channel size, mediated by mutual friction. Upon stopping the heat input, the quantum energy peak quickly decays and the energy at large scales gradually cascades down, forming an inertial range that acquires a classical Kolmogorov form. This feature results in the late stage classical-like decay of the form L(t)∝t−3/2 (69, 70).

In addition, the channel geometry has a profound effect on thermal counterflow. In rectangular channels of high aspect ratio (1:10) with the small dimension less than 100 μm, only one transition has been observed at a critical heat flux q˙c3. This steady-state turbulence was denoted by Tough as T III (64). Metastable laminar states can be observed for heat fluxes considerably larger than q˙c1 or q˙c3. Modern nanofabrication methods have recently enabled the construction of well-defined channels with aspect ratios of several thousand (75). Transition to turbulence in oscillatory flows due to a Helmholtz resonance in these geometries displays a complex hysteretic structure with multiple long-lived metastable states, which are not yet fully understood (76).

Many early investigators have attempted to resolve the puzzle of complex transitions to QT in channel counterflow of He II, probing it by various approaches (see the review by Tough (64) and references therein). Tough describes various transitions to T-I and T-II via seven different dimensionless Re-like parameters, though none was capable of providing a robust phenomenological description. This unsatisfactory state of affairs is at least partly due to the fact that the experimental data possess large scatter for reasons such as the variation of cross-section profiles, surface roughness or entry length. We seriously attempted to take into account also the possibility that, as discussed above for various oscillatory quantum two-fluid flows of He II, instabilities of the normal fluid or the superfluid might be involved, including possible crossover when changing temperature, but did not reach any convincing conclusions.

The phenomenological description of experimental data on the T-I to T-II transition of the channel counterflow, in pipes of about d=1 mm in diameter, is more successful when the vortex line density profiles obtained from the numerical simulations of Baggaley and Laizet (68) are taken into account. Fig. 2 shows the temperature dependence of observed critical heat flux values observed in experiments by Tough et al. (65) and Chase (71)# in comparison with the following simple model assuming that the T-I to T-II transition is triggered by an instability in the T-I state (laminar normal fluid state and turbulent superfluid state), where local vortex line density peaks at about 0.15d from the wall (68). This interpretation is motivated by the experimental information that transition to the T II state is influenced by the shear forces near the channel walls and pre-existing quantized vortices in the flow from T I turbulence. Previous work cited above suggests that the instability will most likely develop upon reaching a critical counterflow velocity, which we estimate by the algebraic sum of the superfluid velocity taken as constant across the channel us(r)=us and the normal fluid velocity un(r). The relevant value of normal fluid velocity is estimated as un(r/d=0.15)≈0.2un. For a critical counterflow velocity of 0.9 cm/s, Fig. 2 shows fair agreement between this simple model and the experimental data over a temperature range within which normal and superfluid densities and velocities vary by two orders of magnitude. The shear arising from mutual friction creates large normal fluid eddies, leading to the peculiar temporal decay of vortex line density (69, 70), eventually acquiring the classical late stage of the Kolmogorov-like form with L(t)∝t−3/2.

Fig. 2. Critical heat flux, q.c2 at the transition from turbulent state T I to T II of thermal counterflow as reported in ref. 65 (open blue circles) and ref. 71 (full black circles), compared to the model described in the text (red line).

D. Rotating Channel Counterflow.

An interesting flow visualization experiment has recently been performed using a spinning glass cryostat by Peretti et al. (77) in Grenoble. Transition to QT has been directly visualized using micron-sized tracer particles of solid hydrogen in a vertical channel rotating about its longitudinal axis. Under slow and steady rotation at an angular frequency Ω, a hexagonal lattice of rectilinear quantum vortices is formed, which at length scales larger than the inter-vortex distance δ=κ/(2Ω) mimics solid body rotation. At the bottom of the vertical channel, a heater is installed that can trigger a normal fluid momentum flux. If driven by an alternating current, a collective wave mode is triggered along the vortex lines. Its frequency is that of thermal excitation, and its transverse amplitude exhibits a resonance when the normal fluid is driven vertically with a frequency equal to the rotation rate Ω. For sufficiently high heater power neighboring quantum vortices strongly interact, reconnect, destroy the original vortex lattice and a rotating counterflow turbulence displaying interesting features (78) is generated. Similar experiments on the generation of QT by perturbing a rotating vortex lattice with well-known structure, representing well-defined starting conditions for the generation of waves or 3D QT (as recently performed with 3He-B in Helsinki (79), and those currently under way with He II in Prague) should shed new light on the generation of QT in helium superfluids.

E. Spherically Symmetric Counterflow of He II.

In order to better understand the underlying physics of transition in counterflow, it is desirable to exclude the crucial influence of the channel walls (80, 81). Indeed, basic features of 1D counterflow are in striking contrast with properties of spherically symmetric 3D counterflow, such as those generated in Prague by a tiny 150 resistor of 2RH≈1.8mm in diameter in the center of a spherical cavity of diameter 20 mm (82). This 3D counterflow represents an unbounded flow. The steady-state profile of local vortex line density is inhomogeneous, roughly L(r)∝r−4, since uns(r)∝r−2. Numerical simulations (83) show that a vortex shell forms in the vicinity of the heater with maximum density in some non-zero distance from the surface. The upper limit of uns evaluated at the surface of the heater where the vortex tangle is first created, is about 1 mm/s, but the expected relationship of the characteristic vortex line density ∝uns2 does not hold in the entire range of counterflow velocities, but remains valid for low uns; data at high uns seem to agree better with a linear relationship.

It is a striking observation that for intermediate uns the observed growth of vortex line density is much weaker or, in some cases, suppressed (82). This region is also the only one showing a systematic temperature dependence, with the plateau-like feature starting first at higher temperatures, while at lower temperatures it happens abruptly as uns increases. The authors argue that for intermediate uns the energy of the vortex tangle is additionally consumed by the emergence of turbulence in the normal component, which is first confined to the vicinity of the heater. This leads to a slower growth rate of vortex line density as the direct energy input of the heater becomes divided between the generation of classical-like turbulence in the normal component and the growth of the vortex tangle in the superfluid. It is remarkable that the instability in the steady 3D counterflow at various temperatures is triggered at the same critical value of the Donnelly number. Upon increasing q˙, the normal component becomes turbulent in the entire cell.

To complete this discussion, we mention an intriguing analogy between the ac flows of He II discussed earlier and the dc pipe/channel flows of He II in view of the multiple transitions observed. Leaving aside the first critical velocity, exceeding which does not result in appreciable change in the drag, the second critical velocity resulted in an abrupt increase in the drag, but still much lower than for the case of developed classical turbulent wake with the drag coefficient of about unity. This might correspond to the Allen–Reekie rule for the T-I state of thermal counterflow (or below the plateau in spherical counterflow), while in the T-II state (or above the plateau in spherical counterflow), both components are turbulent.

F. Pure Superflow.

As a final specific case of the rich variety of two-fluid steady flows of He II, we have to discuss pure superflow, which is similar to the case of 3He-B discussed above: There is no through flow of the normal fluid. It is instructive to consider experiments of Tough’s group (62, 84, 85) measuring the temperature drop over thin tubes of circular and high aspect ratio rectangular cross-sections about 10 cm in length. Pure superflow was induced thermally by a fountain pump, and the superflow rate was accurately measured. Although no special care was taken in order to assure smooth entry lengths, measurements of rather sharp onset of QT have been performed by Baehr & Tough (62) at temperatures increasing in steps of 0.1 K between 1.3 up to 1.9 K for a circular tube 0.134 mm in diameter, while the critical velocity us,cr≈1.5cm/s did not appreciably depend on the temperature when measured as it declined. On the other hand, the authors observed several metastable potential flow states upon slowly increasing the flow velocity up to much higher velocities.

It is striking that the vortex line density in the pure superflow behaves as L1/2≈γsuns with γs the same as in T-II state of thermal counterflow, γ2. This strongly suggests that the normal fluid is turbulent in a single observed state of pure superflow, which is confirmed by the late decay of the form L(t)∝t−3/2; see figure 9 in ref. 86. We note that between 1.3 and 1.9 K νn changes by an order of magnitude, strongly suggesting that the classical Dn-based criteria are unlikely to work. This view is further strengthened by several independent measurements in pure superflow: in pipes of diameters spanning more than three orders of magnitude in size from 10−6 m, the data collected in refs. 87 and 88 suggest that us,cr varies by a factor of about five, perhaps as the fourth root of the channel size with no appreciable effect of the temperature. All these results strongly support the view that the transition to QT in pure superflow occurs via extrinsic vortex nucleation in the superfluid component, which acts as a trigger to transition to turbulence in the normal fluid.

7. Conclusions

We have presented a unified phenomenological description of transition to QT in various dc and ac quantum flows of superfluids He II and 3He-B. Even in the simplest one-component case in the T→0 limit, the transition of ac flow due to an oscillating bluff body displays up to three distinct steps. Even when the transition occurs from potential flow with zero viscosity, there is no classical analogue because of the nucleation of quantized vorticity. In the vast majority of practical cases, the nucleation process of quantized vorticity is extrinsic in character, originating from vortex seeds, almost always existing in macroscopic samples of superfluid helium; however, with special care in He II and especially in 3He-B, both extrinsic and intrinsic nucleation can be generated.

At finite temperatures, we base our approach on the validity of the phenomenological two-fluid model postulating the existence of two velocity fields (5, 6). In 3He-B, the situation is simpler because the normal fluid hardly moves in the frame of reference of the container. This simplification leads to the extraordinary situation in which temperature plays a similar role to the Reynolds number in classical viscous flows; further, the transition to turbulence in the superfluid component of 3He-B does not depend on the flow geometry. In He II, in isothermal flows at low enough flow velocities, the normal component and the superfluid component are independent (or nearly so except via remnant vortices). Upon exceeding certain critical conditions both become turbulent: the normal fluid when reaching the critical Donnelly number, the superfluid by reaching the critical velocity suitable for Donnelly–Glaberson instability. Which instability occurs first depends on the geometry and on whether one deals with the coflow or counterflow, and on the temperature. Because of strong temperature dependence of the kinematic viscosity of the normal fluid, the order of these transitions can, in some cases, be reversed by changing the temperature. Additionally, transition in one fluid may serve as a trigger for transition in the other.

The phenomenology presented here captures many experimental facts and known features of transitions to QT accumulated over many years of research of helium superfluids. We therefore believe that this perspective represents a solid basis for further dedicated studies, both experimental and theoretical. In particular, careful measurements of the density of quantized vortices across a counterflow channel and a stability analysis of the T I flow with mutual friction may bring further important insights into the nature of the T I to T II transition. Direct numerical simulations of channel counterflow in both T I and T II state using the full Biot–Savart integral for the dynamics of quantized vortices and bi-directional coupling with the normal fluid will bring equally important understanding of this puzzling phenomenon, especially if one succeeds in implementing a proper boundary condition for quantized vortices with a finite de-pinning energy.

We acknowledge fruitful discussions on transitions to quantum turbulence over many years with a number of colleagues, too numerous to mention, but wish to single out Carlo Barenghi for sustained collaboration. Support by the Czech Science Foundation under project GAČR 20-00918S and Research Funds from New York University are greatly appreciated.

Author contributions

L.S., D.S., and K.R.S. designed research; performed research; and wrote the paper.

Competing interests

The authors declare no competing interest.

Data, Materials, and Software Availability

All study data are included in the main text.

This article is a PNAS Direct Submission.

*Perhaps by the Kibble-Zurek mechanism (12) due to cosmic rays overheating of 3He-B locally to the normal state as it quickly cools through Tc, or the Kelvin–Helmholtz instability of the interface between 3He-A and 3He-B (29).

†R.J. Donnelly first suggested the use of the “Reynolds number” based on the viscous penetration depth of the normal fluid, but it failed to describe the onset of turbulence for a torsionally oscillating sphere in He II (35). We now know that this failure occurred because the superfluid component of He II underwent an instability first. Proper credit has now been assigned since (37).

‡It was first proposed as an explanation for the observed crossover between two regimes of U-tube oscillations (41).

§Note that the flow of He II due to such thin vibrating wires cannot at any temperature be classified as a high Stokes number flow, and so the analysis discussed above is not applicable.

¶A superleak is a filter with sub-micron-holes that are permeable only to the inviscid superfluid component.

#We consider data above T>1.2 K, as this author himself doubts their validity at lower temperatures, and, very close to the λ-point, where uncertainties in ratios of superfluid to normal densities and velocities are large, the error bars are too large for the heat flux measurements to be reliable.
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