Generation, statistically steady state, and temporal decay of axially rotating thermal counterflow of superfluid ^{4}He (He II) in a square channel is probed using the second sound attenuation technique, measuring the density of quantized vortex lines. The array of rectilinear quantized vortices created by rotation strongly affects the development of quantum turbulence (i.e., turbulence strongly affected by the presence of quantized vortices). At relatively slow angular velocities, the type of instability responsible for the destruction of the laminar counterflow qualitatively changes: the growth of seed vortex loops pinned on the channel wall becomes gradually replaced by the growth due to Donnelly–Glaberson instability, which leads to rapid growth of helical Kelvin waves on vortices parallel with applied counterflow. The initial transient growth of vortex line density that follows the sudden start of the counterflow appears self-similar, linear in dimensionless time, $\Omega t$. We show numerically that Kelvin waves of sufficiently strong amplitude reorient the vortices into more flattened shapes, which grow similarly to a free vortex ring. The observed steady state vortex line density at sufficiently high counterflow velocity and its early temporal decay after the counterflow is switched off are not appreciably affected by rotation. It is striking, however, that although the steady state of rotating counterflow is very different from rotating classical grid-generated turbulence, the late temporal decay of both displays similar features: the decay exponent decreases with the rotation rate $\Omega $ from −3/2 toward approximately −0.7, typical for two-dimensional turbulence, consistent with the transition to bidirectional cascade.

## I. INTRODUCTION

Rotating turbulent flows represent an important physical phenomenon occurring in a wide range of systems from industrial rotating machinery to geophysical flows.^{1} Rotating flows, rather than being driven primarily by the interaction with the boundaries, are strongly affected by the Coriolis body force $Fc=\u22122\rho \Omega \xd7u$, where $\Omega $ is the angular velocity, $u$ is the velocity field, and $\rho $ is the density. In rapidly rotating flows, characterized by low Rossby number $Ro=u/l\Omega $ (with *u* and *l* are the characteristic velocity and the length scale, respectively), the competition between excitation of inertial waves propagating parallel to the axis of rotation and the two-dimensionalization of the flow due to the Taylor–Proudman theorem^{2} leads to a phase transition-like appearance of the inverse cascade,^{3,4} which joins a growing body of intensely studied phase-transition-like phenomena in fluid turbulence.^{5,6} The inverse cascade can lead to the spontaneous formation of large-scale structures^{7} or an intermediate regime with an energy flux loop^{8} with the inverse cascade arrested at an intermediate scale. In this work, we study non-classical vortex instability in a rotating superfluid ^{4}He (He II) in order to shed light on the universal features of rotating turbulence.

^{9,10}This superflow is, thus, potential; however, vorticity is possible in the form of quantized vortices,

^{9,10}line-like topological defects with diameter $a\u22480.15$ nm around which the flow circulation is restricted to a single quantum of circulation $\kappa \u22489.98\xd710\u22128$ m

^{2}/s (corresponding to $2\pi $ phase rotation). Due to the existence of the quantized vortices, the nature of the inertial waves changes drastically superfluids.

^{11}In the steady-state under rotation, the superfluid becomes threaded by a hexagonal lattice of rectilinear singly quantized vortex lines,

^{12}and on length scales larger than the mean distance between quantized vortices, He II mimics classical solid body rotation. The vortex line density (total length of vortices per unit volume or, equivalently, the number of rectilinear vortices per unit area) obeys the Feynman rule:

^{13}the dispersion relation of waves propagating along the axis of rotation is

^{14}

*b*is the typical distance between vortices.

Historically, the most studied type of turbulence in He II is the thermal counterflow,^{10,15–18} which can be set up by applying heat flux $q\u0307$ at a closed end of a channel with its other end open to the bath of He II. The heat flux $q\u0307$ is carried in a convective manner by the normal fluid; by the conservation of mass, a superfluid current arises in the opposite direction, and counterflow velocity is established: $vns=q\u0307/\rho ssT$, where *T* is the temperature and *s* is the specific entropy. For sufficiently low $vns$, the flow of the viscous normal fluid is laminar, and there are no quantized vortices in the superfluid component (except for the remnant ones).^{19} Upon increasing $q\u0307$, on exceeding the small critical counterflow velocity $vnscr$, thermal counterflow becomes turbulent, and a tangle of quantized vortices is generated by extrinsic nucleation and reconnections. Both the superfluid component (as a tangle of quantized vortices) and the viscous normal fluid can become turbulent.^{20–22}

The hexagonal array of vortex lines can become unstable in counterflow oriented along the axis of rotation^{23–25}—the so-called Donnelly–Glaberson instability—which leads to the excitation of helical Kelvin waves above a critical counterflow velocity.^{14,26} A similar instability is believed to be at least partly responsible for the transition to turbulence also in non-rotating thermal counterflow through the so-called vortex mill mechanism.^{27} Experiments^{24} in rotating counterflow at small velocities and vortex filament numerical simulations^{28,29} showed the suppression of the critical velocity for the turbulent growth of vorticity to very small values for rapidly rotating turbulence and possibly a second critical velocity connected to the development of disordered tangle. Recently, the excitation of vortex waves was directly visualized,^{25} where, however, the waves were not forced by Donnelly–Glaberson instability but rather by oscillatory counterflow.

In this work, we experimentally study rotating counterflow probed by the second sound attenuation. We confirm the results of Swanson *et al.*^{24} and extend these results to the dynamics of turbulence growth and decay. With complementary numerical simulations, we relate the initial growth of vortex line density to Kelvin wave growth on individual vortices in the vortex lattice and show that rotating quantum turbulence decays quasiclassically at long decay times consistent with rotating turbulence in elongated domains^{3,30} and previous observations of quasiclassical behavior of quantum turbulence.^{10}

This paper is organized as follows: The experimental setup and protocol are described in Sec. II, and experimental results on steady-state turbulence are shown in Sec. III A. Dynamical transitional behavior is addressed both experimentally and numerically in Sec. III B, after which discussion and conclusions follow.

## II. METHODS

### A. Experimental setup

The geometry of the counterflow channel used in the experiment is illustrated in Fig. 1 and is similar to the past experiments of Swanson *et al.*^{24} The brass flow channel with a square cross section $A=7\xd77$ mm^{2} and length $H=83$ mm is placed vertically on the axis of rotation of the cryostat. A resistive wire heater ( $R\u224816$ Ω) glued to a flat surface is inserted inside the channel from the bottom to provide the counterflow heat flux, keeping the other end open to the liquid helium bath.

The vortex line density in the channel is detected using second sound attenuation.^{31} The channel acts as a semi-open acoustic resonator for the second sound, which is excited and detected at approximately 24 kHz by a pair of oscillating membrane ( $<1$ *μ*m pore size) second sound transducers.^{32}

*L*. Here, however, a complication exists since the attenuation due to vortices depends on the angle $\theta $ between the vortex and direction of sound propagation as $\u2009sin2\theta $, i.e., the second sound is not dissipated when propagating parallel to a straight vortex.

^{33}For a disordered, turbulent tangle, one can assume a random orientation of vortices uniformly distributed in all directions, which results

^{33}in

*B*is a temperature-dependent mutual friction parameter,

^{34}$\Delta 0$ is the width of the unattenuated second sound resonance, and $a0$ and

*a*are the amplitudes on resonance of unattenuated and attenuated peaks, respectively. However, in a situation when all vortices are oriented perpendicular to the direction of sound propagation (i.e., simple rotation without counterflow in the present experiment), on average, $\u2009sin2\theta =1$, and the expression for vortex line density is instead

The typical kinetic and thermal response time^{35} of the present experiment are of the order of 10 ms. The second sound response time (inverse linewidth of the resonance) is, at most, approximately 40 ms, which ensures that none of the transient effects studied, which occur on the timescale of seconds, is significantly affected by the non-turbulent system relaxation times.

The experimental setup was mounted on a custom-made rotating platform, described in detail in the Appendix. In this work, the rotation of the cryostat was restricted to a maximum angular speed of 180° per second.

### B. Vortex filament numerical simulations

Complementary to the second sound attenuation measurements, we studied the rotating counterflow using numerical simulations of a one-way coupled vortex filament model,^{36} in which the movement of the vortices is calculated as the response to the prescribed arbitrary normal fluid velocity and potential superflow.

*B*in (4) and (3) as $\alpha =B\rho n/2\rho $. For the simulations of Kelvin wave growth on a single vortex, Fig. 6, the integral in (6) was calculated directly, while for the local induction approximation, LIA, the integral is neglected. For the rotating counterflow simulations, Fig. 7, the integral was approximated using a Barnes–Hut (tree) algorithm with the expansion of the velocity induced by the tree nodes up to quadrupolar terms.

^{36}

Sections of vortices that approach each other closer than 0.25 mm are reconnected if the reconnection leads to a decrease in total length. For all simulations, $\alpha =0.111$ and $\alpha \u2032=1.437\xd710\u22122$ were used, corresponding to 1.65 K.^{34} The full code of the simulation is open-source and available at https://bitbucket.org/emil_varga/openvort.

## III. RESULTS

To verify the performance of our experimental setup, we first measured the vortex line density in steady rotation, which ought to obey Feynman's rule (1). The vortex line density determined from the frequency sweeps of the second sound resonance is shown in Fig. 1(b) for zero counterflow heat flux and $T=1.65$ K. There is good agreement between theoretical expectation (1) and observed vortex line density, suggesting that negligible turbulence is generated by rotation alone. Note that for the case of simple rotation, all vortices are expected to be aligned parallel to the axis of rotation and perpendicular to the direction of propagation of the second sound; therefore, vortex line density is calculated using (4) rather than (3), which is appropriate for the turbulent cases.

Due to a slight overheating of the flow channel from the heat flux dissipated by the counterflow heater, we restricted our study to 1.65 K, where the second sound velocity has a local maximum.^{34} This significantly reduces the shifting of the second sound resonance, which can result in a spurious decrease in signal strength that can be erroneously interpreted as an increase in vortex line density. This is especially relevant to the transient effects, where it is not possible to sample the full resonance curve. Note that this overheating occurs inside the flow channel, and the bath temperature is stabilized to approximately 1 mK regardless of the heat dissipated in the channel.

### A. Steady-state thermal counterflow

^{33}regardless of the angular velocity of the system. In agreement with Swanson

*et al.*,

^{24}the vortex-free state is suppressed as the rotation rate increases. The initial critical velocity of the rotating vortex array was estimated by Glaberson

*et al.*

^{14}similarly to the Landau critical velocity as

Additionally, the counterflow velocity (heat flux) in Fig. 3 was increased or decreased in steps without being interrupted. At zero rotation, pronounced hysteresis exists between the increasing and decreasing direction of the heat stepping, which is gradually suppressed as the rotation rate increases. In particular, the sharp transition in the increasing heat flux data is either suppressed or absent. This is generally in agreement with the picture that the nature of the transition to turbulence changes even in slowly rotating systems—rather than the growth starting from a random distribution of vortex loops pinned on the channel walls,^{37,38} the flow is linearly unstable at essentially all velocities due to the Donnelly–Glaberson instability.^{24,28,29} Note that the slight offset between the increasing and decreasing velocity at sufficiently high velocities is most likely due to the relaxation of the tangle density and finite heat flux ramp rate.

### B. Transient behavior

The initial growth of the vortex line density after the flow is suddenly switched on (at $t=0$) is shown in Fig. 4 for several rotation speeds. Even the lowest rotation speed studied results in a severe change of the transition behavior, again in line with the expectation that the type of instability is altered by the presence of axially oriented vortex lines. Figure 5 shows the same data with the time axis normalized by the angular velocity. All rotation speeds show a brief rotation-independent delay, after which linear in time growth starts. The linear regime is self-similar for different non-zero angular velocities as all data show the same growth rate in dimensionless time $\Omega t$.

The origin of the linear growth regime is in the growth of Kelvin waves on individual vortices. A vortex filament simulation^{36} (see Sec. II B for the details on the simulations) of an isolated vortex line oriented parallel to counterflow of $vns=1$ cm/s at 1.65 K is shown in Fig. 6. The boundary conditions are open in the directions perpendicular to the counterflow direction (*x* and *y*) and periodic along the counterflow direction (*z*, 1 mm box size). Once the initial helical perturbation grows to a sufficiently large amplitude, the radial expansion of the vortex is mostly due to the imposed counterflow, similar to a planar vortex ring. The approximately linear growth of the helix amplitude was also shown analytically for large times by Van Gorder.^{39} The full non-local interaction, shown in Fig. 6(a), results in a more complex vortex shape than the local induction approximation, shown in Fig. 6(b); however, the growth of the total length of the vortex in the computational domain is essentially unaffected, as can be seen in Fig. 6(c). The total length of each individual vortex in the array, therefore, grows as $L1\u221dt$ and the number of vortices in the channel $nV\u221d\Omega $. It follows that the total vortex line density is $L\u221d\Omega t$ as seen in Fig. 5. As a corollary, the end of the linear growth regime is unlikely to be the result of nonlinear Kelvin wave interactions on individual vortices but rather must occur once the interaction between neighboring vortices in the initial vortex array becomes significant.

To verify this picture, we simulated an axial counterflow with a hexagonal array of quantized vortices as the initial conditions corresponding to 15 °/s [vortex lattice constant 0.44 mm; Fig. 7(a) (Multimedia view)] and 30 °/s [0.31 mm; Fig. 7(c) (Multimedia view)]. The vortex motion was calculated in the laboratory frame of reference, i.e., the imposed normal fluid velocity contains the corresponding solid-body rotation component. Proper treatment of solid walls in a rotating system is rather challenging (due to, i.e., corner vortices) and outside the scope of the present work; therefore, we simplified the simulation to open boundary conditions with the removal of vortices, which are fully outside of a cylinder with radius $rcutoff=4$ mm and centered on the axis of rotation. Along the *z*-axis, periodic boundary conditions with 1 mm box size were used. The snapshots of the developing tangle and the growth of the vortex line density normalized similarly to Fig. 5 are shown in Fig. 7. Similarly to the experimental case, we observe an approximately self-similar linear increase in time of the vortex line density, although the rate of growth [i.e., the tangent of $L(t\u0303)$, $t\u0303=\Omega t$] is about twice as large for the numerical simulation than for the experiment. This is likely a numerical artifact since, as can be seen in Figs. 7(a) and 7(c) (Multimedia view), the growth of the helical Kelvin waves is not uniform throughout the lattice but is more rapid toward the edges. This is most likely due to the artificial treatment of the boundary. Since the number of vortices in the simulations grows rather rapidly, we were unable to reach a steady state due to high computational cost. However, the steady-state that would likely be reached in the simulation would be rather different from the experimental case due to the artificial bounding of the tangle in the in-plane directions and is, thus, not particularly relevant for comparison with the experiment.

*L*given by (1). The initial decay up to 1–2 s appears unaffected by rotation, and late-time decay follows a power law of the form

^{40,41}to approximately $\mu \u22480.75$, although we note that the power-law behavior for the non-rotating data is only approximate. The decay exponent depends on the fitting range—the error bars shown in Fig. 9 were calculated as standard deviations from fits of the decay power law for times $t>ti$, where the initial $ti$ was varied from 3 to 7 s in 10 steps. The resulting fit parameters were averaged for all $ti$, and the standard deviation of this set was used as an estimate of uncertainty. For rotating decaying classical turbulence, Morize and Moisy

^{30}proposed an interpolating turbulent energy spectrum $E(k)=Cp\Omega (3p\u22125)/2\epsilon (t)(3\u2212p)/2k\u2212p$ ( $p=5/3$ for zero solid body rotation, i.e., stationary cryostat) based on dimensional analysis. Using the quasi-classical approximation of superfluid vorticity $|\omega |=\kappa L$, the exponent of the energy spectrum can be related

^{40,41}to the vorticity decay exponent as $\mu =1/(p\u22121)$. The decay exponent $\mu =0.75$, thus, indicates an energy spectrum exponent $p\u22482.3$, which is in good agreement with the rapidly rotating decaying classical turbulence.

^{42}Note that the vortex filament simulations presented here are not suitable for the calculation of energy exponents due to unresolved motion of the normal fluid (one-way coupling).

^{43,44}

We also note that the number of coherent vortices in classical two-dimensional turbulence was observed to decay with exponent $\mu \u22480.7$.^{45} Two-dimensionalization of rapidly rotating turbulence is a well-established property of classical turbulence.^{2} Two-dimensional oscillatory quantum turbulence in confined geometries is known to display rather complex transitional behavior with multiple stable states of the large-scale flow,^{46,47} which are not present in the current data suggesting a simpler structure of the flow. The Rossby number defined using the quasi-classical energy dissipation rate^{40}^{,} $\epsilon =\nu eff(\kappa L)2$ with the effective viscosity^{48,49} $\nu eff\u22480.2\kappa $ is $Ro=(\epsilon \u2113)1/3/(D\Omega )\u22480.08$ for $\Omega =60$ °/s in Fig. 8 ( $\u2113=L\u22121/2$), suggesting that the flow will be strongly affected by Coriolis forces. Assuming that the energy injection scale is close to the intervortex distance $\u2113\u224810\u22122$ cm, for the 60 °/s decay, the dimensionless parameter in the steady state is $\lambda =\u2113/HRo\u22480.033>\lambda c\u22480.03$, which was shown by van Kan and Alexakis^{3} to be a critical value for transition to bidirectional cascade, suggesting that for $\Omega >60$ °/s, the inverse cascade is present already in the steady state.

## IV. DISCUSSION AND CONCLUSIONS

We have shown that the array of vortex lines created by rotation strongly affects the development of quantum turbulence in axially oriented thermal counterflow. The type of instability primarily responsible for the destruction of the laminar flow qualitatively changes at even relatively slow angular velocities from, presumably, the growth of random vortex loops pinned on the inner surfaces of the counterflow channel to the Donnelly–Glaberson instability, which leads to rapid growth of helical Kelvin waves on vortices parallel with applied counterflow. This type of transition to turbulence no longer relies on the stochastic distribution of pinned seed vortices, which is likely the reason for the suppression hysteresis at small counterflow velocities seen in Fig. 3.

Furthermore, we find that the initial transient growth of vortex line density after the sudden start of the counterflow proceeds in self-similar linear-in-time $L\u221d\Omega t$, which differs quite strongly from the exponential growth expected for linear instability. Numerically, we have shown that Kelvin waves of sufficiently strong amplitude (mostly local) nonlinear interactions reorient the vortex into a more flattened shape, which grows similarly to a free vortex ring. This free vortex ring regime is observed numerically also for an array of rotating vortices, although with a total rate of growth of vortex line density about twice the experimentally observed rate. This suggests either improperly accounted-for wall effects or the presence of normal fluid turbulence.

Finally, the observed steady-state vortex line density at sufficiently high counterflow velocities is neither strongly affected by the rotation nor the early stage of decay after the counterflow is suddenly switched off. However, rather than quasi-classical decay characterized by $L\u221dt\u22123/2$ typically observed for the decay of strongly excited counterflow turbulence, the exponent decreases toward approximately −0.75, which is in good agreement with rotating classical decaying turbulence^{30,42} and is possibly related to the decay of the number of coherent vortices in classical two-dimensional turbulence.^{45}

Despite the fact that the thermal counterflow is a type of turbulence specific to superfluid helium, the naturally small forcing length scale set by the intervortex distance enables the study of rotating turbulence in new parameter regimes, helping in the understanding the universal features of rotating turbulence. While further experiments are required (especially locally probed velocity, flow visualization, higher rotation rates, and counterflow velocities, which were not possible in the current experimental run), rotating counterflow turbulence might prove a valuable system capable of achieving flow characteristics (e.g., the ratio of forcing scale to system size) comparable to large-scale geophysical flows.

## ACKNOWLEDGMENTS

This work was supported by Charles University under PRIMUS/23/SCI/017. We are grateful to D. Schmoranzer for assistance with the initial construction of the rotating platform, B. Vejr for the machining of the experimental cell, and to J. Boháč and D. Nazarenko for ensuring liquid helium supply.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**R. Dwivedi:** Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (supporting). **T. Dunca:** Formal analysis (supporting); Investigation (equal); Methodology (equal); Software (supporting). **F. Novotný:** Investigation (supporting); Methodology (supporting). **M. Talíř:** Investigation (supporting); Methodology (supporting). **L. Skrbek:** Conceptualization (equal); Methodology (equal); Writing – original draft (equal). **P. Urban:** Resources (equal). **M. Zobac:** Resources (equal). **I. Vlček:** Resources (equal). **E. Varga:** Conceptualization (lead); Formal analysis (equal); Investigation (equal); Methodology (lead); Software (lead); Writing – original draft (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: ROTATING PLATFORM

We use a custom-made rotating platform that supports an experimental cryostat as well as all necessary electronic equipment (see Ref. 50 for a photograph). The platform rests on a central bearing with mains power for the experimental setup provided through a slip ring and the bath pump connected via a KF50 rotating feedthrough attached to the platform support structure above the cryostat. The vacuum feedthrough is double-sealed with an additional sealed compartment pressurized with ^{4}He gas slightly above atmospheric pressure, separating room air from the low-pressure helium bath to avoid contamination due to possible leaks in the rotating rubber seals.

The rotation is driven by a 1.5 kW permanent magnet synchronous brushless three‐phase AC stepper motor powered by a variable frequency drive controlled by a programable logic controller. The motor speed is reduced by a gearbox and flat belt transmission, which also serves as an emergency torque clutch. The available output torque ensures a braking capability from the maximum velocity of 60 rpm to a full stop in less than a second.

## REFERENCES

*Atmospheric and Oceanic Fluid Dynamics*

*Turbulence: An Introduction for Scientists and Engineers*

*Superfluidity and Superconductivity*

*Quantum Turbulence*

*Progress in Low Temperature Physics*

^{4}He

^{4}He: A vortex mill that works

^{4}He superflow: Steady state