Observation of Remanent Vortices Attached to Rough Boundaries in Superfluid 4 He

Journal of Low Temperature Physics - QFS9 manuscript No. (will be inserted by the editor) Observation of Remanent Vortices Attached to Rough Boundaries in Superfluid 4 He Y. Nago T. Ogawa A. Mori Y. Miura K. Obara H. Yano O. Ishikawa T. Hata Received: date / Accepted: date Abstract We report the study on remanent vortices attached to rough boundaries in superfluid 4 He after the turbulent transition. We used 2.6 µm vibrating wires with smooth surfaces and rough surfaces, a cover box and slow cooling method, in order to investigate the effect of surface roughness on the condition and the number of vortices attached to a wire. The responses of the wire with smooth surfaces show large hysteresis at the turbulent transition. This result indicates that remanent vortices c the wire and surrounding boundaries cause turbulence. At first sweep of driving force of the wire with rough surfaces, we also observed hysteresis as large as the case of the smooth wire: at the other sweeps, however, small hysteresis was observed. These results indicate that once turbulence is generated at a wire velocity during first sweep, vortex lines newly attach to rough surfaces of the wire, which easily cause turbulence at a low wire velocity. Therefore, we conclude that a smooth wire can reduce the number of vortices attached to a wire. 1 Introduction In recent years, quantum turbulence in superfluid 4 He generated by oscillating obstacles has been attracted research interest [1 4]. Turbulence is generated due to vortices attached to an oscillating obstacle: vortices nucleate after cooling through the superfluid transition and remain forming bridges between rough surfaces of an oscillating obstacle [2] or between an oscillating obstacle and surrounding boundaries [5]. It is considered that those attached vortices become unstable to expand above a critical velocity of oscillation, entangle and reconnect themselves, forming turbulence. According to the numerical simulations [6], the Kelvin waves are generated on the vortices attached between an obstacle and surrounding boundaries by an oscillatory flow and become unstable to emit many vortex rings above a critical amplitude of oscillation. However, those mechanisms have remained to be clarified experimentally for years. It is necessary to reduce the number of vortices attached to an obstacle to study the motion of a vortex at the turbulent transition. Y. Nago T. Ogawa A. Mori Y. Miura K. Obara H. Yano O. Ishikawa T. Hata Graduate School of Science, Osaka City University, Osaka 558-8585, Japan E-mail: ynago@sci.osaka-cu.ac.jp

2 (a) Smooth wire 2.6 µm (b) Rough wire 2.6 µm Fig. 1 Electron micrographs of vibrating wires with a 2.6µ m diameter: a wire with smooth surfaces(a) and a wire with rough surfaces(b). There are a lot of ditches parallel to the wire axis and a protuberance on the left side of the wire in photograph (b), while almost no remarkable ditches and protuberances are seen in photograph (a) (see text). Recently, we have been studying the transition to quantum turbulence in superfluid 4 He using a vibrating wire, which is fabricated from a commercial multi-filaments superconducting wire with dies. There are some wires with relatively smooth surfaces and rough surfaces in the multi-filaments, as shown in Fig. 1. In a previous work [7], we reported the experimental results in superfluid 4 He at 1.2 K using vibrating wires with smooth surfaces with different thicknesses, a cover box and slow cooling method to control the number of vortices attached to a wire. It is found that small hysteresis appears in the velocity of the 2.6 µ m wire between the up and down sweeps of driving force, while no hysteresis for the thicker wires. Furthermore, we covered a vibrating wire with a copper box and cooled the helium liquid slowly from above the lambda temperature to 1.2 K in 2 h, twice the previous cooling time, and found that hysteresis becomes much larger. This large hysteresis indicates that the velocity of laminar-to-turbulent transition becomes larger for the 2.6 µ m vibrating wire using a cover box and slow cooling method. We found that these conditions can reduce the number of vortices attached to a wire and prevent surrounding vortices and counterflow from triggering turbulence at a low wire velocity. In this work, we prepared a vibrating wire with smooth surfaces which was used in a previous work [7] and a vibrating wire with rough surfaces, in order to investigate the effect of surface roughness on the condition and the number of vortices attached to a wire. As mentioned in first paragraph of this section, vortices can remain attached to rough surfaces of an oscillating obstacle. Therefore, the surface roughness is likely to affect the number of vortices attached to boundaries. A vibrating wire with rough surfaces is expected to capture more vortices than a vibrating wire with smooth surfaces. We discuss the experimental results for two thin vibrating wires with different roughnesses using a cover box and slow cooling method. 2 Experimental details We prepared a very thin vibrating wire with a diameter of 2.6 µ m to establish the condition of the wire with less attached vortices. The wire is made from a superconducting(nbti) wire fabricated from a commercial multi-filaments superconducting wire with dies. We can produce relatively smooth wires using this method. We set the vibrating wire into a semicircle with two legs fixed to pillars on a copper plate. Resonance frequency of the wire is

3 determined by adjusting the distance between the legs. The wire is applied a magnetic field of 25 mt, driven by ac electric currents. We measured the amplitudes of the wire using a phase locked loop technique as used in a previous work [3] and estimated the velocity at the apex of the wire. We put a vibrating wire in a cover box to reduce the number of vortices attached to the wire. The cover box is cylindrical in shape with 5mm in diameter and 3mm in height, made of copper. There is a pinhole on the cover box to fill the helium liquid into the box. Helium liquid is cooled up to 1.2 K using 4 He evaporation cryostat, and cooling speed is adjusted by pumping valve. Based on a previous work [7], we cooled the helium liquid slowly from above the lambda temperature to 1.2 K in 2 h, to reduce the number of vortices attached to the wire. In this work, we used two 2.6 µm vibrating wires with different roughnesses. There are some wires with different roughnesses in the fabricated multi-filaments, as seen in Fig. 1. We selected the wire with smooth surfaces (Fig. 1 (a)) and the wire with rough surfaces (Fig. 1 (b)) from 2.6 µm multi-filaments. The wire with smooth surfaces has almost no roughness of more than 5 nm. The wire with rough surfaces has a lot of ditches parallel to the wire axis whose width is.14 ±.2 µm and some protuberances whose width is up to 1 µm, as seen in Fig. 1 (b). The velocity of the turbulent transition for the rough wire is expected to decrease, compared with the case of the smooth wire, because these ditches and protuberances are likely to capture more vortices. Resonance frequency is 98 Hz for the smooth wire and 146 Hz for the rough wire in vacuum. We measured the velocity response of a 2.6µm wire with smooth surfaces in a cover box and a 2.6µm wire with rough surfaces in a cover box, after slow cooling. 3 Results and discussion 3.1 Vibrating wire with smooth surfaces Figure 2 shows the velocity response of a vibrating wire with smooth surfaces in a cover box after slow cooling. We find the velocity of laminar-to-turbulent transition to be about mm/s and the large hysteresis at the turbulent transition between the up and down sweeps of driving force. It is considered that there are some possible causes of the turbulent transition in boundary flow of superfluid 4 He at 1.2 K: e.g. remanent vortices attached to an oscillating obstacle [2, 5], free vortex rings [8], and counterflow. Attached vortices can cause turbulence at a sufficiently high oscillation velocity. In the case of a vibrating wire, if there are less attached vortices, we may observe a much higher velocity of the turbulent transition because of the absence of vortices in the top part of the wire. Free vortex rings can easily trigger turbulence at a low critical velocity at which a turbulent flow can be maintained. Also counterflow may be able to grow vortices in the vicinity of an oscillating obstacle, easily causing turbulence at a low critical velocity. However we observed a high velocity of the turbulent transition for the vibrating wire with smooth surfaces as shown in Fig. 2. Consequently, this result indicates that a cover box and slow cooling prevent counterflow and free vortex rings from triggering turbulence in the vicinity of the wire. Also, slow cooling reduces vortex nucleation at the superfluid transition and a cover box prevents surrounding vortices from propagating to the wire, resulting in the wire with less attached vortices which shows a high velocity of the turbulent transition. A cover box and slow cooling can reduce the number of vortices attached to the wire. This is consistent with a previous result [7]. Therefore the turbulent transition at the velocity of about mm/s can be due to the instability of vortices initially attached to the wire.

4 velocity [mm/s] 4 8 12 16 2 driving force [nn] transition velocity [mm/s] 1 2 3 4 5 Series Fig. 2 (Color online) Response of a vibrating wire with smooth surfaces in a cover box after slow cooling to 1.2 K. Large hysteresis appears at the turbulent transition between the up and down sweeps of driving force. Fig. 3 The velocity of the laminar-to-turbulent transition for a vibrating wire with smooth surfaces as a function of n-th profile. The transition velocity is almost constant of about mm/s (see Fig. 2) up to 2th profile. From 2th profile, however, the velocity shifts slightly and also the low transition velocity, i.e. small or no hysteresis are seen. The numerical simulations [6] indicate that the Kelvin waves on vortex lines attached to a stationary sphere arises in oscillatory superfluid flow, expanding and creating many vortex rings at high flow velocities. In a case of a moving object, a similar behaviour is expected to occur, because an oscillating motion of an object can generate the Kelvin waves on vortex lines attached to it. Consequently, we conclude that vortex lines forming bridges between the wire and the walls of the cover box cause turbulence at a velocity of about mm/s. We performed the velocity profile for many times and measured the velocity of laminarto-turbulent transition, as shown in Fig. 3. It is found that the transition velocity is almost constant up to 2th profile. These results indicate that the bridged vortices attached to the wire are firmly pinned even after the turbulent transition. However, one can see that the transition velocity shifts by about 5 mm/s during from 2th to 3th profile. During measuring these profiles, we observed that the temperature decreased slightly by the pumping cooling. Therefore, the shift of the transition velocity implies that the temperature shift causes counterflow in the cover box, which shifts the pinning cites of vortices. Furthermore, small or no hysteresis were sometimes observed, as seen in Fig. 3. These results also imply the existence of counterflow in the cover box, which can grow vortices and cause turbulence in the vicinity of the wire even at a low wire velocity. 3.2 Vibrating wire with rough surfaces We measured the response of a vibrating wire with rough surfaces using a cover box and slow cooling method as shown in Fig. 4. At first sweep of driving force, we observed hysteresis as large as the case of the smooth wire (see Fig. 2), as shown in Fig. 4 (a). Therefore we obtained the wire with less attached vortices as well as the case of the smooth wire, though its roughness is different. At and after second sweep of driving force, however, small hysteresis was observed as shown in Fig. 4 (b): we can no longer observe large hysteresis. The possible reason is as follows. There are initially vortices forming bridges between the surface of the wire and walls of the cover box, the similar condition to the case of the smooth wire. However, once turbulence is generated at a critical velocity during first sweep, vortex

[mm/s] velocity [ (a) First profile 5 1 15 driving force [nn] [mm/s] velocity [ (b) Second profile 5 1 15 driving force [nn] 5 Fig. 4 (Color online) Responses of a vibrating wire with rough surfaces in a cover box after slow cooling to 1.2 K: (a) first profiles and (b) second profile. Though hysteresis as large as the case of a smooth vibrating wire appears at the turbulent transition at first profiles, that becomes smaller at and after second profile. lines newly attach to rough surfaces of the wire, which remain pinned to the surface of the wire even after turbulence ceases. These vortices easily cause turbulence at a wire velocity lower than the velocity that the initially attached vortices become unstable. This is why hysteresis becomes smaller from a second sweep of driving force. The velocity of laminar-to-turbulent transition observed at and after second sweep of driving force is estimated to be about 15 mm/s. Now let us assume that vortices are newly attached to the surfaces of the wire after the turbulent transition at first sweep and cause the instability of the Kelvin wave at a wire velocity of 15 mm/s. The dispersion relation of the Kelvin wave with wave number k is given by [9] v c (κk 2 /4π)ln(1/ka ), where κ and a is the quantum of circulation and vortex core radius (.1 nm in superfluid 4 He), respectively. We estimated half wave length of the Kelvin wave from the frequency of the wire to be 1 µm. Thus a vortex line should be attached to the wire surface with its edges apart by integral multiples of 1 µm,, much longer than the thickness of the wire of 2.6 µm. It is therefore plausible that the turbulent transition is caused by the instability of the Kelvin wave on the vortices attached not between the surfaces of the wire but between the wire and the walls of the cover box. Besides, in stationary superfluid flow, vortices attached between the surfaces with its edges apart by l are considered to cause unstable expansion because of the Glaberson-Donnelly instability [1] at the critical flow velocity given by v c (κ/2πl)[ln(4l/a ) 1/4]. If vortices are attached between the rough surfaces of the wire and cause this instability at the wire velocity of 15 mm/s, the distance between the edges of the vortices is estimated to be l 1 µm. As we found the two protuberances apart from each other by the order of 1 µm in the electron micrographs, it is also possible that vortices are newly attached between the two protuberances on the wire surface after generating turbulence at first sweep of the driving force, which cause the Glaberson-Donnelly instability at the wire velocity of about 15mm/s. It is not clear at present which story is correct: further experiments and discussion are necessary to clarify that. In the case of the wire with smooth surfaces, the distance between the edges of the vortex is estimated to be l.3 µm, based on the model of the Glaberson-Donnelly instability. This value is, however, larger than the maximum roughness of 5 nm on the wire surface as seen in Fig. 1 (a). Therefore the turbulent transition observed for the case of the smooth wire cannot be caused by the Glaberson-Donnelly instability on remanent vortices attached between the surfaces of the wire. Consequently, we conclude that a vibrating wire with rough surfaces can generate

6 turbulence at a low velocity because vortices easily attach to a rough surface. A vibrating wire with smooth surfaces can reduce the number of vortices attached to a wire. 4 Conclusions We performed the experiments in superfluid 4 He using thin vibrating wires with different roughnesses, in order to investigate the effect of surface roughness on the condition and the number of vortices attached to a wire. We measured the responses of vibrating wires with smooth surfaces and rough surfaces in a cover box after slow cooling. The responses of the wire with smooth surfaces show the laminar-to-turbulent transition at a almost constant velocity and large hysteresis between the up and down sweeps of the driving force. These results indicate that remanent vortices forming bridges between the wire and surrounding boundaries causes turbulence and are firmly pinned even after turbulence ceases. At first sweep of driving force of the wire with rough surfaces, we also observed hysteresis as large as the case of the smooth wire: at the other sweeps, however, small hysteresis was observed. These results indicate that once turbulence is generated at a wire velocity during first sweep, vortex lines newly attach to rough surfaces of a wire, which easily cause turbulence at a low wire velocity. Therefore, we conclude that a vibrating wire with smooth surfaces can reduce the number of vortices attached to a wire. It is very useful to use a thin vibrating wire with smooth surfaces with a cover box and slow cooling method for studying the motion of a vortex attached to a vibrating wire at the turbulent transition in superfluid 4 He. Acknowledgements We acknowledge support from a Grant-in-Aid for Scientific Research on Priority Areas (Grant No. 1778) from The Ministry of Education, Culture, Sports, Science and Technology of Japan. References 1. J. Jäger, B. Schuderer, and K.W. Schoepe, Phys. Rev. Lett. 52, 49, (1995). 2. D. Charalambous, L. Skrbek, P.C. Hendry, P.V.E. McClintock, and W.F. Vinen, Phys. Rev. E 74, 3637, (6). 3. H. Yano, N. Hashimoto, A. Handa, M. Nakagawa, K. Obara, O. Ishikawa, and T. Hata, Phys. Rev. B 75, 1252, (7). 4. M. Blažková, M. Človečko, E. Gažo, L. Skrbek, and P. Skyba, J. Low Temp. Phys. 148, 35, (7). 5. N. Hashimoto, R. Goto, H. Yano, K. Obara, O. Ishikawa, and T. Hata, Phys. Rev. B 76, 254(R), (7). 6. R. Hänninen, M. Tsubota, and W.F. Vinen, Phys. Rev. B 75, 6452, (7). 7. H. Yano, T. Ogawa, A. Mori, Y. Miura, Y. Nago, K. Obara, O. Ishikawa, and T. Hata, J. Low Temp. Phys. DOI 1.7/s199-9-9888-9. 8. R. Goto, S. Fujiyama, H. Yano, Y. Nago, N. Hashimoto, K. Obara, O. Ishikawa, M. Tsubota, and T. Hata, Phys. Rev. Lett., 4531, (8). 9. H.A. Nichol, L. Skrbek, P.C. Hendry, and P.V.E. McClintock, Phys. Rev. E 7, 5637, (4). 1. W.I. Glaberson and R.J. Donnelly, Phys. Rev. 141, 28, (1966).