Turbulent Dynamics in Rotating Helium Superfluids

V.B. Eltsov*; R. de Graaf*; R. Hänninen*; M. Krusius*; R.E. Solntsev*; V.S. L’vov†; A.I. Golov‡; P.M. Walmsley‡    * Low Temperature Laboratory, Helsinki University of Technology, P.O.Box 5100, FI-02015-TKK, Finland
† Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
‡ School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK

1 INTRODUCTION

The transition to turbulence is the most well known example of all hydrodynamic transitions. It has been marveled for centuries since dramatic demonstrations can be seen everywhere where a sudden change in the flow occurs, owing to a constriction in the flow geometry, for instance. For 50 years, it has been known that turbulence also exists in superflu-ids (Vinen and Donnelly, 2007), although by its very nature a superfluid should be a dissipation-free system. In many situations, it is found on the macroscopic level that superfluid vortex dynamics mimics the responses of viscous hydrodynamics. This is one of the reasons why it has been thought that superfluid turbulence might provide a shortcut to better understanding of turbulence in general. From the developments over the past 50 years we see that this has not become the case, superfluid turbulence is a complex phenomenon where experiments have often been clouded by other issues, especially by vortex formation and vortex pinning. Nevertheless, the topic is fascinating in its own right: when the flow velocity is increased, the inherently dissipation-free superfluid is observed to become dissipative and eventually turbulent. This is particularly intriguing in the zero temperature limit where the density of thermal excitations approaches zero and vortex motion becomes undamped down to very short wave lengths (of the order of the vortex core diameter).

There are two isotropic helium superfluids in which turbulence has been studied, namely the B phase of superfluid 3He (3He-B) and superfluid 4He (4He II). In the anisotropic A phase of superfluid 3He (3He-A), dissipation is so large that conventional superfluid turbulence is not expected at the now accessible temperatures above 0.1Tc (Finne et al., 2003). Instead rapid dynamics and large flow velocities promote in 3He-A a transition in the topology and structure of the axially anisotropic superfluid order parameter field, a transition from linear line-like vortices to planar sheet-like vortices (Eltsov et al., 2002). Turbulence has also been studied in laser-cooled Bose-Einstein condensed cold atom clouds although so far only theoretically (Kobayashi and Tsubota, 2008; Parker and Adams, 2005), but it is expected that experiments will soon follow. Here, we are reviewing recent work on turbulence in rotating flow in both 3He-B and 4He II, emphasising similarities in their macroscopic dynamics.

A number of developments have shed new light on superfluid turbulence. Much of this progress has been techniques driven in the sense that novel methods have been required, to make further advances in a field as complex as turbulence, where the available techniques both for generating and detecting the phenomenon are not ideal. Three developments will be discussed in this review, namely (i) the use of superfluid 3He for studies in turbulence, which has made it possible to examine the influence of a different set of superfluid properties in addition to those of superfluid 4He, (ii) the study of superfluid 4He in the zero temperature limit where the often present turbulence of the normal component does not complicate the analysis and (iii) the use of better numerical calculations for illustration and analysis.

From the physics point of view, three major advances can be listed to emerge: in superfluid 3He, one can study the transition to turbulence as a function of the dissipation in vortex motion (Eltsov et al., 2006a), known as mutual friction. The dissipation arises from the interaction of thermal excitations with the superfluid vortex when the vortex moves with respect to the normal component. Inclassical viscous flow, such a transition to turbulence would conceptually correspond to one as a function of viscosity. This is a new aspect, for which we have to thank the 3He-B Fermi superfluid where the easily accessible range of variation in mutual friction dissipation is much wider than in the more conventional 4He II Bose superfluid. We are going to make use of this feature in Section 2 where we examine the onset of superfluid turbulence as a function of mutual friction dissipation (Finne et al., 2006a).

Second, in Section 3, we characterise the total turbulent dissipation in superfluid 3He as a function of temperature, extracted from measurements of the propagation velocity of a turbulent vortex front (Eltsov et al., 2007). A particular simplification in this context is the high value of viscosity of the 3He normal component, which means that in practice the normal fraction always remains in a state of laminar flow.

Finally, our third main topic in Section 4 are the results from recent ion transmission measurements in superfluid 4He (Walmsley et al., 2007a; Walmsley and Golov, 2008a), where the decay of turbulence is recorded from 1.6K to 0.08K.Here turbulent dissipation can be examined in the true zero temperature limit with no normal component. As a result we now know that turbulence and dissipation continue to exist at the very lowest temperatures. Although the dissipation mechanisms of 4He or 3He-B in the T → 0 limit are not yet firmly established (Kozik and Svistunov, 2005b; Vinen, 2000, 2001), it is anticipated that these questions will be resolved in the near future (Kozik and Svistunov, 2008b; Vinen, 2006).

Phrased differently, our three studies address the questions (i) how turbulence starts from a seed vortex which is placed in applied vortexfree flow in the turbulent temperature regime (Section 2), (ii) how vortices expand into a region of vortex-free flow (Section 3) and (iii) how the vorticity decays when the external pumping is switched off (Section 4). The common feature of these three studies is the use of uniformly rotating flow for creating turbulence and for calibrating the detection of vorticity. Turbulence can be created in a superfluid in many different ways, but a steady state of constant rotation does generally not support turbulence. Nevertheless, at present rotation is the most practical means of applying flow in a controlled manner in the millikelvin temperature range. In this review, we describe a few ways to study turbulence in a rotating refrigerator. Superfluid hydrodynamics supports different kinds of flow even in the zero temperature limit so that turbulent losses can vary greatly both in form and in magnitude, but generally speaking, the relative importance of turbulent losses tends to increase with decreasing temperature. Two opposite extremes will be examined: highly polarised flow of superfluid 3He-B when a vortex front propagates along a rotating cylinder of circular cross-section (Section 3), and the decay of a nearly homogeneous isotropic vortex tangle in superfluid 4He (Section 4), created by suddenly stopping the rotation of a container with square cross-section.

Turbulent flow in superfluid 3He-B and 4He is generally described by the same two-fluid hydrodynamics of an inviscid superfluid component with singly-quantised vortex lines and a viscous normal component. The two components interact via mutual friction. There are generic properties of turbulence that are expected to be common for both superfluids. However, there are also interesting differences which extend the range of the different dynamic phenomena which can be studied in the He superfluids:

• In typical experiments with 3He-B, unlike with 4He, the mutual friction parameter α can be both greater and smaller than unity (Figure 1) – this allows the study of the critical limit for the onset of turbulence at α ∼ 1 (Section 2).