Two recent experiments have highlighted the links between low-temperature physics and fluid dynamics
When the pilot gives the order to fasten seat belts during a flight, and the other passengers start to worry about spilling their coffee, physicists might console themselves by recalling that Richard Feynman once described turbulence as the last great unsolved problem of classical physics. Although a complete understanding of turbulence in ordinary classical fluids remains elusive, physicists are making real progress in a different direction – the study of turbulence in quantum fluids.
In an ordinary fluid, such as water or air, the flow can be described by a dimensionless number known as the Reynolds number. This number expresses the ratio of inertial and viscous forces in the fluid, and is defined as Re = UL/ν, where U and L are the typical speed and size of the system, and ν is the kinematic viscosity. Flows with small Re are dominated by viscosity and tend to be smooth and laminar. Instabilities appear in the flow as Re increases, and all flows become turbulent at sufficiently large Reynolds numbers.
In a quantum fluid, however, turbulence acquires new features. According to Landau’s two-fluid theory, a superfluid consists of two components: the viscous normal fluid, which is related to the thermal excitations, and the frictionless superfluid associated with the quantum ground state. And whereas the eddies in the normal fluid can be of all sizes and strengths, quantum mechanics restricts the rotational motion of the superfluid to quantized vortex filaments that are like tiny tornadoes. In particular, the circulation of the superfluid – which is defined in terms of a contour integral of the velocity field, vs, around the vortex core – is quantized in units of h/M, where h is Planck’s constant and M is the mass of a superfluid particle (see figure 1).
Turbulent transitions
Naturally occurring helium is in the main helium-4, but it also contains about one part in 107 of helium-3. The two isotopes behave quite differently near absolute zero because helium-4 atoms are bosons while helium-3 atoms are fermions and must therefore obey the Pauli exclusion principle. For instance, helium-4 becomes a superfluid at a critical temperature, Tc, of 2.2 K, whereas helium-3 only displays superfluidity below 2.7 mK. Moreover, the lighter isotope undergoes a second transition when it is cooled below 2.2 mK to produce the so-called B-phase of helium-3.
While studying this B-phase of helium-3, Antti Finne of the Helsinki University of Technology and colleagues in Japan, Finland, the Netherlands, Russia and the Czech Republic have now observed another transition at 0.6 Tc. This transition from turbulent flow at low temperatures to regular laminar flow at higher temperatures has never been seen before (A P Finne et al. 2003 Nature at press; arXiv.org/abs/cond-mat/0304586).
Finne and co-workers start with a rotating sample of helium-3 that does not contain any vortices. Next they inject a few “seed” vortex loops into the sample and then use a non-invasive nuclear magnetic resonance technique to measure how the total length of the vortices – which is proportional to the total energy contained in them – changes as a function of temperature and angular velocity. The results are surprising (see figure).
The total length of the vortices increases with time, with the energy needed to do this coming from the normal fluid (and ultimately from the motor being used to rotate the sample). If the temperature is greater than at 0.6 Tc, the vortices became rectilinear and align themselves along the axis of rotation. However, if the temperature is less than at 0.6 Tc, they evolve into a tangled network that fills the entire superfluid sample (although this eventually relaxes into an ordered array of rectilinear vortices).
What is so special about the temperature at 0.6 Tc? The answer lies in the behaviour of Kelvin waves on the vortex filaments (see part (d) of figure). Finne and co-workers showed that, at high rotations, the superfluid vorticity is determined by the ratio of dissipative and inertial forces in the superfluid. This ratio can be expressed as q = α/(1 – α´), where α and α´ are related to the strength of the interaction between the vortex and the normal fluid. For a single vortex filament, q = 1 represents the crossover from waves that propagate (i.e. Kelvin waves) to waves that are overdamped (q > 1).
Numerical simulations show that an injected vortex loop will expand and become rectilinear when q > 1. However, if q q depends on the temperature, and q is almost equal to one (1.3) when the temperature is 0.6 Tc. This suggests that Finne and co-workers have discovered a dimensionless quantity, q, which plays a role in superfluid turbulence similar to that played by the Reynolds number in classical hydrodynamics.
Moreover, the experiment suggests that we may be close to a breakthrough in understanding all the individual steps that lead to the formation of a turbulent state in a superfluid, and its subsequent decay. Indeed, we are rapidly approaching a time when we can numerically model these phenomena with reasonable accuracy. This is in marked contrast to our understanding of classical turbulence.
Superfluidity at the double
The superfluid states of helium-3 and helium-4 have both proved to be incredibly rich sources of new physics, so it seems only natural to wonder what would happen if both states could co-exist. It is possible to dissolve small amounts of helium-4 in liquid helium-3, which, in principle, provides us with a starting point for such experiments. Unfortunately this reduces the critical temperature of the helium-3 to a few microkelvin, which is currently beyond the reach of experimenters.
However, Gavin Lawes from Cornell University, and colleagues at Manchester University and the University of Delaware have now overcome this problem by studying the properties of helium-3-helium-4 mixtures in aerogels (Phys. Rev. Lett. 90 195301). An aerogel is a porous network of silica strands with diameters of a few nanometres. The 98% porous aerogel used by Lawes and colleagues increased the Tc of the helium-3 in the mixture to a few millikelvin, which is well within the range of current experimental techniques.
So how did the team prove that the superfluid phases of both helium isotopes were co-existing? Landau’s two-fluid theory predicts the existence of two sound modes in liquid helium: in “first sound” the superfluid and normal fluid components move in phase, while in “second sound” they move in antiphase. First sound corresponds to ordinary sound in a classical fluid, and can therefore be observed both above and below Tc. Second sound, on the other hand, is only found in the superfluid state below Tc. The researchers were able to exploit the link between second sound and superfluidity to prove that superfluid helium-3 and superfluid helium-4 were present in their experiment at the same time.
Quantum fluids can display remarkable behaviour in porous materials and other confined geometries. In a rigid porous material, the normal component is clamped by viscosity and cannot move. But an aerogel is not rigid, so it is possible for both the normal fluid and the aerogel to move together, either in phase with the superfluid in a “fast” mode, or against it in a “slow” mode. This slow mode is the key because it depends on the existence of superfluidity: if two different superfluid components are present in a mixture, we would expect to observe two different slow modes.
Lawes and colleagues first filled the aerogel with pure helium-3 and tracked this slow mode at increasing temperatures until it vanished at Tc = 1.62 mK. Then they repeated the experiment with increasing amounts of helium-4 in the mixture, and when the overall concentration of helium-4 reached 10.5%, they observed not one but two slow modes below the Tc for helium-3. This clearly demonstrated that superfluid phases of both isotopes were co-existing inside the aerogel. Above this temperature one of the slow modes vanished, while the researchers tracked the other up to 337 mK.
Superfluidity was first observed in helium-4 in 1938, and these new experiments show that the subject is still going strong some 65 years later. Moreover, they demonstrate that the barriers between different areas of physics – such as low-temperature physics and fluid dynamics – are slowly coming down.
