Quantum turbulence is the name given to the turbulent flow – the chaotic motion of a fluid at high flow rates – of quantum fluids, such as superfluids. The idea that a form of turbulence might be possible in a superfluid via the quantized vortex lines was first suggested by Richard Feynman. The dynamics of quantum fluids are governed by quantum mechanics, rather than classical physics which govern classical (ordinary) fluids. Some examples of quantum fluids include superfluid helium (4He and Cooper pairs of 3He), Bose–Einstein condensates (BECs), polariton condensates, and nuclear pasta theorized to exist inside neutron stars. Quantum fluids exist at temperatures below the critical temperature at which Bose-Einstein condensation takes place.
General properties of superfluids
The turbulence of quantum fluids has been studied primarily in two quantum fluids: liquid helium and atomic condensates. Experimental observations have been made in the two stable isotopes of helium, the common 4He and the rare 3He. The latter isotope has two phases, named the A-phase and the B-phase. The A-phase is strongly anisotropic, and although it has very interesting hydrodynamic properties, turbulence experiments have been performed almost exclusively in the B-phase. Helium liquidizes at a temperature of approximately 4 K. At this temperature, the fluid behaves like a classical fluid with extraordinarily small viscosity, referred to as helium I. After further cooling, helium I undergoes Bose–Einstein condensation into a superfluid, referred to as helium II. The critical temperature for Bose–Einstein condensation of helium is 2.17 K (at the saturated vapour pressure), while only approximately a few millikelvin for 3He–B.
Although in atomic condensates there is not as much experimental evidence for turbulence as in helium, experiments have been performed with rubidium, sodium, caesium, lithium and other elements. The critical temperature for these systems is of the order of micro-Kelvin.
There are two fundamental properties of quantum fluids that distinguish them from classical fluids: superfluidity and quantized circulation.
Superfluidity Superfluidity arises as a consequence of the dispersion relation of elementary excitations, and fluids that exhibit this behaviour flow without viscosity. This is a vital property for quantum turbulence as viscosity in classical fluids causes dissipation of kinetic energy into heat, damping out motion of the fluid. Landau predicted that if a superfluid flows faster than a certain critical velocity (or alternatively an object moves faster than in a static fluid) thermal excitations (rotons) are emitted as it becomes energetically favourable to generate quasiparticles, resulting in the fluid no longer exhibiting superfluid properties. For helium II, this critical velocity is .
Quantized circulation The property of quantized circulation arises as a consequence of the existence and uniqueness of a complex macroscopic wavefunction , which affects the vorticity (local rotation) in a very profound way, making it crucial for quantum turbulence.
The velocity and density of the fluid can be recovered from the wavefunction by writing it in polar form , where is the magnitude of and is the phase. The velocity of the fluid is then , and the number density is . The mass density is related to the number density by , where is the mass of one boson.
The circulation is defined to be the line integral along a simple closed path within the fluid
For a simply-connected surface , Stokes theorem holds, and the circulation vanishes, as the velocity can be expressed as the gradient of the phase. For a multiply-connected surface, the phase difference between an arbitrary initial point on the curve and the final point (same as initial point as is closed) must be , where in order for the wavefunction to be single-valued. This leads to a quantized value for the circulation
where is the quantum of circulation, and the integer is the charge (or winding number) of the vortex. Multiply charged vortices () in helium II are unstable and for this reason in most practical applications . It is energetically favourable for the fluid to form singly-charged vortices rather than a single vortex of charge , and so a multiply-charged vortex would split into singly-charged vortices. Under certain conditions, it is possible to generate certain vortices with a charge higher than 1.
Properties of vortex lines Vortex lines are topological line defects of the phase. Their nucleation makes the quantum fluid's region to become a multiply-connected region. As given by Fig 2, density depletion can be observed near the axis, with on the vortex line. The size of the vortex core varies between different quantum fluids. The size of the vortex core is around for helium II, for 3He–B and for typical atomic condensates . The simplest vortex system in a quantum fluid consists of a single straight vortex line; the velocity field of such configuration is purely azimuthal given by . This is the same formula as for a classical vortex line solution of the Euler equations, however, classically, this model is physically unrealistic as the velocity diverges as . This leads to the idea of the Rankine vortex as shown in fig 2, which combines solid body rotation for small and vortex motion for large values of , and is a more realistic model of ordinary classical vortices.
Many similarities can be drawn with vortices in classical fluids, for example the fact that vortex lines obey the classical Kelvin circulation theorem: the circulation is conserved and the vortex lines must terminate at boundaries or exist in the shape of closed loops. In the zero temperature limit, a point on a vortex line will travel accordingly to the velocity field that is generated at that point by the other parts of the vortex line, provided that the vortex line is not straight (an isolated straight vortex does not move). The velocity can also be generated by any other vortex lines in the fluid, a phenomenon also present in classical fluids. A simple example of this is a vortex ring (a torus-shaped vortex) which moves at a self-induced velocity inversely proportional to the radius of the ring , where . The whole ring moves at a velocity
Kelvin waves and vortex reconnections
Vortices in quantum fluids support Kelvin waves, which are helical perturbations of a vortex line away from its straight configuration that rotate at an angular velocity , with
Here where is the wavelength and is the wave vector.
Travelling vortices in quantum fluids can interact with each other, resulting in reconnections of vortex lines and ultimately changing the topology of the vortex configuration when they collide as suggested by Richard Feynman. At non-zero temperatures the vortex lines scatter thermal excitations, which creates a friction force with the normal fluid component (thermal cloud for atomic condensates). This phenomenon leads to the dissipation of kinetic energy. For example, vortex rings will shrink, and Kelvin waves will decrease in amplitude.
Vortex lattice
Vortex lattices are laminar (ordered) configurations of vortex lines that can be created by rotating the system. For a cylindrical vessel of radius , a condition can be derived for the formation of a vortex lattice by minimising the expression , where is the free energy, is the angular momentum of the fluid and is the rotation, with magnitude and axial direction. The critical velocity for the appearance of a vortex lattice is then
Exceeding this velocity allows for a vortex to form in the fluid. States with more vortices can be formed by increasing the rotation further, past the next critical velocities . The vortices arrange themselves into ordered configurations that are called vortex lattices.
Two fluid nature
At non-zero temperature , thermal effects must be taken into account. For atomic gases at non-zero temperatures, a fraction of the atoms are not part of the condensate, but rather form a rarefied (large free mean path) thermal cloud that co-exist with the condensate (which, in the first approximation, can be identified with the superfluid component). Since helium is a liquid, not a dilute gas like atomic condensates, there is a much stronger interaction between atoms, and the condensate is only a part of the superfluid component. Thermal excitations (consisting of phonons and rotons) form a viscous fluid component (very short free mean path, analogous to classical viscous fluid governed by the Navier–Stokes equations), called the normal fluid which coexists with the superfluid component. This forms the basis of Tisza's and Landau's two-fluid theory describing helium II as the mixture of co-penetrating superfluid and normal fluid components, with a total density dictated by the equation . The table displays the key properties of the superfluid and normal fluid components: Componentvelocitydensityentropyviscositysuperfluidzerozeroclassical fluid The relative proportions of the two components change with temperature, from an all normal fluid flow at the transition temperature ( and ), to a complete superfluid flow in the zero temperature limit ( and ). At small velocities, the two-fluid equations are
where is the pressure, is the entropy per unit mass and is the viscosity of the normal fluid component as given by the table above. The first of these equations can be identified as being the conservation of mass equation, while the second equation can be identified as the conservation of entropy. The results of these equations give rise to the phenomena of second sound and thermal counterflow. At large velocities the superfluid becomes turbulent and vortex lines appear; at even larger velocities both normal fluid and superfluid become turbulent.
Classical vs quantum turbulence
Experiments and numerical solutions show that quantum turbulence is an apparently random tangle of vortex lines inside a quantum fluid. The study of quantum turbulence aims to explore two main questions: Are vortex tangles really random, or do they contain some characteristic properties or organised structures? How does quantum turbulence compare with classical turbulence?
To understand quantum turbulence it is useful to make connection with the turbulence of classical fluids. The turbulence of classical fluids is an everyday phenomenon, which can be readily observed in the flow of a stream or river as was first done by Leonardo da Vinci in his famous sketches. When turning on a water tap, one notices that at first the water flows out in a regular fashion (called laminar flow), but if the tap is turned up to higher flow rates, the flow becomes decorated with irregular bulges, unpredictably splitting into multiple strands as it spatters out in an ever-changing torrent, known as turbulent flow. Leonardo da Vinci first observed and noted in his private notebooks that turbulent flows of classical fluids include areas of circulating fluid called vortices (or eddies).
The simplest case of classical turbulence is that of homogeneous isotropic turbulence (HIT) held in a statistical steady state. Such turbulence can be created inside of a wind tunnel, for example a channel with air flow propelled by a fan from one side to the other. It is often equipped with a mesh to create a turbulent flow of air. A statistically steady state ensures that the main properties of the flow stabilises even though they fluctuate locally. Due to presence of viscosity, without the continuous supply of energy the turbulence of the flow will decay because of frictional forces. In the wind tunnel, energy is consistently provided by the fan. It is useful to introduce the concept of energy distribution over the length scales, the wavevector , and the wavenumber . In one dimension, the wavenumber can be related to the wavelength simply using . The total energy per unit mass is given by