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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 are as follows

Quantum fluids exist at temperatures below the critical temperature $$T_c$$ at which Bose-Einstein condensation takes place.
 * Superfluid Helium (4He and Cooper pairs of 3He)
 * Atomic Bose-Einstein condensates (BECs)
 * Polariton condensates
 * Interior of Neutron Stars

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 4K. 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 $$T_c$$ for Bose-Einstein condensation of helium is 2.17K (at the saturated vapour pressure), while only approximately a few mK for 3He-B.

Although there is not as much experimental evidence for atomic condensates, experiments have been performed with rubidium, sodium, caesium and lithium. 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 flowed faster than a certian critical velocity $$v_c$$ (or alternatively an object moved faster than $$v_c$$ 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 $$v_c \approx 60 \text{m/s}$$.

Quantized circulation
The property of quantized circulation arises as a consequence of the existence and uniqueness of a complex macroscopic wavefunction $$\Psi$$, 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 $$\Psi(\mathbf{x},t)$$ by writing it in polar form $$\Psi(\mathbf{x},t) = |\Psi(\mathbf{x},t)|e^{i\phi(\mathbf{x},t)}$$, where $$|\Psi|$$ is the magnitude of $$\Psi$$ and $$\phi$$ is the phase. The velocity of the fluid is then $$\mathbf{v}(\mathbf{x},t) = (\hbar/m)\nabla \phi$$, and the number density is $$n(\mathbf{x},t) = |\Psi|^2$$. The mass density is related to the number density by $$\rho(\mathbf{x},t) = mn$$, where $$m$$ is the mass of one boson.

The circulation $$\Gamma$$ is defined to be the line integral along a simple closed path $$C$$ within the fluid

$$\Gamma = \oint_C \mathbf{v} \cdot \mathbf{dr}$$

For a simply-connected surface $$S$$, Stokes theorem holds, and the circulation vanishes, as the velocity can be expressed as the gradient of the phase. For a multiply-connected surface, by the uniqueness property of the wavefunction, the phase difference between an arbitrary initial point on the curve $$C$$ and the final point (same as initial point as $$C$$ is closed) must be $$2\pi q$$, where $$q=0,1,2,\cdots$$ in order for the wavefunction to be single-valued. This leads to a quantized value for the circulation

$$\Gamma = \frac{\hbar}{m} \oint_C \nabla S \cdot \mathbf{dr} = q\kappa$$

where $$\kappa = h/m $$ is the quantum of circulation, and the integer $$q$$ is the charge (or winding number) of the vortex. Multiply charged vortices ($$q>1$$) in helium II are unstable and for this reason in most practical applications $$q=1$$. It is energetically favourable for the fluid to form $$q$$ singly-charged vortices rather than a single vortex of charge $$q$$, and so a multiply-charged vortex would split into singly-charged vortices. Under certain conditions, it is possible to generate certain vortices with a charger higher than 1.

Properties of vortex lines
Vortex lines are topological line defects of phase space, the presence which makes the quantum fluid into a multiply-connected region. As given by Fig 2, density depletion can be observed near the axis, with $$\rho = 0 $$ on the vortex line. The size of the vortex core varies between different quantum fluids. The size of the vortex core is around $$a_0 = 10^{-10}\text{m} $$ for helium II, $$a_0 = 10^{-8}\text{m} $$ for 3He-B and for typical atomic condensates $$a_0 = 10^{-4}\text{m} $$. 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 $$v_{\theta} = \kappa/{2\pi r}$$. This is the same formula as for a classical vortex line solution of the Euler equation, however classicaly this model is physically unrealistic as the velocity diverges as $$r \rightarrow 0$$. This leads to the development of the Rankine vortex as shown in fig 2, which combines solid body rotation for small $$r$$ and vortex motion for large values of $$r$$.

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 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. 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 $$v_R$$ inversely proportional to the radius of the ring $$R$$, where $$R >> a_0$$. The whole ring moves at a velocity

$$v_R = \frac{\kappa}{4\pi R}\left[\ln{\left(\frac{8R}{a_0}\right)} - \frac{1}{2}\right]$$

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 $$\omega$$, with

$$\omega \approx \frac{\kappa k^2}{4 \pi}\ln{\left(\frac{1}{ka_0}\right)}$$

Here $$k=2\pi/\lambda$$ where $$\lambda$$ is the wavelength and $$k$$ is the wavevector.

Travelling vortices in quantum fluids can interact with each other, resulting in reconnections of vortex lines and ultimately exchanging the topology when they collide as suggested by Richard Feynmann. For 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 introduces a dissipation of energy, and structures such as vortex rings will shrink, and Kelvin waves will in turn decrease in amplitude.

Vortex lattice
Vortex lattices are laminar (ordered) configurations of vortex lines of vortex lines that can be created by rotating the system. For a cylindrical vessel of radius $$R$$, a condition can be derived for the formation of a vortex lattice by minimising the expression $$F' = F - \mathbf{L}_v\cdot\mathbf{\Omega}$$, where $$F$$ is the free energy, $$\mathbf{L}_v$$ is the angular momentum of the fluid and $$\mathbf{\Omega}$$ is of magnitude $$\Omega$$ in the azimuthal direction. The lower critical velocity is then $$\Omega^1_c = \frac{\kappa}{R^2}\ln{\left(\frac{R}{a_0}\right)}$$.

Exceeding this velocity allows for a vortex to form in the fluid. States with more vortices can be formed by increasing the rotation velocity further, past the next critical velocities $$\Omega^2_c,\,\Omega^3_c, \cdots$$. The vortices arrange themselves into ordered configurations that are called vortex lattices.

Two fluid nature
For a non-zero temperature $$T$$, 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 surround the condensate (which can be identified to the superfluid component). For liquid helium II and 3He-B, there is a much stronger interaction between atoms, with the condensate only being 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 equation), 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 having a density dictated by the equation $$\rho = \rho_n + \rho_s $$. The table displays some of the properties of the components of a quantum fluid such as helium II. The proportions of the two components change continuously from an all normal fluid flow at the transition temperature $$T_c $$ ($$\rho_n/\rho \rightarrow 1 $$ and $$\rho_s/\rho \rightarrow 0 $$), to a complete superfluid flow in the zero temperature limit ($$\rho_s/\rho \rightarrow 1 $$ and $$\rho_n/\rho \rightarrow 0 $$). In full generality, the two-fluid equations can be written in the following way

$$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho_s \mathbf{v}_s + \rho_n \mathbf{v}_n) = 0$$

$$ \frac{\partial(\rho S)}{\partial t} + \nabla \cdot (\rho S \mathbf{v}_n) = 0$$

$$ \rho_s\left[ \frac{\partial \mathbf{v}_s}{\partial t} + (\mathbf{v}_s \cdot \nabla)\mathbf{v}_s\right] = -\frac{\rho_s}{\rho}\nabla P + \rho_s S \nabla T$$

$$ \rho_n\left[ \frac{\partial \mathbf{v}_n}{\partial t} + (\mathbf{v}_n \cdot \nabla)\mathbf{v}_n\right] = -\frac{\rho_n}{\rho}\nabla P - \rho_s S \nabla T + \mu \nabla^2 \mathbf{v}_n$$

where here $$P $$ is the pressure, $$S $$ is the entropy per unit mass and $$\mu $$ 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.

Classical vs quantum turbulence
From physical experiments and numerical solutions, quantum turbulence manifests itself as an apparently random tangle of vortex lines inside a quantum fluid. The study of quantum turbulence aims to explore two main questions:

To understand the idea of quantum turbulence it is useful to make a 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 Vinc i 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 comprise of areas of circulating fluid called vortices (or eddies).
 * 1) Are vortex tangles really random, or do they contain some characterstic properties that would obey physical laws and contain some organisation within the fluid?
 * 2) How does quantum turbulence compare with classical turbulence?

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, a channel with air flow propelled by a fan from one side to the other. It is often equipped with a meshgrid to create a turbulent flow of air. A statistically steady stateensures that on average the velocities of particles will average out over time and length scales, even though they will fluctuate locally. Due to presence of viscosity, without the continous supply of energy the turbulence of the flow is expected to decay because of frictional forces. In the wind tunnel, energy is consistantly provided by the fan. It is useful to introduce the energy distribution over the length scales, the wavevector $$\mathbf{k} = (k_x,k_y,k_z)$$, and the wavenumber $$k = |\mathbf{k}|$$. In one dimension, the wavenumber can be related to the wavelength simply using $$k = 2\pi/\lambda$$. The total energy $$E_{tot}$$ per unit mass is given by

$$E_{tot} = \frac{1}{\mathcal{V}}\int_{\mathcal{V}}\frac{\mathbf{v}^2}{2}\, \mathrm{d}^3\mathbf{r} = \int_{0}^{\infty}E(k)\, \mathrm{d}k$$

where $$E(k)$$ is the energy spectrum. The notion of an energy cascade, where an energy transfer takes place from large scale vortices to smaller scale vortices, which eventually lead to viscous dissipation was memorably noted by Lewis Fry Richardson. Dissipation occurs at the dissipation length scales $$\eta$$ (termed the Kolmogorov length scale), where $$\eta = (\nu^3/\varepsilon)^{1/4}$$ where $$\nu$$ is the kinematic viscosity. By the pioneering work of Andrey Kolmogorov, the energy spectrum was found to take the form

$$E(k) = C \varepsilon^{\frac{2}{3}}k^{-\frac{5}{3}}$$

where $$\varepsilon$$ is the energy dissipation rate per unit volume $$-\mathrm{d}E/\mathrm{d}t$$. The constant $$C$$ is a dimensionless constant, that takes the value $$C \approx 1.5$$. In k-space the value associated to the Kolmogorov length scale is the Kolmogorov wavenumber $$k_{\eta}$$, where viscous dissipation occurs.

Kolmogorov cascade in quantum fluids
For temperatures low enough for quantum mechanical effects to govern the fluid, quantum turbulence is a seemingly chaotic tangle of vortex lines with a highly knotted topology, which move each other and reconnect when they collide. In a pure superfluid, there is no normal component to carry the entropy of the system and therefore the fluid flows without viscosity, resulting in the lack of a dissipation scale $$\eta$$. Analogously to classical fluids, a quantum length scale $$\ell$$ (and the corresponding value in k-space $$k_{\ell}$$) can be introduced by replacing the the kinematic viscosity in the Kolmogorov length scale with the quantum of circulation $$\kappa$$. For scales larger than $$\ell$$, a small polarisation of the vortex lines allows the stretching required to sustain a Kolmogorov energy cascade.

Experiments have been performed in superfluid Helium II to create turbulence, that behave according to the Kolmogorov cascade. One such example of this is the case of two counter-rotating propellers, where both above and below the critical temperature $$T_c$$ a Kolmogorov energy spectrum was observed that is indistinguishable from those observed in the turbulence of classical fluids. For higher temperatures, the existence of the normal fluid component leads to the presence of viscous forces and eventual heat dissipation which warms the system. As a consequence of this friction the vortices become smoother, and the Kelvin waves that arise due to vortex reconnections are dampened. Kolmogorov turbulence arises in quantum fluids for energy input at large length scales, where the energy spectrum follows $$k^{-5/3}$$ in the inertial range $$k_D < k < k_{\ell}$$. For length scales smaller than $$\ell$$, instead the energy spectrum follows a $$k^{-1}$$ regime.

For temperatures in the zero limit, the perturbations of Kelvin waves result in more kinks appearing in the topology of the tangle as they are not dampened during vortex reconnections. For large length scales the quantum turbulence manifests as a Kolmogorov energy cascade (numerical simulations using the Gross-Pitaevskii equation and the vortex-filament model confirmed this effect ), with the energy spectrum following $$k^{-5/3}$$. Lacking thermal dissipation, it is intuitive to assume that quantum turbulence in the low temperaturem limit does not decay as it would for higher temperatures, however experimental evidence showed that this was not the case. A tangle with more kinks has shorter vortex lines which result in faster rotations. Upon exceeding the critical velocity, an emission of phonons occurs, which results in the conversion of kinetic energy into sound and thus the dissipation of energy. At such small length scales, a Kelvin wave cascade arises which proceeds on individual vortices. The energy spectrum of this turbulence follows a Kolmogorov regime for the inertial range $$k_D < k < k_{\ell}$$, and forms a bottle-neck plateau before continuing as a Kelvin wave cascade that obeys the $$k^{-5/3}$$ law. Numerical computations have suggested that the Kelvin wave cascade exists, however there is currently no experimental evidence due to the diffculty of observing and measuring at such small length scales.

Vinen turbulence
Vinen turbulence can be generated in a quantum fluid by the injection of vortex rings into the system, which has been observed both numerically and experimentally. It can also be generated in numerical simulations of turbulent Helium II driven by a small heat flux and of trapped atomic Bose-Einstein condensates, and has been found in superfluid models of the early universe. Unlike the Kolmogorov regime which appears to have a classical counterpart, no analogous regime can be observed in classical turbulence.

Vinen turbulence occurs for very low energy inputs into the system, which prevents the formation of large scale structures that is prevelant in Kolmgorov turbulence, as is shown in Fig 9a. Large scales structures contribute strongly to the amount of non-local interaction, which can be seen in the figure. In stark contrast, Fig 9b displays the vinen turbulence regime, where there is very little non-local interaction and the turbulent flow appears much kinkier than that of the Kolmogorov regime. The energy spectrum of this turbulence peaks at the intermediate scales around $$k_{\ell}$$, rather than at large length scales $$k_{D}$$. From Fig 10, it can be seen that for small length scales the turbulence follows a $$k^{-1}$$ behaviour of an isolated vortex. As a result of these properties vinen turbulence appears as an almost completely random flow with a very weak or negligible energy cascade.

Decay of quantum turbulence
Stemming from the different signatures, Kolmogorov and vinen turbulence both follow power laws relating to their temporal decay. For the Kolmogorov regime, after removing the forcing which sustains the turbulence in a statistical steady-state a decay of $$E(t) \sim t^{-2}$$ for the energy and $$L(t) \sim t^{-3/2}$$ for the vortex line density, which is defined as the vortex length per unit volume. Vinen turbulence decays temporally at a slower rate than Kolmogorov turbulence, the energy decays at a rate $$E(t) \sim t^{-1}$$ and the vortex line density at a rate $$L(t) \sim t^{-1}$$.

Turbulence in atomic condensates
Computer simulations have played a particularly important role in the development of a theoretical understanding of quantum turbulence; they have allowed theoretical results to be checked, and simulations of vortex dynamics to be developed.

Turbulence in atomic condensates has only been studied very recently meaning that there is less information available. A much lower quantity of vortices is produced in condensate systems due to the small system size. As a consequence of this there is not a large length scale seperation between the system size and the inter-vortex size, and therefore a restricted k-space. Numerical simulations suggest that turbulence is more likely to appear to be undertake the vinen regime. Experiments performed in Cambridge have found the emergence of wave turbulence scaling appearing.

Physical generation of quantum turbulence
There are a plethora of methods that can be used to generate a vortex tangle physically in a lab (visualised in fig 11). Here they are listed by the quantum fluid that they can be generated in.

QT in helium II

 * Suddenly towing a meshgrid in the sample of fluid at rest
 * Moving the fluid along pipes or channels using bellows or pumps, creating a superfluid wind tunnel (the TOUPIE experiment in Grenoble )
 * Rotating one or two propellers inside a container ; the configuration of two counter-rotating propellers is called the "von Karman flow" (e.g the SHREK experiment in Grenoble)
 * Creating shockwaves and cavitation by locally focusing ultrasound (this allows for the generation of quantum turbulence away from the boundaries)
 * Oscillating/vibrating forks or wires
 * Applying a heat flux (also termed the "thermal counterflow" ): the prototype experiment is a channel which is open to a helium bath at one end and the opposite is closed and has a resistor. A electrical current is passed through the that is generating ohmic heat; the heat is carried away from the heater towards the bath by the normal fluid component, while the superfluid moves towards the heater so that the net mass flux is zero as the channel is closed. A relative velocity $$v_{ns} = |v_n - v_s|$$ (counterflow) of the two fluid components is set up in this way which is proportional to the applied heat.
 * Injecting vortex rings (rings are generated by injecting electrons which form a small bubble of about 16 Angstroms in size that are accelerated by an electric field, until, upon exceeding the critical velocity, the vortex ring is nucleated)

QT in 3He-B and atomic condensates
In 3He-B, quantum turbulence can be generated by the vibration of wires. For atomic condensates, quantum turbulence can be generated by shaking or oscillating the trap which confines the BEC and by phase imprinting the quantum vortices.

Detection of quantum turbulence
In classical turbulence, one would usually want to measure the velocity, either at a fixed position against time (which is often typical of physical experiments) or at the same time at many positions (often typical of numerical simulations). Quantum turbulence is characterised by the apparant random nature of quantum vortices, and therefore the natural attention is focused directly on vortex lines. In helium II techniques such as the second sound attenuation determine the vortex line density $$L$$. The average distance between vortex lines $$\ell$$ can be found in terms of the vortex line density $$\ell = 1/\sqrt{L}$$.

Detection in helium II

 * Measuring the attenuation of second sound waves
 * Measuring temperature or pressure gradients
 * Measuring ions trapped in the vortices
 * Using tracer particles (small glass or plastic spheres/solid hydrogen snowballs) of size order of a micron, and then imaging them using lasers. Techniques that can be used are PIV (particle image velocimetry) or PTV (particle tracking velocimetry). Most recently, excimer helium molecules have been used
 * Using oscillating forks
 * Using cantilevers
 * Using cryogenic hot wires

Detection in 3He-B and atomic condensates
Quantum turbulence can be detected in 3He-B in two ways: nuclear magnetic resonance (NMR) and by Andreev scattering of thermal quasiparticles. For atomic condensates, it is typical that the condensate must be expanded (by switching off the trapping potential) so that is sufficiently large for an image to be taken. This procedure has a disadvantage as it leads to the condensate being destroyed. The outcome leads to a 2-dimensional image which allows for the study of 2-dimensional quantum turbulence, but imposes a constraint studying 3-dimensional quantum turbulence using this method. Individual quantum vortices have been observed in 3-dimensions, moving and reconnecting using a technique which extracts small fractions of the condensate at a time, allowing for the observation of a time sequence of the same vortex configuration.