Editing Quantum turbulence (section)

== General properties of superfluids ==
[[File:Circulation_in_connected_regions_-_schematic.png|alt=Schematic of circulation in connected regions|thumb|384x384px|Fig 1. The schematic diagram of a fluid (blue) in a cylindrical container. Left: The curve <math>C</math> traces out a closed path in a simply connected region. The path can be shrunk down to the point <math>P</math>, and therefore Stokes theorem can be applied. For a quantum fluid, this indicates that the circulation vanishes. Right: The curve <math>C</math> traces out a closed path in a multiply-connected region (i.e. with holes). The path cannot be shrunk down due to the hole, and therefore Stokes theorem does not hold, leading to a non-zero, quantized circulation. For a quantum fluid, this suggests that vortex structures act like 'holes'.]]
[[File:Rankine_vortex_and_number_density_of_quantum_vortices.png|thumb|381x381px|Fig 2. Left: Simple schematic of a straight vortex line in 3-dimensional space, with positive circulation. Middle: Azimuthal velocity against the radius. (i) shows the fluid speed of a solid-body rotation. (ii) shows the fluid speed of a vortex in both classical and quantum fluids. (iii) a combination of (i) and (ii) to form a Rankine vortex model for a tornado with core of size <math>a_0</math>. Right: Number density against radius of a quantum fluid with vortex <math>a_0</math>. Density depletion can be observed for a small radius <math>r < a_0</math>. The quantity <math>n_s</math> represents the density of the fluid sufficiently far away from the vortex core <math>r > a_0</math>.]]
[[File:Vortex_ring_motion.png|thumb|379x379px|Fig 3. Left: Schematic of a vortex ring of radius <math>R</math> moving at a speed <math>v_R</math>. Middle: 3-dimensional schematic of a quantum vortex ring. The velocity of the ring is generated by the ring itself, which propels itself at a velocity that is inversely proportional to the radius of the ring. The thickness of the ring is greatly exaggerated for the purpose of being able to view the torus-like shape. In reality, for helium II the thickness is approximately <math>10^{-10}\text{m} </math> . Right: The velocity profile of the vortex ring against its size. An inverse relationship can be viewed. This suggests that smaller rings move at a much faster speed, while larger rings move at a much slower speed.]]
The turbulence of quantum fluids has been studied primarily in two quantum fluids: [[Liquid helium|liquid Helium]] and atomic condensates. Experimental observations have been made in the two stable [[isotope]]s of Helium, the common <sup>4</sup>He and the rare <sup>3</sup>He. The latter isotope has two phases, named the A-phase and the B-phase. The A-phase is strongly [[Anisotropy|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 <math>T_c</math> for Bose-Einstein condensation of helium is 2.17K (at the [[Vapor pressure|saturated vapour pressure]]), while only approximately a few mK for <sup>3</sup>He-B.<ref>{{Cite book|last1=Barenghi|first1=C. F.|url=http://link.springer.com/10.1007/978-3-319-42476-7|title=A Primer on Quantum Fluids|last2=Parker|first2=N. G.|date=2016|publisher=Springer International Publishing|isbn=978-3-319-42474-3|series=SpringerBriefs in Physics|location=Cham|doi=10.1007/978-3-319-42476-7|arxiv=1605.09580|bibcode=2016pqf..book.....B |s2cid=118543203}}</ref>

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. [[Lev Landau|Landau]] predicted that if a superfluid flows faster than a certain critical velocity <math>v_c</math> (or alternatively an object moves faster than <math>v_c</math> 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 <math>v_c \approx 60 \text{m/s}</math>.

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

The [[Circulation (physics)|circulation]] <math>\Gamma</math> is defined to be the line integral along a simple closed path <math>C</math> within the fluid

<math>\Gamma = \oint_C \mathbf{v} \cdot \mathbf{dr}</math>

For a [[Simply connected space|simply-connected]] surface <math>S</math>, [[Generalized Stokes theorem|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 <math>C</math> and the final point (same as initial point as <math>C</math> is closed) must be <math>2\pi q</math>, where <math>q=0,1,2,\cdots</math> in order for the wavefunction to be single-valued. This leads to a quantized value for the circulation

<math>\Gamma = \frac{\hbar}{m} \oint_C \nabla S \cdot \mathbf{dr} = q\kappa</math>

where <math>\kappa = h/m </math> is the ''quantum of circulation'', and the integer <math>q</math> is the charge (or winding number) of the vortex. Multiply charged vortices (<math>q>1</math>) in helium II are unstable and for this reason in most practical applications <math>q=1</math>. It is energetically favourable for the fluid to form <math>q</math> singly-charged vortices rather than a single vortex of charge <math>q</math>, 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 <math>\rho = 0 </math> on the vortex line. The size of the vortex core varies between different quantum fluids. The size of the vortex core is around <math>a_0 = 10^{-10}\text{m} </math> for helium II, <math>a_0 = 10^{-8}\text{m} </math> for <sup>3</sup>He-B and for typical atomic condensates <math>a_0 = 10^{-4}\text{m} </math>. 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 <math>v_{\theta} = \kappa/{2\pi r}</math>. This is the same formula as for a classical vortex line solution of the Euler equation, however, classically, this model is physically unrealistic as the velocity diverges as <math>r \rightarrow 0</math>. This leads to the idea of the [[Rankine vortex]] as shown in fig 2, which combines solid body rotation for small <math>r</math> and vortex motion for large values of <math>r</math>, 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's circulation theorem|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 <math>v_R</math> inversely proportional to the radius of the ring <math>R</math>, where <math>R >> a_0</math>.<ref>{{Cite journal|last1=Barenghi|first1=C. F.|last2=Donnelly|first2=R J|date=October 2009|title=Vortex rings in classical and quantum systems|url=https://iopscience.iop.org/article/10.1088/0169-5983/41/5/051401|journal=Fluid Dynamics Research|volume=41|issue=5|pages=051401|doi=10.1088/0169-5983/41/5/051401|s2cid=123246632 |issn=0169-5983}}</ref> The whole ring moves at a velocity

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

=== Kelvin waves and vortex reconnections ===
[[File:Kelvin_waves_in_quantum_vortices.png|thumb|377x377px|Fig 4. Left: Schematic of a Kelvin wave with amplitude <math>A</math> and wavelength <math>\lambda</math>. Right: A straight vortex configuration that has been perturbed into a bent vortex configuration.]]
[[File:Vortex-reconnection-schematic.png|thumb|377x377px|Fig 5. Schematic of vortex reconnection of two vortices. The arrows on the vortices represent the direction of the vorticity in the vortex line. Left: Before the reconnection. Middle: The vortex reconnection is taking place. Right: after the reconnection.]]
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 <math>\omega</math>, with

<math>\omega \approx \frac{\kappa k^2}{4 \pi}\ln{\left(\frac{1}{ka_0}\right)}</math>

Here <math>k=2\pi/\lambda</math> where <math>\lambda</math> is the wavelength and <math>k</math> is the wavevector.

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.<ref>{{cite book|author=R.P. Feynman|title=II. Progress in Low Temperature Physics|publisher=North-Holland Publishing Company|year=1955|volume=1|place=Amsterdam|chapter=Application of quantum mechanics to liquid helium}}</ref> 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 ===
[[File:Vortex_lattice_schematic.png|thumb|377x377px|Fig 6. Schematic of a cylindrical container rotating at a speed of <math>\Omega</math>, forming a vortex lattice of six straight vortex lines.]]
Vortex lattices are laminar (ordered) configurations of vortex lines that can be created by rotating the system. For a cylindrical vessel of radius <math>R</math>, a condition can be derived for the formation of a vortex lattice by minimising the expression <math>F' = F - \mathbf{L}_v\cdot\mathbf{\Omega}</math>, where <math>F</math> is the free energy, <math>\mathbf{L}_v</math> is the angular momentum of the fluid and <math>\mathbf{\Omega}</math> is the rotation, with magnitude <math>\Omega</math> and axial direction. The critical velocity for the appearance of a vortex lattice is then <math>\Omega^1_c = \frac{\kappa}{R^2}\ln{\left(\frac{R}{a_0}\right)}</math>.

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 <math>\Omega^2_c,\,\Omega^3_c, \cdots</math>. The vortices arrange themselves into ordered configurations that are called vortex lattices.

=== Two fluid nature ===
[[File:Helium-II_component_densities.png|thumb|375x375px|Fig 7. Component fractions plotted against the temperature, displaying the mixture of normal fluid and superfluid in helium II, where <math>\rho_s/\rho </math> is the superfluid fraction, and <math>\rho_n/\rho </math> is the normal fluid fraction. For temperatures above the critical temperature, the normal fluid makes up the whole fluid.]]
At non-zero temperature <math>T</math>, 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 [[phonon]]s and rotons) form a viscous fluid component (very short free mean path, analogous to classical viscous fluid governed by the [[Navier–Stokes equations|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 the mixture of co-penetrating superfluid and normal fluid components, with a total density dictated by the equation <math>\rho = \rho_n + \rho_s </math>. The table displays the key properties of the superfluid and normal fluid components:.
{| class="wikitable"
!Component
!velocity
!density
![[entropy]]
!viscosity
|-
|superfluid
|<math>\mathbf{v}_s</math>
|<math>\rho_s</math>
|zero
|zero
|-
|classical fluid
|<math>\mathbf{v}_n</math>
|<math>\rho_n</math>
|<math>S</math>
|<math>\mu</math>
|}
The relative proportions of the two components change with temperature, from an all normal fluid flow at the transition temperature <math>T_c </math> (<math>\rho_n/\rho \rightarrow 1 </math> and <math>\rho_s/\rho \rightarrow 0 </math>), to a complete superfluid flow in the zero temperature limit (<math>\rho_s/\rho \rightarrow 1 </math> and <math>\rho_n/\rho \rightarrow 0 </math>). At small velocities, the two-fluid equations are

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

<math>  \frac{\partial(\rho S)}{\partial t} + \nabla \cdot (\rho S \mathbf{v}_n) = 0</math>

<math> \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</math>

<math> \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</math>

where here <math>P </math> is the pressure, <math>S </math> is the entropy per unit mass and <math>\mu </math> 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.