Editing Quantum turbulence (section)

== Classical vs quantum turbulence ==
[[File:Kolmogorov.png|thumb|377x377px|Fig 8. Schematic diagram of the Kolmogorov energy cascade inside of a wind tunnel. The injection of air occurs at <math>k_D = 2\pi/D</math> where <math>D</math> is the size of the wind tunnel. The quantity <math>k_{\eta}</math>, the Kolmogorov wavenumber, is the value in k-space associated to the [[Kolmogorov microscales|Kolmogorov length scale]], the point at which the turbulent kinetic energy is dissipated into heat.]]
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 [[Isotropy|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 scale]]s, the wavevector <math>\mathbf{k} = (k_x,k_y,k_z)</math>, and the wavenumber <math>k = |\mathbf{k}|</math>. In one dimension, the wavenumber can be related to the wavelength simply using <math>k = 2\pi/\lambda</math>. The total energy <math>E_{tot}</math> per unit mass is given by

<math>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</math>

where <math>E(k)</math> is the ''energy spectrum'', essentially representing the distribution of turbulent kinetic energy over the wavenumbers. 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 <math>\eta</math> (termed the Kolmogorov length scale), where <math>\eta = (\nu^3/\varepsilon)^{1/4}</math> where <math>\nu</math> is the kinematic viscosity. By the pioneering work of [[Andrey Kolmogorov]], the energy spectrum was found to take the form

<math>E(k) = C \varepsilon^{\frac{2}{3}}k^{-\frac{5}{3}}</math>

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

=== Kolmogorov cascade in quantum fluids ===
[[File:Vortex-tangle.png|thumb|374x374px|Fig 9a. Numerically simulated vortex tangle representing Kolmogorov quantum turbulence. The thin lines represent vortex lines inside of a cubic container. The colorbar<ref name=":3">{{Cite journal|last1=Galantucci|first1=L.|last2=Barenghi|first2=C. F.|last3=Parker|first3=N. G.|last4=Baggaley|first4=A. W.|date=2021-04-06|title=Mesoscale helicity distinguishes Vinen from Kolmogorov turbulence in helium-II|url=https://link.aps.org/doi/10.1103/PhysRevB.103.144503|journal=Physical Review B|volume=103|issue=14|pages=144503|doi=10.1103/PhysRevB.103.144503|arxiv=1805.09005|bibcode=2021PhRvB.103n4503G |s2cid=234355425}}</ref><ref name=":4">{{Cite journal|last1=Sherwin-Robson|first1=L. K.|last2=Barenghi|first2=C. F.|last3=Baggaley|first3=A. W.|date=2015-03-23|title=Local and nonlocal dynamics in superfluid turbulence|url=https://link.aps.org/doi/10.1103/PhysRevB.91.104517|journal=Physical Review B|volume=91|issue=10|pages=104517|doi=10.1103/PhysRevB.91.104517|arxiv=1409.1443|bibcode=2015PhRvB..91j4517S |s2cid=118650626}}</ref> represents the amount of non-local interaction, i.e. the amount by how much a section of the vortex line is affected by the other vortex lines surrounding it. (Credit AW Baggaley)]]
[[File:Kolmogorov-in-qf.png|thumb|373x373px|Fig 9. Schematic diagram of the energy spectrum for Kolmogorov turbulence at very small temperatures. The <math>k^{-5/3}</math> energy cascade is present for large length scales, and a Kelvin wave cascade can be observed for very small length scales which undergoes sound emission. A bottleneck pile up occurs around the quantum length scale <math>\ell</math>.<ref>{{Cite journal|last=Krstulovic|first=G.|date=2012-11-09|title=Kelvin-wave cascade and dissipation in low-temperature superfluid vortices|url=https://link.aps.org/doi/10.1103/PhysRevE.86.055301|journal=Physical Review E|volume=86|issue=5|pages=055301|doi=10.1103/PhysRevE.86.055301|pmid=23214835|arxiv=1209.3210|bibcode=2012PhRvE..86e5301K |s2cid=31414715}}</ref>]]
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 <math>\eta</math>. Analogously to classical fluids, a quantum length scale <math>\ell</math> (and the corresponding value in k-space <math>k_{\ell}</math>) can be introduced by replacing the kinematic viscosity in the Kolmogorov length scale with the quantum of circulation <math>\kappa</math>.<ref name=":2" /> For scales larger than <math>\ell</math>, 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,<ref>{{Cite journal|last1=Maurer|first1=J.|last2=Tabeling|first2=P.|date=1998-07-01|title=Local investigation of superfluid turbulence|url=https://iopscience.iop.org/article/10.1209/epl/i1998-00314-9/meta|journal=EPL (Europhysics Letters)|language=en|volume=43|issue=1|pages=29|doi=10.1209/epl/i1998-00314-9|bibcode=1998EL.....43...29M |s2cid=250831521 |issn=0295-5075}}</ref> where both above and below the critical temperature <math>T_c</math> 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 smoother than in low-temperature quantum turbulence. Kolmogorov turbulence arises in quantum fluids for energy input at large length scales, where the energy spectrum follows <math>k^{-5/3}</math> in the inertial range <math>k_D < k < k_{\ell}</math>. For length scales smaller than <math>\ell</math>, instead the energy spectrum follows a <math>k^{-1}</math> regime.<ref name=":5">{{Cite journal|last1=Baggaley|first1=A. W.|last2=Laurie|first2=J.|last3=Barenghi|first3=C. F.|date=2012-11-14|title=Vortex-Density Fluctuations, Energy Spectra, and Vortical Regions in Superfluid Turbulence|url=https://link.aps.org/doi/10.1103/PhysRevLett.109.205304|journal=Physical Review Letters|volume=109|issue=20|pages=205304|doi=10.1103/PhysRevLett.109.205304|pmid=23215501|arxiv=1207.7296 |bibcode=2012PhRvL.109t5304B }}</ref>

For temperatures in the zero limit, the undamped Kelvin waves result in more kinks appearing in the shapes of the vortices. For large length scales the quantum turbulence manifests as a Kolmogorov energy cascade (numerical simulations using the [[Gross–Pitaevskii equation|Gross-Pitaevskii equation]]<ref>{{Cite journal|last1=Nore|first1=C.|last2=Abid|first2=M.|last3=Brachet|first3=M. E.|date=1997-05-19|title=Kolmogorov Turbulence in Low-Temperature Superflows|url=https://link.aps.org/doi/10.1103/PhysRevLett.78.3896|journal=Physical Review Letters|volume=78|issue=20|pages=3896–3899|doi=10.1103/PhysRevLett.78.3896|bibcode=1997PhRvL..78.3896N }}</ref> and the vortex-filament model confirmed this effect <ref name=":6">{{Cite journal|last1=Tsubota|first1=M.|last2=Araki|first2=T.|last3=Nemirovskii|first3=S. K.|date=2000-11-01|title=Dynamics of vortex tangle without mutual friction in superfluid ${}^{4}\mathrm{He}$|url=https://link.aps.org/doi/10.1103/PhysRevB.62.11751|journal=Physical Review B|volume=62|issue=17|pages=11751–11762|doi=10.1103/PhysRevB.62.11751|arxiv=cond-mat/0005280|s2cid=118937769}}</ref><ref name=":7">{{Cite journal|last1=Araki|first1=T.|last2=Tsubota|first2=M.|last3=Nemirovskii|first3=S. K.|date=2002-09-16|title=Energy Spectrum of Superfluid Turbulence with No Normal-Fluid Component|url=https://link.aps.org/doi/10.1103/PhysRevLett.89.145301|journal=Physical Review Letters|volume=89|issue=14|pages=145301|doi=10.1103/PhysRevLett.89.145301|pmid=12366052|arxiv=cond-mat/0201405|bibcode=2002PhRvL..89n5301A |s2cid=39668537}}</ref>), with the energy spectrum following <math>k^{-5/3}</math>. Lacking thermal dissipation, it is intuitive to assume that quantum turbulence in the low temperature limit does not decay as it would for higher temperatures, however experimental evidence showed that this was not the case: quantum turbulence decays even at very low temperatures. The Kelvin waves interact and create shorter Kelvin waves, until they are short enough that emission of sound (phonons), which results in the conversion of kinetic energy into heat, thus dissipation of energy.  This process which shifts energy to smaller and smaller length scales at wavenumbers larger than <math>k_{\ell}</math>  is called the  Kelvin wave cascade and proceeds on individual vortices.<ref name=":8">{{Cite journal|last1=Kivotides|first1=D.|last2=Vassilicos|first2=J. C.|last3=Samuels|first3=D. C.|last4=Barenghi|first4=C. F.|date=2001-04-02|title=Kelvin Waves Cascade in Superfluid Turbulence|url=https://link.aps.org/doi/10.1103/PhysRevLett.86.3080|journal=Physical Review Letters|volume=86|issue=14|pages=3080–3083|doi=10.1103/PhysRevLett.86.3080|pmid=11290112|bibcode=2001PhRvL..86.3080K }}</ref><ref>{{Cite journal|last1=di Leoni|first1=P. C.|last2=Mininni|first2=P. D.|last3=Brachet|first3=M. E.|date=2017-05-26|title=Dual cascade and dissipation mechanisms in helical quantum turbulence|url=https://link.aps.org/doi/10.1103/PhysRevA.95.053636|journal=Physical Review A|volume=95|issue=5|pages=053636|doi=10.1103/PhysRevA.95.053636|arxiv=1705.03525 |bibcode=2017PhRvA..95e3636C |hdl=11336/52186|s2cid=119217270|hdl-access=free}}</ref> Low temperature quantum turbulence should thus consist of a double cascade: a Kolmogorov regime (a cascade of eddies) in the inertial range <math>k_D < k < k_{\ell}</math>, followed by a bottle-neck plateau, followed by the Kelvin wave cascade (a cascade of waves) that obeys the same <math>k^{-5/3}</math> law but with different physical origin. This is at current consensus, but it must be stressed that it arises from theory and numerical simulations only:  there is currently no direct experimental evidence for the Kelvin wave cascade due to the difficulty of observing and measuring at such small length scales.

=== Vinen turbulence ===
[[File:Vortex-tangle-vinen.png|alt=Quantum turbulence-vinen regime|thumb|373x373px|Fig 9b. Numerically simulated vortex tangle representing vinen quantum turbulence. The thin lines represent vortex lines inside of a cubic container. The colorbar<ref name=":3" /><ref name=":4" /> represents the amount of non-local interaction, i.e. the amount by how much a section of the vortex line is affected by the other vortex lines surrounding it. (Credit AW Baggaley)]]
[[File:Vinen-in-qf.png|thumb|373x373px|Fig 10. Schematic diagram of the energy spectrum for vinen turbulence. A <math>k^{-1}</math> regime can be observed for very large wavenumbers, with the peak of the energy spectrum occurring at the wavenumber <math>k_{\ell}</math> associated to the quantum length scale <math>\ell</math>. The green line represents a <math>k^{-5/3}</math> regime for comparison.]]
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 has been observed also in numerical simulations of turbulent Helium II driven by a small heat flux and in numerical simulations of trapped atomic Bose-Einstein condensates; it has been found even in numerical studies of superfluid models of the early universe.<ref name=":3" /> Unlike the Kolmogorov regime which appears to have a classical counterpart, Vinen turbulence has not been identified in classical turbulence..

Vinen turbulence occurs for very low energy inputs into the system, which prevents the formation of the large scale partially polarised structures that are prevalent in Kolmogorov turbulence, as is shown in Fig 9a. The partial polarization contributes strongly to the amount of non-local interactions between the vortex lines, 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. The energy spectrum of Vinen turbulence peaks at the intermediate scales around <math>k_{\ell}</math>, rather than at large length scales <math>k_{D}</math>. From Fig 10, it can be seen that for small length scales the turbulence follows the typical <math>k^{-1}</math> 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 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 <math>E(t) \sim t^{-2}</math> for the energy and <math>L(t) \sim t^{-3/2}</math> for the vortex line density (defined as the vortex length per unit volume) are observed. Vinen turbulence decays temporally at a slower rate than Kolmogorov turbulence: the energy decays as <math>E(t) \sim t^{-1}</math> and the vortex line density as <math>L(t) \sim t^{-1}</math>.

=== Turbulence in atomic condensates ===
Computer simulations have played a particularly important role in the development of the theoretical understanding of quantum turbulence<ref>{{cite journal|author=Schwartz|first=K.W.|year=1983|title=Critical Velocity for a Self-Sustaining Vortex Tangle in Superfluid Helium|journal=Physical Review Letters|volume=50|issue=5|page=364|bibcode=1983PhRvL..50..364S|doi=10.1103/PhysRevLett.50.364}}</ref><ref>{{cite journal|last1=Aarts|first1=R.G.K.M.|last2=de Waele|first2=A.T.A.M.|name-list-style=amp|year=1994|title=Numerical investigation of the flow properties of He II|journal=Physical Review B|volume=50|issue=14|pages=10069–10079|bibcode=1994PhRvB..5010069A|doi=10.1103/PhysRevB.50.10069|pmid=9975090}}</ref><ref>{{cite journal|last1=de Waele|first1=A.T.A.M.|last2=Aarts|first2=R.G.K.M.|name-list-style=amp|year=1994|title=Route to vortex reconnection|journal=Physical Review Letters|volume=72|issue=4|pages=482–485|bibcode=1994PhRvL..72..482D|doi=10.1103/PhysRevLett.72.482|pmid=10056444}}</ref>

Turbulence in atomic condensates has only been studied very recently meaning that there is less information available.<ref>{{Cite journal|last1=White|first1=A. C.|last2=Anderson|first2=B. P.|last3=Bagnato|first3=V. S.|date=2014-03-25|title=Vortices and turbulence in trapped atomic condensates|journal=Proceedings of the National Academy of Sciences|language=en|volume=111|issue=Supplement_1|pages=4719–4726|doi=10.1073/pnas.1312737110|issn=0027-8424|pmc=3970853|pmid=24704880|doi-access=free}}</ref><ref>{{Cite journal|last1=Tsatsos|first1=M. C.|last2=Tavares|first2=P. E.S.|last3=Cidrim|first3=A.|last4=Fritsch|first4=A. R.|last5=Caracanhas|first5=M. A.|last6=dos Santos|first6=F. E. A.|last7=Barenghi|first7=C. F.|last8=Bagnato|first8=V. S.|date=March 2016|title=Quantum turbulence in trapped atomic Bose–Einstein condensates|url=https://linkinghub.elsevier.com/retrieve/pii/S037015731600065X|journal=Physics Reports|language=en|volume=622|pages=1–52|doi=10.1016/j.physrep.2016.02.003|arxiv=1512.05262 |bibcode=2016PhR...622....1T |s2cid=55570454}}</ref> Turbulent atomic condensates contain a much smaller number of vortices compared to turbulence in helium. Because of the small size of typical atomic condensates, there is not a large length scale separation between the system size and the inter-vortex size, and therefore k-space is restricted. Numerical simulations suggest that turbulence is more likely to appear in the Vinen regime.<ref name=":5" /><ref name=":6" /><ref name=":8" /><ref>{{Cite journal|last1=Cidrim|first1=A.|last2=White|first2=A. C.|last3=Allen|first3=A. J.|last4=Bagnato|first4=V. S.|last5=Barenghi|first5=C. F.|date=2017-08-21|title=Vinen turbulence via the decay of multicharged vortices in trapped atomic Bose-Einstein condensates|url=https://link.aps.org/doi/10.1103/PhysRevA.96.023617|journal=Physical Review A|volume=96|issue=2|pages=023617|doi=10.1103/PhysRevA.96.023617|arxiv=1704.06759|bibcode=2017PhRvA..96b3617C |s2cid=119079470}}</ref> Experiments performed in Cambridge have also found the emergence of wave turbulence scaling appearing.<ref name="Navon 72–75">{{Cite journal|last1=Navon|first1=N.|last2=Gaunt|first2=A. L.|last3=Smith|first3=R. P.|last4=Hadzibabic|first4=Z.|date=November 2016|title=Emergence of a turbulent cascade in a quantum gas|url=http://www.nature.com/articles/nature20114|journal=Nature|language=en|volume=539|issue=7627|pages=72–75|doi=10.1038/nature20114|pmid=27808196|arxiv=1609.01271 |bibcode=2016Natur.539...72N |s2cid=4449347|issn=0028-0836}}</ref><ref name=":9">{{Cite journal|last1=Henn|first1=E. A. L.|last2=Seman|first2=J. A.|last3=Roati|first3=G.|last4=Magalhães|first4=K. M. F.|last5=Bagnato|first5=V. S.|date=2009-07-20|title=Emergence of Turbulence in an Oscillating Bose-Einstein Condensate|url=https://link.aps.org/doi/10.1103/PhysRevLett.103.045301|journal=Physical Review Letters|volume=103|issue=4|pages=045301|doi=10.1103/PhysRevLett.103.045301|pmid=19659367|arxiv=0904.2564 |bibcode=2009PhRvL.103d5301H |s2cid=25507605}}</ref><ref name=":7" />