Editing Quantum turbulence (section)

=== 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.