This document summarizes research on quantum turbulence in superfluids like helium-4. Key points include: - Turbulence involves a tangle of quantized vortex filaments. Dissipation occurs through reconnections and kelvin wave cascades. - Numerical simulations show fluctuations in vortex line density follow a f^-5/3 scaling, matching experiments. - Velocity statistics are non-Gaussian at small scales due to the quantum nature of vortices, but become Gaussian at larger scales. - The decay of quantum turbulence can follow either a quasiclassical t^-3/2 or ultraquantum t^-1 scaling depending on conditions.

1. Quantum Turbulence
Andrew W. Baggaley Carlo F. Barenghi Yuri Sergeev
28th November, 2011
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2. Outline
Introduction
Quantum Fluids
Quantised vortices
Vortex reconnections
Kelvin wave cascade
Fluctuations in vortex density
A suprising result?
Numerical simulations
Velocity statistics
Classical picture
Deviation from Gaussianity
A U-turn?
Decay of quantum turbulence
Experimental results
Ultra-quantum and quasiclassical regimes
An explanation?
Current/Future Work
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3. Classical picture
Turbulence in a normal (classical) fluid,
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4. Classical vortices
Velocity v,
vorticity ω = × v.
In a classical fluid,
vortices are
unconstrained,
they can be big or
small weak or strong.
In quantum fluids · · ·
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5. Superfluidity
Below a critical temperature ( 2.2K) 4 He becomes superfluid,
a component of the fluid can flow without viscosity.
What is the nature of turbulence in the superfluid (quantum
turbulence)?
How is energy distributed between scales?
Here we focus on superfluid 4 He, QT also possible in 3 He and
Bose-Einstein condensates. 5 / 57

6. Quantised vortices
Gross-Pitaevskii Equation (weakly
interacting gas)
∂Ψ 2
2
i =− Ψ + gΨ |Ψ |2 − µΨ.
∂t 2m
√
Macroscopic wave function Ψ = neiφ
Velocity v = ( /m) φ (irrotational)
Vorticity constrained to filaments of fixed
circulation Γ
v · dr = Γ
C
Vortex core size dictated by atom size,
hence thin! (∼ 10−8 cm)
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7. Quantum Turbulence
Quantum turbulence (QT) is a tangle of these quantised
vortices.
Hence QT is more simple than classical turbulence as vortices
are well defined elements.
But turbulence requires an energy sink, what is the dissipation
mechanism?
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8. Quantum vortex reconnections
If vortices become close (core size) they can reconnect.
Directly observed (Paoletti, Lathrop & Sreenivasan, [2008]).
Numerically modelled used GPE (Koplik and Levine, PRL,
[1993]),
No violation of Kelvin’s circulation theorem (inside vortex
core, ρ → 0)
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9. Kelvin waves
Quantised vortices
reconnect, creating
pronounced cusps.
Also the interaction of
vortices can lead to
perturbations along the
vortices,
these propagate as Kelvin
waves.
Nonlinear (3 wave)
interactions lead to the
creation of smaller scales. Simulations taken from Kivotedes et al. [2003]
At high k energy dissipated
as phonon (sound) emission.
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10. 10 / 57

11. Energy spectrum
At scale larger than intervortex spacing - classical
(Kolmogorov) regime. Experimentally (Maurer and Tabeling
[1998]) and numerically (Nore et al. [1997], Araki
et al. [2002]) veryfied.
At small (< ) scales quantum regime - Kelvin wave cascade.
At crossover scale arguments for, L’vov et al. [2007] (above),
and against (Kozik & Svistunov [2008]) bottleneck .
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12. Vortex filament model
Helium experiments: average distance between lines ( ≈ 10−1 to
10−4 cm) is much bigger than core-size 10−8 cm ∴ Model vortex
lines as reconnecting space curves s(ξ, t)
Biot-Savart law:
ds Γ (s − r) × dr
=−
dt 4π |s − r|3
LIA:
ds
≈ βs × s
dt
N = number of discretization points
Biot-Savart is slow: CPU ∼ N 2
Tree algorithm is faster: CPU ∼ N log N - (AWB & Barenghi,
JLTP [2012]) 12 / 57

13. Tree approximation
Biot-Savart law scales as N 2 .
Use tree methods (Barnes and Hut [1983]) used in
astrophysical simulations - N log N scaling.
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14. Outline
Introduction
Quantum Fluids
Quantised vortices
Vortex reconnections
Kelvin wave cascade
Fluctuations in vortex density
A suprising result?
Numerical simulations
Velocity statistics
Classical picture
Deviation from Gaussianity
A U-turn?
Decay of quantum turbulence
Experimental results
Ultra-quantum and quasiclassical regimes
An explanation?
Current/Future Work
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15. Grenoble experiment
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16. A suprising result?
Intensity of quantum turbulence is characterized by the vortex
line density L (vortex length per unit volume).
Roche et al. [2007] measured the fluctuations of L in
turbulent 4 He.
They observed that the frequency spectrum scales as f −5/3 ,
Interpret L as a measure of the rms superfluid vorticity
(ωs = Γ L).
Contradiction with the classical scaling of vorticity expected
from the Kolmogorov energy spectrum.
Can numerical simulations shed light on the problem?
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17. Finite temperature effects
0 < T < Tλ - two fluid system,
normal (viscous) fluid and superfluid Helium.
mutual friction (scattering of quasi-particles) means we must
modify equations of motion.
At high T , ρn > ρs : no back reaction from superfluid on
normal fluid,
ds
= vs + αs × (vn − vs ) − α s × s × (vn − vs ) ,
dt
Γ (s − r)
vs = − × dr.
4π L |s − r|3
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18. Modelling the normal fluid
We wish to avoid DNS of turbulent normal fluid,
we use Kinematic Simulations (KS) model, Fung et al. [1992]
M
vn (s, t) = (Am × km cos φm + Bm × km sin φm ) ,
m=1
φm = km · s + ωm t, where km and ωm = 3
km E(km ) are
wavevectors and frequencies.
Easily enforce energy spectrum of vn to reduce to
−5/3
E(km ) ∝ km ,
simple to impose periodic boundary conditions.
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19. 4/3
k1
|vn |, Re = ≈ 180
kM
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20. Numerical results
Frequency spectrum
Vortex line density L = Λ/D3
Energy Spectrum
Quantised vortex dynamo:
rapid initial growth in L,
saturation due to increasing
dissipation (reconnections).
(AWB & Barenghi PRB [2011])
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21. An explanation
Roche et al. argued randomly oriented vortex lines have some
of the statistical properties of passive scalars.
These randomly oriented vortices contribute to vortex line
density,
and so second sound attenuation (temperature waves), which
is experimental method to detect quantised vortices.
We test with by modelling passive lines, ds/dt = vKS .
Must continue to reconnect lines or line length (density) will
never saturate.
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22. Frequency spectrum of passive line
Line density grows and saturates as before but at larger values.
Understandable, consider mutual friction term:
αs × (vn − vs ).
Normal fluid cannot contribute to stretching of vortices.
Frequency spectrum of passive line seems to agree with f −5/3 ,
Result also similar if ds/dt = αs × (vKS − vs )
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23. Outline
Introduction
Quantum Fluids
Quantised vortices
Vortex reconnections
Kelvin wave cascade
Fluctuations in vortex density
A suprising result?
Numerical simulations
Velocity statistics
Classical picture
Deviation from Gaussianity
A U-turn?
Decay of quantum turbulence
Experimental results
Ultra-quantum and quasiclassical regimes
An explanation?
Current/Future Work
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24. Velocity statistics
Ordinary viscous flows
Homogeneous isotropic turbulence
Turbulent velocity v(r, t)
PDFs of components of v are
Gaussian
Flow past a grid
Wind speed
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25. Classical velocity statistics
Experiment: Theory:
Noullez & al Vincent & Meneguzzi
(JFM 1997) (JFM 1991)
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26. Classical velocity statistics
Velocity v at point r is determined by vorticity ω = × v via
Biot-Savart law:
1 ω(r , t) × (r − r )
v(r, t) = dr ,
4π |r − r |3
If r is surrounded by many randomly oriented eddies,
Gaussianity results from Central Limit Theorem
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27. Maryland experiment
Measurement of velocity PDF in superfluid 4 He
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28. Maryland experiment
Turbulent superfluid 4 He,
solid hydrogen tracers radius
≈ 10−4 cm.
Paoletti et al found
−3
PDF(vx ) ∼ vx unlike
ordinary turbulence.
They say vortex
reconnections are
responsible for tails.
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29. Velocity statistics in quantum turbulence
Calculation of velocity PDF in various vortex configurations:
Always power-laws
(even without vortex reconnections)
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30. Velocity statistics in quantum turbulence
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31. Velocity statistics in quantum turbulence
Vortex filament method
Kolmogorov turbulence in 4 He (AWB & Barenghi, PRB [2011])
Heat flow turbulence in 4 He (Adachi & Tsubota, PRB [2011]) 31 / 57

32. Grenoble experiment
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33. Grenoble experiment
10−3
p(v)
10−4
10−5
−4 −2 0 2 4
v− v
σ
Superfluid wind tunnel
Kolmogorov energy
spectrum
Gaussian velocity statistics
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34. Why power-law PDFs rather than classical Gaussian ?
Paoletti, Lathrop & Sreenivasan:
vortex reconnections are responsible for power-law.
Additional interpretation based on Min & Leonard (1996):
velocity statistics of singular and non-singular vortices are
qualitatively different.
For N singular vortices:
−3
N = 1: straight vortex: P DF (vj ) = vj
N > 1: provided that the velocity contribution of each vortex
can be considered an independent random variable, the PDF
converges to Gaussian but extremely slowly, e.g. N ∼ 106
vortices has still significant tails (Weiss, & al, PoF [1998])
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35. From power law to Gaussian
Solution of the Maryland-Grenoble puzzle:
= average vortex spacing
∆ = size of region over which the velocity is averaged
∆=2 ∆= ∆ = /6
• Grenoble: nozzle a = 0.06 cm a >> ≈ 0.5 to 2 × 10−3 cm
• Maryland: tracer a = 10−4 cm tracer a << ≈ 10−3 to 10−2 cm
(AWB & Barenghi, PRE [2011])
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36. Conclusion
= average intervortex spacing
At scales >> quantum turbulence is similar to ordinary
turbulence (same Kolmogorov energy spectrum, same
Gaussian velocity statistics)
At scales << the quantum nature of vortices causes
differences (Kelvin energy spectrum, non-Gaussian velocity
statistics)
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37. Outline
Introduction
Quantum Fluids
Quantised vortices
Vortex reconnections
Kelvin wave cascade
Fluctuations in vortex density
A suprising result?
Numerical simulations
Velocity statistics
Classical picture
Deviation from Gaussianity
A U-turn?
Decay of quantum turbulence
Experimental results
Ultra-quantum and quasiclassical regimes
An explanation?
Current/Future Work
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38. Classical uniform and isotropic turbulence
∞
1 v2
E= dV = E(k) dk
V 2
V 0
Kolmogorov spectrum:
E(k) = C 2/3 k−5/3
Decay of classical turbulence
Energy (per unit mass) is concentrated at scale D with velocity v:
E ∼ v 2 /2
Characteristic timescale (lifetime of largest eddies): τ = D/v
Dissipation rate:
dE E v3 E 3/2
=− ∼ ∼ ∼
dt τ D D
Decay:
D D
E∼ 2 ∼ 3
t t
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39. Decay of vortex line density in T → 0 limit
Two regimes have been observed:
1) Quasiclassical (Kolmogorov), L ∼ t−3/2
spindown (Walmsley, Golov, et al., PRL [2008])
injected ions which form rings (Walmsley & Golov, PRL [2008])
oscillating grid in 3 He-B (Bradley et al., PRL [2006] and Nature Phys.
[2008])
2) Ultraquantum (Vinen), L ∼ t−1
injected ions (Walmsley & Golov PRL [2008])
oscillating grid in 3 He-B (Bradley et al., PRL [2006] and Nature Phys.
[2008])
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40. Decay regimes
Assumptions: Energy is distributed over scales (hence
t−3/2 law:
[Quasiclassical k−5/3 spectrum) ≈ ν ω 2 (in analogy with classical
(“Kolmogorov”) turbulence) ω ≈ κL (this implies existence of coherent
turbulence] structures!)
=⇒ L ∼ t−3/2
The only length-scale is the intervortex distance, = L−1/2
=⇒ the only velocity scale v = κ/(2π )
t−1 law: dE v3
[Ultraquantum =⇒ =− ∼ = ν (κL)2
dt
(“Vinen”)
turbulence] dL ν κ2 2
If E = cL, then ∼− L
dt c
=⇒ L ∼ t−1
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41. Ion injection experiment (Walmsley & Golov, PRL 2008)
Negative ions (electron bubbles) generate vortex rings (Winiecki &
Adams, EPL [2000])
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42. Ion injection experiment – decay regimes
Ultraquantum, L ∼ t−1 (relatively Quasiclassical, L ∼ t−3/2 (prolonged
short injection time) Energy contained ion injection) Kolmogorov spectrum at
at small scales, k 1/ wavenumbers k 1/
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43. Numerical calculation
Biot-Savart calculation (Tree), periodic box D = 0.03 cm.
Beam of vortex rings, R = 6 × 10−4 cm, confined within π/10
angle
Vortex rings interact, reconnect, and form a tangle,
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44. Vortex line density, L vs time
Small ion injection rate: Large ion injection rate:
Ultraquantum decay Quasiclassical decay
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45. Local curvature of the vortex lines
Probability distribution function
of the local curvature, C (cm−1 )
solid line – ultraquantum
dashed line – quasiclassical
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46. Time-dependent spectra: formation and decay
Quasiclassical Ultraquantum
solid: t = 0.1 s solid: t = 0.07 s
dot-dashed: t = 1.1 s dashed: t = 0.12 s
dashed: t = 3.4 s dot-dashed: t = 0.6 s
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47. Generation of large scale motion
Energy input at k = k∗ (2π/k∗ ≈ 2πR, where R is the ring’s
radius)
∞
E= Ek dk = EL + ES
0
k∗ ∞
EL = Ek dk – transferred to large scales EL = Ek dk –
0 k∗
transferred to small scales
Quasiclassical Ultraquantum
ES ES
1 (< 0.13 at all times) 1 (at least > 1)
EL EL
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48. General scenario: ultraquantum decay
Energy is concentrated at small scales
(k > k∗ ), “Kelvulence”
Kelvin wave cascade, phonon emission at
a very small scale (k = kc ) =⇒ decay
E ∼ t−1 and L ∼ t−1
consistent with the rate of phonon
emission
dEtot 2
∼ Etot
dt
(Vinen & Niemela, JLTP [2002])
Precise form of the KW spectrum is not
important
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49. General scenario: quasiclassical decay
Full time-dependent spectrum, consisting
of the Kolmogorov and ultraquantum
parts.
Most of the energy is contained at large
scales,
decay is determined by the quasiclassical
(“Kolmogorov”) part of the spectrum
(from the viewpoint of the Kolmogorov
part, sink = KW + phonon emission)
dE dL
∼ t−2 , ∼ t−3/2
dt dt
The precise form of the KW spectrum as
well as the possible bottleneck are not
important for quasiclassical decay.
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50. Mechanism of formation of the large scale motion
Isotropic initial conditions
Anisotropic initial ‘beam’
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51. Mechanism of formation of the large scale motion
PDF(R) of loop sizes, R (cm)
Reconnection of vortex rings
Bottom: head-on reconnection
Dashed: initial PDF Top: reconnection of rings travelling
Resulting PDFs: in the same direction
Grey: isotropic injection of rings =⇒ formation of large loops
Black: anisotropic beam
Anisotropy of the beam is important!
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52. Outline
Introduction
Quantum Fluids
Quantised vortices
Vortex reconnections
Kelvin wave cascade
Fluctuations in vortex density
A suprising result?
Numerical simulations
Velocity statistics
Classical picture
Deviation from Gaussianity
A U-turn?
Decay of quantum turbulence
Experimental results
Ultra-quantum and quasiclassical regimes
An explanation?
Current/Future Work
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53. Bundling of filaments =⇒ Coherent structures.
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54. Bundling of filaments =⇒ Coherent structures.
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55. Relationship between vortex length and energy
Energy is constant throughout the run =⇒ Energy per unit length
is not constant.
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56. Open Questions
Bottleneck at crossover length scale?
Kelvin wave cascade in both strong and weak regimes.
Topology of structures in QT, define a knot ‘spectrum’ ?
Tracer particles for flow visualisation.
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57. Thank you for listening
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