Joe Vinen

1
Symposium on Universal Features of Turbulence
Warwick, December 2005
An Introduction to Quantum Turbulence
Joe Vinen
School of Physics and Astronomy University of
Birmingham
General introductory review Vinen Niemela J
Low Temp Physics 128, 167 (2002)
2
The aims of this presentation

• Quantum turbulence is the name we give to
  turbulence in a superfluid, in which fluid
  motion is strongly influenced by quantum effects.

• It is an old subject, first discussed about 50
  years ago. But recently interest in it has grown
  strongly, and much of this symposium is concerned
  with it.

• My aims are

• to describe a little of the relevant history

• to provide a background and introduction to many
  of the papers on quantum turbulence that will be
  presented at this symposium.

• to emphasize links with problems in classical
  turbulence

3
History I

• Two parallel and independent developments in the
  1950s

• Feynmans suggestion that superfluids rotate
  through the presence of quantized vortex lines,
  and that these lines could allow a form of
  turbulence in a superfluid.

• Experiments (Hall and Vinen) showing that the
  Gorter-Mellink mutual friction accompanying
  thermal counterflow in superfluid 4He was due to
  turbulence (rotational motion) in the superfluid
  component.

• These experiments were followed by the
  successful search for mutual friction in
  uniformly rotating 4He.

• These developments merged when Hall and I
  realized that vortex lines can give rise to
  mutual friction, due to scattering of the
  thermal excitations that constitute the normal
  fluid by vortex lines. This led directly to a
  quantitative theory of mutual friction in the
  uniformly rotating superfluid a comparison of
  this theory with experiment provided the first
  evidence in favour of the existence of quantized
  vortex lines.

4
History II

• Between 1955 and 1995 attention focussed on

• The theory of thermal counterflow turbulence.

• Problems relating to the nucleation of quantized
  vortex lines.

• The structure and properties of vortices in
  superfluid 3He.

• Thermal counterflow turbulence has no classical
  analogue. Strangely, there were no serious
  experiments on types of superfluid turbulence
  that do have classical analogues until the
  mid-1990s. These experiments opened up a whole
  new range of interesting questions, which have
  been responsible for much of the recent
  renaissance of interest in quantum turbulence.

• At the same time experiments started to appear
  on turbulence in superfluid 3He-B, which again
  raised new and interesting questions. And of
  course experiments started on Bose-condensed cold
  atoms, stimulating still further interest.

• I want to focus especially on these various new
  questions, which will be taken up in more detail
  by other speakers.

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Simple superfluids I

• Simple superfluids (4He 3He-B cold atoms)
  exhibit

• Two fluid behaviour a viscous normal component
  an inviscid superfluid component. Normal
  component disappears at lowest temps.

• Quantization of rotational motion in the
  superfluid component.

(Consequencies of Bose or BCS condensation.)

• Quantization of rotational motion
  , except on quantized vortex lines, each with
  one quantum of circulation

round a core of radius equal to the coherence
length ?
(? 0.05 nm for 4He 80nm for 3He-B larger
for Bose gases).

• Viscosity of normal fluid 4He very small
  3He-B very large. Turbulence in normal
  fluid? 4He YES 3He-B NO.

6
Calculating vortex motion

• In absence of normal fluid an element of vortex
  generally moves with the local superfluid
  velocity, often calculated with the vortex
  filament model using either the full non-local
  Biot-Savart law

or in appropriate cases the local induction
approximation (LIA)

• In the presence of normal fluid we must add the
  force of mutual friction

( a transverse component)
This force causes the vortex to move relative to
the local superfluid velocity in accord with to
the classical Magnus effect.
A useful dimensionless parameter is
.
If (4He) motion of vortex is only
weakly perturbed by mutual friction..
If (3He-B at high temperatures)
motion is strongly perturbed.
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The Non-Linear Schrodinger Equation and vortex
reconnections

• Our description of vortex motion has been
  essentially classical. It will fail on length
  scales where quantum effects become important,
  notably on scales of order ? . The only
  available quantum theory is that leading to the
  NLSE

• describes the static and dynamic behaviour of
  the condensate,

• but applies quantitatively only to the
  weakly-interacting Bose gas.

• In the context of quantum turbulence the most
  important effect not described by the classical
  theory is the vortex reconnection, first
  described in terms of the NLSE by Koplik Levine.

• Reconnections can be included in the vortex
  filament model, but

• inclusion is artificial

• real reconnections are dissipative, which can be
  important (Barenghi, Adams et al)

• Therefore NLSE is often used in connections with
  quantum turbulence, in spite of its shortcomings
  for real helium.

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Thermal counterflow turbulence

• 4He above 1K.

• Experiments indicated homogeneous turbulence in
  the superfluid component, maintained by the
  relative motion of the two fluids. No classical
  analogue.

• A understanding was provided by the pioneering
  simulations of Schwarz, based on the vortex
  filament model and the LIA. He showed that
  self-sustaining tangles of lines could arise from
  the mutual friction, provided that one allows
  for reconnections (artificially introduced).

• Schwarz provided us with a quantitative theory,
  but some problems remain

• Is the normal fluid turbulent (Melotte
  Barenghi)?

• Artificial introduction of reconnections
  (correct criteria?).

• Is the LIA adequate (high values of ?)?

9
Vortex nucleation

• An energy barrier opposes the creation of
  vortices, except at the highest velocities.

• Of course such a barrier is crucial to the
  existence of superfluidity!

• But at typical velocities and temperatures at
  which turbulence is observed the barrier is often
  too large to be overcome thermally or by
  tunnelling, especially in 4He .

• Therefore in these cases the nucleation must be
  extrinsic i.e. dependent on remanent vortices.

• Until recently, study of extrinsic nucleation
  was hampered by ignorance of, and lack of
  control over, the configuration of the nucleating
  vortex. However, recent experiments in Helsinki
  have shown that with 3He-B there can be better
  control, and this work will be described in
  later papers.

• Both nucleation and subsequent propagation of
  turbulence have been studied in 3He-B in a
  rotating vessel (analogous to classical spin-up
  experiments).

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Intrinsic vortex nucleation

• Intrinsic nucleation can be observed

• with some care in 3He-B

• with much care in 4He (small ? makes it
  difficult to remove remanent vortices)

• An early demonstration with 4He involved vortex
  nucleation by a small sphere in the form of a
  negative-ion bubble (McClintock et al)

• an energy barrier of about 3 K was observed at a
  bubble velocity 40 ms-1

Also seen in later work on flow through small
apertures by Varoquaux et al

• at higher temperatures there was thermal
  activation

• at lower temperatures there was quantum
  tunnelling

• The 3K barrier was correctly predicted by a
  modified vortex filament model (Muirhead, Vinen
  Donnelly).

• Modelling of this type of nucleation with the
  NLSE (Roberts Berloff Frisch, Pomeau Rica
  Huepe Brachet Winiecki, McCann Adams), can
  be very instructive, but the NLSE may not provide
  a good enough model for real helium (is there an
  energy barrier?).

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Quasi-classical quantum turbulence I

• It is strange that for many years the only form
  of quantum turbulence to be studied seriously was
  that produced by thermal counterflow in 4He,
  which does not have a classical analogue.

• An obvious question

• What happens if you replace the classical liquid
  in a typical example of classical turbulence by a
  superfluid?

• For example in flow through a grid, which
  classically produces the much-studied case of
  homogeneous isotropic turbulence.

• Do we get analogues of Richardson cascades
  Kolmogorov energy spectra etc.?

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Quasi-classical quantum turbulence II

• Even now there are only two detailed
  experiments, both on 4He above 1K

• Observation of the spectrum pressure
  fluctuations in turbulence produced by
  counter-rotating discs (Maurer Tabeling).

• Observation of the decay of vortex-line density
  in the wake of a steadily moving grid (Stalp,
  Skrbek Donnelly).

• The pressure fluctuations are observed with a
  pressure transducer with size 0.5mm. They show
  that on scales ? 0.5 mm there is a Kolmogorov
  spectrum,

indistinguishable from that above the superfluid
transition

• The moving grid experiments are more difficult
  to interpret, but are consistent with

• a similar Kolmogorov spectrum on scales gtgt mean
  vortex spacing ?

• dissipation, on a scale ?, given by the
  quasi-classical expression

Lvortex line density
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Why quasi-classical behaviour? (Vinen 2000)

• Start by thinking about the probable outcome of
  a grid-flow experiment at a very low temperature
  (no normal fluid).

• There are no detailed experiments at these
  temperatures, although it is known that
  turbulence can be created by a grid and does
  decay.

• On small length scales (lt?) the turbulence must
  be very different from any classical type.

• But on large scales (gtgt?, containing many
  vortices) the vortex lines can be arranged, with
  local polarization, to mimic classical turbulent
  flow, including, probably, the time-evolution of
  this flow.

• So we can argue that on scales gtgt ? there could
  be a Richardson cascade and Kolmogorov energy
  spectrum. This is provided that, as seems to be
  the case, there is dissipation on a small scale.
  We return to the origin of this dissipation
  later.

• All this could apply equally to 4He and 3He-B.

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Why quasi-classical behaviour? II

• Now raise the temperature, to produce some
  normal fluid.

• We must now distinguish between 4He and 3He-B.

• In 4He the normal fluid has a very small
  viscosity. Therefore it too becomes turbulent in
  the wake of the grid, with a Richardson cascade
  and Kolmogorov energy spectrum. Thus the flow in
  each fluid is likely to display Kolmogorov
  spectra. But the two fluids are coupled by mutual
  friction. The two velocity fields become locked
  together, and we get a single velocity field
  with a single Kolmogorov spectrum, as observed.

• In 3He-B the normal fluid is too viscous to
  become turbulent. Therefore its effect is the
  damp the turbulence in the superfluid, through
  the effect of mutual friction. The result can be
  predicted (Vinen 2005 Lvov et al 2005) it
  turns out that

• a small mutual friction (? ltlt 1) damps only the
  largest quasi-classical eddies

• a large mutual friction (? ? 1) will kill the
  turbulence in the superfluid.

(?-1 acts as a kind of Reynolds number)
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Experimental and computational evidence?

• Evidence, already noted, that quasi-classical
  behaviour can be seen in 4He at high
  temperatures.

• BUT, no detailed experimental evidence yet for
  quasi-classical behaviour at very low
  temperatures.

• There is evidence from the spin-up experiments
  that 3He-B does behave at high temperatures in
  the way suggested (importance of the parameter
  ?), but no experiments yet on homogeneous
  turbulence in 3He-B .

• Computational evidence for behaviour at T 0.
  Eg Kobayashi Tsubota, based on NLSE.

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Dissipation in quantum turbulence

• When there is normal fluid this is easy

• there is viscous dissipation in the normal fluid

• there is dissipation in the superfluid due to
  mutual friction. In 4He this occurs only on
  length scales ? ?, where the two velocity fields
  cannot match, but this is sufficient to provide
  high-k dissipation required for the Kolmogorov
  spectrum. Indeed it is possible to predict the
  effective kinematic viscosity at
  temperatures above 1K.

17
Dissipation in quantum turbulence at very low
temperatures

• No normal fluid no viscous dissipation no
  mutual friction. What other mechanisms can there
  be?

• Vortex motion can radiate sound. But typical
  frequencies associated with this motion on a
  scale ? are too small to produce significant
  radiation.

• We need energy flow to smaller length scales.
  Look at a simulation the evolution of a tangle
  of vortex lines at very low temperatures.

• The kinks involve smaller length scales and are
  produced by large numbers of reconnections.

Tsubota et al
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Dissipation associated with reconnections I

• Two sources of dissipation

• phonon emission during reconnections.

• phonon emission from high-frequency Kelvin waves
  produced by reconnections.

• It turns out that phonon emission during
  reconnections is likely to be very important in
  cases where the vortex spacing ? is not much more
  than the vortex core size ?. This is the case in
  Bose gases modelled by the NLSE simulations by
  Nore Brachet and by Kobayashi Tsubota.

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Dissipation associated with reconnections II

• But in helium (especially 4He) dissipation
  during reconnections is relatively unimportant
  owing to the small size of the vortex core.

• In that case we note that repeated reconnections
  lead to the continual generation of Kelvin waves
  on each length of vortex (cf plucking of a
  string).

• Some of these Kelvin waves have a very high
  frequency and can generate phonons very
  efficiently

• Others have a lower frequency, but non-linear
  interactions can lead to transfer of energy (in a
  cascade?) to the required high frequencies
  (Svistunov Vinen Vinen, Tsubota Mitani
  Kozik Svistunov numerical work and weak
  turbulence theory).

• This transfer process involves a form of wave
  turbulence (again a link with classical fluid
  mechanics), which will be discussed rather fully
  in later papers, along with the form of energy
  spectra associated with this process and
  questions about direct and inverse cascades.

20
The overall picture?

• So perhaps we have the following picture of the
  evolution of turbulence in superfluid 4He at a
  very low temperature. Energy flows to smaller
  and smaller length scales

• First in a classical Richardson cascade

• Followed by a Kelvin-wave cascade

• With final dissipation by radiation of phonons

• The length scale ( vortex spacing) at which we
  change from Richardson to Kelvin-wave cascades
  adjusts itself automatically to achieve the
  correct dissipation.

• 3He-B may be similar except that energy can be
  lost from the Kelvin waves into quasi-particle
  bound states in the cores of the vortices
  (Caroli-Matricon states), which do not exist in
  4He. This occurs at a frequency much smaller
  than that required for phonon radiation.

phonons
21
Oscillating wires and grids at very low
temperatures

• Recent experiments on both 4He and 3He-B
  (Lancaster Osaka).

• Turbulence produced is inhomogeneous.

• No systematic classical results with which to
  compare.

• Too early to draw conclusions?

22
Comments and conclusions

• We have focussed on cases of homogeneous
  turbulence, because it seems best to try to
  understand these cases first.

• Much of our discussion has been speculative,
  although it has thrown up many interesting
  theoretical questions. We have also ignored
  potentially interesting details, such as
  deviations from Kolmogorov scaling and the
  existence of analogues of coherent structures in
  classical turbulence.

• There is still a serious shortage of
  experimental data, especially at very low
  temperatures, and the data we do have are based
  on techniques that do not provide the kind of
  detailed information (about eg velocity fields)
  available to those studying classical fluid
  mechanics. Simulations provide some kind of
  experimental data. But are they reliable and
  can they extend over the wide ranges of length
  scale that seem to be important in quantum
  turbulence?

• Major problems and challenges face us in the
  development of new techniques relating to very
  low temperatures and to the acquisition of more
  sophisticated data. Papers by Carlo Barenghi,
  Gary Ihas, and the Lancaster 3He Group will
  address some of these questions.

• Finally I have emphasized relationships between
  quantum turbulence and classical turbulence
  (including wave turbulence). Other links will be
  emphasized later in the symposium.

23
Acknowledgements

• Many friends, colleagues and organizations.

Tsunehiko Araki Carlo Barenghi, Demetris
Charalambous Russell Donnelly, Marie
Farge, Shaun Fisher. Andrei Golov Henry
Hall Demos Kivotides, Matti Krusius, Akira
Mitani Peter McClintock, Joe Niemela, Alastair
Rae, David Samuels, Ladik Skrbek, Edouard
Sonin, Steve Stalp, Boris Svistunov, Makoto
Tsubota, Grisha Volovik.

• Cryogenic Turbulence Laboratory, University of
  Oregon (NSF Grant D MR-9529609)
• Newton Institute for Mathematical Sciences,
  Cambridge
• The Royal Society.

• Grant support from EPSRC

24
Thank you
25
Comments and conclusions

• We have focussed on cases of homogeneous
  turbulence, because it seems best to try to
  understand these cases first.

• Much of our discussion has been speculative,
  although it has thrown up many interesting
  theoretical questions. We have also ignored
  potentially interesting details, such as
  deviations from Kolmogorov scaling and the
  existence of analogues of coherent structures in
  classical turbulence.

• There is still a serious shortage of
  experimental data, especially at very low
  temperatures, and the data we do have are based
  on techniques that do not provide the kind of
  detailed information (about eg velocity fields)
  available to those studying classical fluid
  mechanics. Simulations provide some kind of
  experimental data. But are they reliable and
  can they extend over the wide ranges of length
  scale that seem to be important in quantum
  turbulence?

• Major problems and challenges face us in the
  development of new techniques relating to very
  low temperatures and to the acquisition of more
  sophisticated data. Papers by Carlo Barenghi,
  Gary Ihas, and the Lancaster 3He Group will
  address some of these questions.

• Finally I have emphasized relationships between
  quantum turbulence and classical turbulence
  (including wave turbulence). Other links will be
  emphasized later in the symposium.