Transcript Document

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)
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
History I
 Two parallel and independent developments in the 1950s:
• Feynman’s 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.
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.
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: curl vs  0 , except on
quantized vortex lines, each with one quantum of
circulation
   vs  dr  h m4 or h 2m3
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.
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
v s r0  
 r  r0   dr

3
4
r  r0
or in appropriate cases the local induction approximation (LIA)
  1 
(loc )
r  rr r
v s (r0 ) 
ln
4  r  
 In the presence of normal fluid we must add the force of mutual friction
fd ~  v n  v L 
(+ 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    s .
If   1 (4He) motion of vortex is only weakly perturbed by mutual friction..
If   1 (3He-B at high temperatures) motion is strongly perturbed.
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.
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 selfsustaining 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 )?
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).
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
• at higher temperatures there was thermal activation
• at lower temperatures there was quantum tunnelling
Also seen in later
work on flow through
small apertures by
Varoquaux et al
 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?).
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.?
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,
E k   C 2 3k  5 3
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 >> mean vortex spacing 
• dissipation, on a scale ~ , given by the quasi-classical expression
     L     effective mean square vorticity
2 2
L=vortex line
density
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 (<~) the turbulence must be
very different from any classical type.
 But on large scales (>>, 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 >>  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.
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 ( << 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)
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.
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.
     2L2
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
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.
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.
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 quasiparticle 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
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?
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.
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
Thank you
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.