Probing of quantum turbulence with the emitting vortex loops
The statistics of vortex loops emitted from the domain with quantum turbulence is studied. The investigation is performed on the supposition that the vortex loops have the Brownian or random walking structure with the generalized Wiener distribution. The main goal is to relate the properties of the...
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irk-123456789-1196982017-06-09T03:02:30Z Probing of quantum turbulence with the emitting vortex loops Nemirovskii, S.K. Письма pедактоpу The statistics of vortex loops emitted from the domain with quantum turbulence is studied. The investigation is performed on the supposition that the vortex loops have the Brownian or random walking structure with the generalized Wiener distribution. The main goal is to relate the properties of the emitted vortex loops with the parameters of quantum turbulence. The motivation of this work is connected with recent studies, both numerical and experimental, on study of emitted vortex loops. This technique opens up new opportunities to probe superfluid turbulence. We demonstrated how the statistics of emitted loops is expressed in terms of the vortex tangle parameters and performed the comparison with numerical simulations. 2014 Article Probing of quantum turbulence with the emitting vortex loops / S.K. Nemirovskii // Физика низких температур. — 2014. — Т. 40, № 12. — С. 1436-1438. — Бібліогр.: 10 назв. — англ. 0132-6414 PACS 67.25.dk, 47.37.+q http://dspace.nbuv.gov.ua/handle/123456789/119698 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Письма pедактоpу Письма pедактоpу Nemirovskii, S.K. Probing of quantum turbulence with the emitting vortex loops Физика низких температур |
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The statistics of vortex loops emitted from the domain with quantum turbulence is studied. The investigation is performed on the supposition that the vortex loops have the Brownian or random walking structure with the generalized Wiener distribution. The main goal is to relate the properties of the emitted vortex loops with the parameters of quantum turbulence. The motivation of this work is connected with recent studies, both numerical and experimental, on study of emitted vortex loops. This technique opens up new opportunities to probe superfluid turbulence. We demonstrated how the statistics of emitted loops is expressed in terms of the vortex tangle parameters and performed the comparison with numerical simulations. |
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Article |
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Nemirovskii, S.K. |
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Nemirovskii, S.K. |
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Nemirovskii, S.K. |
| title |
Probing of quantum turbulence with the emitting vortex loops |
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Probing of quantum turbulence with the emitting vortex loops |
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Probing of quantum turbulence with the emitting vortex loops |
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Probing of quantum turbulence with the emitting vortex loops |
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Probing of quantum turbulence with the emitting vortex loops |
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probing of quantum turbulence with the emitting vortex loops |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2014 |
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Письма pедактоpу |
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Probing of quantum turbulence with the emitting vortex loops / S.K. Nemirovskii // Физика низких температур. — 2014. — Т. 40, № 12. — С. 1436-1438. — Бібліогр.: 10 назв. — англ. |
| series |
Физика низких температур |
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AT nemirovskiisk probingofquantumturbulencewiththeemittingvortexloops |
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2025-07-08T16:26:44Z |
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2025-07-08T16:26:44Z |
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1837096773955354624 |
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© S.K. Nemirovskii, 2014
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12, pp. 1436–1438
Letters to Editor
Probing of quantum turbulence with the emitting vortex loops
S.K. Nemirovskii
Institute of Thermophysics, 1 Lavrentyev Ave., Novosibirsk 630090, Russia
Novosibirsk State University, 2 Pirogova Str., Novosibirsk 630090, Russia
E-mail: nemir@itp.nsc.ru
Received August 15, 2014, revised September 10, 2014, published online October 22, 2014
The statistics of vortex loops emitted from the domain with quantum turbulence is studied. The investigation
is performed on the supposition that the vortex loops have the Brownian or random walking structure with the
generalized Wiener distribution. The main goal is to relate the properties of the emitted vortex loops with the pa-
rameters of quantum turbulence. The motivation of this work is connected with recent studies, both numerical
and experimental, on study of emitted vortex loops. This technique opens up new opportunities to probe super-
fluid turbulence. We demonstrated how the statistics of emitted loops is expressed in terms of the vortex tangle
parameters and performed the comparison with numerical simulations.
PACS: 67.25.dk Vortices and turbulence;
47.37.+q Hydrodynamic aspects of superfluidity; quantum fluids.
Keywords: quantum turbulence, vortex loops, superfluid turbulence.
Scientific background and motivations
Quantum turbulence (QT) in superfluids is one of the
most fascinating phenomena in the theory of quantum
fluids [1]. In general, the vortex tangle composing QT,
consists of a set of vortex loops of different lengths and
having a random structure. Question of an arrangement of
the vortex tangle is a key problem of the theory of QT.
Recently, a number of works devoted to the study of the
vortex tangle structure with the use of loops emitted from
the turbulent domain have been performed [2–5]. This
technique opens up new opportunities for research QT.
One of the main problem in this activity is to relate the
properties of the emitted vortex loops with the parameters
of QT. In the present work we propose an analytical ap-
proach that allows to relate the statistics of vortex loops
with the parameters of the real vortex tangle. This ap-
proach is based on the Gaussian model of the vortex tan-
gle, which describes the latter as a set of vortex loops
having a random walking structure with the generalized
Wiener distribution [6]. Then we apply our results for
data processing of numerical work [2] who studied QT
driven by an oscillating sphere. Generation of QT by os-
cillating objects is very important topic in this field (see,
e.g., [7–9]).
Flux of vortex loops emitted from quantum turbulence
In this paragraph we very briefly describe main ideas
leading to the theory of emission of vortex loops, details
can be found in paper by the author [10]. Vortex loops
composing the vortex tangle can move as a whole with a
drift velocity lV depending on their structure and their
length l. The flux of the line length, energy, momentum
etc., executed by the moving vortex loops takes place. In
the case of inhomogeneous vortex tangle the net flux J of
the vortex length due to the gradient of concentration of
the vortex line density ( , )x t appears. The situation here
is exactly the same as in classical kinetic theory with the
difference being that the “carriers” are not the point parti-
cles but the extended objects (vortex loops), which possess
an infinite number of degrees of freedom with very in-
volved dynamics.
To develop the theory of the transport processes ful-
filled by vortex loops (in spirit of classical kinetic theory)
we need to know the drift velocity lV and the free path
( )l for the loop of size l. Referring to the paper [10] we
write down here the following result. The drift velocity lV
and the ( )l for the loop of size l are
0= / , ( ) =1/2l mV C l l lbv . (1)
Probing of quantum turbulence with the emitting vortex loops
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12 1437
Quantity is 1/2
0( /4 )ln( / ),a where is the
quantum of circulation, 0a is the core radius, Cv is nu-
merical factor of the order of unity, and mb is the numeri-
cal factor , approximately equal to 0.2.mb The 0 is
the parameter of the generalized Wiener distribution, it is
of order of the interline space 1/2 . The probability ( )P x
for the loop of length l to fly the distance x without colli-
sion is ( ) = (1/ ( ))exp( / ( )).P x l x l Knowing the aver-
aged velocity lV of loops, and the ( )P x (both quantities
are l-dependent), we can evaluate the spatial flux J of the
vortex loops.
Let us consider the small area element placed at some
point of the boundary of domain containing QT and ori-
ented perpendicularly to axis x (see for details Fig. 2 of
paper [10]). The x-component of flux J of the number of
loops executed by loops of sizes ,l placed in , direc-
tion, and remote from the area element at distance ,R can
be written as
1
( ) = ( , , , )( cos ) ( )sin .
4
x lJ l n l R V P R d d dR (2)
Here the quantity coslV is just the x component of
the drift velocity, the factor ( )P R dR is introduced to con-
trol an attenuation of flux, due to collisions. In the spirit of
classical kinetic theory, we assume the local equilibrium is
established.
In paper [10] Eq. ( 2) was the starting point to develop
the theory of diffusion of vortex loops in QT, therefore the
density of loops ( )n l was supposed to depend on spacial
position, in spherical coordinates = ( , , , ).n n l R Here
we put another goal to study radiation of loops from the
domain with uniform vortex tangle. Supposing that inside
domain the density of loops does not depend on spatial
coordinates ( ( , , , ) = ( )),n l R n l and integrating out over
solid angle d d and over position of loops dR we ob-
tain the x-component of the loop flux through the domain
boundary
0
1
= ( ) .
4
xJ n l dl
l
(3)
The flux ,xJ described by formula (3) carries the loops
of different sizes in the normal to boundary direction and
provides information on the distribution of loops inside of
the turbulent domain. However, since the speed of loops
depends on their sizes the initial distribution changes as the
vortex “cloud” propagates. For instance, in some time the
small vortex loops (practically rings) get ahead, and the
measurements in a short time will detect only small loops.
That means that the experimental data essentially depends
on the position of detector and time time of registration. In
a stationary situation when the turbulence is maintained by
some means (counterflow or oscillating structures), the
intensity of detected loops should be steady in time.
Processing of experimental and numerical works
As an illustration let’s treat the numerical work [2] on
the statistics of emitted loops. Unfortunately, the pure ex-
perimental works are not too reliable for a proper quantita-
tive treatment. In work [2] the authors have studied the
statistics of vortex loops emitted from QT driven by an
oscillating sphere of radius 1 m with the frequency
3 kHz, and the amplitude 5.31 m. Counting of loops was
made on the spherical boundary of radius 30 m (centered
at the center of the oscillating sphere). The result of this
counting is illustrated in Fig. 1(a) ,where the a probability
density function (PDF) Prnum(l) of the length of the emit-
ted vortex loops is presented.
Let us consider this result from the position of the theory
stated above. Formula (3) describes the flux of loops per unit
area. The integrand in (3) is distribution of loops over their
lengths in the propagating loop “jet”. The total number of
emitted loops (per unit time) is ( ) = .N l dJ S The Gaussi-
an model, used in our consideration predicts that the density
of loops ( )n l inside the domain with vortex tangle is the
power-like function
5/2( ) = .n l Al This behavior, however,
has a low cutoff near the length 0.l Below the cutoff
there are a few loops of smaller sizes which do not essential-
ly affect the whole theory. From the graph for PDF Prnum(l)
Fig. 1. (Color online) The PDF of the length of emitted vortex
loops obtained in [2] (a). Lower line is the PDF from picture (a)
depicted in logarithmic coordinates. The upper curve is the ana-
lytical PDF Pran(l) ≈ 50l
–3
, obtained in this work (b).
S.K. Nemirovskii
1438 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12
in Fig. 1 it is seen that the cutoff is l ≈ 5 m. The size l ≈
≈ 5 m is the point of the maximum of the PDF Prnum(l),
where the power-like behavior ceases. Using (3) we get that
the analytical PDF is
3
an( ) = ( )/ ( ) .l N l N l dl lPr Fur-
thermore, taking into account that in the normalization factor
( )N l dl the value of integral is accumulated near the lower
limit l ≈ 5 m, we get that the analytical PDF approximatel
is Pran(l) ≈ 50l
–3
.
In Fig 1(b) we depicted PDF Prnum(l) obtained in work
[2] and our Pran(l) ≈ 50l
–3
in the logarithmic coordinates.
It can be seen that the power –3 is indeed close to the nu-
merical data, deviations appear near the low cutoff where
the Gaussian model does not work properly. As for abso-
lute values of the PDFs (numerical and analytical) they
differ by a factor about 2–3 (for large loops). This can be
explained by the fact that the vortex turbulence, produced
in numerical simulation is not the dense and uniform struc-
ture, which was required in the analytical consideration. In
fact, the interline space obained from the data on total
length (see Fig. 2 of paper [2]) is about 20 m which is
comparable with size of largest loops. This implies that the
QT is rather dilute.
Conclusion
Summarizing we can conclude that the “clouds” of vor-
tex loops emitted from the turbulent superfluid helium
provides an important information on the structure of QT.
However, due to very complicated dynamics of the net-
work of vortex loops, this information is hidden and should
be extracted with the use of appropriate formalism. It
should be understood that the described above procedure is
greatly simplified, many real features such as a possible
anisotropy, the mutual friction, the specific conditions of
the turbulence generation have not been considered. This is
supposed to be performed in future.
The study was performed by grant from the Russian
Science Foundation (project N 14-29-00093).
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