Energy spectrum of the quantum vortices configurations
The energy spectra of the 3D velocity field, induced by various vortex filaments configurations are reviewed. The especial attention is paid to configurations generating the Kolmogorov type energy spectrum E(k) ∝ k⁻⁵/³. The motivation of this work is related to the problem of modeling classical tu...
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irk-123456789-1279372018-01-01T03:02:50Z Energy spectrum of the quantum vortices configurations Nemirovskii, S.K. 10th International Conference on Cryocrystals and Quantum Crystals The energy spectra of the 3D velocity field, induced by various vortex filaments configurations are reviewed. The especial attention is paid to configurations generating the Kolmogorov type energy spectrum E(k) ∝ k⁻⁵/³. The motivation of this work is related to the problem of modeling classical turbulence with a set of chaotic vortex filaments. The quantity <v(k)v(–k)> can be exactly calculated, provided that we know the probability distribution functional P({s(ξ,t)}) of vortex loops configurations. The knowledge of P({s(ξ,t)}) is identical to the full solution of the problem of quantum turbulence and, in general, P is unknown. One of the simplifications is to investigate various truthful vortex configurations which can be elements of real vortex tangles. These configurations are: the uniform and nonuniform vortex arrays, the straight lines with excited Kelvin waves on it and the reconnecting vortex filaments. We demonstrate that the spectra E(k), generated by the these configurations, are close to the Kolmogorov dependence ∝ k⁻⁵/³, and discuss the reason for this as well as the reason for deviation. 2015 Article Energy spectrum of the quantum vortices configurations / S. K. Nemirovskii // Физика низких температур. — 2015. — Т. 41, № 6. — С. 608-614. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 67.25.dk, 47.37.+q, 03.75.Kk http://dspace.nbuv.gov.ua/handle/123456789/127937 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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10th International Conference on Cryocrystals and Quantum Crystals 10th International Conference on Cryocrystals and Quantum Crystals |
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10th International Conference on Cryocrystals and Quantum Crystals 10th International Conference on Cryocrystals and Quantum Crystals Nemirovskii, S.K. Energy spectrum of the quantum vortices configurations Физика низких температур |
| description |
The energy spectra of the 3D velocity field, induced by various vortex filaments configurations are reviewed.
The especial attention is paid to configurations generating the Kolmogorov type energy spectrum E(k) ∝ k⁻⁵/³. The
motivation of this work is related to the problem of modeling classical turbulence with a set of chaotic vortex filaments.
The quantity <v(k)v(–k)> can be exactly calculated, provided that we know the probability distribution functional P({s(ξ,t)}) of vortex loops configurations. The knowledge of P({s(ξ,t)}) is identical to the full solution of
the problem of quantum turbulence and, in general, P is unknown. One of the simplifications is to investigate various
truthful vortex configurations which can be elements of real vortex tangles. These configurations are: the uniform
and nonuniform vortex arrays, the straight lines with excited Kelvin waves on it and the reconnecting vortex
filaments. We demonstrate that the spectra E(k), generated by the these configurations, are close to the Kolmogorov
dependence ∝ k⁻⁵/³, and discuss the reason for this as well as the reason for deviation. |
| format |
Article |
| author |
Nemirovskii, S.K. |
| author_facet |
Nemirovskii, S.K. |
| author_sort |
Nemirovskii, S.K. |
| title |
Energy spectrum of the quantum vortices configurations |
| title_short |
Energy spectrum of the quantum vortices configurations |
| title_full |
Energy spectrum of the quantum vortices configurations |
| title_fullStr |
Energy spectrum of the quantum vortices configurations |
| title_full_unstemmed |
Energy spectrum of the quantum vortices configurations |
| title_sort |
energy spectrum of the quantum vortices configurations |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2015 |
| topic_facet |
10th International Conference on Cryocrystals and Quantum Crystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/127937 |
| citation_txt |
Energy spectrum of the quantum vortices configurations / S. K. Nemirovskii // Физика низких температур. — 2015. — Т. 41, № 6. — С. 608-614. — Бібліогр.: 34 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT nemirovskiisk energyspectrumofthequantumvorticesconfigurations |
| first_indexed |
2025-07-09T08:02:38Z |
| last_indexed |
2025-07-09T08:02:38Z |
| _version_ |
1837155652237000704 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6, pp. 608–614
Energy spectrum of the quantum vortices configurations
Sergey K. Nemirovskii
Institute of Thermophysics, 1 Lavrentyev Ave., Novosibirsk 630090, Russia
Novosibirsk State University, Novosibirsk, Russia
E-mail: nemir@itp.nsc.ru
Received September 7, 2014, published online April 23, 2015
The energy spectra of the 3D velocity field, induced by various vortex filaments configurations are reviewed.
The especial attention is paid to configurations generating the Kolmogorov type energy spectrum E(k) ∝ k–5/3. The
motivation of this work is related to the problem of modeling classical turbulence with a set of chaotic vortex fila-
ments. The quantity <v(k)v(–k)> can be exactly calculated, provided that we know the probability distribution func-
tional ({ ( , )})tξs of vortex loops configurations. The knowledge of ({ ( , )})tξs is identical to the full solution of
the problem of quantum turbulence and, in general, is unknown. One of the simplifications is to investigate var-
ious truthful vortex configurations which can be elements of real vortex tangles. These configurations are: the uni-
form and nonuniform vortex arrays, the straight lines with excited Kelvin waves on it and the reconnecting vortex
filaments. We demonstrate that the spectra E(k), generated by the these configurations, are close to the Kolmogorov
dependence ∝ k–5/3, and discuss the reason for this as well as the reason for deviation.
PACS: 67.25.dk Vortices and turbulence;
47.37.+q Hydrodynamic aspects of superfluidity; quantum fluids;
03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow.
Keywords: superfluidity, vortices, quantum turbulence.
1. Introduction and scientific background
The energy spectra of the 3D velocity field produced by
various configurations of the quantized vortex filaments are
described. Especial attention is paid to configurations generat-
ing the Kolmogorov type energy spectrum 5/3( ) .E k k−∝
Such statement of problem is motivated by the problem of
modeling classical turbulence with a set of chaotic vortex
filaments. This idea has been discussed for quite a long
time.(For details see, e.g., [1–3]). In classical fluids thin
vortex tubes do not exist because they spread due to vis-
cosity, so the concept of vortex filaments should just be
considered as a model. Quantum fluids, where the vortex
filaments are real objects, give an excellent opportunity for
developing the study of the question of whether the dy-
namics of a set of vortex lines is able to reproduce (at least
partially) the properties of real hydrodynamic turbulence.
Among various arguments supporting the idea of quasi-
classic behavior of quantum turbulence, the strongest is the
k dependence of the spectra of energy ( )E k obtained in
numerical simulations and experiments. There are many
works, that demonstrate a dependence of ( )E k close to the
Kolmogorov law 5/3( ) .E k k−∝ These are works, based on
both the vortex filament method [4–6] and the Gross-
Pitaevskii equation [7,8]. The most common view of quasi-
classical turbulence is the model of vortex bundles. The
point is that the quantized vortices have a fixed core radius,
so they do not possess the very important property of clas-
sical turbulence — stretching vortex tubes with a decrease
in core size. The latter is responsible for the turbulence
energy cascade from large scales to the small scales. Col-
lections of near-parallel quantized vortices (vortex bun-
dles) do possess this property, so the idea that the quasi-
classical turbulence in quantum fluids is realized via vortex
bundles of different sizes and intensities (number of
threads) seems quite natural. Meanwhile, the concept of
the bundle structure is vague and up to now it has not been
definitely confirmed. It is unclear how the bundles can
spontaneously appear (at low temperature, when the cou-
pling with normal component is small). Moreover, even if
they are prepared artificially, they are extremely unstable
(see, [3,9]), can be easily destroyed in a result of reconnec-
tion either between the neighboring threads or in collisions
with the other bundles, with the formation of the “bridg-
ing”. Therefore it quite tempting to find another alternative
mechanism for the appearance of the Kolmogorov type
spectrum, and we are looking for the various vortex fila-
ments configurations as a candidate for this purpose.
© Sergey K. Nemirovskii, 2015
Energy spectrum of the quantum vortices configurations
Of course, the more honest way is to resolve the general
problem and find the vortex configuration studying the
vortex dynamics on the basis of the motion equations. But
this way is practically impossible due to the incredible
complexity of the problem. In general, superfluid turbu-
lence (ST) can be regarded as the kinetics of merging and
splitting vortex loops, which, in addition, possess their own
very involved dynamics, with an infinite number of de-
grees of freedom. We disregard for a time the presence of
the very long lines stretching from wall to wall (“infinite
lines” in the context of cosmic strings). Thus, in the full
statement of the problem we have to deal with a set of ob-
jects (vortex loops), which does not have a fixed number of
elements — the elements can be born and die. Thus, an
analog of the secondary quantization method is required,
with the difference being that the objects (vortex loops)
themselves possess an infinite number of degrees of free-
dom with very involved dynamics. Clearly, this problem
can hardly be resolved in the near future, and substantial
simplifications are required. One of these simplifications is
to investigate various truthful vortex configurations which
can be elements of real vortex tangles.
In the work we introduce the general method for calcu-
lation of the energy spectrum via the vortex line configura-
tion, then we consider the couple of simple but useful ex-
amples -the straight line and vortex ring of radius .R Then
we study uniform and nonuniform vortex arrays, the
straight line with excited Kelvin wave on it and then we
study the case of the reconnecting vortex filaments. We
demonstrate that the spectra ( ),E k generated by the these
configurations, is very close to the Kolmogorov depend-
ence 5/3,k−∝ and discuss the reason for this as well as the
reason for deviation.
2. Calculation of spectrum
Let us discuss the energy of the flow created by a vor-
tex structure [10–12]. The energy of the vortex tangle is
defined as
2
2 3
, 0 0
( ) ( )1= = .
2 8 | ( ) ( ) |
LL ji
i i j js
s s i j
i i j jj i
E d d d
′ ′ξ ⋅ ξρ κ
ρ ξ ξ
π ξ − ξ∑∫ ∫ ∫
s s
v r
s s
(1)
Here we used that formally the vorticity field ( )rω can
be written as
( ) = = ( ) ( ( , )) .s t d′∇× κ ξ δ − ξ ξ∫ω r v s r s (2)
In relations (1)–(2) we used ( ) = ( )i iξ ξs s
is the union of
lines ( )i iξs where ( )i iξs describes the i-vortex line posi-
tion parameterized by the label variable ,iξ ( )i is′ ξ deno-
tes the derivative with respect to variable iξ (the tangent
vector) and = j
C C
∑∫ ∫ .
By the use of the formula from the theory of general-
ized functions (see [13])
3
2 2
1 = e ,
| | 2
id
kπ∫
kr
k
k
r
(3)
the average energy E (1) can be rewritten as
3
3=
(2 )
dE ×
π∫
k
k
2 ( ) ( ))
2
, 0 0
1 ( ) ( ) e .
2
LL ji i i i j js
j i j j i j
i j
d d
k
ξ − ξ
ρ κ ′ ′× ξ ⋅ ξ ξ ξ
∑∫ ∫
k(s s
s s
(4)
Equation (4) is of crucial importance, since it offers a sim-
ple way of calculating the energy spectrum ( )E k of the
three-dimensional velocity field induced by the vortex tan-
gle. According to Eq. (4), the quantity ( )E k is just the
expression inside the brackets. Note that the energy spec-
trum ( )E k can be also derived from the expression for the
Fourier transform of the velocity field 2= / .k×ωk kkv
Taking into account that due to incomressibility = 0,ωkk
one easily obtains 2= / .k− −⋅ ω ⋅ωk k k kv v
For the isotropic case, the spectral density depends on
the absolute value of the wave number .k Integrating over
solid angle lead to formula (see [14]):
2
2( ) =
(2 )
sE k
ρ κ
×
π
, 0 0
sin( ( ) ( ) )
( ) ( ) .
( ) ( )
LL ji i j
j i j j i j
i ji j
k
d d
k
ξ − ξ
′ ′× ξ ⋅ ξ ξ ξ
ξ − ξ
∑ ∫ ∫
s s
s s
s s
(5)
For anisotropic situations, formula (5) is understood as
the angular average. Thus, for calculation of the energy
spectrum ( )E k of the 3D velocity field, induced by the
collapsing vortex filament we need to know the exact con-
figuration { ( )}s ξ of vortex lines. Further we will apply
these formulas to study some particular configurations
{ ( )}s ξ which probably induce the Kolmogorov spectrum
5/3( ) .E k k−∝ Before we do it, let’s describe the couple of
simple but useful examples — the straight line and vortex
ring of radius .R
The angle averaged spectrum created by straight vor-
tex line is directly evaluated from (5) to give result
2 1
straight ( ) = ( /4 )sE k k−ρ κ π (per unit length). This spec-
trum is discussed early (see, e.g., [7,15] proposed the 1k−
spectrum on the basis of dimensional consideration. This
result is important, since it states that for any vortex system
the high wave numbers larger than inverse curvature,
( )E k should scale as 1.k−
Now we will consider a vortex ring with radius R ly-
ing in the x-y plane. The line ( )s ξ can be parameterized as
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 609
Sergey K. Nemirovskii
( ) = ( cos , sin , 0)s x R Rϕ ϕ with [0,2 ].ϕ∈ π Applying it to
Eq. (5) we get (see also [7])
2
ring 2=
(2 )
s R
E
ρ κ
×
π
2 2
1 2 1 2
1 2
1 20 0
cos( )sin(2 sin(( )/2))
.
2 sin(( )/2)
kRd d
k
π π ϕ −ϕ ϕ −ϕ
× ϕ ϕ
ϕ −ϕ∫ ∫
(6)
Evaluating the integral numerically shows that the spec-
trum ring ( )E k scales like 2
ring ( )E k k for 1.kR << Fre-
quently, the 2k is referred to as a proof for the thermody-
namical equilibrium state. We would like to stress,
however, that this distribution of the energy (valid far from
the ring) has nothing to do with the equipartition law. It is
a consequence of the fact that closed vortex loops induce a
far field flow scaling as 31/r (Fig. 1). That, in turn, gener-
ates a spectrum 2( ) .E k k∝ This fact was established for
classical turbulence (see, e.g., [18]). For quantum turbu-
lence this result was discussed by Stalp, Skrbek and Don-
nelly [19]. For the large ,k namely for 1,kR >> spectrum
ring ( )E k scales like 1k− as for straight line.
3. Vortex array
3.1. Uniform lattice
In Introduction we discussed, that one of ways to sup-
port the quasi-classic behavior of quantum turbulence is
the vortex bundles structure. Let’s study what is the energy
spectrum of 3D flow induced by the array of firtex fila-
ments, imitating the bundle. First we consider a set of
straight vortex filaments forming the square lattice
( ) = ( , , )i i js x y zξ
. Points ,i jx y are coordinates for
vortices on the xy-plane, indices ,i j runs from 1 to .N
The neighboring lines are separated by distance b , i.e.,
1 = .i ix x b+ − In case of different straight lines we have to
perform the integration between different lines and
2 2 2
1 1 2 2 1 2 1 2( ) ( ) = ( ) ( ) ( )i i j j i jx x y y z zξ − ξ − + − + − =s s
2 2= ( )ij i jd z z+ −
where 2 2
1 2 1 2= ( ) ( )ij i i j jd x x y y− + − distances between
vortices on the xy-plane. Then Eq. (5) can be rewritten as
2
( ) =
s
E k
Lρ κ
2 2
1 22 2, =1 , =10 01 2 1 2
sin( ( ) )1 ( ).
4 ( ( ) )
LLN N ij i j
i i j j ij i j
k d z z
d z z
k d z z
+ −
= −
π + −
∑ ∑ ∫∫
Integral over z is in the table by Ryzhik–Gradshtein
((3.876), see [20])
02
, =1 , =11 2 1 2
( ) 1= ( ).
4
N N
ij
i i j js
E k J kd
kL πρ κ
∑ ∑ (7)
Thus, determination of the spectrum on the basis (7)
should be done with the use of the quadruple summation
(over ( , , , )),i j i jx x y y which requires large computing re-
sources. Clear, however, that for very small ,k which cor-
responds to very large distance, the whole array can be
considered as large single vortex with the circulation 2 .N κ
Accordingly, the spectrum (per unit height) should be
4 2 1( /4 ) .s N k−ρ κ π For large ,k which corresponds to very
small distance from each line, the spectrum (per unit
height) should be 2 1( /4 ) .s k−ρ κ π In the intermediate re-
gion <<1,kb and >>1Nkb (this condition implies that
inverse wave number 1k− is larger intervortex space be-
tween neighboring lines, but smaller then the size of the
whole array ),Nb we can replace the quadruple summation
by the quadruple integration with infinite limits. This pro-
cedure corresponds that we exclude the fine-scale motion
near each of vortex, and are interested in the only large-
scale, coarse-grained motion. After obvious change of var-
iables ,i i i ix kx y ky→ → etc. we get that the whole integral
should scale as 41/ ,k and accordingly 5( ) 1/E k k∝ (com-
pare with [17], Appendix A.3 and Subsection III.a ).
The obtained result can be illustrated in the 2D case.
Noting that the full 2= ( ) = ( ) ,E E k dk E d∫ ∫ k k and in the
isotropic case, it is ( )2 .E kdkπ∫ k This implies that
( )2 = ( ),E k E kπk and, correspondingly, 2( ) = /E dE dk k
should behave as 61/ .k− Since, further, ( ) = ( ) ( ),E −k v k v k
we see that ( )kv scales as 31/ .k− The latter means (see,
e.g., [2], Eqs. (4.60), (4.61)) that the velocity ( )rv scales
as 1.r Indeed, in the 2D case we have the following chain
of relations
Fig. 1. (Color online) The spectrum induced by ring of unit radius
(see text for explanations).
610 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
Energy spectrum of the quantum vortices configurations
2 cos
3
1 1 1( ( )e e =
2 2
i ikrd kdkd
k
θ∝ θ
π π∫ ∫krv r) = v k k
03
1= ( ) .J kr kdk
k∫ (8)
Thus, the velocity field (v r) scales with distance r ac-
cording to the following rule:
02
1( ( ) ( ).
( )
r J kr d kr
kr
∝ ∫v r) (9)
Thus, the uniform vortex array creates the course-
grained motion, which is rotation (velocity is proprtional to
the distance from center), as it should be. Moreover, the
coefficient is proportional to 2/2 ,bκ which corresponds to
the Feynman rule. Integral in (9) does not converge, since
it corresponds situation when all infinite plain rotates, but
we have to take into account that this consideration (in
addition approximate) concerns the domain inside of the
vortex array and we ignorred all boundary effects. Con-
cluding this subsection we state that the uniform vortex
bundles do nt generate the Kolmogorov spectra.
3.2. Noninform lattice
Let’s now consider the nonuniform vortex bundle. To
model this situation we just can choose that the distance b
between lattice points (see previous subsection) is not con-
stant, but depends on the numbers ,i j of the cell nodes.
We have to realize that the problem of the spontaneous
formation of vortex bundles is only son the stage of discus-
sion so far, and there is no ideas concerning the arrange-
ment of these bundles. We will choose the power law de-
pendence for the distance between the lattice points.
1 0 1 0= , = .i i j jx x b i y y b jλ λ
+ +− − (10)
We do not ascertain the quantaty λ it is free parameter of our
approach. Under condition (10) the expression (7) turns into
0 02
=1 =1
( ) 1= ( ).
4
N N
i js
E k J kb i
kL
λ
πρ κ
∑∑ (11)
That means that the change the summation by integra-
tion we have to take 1/ 1/,i i i ix k x y k yλ λ→ → (instead of
the change of variables ,i i i ix kx y ky→ → made previous
subsection), etc. we get that the whole integral should scale
as 1 4/1/ .k + λ It is easy to see that for 6,λ = the spectrum
5/3( ) .E k k−∝
Currently, in classic hydrodynamics, the highly important
topic — the role of hydrodynamic collapse and its role in the
formation of turbulent spectra. is being intensively discussed
(see, e.g., [21,22]). Briefly this phenomena can be described
as spontaneous infinite grows of vorticity with formation of
singularity ( ).rω In particular, in the continuously distributed
vortex field the vortex lines (not quantized vortex filaments,
just hydrodynamic vortex lines!) approach each other and
accumulates at some points forming singular distribution
2/3( ) .r r−ω The latter results in the increment for velocity
field 1/3( ,r+v r δr) which, in turn results it to the Kol-
mogorov spectrum 5/3( ) .E k k−∝
Let’s find 2D density of vortices on the xy plane under
condition (10), or, according to the Feynman rule, the dis-
tribution of vorticity ( ).rω In the “space” of indices { , }i j
vortices are distributed uniformly (one vortex per lattice
site { , }),i j but since the distances between the sites vary,
the distribution of vortices in the real xy space is
nonuniform. Let us consider “the ring” of radius from I to
I I+∆ in { , }.i j Then, the number of points N∆ in ring is
just 2 ,I Iπ ∆ the radius of ring in real xy space is
0= ,r b Iλ and the thickness of ring is 1
0= .r b I Iλ−∆ λ ∆
From these relations it follows that ( )n r scales with r as
1 2/
1( ) = .Nn r
r r − λ
∆
∝
∆
If 6,λ = then 2/3( ) = ( ) = ,n r r r−ω as it should be for
the classical turbulence [21].
4. 1D Kelvin waves spectrum and 3D velocity spectrum
In the literature there is discussed the idea of obtaining
the 3D velocity spectrum just by putting it equal to the
spectrum of 1D Kelvin waves. For instance, as stated in
[23]. “We notice that, because the fluctuations of the ve-
locity field are induced by the Kelvin wave fluctuations on
the filaments, it is reasonable to expect that
( ) ( ).”KWE k E k (12)
The same idea was used in papers by Nazarenko et al. (see,
e.g., [24]). Details of this activity can be read in a series of
papers by L’vov, Nazarenko and coauthors [8,24–27].
Let us consider this problem on the basis of general
formula (5). We take ( , ) = ( ( , ), ( , ), )t x z t y z t zξs and denote
the two-dimensional vector ( ( , ), ( , ))x z t y z t as ( , )a z tρ
(where the dimensionless amplitude 1).a << Substituting it
into (5) and expanding in powers of a, we get,
_____________________________________________________
( ) 22
2 1 2 12
straight 1 22 2
2 10 0
cos ( ( ) ( ))
( ) = ( )
(2 ) 2
LL
s
KW
k z z z z
E k E k a dz dz
z z
− ρ −ρρ κ + −
π −
∫∫ (13)
( ) ( )2
2 1 2 1 1 2 2 1
3
2 12 1
sin ( ( ) ( )) ( ( ) ( )sin
.
2
k z z z z z z k z z
k z zk z z
′ ′− ρ −ρ ρ ⋅ρ − − +
−−
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 611
Sergey K. Nemirovskii
To move further we have to find the correlation charac-
teristics for the fluctuating vector of displacement 2( )zρ .
We accept that the ensemble of Kelvin waves has a follow-
ing power-like spectrum:
( ) ( ) = .sp p Ap−ρ ⋅ρ − (14)
We take here the notation p for the one-dimensional
vector, conjugated to ,z reserving the notation k for the
absolute value of the wave vector of the 3D field. The
formula (14) implies that (see, e.g., [2], Eqs. (4.60),
(4.61)) the squared increment for the vector of displa-
cement scales as, 2 1
2 1 2 1( ( ) ( )) ( ) .sz z z z −ρ −ρ ∝ − Then
the second order correlator 2 1( ( ) ( ))z z′ ′ρ ρ scales as
3
2 1 2 1( ( ) ( )) ( ) .sz z z z −′ ′ρ ρ ∝ − Substituting it into (13) and
counting the powers of quantity ,k we conclude that the
correction ( )E kδ to the spectrum ( ),E k due to the en-
semble of Kelvin waves has a form:
2 2( ) .sE k a k− +δ ∝ (15)
It is remarkable fact that this quantity coincides formally with
the one-dimensional spectrum of KW 2 2( ) sE p a p− +δ ∝
however this contribution is small, by virtue the smallness
of the wave amplitudes ,a and disappears with the KW.
In series papers L’vov, Nazarenko and coauthors
[8,24–26] it was proposed the spectrum for Kelvin waves
of shape (14) with =11/3,s therefore the 3D spectrum of
velocity field 5/3( ) ,E k k−δ ∝ i.e., the Kolmogorov type
energy spectrum.
5. Reconnecting lines
Let’s now describe the energy spectrum of the 3D ve-
locity field, induced by vortex filaments collapsing to-
wards reconnection [28]. The first obstacle is the exact
determination of the vortex lines configurations. Despite
the huge number of works devoted to the dynamics of col-
lapsing lines both in classic and quantum fluids [29,30]
(this list is far from full) the exact solution ( )ξs for the
shape of curves has not yet obtained. The main results are
obtained by different approaches, combining analytical and
numerical methods, such as the vortex filament method (in
both the local induction approximation and the Biot–Savart
law), as well as the full 3D studies of Navier–Stocks equa-
tions or the Nonlinear Schrödinger equation for vortices in
the Bose–Einstein condensate.
Qualitatively, the results of these investigations are
quite similar and can be described as follows. Due to long
range interaction in the Biot–Savart integral, the initially
arbitrarily oriented vortices, when they approach each oth-
er, start by reorienting their close segments so as to bring
them into an antiparallel position. Further, cusps may ap-
pear on the approaching segments of two vortex lines. The
curvature of these cusps may be so large that the self-
induced velocity of each perturbation overcomes the repul-
sion from the adjoining vortex line. Further the cusps grow
and approach each other closer; this increases their curva-
ture and, correspondingly, their self-induced velocities and
this process is repeated faster and faster. It is important that
this process grows explosively, since the distance between
the two perturbed segments, ,∆ decreases according to the
relation 1/2( ) ,t t∗∆ − where t∗ is some quantity depend-
ing on the relevant parameters and initial conditions. Thus,
in a finite time the vortex lines collapse. Asymptotic lines
are two hyperbolic curves lying on opposite sides of the
pyramid (see, e.g., [30,33]). However, in a recent study,
[31] it was shown that the curves are not exact hyperbolas,
but slightly different lines (the authors call these curves
quasi-hyperbolae) of type
2 2 2 2 2 2( ) = /( ) ,h a a aξ ξ +ξ + +ξ
and that they lie not in the planes of the pyramid sides but
on the curved surfaces, bent inwards. In the moments just
before the collapse, when the vortex cores are nearly touch
each other, the very acute kink appears. This curves may
be written in parametric form (cf. formula (16) of [31])
( ) ( )1,2 ( ) = ( ) , , ( ( ) ) .h c h h bξ ± ξ − ±ξ ξ − s (16)
The described configuration is shown in Fig. 2. The signs
are chosen so that 1 2(0) (0) = 1′ ′⋅ −s s (the vortices are anti-
parallel). Quantity a is of the order of the curvature radius
on the tip of the kink of the curve; quantity b (related to
a , see [31]) is responsible for bending the surfaces on
which the quasi-hyperbolae lie. Quantity c is also of the
order of a and is responsible for closeness of the fila-
ments. All three quantities are smaller than the intervortex
space 1/2= −δ (where is the vortex line density). This
vision is consistent with the results of numerous numerical
works, studying the collapse of vortex lines (see, e.g.,
[22,23] and references therein, the decisive picture ob-
tained in [32] is shown in the inset of Fig. 2).
Fig. 2. (Color online) The touching quasi-hyperbolae describing
the collapsing lines (see Eq. (16)) obtained in [31]. In the inset we
set (as an example) the kinks on the anti-parallel collapsing vor-
tex tubes obtained in numerical simulation [32].
612 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
Energy spectrum of the quantum vortices configurations
In the Fig. 3(a) we presented the results of numerical
calculation of spectrum ( )E k on the basis of formula (5)
(without prefactor before integral) using a configuration
{ ( )}ξs of vortex lines described by (16). We chose the
following parameters: a = 0.1, b = 0.099, c = 0.1 (the case
a c≈ corresponds to nearly touching curves). It is seen
that in the interval of wave numbers k between 1–50 the
slope of ( )E k is indeed close to –5/3. Because of the rap-
idly oscillating function, the evaluation of integral (5) is
difficult, even numerically. In addition, numerical results
obscure underlying physics. Therefore, to explain appear-
ing of the Kolmogorov-type spectrum we performed an
analytical study [28]. Referring the reader for details to the
author’s work [28], we briefly describe main speculations.
The integral (5) can be approximately evaluated for large
k using the method of asymptotic expansion [34]. When
k is large the function 1 2sin( ( ) ( ) )k ξ − ξs s is a rapidly
varying function; therefore, the main contribution into in-
tegral comes from points of minimal value of the separa-
tion function between points of the curves 1 2( , ) =D ξ ξ
1 2( ) ( ) .= ξ − ξs s This is enhanced by the fact that the dis-
tance is included in the denominator in the integrand of (5).
Thus, the behavior of the phase function 1 2( , )D ξ ξ near
minimum is crucial for value of the integral and for its k
dependence. Let us study the phase function 1 2( , )D ξ ξ for
the vortex configuration described by Eq. (16) just before
collapse when .c a≈ It is convenient to introduce variables
1 2=ρ ξ −ξ and 1 2= ( )/2R ξ +ξ and recast the double inte-
gral 1 2
C C
d dξ ξ∫ ∫ as multiple integral dR dρ∫ ∫ in the do-
main bounded by lines = 2Rρ and = 2 .Rρ − Analysing
properly the phase function ( , )D Rρ and using of the
methods of asymptotic expansion we can show that the
spectrum ( )E k is
2
2 2
=00 =0
sin( ( , ))( ) = .
( , )/
s
kD RE k dR
kD Rk D ρ
ρ
π ρ
ρ κ
ρ∂ ∂ρ
∫ (17)
In the Fig. 3(b) we present ( ),E k calculated on the
basis of formula (17). First, please note, that spectrum
calculated with the use of (17) very close to the spectrum
calculated on basis (5); this justifies the approximated pro-
cedure, as described above. Second, and more importantly,
the fact is that again in the interval of wave numbers k
between 1–50 the slope of ( )E k is close to –5/3. Analyz-
ing the procedure one can see that the 5/3k−≈ depend-
ence appears because of in the most important interval of
scale from the curvature of kinks to interline distance δ
the quantity (0, )D R is close to 3/2.R In fact, the authors
of the numerical works [4–8] cited in the introduction,
obtained the spectrum 5/3( )E k k−≈ for the wave numbers
,k around 2 / .k ≈ π δ
6. Conclusions
Summarizing, it can be concluded that the 3D energy
spectrum ( )E k consists of several parts. At small ,k asso-
ciated with the large scales, on the order of the size of the
system, or on the scale of the stirring forcing (grids, pro-
pellers, vibrating objects, etc.), the spectrum behaves as
2( ) .E k k∝ We recall again that this is the consequence of
the asymptotic behavior ( )r →∞ of the velocity field, not
of thermodynamical equilibrium. For large wave numbers,
exceeding the inverse intervortex space > 2 / ,k π δ the en-
ergy spectrum ( )E k should be close to 1,k− again regard-
less of the specific model. The region of intermediate k is
the most intriguing and exciting, it depends on the detailed
structure of the vortex tangle. Coming back to the aims of
the work stated in the introduction we can suggest that the
spectrum ( )E k close to the Kolmogorov dependence
5/3,k−∝ which was observed in many numerical simula-
tions on the dynamics of quantized vortex filaments [4–8],
can appear from the some “standard” configurations com-
posing the real vortex tangle.
The work was supported by the grant N 14-19-00352 from
RSCF (Russian Scientific Foundation).
Fig. 3. (Color online) The same spectrum obtained on the basis of procedure described in the text (see Eq. (17)) (a). The spectrum E(k),
obtained numerically on the base formula (5). The straight line has a slope –5/3 (b).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 613
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614 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
1. Introduction and scientific background
2. Calculation of spectrum
3. Vortex array
3.1. Uniform lattice
3.2. Noninform lattice
4. 1D Kelvin waves spectrum and 3D velocity spectrum
5. Reconnecting lines
6. Conclusions
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