Subensemble concept in 2D magnetized particle transport model
Transport of magnetized particle undergoing random frozen isotropic electrostatic field is considered. Because of infinitely long field correlation time and particle trapping this problem is of particular interest as a test for closure of statistical equations. The concept of subensembles is incor...
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nasplib_isofts_kiev_ua-123456789-1153212025-02-09T14:48:29Z Subensemble concept in 2D magnetized particle transport model Концепция подансамблей в двумерной модели переноса замагниченных частиц Концепція підансамблів у двовимірній моделі перенесення замагнічених частинок Zasenko, V.I. Cherniak, O.M. Zagorodny, A.G. Basic plasma Transport of magnetized particle undergoing random frozen isotropic electrostatic field is considered. Because of infinitely long field correlation time and particle trapping this problem is of particular interest as a test for closure of statistical equations. The concept of subensembles is incorporated here in the analytical approach we developed earlier. The Lagrangian velocity correlation function is calculated in the drift approximation. To verify validity of the analytical method the results are compared with ones found from a direct numerical simulation. А better quantitative agreement obtained with the use of subensemble concept is shown. Рассмотрен перенос замагниченных частиц под действием случайного замороженного изотропного электростатического поля. Из-за бесконечно большого времени корреляции поля и захвата частиц эта проблема имеет особый интерес для проверки замыкания статистических уравнений. Аналитический подход, развитый нами ранее, дополнен концепцией подансамблей. Рассчитана лагранжева корреляционная функция скорости в дрейфовом приближении. Для проверки достоверности аналитического метода его предсказания сравниваются с результатами прямого численного моделирования. Показано, что использование концепции подансамблей улучшает количественное согласование результатов. Розглянуто перенесення замагнічених частинок під дією випадкового замороженого ізотропного електростатичного поля. Через нескінченно довгий час кореляції поля та захоплення частинок ця проблема має особливий інтерес для перевірки замикання статистичних рівнянь. Аналітичний підхід, розвинутий нами раніше, доповнено концепцією підансамблів. Розраховано лагранжеву кореляційну функцію швидкості в дрейфовому наближенні. Для перевірки достовірності аналітичного методу його передбачення порівнюються з результатами прямого числового моделювання. Показано, що використання концепції підансамблів поліпшує кількісне узгодження результатів. The work is supported by the STCU project 6060. 2016 Article Subensemble concept in 2D magnetized particle transport model / V.I. Zasenko, O.М. Cherniak, A.G. Zagorodny // Вопросы атомной науки и техники. — 2016. — № 6. — С. 77-80. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.65.Cc https://nasplib.isofts.kiev.ua/handle/123456789/115321 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Basic plasma Basic plasma |
| spellingShingle |
Basic plasma Basic plasma Zasenko, V.I. Cherniak, O.M. Zagorodny, A.G. Subensemble concept in 2D magnetized particle transport model Вопросы атомной науки и техники |
| description |
Transport of magnetized particle undergoing random frozen isotropic electrostatic field is considered. Because of
infinitely long field correlation time and particle trapping this problem is of particular interest as a test for closure of
statistical equations. The concept of subensembles is incorporated here in the analytical approach we developed
earlier. The Lagrangian velocity correlation function is calculated in the drift approximation. To verify validity of
the analytical method the results are compared with ones found from a direct numerical simulation. А better
quantitative agreement obtained with the use of subensemble concept is shown. |
| format |
Article |
| author |
Zasenko, V.I. Cherniak, O.M. Zagorodny, A.G. |
| author_facet |
Zasenko, V.I. Cherniak, O.M. Zagorodny, A.G. |
| author_sort |
Zasenko, V.I. |
| title |
Subensemble concept in 2D magnetized particle transport model |
| title_short |
Subensemble concept in 2D magnetized particle transport model |
| title_full |
Subensemble concept in 2D magnetized particle transport model |
| title_fullStr |
Subensemble concept in 2D magnetized particle transport model |
| title_full_unstemmed |
Subensemble concept in 2D magnetized particle transport model |
| title_sort |
subensemble concept in 2d magnetized particle transport model |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2016 |
| topic_facet |
Basic plasma |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/115321 |
| citation_txt |
Subensemble concept in 2D magnetized particle transport model / V.I. Zasenko, O.М. Cherniak, A.G. Zagorodny // Вопросы атомной науки и техники. — 2016. — № 6. — С. 77-80. — Бібліогр.: 6 назв. — англ. |
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Вопросы атомной науки и техники |
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BASIC PLASMA
ISSN 1562-6016. ВАНТ. 2016. №6(106)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2016, № 6. Series: Plasma Physics (22), p. 77-80. 77
SUBENSEMBLE CONCEPT IN 2D MAGNETIZED PARTICLE
TRANSPORT MODEL
V.I. Zasenko, O.М. Cherniak, A.G. Zagorodny
Bogolyubov Institute for Theoretical Physics, Kiеv, Ukraine
E-mail: zasenko@bitp.kiev.ua
Transport of magnetized particle undergoing random frozen isotropic electrostatic field is considered. Because of
infinitely long field correlation time and particle trapping this problem is of particular interest as a test for closure of
statistical equations. The concept of subensembles is incorporated here in the analytical approach we developed
earlier. The Lagrangian velocity correlation function is calculated in the drift approximation. To verify validity of
the analytical method the results are compared with ones found from a direct numerical simulation. А better
quantitative agreement obtained with the use of subensemble concept is shown.
PACS: 52.65.Cc
INTRODUCTION
Understanding of basic mechanisms of particle
transport is important for many problems in plasma
physics as well as for prediction of impurity migration
in various media with random velocity fields such as the
atmosphere and the ocean. Description of turbulent
transport can be carried out on different levels – from
phenomenological models to comprehensive numerical
simulation on a basis of kinetic equations. The most
direct method to formulate transport equations is
statistical averaging of equations of microscopic particle
motion in random fields. On this way arise a problem of
closure, i.e. expression of higher statistical moments via
lower ones, inherited from a statistical nonlinearity of
basic equations. There is no regular procedure to make
this closure in general case. A closure depends on a
particular problem and is a key element in the
formulation of statistical approach.
To describe particles in a random field of force or
velocity characterized by small Lagrangian correlation
time, and respectively small Kubo number, the known
Corrsin approximation can be applied. However for
trapped resonance particles which for a long time
interact with a locally regular field the Corrsin
approximation cannot be used. These particles play an
important role in transport processes and for their
statistical description other closure should be found.
In this paper a spread of magnetized particles in a
random electrostatic field is considered. The slower is a
field variation over time the more apparent becomes the
effect of their trapping. The effect is most pronounced
in the limit of frozen turbulence, i.e. when fields are
constant in time. Respectively this case is the most
difficult for the theoretical description when the result
of the calculations is very sensitive to a method of
closure. Thus recovering of particle spread in a frozen
random field with infinite trapping time is a test for
closure procedure. If the procedure is found to be
justified for a frozen field, the more reliable it can be
applied to a field varying in time.
Particle diffusion caused by external fields was
studied analytically and numerically in a number of
works. However analytical description of particle
trapping effect is not properly built. Relatively new
semianalytical approach to statistical description of
magnetized particles in external field of electrostatic
random waves was given in the works [1] where the
authors proposed and developed the decorrelation
trajectories method. The main elements of this method
are the assumption about subensembles – groups of
particles with similar initial conditions that are
characterized by specific behavior, and the closure
procedure formulated for partial diffusion coefficients in
subensembles.
In our paper [2] a different closure procedure based
on the equation for mean square displacement of
particles was proposed. In the paper [3] the concept of
particle subensembles formed in accordance with values
of potential and velocity in the initial particle positions
was analyzed in detail, and effectiveness of two
methods of closure was compared. Our closure gives a
qualitative agreement with simulation without use of
fitting parameters. Specific behavior was observed in
simulation for groups of particles (subensembles) with
the same value of potential in the initial particle
positions. On contrary for subensembles arranged by the
initial particle velocities no specific behavior was
observed in simulation.
In this paper we combine elements of both
approaches. These are splitting of a particle ensemble
on subensembles labeled by a value of potential, and
closure on mean square particle displacement. As before
no free parameters will be used. Note that a rigorous
proof of closure procedure is a complex problem and
remains beyond a scope of this work. Nevertheless,
some particular steps in a course of formulation of the
description as well as the final results were checked by
direct numerical simulation. Although the transition to
continuum in simulation remains a problem, in other
aspects the model for numerical simulation is adjusted
with the analytical one.
1. MODEL
We consider motion of strongly magnetized particles
in a random electrostatic field. The electric field is
statistically uniform and isotropic. To focus on a
particle trapping effect we consider frozen fields.
Particle motion will be described in a drift
approximation neglecting the effects of finite Larmor
radius, and this enhances a trapping effect.
78 ISSN 1562-6016. ВАНТ. 2016. №6(106)
A simple model allows establishing links between
the assumptions and the results, avoiding interference
with less important factors. Isotropy at this stage helps
to avoid mutual influence of diffusion in different
directions. The effects of the finite Larmor radius are
important and will be considered elsewhere.
Drift motion of magnetized particles in constant
magnetic and electric fields is governed by the
equations
( ), ( ).x yv v
y x
r r (1)
With accuracy to a coefficient a potential is a stream
function (r).
Using the Kraichnan approach [4] the equation for
distribution function of particles F(r,t) averaged over
realizations of random field (r) can be obtained in a
form
( , ) ( , ; ,0) ( ,0),F t d W t F r r r r r (2)
where W (r, t, r´, t´) is the averaged transition
probability between two points of coordinate space
( , ; , )
( , ; , ) ( ) ( )
( , ; , ),
'
t
i j
i t
j
W t t
t
d d W t V V
r
r
W t
r r
r r
r
ρ ρ ρ
ρ
(3)
( , ; , ) ( ).W t t r r r r (4)
It is problematic to find a solution of the nonlinear
integro-differential equation (3) for W (r, t, r´, t´) even
numerically; therefore it requires a reduction to a
simpler form. Along with this a choice of Eq. (3) as a
basic one – instead of immediate use of a reduced
equation (will be given later) – makes evident what
assumptions should be checked out in a course of
reduction. Verification of the assumptions can be
carried out by means of numerical simulation.
One of the important steps concerns the transition
from the nonlocal description to partially local one. For
a problem with trapped particles such transition is not
obvious, and without proper justification it looks
doubtful.
Under conditions of spatial and temporal locality
there are integral relations between the Lagrangian
correlation function, the diffusion coefficient, and the
mean square displacement. We found the Lagrangian
velocity correlation function and the mean square
displacement from direct simulation. It was shown that
by double numerical integration of the Lagrangian
correlation function the mean square displacement is
recovered with sufficient accuracy. This means that the
Taylor relation between the diffusion coefficient D(t)
and the Lagrangian velocity correlation function VL(t)
can be applied in the case of trapped particles under
consideration
0
( ) ( ) .
t
LD t V d (5)
The Lagrangian velocity correlation function ( )LV t can
be given through the Eulerian correlation function
2( )EV r , which for a frozen uniform isotropic field
depends only on a distance between two points, and the
transition probability
2
2
( ) ( , ; ,0) ( ),
( ) ( ) ( ) ( ) ( ) .
L E
E x x y y
V t d W t V r
V r V V V V
r r 0
r 0 r 0
(6)
To calculate the Lagrangian correlation function
VL(t) according to Eq. (6) we have to know the solution
of Eqs. (3). As far as it is unknown we need some
assumption to establish relation between the Eulerian
and Lagrangian correlation functions.
To formulate this assumption we were guided by the
following considerations. In the lowest zero
approximation a transition from the Eulerian correlation
function to the Lagrangian one may be performed by
coordinate transformation to the reference frame of free
particle. In the problem under consideration in absence
of electric field there is no drift motion, so the transition
probability is zero. Nevertheless, we will use this
analogy taking into account that a particle displacement
in the lowest order of approximation is caused by a
random field.
Using Eqs. (3), (5), and (6) we obtain for the second
moment of the transition probability
2r
2
0
2 ( ) .
t
r D t dt (7)
The closure proposed for the problem is given by the
relation
2( ) ( ( ) )L EV t V r t . (8)
Eqs. (5), (7), and (8) makes a closed set of equations
and can be solved numerically. A satisfactory
quantitative agreement of the results obtained from this
approach and direct simulation of particle motion is
shown in papers [2, 3].
2. SUBENSEMBLES
The method can be improved by more detailed
description of particle motion. For this we take into
account that their motion governed by Eq. (1) occurs
along the equipotential lines (r) = const. Simulation
shows a specific behavior of different groups of
particles which belong to equipotential surfaces of
different levels [3]. The difference is noticeable between
individual particle trajectories; they are shorter at peaks
or hollows of potential and longer in valleys. As well a
statistical analysis shows that the mean square
displacement of particles in subensembles grows, and
after some time comes to saturation. Distinction
ISSN 1562-6016. ВАНТ. 2016. №6(106) 79
between subensembles is manifested by difference in
saturation times of mean square displacement and also
its levels. Kurtosises of particle displacement in
subensembles are different as well. Along with this, no
specific behavior for groups of particles with different
initial velocities was found in numerical simulation.
Fig. 1. The Lagrangian velocity correlation function
found from numerical simulation (NE), in the basic
model without splitting on subensembles (MA), and in
the model with subensembles (SubEns)
Amplitude of the partial Euler correlation function in
each subensemble is determined by the level of the
equipotential surface. The contribution of subensembles
to statistical characteristics of the whole ensemble of
particles is proportional to their weights. According to
the central limit theorem distribution of particle over
subensembles is Gaussian. Direct numerical test shows
that for finite number of particles in numerical
simulation the Gaussian distribution over initial values
of potential is a good approximation.
It should be noted that the form of Lagrangian
correlation functions obtained from direct simulation
show diversity for those subensembles which are
characterized by a large absolute value of a potential,
close to the maximum. These subensembles consist
from strongly trapped particles which are moving on
contour lines of small size. Thus because of a small
dispersion and relatively small number of strongly
trapped particles distinction of their contribution can be
neglected.
3. RESULTS
The random field (r) in numerical simulation was
taken as a superposition of harmonics with Nk wave
numbers oriented over N directions, and with random
phases ai and aj
1 1
( ) ( , ) cos cos .
k
NN
i i j i j
i j
r k r
r (9)
The wave intensity is distributed over wave numbers
according to
2 2
2
2
0 max e
2
xp .
k
i
i
k
k k
k
N N
Then corresponding Eulerian correlation function is of
the form
2
0 0
22 1
exp(( ),) )( ) (
8
p kI p p r r 0 , (10)
where I0 is the modified Bessel function. As it was
mentioned the distribution of amplitude φ0 over
subensembles is Gaussian.
Velocity correlation function is obtained by double
differentiation of the potential correlation function over
coordinates in a two-dimensional space
2( ) ( ) ( )EV r r 0 . (11)
Eqs. (10), (11) gives the Eulerian velocity correlation
function for Eqs. (5), (7), (8). This system of equations
was solved numerically. In results the Lagrangian
velocity correlation function, the running diffusion
coefficient and evolution of the particle mean square
displasement were obtained. On the other hand the
Lagrangian velocity correlation function can be found
from a direct simulation of particle motion governed by
Eqs. (1), (9).
Fig. 2. The same in the other scale
In Fig. 1 the Lagrangian velocity correlation functions
found from numerical simulation, in the basic model
without splitting on subensembles, and in the model
which account for specific particle motion in
subensembles are compared.
The effect of particle trapping observed in numerical
simulation is reflected by the negative values of the
Lagrangian correlation function on a large time interval.
As could be seen the correlation decreases in time but
remains finite and tends slowly to zero. In particular
realization of a frozen field particle motion is
completely deterministic, thus the correlation time goes
to infinity. These features are reflected by the basic
model; however it gives an exaggerated negative value
of the velocity correlation function. Calculations based
on the subensemble concept improve the quantitative
agreement with the results of numerical simulation as it
is shown in different scale in Fig. 2.
80 ISSN 1562-6016. ВАНТ. 2016. №6(106)
The proposed model allows generalization to a
variable random field with a finite correlation time. It
also can be expanded to account for the effects of the
finite Larmor radius.
ACKNOWLEDGЕMENTS
The work is supported by the STCU project 6060.
REFERENCES
1. M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu.
Diffusion with intrinsic trapping in two-dimensional
incompressible stochastic velocity fields // Phys. Rev. E.
1998, v. 58, p. 7359-7368.
2. R. Balescu. Aspects of Anomalous Transport in
Plasmas. IOP Publ., 2005.
M. Vlad, F. Spineanu. Trajectory structures and
transport // Phys. Rev. E. 2004, v. 70, р. 056304.
3. T. Hauff and F. Jenko. Turbulent E×B advection of
charged test particles with large gyroradii // Phys.
Plasm. 2006, v. 13, p. 102309.
4. V.I. Zasenko, A.G. Zagorodny, O.M. Chernyak.
Diffusion in a frozen random velocity field
// Ukr. J. Phys. 2011, v. 56, p. 1007-1010.
5. O.M. Cherniak. Particle's diffusion in a two-
dimensional random velocity field // Ukr. J. Phys. 2015,
v. 60, p. 1196-1210.
6. R.H. Kraichnan. Diffusion by a random velocity field
// Phys. Fluids. 1970, v. 13, p. 22-31.
Article received 10.10.2016
КОНЦЕПЦИЯ ПОДАНСАМБЛЕЙ В ДВУМЕРНОЙ МОДЕЛИ ПЕРЕНОСА
ЗАМАГНИЧЕННЫХ ЧАСТИЦ
В.И. Засенко, А.Н. Черняк, А.Г. Загородний
Рассмотрен перенос замагниченных частиц под действием случайного замороженного изотропного
электростатического поля. Из-за бесконечно большого времени корреляции поля и захвата частиц эта
проблема имеет особый интерес для проверки замыкания статистических уравнений. Аналитический
подход, развитый нами ранее, дополнен концепцией подансамблей. Рассчитана лагранжева корреляционная
функция скорости в дрейфовом приближении. Для проверки достоверности аналитического метода его
предсказания сравниваются с результатами прямого численного моделирования. Показано, что
использование концепции подансамблей улучшает количественное согласование результатов.
КОНЦЕПЦІЯ ПІДАНСАМБЛІВ У ДВОВИМІРНІЙ МОДЕЛІ ПЕРЕНЕСЕННЯ ЗАМАГНІЧЕНИХ
ЧАСТИНОК
В.І. Засенко, О.М. Черняк, А.Г. Загородній
Розглянуто перенесення замагнічених частинок під дією випадкового замороженого ізотропного
електростатичного поля. Через нескінченно довгий час кореляції поля та захоплення частинок ця проблема
має особливий інтерес для перевірки замикання статистичних рівнянь. Аналітичний підхід, розвинутий
нами раніше, доповнено концепцією підансамблів. Розраховано лагранжеву кореляційну функцію швидкості
в дрейфовому наближенні. Для перевірки достовірності аналітичного методу його передбачення
порівнюються з результатами прямого числового моделювання. Показано, що використання концепції
підансамблів поліпшує кількісне узгодження результатів.
http://scitation.aip.org/content/aip/journal/pop/13/10/10.1063/1.2360173
http://scitation.aip.org/content/aip/journal/pop/13/10/10.1063/1.2360173
|