Finite larmor radius effects on turbulent transport of test-particles
A test-particle approach to study transport processes in two-dimensional random electric field is proposed. Despite such approach is not as complete as self-consistent one it allows a better control of problem parameters and makes results more tractable. A frozen electric field is considered. Beca...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2016
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nasplib_isofts_kiev_ua-123456789-1153262025-02-09T22:26:23Z Finite larmor radius effects on turbulent transport of test-particles Влияние конечного ларморовского радиуса на турбулентный перенос пробных частиц Вплив скінченного ларморівського радіусу на турбулентне перенесення пробних частинок Cherniak, O.M. Zasenko, V.I. Zagorodny, A.G. Basic plasma A test-particle approach to study transport processes in two-dimensional random electric field is proposed. Despite such approach is not as complete as self-consistent one it allows a better control of problem parameters and makes results more tractable. A frozen electric field is considered. Because of a strong particle trapping effect the problem is a difficult test for statistical methods. Earlier in our previous works particle transport was examined in a drift approximation; here we study finite Larmor radius effects on particle transport. Some methods to account for a finite Larmor radius are considered as generalization of our moment approximation. Results of analytical approximations and direct numerical simulation are compared, and most accurate method is found. The difference between dispersion of particle and gyrocentre displacement is discussed. Предложен метод изучения процессов переноса пробных частиц в двухмерном случайном електрическом поле. Несмотря на его неполноту в сравнении с самосогласованным описанием он позволяет лучше управлять параметрами задачи и делает результаты исследования более объяснимыми. Рассмотрено замороженное электрическое поле. Поскольку эффекты захвата частиц сильны, эта задача служит хорошим тестом для проверки статистических методов. Ранее перенос частиц был рассмотрен нами в дрейфовом приближении; в данной работе мы исследуем влияние конечного ларморовского радиуса на этот процесс. Рассмотрено несколько способов обобщения развитого ранее дрейфового приближения. Сравнение результатов, полученных на основе аналитических приближений и прямого численного моделирования, позволило определить наиболее точный метод учёта конечного ларморовского радиуса. Уделено внимание различию в дисперсии смещения частиц и соответствующих ведущих центров. Запропоновано метод дослідження процесів перенесення пробних частинок у двомірному випадковому електричному полі. Попри його неповноту в порівнянні із самоузгодженим описом він дозволяє краще керувати параметрами задачі, сприяє кращому розумінню результатів дослідження. Розглянуто заморожене електричне поле. Через сильний ефект захоплення частинок ця задача є добрим тестом для перевірки статистичних методів. Раніше перенесення частинок було розглянуто нами у дрейфовому наближенні; тут ми досліджуємо вплив скінченного ларморівського радіусу на цей процес. Розглянуто декілька шляхів узагальнення розвинутого раніше дрейфового наближення. Порівняння результатів, отриманих на основі аналітичних наближень та прямого числового моделювання, дозволило визначити найбільш точний метод врахування скінченного ларморівського радіусу. Приділено увагу різниці в дисперсії зміщення частинок та відповідних ведучих центрів. This work is supported by STCU project 6060. 2016 Article Finite larmor radius effects on turbulent transport of test-particles / O.M. Cherniak, V.I. Zasenko, А.G. Zagorodny // Вопросы атомной науки и техники. — 2016. — № 6. — С. 96-99. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.65.Cc https://nasplib.isofts.kiev.ua/handle/123456789/115326 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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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 Cherniak, O.M. Zasenko, V.I. Zagorodny, A.G. Finite larmor radius effects on turbulent transport of test-particles Вопросы атомной науки и техники |
| description |
A test-particle approach to study transport processes in two-dimensional random electric field is proposed.
Despite such approach is not as complete as self-consistent one it allows a better control of problem parameters and
makes results more tractable. A frozen electric field is considered. Because of a strong particle trapping effect the
problem is a difficult test for statistical methods. Earlier in our previous works particle transport was examined in a
drift approximation; here we study finite Larmor radius effects on particle transport. Some methods to account for a
finite Larmor radius are considered as generalization of our moment approximation. Results of analytical
approximations and direct numerical simulation are compared, and most accurate method is found. The difference
between dispersion of particle and gyrocentre displacement is discussed. |
| format |
Article |
| author |
Cherniak, O.M. Zasenko, V.I. Zagorodny, A.G. |
| author_facet |
Cherniak, O.M. Zasenko, V.I. Zagorodny, A.G. |
| author_sort |
Cherniak, O.M. |
| title |
Finite larmor radius effects on turbulent transport of test-particles |
| title_short |
Finite larmor radius effects on turbulent transport of test-particles |
| title_full |
Finite larmor radius effects on turbulent transport of test-particles |
| title_fullStr |
Finite larmor radius effects on turbulent transport of test-particles |
| title_full_unstemmed |
Finite larmor radius effects on turbulent transport of test-particles |
| title_sort |
finite larmor radius effects on turbulent transport of test-particles |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2016 |
| topic_facet |
Basic plasma |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/115326 |
| citation_txt |
Finite larmor radius effects on turbulent transport of test-particles / O.M. Cherniak, V.I. Zasenko, А.G. Zagorodny // Вопросы атомной науки и техники. — 2016. — № 6. — С. 96-99. — Бібліогр.: 4 назв. — англ. |
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Вопросы атомной науки и техники |
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ISSN 1562-6016. ВАНТ. 2016. №6(106)
96 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2016, № 6. Series: Plasma Physics (22), p. 96-99.
FINITE LARMOR RADIUS EFFECTS ON TURBULENT TRANSPORT OF
TEST-PARTICLES
O.M. Cherniak, V.I. Zasenko, А.G. Zagorodny
Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
E-mail: anchernyak@bitp.kiev.ua
A test-particle approach to study transport processes in two-dimensional random electric field is proposed.
Despite such approach is not as complete as self-consistent one it allows a better control of problem parameters and
makes results more tractable. A frozen electric field is considered. Because of a strong particle trapping effect the
problem is a difficult test for statistical methods. Earlier in our previous works particle transport was examined in a
drift approximation; here we study finite Larmor radius effects on particle transport. Some methods to account for a
finite Larmor radius are considered as generalization of our moment approximation. Results of analytical
approximations and direct numerical simulation are compared, and most accurate method is found. The difference
between dispersion of particle and gyrocentre displacement is discussed.
PACS: 52.65.Cc
INTRODUCTION
A drift approximation is widely used to describe
transport of particles in plasmas. It is often accepted that
the drift approximation gives a main contribution to
transport, and the effects of finite Larmor radius can be
taken into account as corrections. Along with this,
account for exact motion of particles can influence the
results significantly.
Recently the decorrelation trajectory method was
proposed to study two-dimensional diffusion of particles
in constant magnetic and random electric fields [1, 2].
In our work [3] an alternative approach to this problem
in drift approximation was developed. In the paper [4]
the decorrelation trajectory method was compared with
our approach and some advantages of latter were
shown. Here we generalize our moment approximation
to take into account finite Larmor radius effects. Few
different procedures of gyroaveraging are considered.
Direct numerical simulation of particle motion in
constant magnetic and random electric field is
performed. Statistical characteristics of particle
ensemble obtained in simulation are used to check the
predictions of analytical approaches. The most effective
gyroaveraging procedure is found.
1. MODEL
We consider motion of test particles in a constant
magnetic field perpendicular to a frozen random electric
field given by the potential
2exp cos 2i i iA χ κ χ , (1)
where A is a normalized amplitude
3/2
0 max /A N N , (2)
and set of Nr = N N wave vectors are
max / cos 2 / ,sin 2 /i n N m N m N κ . (3)
Here n = 1,…, N, m = 1,…, N, and {i} is the set of
random phases that determines a realization of random
potential (3).
The Eulerian correlation function of potential (1) in
a laboratory frame is
1 2 2 2 2
02 exp / 2 / 2EC I
χ χ χ . (4)
Corresponding correlation function of velocity is
obtained as derivative of the potential correlation
function (4)
2 2 2 2/ /
d d
E E
x yC C χ χ . (5)
Particle motion is governed by equations for coordinate
of gyrocentre d and gyroradius
/ / , , ,i ij d djd d i j x y χ ρ , (6)
0/ / 2 /i ij d dj jd d χ ρ , (7)
where xy = - yx = 1. The solutions of 2Nr
equations (6, 7) are found numerically by using the
Runge-Kutta method of the 5-th order. Then obtained
trajectories are averaged over Nr realizations, and a
mean square displacement is calculated. Further it is
compared with a prediction of the analytical models.
Parameters of numerical models were the following:
max = 2, dimensionless amplitude of potential 0 = 0.1,
and N = N N = 1440 (N = 20, N = 72).
2. ANALYTICAL APPROXIMATION
The analytical approximation is based on the Taylor
relation
1
2
Ld
D d C
d
, (8)
that gives a diffusion coefficient D() and mean square
displacement () as an integral over time of the
Lagrangian correlation function of velocity components
ISSN 1562-6016. ВАНТ. 2016. №6(106) 97
along particle trajectories CL
(). This correlation
function is unknown and should be derived from the
Eulerian correlation function (5). There is no
mathematically direct way to obtain it in general case,
so various approximations are used.
We start from the moment approximation for a drift
motion proposed in [3]. It was analyzed and compared
with the decorrelation trajectory method [1, 2] in [4].
According to the moment approximation the relation
between the Eulerian and the Lagrangian correlation
functions for an isotropic field is given by
,
d d d d
L E
i iC C Χ . (9)
Here we generalize our previous approach [3, 4] to
account for a finite Larmor radius. Two methods are
considered: averaging over the Larmor gyration of the
Eulerian correlation function (5)
1
02 exp
d dd d
E A E
d dC d i C J
χ κ κχ κ ,(10)
and averaging of the random field (1) that gives for the
Eulerian correlation function (5) the different expression
1 2
02 exp
d dd d
E B E
d dC d i C J
χ κ κχ κ .(11)
These expressions were used in the works [1, 2] in
application to the decorrelation trajectories method.
Combining the assumption (9) with equations
(4, 5, 8) and (10) or (11) we obtain the final equation for
a mean square displacement in the moment
approximation in a form
,2 2/
d dd d
E A Bd d C . (12)
The correlation functions (10, 11) are calculated by
means of numerical integration. Expansion of Bessel
function are taken as
2 2 4 4 6 6
0( ) 1 / 4 / 64 / 2304 ...J ,
for < 1, or asymptotic
0 ( ) 1 /J ,
for > 1. Results of these approximations are
compared with direct numerical simulation in the next
section.
3. RESULTS OF SIMULATION
The results are presented in Figs. 1-6. The mean
square displacement calculated from numerical
simulation (NS) is obtained for ensemble of Nr = 104
realizations of random potential (1). Results of
calculation are given for four values of initial Larmor
radius (0) = 0, 0.1, 1, 10.
In Fig. 1 temporal evolution of mean square
displacement of gyrocentres obtained from numerical
simulation for a various initial Larmor radius is shown.
The results for initial radius (0) = 0.1 are found to be
very close to (0) = 0, the difference is of the order of
fluctuations. For larger initial radius difference becomes
significant, thus the mean square displacement for
(0) = 10 drops almost in four times in compare to
(0) = 0.
Fig. 1. Mean square displacement of gyrocentres
obtained from numerical simulation (NS, Nr = 104)
for initial Larmor radius (0) = 0, 0.1, 1, 10
Fig. 2. Mean square displacement of particle
gyrocentres for initial Larmor radius (0) = 0.
Numerical simulation (NS, Nr = 104), and the moment
approximation (MA)
Fig. 3. Mean square displacement of gyrocentres
for (0) = 0.1. Numerical simulation (NS, Nr = 104)
and analytical model (MA). Different methods
of account for Larmor radius effect give
the same results
98 ISSN 1562-6016. ВАНТ. 2016. №6(106)
The comparison of mean square displacements of
gyrocentres found from numerical simulation and
using the moment approximation (MA) for initial
Larmor radius (0) = 0 is demonstrated in Fig. 2. The
moment approximation recovers the same subdiffusive
behaviour as direct numerical simulation with a
quantitative agreement.
The results for a small initial Larmor radius (0) =
0.1 are shown in Fig. 3. The difference with the results
for radius (0) = 0 (see Fig. 2) is small, that is in
agreement with the results given in Fig. 1. All
approximations of equation (10) – A, and equation
(11) – B (using series (Exp) and numerical integration
(NI)) give a similar curves; difference between
methods of gyroaveraging is negligible for a small
gyroradii.
Mean square displacement of gyrocentres for
gyroradius (0) = 10 is shown in Fig. 4. Temporal
evolution of mean square displacement obtained by the
moment approximation B (both with asymptotic (Asm)
and by numerical integration (NI)) is found to be in
quantitative agreement with direct numerical
simulation. On the contrary the results obtained with
approximation A, Eqs. (10), (by numerical integration
(NI) and with asymptotic (Asm)) are inconsistent with
results of direct numerical simulation in a range of
large gyroradii.
We may conclude that the moment approximation
with gyroaveraging based on equation (11) by means
of numerical integration, series expansion and
asymptotyc – quantitatively recovers the evolution of
mean square displacement of gyrocentres obtained
from direct numerical simulation. And thus it can be
considered as the sufficiently accurate method to
account for finite Larmor radius effects.
Now we switch from consideration of gyrocenters
d statistics to examination of exact particle
trajectories = d + . In Fig. 5 the mean square
displacement of particles obtained from direct
numerical simulation for Nr = 104 realizations is given.
It demonstrates a difference between gyrocentre (see
Fig. 1) and particle trajectory statistics. For a small
initial Larmor radius, (0) < 1, there is no significant
difference between dispersion of gyrocentres and exact
particle trajectories. But with increase of the initial
radius (0) > 1 the difference becomes noticeable. For
(0) = 10 initial evolution of dispersion of guiding
centre and exact particle position is completely
different. The reason is a contribution from a mean
square displacement of Larmor radius, its temporal
evolution is shown in Fig. 6. It saturates with time: for
a small initial Larmor radius a saturation value is
negligible. Whether initial value is not small it grows
to large magnitude. For any initial value Larmor radius
reaches a saturation value within a finite time interval.
Consequently a mean square displacement of particle
trajectories would be shifted against a curve for
gyrocentre dispersion.
Fig. 4. Mean square displacement of gyrocentres for
(0) = 10. Numerical simulation (NS, Nr = 104) and
analytical model (MA) with approximation given by
Eq.(10) – (A), and Eq. (11) – (B); (Asm) – analytical
asymptotic, (NI) – numerical integration
Fig. 5. Mean square displacement of particles
trajectories obtained by numerical simulation (NS,
Nr = 104) for initial Larmor radius (0) = 0, 0.1, 1, 10
Fig. 6. Mean square displacement of Larmor radius
obtained by numerical simulation (NS, Nr = 104)
for initial Larmor radius (0) = 0, 0.1, 1, 10
CONCLUSIONS
Temporal evolution of a mean square displacement
of gyrocentres found from the direct numerical
simulation and from the moment approximation based
on a gyroveraging of random potential were compared.
ISSN 1562-6016. ВАНТ. 2016. №6(106) 99
For initial Larmor radius in wide range, (0) = 0, 0.1,
1, 10, analytical method based on Eq. (11) gives a
satisfactory quantitative agreement. On the contrary the
moment approximation using other gyroaveraging of
correlation function (10) is inconsistent with results of
direct numerical simulation for large gyroradii.
The numerical simulation shows that a mean square
displacement of particles trajectories is shifted from
mean square displacement of gyrocentres by a value of
Larmor radius dispersion. For any initial value of
Larmor radius its dispersion is saturated within a finite
time interval.
ACKNOWLEDGEMENTS
This work is supported by STCU project 6060.
REFERENCES
1. M. Vlad and F. Spineanu, S-I Itoh, M. Yagi, K. Itoh.
Turbulent transport of ions with large Larmor radii
// Plasma Phys. Cntrl. Fusion. 2005, v. 47, p. 1015-
1029.
2. T. Hauff and F. Jenko. Turbulent ExB advection of
charged test particles with large gyroradii
// Plasma Phys. Cntrl. Fusion. 2006, v. 13, p. 102309-
102309-11.
3. 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.
4. O.M. Cherniak. Particle's diffusion in a two-
dimensional random velocity field // Ukr. J. Phys. 2015,
v. 60, p. 1196-1210.
Article received 05.10.2016
ВЛИЯНИЕ КОНЕЧНОГО ЛАРМОРОВСКОГО РАДИУСА НА ТУРБУЛЕНТНЫЙ ПЕРЕНОС
ПРОБНЫХ ЧАСТИЦ
А.Н. Черняк, В.И. Засенко, А.Г. Загородний
Предложен метод изучения процессов переноса пробных частиц в двухмерном случайном
електрическом поле. Несмотря на его неполноту в сравнении с самосогласованным описанием он позволяет
лучше управлять параметрами задачи и делает результаты исследования более объяснимыми. Рассмотрено
замороженное электрическое поле. Поскольку эффекты захвата частиц сильны, эта задача служит хорошим
тестом для проверки статистических методов. Ранее перенос частиц был рассмотрен нами в дрейфовом
приближении; в данной работе мы исследуем влияние конечного ларморовского радиуса на этот процесс.
Рассмотрено несколько способов обобщения развитого ранее дрейфового приближения. Сравнение
результатов, полученных на основе аналитических приближений и прямого численного моделирования,
позволило определить наиболее точный метод учёта конечного ларморовского радиуса. Уделено внимание
различию в дисперсии смещения частиц и соответствующих ведущих центров.
ВПЛИВ СКІНЧЕННОГО ЛАРМОРІВСЬКОГО РАДІУСУ НА ТУРБУЛЕНТНЕ ПЕРЕНЕСЕННЯ
ПРОБНИХ ЧАСТИНОК
О.М. Черняк, В.І. Засенко, А.Г. Загородній
Запропоновано метод дослідження процесів перенесення пробних частинок у двомірному випадковому
електричному полі. Попри його неповноту в порівнянні із самоузгодженим описом він дозволяє краще
керувати параметрами задачі, сприяє кращому розумінню результатів дослідження. Розглянуто заморожене
електричне поле. Через сильний ефект захоплення частинок ця задача є добрим тестом для перевірки
статистичних методів. Раніше перенесення частинок було розглянуто нами у дрейфовому наближенні; тут
ми досліджуємо вплив скінченного ларморівського радіусу на цей процес. Розглянуто декілька шляхів
узагальнення розвинутого раніше дрейфового наближення. Порівняння результатів, отриманих на основі
аналітичних наближень та прямого числового моделювання, дозволило визначити найбільш точний метод
врахування скінченного ларморівського радіусу. Приділено увагу різниці в дисперсії зміщення частинок та
відповідних ведучих центрів.
|