Ions cross-B collisional diffusion and electromagnetic wave scattering
The calculation is presented of the averaged quadratic displacement of a collisional charged particle in a magnetic field. This calculation is used to obtain the statistical presentation of the electromagnetic field scattered by these particles. These results extend the previous calculations that we...
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nasplib_isofts_kiev_ua-123456789-816702025-02-23T18:18:30Z Ions cross-B collisional diffusion and electromagnetic wave scattering Tomchuk, B.P. Grésillon, D. Нелинейные процессы The calculation is presented of the averaged quadratic displacement of a collisional charged particle in a magnetic field. This calculation is used to obtain the statistical presentation of the electromagnetic field scattered by these particles. These results extend the previous calculations that were restricted to non-magnetized particles (Ornstein equation, Einstein diffusion, etc.). In addition this calculation foresees effects that are absent of the Ornstein equation: a modulation of the averaged quadratic displacement function at the cyclotron frequency and a maximum of the Cross-B diffusion coefficient when the cyclotron frequency is equal to the collision frequency (Bohm diffusion). 2000 Article Ions cross-B collisional diffusion and electromagnetic wave scattering / B.P. Tomchuk and D. Grésillon // Вопросы атомной науки и техники. — 2000. — № 1. — С. 205-208. — Бібліогр.: 4 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/81670 533.9 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы Нелинейные процессы |
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Нелинейные процессы Нелинейные процессы Tomchuk, B.P. Grésillon, D. Ions cross-B collisional diffusion and electromagnetic wave scattering Вопросы атомной науки и техники |
| description |
The calculation is presented of the averaged quadratic displacement of a collisional charged particle in a magnetic field. This calculation is used to obtain the statistical presentation of the electromagnetic field scattered by these particles. These results extend the previous calculations that were restricted to non-magnetized particles (Ornstein equation, Einstein diffusion, etc.). In addition this calculation foresees effects that are absent of the Ornstein equation: a modulation of the averaged quadratic displacement function at the cyclotron frequency and a maximum of the Cross-B diffusion coefficient when the cyclotron frequency is equal to the collision frequency (Bohm diffusion). |
| format |
Article |
| author |
Tomchuk, B.P. Grésillon, D. |
| author_facet |
Tomchuk, B.P. Grésillon, D. |
| author_sort |
Tomchuk, B.P. |
| title |
Ions cross-B collisional diffusion and electromagnetic wave scattering |
| title_short |
Ions cross-B collisional diffusion and electromagnetic wave scattering |
| title_full |
Ions cross-B collisional diffusion and electromagnetic wave scattering |
| title_fullStr |
Ions cross-B collisional diffusion and electromagnetic wave scattering |
| title_full_unstemmed |
Ions cross-B collisional diffusion and electromagnetic wave scattering |
| title_sort |
ions cross-b collisional diffusion and electromagnetic wave scattering |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2000 |
| topic_facet |
Нелинейные процессы |
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https://nasplib.isofts.kiev.ua/handle/123456789/81670 |
| citation_txt |
Ions cross-B collisional diffusion and electromagnetic wave scattering / B.P. Tomchuk and D. Grésillon // Вопросы атомной науки и техники. — 2000. — № 1. — С. 205-208. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT tomchukbp ionscrossbcollisionaldiffusionandelectromagneticwavescattering AT gresillond ionscrossbcollisionaldiffusionandelectromagneticwavescattering |
| first_indexed |
2025-11-24T08:23:26Z |
| last_indexed |
2025-11-24T08:23:26Z |
| _version_ |
1849659339997970432 |
| fulltext |
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ 2000. №1.
Серия: Плазменная электроника и новые методы ускорения (2), с. 205-208.
205
UDK 533.9
IONS CROSS-B COLLISIONAL DIFFUSION AND ELECTROMAGNETIC
WAVE SCATTERING
B.P. Tomchuk and D. Grésillon
Laboratoire de Physique des Milieux Ionisés, Ecole Polytechnique, Palaiseau, France
bogdan.tomchuk@polytechnique.fr
dominique.gresillon@polytechnique.fr
The calculation is presented of the averaged quadratic displacement of a collisional charged particle in a magnetic
field. This calculation is used to obtain the statistical presentation of the electromagnetic field scattered by these parti-
cles. These results extend the previous calculations that were restricted to non-magnetized particles (Ornstein equation,
Einstein diffusion, etc.). In addition this calculation foresees effects that are absent of the Ornstein equation: a modula-
tion of the averaged quadratic displacement function at the cyclotron frequency and a maximum of the Cross-B diffusion
coefficient when the cyclotron frequency is equal to the collision frequency (Bohm diffusion).
Introduction
Collective scattering (CS) is a common diagnostics
of plasma turbulence that provides information on
plasma irregularities and motions. This information is
especially valuable for magnetized ionosphere as well as
for fusion plasmas. In some cases, plasma cross-B diffu-
sion coefficients can be obtained from the CS signal
spectrum analysis (1). The information given by the CS
signal, however, is constituted not only of macroscopic
plasma parameters but also of some particle macro-
scopic parameters at a given spatial scale. These macro-
scopic features are expected when the observed scale is
of the order of the ion Larmor radius.
This analysis is the objective of the present contri-
bution. A theoretical examination of the CS physical
process leads to a calculation of the scattering signal
issued from a magnetized plasma with taking into ac-
count effects of collisions with neutrals.
Magnetized ion motion and the scattering signal
If we have a set of particles exposed to an incident
electromagnetic field, the scattered electromagnetic field
is sum of individual field scattered from each elementary
particles. The scattering experiment geometry defines a
scale. When this scale is larger then the Langmuir length
Dλ the elementary building blocks are the dressed ions.
The total scattered field is the sum of fields scattered by
all of observed dressed particles:
( ) ( ) ( )0
1
, 0, j
N
ik r t
s i
j
rE R t E t e
R
⋅
=
= ∑
! !! !
, (1.1)
where ( )0,iE t
!
is the incident field complex amplitude
at a reference origin near to the position of the scattering
particle jr! , 0r is the classical Thomson radius, R is
the distance between scattering position and observer,
and k
!
is the analyzing wave-vector which appear as
difference between scattered wave-vector ik
!
and inci-
dent wave-vector sk
!
:
s ik k k= −
! ! !
. (1.2)
We assumed 1Dkλ " . To form the field at time
t τ+ , the particle trajectories are spit in two terms: the
mean fluid displacement ( )( ),r t τ∆
! ! (defined as the dis-
placement averaged over a small scale cloud of particles)
and the additional individual particle displacement ( ),i tδ τ
!
(relative to the center of mass of our local ''cloud''):
( ) ( ) ( )( ) ( ), ,j j j jr t r t r t tτ τ δ τ+ = + ∆ +
!!! ! ! (1.3)
The product of both fields at time t τ+ and at ini-
tial time t provides the scattered field correlation func-
tion. If statistical independence is assumed between den-
sity, macroscopic and microscopic displacements, the
correlation function will take the following form:
( ) ( ) ( )
2
0 0, ik ik
i
rC E t NS k e e
R
τ τδτ ⋅∆ ⋅≅
! ! !!!! (1.4)
where ( )S k
!
is the “form factor”.
( )
2
0
1 j
N
ikr
j
S k e
N
−
=
= ∑
!!!
(1.5)
The next two terms are the “statistical characteris-
tics” of the macroscopic displacement τ∆
!
and of the
microscopic random displacement, both occurred during
time τ along k
!
. If these displacements are gaussian
random processes:
2 2
,
2 2
,
1
2
1
2
k
k
kik
kik
e e
e e
ττ
ττ
δδ
− ∆− ⋅∆
−− ⋅
=
=
!
!
! !
! !
(1.6)
The macroscopic random mean square (turbulent
motion) can usually be described by the Ornstein
dispersion (see e.g. (2)):
2
, 2 1 c
t
M ck t
c
tD e ττ
τ
−
∆ = − +
! , (1.7)
where MD and cτ are respectively the turbulent fluid
diffusion coefficient and correlation time.
For large time t , the Ornstein dispersion tends to the
Einstein diffusion, proportional to the time.
206
2 2 MD t∆ # (1.8)
Consequently the “characteristic” (1.6) is a decreas-
ing exponential at the inverse time rate 2
Mk D .
Ion motion in a plane perpendicular to a
uniform magnetic field
The scattering electric field time correlation (1.4)
is thus made of the product of two different “statisti-
cal characteristics”. One corresponds to the mean
motion, and the second one to the microscopic mo-
tion. The microscopic motion is the cyclotron motion
at angular frequency ω , with random ion-neutral
collisions at frequency ν . Using an appropriate mi-
croscopic, individual particle model, the microscopic
diffusion can be calculated (3).
The motion of a charged particle in a magnetic filed
between two collisions is along a Larmor circular orbit
(See Figure 1). A sample trajectory in Figure 1 is shown
for a positively charged particle. The mean square dis-
placement does not depend on particle charge.
Figure 1. Projection of a charged particle sample tra-
jectory on a surface perpendicular to the magnetic field.
Bold line — trajectory; positions 0,1,2,3,4,5,6 — show
points of collision of our particle with neutrals.
For the uniform magnetic filed, charge particle tra-
jectory is circle of Larmor radius:
uρ
ω
= , (1.9)
where u is the particle velocity and ω is the cyclo-
tron frequency.
The cyclotron motion is interrupted by collision with
a neutral atom. After collision the particle continues a
rotation at cyclotron frequency but around a new center
of rotation and with a different radius.
The particle motion between collisions 1i − and i
is shown in Figure 2. It is a circular motion of cyclo-
tron radius iρ . The projection of this displacement
along a given direction (perpendicular to the mag-
netic field) from position 1i − to position, can be
written as:
( ) ( )
( )( ) ( )
1
1 1 1
cos cos
cos cos
i i i i i
i i C i i it t
δ ρ ϕ ρ ϕ
ρ ϕ ω ϕ
−
− − −
= −
= − − −
(1.10)
Figure 2. Sample of trajectory (projection on face per-
pendicular to magnetic filed) between two collisions.
The total displacement after N collision is the sum
of elementary displacements:
1
1
N
N i
i
δ
+
=
∆ = ∑ , (1.11)
when the number of collisions is large, a statistical
average has to be made over the velocity probabilities
distribution (or equivalently over the Larmor radius
iρ probability distribution) and over the collision
phase angle iϕ . We assume that radius length before
and after collision not correlated and that after colli-
sion phases probability distribution iϕ is uniform
between 0 and 2π .
Then the mean square of (1.11) can be written as:
( )
( )
2 2 2
2 12 2 2 2 1
4 sin ...
2
4 sin 4 sin .
2 2
C N
N
C C
t t
t t t
ω
ρ
ω ωρ ρ
−
∆ = + +
− + +
(1.12)
The average over the different possible number of
collisions can be calculated using the probability func-
tion of a number n of collision in a time interval t :
n t
nP e νν −= , (1.13)
where ν is the collision frequency.
207
The mean of collision number takes the follow-
ing form:
( )2 1
12 2 2
1
0 10 0
4 sin
2
tt N
C i iN vt
Nt
N i
t t
v e dt dt
ω
ρ
∞ +
−−
= =
−
∆ =
∑ ∑∫ ∫$
(1.14)
After integration and summation, the microscopic
mean square random motion (diffusion) normalized to
the root mean square ion cyclotron radius ρ across the
magnetic field is found as:
( )
( ) ( )
22
2 22
2 2 22
2 2 2
1 cos 2 sin
22 1
1 1 1
C CC C C
vtt
C C C
t t
v vv t e
v v v
ω ωω ω ω
ν
ω ω ωρ
−
− − ∆ = + − +
+ + +
(1.15)
The resulting diffusive motion (normalized to the
mean cyclotron radius) is a function of the time t , of
the cyclotron frequency Cω and of the collision fre-
quency ν . This diffusive motion contains a modu-
lated part at the cyclotron frequency that is dumped
at the collision rate.
Variation of this mean square motion as a function of
time (normalized to the collision frequency) and of the
collision rate ν (normalized to the collision frequency)
is shown in Figure 3.
Figure 3. Variation of mean square motion as a func-
tion of time and magnetic field intensity.
Both variables: time t and cyclotron frequency cω
(magnetic field) are normalized to the
collision frequency ν
This figure clearly shows that for fixed value of
magnetic field the initial part of time variation is
significantly modulated at the cyclotron frequency.
This modulation dumped as the time increases
above the collision time ( 1tν > ) and at longer time
the mean square motion recovers the classical be-
havior of linear incense at the Einstein’s rate for
Brownian motion:
2
2 2 2v
t
t Dt
v
σ∆ = = (1.16)
Another presentation of function (1.15) is
shown in Figure 4 where time and collision fre-
quency are normalized to the cyclotron. This nor-
malization used in Figure 4 is especially appropri-
ate for auroral plasmas. In this case, the magnetic
field is approximately uniform, while the collision
frequency is a rapidly varying function of altitude.
In looking at Figure 4, the collision frequency ν
can be thought as the inverse altitude (large colli-
sion frequency are down earth, low to zero colli-
sion frequency are in the high ionosphere). In
laboratory plasma instead the collision frequency
is uniform but not the magnetic field and this is
why the normalization choice of Figure 3 might be
more appropriate.
In the no-collision case, the diffusion is seen to
oscillate and come back to zero each cyclotron
period. This modulation is kept as long as the col-
lision rate is small. For large collision frequency
instead, the diffusion behavior is that of Brownian
motion, linearly increasing with time.
Figure 4. Variation of diffusion as a function of time
and collision frequency.
Both time t and collision frequency ν variable are
normalized to the cyclotron frequency cω
A maximum of the displacement rate of increase
with time is observed for the case where the collision
frequency ν is equal to the cyclotron angular frequency
Cω : Cν ω= . This rate of increase is known as the
Bohm diffusion.
The similar result could be obtained with a more
elaborate Fokker-Plank model of collision.
Microscopic “characteristic function” and
scattered electric field time correlation
The result (1.15) obtained for microscopic mean
square random motion could be used in (1.6) to ob-
tain the “characteristic function” and thus the scat-
tered electric field time correlations functions. This
time correlation is:
208
( ) ( )
( ) ( )
22
222 2
2 2 2
2 2 2
1 cos 2 sin
2exp 1
1 1 1
r
C CC C C
ik t vt
C C C
t t
v vve k t e
v v v
ω ωω ω ω
ρ ν
ω ω ω
− ⋅∆ −
− −
= − + − +
+ + +
! !
(1.17)
Figure 5. Variation of microscopic part of the scat-
tered electric field correlation function with time and
collision frequency (both normalized to the cyclotron
frequency) for different observed scale kρ
This calculated time correlation functions is shown in
Figure 5 for several different value of kρ , as a function
of time Ctω and of the collision frequency Cν ω , for
cases where 0.5;1;1.5kρ = , i.e. when the observed
scale is of the order of 2π times the cyclotron radius.
These parameters are in the range observation done by HF
radar, of the backscattered electric field in auroral iono-
sphere. In these plasmas the atomic nitrogen Lamoure
radius is about 3m and the scattering wavelength is 10 to
15 meters. The height at witch the collision ion – neutral
is equal to the angular ion-cyclotron frequency is 100km,
at the upper border of the E-region of ionosphere.
For low collision rate, the correlation function is
seen to be modulated at the cyclotron frequency and
the modulation depth is larger for large kρ .
For higher collision frequencies particle forget
rapidly about its cyclotron characteristics and the
modulation disappears.
The fastest time decrease of the correlation func-
tion is observed when Cν ω= .
We should notice again this is not the “complete” scat-
tered electric field correlation function but only the “micro-
scopic” part of it. The other components in equation (1.4)
should also contribute. However some of microscopic func-
tion properties should still hold sins they may apply for
different characteristic time and scale. Namely, experiment
done for different values of the observation scale kρ
might show the modulation at the cyclotron frequency (Fig-
ure 5).
The modulation amplitude of the characteristic
function should be especially sensible for large k ,
where 1kρ > .
Conclusion
The microscopic mean square random motion of a
charged particle in a uniform magnetic field was calcu-
lated and used to obtain the microscopic part of the scat-
tered electric field time correlation function. The scattered
electric field is a common observations in natural (iono-
sphere) and fusion plasmas. Since the calculated micro-
scopic time correlation function is a multiplier of the total
correlation function, one might expect that some of its
features could be observed experimentally. The cyclotron
modulation should especially be expected when frequency
is small, i.e. for fusion plasma or for high altitude iono-
sphere. It also predict a maximum of the diffusion coeffi-
cient in the ionosphere at the E-layer were Cν ω= .
References
1. D. Grésillon, B. Cabrit, J.-P. Villain, C. Hanuise,
A. Truc, C. Laviron, P. Hennequin, F. Gervais,
A. Quemeneur, X. Garbet, J. Payan and P. Devynck Col-
lective scattering of electromagnetic wave and cross-B
plasma diffusion// Plasma Physics and Controlled Fusion
1992, vol.34, p. 1985-1991.
2. S. Chandrasekhar// Review of Modern Physics, 1943,
vol. 15, p. 1-89.
3. B.P. Tomchuk, PhD thesis, “Diffusion des particules
transversalement au champ magnétique
et diffusion exacerbée des ondes électromagnétiques”,
Ecole Polytechnique, January 2000.
4. PPJM. Schram, AG Sitenko, AG Zagorodniy.
Turbulent diffusion Influence on Large-Scale Fluctuations
in Plasmas // EPS Maastricht, 1999.
Laboratoire de Physique des Milieux Ionises, Ecole Polytechnique, Palaiseau, France
Introduction
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