Nonquasineutral current structures in plasmas with a zero net current
A nonquasineutral vortex structure with a zero net current is described that arises as a result of electron drift in
 crossed magnetic and electric fields, the latter being produced by charge separation on a spatial scale of about the
 magnetic Debye radius. In such a structure wit...
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| Veröffentlicht in: | Вопросы атомной науки и техники |
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| Datum: | 2006 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Zitieren: | Nonquasineutral current structures in plasmas with a zero net current / A.V. Gordeev // Вопросы атомной науки и техники. — 2006. — № 6. — С. 118-120. — Бібліогр.: 5 назв. — англ. |
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| author | Gordeev, A.V. |
| author_facet | Gordeev, A.V. |
| citation_txt | Nonquasineutral current structures in plasmas with a zero net current / A.V. Gordeev // Вопросы атомной науки и техники. — 2006. — № 6. — С. 118-120. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | A nonquasineutral vortex structure with a zero net current is described that arises as a result of electron drift in
crossed magnetic and electric fields, the latter being produced by charge separation on a spatial scale of about the
magnetic Debye radius. In such a structure with radius r ~ rB, the magnetic field maintained by a
drift current on the order of the electron Alfven current. The system with closed current that is considered in the present paper can also serve as a model of hot spots in the channel of a Z-pinch.
|
| first_indexed | 2025-12-07T18:36:26Z |
| format | Article |
| fulltext |
118 Problems of Atomic Science and Technology. 2006, 6. Series: Plasma Physics (12), p. 118-120
NONQUASINEUTRAL CURRENT STRUCTURES IN PLASMAS
WITH A ZERO NET CURRENT
A.V. Gordeev
Russian Research Centre “Kurchatov Institute”Moscow, Russia, e-mail: gordeev@dap.kiae.ru
A nonquasineutral vortex structure with a zero net current is described that arises as a result of electron drift in
crossed magnetic and electric fields, the latter being produced by charge separation on a spatial scale of about the
magnetic Debye radius ( )eB en4Br π=
r
. In such a structure with radius r ~ rB, the magnetic field maintained by a
drift current on the order of the electron Alfven current JAe =me c3/(2e) and can become as strong as
2
ee cmn4B π≅ . The system with closed current that is considered in the present paper can also serve as a model
of hot spots in the channel of a Z-pinch.
PACS: 52.25.Xz, 52.30.-q, 52.30.Ex, 52.55.-s
1. INTRODUCTION
In the resent years, investigations have been carried out
with the nonquasineutral current structures whose size
varies from a few microns in pinches to billions of
kilometers in cosmic space and in which charges are
separated on spatial scales of about the magnetic Debye
radius rB ~ B/(4 ene) and an electric field is generated due
to the Hall effect – the factor that set electrons into
relativistic drift motion [1-3]. An important feature of the
resulting quasi-equilibrium is the onset of crossed electric
and magnetic fields B~E
rr
. However, in the structures
considered theoretically and numerically in [1-3], the net
current in the quasi-equilibrium states under analysis was
nonzero. The question to be answered is then how laser
pulses or other extreme energy inputs (e.g., in Z-pinches)
can drive a nonzero net current in such isolated structures,
The obtained in the further investigations result that the
net current in such quasi-neutral structures is zero
substantially simplifies the construction of the scenario
for relaxation to them. The nonquasineutral current
structures in question could serve as a model of X-ray-
emitting hot plasma spots on spatial scales of c/ pe at
electron densities of ne ~ 1020 – 1023 cm-3, which have
been achieved in experiments with Z – pinches [4].
2. THE MAIN EQUATIONS
We describe the electron plasma by equation of motion
for cold relativistic electrons in the following modified
form [1-3]
]v[
c
eEe)cm(
t
p
e
2
e
e Ω×−−=γ∇+
∂
∂ rrr
r
, (1)
]p[
e
cB e
rrr
×∇−=Ω , (2)
and Maxwell equations
t
E
c
1)vnvnz(
c
e4]B[ eeiii ∂
∂
+−
π
=×∇
r
rrr
, (3)
]E[
t
B
c
1 r
r
×∇=
∂
∂
− , )nnz(e4E eii −π=⋅∇
r
. (4)
Here ie vv ,
r
are the electron and the ion velocities,
eee vmp rr
γ= , 22
e cv11 r
−=γ , ne and ni are the
electron and the ion densities, zi is the ion charge
number, E
r
is the electric field, B
r
is the magnetic field.
At first, we use the spherical coordinates (r, , ).
Making use the stationary equations (1) and introducing
the vector potential A
r
, one can obtain the expressions
for the electric and magnetic fields components
( )
−
∂
∂
−γ
∂
∂
−= ϕϕ
ϕ
e
e2
er p
e
cAr
rr
v
c
ecm
r
eE (5)
( )−γ
θ∂
∂
−=θ
2
ecm
r
1eE
−θ
θ∂
∂
θ
− ϕϕ
ϕ
e
e p
e
cAsin
sinr
v
c
e
, (6)
( )ϕθ
θ∂
∂
θ
= Asin
sinr
1B r , ( )ϕθ ∂
∂
−= rA
rr
1B . (7)
Further on, it will be assumed that the ion velocity is
equal to zero and the ion density is constant.
Now, inserting expressions (5) – (7) into the stationary
Eq.(3) and the second Eq.(4) and eliminating the
electron density ne, one can obtain, after going to the
function b = a r sin and the variable s = r sin , the
following final equation
( )γββγ−βγ=
γ
−
γ
2
i ds
dsN
ds
db
sds
db1
ds
d
. (8)
Here 2
e cm
eA
a ϕ= ,
c
v eϕ=β ,
21
1
β−
=γ ,
2
e
22
i
i cm
nez4
N
π
= .
mailto:gordeev@dap.kiae.ru
119
At last, the potentiality condition for the electric field E
r
results in the additional connection between functions b
and
β
=βγ−
s
fsb , (9)
where f(x) is the arbitrary function of its argument.
In order to obtain the localized configuration, it
is necessary that (s) should vanish both for s = 0 and
for ∞→s . This is why it is necessary to further
analyze Eq. (8).
The asymptotic behavior of the function (s) at
s = 0 and ∞→s can be examined by rewriting Eq.(8)
in a somewhat different form. We substitute the
expression b=f+ s, which follows from (9), into Eq.(8)
and differentiate the function f( ) in the resulting equation
with respect to the argument = /s to obtain
−
β
γ+
γ
β−βγ+
ξ
+
ξ ds
d
s
1
ds
d)(
ds
d
d
fdF
d
dfF 2
2
2
2
2
2
21
β=βγ− i2 N
s
1
, (10)
−β
γ
++
β
γ
+−
β
γ
= 24232
2
221
21
s
1
ds
d21
s
1
ds
d
s
1F
ds
d
s
1
ds
d
s
1 2
3
2
2
β
β+
β
β ,
2
222 sds
d
s
1
s
1F
β
−
β
γ
= .
For f=0, i.e. for an electron fluid that is free of vorticity,
there not exists the solution which meets the physically
reasonable conditions at s = 0 and ∞→s together. For
0f ≠ , the asymptotic behavior of the function (s) at s
= 0 differs radically from that in the previous case. If
0dsdf ≠ , then, in the vicinity of the point s = 0, the
function (s) satisfies the equation
0
s
3
ds
d
s
3
ds
d
22
2
=β+
β
−
β
, (11)
which has the solutions
3
21 sCsC +=β . (12)
Now, these two solutions near the point s = 0 are
physically allowed. In addition, the constant C1 is an
eigenvalue of nonlinear equation (6) and should be found
from the condition for to vanish at ∞→s by the
taking into account the asymptotic (12).
3. THE FILAMENT STRUCTURE AND THE
ESTIMATE OF THE MAGNETIC FIELD
VALUE
Let us examine in more detail the structure of the current
equilibrium state under consideration, namely, the state
that arises as a result of the balance between the electric
and magnetic forces and also the centrifugal force
2
e
z
2
e
2
cm
eB
cm
eE
s
β+=
γβ ρ . (13)
Here
ds
db
s
1
cm
eB
2
e
z = (14)
is the z component of the magnetic field. The expression
for the radial electric field component in the (x,y) plane
follows from Eq.(13)
−βγ
β
=ρ
ds
db
scm
eE
2
e
. (15)
Now, we change the spherical coordinates to the
cylindrical. The use of the spherical coordinates at the
first stage of the equation transformation allows to obtain
the additional condition (9).
In the stationary case, it is convenient to convert
Eq. (3) and the second Eq.(4) into the form:
=β
ds
db
s
1
ds
dN e , 2
e
e
2
e cm
ne4
N
π
= , (16)
βγ−β=−
ds
db
ds
d
s
1NN ie . (17)
From Eq. (16), we can see that the dimensionless electron
current density Ne is expressed in terms of the
derivative of the magnetic field component Bz with
respect to s. At the point s0 , at which the magnetic field
has maximum, the dimensionless current density Ne
vanishes, which corresponds to the change in the sign of
the velocity component v .
Hence, since the electron velocity equals zero at the axis
of the vortex structure and at the point s = s0, it has a
maximum at a certain intermediate point s = s1 for C1 > 0.
According to Eq.(15), the electric field component E
vanishes at the point s = s0. Therefore, by integrating
Eq.(17), we can show that the total charge in the region
0ss0 ≤≤ is equal to zero.
It is easy to see that, in the region s > s0, the electron
velocity is negative, has a minimum at a certain point
s = s2 , and tends to zero at infinity.
For C1 < 0 the signs of the velocity and the magnetic field
are changed, but the sign of the electric field is conserved
according to Eq. (13).
From Eq. (17), if the estimate of 3sds/db ∝ is taken
into account, over the region of the small values s one can
obtain
2
1ie C2N)0(N −= . (18)
Thus, near the axis there exists the excess of ions and the
electric field is positive. Therefore, these ions expand
towards the periphery in the considered quasi-
equilibrium.
The estimation of the value of the magnetic field in the
filament from Eq. (14) by the account Ni s2 ~ 1 gives:
120
2
eei
2
e cmn4N
e
cm
B π≅≅ . (19)
In accordance with [5], we set 322
e cm105.1n −⋅≅ to
obtain 8104B ⋅≅ G. This estimate is reasonable close
to the value 9107.0B ⋅≅ G, which was measured in
the experiments on the irradiation of a plasma by the
high-power laser pulses [5]. Note that such strong
magnetic field is maintained by the drift current that
flows on micron scales and whose magnitude is on the
order of the electron Alfven current
( ) 5.8e2cmJ 3
eAe ≅= kA.
ACKNOWLEDGEMENTS
This work supported in part by the “Russian Research
Centre Kurchatov Institute Program for the Support of
Initiative Projects” and the “RF Presidential Program for
State Support of Leading Scientific Schools”, grant no.
NSh-2292.2003.2.
REFERENCES
1. A.V. Gordeev and T.V. Losseva// JETP Lett. 1999,
N 70, p.684.
2. A.V. Gordeev and T.V. Losseva// Plasma Physics
Reports. 2003, v.29, p.748.
3. A.V. Gordeev and T.V. Losseva // Plasma Physics
Reports, 2005, v.31, p.26.
4. A.V. Aglitskii, V.V. Vichrev, A.V. Gulov et al.
Spectroscopy of Multicharged Ions in Hot Plasmas.
Moscow: “Nauka”, 1981. (In Russian).
5. U. Wagner, M. Tatarakis, A. Gopal et al.// Phys. Rev.E,
2004, p.026401.
.
,
,
( )eB en4Br π=
r
. r ~ rB
JAe =me c3/(2e) 2
ee cmn4B π≅ .
,
Z- .
.
,
,
( )eB en4Br π=
r
.
r ~ rB
JAe =me c3/(2e) 2
ee cmn4B π≅ .
, , , Z- .
|
| id | nasplib_isofts_kiev_ua-123456789-81801 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:36:26Z |
| publishDate | 2006 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Gordeev, A.V. 2015-05-20T17:26:42Z 2015-05-20T17:26:42Z 2006 Nonquasineutral current structures in plasmas with a zero net current / A.V. Gordeev // Вопросы атомной науки и техники. — 2006. — № 6. — С. 118-120. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.25.Xz, 52.30.-q, 52.30.Ex, 52.55.-s https://nasplib.isofts.kiev.ua/handle/123456789/81801 A nonquasineutral vortex structure with a zero net current is described that arises as a result of electron drift in
 crossed magnetic and electric fields, the latter being produced by charge separation on a spatial scale of about the
 magnetic Debye radius. In such a structure with radius r ~ rB, the magnetic field maintained by a
 drift current on the order of the electron Alfven current. The system with closed current that is considered in the present paper can also serve as a model of hot spots in the channel of a Z-pinch. This work supported in part by the “Russian Research Centre Kurchatov Institute Program for the Support of Initiative Projects” and the “RF Presidential Program for State Support of Leading Scientific Schools”, grant no. NSh-2292.2003.2. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Nonquasineutral current structures in plasmas with a zero net current Article published earlier |
| spellingShingle | Nonquasineutral current structures in plasmas with a zero net current Gordeev, A.V. Basic plasma physics |
| title | Nonquasineutral current structures in plasmas with a zero net current |
| title_full | Nonquasineutral current structures in plasmas with a zero net current |
| title_fullStr | Nonquasineutral current structures in plasmas with a zero net current |
| title_full_unstemmed | Nonquasineutral current structures in plasmas with a zero net current |
| title_short | Nonquasineutral current structures in plasmas with a zero net current |
| title_sort | nonquasineutral current structures in plasmas with a zero net current |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81801 |
| work_keys_str_mv | AT gordeevav nonquasineutralcurrentstructuresinplasmaswithazeronetcurrent |