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Similarities between some processes in tokamaks and plasma opening switches
PHYSICS ELSEWER
Physics
Reports
REPORTS
283 (1997) 177-184
Similarities between some processes in tokamaks and plasma opening switches L.I. Rudakov Kurchatov Institute, Moscow, Russian Federation
Abstract Some consequences of formal similarities between the equations of the non-linear pulsed magnetic field dynamics in magnetized plasma are analyzed. PACS:
drift wave theory
and those for
52.35.Ra
1. Introduction
Drift instabilities and turbulence of non-uniform magnetized plasmas have been subject of investigation for more than 30 yr [l, 21. The subject has received renewed interest in recent years in connection with the anomalous heat and particle transport in tokamaks, especially in the edge plasma. It is less well known that pulsed plasma research has also been intensified during the last decade and that, once again, the Z-pinch has become an object of investigation. In these experiments a plasma cylinder 1-2 cm in length, of initial radius m 1 cm and density 10’8-1020 cme3 was compressed with velocity (2-3) x 10’ cm/s by the magnetic field of an axial current rising to (l-10) MA in lo-’ s [3,4]. It was found that the radial compression of a strongly radiative plasma was very unstable and had to be described by the magnetohydrodynamic equations including the Hall term in Ohm’s law (which we denote as HMHD). This means that the magnetic field is frozen not into the ion component of the plasma but is frozen into the electron component [4]. Another object currently under intense investigation is the Plasma Opening Switch (POS). POS is a plasma bridge between two electrodes shortening the current from a pulse generator for a certain time interval [S-7]. It has been found that a magnetic field can penetrate into the plasma as a shock wave with a velocity exceeding the Alfven velocity. This phenomenon is also described by HMHD [S-lo, 111. It will be shown below that drift turbulence and pulse plasmas are described by very similar non-linear equations (see the reviews [ 11,121). The difference is in the formulation of the problems. 0370-1573/97/$32.00 0 1997 PII SO370-1573(96)00059-2
Elsevier
Science B.V. All rights reserved
178
L.I. Rudakov 111 Ph_vsics Reports 283 (1997) 177-184
The drift turbulence is usually considered with the assumption that its properties are defined by the local gradients of density and temperature. In contrast, the penetration of a pulse magnetic field into a plasma is defined by boundary conditions at the outer boundary and the electrodes. The aim of the current paper is to analyze some theoretical achievements in the two above-mentioned fields to facilitate an exchange of ideas between them.
2. Selection of the physical models and the equations describing them We will restrict ourselves to a hydrodynamic approximation in the description of drift waves. We consider a plasma layer between two toroidal magnetic surfaces to be planar, operating with Cartesian coordinates (x, y, z) instead of toroidal coordinates (I^,1-0,Rq) while allowing for a ydependence of BZ. We consider motion of the plasma due to a three-dimensional excitation of potential. Electrons are distributed according to the Boltzmann law in such a motion, if they pass an excitation region faster than it changes, co < v,/l: n = n0(x)exp(ecplT&)) Here no(x) and a typical frequency field and v, is the T,(d In n/dx) could plasma motion is (4 Discontinuity
an
-
at
+ div
(b) Equation
au,
-at
(4
.
T,,(x) are plasma density and temperature outside the excitation region, (r) is of the plasma motion, 1 is the length of the excitation region along the magnetic electron thermal velocity. We will assume that Ti < T,, but that (dTi/dx) and be comparable. Under these conditions the potential excitation eq < T, and the described by the following set of equations: equation: a
VqxB
- c~
B2
VqxB
CTVV,
+ vz2=
VqxB
+&((VqxB)xV)Vq
along the magnetic
co*--k
=
,p;, cs
Mc
1 -=-a _I
,
(2)
ions: (B, < B,):
excitation:
VT,, .
iOk;) = (k~c~/o’)(o
eB (OBi
=Ddn.
+ n6Ti).
In the linear approximation eq < T, for an excitation where 1 B pk, 9 p/a, this set of equations can be reduced
cm - co*-
n +&nc,
field for the magnetized
-&-(nT,
for the ion temperature
idTi=CT
1
c
-~Bw~;V(P
of motion
Equation
(1)
C$
=
- “*vi)
dlnn dx T,
Y$
.
of the type exp( - iwt + ik,y + ik,z), to the following dispersion relation:
’
yi=
(5) dln Ti Ti __ dlnn T,’
p=cI, UBi
L.I. Rudakov / Physics Reports 283 (1997) 177-184
179
Eq. (5) has an unstable solution if dT;/dx > T,d lnn/dx. This is an ion gradient-temperature instability, which was first discovered by Rudakov and Sagdeev in 1962 [l]. Within the range of stable values of qi we will consider an equation for non-linear drift waves
>= . AY
ViAY
(6)
Here Y = ecp/T,, vi = D/‘p2WBi,v, = - p/a and the time and space coordinates are dimensionless: t = CL)Bir,(x, y) = (x, y)/p. The third term in Eq. (6) is associated with a toroidal field dependence on the poloidal coordinate (y in our case). Eq. (6) and its modifications describe not only the drift waves and plasma turbulence, but also Rossby waves in planet atmospheres. This equation is a subject of many theoretical investigations and computer simulations. The works of Galeev and Rudakov [13], Hasegawa and Mima [14], Petviashvily [15] and Horton [16] must be mentioned here. Let us now consider an equation describing the penetration of a pulsed magnetic field into a plasma. Let the magnetic field have one component, B, (in real experiments it is the magnetic field of an axial current having two components, j, and j,). The equation for B,, which is of interest to us, can be derived from the following equations: (a) Equation for a two-dimensional (nX, v,) motion of a cold electron component of the plasma:
+
(uev)ue = - f
(b) The Maxwell
VxB=
- ue x
uBe
-
vei(“c
-
ui)
.
(7)
equations:
_4’en 7 t”e -
ui)
.
In usual MHD the velocity of the plasma ions due to the magnetic pressure is supposed to be much greater than the current velocity (t’= - Vi). We are interested in the opposite case (v, - Vi) 9 Vi. It could be shown, that this corresponds to the case when a typical time, to, and length scale of the problem, a, satisfy the following conditions:
As follows from Eqs. (7)-(9): ;(h+[Vxu,],)+
g&-&i
h+c;xuely=v,Ab. >
(10)
180
L.I. Rudakov / Ph_vsics Reports 283 (1997) 177-184
where time and coordinates
t =
O&t
)
are normalized:
(x, 4 = b, 4 ?
47we2 2 Cope = m ’
,
l’ei
v, = WBfL )
h=+, 0
n2 n0
eB
UBe
=
-
mc ’
For the case of small deviations
(B. - B,) Q Bo, (no - n) Q no, Eq. (10) can be rewritten EL-;;
$(b-Ab)+;;(l-(I-b))-
(11)
Ab=v,Ab. >
For (1 - b) < 1 Eq. (11) is similar to Eq. (6), for the excitation of the potential ye = 1. We can take into account small density changes due to ion motion: a26n -=;Ab2. at2
as
Y if R -+ a~,
(12)
It then follows that a plasma slab with a magnetic field and density gradient relative to excitations as exp( - iot + ik,x), d In n/dx < k, 4 ~l)Jc: (c~ - w,* + iv,kz) = 9
along z is unstable
(o - og) , (13)
cB dn LO* n = k -x 4rcn2 dz ’
c$=k
--
c dB
x 4xn dz ’
J/J:,=-...-
B2
4znM
’
The last formulas are in the usual units. The instability of a distributed current layer for dln B/din n > 1 described by Eq. (13) corresponding to Eq. (5) for the ion gradient temperature instability, was discovered by Gordeev and Grechikha [17]. Thus we have demonstrated that such different physical processes as the drift waves in magnetized plasma and the fast magnetic field penetration into a plasma are described by virtually the same equations.
3. Fast shock waves in tokamaks The set of Eqs. (l)-(4) is usually studied with the assumption that the properties of the turbulence and particle transport are defined by local values of density and temperature gradients. Computer simulation of Eq. (6) for the case R + 00, v], = 0, D = 0 shows that, due to the modulational instability discovered in [IS], an initially homogeneous drift turbulence evolves to a new turbulent state. This new state is a set of coherent structures mainly vortices [19].
181
L.I. Rudakou 1 Physics Reports 283 (1997) 177-184
Fig. 1
In contrast to this approach, A. Kingsep, Yu. Mokhov and K. Chukbar solution for Eq. (10) in the form of shock wave. For the case
have considered
another
-&k=Const, a planar
shock wave is described
aB
c
at
8zeni
Its asymptotic
by the following
equation:
c2 a2B dnO aB2 = vei z p . dx az solution
(14)
is
B=B,tanhy, cBO dlnn ’
=
-
then
dx
6=7.
’
(15)
C2V,i
qx u
Solution (15) has been obtained for motionless ions. More general consideration of twodimensional shock waves can be found in [9]. Unlike the case for solitons or vortices, energy carried into a plasma by shock wave is supplied by ion and electron flows from the plasma boundary. Solution of this type may be found for tokamak plasma as well. If for some reason, a potential jump arises at the plasma boundary or at one of magnetic surfaces, it would penetrate into the plasma in the form of a shock wave similar to Fig. 1 where the arrows show the direction of ion flow. The structure of the rarefaction shock wave between the points 1 and 2 is described by the following equation resulting from Eq. (6) if v $ u/b, a/R < 1: 2a a2y <=y-_RX’ ap ’
asv
^Cf+a*g-L.*)J.$~=vwhich has the solution: y:-__
2 u - v* tanh r?,
u
61’ (jv
-4 - ut [
u - v* .
Between the point 2 and 3 ions drift along the line n/B2 = Const. The displacement of the shock wave in x-direction may be about a2/R.
(16)
(17)
182
L.I. Rudakov / Physics Reports 283 (‘I997) 177-184
Eq. (6) does not take into account a plasma motion along the magnetic field lines resulting in plasma flowing into the rarefied region after shocks. The length of the region (2))(3) may be estimated from mass balance considerations:
Here 1, is the length of the non-linear structure along the magnetic field. The diffusion coefficient, D,in a slightly collisional plasma may be mainly turbulence induced transport. In this case Eqs. (16) and (17) apply if
D>
due to small-scale
p2c, Y li2/a .
The solution under consideration describes long-distance ion flows across the magnetic surfaces. In contrast to vortices, these flows are not closed locally and can result in significant particle transport. A similar problem concerning stochastic transport of magnetic field within the framework of Eq. (11) has been considered by Isichenko and Kalda [20].
4. Magnetic field transport by vortices in non-uniform plasmas Soliton and vortex type solutions for Eqs. (6) and (11) have been found by a number of authors [ 11,121. The Red Spot on Jupiter is an example of such a vortex. We will apply a technique used in Ref. [15] to Eq. (10). Let us consider a solution of the type B = B(x, z - ut)in a collisionless case, v, + 0, with small density variations:
n=(l-F),
x
The plasma occupies the region z > 0, while an electric current rewrite Eq. (10) for this case as follows:
is flowing in x direction.
We will
(18) Eq. (18) has an integral: /lb - h = n(x)G b + u ndx
e n(x)G(b
+ ux) ,
(19)
(S)
where G is an arbitrary function. Eq. (19) has a wide range of localized solutions of vortex and solution type, i.e. solutions, where the magnetic field is constant away from the soliton. It is obvious that, providing B, = 0,the vortices can be only monopoles. We consider a particular function G: G = - (h + u(xO + x)) - A2[(b - b,) + ux12 .
L.I. Rudakoa ,! Physics Reports 283 (I 997) 177-184
From the condition db + 0 when h + b, we obtain an equation an equation describing its form: u = h,la
= - (cB,/4nen2)
Ah = (1 - n(x))(b
- h,)
dnldx , - A*(h - b,)’
for the velocity
183
of the vortex and
(20) .
(21)
Here A2 is arbitrary constant. The maximum size of the vortex is limited by (ac2/o.$,)‘13. According to Eq. (20) the vortices cannot transport the magnetic field into the plasma where B, + 0. Hence, a formation of localized structures due to the instability of a current layer of the magnetic shock wave described by Eq. (10) and its solution (15) should not considerably change the velocity of the wave. Penetration of a magnetic field into a non-magnetized plasma is thus possible only if the integral (19) is changed due to Coulomb or turbulent diffusion:
5. Conclusion The formal similarities in the equations of drift field dynamics in magnetized plasmas, which were for example, [ll, 12]), has allowed us to apply the to the other. We suppose that such a cross-reference
wave turbulence and those of pulsed magnetic noticed and discussed before in literature (see, methods and solutions for one of the problems analysis could be useful for future applications.
Acknowledgements The present work has been done by the author is AugusttSeptember 1994 during the visit to the Plasma Research Laboratory, Research School of Physical Sciences & Engineering, the Australian National University, where he had spent two months as a Visiting Fellow. The author would like to express his gratitude to the colleagues from PRL for interesting discussions and hospitality. This work was also reported and discussed during workshop in ITP UCSB.
References [l] L.I. Rudakov and R.Z. Sagdeev, Sov. Phys. Doklady 6 (1963) 415. 123 W. Horton in: Handbook of Plasma Physics, Vol. II, eds. A. Galeev and R. N. Sudan (North-Holland, New York, 1984 p. 383. [3] See, for example, Dense Z-pinches, AIP Conf. Proc., No. 195, eds., N. Pereira, J. Davis and N. Rostoker (American Inst. Phys., New York, 1989). [4] L. Rudakov, Magnetically Accelerated Plasma Stability, ibid. p. 290. [S] Yn. Golovanov, G. Dolgachev, L. Zakatov and V. Skoryupin, Sov. J. Plasma Phys. 19 (1988) 519. [6] D. Hinshelwood, B. Weber, J. Grossman and R. Commisso, Phys. Rev. Lett. 68 (1992) 3567. [7] M. Sarfaty, Y. Maron, Ya. Krasik et al., in: Beams’94 Conf. Proc. San-Diego, USA (1994). [8] A. Kingsep, Yu. Mokhov and K. Chukbar, Sov. J. Plasma Phys. 10 (1984) 495.
184
L.I. Rudakov / Physics Reports 283 (1997) 177-184
[9] A. Fruchtman and L. Rudakov, Phys. Rev. Lett. 69 (1992) 2070. [lo] A.V. Gordeev, AS. Kingsep and L.I. Rudakov, Physics Reports: Electron Magnetohydrodynamics 243 (5) (July 1994). [11] See V. Petviashvily and V.V. Yankov, in: Review of Plasma Physics, Vol. 14, ed. B. Kadomtsev (Consultants Bureau, New York 1989). [12] See A. Kingsep, K. Chukbar and V.V. Yankov in: Review of Plasma Physics, Vol. 16, ed. B. Kadomtsev (Consultants Bureau, New York, 1990). 1131 A. Galeev and L. Rudakov, Sov. Phys. JETP 18 (1963) 444. [14] A. Hasegawa and K. Mima, Phys. Fluids 21 (1978) 87. [15] V. Petviashvily, JETP, 32 (1980) 632. 1161 W. Horton, Phys. Fluids, 1 (1989) 524. [17] A. Gordeev and A. Grechikha, Fis. Plasmy, 18 (1992) 3. [lS] L. Rudakov, Sov. Phys. JETP 21 (1965) 917. [19] See Transport, Chaos and Plasma Physics, eds., S. Benkadda, F. Doveil and Y. Elskens (World Scientific, Singapore, 1994). [20] M. Isichenko and J. Kalda, JETP 99 (1991) 224.
Physics
Reports
REPORTS
283 (1997) 177-184
Similarities between some processes in tokamaks and plasma opening switches L.I. Rudakov Kurchatov Institute, Moscow, Russian Federation
Abstract Some consequences of formal similarities between the equations of the non-linear pulsed magnetic field dynamics in magnetized plasma are analyzed. PACS:
drift wave theory
and those for
52.35.Ra
1. Introduction
Drift instabilities and turbulence of non-uniform magnetized plasmas have been subject of investigation for more than 30 yr [l, 21. The subject has received renewed interest in recent years in connection with the anomalous heat and particle transport in tokamaks, especially in the edge plasma. It is less well known that pulsed plasma research has also been intensified during the last decade and that, once again, the Z-pinch has become an object of investigation. In these experiments a plasma cylinder 1-2 cm in length, of initial radius m 1 cm and density 10’8-1020 cme3 was compressed with velocity (2-3) x 10’ cm/s by the magnetic field of an axial current rising to (l-10) MA in lo-’ s [3,4]. It was found that the radial compression of a strongly radiative plasma was very unstable and had to be described by the magnetohydrodynamic equations including the Hall term in Ohm’s law (which we denote as HMHD). This means that the magnetic field is frozen not into the ion component of the plasma but is frozen into the electron component [4]. Another object currently under intense investigation is the Plasma Opening Switch (POS). POS is a plasma bridge between two electrodes shortening the current from a pulse generator for a certain time interval [S-7]. It has been found that a magnetic field can penetrate into the plasma as a shock wave with a velocity exceeding the Alfven velocity. This phenomenon is also described by HMHD [S-lo, 111. It will be shown below that drift turbulence and pulse plasmas are described by very similar non-linear equations (see the reviews [ 11,121). The difference is in the formulation of the problems. 0370-1573/97/$32.00 0 1997 PII SO370-1573(96)00059-2
Elsevier
Science B.V. All rights reserved
178
L.I. Rudakov 111 Ph_vsics Reports 283 (1997) 177-184
The drift turbulence is usually considered with the assumption that its properties are defined by the local gradients of density and temperature. In contrast, the penetration of a pulse magnetic field into a plasma is defined by boundary conditions at the outer boundary and the electrodes. The aim of the current paper is to analyze some theoretical achievements in the two above-mentioned fields to facilitate an exchange of ideas between them.
2. Selection of the physical models and the equations describing them We will restrict ourselves to a hydrodynamic approximation in the description of drift waves. We consider a plasma layer between two toroidal magnetic surfaces to be planar, operating with Cartesian coordinates (x, y, z) instead of toroidal coordinates (I^,1-0,Rq) while allowing for a ydependence of BZ. We consider motion of the plasma due to a three-dimensional excitation of potential. Electrons are distributed according to the Boltzmann law in such a motion, if they pass an excitation region faster than it changes, co < v,/l: n = n0(x)exp(ecplT&)) Here no(x) and a typical frequency field and v, is the T,(d In n/dx) could plasma motion is (4 Discontinuity
an
-
at
+ div
(b) Equation
au,
-at
(4
.
T,,(x) are plasma density and temperature outside the excitation region, (r) is of the plasma motion, 1 is the length of the excitation region along the magnetic electron thermal velocity. We will assume that Ti < T,, but that (dTi/dx) and be comparable. Under these conditions the potential excitation eq < T, and the described by the following set of equations: equation: a
VqxB
- c~
B2
VqxB
CTVV,
+ vz2=
VqxB
+&((VqxB)xV)Vq
along the magnetic
co*--k
=
,p;, cs
Mc
1 -=-a _I
,
(2)
ions: (B, < B,):
excitation:
VT,, .
iOk;) = (k~c~/o’)(o
eB (OBi
=Ddn.
+ n6Ti).
In the linear approximation eq < T, for an excitation where 1 B pk, 9 p/a, this set of equations can be reduced
cm - co*-
n +&nc,
field for the magnetized
-&-(nT,
for the ion temperature
idTi=CT
1
c
-~Bw~;V(P
of motion
Equation
(1)
C$
=
- “*vi)
dlnn dx T,
Y$
.
of the type exp( - iwt + ik,y + ik,z), to the following dispersion relation:
’
yi=
(5) dln Ti Ti __ dlnn T,’
p=cI, UBi
L.I. Rudakov / Physics Reports 283 (1997) 177-184
179
Eq. (5) has an unstable solution if dT;/dx > T,d lnn/dx. This is an ion gradient-temperature instability, which was first discovered by Rudakov and Sagdeev in 1962 [l]. Within the range of stable values of qi we will consider an equation for non-linear drift waves
>= . AY
ViAY
(6)
Here Y = ecp/T,, vi = D/‘p2WBi,v, = - p/a and the time and space coordinates are dimensionless: t = CL)Bir,(x, y) = (x, y)/p. The third term in Eq. (6) is associated with a toroidal field dependence on the poloidal coordinate (y in our case). Eq. (6) and its modifications describe not only the drift waves and plasma turbulence, but also Rossby waves in planet atmospheres. This equation is a subject of many theoretical investigations and computer simulations. The works of Galeev and Rudakov [13], Hasegawa and Mima [14], Petviashvily [15] and Horton [16] must be mentioned here. Let us now consider an equation describing the penetration of a pulsed magnetic field into a plasma. Let the magnetic field have one component, B, (in real experiments it is the magnetic field of an axial current having two components, j, and j,). The equation for B,, which is of interest to us, can be derived from the following equations: (a) Equation for a two-dimensional (nX, v,) motion of a cold electron component of the plasma:
+
(uev)ue = - f
(b) The Maxwell
VxB=
- ue x
uBe
-
vei(“c
-
ui)
.
(7)
equations:
_4’en 7 t”e -
ui)
.
In usual MHD the velocity of the plasma ions due to the magnetic pressure is supposed to be much greater than the current velocity (t’= - Vi). We are interested in the opposite case (v, - Vi) 9 Vi. It could be shown, that this corresponds to the case when a typical time, to, and length scale of the problem, a, satisfy the following conditions:
As follows from Eqs. (7)-(9): ;(h+[Vxu,],)+
g&-&i
h+c;xuely=v,Ab. >
(10)
180
L.I. Rudakov / Ph_vsics Reports 283 (1997) 177-184
where time and coordinates
t =
O&t
)
are normalized:
(x, 4 = b, 4 ?
47we2 2 Cope = m ’
,
l’ei
v, = WBfL )
h=+, 0
n2 n0
eB
UBe
=
-
mc ’
For the case of small deviations
(B. - B,) Q Bo, (no - n) Q no, Eq. (10) can be rewritten EL-;;
$(b-Ab)+;;(l-(I-b))-
(11)
Ab=v,Ab. >
For (1 - b) < 1 Eq. (11) is similar to Eq. (6), for the excitation of the potential ye = 1. We can take into account small density changes due to ion motion: a26n -=;Ab2. at2
as
Y if R -+ a~,
(12)
It then follows that a plasma slab with a magnetic field and density gradient relative to excitations as exp( - iot + ik,x), d In n/dx < k, 4 ~l)Jc: (c~ - w,* + iv,kz) = 9
along z is unstable
(o - og) , (13)
cB dn LO* n = k -x 4rcn2 dz ’
c$=k
--
c dB
x 4xn dz ’
J/J:,=-...-
B2
4znM
’
The last formulas are in the usual units. The instability of a distributed current layer for dln B/din n > 1 described by Eq. (13) corresponding to Eq. (5) for the ion gradient temperature instability, was discovered by Gordeev and Grechikha [17]. Thus we have demonstrated that such different physical processes as the drift waves in magnetized plasma and the fast magnetic field penetration into a plasma are described by virtually the same equations.
3. Fast shock waves in tokamaks The set of Eqs. (l)-(4) is usually studied with the assumption that the properties of the turbulence and particle transport are defined by local values of density and temperature gradients. Computer simulation of Eq. (6) for the case R + 00, v], = 0, D = 0 shows that, due to the modulational instability discovered in [IS], an initially homogeneous drift turbulence evolves to a new turbulent state. This new state is a set of coherent structures mainly vortices [19].
181
L.I. Rudakou 1 Physics Reports 283 (1997) 177-184
Fig. 1
In contrast to this approach, A. Kingsep, Yu. Mokhov and K. Chukbar solution for Eq. (10) in the form of shock wave. For the case
have considered
another
-&k=Const, a planar
shock wave is described
aB
c
at
8zeni
Its asymptotic
by the following
equation:
c2 a2B dnO aB2 = vei z p . dx az solution
(14)
is
B=B,tanhy, cBO dlnn ’
=
-
then
dx
6=7.
’
(15)
C2V,i
qx u
Solution (15) has been obtained for motionless ions. More general consideration of twodimensional shock waves can be found in [9]. Unlike the case for solitons or vortices, energy carried into a plasma by shock wave is supplied by ion and electron flows from the plasma boundary. Solution of this type may be found for tokamak plasma as well. If for some reason, a potential jump arises at the plasma boundary or at one of magnetic surfaces, it would penetrate into the plasma in the form of a shock wave similar to Fig. 1 where the arrows show the direction of ion flow. The structure of the rarefaction shock wave between the points 1 and 2 is described by the following equation resulting from Eq. (6) if v $ u/b, a/R < 1: 2a a2y <=y-_RX’ ap ’
asv
^Cf+a*g-L.*)J.$~=vwhich has the solution: y:-__
2 u - v* tanh r?,
u
61’ (jv
-4 - ut [
u - v* .
Between the point 2 and 3 ions drift along the line n/B2 = Const. The displacement of the shock wave in x-direction may be about a2/R.
(16)
(17)
182
L.I. Rudakov / Physics Reports 283 (‘I997) 177-184
Eq. (6) does not take into account a plasma motion along the magnetic field lines resulting in plasma flowing into the rarefied region after shocks. The length of the region (2))(3) may be estimated from mass balance considerations:
Here 1, is the length of the non-linear structure along the magnetic field. The diffusion coefficient, D,in a slightly collisional plasma may be mainly turbulence induced transport. In this case Eqs. (16) and (17) apply if
D>
due to small-scale
p2c, Y li2/a .
The solution under consideration describes long-distance ion flows across the magnetic surfaces. In contrast to vortices, these flows are not closed locally and can result in significant particle transport. A similar problem concerning stochastic transport of magnetic field within the framework of Eq. (11) has been considered by Isichenko and Kalda [20].
4. Magnetic field transport by vortices in non-uniform plasmas Soliton and vortex type solutions for Eqs. (6) and (11) have been found by a number of authors [ 11,121. The Red Spot on Jupiter is an example of such a vortex. We will apply a technique used in Ref. [15] to Eq. (10). Let us consider a solution of the type B = B(x, z - ut)in a collisionless case, v, + 0, with small density variations:
n=(l-F),
x
The plasma occupies the region z > 0, while an electric current rewrite Eq. (10) for this case as follows:
is flowing in x direction.
We will
(18) Eq. (18) has an integral: /lb - h = n(x)G b + u ndx
e n(x)G(b
+ ux) ,
(19)
(S)
where G is an arbitrary function. Eq. (19) has a wide range of localized solutions of vortex and solution type, i.e. solutions, where the magnetic field is constant away from the soliton. It is obvious that, providing B, = 0,the vortices can be only monopoles. We consider a particular function G: G = - (h + u(xO + x)) - A2[(b - b,) + ux12 .
L.I. Rudakoa ,! Physics Reports 283 (I 997) 177-184
From the condition db + 0 when h + b, we obtain an equation an equation describing its form: u = h,la
= - (cB,/4nen2)
Ah = (1 - n(x))(b
- h,)
dnldx , - A*(h - b,)’
for the velocity
183
of the vortex and
(20) .
(21)
Here A2 is arbitrary constant. The maximum size of the vortex is limited by (ac2/o.$,)‘13. According to Eq. (20) the vortices cannot transport the magnetic field into the plasma where B, + 0. Hence, a formation of localized structures due to the instability of a current layer of the magnetic shock wave described by Eq. (10) and its solution (15) should not considerably change the velocity of the wave. Penetration of a magnetic field into a non-magnetized plasma is thus possible only if the integral (19) is changed due to Coulomb or turbulent diffusion:
5. Conclusion The formal similarities in the equations of drift field dynamics in magnetized plasmas, which were for example, [ll, 12]), has allowed us to apply the to the other. We suppose that such a cross-reference
wave turbulence and those of pulsed magnetic noticed and discussed before in literature (see, methods and solutions for one of the problems analysis could be useful for future applications.
Acknowledgements The present work has been done by the author is AugusttSeptember 1994 during the visit to the Plasma Research Laboratory, Research School of Physical Sciences & Engineering, the Australian National University, where he had spent two months as a Visiting Fellow. The author would like to express his gratitude to the colleagues from PRL for interesting discussions and hospitality. This work was also reported and discussed during workshop in ITP UCSB.
References [l] L.I. Rudakov and R.Z. Sagdeev, Sov. Phys. Doklady 6 (1963) 415. 123 W. Horton in: Handbook of Plasma Physics, Vol. II, eds. A. Galeev and R. N. Sudan (North-Holland, New York, 1984 p. 383. [3] See, for example, Dense Z-pinches, AIP Conf. Proc., No. 195, eds., N. Pereira, J. Davis and N. Rostoker (American Inst. Phys., New York, 1989). [4] L. Rudakov, Magnetically Accelerated Plasma Stability, ibid. p. 290. [S] Yn. Golovanov, G. Dolgachev, L. Zakatov and V. Skoryupin, Sov. J. Plasma Phys. 19 (1988) 519. [6] D. Hinshelwood, B. Weber, J. Grossman and R. Commisso, Phys. Rev. Lett. 68 (1992) 3567. [7] M. Sarfaty, Y. Maron, Ya. Krasik et al., in: Beams’94 Conf. Proc. San-Diego, USA (1994). [8] A. Kingsep, Yu. Mokhov and K. Chukbar, Sov. J. Plasma Phys. 10 (1984) 495.
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L.I. Rudakov / Physics Reports 283 (1997) 177-184
[9] A. Fruchtman and L. Rudakov, Phys. Rev. Lett. 69 (1992) 2070. [lo] A.V. Gordeev, AS. Kingsep and L.I. Rudakov, Physics Reports: Electron Magnetohydrodynamics 243 (5) (July 1994). [11] See V. Petviashvily and V.V. Yankov, in: Review of Plasma Physics, Vol. 14, ed. B. Kadomtsev (Consultants Bureau, New York 1989). [12] See A. Kingsep, K. Chukbar and V.V. Yankov in: Review of Plasma Physics, Vol. 16, ed. B. Kadomtsev (Consultants Bureau, New York, 1990). 1131 A. Galeev and L. Rudakov, Sov. Phys. JETP 18 (1963) 444. [14] A. Hasegawa and K. Mima, Phys. Fluids 21 (1978) 87. [15] V. Petviashvily, JETP, 32 (1980) 632. 1161 W. Horton, Phys. Fluids, 1 (1989) 524. [17] A. Gordeev and A. Grechikha, Fis. Plasmy, 18 (1992) 3. [lS] L. Rudakov, Sov. Phys. JETP 21 (1965) 917. [19] See Transport, Chaos and Plasma Physics, eds., S. Benkadda, F. Doveil and Y. Elskens (World Scientific, Singapore, 1994). [20] M. Isichenko and J. Kalda, JETP 99 (1991) 224.