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Admissible Equilibria of Non-neutral Plasmas in a Malmberg-Penning Trap

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Admissible Equilibria of Non-neutral Plasmas in a Malmberg-Penning Trap
Igor Kotelnikov
Budker Institute of Nuclear Physics, Lavrentyev Avenue 11, Novosibirsk, 630090, Russia
Massimiliano Rome´
I.N.F.N. Sezione di Milano and Dipartimento di Fisica, Universita` degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy
(Received 5 May 2008; published 22 August 2008)A ‘‘parallel current constraint’’ is derived, that in combination with the Poisson equation allows one toselect admissible equilibria of non-neutral plasmas in a Malmberg-Penning trap in the presence of anonuniform and nonaxisymmetric magnetic ﬁeld. Asymmetry-induced currents (analogous to the Pﬁrsch-Schlu¨ter currents in Tokamaks) appearing in a non-neutral plasma even in the absence of magnetic driftsare explicitly computed in the case of a uniformly tilted magnetic ﬁeld.
DOI: 10.1103/PhysRevLett.101.085006 PACS numbers: 52.27.Jt, 52.27.Aj
The radial conﬁnement of non-neutral plasmas inMalmberg-Penning (MP) traps is provided by a strongaxial magnetic ﬁeld. This ﬁeld is assumed to be uniformin most theories that deal with plasma conﬁnement.However, small perturbations of the magnetic ﬁeld mayplay a crucial role in the transport of non-neutral plasmasin this kind of conﬁnement devices [1]; see also the reviewpaper [2] and references therein for further discussion of the problem of non-neutral plasma transport. On the otherhand, it is well known that an accurate treatment of theplasma transport requires at ﬁrst an analysis of the plasmaequilibrium, as it is demonstrated by established theoriesfor quasineutral plasma conﬁned, e.g., in tandem mirrors[3].It may be wondered whether an equilibrium of a non-neutral plasma exists in an asymmetric magnetic ﬁeld,since an asymmetry leads in general to plasma expansion.A positive answer to this question implies that the equilib-rium is referred to a time interval shorter than the expan-sion time
m
. If the asymmetry is small, the latter isexpected to be at least greater then the axial bounce timeof the particles inside the trap
b
and the plasma azimuthalrotation time
2
=!
E
, i.e.,
m
ð
b
;
2
=!
E
Þ
. In general,
m
/
2
, where the parameter
characterizes the small-ness on the magnetic ﬁeld inhomogeneity. At this stage itcan be assumed that
B=B
, where
B
represents thedifference of the actual magnetic ﬁeld from an ideal uni-form magnetic ﬁeld
B
¼
B
e
z
directed along the symme-tryaxisofthecylindricalconﬁnementdevice.Forthesmall
values achieved in existing devices the expansion timecan therefore be quite large, and for a shorter time interval,
t
m
, it is possible to consider a slowly evolving plasmacolumn as being in a static equilibrium.Systematic studies of nonaxisymmetric equilibria in aMP trap have been started in Refs. [4,5]. In Ref. [4] the
equilibrium of a non-neutral plasma column in a weaklytilted magnetic ﬁeld was simulated numerically. In Ref. [5]an electrostatic asymmetry was introduced by azimuthallysectored electrodes, and the analytical treatment was lim-ited to the case of a cold plasma with a stepwise radialdensity proﬁle. Later on, three-dimensional numericalparticle-in-cell simulations of the non-neutral plasma equi-librium with quadrupole or mirror magnetic perturbationshave been reported in Ref. [6]. However, similar numericalsimulationsarehardlyabletouncoverﬁne-structureeffectsthat limit plasma lifetime in existing and future facilitiesdesigned to achieve improved conﬁnement of non-neutralplasmas.In Ref. [7] the equilibrium of non-neutral plasmas on aset of nested toroidal magnetic surfaces has been recentlyconsidered. This work together with the theory of quasi-neutral plasma equilibria in tandem mirrors [3] bestows aguideline of how to establish a constraint on the shape of admissible plasma equilibria. Together with Poisson’sequation, rewritten in ﬂux coordinates, this constraint con-stitutes a self-consistent method for determining asymmet-ric equilibria of non-neutral plasmas in a MP trap.The approach is based on the use of curvilinear ﬂuxcoordinates for the magnetic ﬁeld. As it was argued inRef. [8], performing the calculations in ﬂux coordinatesmakes the interpretation of the plasma equilibrium mucheasier and provides the best approach to the problem of theerror ﬁeld mediated transport.The electric current produced by the ﬂowing electronsconﬁned in a MP trap produces a negligible change of themagnetic ﬁeld, if the electron density
n
is far below theBrillouin limit [9]
n
B
B
2
=
8
mc
2
(with
m
the particlemass and
c
the speed of light) except for the case of a fastrotating non-neutral plasma equilibrium [10]. The mag-netic ﬁeld can then be described by a scalar magneticpotential
such that
B
¼
r
:
(1)Alternatively, any divergence-free ﬁeld can be written as
B
¼
r
r
#;
(2)PRL
101,
085006 (2008) PHYSICAL REVIEW LETTERS
week ending22 AUGUST 2008
0031-9007
=
08
=
101(8)
=
085006(4) 085006-1
2008 The American Physical Society
where
and
#
are ﬂux coordinates [11], which are con-stant along the magnetic ﬁeld lines. The relations
¼
ð
x;y;z
Þ
,
#
¼
#
ð
x;y;z
Þ
,
¼
ð
x;y;z
Þ
deﬁne a systemof curvilinear coordinates.The momentum balance equation for a pure electronplasma is
mn
@
v
@t
þ
v
r
v
¼
en
1
c
v
B
r
r
p;
where
e
is the particle charge,
v
the ﬂuid velocity,
theelectrostatic potential, and
p
the scalar pressure. In theequilibrium state, the time derivative vanishes. If the elec-tron density is far below the Brillouin limit,
n
n
B
, andthe plasma column is in a slow rotation state [10], then the
v
r
v
term is negligible in comparison with the otherterms and the force balance equation reduces to
r
p
¼
en
1
c
v
B
r
:
(3)Dotting Eq. (3) with
B
, one ﬁnds that
B
r
p
¼
en
B
r
. The electron temperature
T
tends to be constant alongthe magnetic ﬁeld,
B
r
T
¼
0
. When this situation isreached, the electron density must have the form
n
¼
N
ð
;#
Þ
exp
eT
ð
;#
Þ
(4)and must also be consistent with the Poisson equation.Therefore, the fundamental equilibrium equation for apure electron plasma is
r
2
ð
;#;
Þ ¼
4
eN
ð
;#
Þ
exp
eT
ð
;#
Þ
:
(5)This equation contains two functions of
and
#
,
N
ð
;#
Þ
and
T
ð
;#
Þ
which are subject to a constraint derivedbelow. The dependence of
N
and
T
on
#
makes the plasmaequilibria in MP traps very different from those obtained intoroidal devices [7], where the functions
N
ð
Þ
and
T
ð
Þ
are entirely determined by the experimental conditions andby the plasma transport processes.The equilibrium equation (3) implies
v
¼
v
k
B
B
c
r
pen
þ
r
B
B
2
;
(6)leaving undeﬁned the velocity
v
k
parallel to the magneticﬁeld. According to Eq. (4), the pressure
p
ð
;#;
Þ ¼
T
ð
;#
Þ
N
ð
;#
Þ
exp
½
e=T
ð
;#
Þ
;
(7)is a function of
,
#
, and
. Thus,
r
p
¼
@p@
r
þ
@p@#
r
#
en
r
:
(8)Combining Eqs. (6) and (8), the
r
terms cancel, and
v
¼
v
k
B
B
c@p@
r
B
enB
2
c@p@#
r
#
B
enB
2
:
(9)This relation reveals that
@p=@
and
@p=@#
cannot beneglected even when
p
vanishes in the zero temperaturelimit, as explained in Ref. [7].The parallel component
v
k
of the plasma ﬂow, Eq. (9),must be consistent with the steady-state constraint
r
ð
en
v
Þ ¼
0
:
(10)This constraint leaves a net parallel electric current of thenon-neutral plasma undetermined in a toroidal conﬁne-ment conﬁguration [7], but it leads to a closure conditionfor the MP trap geometry.Combining Eq. (9) with Eq. (10) yields
r
env
k
cB
B
¼
r
1
B
2
@p@
r
r
þ
r
1
B
2
@p@#
r
#
r
:
(11)The left-hand side is transformed according to
r
env
k
cB
B
¼
B
2
@@env
k
cB:
(12)Computing the gradients in the right-hand side, one has totake into account that triple products with two gradients of the same function are equal to zero, for example,
r
r
r
¼
0
. Reminding that
B
is considered here as afunction of
,
#
, and
, and
p
is a function of
,
#
, and
,one obtains
B
2
@@env
k
cB
¼
@@#
1
B
2
@p@
r
#
r
r
þ
@@
1
B
2
@p@#
r
r
#
r
þ
1
B
2
@
2
p@#@
r
#
r
r
þ
1
B
2
@
2
p@@#
r
r
#
r
þ
1
B
2
@
2
p@@
r
r
r
þ
1
B
2
@
2
p@@#
r
r
#
r
:
Since
r
r
#
r
¼
B
2
and
r
#
r
r
¼
B
2
, the third and fourth terms in the last equation cancel each other.In the ﬁfth term one can change the order of the partial derivatives over
and
and then make use of the equality
@p=@
¼
ep=T
. The triple product
r
r
r
is equal to
ð
@=@#
Þ
r
#
r
r
¼
B
2
ð
@=@#
Þ
. The sixthterm is transformed in a similar way. Dividing both sides of the last equation by
B
2
leads toPRL
101,
085006 (2008) PHYSICAL REVIEW LETTERS
week ending22 AUGUST 2008
085006-2
@@env
k
cB
¼
@p@ @@#
1
B
2
þ
@p@# @@
1
B
2
þ
1
B
2
@e@# @@ pT
1
B
2
@e@ @@# pT :
(13)This equation allows one to calculate the plasma currentalong the magnetic ﬁeld lines. Since the current vanishes atthe ends of the plasma column, the integral of the right-hand side over the entire range of
must be equal to zero.This yields the ‘‘solvability condition’’
0
¼
Z
11
@p@ @@#
1
B
2
þ
@p@# @@
1
B
2
þ
1
B
2
@e@# @@ pT
1
B
2
@e@ @@# pT
d;
(14)where the integration is formally extended over an inﬁniteinterval (actually it covers the interval of a magnetic ﬁeldline where the plasma pressure
p
is nonzero).The constraint (14) interrelates two functions of
and
#
, namely
N
and
T
, and, in general, it allows determining
N
ð
;#
Þ
if
T
ð
;#
Þ
is given or vice versa. One can argue,however, that
T
ð
;#
Þ
is not completely independent of
N
ð
;#
Þ
.Indeed,adifferentialplasmarotationwouldresultin a fast sharpening of the temperature gradient across theplasma streamlines so that even a weak transverse thermalconductivity effectively ﬂattens the temperature along thestreamlines. Therefore one can assume that
v
r
T
¼
0
inaddition to the condition
B
r
T
¼
0
used in the derivationof Eq. (4). Dotting Eq. (9) with
r
T
one ﬁnally concludesthat
T
depends on
and
#
through the dependence of
N
onthese coordinates, i.e.,
T
ð
;#
Þ ¼
T
ð
N
ð
;#
Þ
Þ
. For the sakeof simplicity it is assumed below that
T
¼
const
. Thisassumption is relevant to the state of global thermal equi-librium [12,13], which is also characterized by a rigid
plasma rotation.The parallel current constraint (14), together with thePoisson equation (5), allows also computing the plasmacurrents induced by a magnetic ﬁeld perturbation in a non-neutral plasma conﬁned in a MP trap. These currents canbe thought of as an analog of the Pﬁrsch–Schlu¨ter currentsin Tokamaks [14] or the Stupakov currents in tandemmirrors [3,15]. However, they appear even in the case of
a uniform magnetic tilt which does not give rise to anymagnetic drift, whereas both Pﬁrsch’s–Schlu¨ter’s andStupakov’s currents srcinate from magnetic drifts.Considering the case of a weak magnetic perturbation,
1
, the unknown functions
and
N
can be sought inthe form
ð
;#;
Þ ¼
0
ð
;
Þ þ
1
ð
;#;
Þ
and
N
ð
;#
Þ ¼
N
0
ð
Þ þ
N
1
ð
;#
Þ
. The linearized versionsof Eqs. (14) and (5) can be readily solved in the region
far from the plasma column ends, where the unperturbedelectric potential
0
¼
0
ð
Þ
does not depend on
. Anexample of solution is shown in Fig. 1.Omitting the details of the calculations, in the case of auniform magnetic ﬁeld
B
tilted by a small angle
withrespecttotheaxisofthetrap,theperturbedpotentialcanbewritten as
1
ð
;#;
Þ ¼
ð
1
Þ
1
ð
Þð
=B
Þ
cos
#;
(15)while
N
1
¼
0
if
¼
0
in the midplane of the plasmacolumn (this can be accomplished with a proper choice
0.0 0.2 0.6 0.8 1.0.00.20.40.60.81.0
FIG. 1 (color online). Unperturbed density
n
0
(solid line),unperturbed electric potential
0
(dashed line), and radial part
ð
1
Þ
1
(dot-dashed line) of the perturbed potential used for calcu-lating the Pﬁrsch-Schlu¨ter currents vs the ﬂux radius
¼ð
2
=B
Þ
1
=
2
normalized over the radius
R
of the MP trap. Thedensity is normalized by its maximal value
n
, and the potentialsby
ð
T=e
Þð
a=
D
Þ
2
, with
D
½
T=
ð
4
e
2
n
Þ
1
=
2
the Debye length.The density proﬁles corresponds to a global thermal equilibriumwith a column radius
a=R
¼
0
:
25
(computed at
1
=
2
of themaximal density), and
D
=R
¼
0
:
05
.FIG. 2 (color online). Level curves of
1
(streamlines of theasymmetry-induced current density) on a ﬂux surface with
=R
¼
0
:
25
for a uniform magnetic ﬁeld with a tilt angle
¼
1
. Solid and dashed lines correspond to a clockwise and acounterclockwise ﬂowing current, respectively. The total lengthof the plasma column is
L
¼
8
R
. Other parameters are indicatedin Fig. 1.
PRL
101,
085006 (2008) PHYSICAL REVIEW LETTERS
week ending22 AUGUST 2008
085006-3
of the srcin of the system of coordinates). The contra-variant components of the electric current
j
¼
en
v
deter-mined from Eqs. (9), (11), and (12) assume the following
elegant form
j
i
¼
c@p@# ;c@p@ ;env
k
B
;
(16)where
env
k
B
¼
c@N
0
@
Z
1
@e
1
@#
exp
e
0
T
d:
(17)Introducing the ﬂux function
1
cT B
2
@N
0
@
Z
1
e
1
T
exp
e
0
T
d;
(18)the asymmetry-induced part of the electric current can becast in the vector form
j
1
¼
r
r
1
:
(19)This equation shows that the radial current density van-ishes,
j
11
¼
0
, and that the streamlines of
j
1
within a givenﬂux surface
¼
const
coincide with the contours
1
¼
const
. The level curves of
1
for a given ﬂux radius
¼ð
2
=B
Þ
1
=
2
in the case of a plasma density
N
0
ð
Þ
corre-sponding to a global thermal equilibrium [12,13] are drawn
in Fig. 2. These level curves show a similar topology ateach radius for the parallel current density
j
k
. It can beshown thatthis feature is nolonger validin thegeneral caseof a variable magnetic tilt. This contour plot shows thatasymmetry-induced currents are almost parallel to themagnetic ﬁeld lines on the majorpart of the plasma columnexcept in the proximity of the column ends where they areshorted out azimuthally. In addition, the radial proﬁles of
j
k
are peaked near the column edge,
ð
2
=B
Þ
1
=
2
a
,as shown in Fig. 3, and the maximum value of
j
k
is roughlyevaluated as
ð
cTn
=
32
D
B
Þð
a=
D
Þ
2
ð
L
2
=R
2
Þ
, where
n
is the peak density.In this Letter a parallel current constraint has beenderived that selects a class of admissible plasma equilibriain the trap in the presence of a nonuniform and a non-axisymmetric magnetic ﬁeld. In combination withPoisson’s equation this constraint provides a full set of equations for determining self-consistent equilibria of non-neutral plasmas in MP traps.This work was started during a visit of I.K. to theDepartment of Physics of the University of Milano thanksto support from the Cariplo Foundation and the LandauNetwork—Centro Volta. The authors are grateful toProfessor R. Pozzoli for useful discussions.
[1] J.H. Malmberg and C.F. Driscoll, Phys. Rev. Lett.
44
, 654(1980).[2] D.H.E. Dubin and T.M. O’Neil, Phys. Plasmas
5
, 1305(1998).[3] D.D. Ryutov and G.V. Stupakov, in
Reviews of PlasmaPhysics
, edited by B.B. Kadomtsev (Consultants Bureau,New York, 1987), Vol. 13, pp. 93–202.[4] G.W. Hart, Phys. Fluids B
3
, 2987 (1991).[5] R. Chu, J.S. Wurtele, J. Notte, A.J. Peurrung, andJ. Fajans, Phys. Fluids B
5
, 2378 (1993).[6] K. Gomberoff, J. Fajans, A. Friedman, D. Grote, J.-L. Vay,and J.S. Wurtele, Phys. Plasmas
14
, 102111 (2007).[7] T.S. Pedersen and A.H. Boozer, Phys. Rev. Lett.
88
,205002 (2002).[8] I. Kotelnikov, M. Rome´, and A. Kabantsev, Phys. Plasmas
13
, 092108 (2006).[9] L. Brillouin, Phys. Rev.
67
, 260 (1945).[10] I. Kotelnikov, M. Rome´, and R. Pozzoli, Phys. Lett. A
372
,1445 (2008).[11] M.D. Kruskal and R.M. Kulsrud, Phys. Fluids
1
, 265(1958).[12] T.M. O’Neil, Comments Plasma Phys. Control. Fusion
5
,213 (1980).[13] I. Kotelnikov, R. Pozzoli, and M. Rome´, Phys. Plasmas
7
,4396 (2000).[14] D. Pﬁrsch and A. Schluter, Max-Planck-Institut ReportNo. MPI/PA/7/62, 1962 (unpublished).[15] G.V. Stupakov, Fiz. Plazmy
5
, 871 (1987) [Sov. J. PlasmaPhys.
5
, 486 (1979)].FIG. 3 (color online). Radial proﬁles of the parallel currentdensity
j
k
for a uniform magnetic tilt along the ray
#
¼
=
2
,
¼
0
(midplane of the trap) for
a=R
¼
0
:
25
(solid lines),
a=R
¼
0
:
5
(dashed lines) and various values of
D
=R
(indicatedon the plot);
j
k
is normalized by
ð
cn
T=B
D
Þð
L=R
Þ
2
ð
a=
D
Þ
2
.
PRL
101,
085006 (2008) PHYSICAL REVIEW LETTERS
week ending22 AUGUST 2008
085006-4

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