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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
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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   eT  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   eT  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   11   @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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