UCRL-JRNL-220196

Current Driven Rotating Kink Mode in aPlasma Column with Non-Line-Tied FreeEnd

I. Furno, T. P. Intrator, D. D. Ryutov, S. Abbate, T.Madziwa-Nussinov, A. Light, L. Dorf, G. Lapenta

March 28, 2006

Physical Review Letters

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Current driven rotating kink mode in a plasma column with a non-line-tied free end

I. Furno1, T.P. Intrator1, D.D. Ryutov2, S. Abbate1, T. Madziwa-Nussinov1, A. Light1, L. Dorf1, and G. Lapenta1

1 Los Alamos National Laboratory, M.S. E526, Los Alamos, NM 87545, USA2 Lawrence Livermore National Laboratory, Livermore CA 94551

(Dated: March 12, 2006)

First experimental measurements are presented for the kink instability in a linear plasma columnwhich is insulated from an axial boundary by finite sheath resistivity. Instability threshold below theclassical Kruskal-Shafranov threshold, axially asymmetric mode structure and rotation are observed.These are accurately reproduced by a recent kink theory, which includes axial plasma flow and oneend of the plasma column that is free to move due to a non-line-tied boundary condition.

PACS numbers: 52.35.Py, 52.30.Cv, 52.70.Ds, 52.70.Kz

The current driven kink instability is a magnetohydro-dynamic (MHD) instability which a!ects current carry-ing plasmas in Nature and laboratory. The kink modestructure and stability condition are strongly dependenton the system geometry and the boundary conditions(BCs). Kruskal[1] and Shafranov [2] (hereafter referredto as KS) considered first the ideal MHD stability of acylindrical plasma column with magnetic field compo-nents (0, B!, Bz) using cylindrical coordinates (r, !, z).For an infinitely long (equivalent to periodic axial BCs)column, they obtained a linearly unstable helical kinkmode of structure " = ei(!+2"z/L) when the plasma cur-rent Ip exceeds the Kruskal-Shafranov limit

IKS = (2#)2a2Bz/(µ0L) (1)

where a and L are, respectively, the radius and lengthof the current channel, and " is the displacement of theplasma column from the equilibrium position. The KStheory has been quite successful in predicting the behav-ior of toroidal plasmas for which the periodic BCs yield aproper accounting for the finite length of the system. Inlinear systems, however, substantial deviations from KSpredictions can result from di!erent axial BCs.

The importance of the BCs has long been recognized[3,4] and is of particular relevance to the stability of line-tied flux ropes in space physics (c.f. Ref. [5] and sur-vey Ref. [6]), and astrophysical jets[7]. In recent years,there has been a renewed interest in the stability of aline-tied plasma column in laboratory devices (see Refs.[8–10] and references therein). The kink stability ofa plasma column with line-tied ends has been experi-mentally investigated in linear devices, where line-tyingis attributed to the presence of highly conducting endplates[11], and in open systems to a local discontinuityfor the Alfven velocity that forms a virtual boundaryaround the system[12, 13].

In this Letter, we experimentally investigate the exter-nal kink instability in conditions where one end of theplasma column is line-tied to the plasma source, and theother end is not line-tied and therefore free to slide overthe surface of the end-plate. The latter BC is a resultof plasma sheath resistance that insulates, at least par-

pI

Bz

1 m zy

x

B!

5 7

8

plasma column

61 2 3 4

FIG. 1: Experimental setup with diagnostics and coordinatesystem. (1) Vacuum vessel; (2) plasma gun; magnetic probesin (3) an axial array and (4) an azimuthal array; (5) externalcoils; (6) external anode. On the right, an axial cut nearthe external anode. The fast camera (8) is located at themidplane and views the plasma column along the x direction.The bidimensional magnetic probe (8) measures (!Bx, !By)at the edge. Also schematically shown is the plasma columnwhose end rotates at the external anode.

tially, the plasma from the end-plate. Compared to theline-tied case, we find significant di!erences in the kinkmode structure and lower critical current for the onset ofthe kink. The axial flow velocity, the direction of the ax-ial magnetic field, and the rotation frequency of the kinkmode are all correlated. The experimental results agreewith the predictions from a recent theory of the externalkink instability for a slender plasma column[14].

The experiments are conducted in the Reconnec-tion Scaling Experiment (RSX) which uses electrostaticplasma guns to generate the plasma[15]. Figure 1 showsthe experimental setup with a view of the plasma gunlocated at z = 0 and radially inserted into the center ofthe RSX cylindrical vacuum vessel (4 m length and radiusrwall = 0.2 m). A hydrogen arc plasma is ejected to forma cylindrical plasma column in a constant, uniform, axialmagnetic field of Bz = 120 Gauss generated by externalcoils, Fig. 1-(5). Electron density and temperature haveGaussian profiles with half-maximum radius r0 ! 2 cmsuch that r0 " rwall. Central electron density and tem-perature are ne0 = 1 # 3 $ 1019 cm!3 and Te0 = 5 # 14eV, and axially decrease away from the gun. The axial

2

00.2

0.4

0

3

time[ms]I p

[kA

]0.4 1.2 1.4

780 800 820 840time[µs]

-0.4

+0.40

-0.4

+0.40

-0.4

+0.40

azimuth. array z=48cm

-1

+10

-1

+10

-1

+10

z=48cm

z=62cm

axial array !="

#B

[G

auss

]

!="/2

!=0

!="

(a)

(b)

(c)

2 #B!

780 800 820 840time[µs]

!

(d)

z=14cm

#B

[G

auss

]!

FIG. 2: Time histories of (a) the plasma current; (b) az-imuthal magnetic field B! at (z = 48 cm, " = #/2, r = 15cm). Expanded view of the temporal evolution of !B! fromazimuthal (c) and axial (d) arrays of magnetic probes duringthe stationary phase of the discharge.

velocity vz ! 3 # 5 $ 104 m/s of the plasma has beenestimated by solving the momentum balance equationsfor dominantly axial flow constrained by experimentallymeasured axial gradients in pressure and density. A cur-rent is driven in the plasma by negatively biasing the gunwith respect to an external anode (50 cm2 stainless steelplate) which is electrically isolated from the vacuum ves-sel. The external anode can be positioned at di!erentaxial locations z = 0.3#3 m allowing the length L of thecurrent-carrying plasma column to be varied.

The MHD activity is monitored by two arrays of mag-netic probes that include a total of seven coils insertedin the vacuum vessel at r = 15 cm to measure the az-imuthal magnetic field B!. In the axial array, Fig. 1-(3),four magnetic probes are positioned at ! = #/2 (top ofthe vessel) and axially located at z = 0.14, 0.48, 0.62 and0.76 m. In the azimuthal array, Fig. 1-(4), the four mag-netic probes are located at z = 48 cm and azimuthallyequispaced by #/2. One magnetic probe is shared bythe two arrays. Signals are passively integrated beforedigitizing at 20 MHz.

In Fig. 2(a), the time evolution of the plasma currentmeasured at the external anode is shown for a dischargewith L = 0.92 m. The plasma current begins at t = 0when the bias is turned on (arc discharge starts at t = #1ms) and after ! 150 µs reaches a stationary phase whichlasts for ! 1.4 ms. The total plasma current Ip ! 320A during the stationary phase is determined by the biasvoltage and can be varied in the range Ip = 0.05# 1 kA.Figure 2(b) shows the time evolution of B! as measuredby the magnetic probe located at z = 48 cm, ! = #/2.

For an expanded time window during the stationaryphase, measurements of the perturbed azimuthal mag-

80 120 160 200I [A]

30 600 90z[cm]

!B

[a

.u.]

"

(a)

(b) (c)

ext. anode

25

35

45

Ro

t. f

req

. [kH

z]

p

plasma gun

#crit

!B

[G

auss

]"

0

0.2

0.4

0.8

1.2

fcrit

FIG. 3: (a) Amplitude of the perturbed azimuthal magneticfield, !B!, and (b) rotation frequency of the mode duringthe stationary phase as a function of the plasma current Ip.(c) Axial structure of !B! close to the instability thresholdfor Ip/Icrit ! 1.2 showing non-line-tied BC at the externalanode. The dot-dashed line shows the structure of !B! aspredicted from theory.

netic field $B! from the azimuthal, Fig. 2(c), and theaxial, Fig. 2(d), arrays of probes show oscillations withm = 1 azimuthal periodicity at the frequency f ! 50kHz. These oscillations are observed when the plasmacurrent Ip exceeds a current threshold Icrit (see below),but are not detectable for Ip < Icrit. The observed insta-bility growth time %G ! 4 µs is of the order of the axialAlfven time.

The m = 1 azimuthal structure is consistent with aplasma column rotating as a rigid body at frequency f .The phase of the signals from the axial array increaseslinearly along the z direction, Fig. 2(d), indicating arotating helical current channel. The kinked deformationis a right(left) handed helix if Jz · Bz > 0 (< 0), asexpected from a paramagnetic kink[12]. The direction ofrotation reverses when reversing the external magneticfield Bz. In both cases, the mode rotation is such thatthe helix always screws into the external anode. Boththe rotation frequency and the average amplitude of the$B! oscillations at the dominant frequency during thestationary phase scale linearly with Ip as shown in Fig.3(a,b) where the plasma current is varied in the range 86A < Ip < 190 A for a plasma column length L = 0.92 m.The plasma current corresponding to the limit $B! %0 corresponds to the kink stability threshold which forthese data is Icrit = 70 ± 7A. The rotation frequency atthe stability threshold is fcrit = 28 ± 3.5 kHz, Fig. 3(b).

Figure 3(c) shows the structure of the m = 1 modeclose to criticality for Ip ! 1.2 $ Icrit. While a line-tiedBC holds at the gun, the large amplitude of the mode atz & 76 cm can only occur if the plasma slides over theexternal anode and is therefore not line-tied. This is con-firmed by fast camera images of the plasma column at theexternal anode as shown in Fig. 4. Visible H# plasmaemission is imaged with a Cooke DiCam Pro intensifiedcharge-coupled device (CCD) camera[16] that views theexternal anode edge perpendicularly to the z axis, see

3

0

2

4

0 10 18

(d)(c)

1

2

0

(a)

external anode displaced column

equilibrium position

(b)

100 300 500

! [

cm]

L

I [A] ! [m]LpI [A]p

! L

2.5

cmyeq

ymax

FIG. 4: Optical and magnetic measurements near the exter-nal anode demonstrate PNLT BC. As the plasma gun shootsfrom right to left, a fast gated image of visible emission shows(a) the equilibrium, yeq, and (b) the displaced, ymax, posi-tion of the plasma during the rotation. (c) Measured opticaldisplacement $L = |ymax " yeq| for di!erent Ip values. (d)Magnetic field perturbation !B = (!B2

x +!B2y)1/2 versus Ip$L

compared with predictions from Eq. (2) (dashed line). $L

and !Bx,y are measured 1 cm in front of the external anode.

Fig. 1-(7). In Figs. 4(a-b) we show two fast-gated (200ns gate " Alfven time) images per discharge, that showrespectively the plasma column in the equilibrium posi-tion and at the maximum displacement &L later in timeduring the rotation.

The observed sliding of the plasma column over theexternal anode surface is not consistent with a line-tiedBC, which might be presumed at the external anodefor time scales less then a 100 # 200 µs resistive dif-fusion time. Deviations from line-tying may appear ifthe plasma is insulated from the external anode by afinite plasma sheath resistance. The degree of insula-tion is measured by the ratio ' of the Alfven transit toinductive decay times of the current[14]. We estimate' ! 15 from the formula ' ! (cs/vz)[c/(a(pi)]2)

1/2e

where cs = (Te/mi)1/2, )e = neTe/B2z and (pi is the

plasma frequency (derivation in Ref. [14]) for typicalplasma parameters a ! 2 cm, Te ! 10 eV, ne ! 1013

cm!3, vz/cs ! 1. For the present experiments, we con-clude that ' ' 1 and the plasma is insulated from theconducting boundary by finite sheath resistance. Ryu-tov and coauthors [14] have shown that for ' ' 1 theBC can be formulated as (*&/*z + i(k0/2)&)|z=L = 0where k0 = B!/(aBz) = µ0Ip/(2#a2Bz) with Cartesiancomplex displacement & = "x(z, t) + i"y(z, t). In the fol-lowing, we will refer to this condition as to perfect non-line-tying (PNLT) BC and we will show that it holds atthe external anode, after using an equivalent conditionthat can be directly compared with experimental data.

With a thin plasma column approximation (a "rwall, L) and assuming that r = 0 is the equilibriumposition, the Cartesian components of the perturbedmagnetic field at r = rp ' a can be expressed as

$Bx,y = (a2/r2p)Bz(±*"x,y/*z+k0"y,x). Using the PNLT

condition, in the vicinity of the external anode the per-turbed magnetic field can be expressed as

$B = ($B2x + $B2

y)1/2 = Ip&L(3µ0)/(4#r2p) (2)

where &L = ["x(L, t)2 + "y(L, t)2]1/2.We use magnetic measurements combined with plasma

column images near the external anode to test Eq. (2)which is equivalent to the PNLT condition. Gated im-ages such as those in Fig. 4(a,b) determine the maximumdisplacement ymax and equilibrium yeq positions of emis-sivity along a sight line parallel to the external anodesurface and axially spaced 1 cm in front of it. The im-age in Fig. 4(b) is synchronized with a signal from amagnetic probe located at the same axial location butazimuthally separated by #/2 to provide the position ofmaximum column displacement. In Fig. 4(c), the dis-placement &L = |ymax # yeq| is shown for a series of dis-charges that scan the plasma current 100 A< Ip < 500A. In Fig. 4(d), the magnetic perturbations ($Bx, $By)are measured at rp = 17 cm and 1 cm in front of the ex-ternal anode using a bi-dimensional magnetic probe. InFig. 4(d), the experimental data show excellent agree-ment with the PNLT predicted scaling in Eq. (2) (shownas a dashed line). We therefore conclude that PNLT BCapply to the external anode.

For a PNLT BC at z = L and with axial plasmaflow velocity vz, the eigensolution for the MHD equa-tion of motion (Eq. (57) in Ref. [14]) is & = C(eik+z #eik!z)ei!!i$t with axial wave numbers k± = ±(#/2L)[1((1#M2)1/2] where M = vz/(

)2vA) is the Alfven Mach

number, and vA is calculated using the average densityne = (2/a2)

! a0 ne(r)rdr. In Fig. 3(c), the axial struc-

ture of $B! from experimental data for Ip ! 1.2 $ Icrit

(squares) and from the theoretical eigensolution (dot-dashed line) are compared. The good agreement suggestsa robustness of the eigensolution for slightly super-criticalregimes. The total axial phase shift at criticality (notshown here) is also consistent within the experimentaluncertainties with the expected eigensolution for Machnumbers in the range M = 0.25# 0.4.

The critical current in this case is

Icrit =12

(2#a)2Bz

µ0L(1#M2)1/2 * IKS

2(1#M2)1/2 (3)

showing that the kink mode is unstable at half the KScurrent for vanishing flow and smaller currents when flowexists. The eigenfrequency ( at the criticality has a finitereal component

Re(() = #)

2#vA

LM(1#M2)1/2 (4)

The perturbed plasma column rotates at frequencyRe(() = 2#f , constant along the column. The rota-tion is driven by axial plasma flow from the gun to the

4

50

150

250

10

30

50

[k

Hz]

I

[A

]cr

it

0.5 1.0 1.5 2.01/L [m ]-1

f crit

(a)(b)

0.5 1.0 1.5 2.01/L [m ]-1

FIG. 5: For three di!erent column lengths (L =0.48, 0.92, 1.38 m): (a) critical current Icrit (b) rotation fre-quency at the criticality fcrit as a function of the inverse col-umn length 1/L. The grey region indicates predictions fromKS theory. Dotted lines show linear fits.

external anode along the helically-perturbed column aswell as a Doppler shifted k± · vz, such that the helix al-ways screws into the external anode, consistently withthe experimental observations.

To investigate the applicability of Eqs. (3) - (4), figure5 shows (a) the critical current and (b) the rotation fre-quency at the criticality as a function of the inverse col-umn length 1/L which is in agreement with a predictedlinear scaling.

The system comprising Eqs. (3) -( 4) provides a theo-retical relationship between Icrit, fcrit, a, M and the av-erage plasma density (or Alfven speed). This was solvedusing experimental measurements of the electron den-sity profile at criticality together with deduced scalingsIcrit = 59/L [A] and fcrit = 2.4 $ 104/L [Hz] from thedata in Fig. 5. We obtained a radius a = 2 ± 0.2 cmwhich is in agreement with the experiment (see Fig. 6)and Mach number M = 0.25#0.4. The corresponding ax-ial flow velocity vz ! 3# 4.8$ 104m/s is also consistentwith our independent estimate. In Fig. 5, predictionsfrom KS theory using the computed a are compared tothe experimental data.

For the external kink, the plasma radius a is expectedat the position where conductivity, magnetic di!usiontime, and axial current density Jz(r) are negligibly small.This is experimentally tested using profile measurementsat the criticality. Figure 6 shows a Bennet profile least-squares fit of the axial current density Jz(r) at the in-stability threshold, as obtained from azimuthal magneticfield measurements at z = 0.48 m and L = 0.92. Thepreviously determined radius a ! 2 ± 0.2 cm is locatedat the edge of the profile where Jz(a) ! 0.05Jz(r = 0) atthe resolution limit of the measurement. Similar conclu-sions follow from the magnetic di!usivity, derived fromthe measured electron temperature profile and the ob-served instability growth time %G.

In summary, we have presented first experimental re-sults for the current driven kink instability in a plasmacolumn with one free end that is not line-tide to itsboundary due to a finite sheath resistance. Mode struc-ture and instability threshold are significantly di!erent

0 2 4 6 8r[cm]

J [

A/m

/1

0

]2

4z 0

5

10

15

B [

Gau

ss]

!

0

4

8

noise level

FIG. 6: Radial profiles of current density (solid line) frommeasured azimuthal magnetic field (squares) at the criticality.Vertical grey region indicates the error bounded estimate ofa at the edge of the current channel.

from predictions from the Kruskal-Shafranov theory andare accurately reproduced by a recently developed kinktheory. The finite rotation of the mode, which is also ob-served in other laboratory devices[11, 13], is due to theplasma flow along the helically kinked plasma column.

Though not addressed in this Letter, the e!ect of anon-line-tied end should be important in the non-linearstage of the kink dynamics. For the relaxing kinkedplasma column, a destabilizing J! $ Bz force coexistswith a stabilizing curvature force due to axial field linebending. The free end BC allows displacements with re-duced curvature and restoring force which may result ina helical state saturated at larger level compared to theline-tied case. Numerical simulations are being imple-mented for the non-linear evolution of the kink mode.

Support by Los Alamos Laboratory Directed Researchand Development - Exploratory Research program, andAssociazione Sviluppo Piemonte for S. Abbate is grate-fully aknowledged.

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