R.E. Terry and N.R. Pereira- Leakage currents outside an imploding Z pinch

8/3/2019 R.E. Terry and N.R. Pereira- Leakage currents outside an imploding Z pinch

http://slidepdf.com/reader/full/re-terry-and-nr-pereira-leakage-currents-outside-an-imploding-z-pinch 1/9

Leakage currents outside an imploding 2 pinch

R. E. TerryNaval Research Laboratory, Washington, DC 203 75

N. FL PereiraBerkeley Research Associates, Springfield, Virginia 22150

(Received7 June 1989;accepted24 August 1990)

Leakagecurrents outside a pulse-line-drivenZ pinch are considered n two circumstances. n

the initial stageof the pinch a non-neu tral electron flow can arise before magnetic nsulation isestablished.The relative importance of such currents is estimated n terms of diode impedanceand pulse-linedimensions. n the later stagesof the pinch a neutralized current flow can arisein any tenuousplasma hat may be present n the pinch periphery. The effectsof the neutralcurrent a re estimated hrough self-similar solutions to a gyrokinetic equation. The collisionlessplasmacorona can contain an important fraction of the implosion energy.

I. INTRODUCTION

An imploding Z pinch can be used to generate arge

amounts of kilovolt x rays.’ In the region of interest the yieldin kilovolt x rays s roughly proportional to the fourth powerof the peak current into the diode,2 n agreementwith theo-ry.” The modelsassume hat the current density n the pinchis constant, and that all the current that goes nto the diodeflows through the pinch. It is very difficult to confirm experi-mentally that no current flows outside the Z pinch. How-ever, the strong dependence f x-ray yield on the currentimplies that a small current leakage hrough the pinch pe-riphery could reduce he pinch’s x-radiation efficiency sub-stantially.

In this paperwe consider he eakage urrent outsidea Zpinch in two circumstances.One is a non-neutral electronflow in the initial stageof the pinch; the other is a neutralizedcurrent flow in a tenuousplasma hat may be present n thepinch periphery.

The non-neutral electron flow at the start of the pul secomes rom the onset of magnetic insulation. For relevantparameters (e.g., Is 1 MA and pinch radius r=: 1 cm) themagnetic field in the vacuum outside the pinch is approxi-mately 20 T (200 kG), sufficiently strong to provide mag-netic insulation. However, magnetic nsulation needssometime to become established. In high-impedance vacuumtransmission ines a small current pul se4 s seenat the frontend of the pulse, and a similar transient may have been ob-

served n a low-impedanceZ pinch.5 Subsequently he cur-rent becomes ufficiently large to cut off this transient. Ac-cording to our computer simulations the transient current iscomposedof an electron sheath hat is captured by the mag-netic field. When magnetic insulation sets in the electronsheath rolls up into relatively stable vortices.

The neutralized current flow in the pinch peripherywould take place at a later time, during pinch compression.In the initial stage of the pulse the material in the diode israpidly ionized, a complicated processoutside the scopeofthis work. The current compresseshe bulk of the material,which becomes dense ollisional plasma hat is accelerated

toward the axis. The behavior of this plasma s describedbythe resistive radiation-hydromagnetic equations.6However,

in computations with these equations’ the exterior of theplasmasometimesbecomes oo conductive n an unphysicalway-the plasma temperature T as computed can exceed100 keV. In this case he standard Spitzer formula for theconductivity cr n a collisional plasma, (Ta: T3’2, predicts ahighly conductive plasma in the pinch periphery that pre-vents the current from entering the Z pinch. The problemdisappearswhen the currents in the plasma periphery arecomputed by keeping account of the transition from a plas-ma dominated by collisions to a collisionless plasma. Forpractical purposes he plasma becomescollisionless whenthe electron-electron collision time exceeds typical hydro-dynamic time scale.

In the strongly magnetizedcollisionlessexterior plasmathe standard equations of resistive hydrodynamics are not

valid. Instead, the plasma electrons drift in the confiningmagnetic field Be (r) and the combined applied and space-charge electric field E = [& (z,r),E, (z,r) 1. The E, x Be

drift inward generally dominates.The current densityJ, nolonger satisfiesOhm’s law, but can be derived by setting thepower density J, *E, equal to the change n kinetic energydensity.

The collisional core of the Z pinch implodes differentlythan the plasmaperiphery. In the core he colhsionsguaran-tee an isotropic velocity distribution, and an isotropic pres-sure, but in the periphery the electronsconserve heir angu-lar momentum pe as they travel along the magnetic field

lines around the pinch axis. As the radius r decreasesheelectron velocity v, increasesas l/r, and the parallel pres-sure P,, a vi increasesas l/?. Also, the magnetic momentaround the magnetic ield, p a mv: /Be, is conserved.There-fore the perpendicular pressurePL a v: increases inearlywith the magnetic field B, (r), which is proportional to l/r.The ratio of parallel to perpendicular components of thepressure,P,, Pl a r - ‘, can become mportant.

A simple estimate or any peripheral current can be ob-tained from the Bennett relation applied separately o thecore and the periphery. One has 2 NT, with I the current,N the line density of the particles, and T the temperature.Therefore,

in order of magnitude Iperi /rcore

- (N,,,, Z”ri/Nco,e T,,, ) *‘2,where NcoreCperijs the number

195 Phys. Fluids B 3 (I), January 1991 0899-8221 I91 /010195-09$02.00 @ 1990 American Institute of Physics 195

Downloaded 04 Jan 2010 to 132.250.22.9. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp



of particles n the core (periphery) of the pinch, Tcoretperij sthe applicable emperature, and Itoral = Iperi + to,.

A reasonablemassper unit length for the core of the Zpinch is perhaps 100pg/cm. The initial particle density inthe Z-pinch periphery is on the order of n z 10r3/cm3 (for0.1 Pa or 1mTorr), spreadover, perhaps, 100cm*. The massper unit length on the outside of the pinch is thus -0.01,ug/cm. Setting the typical temperatures about equal sug-gestsa relative current loss of - I % in such an equilibrium,

which is negligible. However, when Twri ) T,,, after com-pressionof the periphery, and/or when pressurebalance snot applicable, he current loss through the periphery maybecome arger.

In Sec. III of this paper we treat current conductionthrough the periphery with the gyrokinetic formulation ofBernstein and Catto.’ This model is obtained from the sin-gle-particlekinetic equation by introducing higher-order ac-curate invariants in the velocity spaceand then averagingover the gyrophase.We employ a current density obtainedfrom the velocity moment equation of this gyrokinetic for-mulation, which expresseshe momentum balancebetween

particles and fields. The current density then nvolvesa pres-sure tensor.Analogous to self-similar hydromagnetic flows’ with

scalar pressure, he gyrokinetic momentum equation alsoadmits self-similar distribution function solutions that ex-hibit tensor pressure. If PI, #Pi these special solutions re-main separablebut exhibit three characteristic frequenciesinsteadof two, and, when we retain displacementcurrent, afourth frequency is added. A particular example from theself-similar theory is usedhere o illustrate the effectsof peri-pheral gyrokinetic plasma, future work will discuss thesesolutions n more depth.

The retention of displacementcurrent is required for the

energy ransport from an external circuit through the vacu-um region to the plasma oad. The principal consequencesthat the well-known plasma dielectric constantE-(1 +-c2/&) “2 becomes significant (c, is the Alfvenspeed).The plasmadielectric, the radial accelerationof plas-ma mass,and the two tensor pressurecomponentsprovide afour-fold energy sink between he core plasma oad and theexternal driver circuit.

Ii. ESTABLlSHMENT OF MAGNETIC INSULATIONAROUND a 2 PINCH

In the initial stageof the pinch all material is contained

in the conductor, and outside of the conductor is vacuum.Free electron current in the diode can be computed with aparticle in cell code. One computation uses he geometry ofFig. 1, which is typical for Z pinches,The cathode s a cylin-der with radius 3.4 cm, inside an anodewith radius 5 cm, anda 3 cm wide anode-cathode gap forming the diode. The di-ode s bridgedby an deal conductor that mimics the Z pinch.The Z pinch radius, 0.6 cm, is representativeof multiple wireloadsbut substantially smaller than the initial radius of a gaspuff. Electrons can be emitted from the cathode surfacesasindicated. The emission assumes pace-chargeimited flow,i.e., electron emission continues until the normal electric

tield vanishes. ons are ignored in our computations.

6

FIG. 1. Geometry of the Z-pinch diode as used in the computation. Thecathode is a cylinder of radius 3.4 cm at 3 cm away from an anode withradius 5 cm. The 2 pinch is mocked up as a 0.6 cm radius conductor thatconnectscathode and anode. The electron emission surfaces are indicatedon the cathode. The voltage source s connectedat the entrance of the diode,6 cm away from the anode plane. Initially the diode vacuum contains noelectrons: The figure shows he electron positions at 4.4 nsec nto the simu-

lation.

Figure 2 is the assumed lectrical pulse at the diode. Thevoltage ises inearly on a 5 nsec ime scale o over 1 MV, Thecurrent is quadratic in time in agreementwith i = V/L. Atthe end of the simulation the current exceeds100 kA, for amagnetic field at the Z-pinch edgeup to about 4 T (40 kG) ,The current in free electrons s the lowest line in F ig. 2. Thecurrent starts once he electric field in the diode feedexceedsthe field emission threshold, taken to be 200 kV/cm in thecode. This occurs at about 1.5 nsec.

When the first electronsare emitted the current throughthe Z pinch is only 15 kA, insufficient for magnetic insula-tion, Therefore the free electronssimply cross he diode gap.Later in time theseelectronsbecome rapped by the magnet-ic field from the Z-pinch current. These ntermediate stagesare not shown, but they can be inferred from Fig. 1, which

TlME (ns)

FIG. 2. The voltage Yand the current iof the electrical pulse at the diodeentrance. The dashed line at the bottom gives the leakage current in free

electrons coming off the cathode shank.

196 Phys. Fluids B, Vol. 3, No. ‘I, January 1991 R. E. Terry and N. R. Pereira 196




shows he distribution of free electrons at the end of the run.The magnetically insulated space-charge heath n the cur-rent feed appears o stream off the cathode into the diodealready filled with electrons emitted previously. The sheathis clearly unstable, olling up into vortex-like structures thatpersist as they move toward the center conductor.

There is an exact analogy betweenstrongly magnetizednon-neutral electron beams and inviscid fluid Aow.‘@~~ ntwo-dimensional Cartesian geometry with constant magnet-

ic field B,, the combination of charge density, permittivity,and magnetic field in the form p/e$, corresponds to thevorticity [I = VXU,, where U, is the two-dimensionaldrift velocity. This drift velocity U, can be derived from thestreamfunction II, as U, = ( - @/&,LJ+/&>, wherev’* = gz.

The exact correspondencebreaks down in cylindricalgeometry because he magnetic field varies with radius,

& = pd /27r-r; now

VxU, =-p -E,d(l/Bo)

%&I ar .(1)

The last term is small compared to the first, except perhapsnear the central conductor, where the blob size s compara-ble to the geometrica l scale.Hence n our diode the analogywith inviscid shear low is largely valid.

The literature on magnetic nsulation’“” contains esti-mates along the same ines as the above. In addition, it isshown in detail that the single-particle considerations used

here correspond well to a treatment that includes the self-electric and self-magnetic ields.

The magnetically nsulated space-charge heathcomingoff the cathode s then analogous o a vortex sheath. t is wellknown that such a sheath s unstable,and that the instabilitydevelops into coherent vortex regions reminiscent of thespace-charge lobs seen n our simulation. It is also knownthat the vortex regions are relatively stable, similar to thepersistenceof our blobs. Furthermore, interaction betweenthe blobs can be treated as the motion of two-dimensional

vortex lines.

A typical Z-pinch diode is about 3 cm wide, and theinductance L z 6 nH. A typical pulse rise time may be s 12nsec, or a diode impedanceof about 0.5 a. In contrast, thevacuum impedanceZ,/21-r = 60 a, and the impedance atiois &,. The leakageelectron current 1, is thus always a smallfraction of the total current I, whence , can be ignored.

The electron blobs appear to be self-contained entities,consistent with E X B drift motion for the strongly magne-tized electrons n the blob’s space-charge ield. In this casethe blob electrons remain on equipotential lines, since thedrift is perpendicular to the potential gradient,(ExB)*E = 0. In a cylindrically symmetric blob the chargeconvectedby the drifting electrons s then invariant, and theblob is a stationary entity. Qualitatively similar results arefound in simulations with other parameters, e.g., a slowervoltage rise time, or a smaller Z-pinch inductance more rep-resentativeof a gas puff.

However, for singleexploding wires the vacuum currentmay become comparable to the conduction current. As anexample, he inductance of a single 25pm radius wire load ina 3 cm wide diode is about 45 nH. For a fast ( 10 nsec) rise

time pulse he factor Z,/Z,-0. 1, but a relatively small gapsizepartly compensates.When ri/dz 3, for example, ,/I isof the order fo. In this case here should be a sizable eakagecurrent in free electrons, potentially generating a measura-ble’ bremsstrahlung x-ray signal. If corroborated by furtherwork the bremsstrahlung x-ray signal may be an excellentindicator for the arrival time of the electrical pulse at the Zpinch.

III. COLLISIONLESS PLASMA PERIPHERY

The leakage urrent in free electrons or the run in Fig. 2peaksat I5 kA around 3.5 nsec,and decreaseshereafter. Atthese ater times the current is magnetically insulated, andits magnitude can be estimatedby considering he drift cur-rent in the thin space-charge heath parallel to the cathodeshank (seeFig. 1) . The current in the detachedblobs can beconsidered equal to that flowing along the cathode shanksince here are no sourcesor sinks n the detachment region.

Once magnetic insulation has been established,as dis-cussed n Sec. I, the current flows principally through the

pinch, and the pinch accelerates nward. Eventually thepinch stagnateson axis, forming a dense,current-carryingcylinder of plasma. However, this pinch core could be sur-rounded by a periphery of collisionlessplasma, rom Z-pinchmaterial that did not fully implode or from background gasthat was swept n from the outer regionsof the diode. In thissection we consider he current that might be carried by sucha plasma.

In an insulating magnetic field B, the sheath electrons The peripheral plasma might be different in Z pinchesdrift in the axial direction along the cathode shank with drift that start from an injected gas on one hand, and multiplevelocity U, z Er X BJB i. The electric field varies hrough- wires on the other. The injected gas orms a plume with theout the electron sheath rom zero at the metal surface of the highest density opposite he nozzle opening, and wide wingscathode to E, at the top of the sheath,but the magnetic field of decreasing density. The initial breakdown and currentis mainly from the central current; hence B, zpJ/2nri. conduction should occur along the spatial contour in the

The current in the electron sheath 1, is then approximately1, z2m,aU,. The surface charge density u of the cathodesheath s (T= +Yr, where E, is the normal electric field atthe outside of the electron sheath. Since the sheath is thincompared to the width d of the pulse ine, and d is less hanthe inner radius ri, the electric field can be approximated byE, =: V/d. Usually the rise time r is so fast that the voltagecanbe approximatedby the inductive voltageVZ LI /T, withL the diode i nductance and I the total d iode current.

The relative importance of early free electron leakagecurrent to the total pulse ine current then becomes

$ =: #(g(&)*, (2)

where Z, = (,uJe,,) “’ =pcloc= 377 R, and Z, = V/I

z L /T is the total diode impedance.After magnetic nsula-tion is established n the pulse line this leakagecurrent de-cays away.

197 Phys. Fluids B, Vol. 3, No. 1, January 1991 R. E. Terry and N. R. Pereira 197




diode, which (given the electric field and material profiles) Thornhill. In Spielman’s experiment the puff mass wasis at the minimum of the Paschencurve, and thus should much larger, the front end nductance was reduced, he neu-leavesome gas outside the initial current channel. This gas tral how speedwas ncreased o Mach 8 (rather than Machmay be ionized later to form the periphery. With wires the 4j, and the return current anodeplane wasboth a somewhatexterior plasma would come from the early blowoff as the thinner wire meshand closer o the cathode. t would appearwire load is heated, or it might not exist at all. The ambient that the energy ransfer was quite efficient, perhaps ndicat-gas n the diode can be another source of tenuous plasma, ing that corona1plasma was either absent or if presentnotperhaps esulting in ion densitiesas high as 10L2/cm3 and behaving as a strong energy sink. The best data presentlyelectron densities< 10’3/cm3). available cannot resolve hesepossibilities.

Ironically, sometimes’8 here s current leakage hrougha tenuousplasmaon axis nside the wires: when the wires areheavy the outer part of the wire is shed continuouslythroughout the current pulse. This plasma mplodes on axisbefore he original wires have moved, and diverts part of thecurrent.

More recent wire load implosions’9-2’ involving Ni ar-rays ndicate a fairly weak nductive current notch and radi-ation emission peaking after the plasma starts to expand,When the diode energy low is examined he energy nput tothe load from classical Ohmic heating and inductancechange s apparently insufficient to account for the total en-ergy radiated. Someenergy must be removed rom the vacu-

um magnetic ield in order to account for the discrepancy.Now such an analysis s not supported by an independentmeasure f the current distribution in the pinch; indeed, helocation of the current path is simp1y aken as he boundaryof luminosity in pinhole photos. Should the current flowdeeper n the plasma so hat the peakcompression s greater,then some mprovement in the energy discrepancywould berealized nsofar as he effective nductance changewould belarger. Yet it is difficult to account for the energy by thiseffect alone because he current path would have to be verydeep ( -0.01 mm). Such a current filament would be muchthinner, in fact, than the observed Ni) L-shell emission ila-

ment. If, however, he outer load regionsare n a gyrokineticcoronal imit, as describedbelow, then the work done on thegyrokinetic plasma would be absorbed from the vacuummagnetic ield and at least partially transferred to the interi-or pinch after stagnation. In consonancewith observation,such motion would not provide much ofan inductive currentnotch, because f the ability of this coronal plasma o hold amore constant current while a load stagnation occurs.

Hence there is a good reason o believe hat only a de-tailed examination of the differences n behavior between aspuffs and imploding wires would contain some direct evi-dence or current leakage hroughout a peripheral plasma.The differences o be examined must rest upon cIear mea-sures of the load mass n a gas puff, which is not generaliyavailable, and good measuresof energy flow in the dioderegion. If corona plasma is produced mostly from back-ground gas, then no major wire/puff differenceswould beexpected. f the corona plasma n gas puffs is due to an in-complete sweep-up of the load gas, then wi re/puff differ-enceswould depend on the actual puff load mass (and itsdistribution) as compared to the extent (and distribution)

of any early wire blowoKThe remainder of this paper assumes hat the Z-pinch

periphery contains a collisionlessplasma, with density andtemperature fixed to valueswithin the known experimentaland theoretical limits. For the number and energy densitiesone could reasonablyassociatewith such a “test” plasma,asimple calculation will show how the peripheral plasmacanbecome mportant for Z-pinch implosions.

What is the current density n the collisionlessperipherysurrounding the current carrying Z pinch? In a dense lasmathe collisions are the dominant influence balancing the mo-tion of the plasma electrons against the accelerationby the

electric field. The resulting collisional electron drift, andthus the current density Jis proportional to the electric field23, J = 0%. The electrical power input into the plasmaends

up in increased andom velocities of the plasma particles,viz., Ohmic heating.

Moreover, comparing theoretical and experimental en-ergy coupling to Ar gas puffs provides further indirect evi-denceof exterior energy sinks, which could be tenuousperi-pheral plasma.The work of Thornhil12’ suggests hat lowermass puff loads exhibit radiative K-shell yields lower thanexpected f all the energy ransferred to the load cavity was,in fact, coupled o the central plasma oad. The experimentalyields were more heavily weighted to the L shell, suggestinga lower load temperature at stagnation-even though thepuff gas oad acceptedmore generator energy than the hy-dromagnetic calculations could predict. This is again pre-cisely what would be expectedshould an intervening energysink arise n the load periphery and “soften” the implosion.

In contrast, more recent gas puff implosionsz3show afairly pronounced nductive current notchesas ittle as 5 cmfrom the load. The design changes o the gas puff load werequite substantial n comparison o the experiment studied by

In the absence fcollisions the magnetizedplasma n theZ-pinch periphery can absorb energy from the electricalpulse. In this case he energy doesnot go into random mo-tion of the plasmaparticles, but instead s put into organizedmotion. A strongly magnetizedplasma n an electric field isnot acceleratedalong the field, but drifts perpendicuiar tothe electric and magnetic fields with a velocityW = cU= - (E/B). For constant E the drift speed e-mains constant, but when the electric field increases n timethe plasma accelerates. The kinetic energy densityY- = p W2/2 increases p denotes he plasma massdensity)and electrical power is absorbed. Likewise, electrons thatExB drift into a region of increasedmagnetic field spin up,increasing heir perpendicularenergyw, while the magneticmomentp - w, /B staysconstant. The increase n w, , or theincreasedmagnetization, s another power sink. In a cylin-drically symmetric geometry with B- i/r the particle’s an-gular momentum about the axis, ps = mnt,, is conserved;therefore a drift toward the axis ncreases he parallel energywII as l/3. Further details of the description are developed

198 Phys. Flui ds B, Vol. 3, No. 1, January 1991 R. E. Terry and N. FL Pereira 198




in the Appendix, where it is shown how special solutions of density bounded above by this constraint when the densitythe gyrokinetic equation can be used to model the coronal scale n, = (2.366~ 109/A) [&,,/U(g, )]2(1C,,,/1 MA)*plasma. ( cmh3), and the density profile

In particular, the limit of homogeneouscompressionand special nitial conditions allows the solutions o simplify.The dependentvariablesand their derivatives are then func-tions of the single self-similar invariant 5 = r/r. a (7). Herer. = r. /I, is a dimensionless cale ength that arisesnatural-ly in the theory with ct,, = &,and r = t /&,. The time scale o s

tied to a particular implosion time scale.The variable a ( 7)contains the sole time dependence, e.g., U= irr,,

DU/Dr = 6r,. Four primary frequenciesarise. The oscilla-tory frequency is related to the magnetic field pressure;rA, =wA ’ is the transit time of an AlfvCn wave with velocityc, through the pinch scale length. Two new frequencies,expressedas ratios of thermal velocities to light speed,arerelated to the anisotropy. The parallel pressure requency s0; = cf /c2, and the perpendicular pressure frequency,CUTc:/c*. The displacement current frequency, r$ = w, ’is a measureof the overall dielectric strength characterizingany particular flow.

nw =-g-(& - 1)

x &-T -Jr&- >+1 (3a)are evaluated for an enclosed current

m = Lre *(l/J=-l/J-+1) of a fewmega-amperes.

The selectionof rr to set he density leavesarbitrary thechoice of either wg or wA For this example he selectionof amoderately strong dielectric coefficient,

E(g)/Eo = r&j- (3b)

(peaking at about 150 EJ implies, together with r., a valuefor 0,. Similar expressions or the temperature profiles areeasily derived from Eqs. (A9b) and (A9c).

To assess he role of this sort o f collisionless currentchannel in the evolution of a Z pinch, we select a set of sel f-similar profile functions, which is compatible with the (rath-er sparse) information available on the coronal p lasma andexamine the evolution of the specia l solution. In particular,what fraction of “new” current is absorbed nto such a cor-ona after it is formed under the magnetizing influence ofearly interior currents flowing only in the densepinch? Whatfraction of the incoming energy low from the driving circuitis diverted into this corona, and what circuit waveform be-havior is characteristic of such an external plasma ayer?

We can formulate thesequestions n a realistic contextby specializing he self-similar equation of motion derived nthe Appendix to portray a coronal rundown in a Z-pinchdiode. Each of the four independent requenciesappearing n

42

SC- +wI ___ +&*c-2 a* a a

can be set to produce a desired result in the solution. In theexample developed here w: and of are selected o corre-spond to I( 11) emperatures of 75 60) eV, which is consis-tent with ionization at a few electron volts and subsequentOhmic heating by about a factor 10 before a “collisionless”state is obtained.24 This leaves u&, w:, a radial extent[g< ,<, 1, and an initial velocity U(g, ) to be developed.

Since r. = w&/w, = UC&J, /tr,, the selection of either &,or U(g, ) ( ~7.33 X 10’ cm/set) is, for fixed rr , guided bythe typical collapse time of pinch loads, taken here to beabout 30 nsecover a radial extent of 1.5cm. In keeping withcommon pinch diode dimensions, the radial extent of thecorona is taken to be ~4 cm. Hence the values or 6, and6, will correspond o<, = 1.5cm/&r,, {, = 5.5 cm/&r,;I,, = 30 cm, since the collapse time is in the nanosecondrange.

Such a group of solutions s shown n Fig. 3. To summa-rize, initially the interior boundary of the solution domain is

given a velocity of 2.0 x 10’ (cm/set), which corresponds oa typical peak implosion velocity of Z-pinch loads in ma-chines of moderate energy. The interior coronal (H + ) iondensity is about 1.04~ 1015 m - 3 per MA* of interior cur-rent; it extends or 4 cm outside he interior radius of 1.5cm.[The factor MA* is obtained from Eq. (3a), in analogy tothe Bennett pinch relation.] The initial coronal conductioncurrent is 34% of the core current. The density n(g) decaysnearly a l/g’, while the perpendicu lar nearly h, is nearlya 5; Eq. (A9b) produces a parallel energy profile h,, alsonearly a{*. Since c,, is an azimuthal thermal speedat thescale radius, and since the thermal speedprofile is propor-

tional to the scale radius, the averageangular momentumprofile isp, - r,v, -c,, 6 *r. /, so that L f -6 4( Ior. ) ‘ci asde-manded by the self-similarity constraints. The net result isthat the azimuthal component P,, = pL i /? a novf is nearlyindependent of radius. The perpendicular componentgPl a &I& is also weakly dependenton radius.

The initialization conditions required to produce thissort of energy profile are not precisely known, but the fact

Gyrokinetlc Profiles : R* = 2.7 1 E-00 1

;.I

0.2 0.4 0.6 0.6

5

The value inferred for r* must be guided by the den-sities implied for the profile. With an upper limit of 10”ions/cm3, suggested y the lower limit of schlieren measure-

ments, values of r. on the order of [0.25, 0.3121 keep the

FIG. 3. Examples of initial self-similar profil es are shown. As labeled thecurves are normalized to n,= 1.04X 10’” cm-’ MA-*, I= 1.34X1,,,,,

h = 150, h,, = 60 eV, and h, = 75 eV.





that P,, - {Pl is consistent with a simple model for the earlyphase. f thin peripheral gas s photoionized and Ohmicallyheatedwhile in an azimuthally directed magnetic ield, thenit will evolve o a collisionlessstate principally through elec-tron-neutral collisions. The conductivity will smoothlyevolve o a small value as unaway electronsseparate n ener-gy from the bulk electrons hat interact readily with the neu-trals and ons. Since he runaway electronsare captured n aradial ExB drift they are confined to the r-z componentsof

the velocity space unless scattered into the 19 omponent.The energydumped nto the r-z motion dependson the timehistory of the conductivity and local E, field, but the rate ofscattering into the B direction depends on the number ofscattering neutrals and ions available at any radial location.Since he number ofsuch scattering centers n any azimuthalflux tube s linear in the radial coordinate, the power divert-ed to the azimuthal direction increases cc . Hence, for anyparticular time history of r-z heating, the final energydumped n the 6 direction will be proportionately greater atlarger radius. The energy density in the r-z component isdeposited n each volume independently, but at a fixed radi-us scattering n the whole volume swept out by a gyratingelectron diverts energy to the azimuthal component.

Insofar as these self-similar flows fix the (conduction)current partition between he core and the periphery, theyare only useful as approximate models of the load as t entersthe final rundown and stagnation phase. At this time theinterior load is expected o be hot enough to curtail furthermagneticdiffusion, while the peripheral plasma s forced toconserve lux unless t becomes urbulent as a result of mi-croinstabilities. Since much of the energy transfer to a Zpinch s due o the load’s nductancechange,a fair fraction ofthis energy s transferred in the last phaseof the rundown.The sel f-similar models can therefore offer a simple treat-

ment of this important phase of the motion by initializingtheir parameters o pinch load conditions after the implosionis well underway.

The dynamic consequencesor an imploding l oad sur-roundedby a coronal plasmaof this sort are shown n Fig. 4.The final 30 nsec of the rundown are strongly modified by

Gyrokinetic Solution for R’ = 2.?lE-001

%Sl

FIG. 4. Time histori es of the current Jo,a, /iC,,rCt xterior electric field

E(&, ), radius (a), and absorbedenergyE, (7) over a 50 nsec undown. As

labeled the curves are normalized to I = 1.45XiC,,,e, E, = 1.45MV/cm MA*, E, = 143 kJ/cm MA’, and a = 1.

the collisionlessmagnetoplasma orona. As the radial com-pression ( -a - ’ ) reaches ts maximum, the exterior elec-tric field reversesand the peak in the absorbed energy isobtained. While the conduction current is constant over theinterval, the total current ri ses slightly and exhibits a verymild inductive notch. All this time dependence,about a+ 11% current variation, is due to the displacement cur-

rent, which alternately chargesand discharges he plasmadielectric while a fixed conduction current leaks across he

diode.If the load were a bare slug, its motional impedancewould become more important in the rundown. Furtherpower absorption by the Z pinch would be strongly curtailedand one might expecta 30%-40% “inductive notch” to de-velop in the current waveform. The gyrokinetic corona has

changed his picture substantially. Now further power ab-sorption can occur, but most of the energy goes no furtherthan the corona. The implosion and stagnation occur on anormal time scale but the interior load current is fixed.Whereas he coronal conduction current is fixed as well, thedisplacementcurrent component can vary in strength from5%-20% of rhe core current, depending on the strength of

the plasmadielectric that hasbeen rapped in the periphery,At fixed interior current the stronger dielectric cases end toimplode lesssharply asa result of the larger massand slower

AlfvCn speeds.The slower time scale mplies less displace-ment current because he accelerations are weak. In con-trast, the weaker dielectric casesprovide a more prominentdisplacement current component because he accelerationsare stronger. During the stagnation, energy is deposited nthe coronal fields and matter. The energy given the coronalplasma s releasedwhen the motion reverses,as a result ofthe recoil of the gyrokinetic plasma from the angular mo-mentum and magnetization stresses,which build up as the

radius gets smaller and finally cannot be overcome.In the case llustrated previously (at the time of peak

compression) this diverted energy E, (r) has risen to 143kJ/cm MA’ of interior current. In the absence f any dissi-pative mechanisms his energywould be released ack o thedriving circuit as the plasmaexpands-it is nevercoupled othe denseportions of the load! If we evaluate the peak cor-onal energy or the currents and oad engths discussedn theGamble II analysisby Thornhill, the absorption of this cor-onal energy nto the load region would result in much moreadditional energy removed rom the pulseline, which morethan removes he energycoupling discrepancybetween he-ory and experiment at lower load masses.f anomalousdissi-pation mechanismsare, n fact, active in these coronal plas-mas, then this is also the rough magnitude of energyavailable to the load as additional heating. Such a channelwould provide a path for conversionof energy n the vacuummagnetic field to the load without compression of the cur-rent filament to very small dimensions, as required to ex-plain the observationsof Ref. 19.

IV. S~M~A~~ AND CONCLUSIONS

Current can be lost in the periphery of a Z pinch byvarious mechanisms. nitially the Z pinch carries no current,and the inductive voltage at the edge of the cathode away





from the pinch material can drive a current in free electrons.The free electrons can be the bulk of the current at the verystart, but this current is cut off rapidly once he current startsflowing through the Z pinch.

The tenuous plasma outside the Z pinch can also carrycurrent. How much current is shunted through this plasmadependson the plasma density, and also on parameterssuchas the temperature and the magnetic field in the periphery,i.e., the current in the main Z pinch. The pressure of thecollisionlessplasma n the periphery becomesanisotropic asthe plasma contracts toward the axis, as a result of the sepa-rate conservation of the electron’s magnetic moment, andthe electron’s angular momentum about the axis. Undertheseconditions the plasmacan still oscillate n a self-similarmanner for a constant interior Z-pinch current. The con-straint of constant nterior current is also consistentwith theideal hydromagnetic self-similar flows useful in modelingthe dense nterior plasma. Both interior and peripheral plas-ma are hus modeled n a mutually consistentway. While theapportioning of current between he interior and the coronais an important issue, t is beyond the scope of this work.

Here the model is built to describe he motion after the inte-rior plasma s hot enough to curtail further magnetic diffu-sion, and thus a fixed interior current is the proper boundarycondition.

In contrast to self-similar hydromagnetic oscillator so-lutions, the present gyrokinetic oscillators continue to ab-sorb generator power as they collapse.The coronal plasmamay thus be a common modifying influence on the currentwaveform, reducing or removing the “inductive notch” ex-pected rom simpler treatments. Because f this extra energysink, the comparison of theory and experiment s modified inthe right direction and magnitude.

While there are many potential caveats o the develop-ment presentedhere as we consider the early time develop-ment of the corona, the detailed role of transit time effectsand the microstability of the evolving distribution functions,the collisionlessmagnetoplasma orona treated here may bean important ingredient in many plasma radiation sources,one that deserves lose experimental attention.

ACKNOWLEDGMENTS

We acknowledge many helpful discussions with IraBernstein.

Support by the DefenseNuclear Agency for this work is

gratefully acknowledged, and in particular the use of thePIC code MAGIC.

APPENDIX: DEVELOPMENT OF SELF-SIMILARSOLUTIONS

Because he electrical properties of collisionlessplasmaare generally determined by a few moments of the distribu-tion function, a convenient path to developing he constitu-tive reIation (E) is to examine he moment relations of thegyrokinetic equation. Only four parameters are, in fact, re-quired to determine he response f the system.Each of theseis basedon a particular moment of the distribution function.

For example, using the variables of the main text, energyconservation s expressed y the statement

dY-j.E=-

dt-I- v-F, (AI)

where he pressure orce is F = - V-P, and P is the pressuretensor. Two of the basic parameters are already obvious inthe two independentpressure ensor components, he othersarise as shown below.

The macroscopic quantities that enter into the fluid(moment) equationsare averages ver the particle distribu-

tion function, The distribution functions that solve he gyro-kinetic equationscontain a classof separable unctions of theformg( r,t)F(,u,p,,r,). Here, instead of writing the distribu-tion in terms of the velocity components n cylindrical geom-etry, it is better to use he dynamical invariants, the magneticmoment u, and the angular momentump,. The two equiva-lent representationsof the gyroaverageddistribution func-tion are connectedby

f(r,v,O = [m*r/2nB(r,t)ln(r,t)F(CL,Pe,~~‘o). (AZ)

The factor in [ * * * is the Jacobian of the transformationfrom v top, pe, and 4 the gyrophase.Without collisions the

invariants for each particle do not change, and the velocityfunction F is constant in time along a guiding center trajec-tory, r( r,,t) . The particle density n (r,t) is no longer part ofF, which is normalized as

( 2~~~o))~p 4, FWe,ro)= 1, (A3)

for both electrons and ions. With the assumption of quasi-neutrality the massdensity p is p = ( Zmelec+ mien n (r,t) .

Without collisions the distribution function is not neces-sarily Maxwellian. Instead, the distribution function shouldbe computed by modeling the transition between the colli-sional and the collisionlessplasma.While this is a complicat-

ed problem, outside the limited scopeof this paper, the gen-eral result24 s a tenfold increase in mean kinetic energybefore the electronsbecomecollisionless.Here the distribu-tion function is viewed as an input to the problem.

The parallel pressure s defined hrough the average al-

ue 0 p, ,2

?P,, =pL; =p dpdp, F$,

and the perpendicular pressurebecomes

PL =pBN, =pBs

dp dp, 3.m

(A4b)

Here, and in other moments, the summation over species simplicit.

The axial current density, defined by averaging of theaxial drift velocity of the guiding center

D(E/B)

Dt

is then readily calculated by momentum balance o be

The first term is related to the accelerationof the EXB driftmotion, the secondcomes rom angular momentum conser-





vation, or PII, and the third is the magnetization contribu-tion, from Pi. In the acceleration erm the ions dominate; heelectrons are more important for the two pressure-relatedterms, without thermal equilibrium between electrons andions.

The fluid equation of motion in the radial direction, foran anisotropic pressure, ncludes a centrifugal force L a/3that reflects he conservation of angular momentum,

DU PLi 1 a(rpBM,) -JB

- --‘Z=? r dr ‘* (A61

An isotropic pressure mplies that the collisions are suf-ficiently frequent to keep the velocity components rando-mized. The frequent collisions imply a finite conductivity, incontrast to the assumption of infinite conductivity, in idealhydromagnetic theory, which demandsa collisionlessplas-ma. One might argue that pressure isotropy has beenreachedon time scales onger than those of interest, or as aninitial condition. However, in a cylindrical geometry, as theplasmagets closer to the axis the pressurePII parallel to themagnetic field B = B,(r) increasesas l/?, while the per-pendicular pressurePI increases s l/r. Thus the anisotropyis ntimately connected o the time scaleof interest n a colli-sionless ylindrical plasma.

In this gyrokinetic description the Maxwell-V lasov sys-tem, as completed by the momentum balance ncluding theanisotropic pressureP,can be summarized n the followingequations:

g +V*(nW) =o, (ATa)

C3B- = VX(WXB),at

(A7b)

VxBz-?!@+k!!?, (A7c).c c at

1 .

The displacementcurrent is retained n the anaiysis n orderto make connection with external power sources or sinks.For a cylindrically symmetric geometry the nonvanishingdependentvariables are the fluid density p, the radial com-ponentof the fluid velocity W, = cU, the axial component ofthe current density J, = J, the azimuthal component of themagnetic field B, = B, and the parallel and perpendicularmomentsL,, and ~34~.

In the limit of homogeneous ompression ( Ua r), spe-cial initial conditions in thesevariables allow the equationsto simplify. The dependent variables and their derivatives

are then functions of the single self-similar invariant6 = r/r.a( 7). Here r. is a dimensionlessscale ength thatarises directly in the transformation, viz.,r. mi7og/w, = r, /l,, with ctO= 1, and 7 = t/t,,. The timescale ,, s a free parameter only to the extent hat a particularimplosion time scale or the core plasma remains arbitrary.The variable cy r) contains the sole time dependence, .g.,U = &r,, DU/Dr = &-,.

A self-similar oscillation is possibleonly when the acce-lerations are proportional to the pinch radius. For an iso-tropic pressure he proportionality of J,B,/n to radius im-

plies that the magnetic field is proportional to 6. In theanisotropic caseadditional profiles must be definedself-con-

sistently. Using Eq. (A7c) to eliminateJ, and separating hefour distinct spatial dependencies hat arise, Eq. (A6) be-comes

4 4 0:2

a=-- S&2*

a3 “1-y” a WI

The oscillatory term related to the magnetic field pres-sure is - o: /o, where T+,= o, * is the transit time of anAlfvin wavewith veloci ty c, through the pinch scale ength

r*, VIZ.,WA= c$, $ = B */4n3~?. As expected his term alsooccurs n the correspondingequation for the isotropic idealhydromagnetic case.The two new terms related to the ani-sotropy are the parallel pressure erm, with wi =ci/$, andthe perpendicular pressure erm, with w: z c: /c*. The veloc-ities c,, c,,, and ct are thus alternate measures of thestrengthsof the fundamental moments n the theory, servingto parametrize the corresponding separation constants.Each dimensionless requency (ok =rk ’ ) derives from aseparate erm in the momentum equation, and the velocityratios obtain from the normalization of the moments to thelight speed and the required dimensionali ty, viz.,

L i -Z;Sc~h,, g) and BM, -cfh, (g). The final term, pro-portional to &*, ariseshere only when we keep he full effectsof displacementcurrent; rg = oz I is a measureof the over-all dielectric strength characterizing any particular flow.

The four frequenciesappearing n Eq. (A8) are con-nected to four constraint relations that define the requiredplasma profiles for enclosed currentr(g) = (cZ,r,/2)~B&T), number density n,(5), perpendic-ular (kinetic) temperature hL (Q, and parallel (kinetic)temperature h,, (Q. With the dielectric coefficient ~(6) giv-en by

E(& = If1 + (4rrm,c2)n0(6)/B$(0 ,

the constraints are given by

(A94

@;I (c@2,h,,G)[l - 1/&&31 t (A9b)2w, = - (c’,/cw I- l/&.0 I/

2g22n,ma, @bGM, kc) ] P (A9c)

r? = fl + &Y4 nB&)Ie%*], (Agd)

I= [2 -I- &St In B&)]/t?. WeI

These equations require mE < 1 to obtain positive definitenumber density profiles and offer boundedsolutions for eachof the variables, usually over a finite range [g, ,c, 1. Thespatial range used can be chosen to fit whatever physical

dimensionsare imposed because hese solutions are simplyfollowing the motion of the singleparticle gyrokinetic trajec-tories. In other words, clipping these solutions in radius isadmissibleas ong as he profile valuesand derivativeson theinterior of any such domain limit to the proper valuesas oneapproaches he boundary.

’ N. R. Pereira and J. Davis, J. Appl. Phys. 64, Rl ( 1988).*J. Pearlman and J. C. Rio&n, Proc. SPIE 537,102 ( 19851; alsoseeFig. 5

in Ref. 1.‘J. W. Thornhill, K. G. Whitney, and J. Davis, J. Quant. Spectrosc.Radiat.Transfer 44,251 (1990).

4G. Merket and D. A. Whittaker (private communication).

202 Phys. Fluids B, Vol. 3, No. 1, January 1991 FL E. Terry and N. R. Pereira 202




* F. C. Young (private communication ).OR. W. Clark, J. Davis, and F. L. Cochran, Phys. Fluids 29,197l (1986).‘F. L. Cochran (private communication).*I. B. Bernsteinand P. J. Catto, Phys. Fluids 28, 1342 1985).‘F. S. Feller. Phvs. Fluids 25. 643 ( 1982).loR. C. Dav idson, Phys. Fluids 27, i 804 (‘1984).” R. C. Davidson, The Theoryof Non-Neutml Plosmos Benjamin, Read-

ing, MA, 1974).” R. J. Briggs, J. D. Daugherty, and R. H. Levy, Phys. Fluids 13, 421

(1970).

I3R. H. Levy, Phys. Fluids 8, 1288 1965).

14M. S. di Capua, EEE Trans. PlasmaSci. PS-11,205 (1983).“A. Ron, A. Mondelli, and N. Rostoker, EEE Trans. PlasmaSci.PSI, 85

(1973).

16R.V. Lovelaceand E. Ott, Phys. Fluids 17, 1263 1974).“J. M. Creedon,J. Appl. Phys. 48, 1070 (1977).‘*R. F. Benjamin,J. S. Pearlman,E. Y. Chu , and J. C. Riordan, Appl. Phys.

L&t. 39,848 (1981).

I9M. Krishnan, C. Deeney,T. Nash, P. D. LePell, and K. Childers, n DenseZ Pinches,AIP Conf. Proc. 195 (AIP, New York, 1989), p. 17.

203 Phys. Fluids B, Vol. 3, No. 1, January 1991

“C, Deeney,T. Nash, P. D. LePell , M. Krishnan, and K. Chil ders, n DenseZ Pinches,AIP Conf. Proc. 195 (AIP, New York, 1989), p. 55.

*’ C. Deeney,T. Nash, P. D. LePdl, K. Childers, and M. Krishnan, in DenseZ Pinches,AIP Conf. Proc. 195 (AIP, New York, 1989), p. 62.

**J. W. Thornhill (private communication ); and F. C. Young, S. J. Ste-phanakis, and V. E. Scherrer,Rev. Sci. Instrum. 57,2174 (1986).

23R. B. Spielman,R. J. Dukart, D. L. Hanson,B. A. Hammel, W. W. Hsing,M. K. Matzen, and J. L. Porter, in DenseZ Pinches,AIP Conf. Proc. 195(AIP, New York, 1989), p. 3.

24R. E. Terry, in NRL Memorandum Report &X1,1987, p. 49 ff. SeeAIPDocument No. PAP S PFBPE-03-0195-101or 101 pagesof Naval Re-

search Laboratory Memora ndum Report 6051, “Advanced ConceptsAnnual Report 1 986” from the NRL Plasma Radiation Branch to theDefenseNuclear Agency. Order by PAPS number and ournal referencefrom American Institute of Physics,PhysicsAuxiliary Publication Ser-vice, 335 East 35thSt., New York,NY 10017.Thepriccis%1.50foreachmicrofiche (98 pages)or $5.00 or photo copiesof up to 30 pages,$0.15for eachadditional pa geover 30 pages.Airmail additional. Make checkspayabl e o the American Institute of Physics.

R. E. Terry and N. R. Pereira 203

R.E. Terry and N.R. Pereira- Leakage currents outside an imploding Z pinch

Documents

Transcript of R.E. Terry and N.R. Pereira- Leakage currents outside an imploding Z pinch