S. C. Bott et al- Dynamics of cylindrically converging precursor plasma flow in wire-array Z-pinch experiments

8/3/2019 S. C. Bott et al- Dynamics of cylindrically converging precursor plasma flow in wire-array Z-pinch experiments

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Dynamics of cylindrically converging precursor plasma flow in wire-array Z-pinch experiments

S. C. Bott,1 S. V. Lebedev,1 D. J. Ampleford,2 S. N. Bland,1 J. P. Chittenden,1 A. Ciardi,1 M. G. Haines,1 C. Jennings,2

M. Sherlock,1 G. Hall,1 J. Rapley,1 F. N. Beg,3 and J. Palmer41

The Blackett Laboratory, Imperial College, London SW7 2BW, United Kingdom2

Sandia National Laboratory, Albuquerque, NM USA3Department of Mechanical and Aerospace Engineering, University of California San Diego, California, USA

4AWE Plc., Aldermaston, RG7 4PR, United Kingdom

Received 6 April 2006; revised manuscript received 24 July 2006; published 12 October 2006

This paper summarizes the present understanding of the processes leading to precursor column formation incylindrical wire arrays on the 1 MA MAGPIE generator at Imperial College London. Direct experimentalmeasurements of the diameter variation during the collapse and formation phase of the precursor column arepresented, along with soft x-ray emission, and quantitative radiography. In addition, data from twisted cylin-drical arrays are presented which give additional information on the behavior of coronal plasma generated inwire array z pinches. Three stages in precursor column formation are identifiable from the data: broad initialdensity profile, rapid contraction to small diameter, and slow expansion after formation. The correlation ofemission to column diameter variation indicates the contraction phase is a nonlinear collapse resulting from theincreasing on-axis density and radiative cooling rate. The variation in the minimum diameter is measured forseveral array materials, and data show good agreement with a pressure balance model. Comparison of columnexpansion rates to analytical models allows an estimate of column temperature variation, and estimates of thecurrent in the column are also made. Formation data are in good agreement with both fluid and kineticmodeling, but highlight the need to include collisionless flow in the early time behavior.

DOI: 10.1103/PhysRevE.74.046403 PACS numbers: 52.20.Hv, 52.59.Qy, 52.70.La

I. INTRODUCTION

A wire array z-pinch experiment displays several distinctstages throughout its duration, from wire initiation to implo-sion and x-ray generation. On longer current drive100 ns machines the processes occurring prior to the im-plosion of the main array mass comprise a significant pro-portion of the drive time, which presents an opportunity to

study these in detail. One of the most interesting phenomenato develop during this period is a remarkably uniform andstable precursor plasma column on the axis of the array.Since its first identification 1, precursor column structureshave been observed in a large number of experiments andwithin a wide range of array parameters 13. The arrival ofthis mass fraction on the array axis, preimplosion, is due tothe formation of a heterogeneous plasma structure from eachwire caused by the passing of the drive current: relativelycold wire cores surrounded by a low-density hot coronalplasma 47. The radial JB force accelerates the coronalplasma towards the axis, while the cold wire cores are con-tinuously ablated. It is the stagnation of this flow at the axiswhich forms the observed precursor column.

This uniform and stable plasma object by itself deservessome attention due to an interesting combination of plasmaphysics processes occurring during its formation. These in-clude a transition from collisionless to collisional regimes ofinteraction of the plasma flow, the effect of radiation coolingon the dynamics of the plasma, and the development of ion-ization balance. In wire array z pinches, the convergingplasma flow is often sustained for a long time 100 ns andin the precursor column these processes can develop in aquasistationary situation. The high degree of the azimuthalsymmetry of the plasma flow in this system makes it an

almost ideal test bed for one-dimensional 1D radiation hy-drodynamic codes, and indeed is closely analogous to anexisting problem in hydrodynamics: Nohs problem 8. Theformed column is an interesting candidate for high energydensity physics HEDP experiments, and one can also notea close analogy between the precursor formation in wire ar-ray Z pinches and stagnation of plasma on axis of symmetryin other systems with converging plasma flows, e.g., of the

plasma ablated from the walls of ICF hohlraums 9.The effect of the precursor plasma flow on the implosiondynamics of wire array z pinches and on parameters of thex-ray pulses generated by the implosion is actively debatedat present. Among reasons for this interest is the unexpectedremarkable performance of large diameter wire arrays, whichhave relatively large interwire separations 3 mm 10.The core-corona plasma structure and the resulting inwardflow of coronal plasma is apparently important even for thearrays with very small interwire separations 0.21 mm, asfollows from experiments 11 on the Z facility at SNL, inwhich a very early arrival of the precursor plasma on thearray axis was detected. For these arrays it is difficult toenvision formation of a plasma shell this early in the currentpulse, and hence the discrete nature of the wires should playan important role in their operation. The prefilling of theinterior of an array with the plasma should be taken intoaccount not only for understanding of x-ray pulse generation,where the accretion of mass during implosion may reduceinstabilities at the accelerating plasma front, but also in thedesign of different wire array z-pinch loads. This could beespecially important for the concepts of dynamic 12 andstatic wall 13 hohlraums, where the precursor flow can sig-nificantly change conditions of the foam targets installed onthe array axis 14. To this end, accumulation of experimen-

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tal information on formation and behavior of the precursorplasma in wire array Z pinches is required. Experiments withrelatively low currents and long rise times can help to high-light important processes, and hence evolve a better under-standing of the corresponding physical processes throughcomparison with computer and analytical models, and shouldbe useful for establishing scaling laws.

In this paper we present an experimental study of the

dynamics of the precursor plasma formation in wire array Zpinches at the 1 MA current level. The content of the re-mainder of the paper is as follows. In Sec. II the experimen-tal setup and diagnostics are described. Section III describesthe behavior of the precursor plasma and formation of thecompact precursor column in cylindrical wire arrays. Data ispresented on parameters of the coronal plasma flow in alu-minum and tungsten wire arrays, as are discussions on howthe difference in the collisionality of the coronal plasma flowfor the two materials affects the dynamics of the plasmaaccumulating on the array axis. Data demonstrating forma-tion times, column diameters, stability time scales, and radia-tion profiles for several different materials are presented and

discussed. In addition, an analytical approach is applied toestimate the current in the precursor, based on the analysis ofstability of the precursor column. In Sec. IV data are dis-cussed with reference to current MHD modeling 15 and toa recently published kinetic description of the precursor for-mation process 16, which provide some insight into thecolumn formation mechanism. Section V describes experi-ments with twisted wire arrays, which provide some addi-tional information on the behavior of the coronal plasmagenerated in wire array z pinches. Applications of the precur-sor column are suggested in Sec. VI, and conclusions arepresented in Sec. VII.

II. EXPERIMENTAL SETUP AND DIAGNOSTICS

Experiments were carried out on the MAGPIE generator17 with a peak current of 1 MA and a rise time of 240 ns10% to 90% rise time is 150 ns, which is closely recon-structed by a sin2 wave form. Wire arrays 16 mm in diameterand 23 mm long were used in all of the experiments, withfour current-return posts situated on a diameter of 15.5 cm.The load area design allows both end-on along the axis ofsymmetry and side-on radially diagnosis of the entire ar-ray. Experiments were performed with arrays of 8, 16, 32,and 64 Al or W wires of different diameters 10, 15, 50 mAl, 4, 5, and 7.5, 13 m W, and in some cases with Ni, Cu,Ti, and Mo wires. Over-massed arrays 50 m, Al, and13 m W do not implode on the time scale of the experi-ment, and are used to study the precursor on long timescales. Where imploding loads were used, implosion timesfor arrays with different number of wires were between 180and 320 ns, as determined from the optical streak photo-graphs and the onset of the x-ray pulse from the pinch.

In addition to standard cylindrical arrays, twisted arraysare also investigated. For these, the top electrode holding thewires was rotated by /4, 3/4, or /2 relative to the bot-tom electrode. Small angles of rotation do not disturb thearray behavior, and can be used to measure the azimuthally

averaged radial distribution of the coronal plasma inside thearray. For larger angles of rotation, the magnitude of theaxial magnetic field generated inside the twisted wire arraywas large enough to affect the dynamics of the inward flowof the coronal plasma and the formation of the precursorcolumn on the array axis.

Diagnostics used in the experiments included laser prob-

ing, optical and soft x-ray imaging, a range of filtered x-raydiodes, time and space resolved XUV spectroscopy, andx-ray radiography. Optical probing was performed using afrequency doubled Nd-YAG laser 532 nm with stimulatedBrillioun scattering SBS pulse compression imaging pulseduration 0.4 ns, and images were recorded on CCD cam-eras. The laser beam was split to provide simultaneousend-on and side-on measurement, as well as multiple frametimes. End-on laser probing provides information about theazimuthal structure and radial distribution of the coronalplasma, and was achieved by installing a mirror into the baseof the cathode. The side-on optical diagnostics include laserprobing with interferometer, shadow, and Schlieren channels,

and a streak camera with its slit oriented along the arrayradius. The timing of the laser pulse was registered on theoptical streak camera, which provided synchronization of op-tical streaks with other diagnostics. X-ray diagnostics in-clude three gated x-ray cameras 2 ns gate, 9 ns, or 30 nsinterframe time 18, XRD diodes with an aluminum photo-cathode and photoconducting detectors PCD 19 with vari-ous filters. The gated x-ray cameras had a spatial resolutiontypically of 100200 m, and for measurements during therun-in phase of implosion were filtered with 1.55 m poly-carbonate transmission window between 150290 eV.XUV spectroscopy in the range 50400 was performedusing a grating grazing incidence spectrometer and tempo-ral resolution was achieved by imaging the spectra onto agated MCP camera with an exposure time of 10 ns. Theuse of an additional slit, which formed an image on the inputslit of the spectrometer, provided spatial resolution of0.5 mm in the radial direction.

III. PRECURSOR COLUMN

A. Precursor column formation

The precursor column is a compact, well defined, stronglyemitting object observed to form on the axis of a cylindricalwire array prior to the main implosion phase. Figure 1 shows

FIG. 1. Radial optical streak photograph showing relative tim-ings of precursor column formation, start of main array implosion,and stagnation.

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a radial optical streak photograph of a 16 mm diameter alu-minum array comprising 64 aluminum wires with a 10 mdiameter. The formation of the precursor column on the arrayaxis occurs at 160 ns, the start of the implosion of thearray at 250 ns and the stagnation of the array on the axis at300 ns.

The formation of the column is rapid, and observed on all

diagnostics at similar times. Figure 2 shows two side-onXUV emission images for a tungsten array taken before andafter formation of the compact precursor column from aninitially broad emission region. This initial distribution is aresult of the interaction of the plasma streams from the indi-vidual wires as they converge onto the array axis. The wirecores are stationary at this time and slowly ablate, generatingcoronal plasma which is continuously injected into the array.In experiments, measurements of the inwards radial velocityof plasma streams in aluminum arrays gives 1.5105 ms1 20, with wire arrays made of other materialsyielding comparable values.

The rocket model 21 defines an ablation velocity as a

constant relating force and the mass ablation rate of the wirecores, and sets this equal to observed plasma stream flowvelocity. Data from multiple experiments gives ablation ratesconsistent with an approximately constant ablation velocity,and the rocket model is widely used in analytical diagnosesof wire array behavior. Computational and theoretical mag-netohydrodynamic MHD models suggest that stream flowvelocity varies with several parameters, including radial po-sition, but that the value as plasma reaches the axis is similarto the ablation velocity inferred from experiment. Given this,a constant ablation velocity can be taken as a representativeof the flow velocity where plasma streams interact close tothe array axis and the rocket ablation model will be deemeda good approximation for the present study.

The apparent universality of this ablation velocity sug-gests that the ion kinetic energies Eion=0.5mionVabl

2 for dif-ferent materials vary considerably by virtue of their atomicmasses. Using Vabl=1.510

5 ms1 from above, calculatedvalues are 3 keV for aluminum and 21 keV for tungsten.The mean free path mfp for ion collisions will determinethe extent to which counterpropagating plasma streams inter-act at the array axis, and this varies as the square of the ionkinetic energy. The characteristic scale length of the experi-ment is the array radius, which was 8 mm for all arrays. Amean free path significantly shorter than this would indicate

that plasma streams would stagnate at the array axis, and avalue significantly larger would suggest a low level of inter-action, and that counterstreaming behavior would occur.

Plasma streams ablated from the wire cores first reach theaxis at 105 ns, which is a combination of a 50 ns dwelltime during which the core-corona structure is formed,and the time of flight from the array radius to the axis as-suming fixed ablation velocity. The rocket model gives ionnumber densities of 1017 cm3 for aluminum and1016 cm3 for tungsten. The calculated mean free path valuesat axis, at this time are then 0.5 mm for Al ions assumingz6, and 10 mm for W ions assuming z14. The ab-lated plasma streams for aluminum should therefore behighly collisional on the scale of the array radius, whereastungsten streams should display a much lower degree of col-lisionality. This is in good agreement with formation ofshocks where streams from adjacent wires collide, which areseen on laser probing and XUV emission images of Al ar-rays, and the absence of such features for comparable Warrays Fig. 3. The plasma streams generate a density profilewhich is peaked on axis, as a result of both collisions and the

convergent geometry, and which continues to rise with in-creasing stream density during the experiment. During thisperiod, imaging diagnostics observe a broad object on thearray axis, with a diameter of several millimeters: i.e., a sig-nificant proportion of the initial array diameter Fig. 4. Atsome point, the plasma that has converged onto the axis un-dergoes a rapid contraction in diameter and an associatedx-ray output is recorded. The appearance of a well definednarrow column is observed and this is the compact precursorcolumn. The formation time scale of the precursor column istypically10 ns 22, and diameter of the formed column ischaracteristic of material, varying inversely with atomic

FIG. 2. Side on XUV emission image of a 16 wire tungstenarray showing formation of the compact precursor column.

FIG. 3. End-on XUV emission from 16 mm diameter arrays ofleft 1620 m Al at 134 ns, and right 1613 m W a t134 ns. White circles indicate positions of wires.

FIG. 4. Laser interferogram of a 16 mm diameter, 16 wire Warray at early time and corresponding electron density plot.

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number. The column then remains stationary on the arrayaxis until plasma flow from the wire cores diminishes or themain array mass implodes onto it.

To investigate these processes experimentally, the diam-eter of the on-axis emission region was monitored for over-massed nonimploding aluminum and tungsten cylindricalwire arrays. Gated x-ray cameras were mounted both end-onand side-on, and imaged through 100 m pinholes that wereeither unfiltered h40 eV or filtered with 2 m polycar-bonate film h200280 eV. PCD traces, both unfilteredand filtered with the same polycarbonate film, were also re-corded during the experiments to correlate the array softx-ray emission with the XUV images. The diagnostics wereafforded a full view of the array in each orientation.

The variation of the diameter and typical PCD tracestaken over a series of experiments are given in Fig. 5. PCDtraces were observed to be repeatable to within 5 ns, andthe features discussed below were reproduced in each experi-ment. The variation in diameter occurs similarly for bothmaterials, and three distinct stages in the diameter evolutioncan be identified: i broad initial density profile, ii rapidcontraction to small diameter, iii expansion of column fol-lowing formation. The initial diameters are significantly dif-ferent for aluminum and tungsten. In Fig. 5, aluminum

shows an increasing diameter from 110 ns, similar to thefirst arrival of plasma at the array axis, and reaches a maxi-mum of3 mm before the rapid contraction phase. Tung-sten, however, shows a diameter of 7 mm from earlytimes, which remains relatively constant until collapse oc-curs. This is a direct result of the collisionality differencesdiscussed above. Tungsten ions are less confined to the arrayaxis and counterstream for some time, and so a broader emis-

sion region is observed. Aluminum ions essentially stagnateat the axis, forming a more confined emission region.The contraction phase and appearance of the precursor

column occurs noticeably earlier in the current drive for alu-minum 150160 ns than for tungsten 160175 ns. Thisprocess occurs over approximately 1015 ns for both mate-rials, and this corresponds to collapse rates of the order of theablation velocity, although present results do not allow anexact measurement. The average collapsed diameters are1.6 mm for aluminum and 0.6 mm for tungsten. Followingformation, both aluminum and tungsten columns are ob-served to continually increase in diameter for the remainderof the experiment. This expansion is discussed in more detailin Sec. III B.

The x-ray signal from tungsten arrays shows several fea-tures. A slow start to the emission begins at 100 ns andrises increasingly rapidly to a peak at 175 ns. Following thepeak, the signal then drops rapidly to a low, but non-zerolevel some 40 ns later. The FWHM of the signal is 25 ns.Aluminum shows a similar rise, but has a double peakedstructure, with the first peak at 155 ns and the second at180 ns. The FWHM of this is 70 ns, and the signal levelsubsequently drops to zero. It should be noted that the stag-nation of an imploding array on MAGPIE produces a signalin this energy band 2030 times greater than that of the pre-cursor formation signal shown in Fig. 5.

The appearance of the compact precursor column, i.e.,

when the smallest diameter is observed, occurs at or shortlyafter the first formation x-ray peak within errors in bothmaterials for the data discussed above. The formationmechanism can be further examined by correlating the PCDtraces to end-on XUV images, both filtered with 2 m poly-carbonate film, taken during column formation. This isshown for a tungsten array in Fig. 6. The first filtered XUVimage shows a large diameter 5.7 mm object during theinitial x-ray rise. By the time of the second image, at 186 ns,the x-ray emission is falling and this object has contracted to3 mm. Figure 5 indicates that the time of collapse is of theorder of 10 ns, as shown above, and so this contraction isunlikely to have started prior to the x-ray peak at 170 ns.This is confirmed by the unfiltered image taken at 168 ns onthe same shot, which shows the broad initial density profileis still present at this time, and hence little contraction hasoccurred by the time of peak x-ray emission. The filteredXUV image at 216 ns shows a very narrow object with adiameter of 0.6 mm, which shows only a low level of emis-sion through the filter. This is the fully formed precursorcolumn.

Given that the formation of the precursor column is ob-served to form in similar ways for both materials it is likelythat the radiation signature represents the same mechanismfor aluminum and tungsten, but that the material properties

FIG. 5. The variation of precursor diameter with time fromXUV imaging, correlated to soft x-ray emission for top 16 wiretungsten and bottom 32 wire aluminum arrays.


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vary its exact form. With reference to the data presented inFigs. 26 it is possible to construct a likely mechanism forthe formation process. At early times, the onset of collision-ality between highly energetic ions, and subsequent energytransfer to electrons, begins to heat the plasma on axis. If thethermal pressure of the central plasma object is comparableto the kinetic pressure of the ablated plasma streams, it couldnot be compressed at this time and would remain at some

relatively large diameter. As the density of the incomingstreams increases with the drive current, the increased kineticpressure forces density at the axis to slowly rise. The sub-stantial thermal pressure, however, does not allow a reduc-tion in diameter of the central plasma object. The level ofradiation emitted from the central plasma will increase as thedensity rises. Since the radiation loss rate is proportional tothe square of the ion density, the measured output would riseincreasingly rapidly. Eventually the energy being radiatedwill exceed the energy thermalized by the incoming plasmastreams, and the central plasma object can begin to cool. Thiseffect will be greatest where density is highest, i.e., at theaxis in cylindrical geometry. Where cooling has occurred thekinetic pressure can compress the plasma, generating higherdensity and increasing the radiation loss, and a run-awaycollapse process is initiated. The density profile would becompressed onto the axis, generating the high density pre-cursor column.

These mechanisms will be dependent on the array mate-rial, which may alter the significance or timing of each event.If this physical picture is correct, the total energy radiated bythe time collapse begins, i.e., by the peak of filtered x-raysignal in Fig 7, should be comparable to the accumulatedstagnated kinetic energy of the plasma streams up until thistime. Quantification of the radiated energy loss can be

achieved experimentally by recording the emitted radiationusing unfiltered PCDs. These are sensitive to radiation in therange 4 eV to 3 keV, and so record the total radiatedspectrum. Unfortunately this means that the detector satu-

rates rapidly during an experiment, and signals are reliableonly for the relatively early stages of the current drive. Fortungsten experiments it was possible to obtain unfilteredtraces which proved reliable up to the peak of the filteredx-ray signals discussed above 180 ns. The recorded volt-age traces can be converted to power, using the calibratedsensitivity in Ref. 19, and a scaling for the fraction of ra-diation viewed by the detector gives a total radiated powerassuming isotropic emission. This is shown in Fig. 7. Theunfiltered PCD signal begins to rise at 85 ns and reaches4.5 GW/cm at the time of the peak in the filtered PCDsignal. For comparison, the stagnation of the main array onMAGPIE produces several hundreds of GW/c of peak radi-

ated power. The integral of the power trace gives the abso-lute radiated energy, and up to the peak in the filtered x-raysignal this is 160 J. The calibration of the PCDs is notnecessarily exact for the devices used here, and is affected byusage and damage. This figure may therefore be subject tosubstantial errors, but is sufficient for an order of magnitudeestimation. For comparison to this, the rocket ablation model21 gives a total mass accumulated in the precursor at agiven time see Eqs. 13 below. The use of the constantablation velocity then allows a total accumulated kineticenergy in the initial density profile to be calculated. For thekinetic energy of all ions and assuming no energy losses, thisvalue is 185 J.

The energy loss which allows collapse and precursor col-umn formation will comprise contributions from both theradiation loss measured above and ionization processes. Togive an indication of the possible contribution from ioniza-tion, one can calculate the energy required to ionize all thetungsten on the axis at 180 ns from z=8 in the streams toz =14. This value is 15 J, which is small by comparison tothe total radiated energy measured. Similar calculations foraluminum suggest the contribution from ionization may forma slightly larger proportion of the total expected, but is againsmall by comparison to the radiated energy loss. In theformed column the end-on radiation profile becomes hollow,

FIG. 6. Filtered end-on XUV images correlated to PCD tracewith the same filter during precursor column formation for a 16wire tungsten array, and single image right unfiltered end-on XUVimage at peak x-ray signal, 168 ns, showing broad density profile.

FIG. 7. Total radiated power and filtered PCD signal for a 16wire tungsten array during precursor column formation.


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as discussed in Sec. III B, and a high temperature hencehigher z region is only a fraction of the total column vol-ume. In this case, the average charge state may not be sig-nificantly different from the stream charge state. It is likelythat energy consumed by ionization processes contributes tothe column formation, but the present level of accuracy ofPCD calculations do not allow a clear determination of this.More accurately calibrated radiation measurements are re-quired to evaluate this contribution.

Once the collapse process has begun, compression thencontinues until the final precursor diameter is reached. Thisfinal diameter is likely to be determined by the balance of theradiative loss and thermal pressure of the plasma with thestream kinetic pressure. These parameters will again varywith material, resulting in a different diameter for different

array materials. Further discussions of this are given in Sec.III B below. The first peak in the filtered x-ray signal Fig. 5indicates when the formation process is occurring, giving anindication of when the precursor column is forming duringan experiment. The second peak is likely to be a result ofhydrodynamic bounce, or column compression, both ofwhich would vary with array material. A comparison to mod-eling results is presented in Sec. IV in order to elucidate apossible mechanism for the form of the x-ray pulse.

Experimental results show that the precursor column isnot formed simultaneously at all axial positions. Laser, XUVand visible self-emission imaging all show that the precursorcolumn is initially formed at the anode Fig. 8, beingformed fully to the cathode several nanoseconds later. Thedelay at different axial positions can be quantified during anexperiment by using multiple frame diagnostics to infer aphase velocity of formation in the axial direction. This is ofthe order 106 ms1, assuming linear formation, but experi-mental values are very limited at present. This delay in for-mation with axial position is not strongly reflected in thex-ray pulse measured during the formation of the precursorcolumn. Experiments have been carried out using a narrowhorizontal slit placed close to the pinch, in front of a PCDwith the same filtering and at the same distance as used pre-viously. In this arrangement, the PCD views only a 2 mm

horizontal section of the array. If the axial formation delaywere significant, a narrowing and timing difference would beexpected relative to the x-ray traces presented in Fig. 5,where the full array height is viewed. The resulting traces,however, maintain their relative timings and forms.

It is possible that this difference in formation time is aresult of a variation in the wire ablation rate in the z direc-tion. If it is assumed that mass ablation rate varies continu-

ously along the z axis, we can estimate the variation in ab-lation velocity required to give a zippering of the precursorcolumn of a rate similar to experiments. For a difference ofapproximately 0.5105 ms1, the time of flight to the axisfor the anode end and cathode end of the array would differby approximately 20 ns. So, for example, ablation velocitywould vary linearly between 1.25105 ms1 and 1.75105 ms1 with the cathode end being the lower of thesetwo velocities. The slightly higher velocity at the anode endwould allow formation of the precursor column here first,and the delay of formation at the cathode is then a result ofthe extended flight time to the axis with a lower velocity.This scenario gives a phase velocity of precursor column

formation to be of the order of 106

ms1

, similar to experi-ments. Collisionality issues will also play a role for tungstenarrays, and the lower velocity and greater mass ablated at thecathode end would give a reduced period of collisionlessflow here. The phase velocity may therefore be higher than issuggested by the argument above. In aluminum arrays, theablated plasma streams are collisional on the scale of thearray radius from early times, and the delay in formation isdirectly related to the time of flight to the axis. To give thesame column formation phase velocity for both Al and Warrays, the difference in ablation velocities between cathodeand anode would need to be greater for W than in Al.

B. Characteristics of the precursor column

This section investigates the behavior of the fully formedprecursor column. The rise of the x-ray pulse due to thestagnation of the array on the axis begins when the implod-ing sheath impacts the precursor 11,2325, not the barearray axis, and so the precursor properties are important indefining the final stage dynamics of wire arrays.

Following formation, the precursor column continues toradiate in the range 40200 eV for all the array materialsstudied. The diameter of the precursor column immediatelyafter formation shows a dependence on array material, aswas noted for aluminum and tungsten in the previous sec-tion. Several other array materials have been investigated onthe MAGPIE generator, and the observed minimum columndiameters are plotted in Fig. 9. The average diameter valuesfor each material are shown, with error bars representing thestandard deviation. Some materials have been investigatedonly for a small number of experiments, and so statisticalanalysis would not be meaningful. Values range from 3 mmfor carbon arrays to 0.25 mm for gold arrays. Significantly,the values decrease almost monotonically with increasingatomic number, suggesting that radiative cooling plays animportant role. Previous analysis of the variation in the pre-cursor column diameter is presented in Ref. 22, which

FIG. 8. Laser interferometer and shadow images from the sameshot, showing axial nonsimultaneity of a precursor column forma-tion at 160 ns in a 16 wire tungsten array.


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leads to a description of how the precursor diameter varieswith atomic number. The method is derived from the rocketablation model which evaluates the mass ablated from thewires in an array at time t as

mt =0

4VaR0

0

t

I2tdt, 1

where Va is the ablation velocity, R0 is the array radius, andI is the drive current which varies with time, t. From thisexpression, the mass density inside a given precursor radius,rp, at time t can be described. This is achieved by adjusting

the integral limits to account for the time of flight delay fromthe wires to the required radius, tR0/Va, and dividing bythe precursor volume

pt =0

42rp2

VaR0

0

tR0/VaI2tdt. 2

To recover the precursor column radius, some simplifica-tions of the physical situation must be made. After formation,it is assumed that the column diameter remains at a radiusdetermined by the balance between the kinetic pressure ofthe incoming plasma streams and the thermal pressure of thecolumn itself. To allow this situation to exist it must also beassumed that all kinetic energy from the streams is radiatedaway. The kinetic pressure is given by

Pkin =Va

2rp

dm

dt3

and the thermal pressure is given by z +1NiT, where the iondensity can be replaced by the precursor density in Eq. 2divided by the ion mass. If these expressions are then evalu-ated at the boundary of the precursor column, and so r=rp inboth cases, an expression for the equilibrium radius of thecolumn can be found

Req =

0

tR0/Va

I2tdt

I2t R0/Va

2z + 1T

Vamion. 4

Previous experiments on MAGPIE applied radially re-solved XUV spectroscopy to estimate the plasma tempera-ture for an aluminum precursor column 22. These experi-

ments inferred an electron temperature of 5060 eV, andthis value can be used as a guide in the above analysis. Toevaluate the relative variation in radius for different materi-als, a ratio of the expressions derived from Eq. 4 for twomaterials can be taken. For example, for tungsten and alumi-num this gives

RW

RAl=

ZW + 1TWAAlZAl + 1TAlAW

. 5

For ZW=14, ZAl=6, TW =TAl=60 eV, a ratio of 0.31 iscalculated. This compares well with the value from experi-ments of 0.37 using the minimum radius values of RW0.3 mm and RAl0.8 mm.

This simple analysis can be extended to include all thearray materials investigated on MAGPIE, and this allowscomparison to the experimental data. The calculated varia-tion with atomic number is given by the solid line in Fig. 9.The values for charge states used for the comparison assum-ing T60 eV in all cases are ZC =4, ZAl=6, ZTi,Fe,Ni=7,ZCu=8, ZMo=10, ZW,Au=14, which are consistent with co-rona equilibrium CE ionization values 26. Note that themodel is normalized to tungsten, and so the ratio in Eq. 5equates to unity for this material. The experimental columndiameter observed immediately after formation comparesvery well to the diameters expected from this analysis, al-

though the diameter is slightly overestimated for most mate-rials. Despite this, the similarity of experimental and modelresults suggests that the precursor column is in approximatepressure balance immediately after formation. The expansionat later times discussed in the next section moves experimen-tal values towards, not away from, the model values indicat-ing that the column remains in, or close to, the pressurebalance assumed in this model.

Following the formation of the precursor column at itsequilibrium radius, two processes are observed. The col-umn diameter slowly increases, and the center of the columnviewed end-on begins to show reduced emission. The latterof these is shown in Fig. 10, which comprises a sequence ofend-on XUV images for a tungsten array taken during asingle experiment. The plot shows line outs of the emissionprofile through the array diameter at the image times notethat film contrast has been normalized to the emission ob-served close to the wire position, at 1 mm and 15 mm, foreach image. At 133 ns a broad initial emission profile occu-pies a large portion of the array diameter, as was reported inFig. 5. At 163 ns, the highly emitting well-defined precursorcolumn is seen to develop from the broad emission profile,and appears partially formed at this time. At 193 ns the col-umn has been fully formed for 20 ns and the center of thecolumn radiates less than the region of high emission at its

FIG. 9. Variation of precursor diameter immediately after for-mation with atomic number; Data are shown with error barsabsence indicates too little data for statistical analysis, along withanalytical model estimations solid line.


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perimeter. This process is observed for both tungsten andaluminum arrays.

The appearance of this hollow radiation profile could beexplained in the following way. Once the precursor column

has formed, its density is significantly greater than theplasma streams which continue to impact it, and a classicstanding shock scenario is realized. The high rate of radiationfrom the dense column allows rapid cooling of energetic ionsfollowing impact at the shock front, i.e., the column radius,and the rate of cooling at a given post-shock velocity yieldsa cooling length. If the velocity is relatively constant withtime, the cooling length will then depend on the density ofthe column and incident streams, and the ion energy on im-pact. If the precursor grows in radius and/or density, thecooling length may become shorter than the column radiusitself, and so a hollow radiation profile may be expected.

The XUV cameras used to image the precursor column in

Fig. 10 are sensitive to radiation above 40 eV. If the centerof the column has equilibrated and cooled to below thislevel, it would be expected that emission at this position inthe image would be reduced, as is observed. The regionwhere ions are thermalizing across the shock is typically400 m at times shortly after column formation. Atpresent, there is little data on the variation of this width withtime, although it is clear that this will continue to reduce ifdensity continues to rise and may therefore provide an indi-rect measure of the change in column density with time. Itshould be noted that this process appears analogous to theradiation cooling signature observed in laser-generated jets27.

The experimental results presented in Sec. III A show thatrather than remaining static at its minimum radius, the pre-cursor column continuously expands for both materials in-vestigated in detail. The variation in diameter after columnformation, 165 ns for Al and 180 ns for W, are plotted inFig. 11. The scatter in the data limits analysis to a simplelinear fit to give an average expansion rate over the timescale investigated and fitted rates for tungsten and aluminumare 2.5 m/ ns and 10 m/ns, respectively.

Since the column is expanding with time, the analysespresented above cannot be directly applied for times subse-quent to moment of formation. We can, however, apply simi-

lar methods with slightly different assumptions. Despite therequirement that the column is static, the above discussionsallow a good estimate of the precursor diameter at formation,and suggests thermal and kinetic pressures are balanced. Af-ter column formation, plasma streams continue to arrive atthe precursor column edge, increasing the mass accumulatedin the column. The evolution of the column can thereforeoccur broadly in two ways: with constant radius and increas-ing density, or constant density and increasing radius. Theformer of these is clearly not occurring here and so the as-sumption of constant density will be investigated for com-parison to the observed expansion rates.

For a given precursor column radius an average density

can be calculated at the observed time of formation from Eq.3. If density remains constant from this time on and thecolumn remains in approximate pressure balance, the columnwill grow in radius as mass continues to arrive. Reference16 derives the rate of change of the column radius from therocket ablation model as:

Rp

t= 0

42miniVaR01/2

t

0

tR0/Va

I2tdt1/2

. 6

Note that the equation given in Ref. 16 has been cor-rected by the original author to the above version.. Due tothe lack of a reliable experimental measurement at present,the ion density is taken at column formation from the modeldescribed in the reference to be ni31019 cm3. If thisvalue is used, an expected growth rate can be found whichvaries as the square root of the ion masses for different ma-terials. The use of the model density is not ideal and willclearly influence the calculated growth rate. The additionalexperimental studies of density suggested below would bevaluable here, and would also provide direct experimentalevidence on whether constant density is a reasonable as-sumption.

An alternative approach is to continue the pressure bal-ance scenario resulting in Eq. 2. An expression for the pre-

FIG. 10. Development of hollow radiation profile in tungstenprecursor column produced in a 16 mm diameter, 16 wire array.

FIG. 11. Observed growth rates for top aluminum and bot-tom tungsten precursor columns after formation, with straight linefits.


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cursor column radius is given by Eq. 3, and this is inde-pendent of the column density, which was problematic in themethod above. A temperature must be provided, but aroundthe time of formation this is 60 eV for aluminum, as dis-cussed previously. If we assume this temperature remainsconstant, the column radius will change as a function of themass accumulation described by the first term, and a rate ofchange can be calculated for aluminum

Rp

t=

2z + 1T

Vamion

t0tR0/Va

I2tdt

I2t R0/Va . 7A rate for tungsten can be evaluated if it is assumed that

the tungsten precursor column also remains at 60 eV. Theionization states used for Eq. 5 are again used here. Thesetwo approaches can be compared to the experimentally ob-served rates to examine which of the two cases is more likelyto be occurring in the precursor column expansion phase.This is shown in Fig. 12.

Both models predict that the growth rate for aluminumwill be greater than that for tungsten, as a result of the dif-ference in ion mass, and this is consistent with observations.For aluminum, during the period 180215 ns both modelsshow values which are within the errors of the experimentaldata. Following this, the constant density model remains atthe limit of the errors on the data, while the constant tem-perature model increasingly deviates from this. For tungsten,the constant density approach consistently overestimates thegrowth rate. The constant temperature model agrees at earlytimes, 180200 ns, but again disagrees at late times. Therelatively large errors from the experimental measurementsdo not allow a deduction of the trend in the precursor columngrowth rate, which could feasibly distinguish between thetwo simple models presented here. A more detailed study isrequired to reduce these errors.

The fact that the constant temperature model increasinglydeviates from data in both cases suggests that this assump-tion becomes more inaccurate, and hence that the columnmay be cooling with time. This is consistent with the x-ray

signals and the XUV filtered images presented earlier. If thetungsten precursor temperature is taken as lower than thealuminum precursor, the constant temperature model curvemoves into closer agreement with experiment. If tungstenand aluminum column temperatures are indeed comparable,the agreement with experiment for tungsten is lost at an ear-lier time, perhaps suggesting a greater cooling rate with re-spect to the aluminum column. It seems likely that any in-ferred temperature drop with time is associated with someincrease in the density of the column, and that both of thesefactors determine the growth rate following formation. Thiseffect appears more pronounced in tungsten than in alumi-num, where density may remain close to constant after for-

mation. Both these simple models vary as the inverse of theion mass, and so the majority of array materials would beexpected to display a precursor column expansion rate be-tween that of aluminum and tungsten.

Since the discussion presented above suggests that theprecursor cools with time, an interesting approach is to usethe experimentally measured rate of column expansion pre-sented in Fig. 12 to determine this temperature variation. Ifthese values are substituted for dRp/dt in Eq. 7, and againVa is 1.510

5 ms1, the product z +1T can be recoveredfor both aluminum and tungsten given the correct currentwave form, and a charge state. The variation of z+1T, andthe precursor temperature on MAGPIE assuming both coro-

nal equilibrium CE and local thermal equilibrium LTEionization models are given in Fig. 13.

Both materials show a decreasing temperature with time,as may be expected from the discussion above. The initialtemperatures using LTE ionization are similar to the 60 eVdetermined from aluminum experiments, with results fromCE ionization slightly higher than this. Both are quantita-tively similar despite the use of a simple pressure balancemodel. The difference of20% in temperature between alu-minum and tungsten does not appear unreasonable, and othermaterials would be expected to fall between these two

FIG. 12. Experimental values for top aluminum and bottom

tungsten precursor column growth rates compared to simple modelsassuming constant density and constant temperature. Bars representerror ranges on experimental values.


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extremes. Between 180 ns and 280 ns after current start anapproximately linear temperature decrease occurs, and ap-proximate cooling rates for Al are 0.55 eV/ns CE and0.34 eV/ ns LTE, and for tungsten are 0.65 eV/ ns CE and0.29 eV/ ns LTE. As above, additional experiments to moreaccurately determine the rate of expansion of the columnafter formation as well as additional XUV spectral datawould increase the confidence in this determination of thecolumn temperature variation.

The use of nonimploding arrays with the relatively longrise time and good diagnostic access afforded on MAGPIEallows analysis of the expansion rates of the precursor col-umn without the additional complication of the main arrayimplosion. For imploding loads, the main array mass snowploughs onto the precursor column and a large fraction of thegenerated radiation is trapped inside this converging piston.This will increasingly raise the thermal energy of the precur-sor column, and therefore increase the column expansion rateduring the implosion phase. Cuneo et al. have investigatedthis hohlraum effect on the precursor column for tungstenarrays on the Zmachine 11. The high opacity of an implod-ing 300 wire array causes a significant expansion of the pre-cursor column, the start of which is coincident with the onset

of the main array implosion from its initial diameter. Thisreference also develops an analytical model relating the pre-cursor expansion rate to the radiated power of the implodingpiston, which matches closely with the experimentally ob-served precursor column diameter change during implosion.

Direct measurements of precursor column density andprofile can be achieved using quantitative radiography. An xpinch 28 and references therein, mounted in one of thecurrent return posts, provides high spatial resolution pointprojection backlighting of the array axis. This image is pro-

jected onto a film pack comprising a radiation filter12.5 m titanium transmitting 35 keV radiation and afilm Kodak M100 placed immediately behind the filter.Quantitative measurements were taken for a tungsten array,by comparing transmission through the precursor column tothat of a step wedge comprising 15 nm, 30 nm, and 45 nmthicknesses of tungsten deposited directly onto the titaniumradiation filter. This is presented in Fig. 14. Spatial resolution

in this image is better than 5 m, and exposure time is1 2 ns.The timing of the radiograph is unclear from this experi-

ment, however a statistical analysis of similar x pinches onMAGPIE gives an average value of 20020 ns. Line outs ofthe image at the indicated positions are also given in Fig. 14,and the absorption profile gives an average areal density of2104 kg/m across the precursor column. The areal den-sity sensitivity limit for the diagnostic is 1104 kg m2

equivalent to 5 nm of tungsten at solid density. The integralof these mass density profiles yields a mass per unit length of3.6107 kg/m at this time, which represents approxi-mately 3% of a typical imploding array 325 m tungstenwires. Analytical calculations from the rocket model 22suggest the mass fraction in the precursor over this timerange is 618 %. This discrepancy may be due to severalfactors. The lack of an accurate time for the radiograph isproblematic, and it is possible that the column is still at theearly stages of formation. This may account for the observeddifferences in the radial profile of the column at differentaxial positions. If the column is indeed fully formed at thetime of the radiograph in Fig. 14, some regions may be be-low the sensitivity limit indicating a radial variation in den-sity in the column. A study of the density profile of the pre-cursor column would increase the current understanding of

FIG. 13. Variation of the product z + 1T and precursor columntemperature with time using coronal equilibrium CE and localthermal equilibrium LTE ionization models for top aluminum

and bottom tungsten.

FIG. 14. Radiograph of precursor column for a 16 wire tungstenarray with Line outs at indicated positions.


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wire ablation and precursor column formation, and providedensity values to compare to computational codes.

C. Current in the precursor

Previous work 22 noted the remarkable stability of theprecursor column over the duration of the experiment. Datafrom the laser probing and soft x-ray imaging showed thatdevelopment of the MHD instabilities M=0,1 in the pre-cursor column are not seen for either aluminum or tungsten,and the plasma column remains stable for 100150 ns. Morerecent studies show that at very late time W arrays do displaysome level of instability. The filtered x-ray pulse as in Fig.5 for nonimploding tungsten arrays show a foot pulsefeature, at 320 ns. Laser shadowgrams taken at the peak of

this feature apparently show M=0,1 type-structures, asshown in Fig. 15. Similar features are not observed in alu-minum precursor columns, either in x-ray signals or imagingdiagnostics, at times as late as 300 ns after column forma-tion. This information can be used to obtain an upper esti-mate of the current which might flow through this plasma,assuming that the growth time of the classical MHD insta-bilities would be given by these time scales.

Characteristic growth rates of m =0,1 instabilities in a Zpinch can be written as 29

= cAr0, 8

where cA =B

is the Alfven speed; r0 is the plasma radius;and is a scaling factor which depends on the instabilitymode, kr0 and the distribution of current. Substituting forB =I/ 2r0 and mass accumulated in the precursor per unitlength mpr=r0

2 gives

= cAr0 = I

2r

mpr. 9

The upper estimate of the current which might flowthrough the aluminum precursor column could be obtainedusing 1/300 ns, r0 =1 mm, and mpr410

6 kg/m,

which is 50% of the initial mass of a 1615 m Al wirearray. Taking the wavelength of any instabilities in the pre-cursor to be determined by the coronal streams as 0.5 mm,the minimum value of is 0.9 corresponding to a m =1mode with uniform current distribution. Hence, maximumcurrent through the precursor column is less than 23 KA or2.3% of the total current through the array. Here themass accumulated in the precursor is deliberately overesti-

mated, and the use of smaller mass will give a smaller upperlimit of the current. The laser probing measurements pre-sented earlier gave the precursor mass equal to 2% of thearray mass at 150 ns, while later in time, this mass couldbe as high as 10% 22. Use of the smaller mass of theprecursor, corresponding to 10% of the mass of 1615 m Al array, reduces the above estimate of the upperlimit of current in the precursor to 1% of the total current.This is in good agreement with direct experimental measure-ments of the current on-axis during 16 wire experiments, inwhich a sensitivity limited value of2% of the drive currentwas obtained 30.

In tungsten, the development of MHD instability struc-tures are observed at late time, some 130 ns after column

formation. If we take the growth time to be 100 ns, r=0.3 mm, and use 50% and 10% of the mass of an array, weobtain 2% and 1% of the drive current respectively.These current estimations are not sufficient to provide con-tainment of the column, which is consistent with the assump-tions in the pressure balance condition used in the previoussection.

The situation, however, could be different for some othermaterials, resulting in a higher current fraction. This followsfrom our preliminary experiments with Ni wire arrays com-posed of 8 wires 31, in which development of m =1 insta-bilities with characteristic growth time of 50 ns was ob-served in the precursor. This indicates that as much as 30%of the total current could flow through the precursor column

in this case. This is consistent with the delay of the implosionobserved for Ni wire arrays, and the implosion time mea-sured from optical streak photographs for Ni arrays was50 ns or 20% longer than it should be according to zero-dimensional 0D analysis. In contrast, both W and Al arraysshowed implosion times were in good agreement with 0Dprediction. Other factors may also influence the current frac-tion present in the precursor. In 4-wire tungsten arrays onMAGPIE, rapid development of instabilities in the precursorcolumn was observed after formation, in a similar fashion toNi arrays, indicating a significant portion of the drive currenton axis during the experiment 32, in contrast to the 16 wireresults discussed in this work. When MAGPIE is operatedwith a pre-pulse, comprising a linear ramp over500 ns to 35 kA, current driven instabilities in the precursorcolumn are also observed 33, and the variation in current inthe precursor column for nested 34 and mixed wire 35arrays is the subject of future investigations. It should benoted that much work is currently being carried out to deter-mine the current distribution in the various array configura-tions, notably on the Zebra generator at the University ofNevada, Reno 36 and on the Angara 5-1 facility at Troitsk,Moscow 37.

In summary, for 16 and 32 wire Al and W arrays thecurrent fraction in the precursor during experiment is low

FIG. 15. X-ray signals and laser shadowgraphy showing devel-opment of MHD instability structures in tungsten precursor at latetime.


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3% , and this fraction appears to be slightly higher fortungsten than aluminum. It is clear, however, that furtherwork must be carried out to determine the experimental arrayparameters that control the transfer of current to the arrayaxis.

IV. COMPARISON TO COMPUTATIONAL MODELS

Both hydrodynamic HD and kinetic models were ap-plied in order to further elucidate the formation mechanismof the precursor column. The hydrodynamic model is a sim-plified version of the three-dimensional 3D Eulerian resis-tive magnetohydrodynamic Gorgon code developed byChittenden et al. A full description can be found in Ref. 15.The code performs explicit hydrodynamics on a Cartesianx ,y ,z Eulerian grid, using second order Van-Leer advec-tion. The thermal and magnetic field diffusion equations aresolved implicitly by iterative solution of matrix equations.The model is two temperature electrons and ions with localthermal equilibrium LTE ionization and a simple recombi-nation radiation loss model. The radiation loss model is

modified to include a probability of escape that allows asmooth transition to blackbody emission in the dense plasmaregions. A cut-off density of 104 kg m3 is used in order torepresent the vacuum surrounding the pinch.

For the current study, this code is used in 1D cylindricalcoordinates and magnetic field effects are neglected, and themodel is therefore purely hydrodynamic HD. A continuousflow of plasma is introduced at the boundary with a flowvelocity and density consistent with the rocket model. Anartificial filtered radiation output can be generated by sum-ming radiation from regions of plasma with a temperature45 eV, and this allows some comparison to experimentalx-ray signals. The temperature chosen here is rather arbitrary

in an attempt to more clearly discern the mechanisms atwork, but is not in significant disagreement with the experi-mental measurements made at photon energies of200280 eV presented in Sec. III B. Cases for over-massedaluminum and tungsten were investigated.

The electron and ion temperatures and mass density foraluminum at selected times are given in Fig. 16. A 2 mmradius from the array axis is shown in each case, and thefiltered radiation output for this run is given in the samefigure note that in this configuration the array does not im-plode. At 130 ns, the plasma shows a relatively broad lowdensity profile as a standing shock develops. The electronand ion temperatures towards the axis axis at 0 mm haveequilibrated, but the thermalization of streams at the shockgive a high ion temperature here. The electron temperaturetowards the axis 70 eV and the radiation output is increas-ing, as density slowly rises, towards the first peak in thefiltered x-ray signal.

At 155 ns, the first x-ray peak has been passed, and theelectron and ion temperatures at the axis have dropped to20 eV as a result of radiative loss. This cooling reduces theplasma temperature to below the energy cutoff for the radia-tion output, and so causes this signal to drop. The densityprofile is then compressed by the incoming plasma streamsand begins to reduce in radius. The plasma on axis continues

to be compressed to higher density, and the precursor columnis present at 185 ns. As the column formation completes, theincoming plasma streams compress it on the axis, resultingin a small rise in the electron temperature, and this generatesthe second x-ray pulse. At 250 ns the column is observed asa stable shock, giving a high density object on the array axis.The electron temperature on axis is low, 15 eV, and thecontinued thermalization of plasma at the column edge gives

a temperature of30 eV here. This is below the filter forthe radiation output, and so this remains at zero. Note thatthis radiation profile will be hollow, i.e., low temperatureat the column center with higher temperature surroundingregion, in qualitative agreement with experimental observa-tions.

The filtered x-ray output describes a two stage process foraluminum. The first peak is a result of increasing radiationloss due to a slow density rise. It is only following this peakthat the broad initial density profile contracts to form theprecursor column. The second peak is a result of compres-sion of the column immediately after its formation by theincoming plasma streams. The timing of the first x-ray peak,at 142 ns, is similar to the experimental value of 155 ns, but

the separation of the peaks is far greater in the simulationthan observed in the experiment. It should be noted that theexact timing of the simulated x-ray peaks are heavily depen-dent of the radiation loss rate used in the model.

For a tungsten array simulation, the filtered x-ray outputsignal is significantly different Fig. 17. The two stage sig-nal from aluminum is not seen, and instead a single peak at128 ns is generated. However, analysis of the density andtemperature profiles shows that the processes which deter-mine the x-ray pulse for aluminum are indeed occurring in asimilar fashion for tungsten. The greater radiation loss ratefor the heavier material causes these two processes to bealmost simultaneous, and only a single peak is produced

which comprises contributions from both. The timing of thissignal is 50 ns earlier than is observed experimentally,which is a direct result of the inability of a fluid code tomodel the collisionless flow conditions in tungsten at earlytimes. Note that the timing for aluminum of this first peak ismuch closer to the experiment. The aluminum plasmastreams are collisional from early times, and so the fluidapproximation is more appropriate and a reasonable agree-ment results. To investigate the effect of collisionality com-putationally, the application of a particle code is discussednext.

The kinetic approach uses a hybrid model comprising aparticle description for ions and an electron fluid, and a fulldescription can be found in 16. Ions are injected at thearray radius with a number density consistent from the rocketmass ablation model, and the experimental velocity of 1.5105 ms1, which is assumed to be constant. Again, over-massed array configurations are used. Ions are advanced inconfiguration and velocity space in the usual particle-in-cellmanner, and using a Monte Carlo collision algorithm. Inter-action with the electron fluid is via a frictional force, whichincorporates the temperature inequality and the electron-ionenergy transfer time, representing the multiple scattering ofions from a Maxwellian distribution of field electrons. Theradiation model used for the fluid code is utilized again here


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to determine cooling rates for aluminum. Due to the lack ofa simple reliable radiation loss model for tungsten, the samemodel is used to calculate cooling rates in tungsten.

Figure 5 of Ref. 16 shows the evolution of density pro-files for aluminum and tungsten arrays. Aluminum shows a

relatively well defined shock profile even at early times, re-sulting from the stagnation of the ions streams as they reachthe array axis. Tungsten, however, shows a very broad flatprofile for a long time. This is a result of the collisionlessflow conditions of the ions streams at early times, and den-

FIG. 16. Plots from hydrodynamic model results showing a Mass density profiles for selected times, along with filtered x-ray output,and b Electron and ion temperature profiles at selected times for simulated aluminum array. Array axis is at 0 mm and the only first 2 mmradially from axis are simulated..


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sity is peaked on axis due to the convergent flow configura-tion, rather than stagnation of the plasma streams.

The collapse of this initial profile occurs earlier in alumi-

num, and the column is fully formed by 160 ns at a diameterof1 mm. This does not occur in tungsten until 175 ns, andthe formed column diameter is 0.5 mm. The maximumdensity in both cases remains similar after formation, andboth columns grow in radius as a result of the accruement ofmass from the plasma streams. The on-axis ion temperatureis seen to reduce as column formation occurs for both mate-rials. At late times, the ion temperature in low at the columncenter, and a peak in the ion temperature at the edge of thecolumn is observed where ions continue to thermalize, in asimilar fashion to the fluid code results.

Both models show formation of a high density columnforming on axis from a broad initial density profile for both

aluminum and tungsten. In addition, after formation both de-scriptions show a peak in ion temperature at the edge of theshock which defines the formed column, while the columncenter has equilibrated to the electron temperature. This sug-gests a hollow emission profile as observed in the experi-ment. The macroscopic dynamics of precursor column for-mation appear to be well described in both cases.

The formation times, however, differ between the ap-proaches. Aluminum column formation times agree reason-ably well, but the HD code predicts tungsten column forma-tion significantly earlier than observed either experimentallyor in the kinetic model. The kinetic description shows amuch broader initial density profile for W than Al, in linewith the experimental observations. The fluid model showsthe opposite case, the limiting factor being the onset of ra-diative loss, which is more delayed for Al than W, allowingthe standing shock to grow to larger radii. These differencesare a result of collisionless flow in tungsten plasma streamsat early times, which fluid modeling is unable to account for.This occurs for some 40 ns after tungsten plasma streamsreach the array axis, delaying the formation of the precursorcolumn and the peak in the x-ray signal. If this approximatedelay is simply added to the formation times and x-ray signalfor the HD model results, a far better agreement with experi-ment is achieved. There is currently no x-ray signal compari-

son from the kinetic code, but column formation times for Aland W agree very well with the experimental values of150160 ns and 170180 ns respectively.

The formation mechanism, described in detail above forthe HD model, shows an inherently two stage process, thetimings of which are determined by the array material. Col-lapse of the initial broad density profile occurs only after thefirst peak of the x-ray signal, as is observed experimentally.

The second stage is a compression of the formed column,resulting in a small temperature rise. The delay between thetwo events is significantly smaller for tungsten than for alu-minum, again in qualitative agreement with experiment. Ma-terials of different atomic masses would be expected to fallbetween these two extremes, and show an x-ray signaturewith two peaks separated by a distance related to the radia-tion loss rate of the material. Preliminary investigations us-ing the HD model show this is indeed the case. Nickel, forexample, gives two filtered x-ray peaks separated by lessthan is seen for aluminum but greater than for tungsten. Theabsolute timing differences, however, are heavily dependenton the details of the radiation loss routines applied in the

codes. These are notoriously difficult areas to clearly define,and further work on both experiment and computational ra-diation loss models is required.

The fluid approach is reasonable for the aluminum arraysinvestigated in this work, but the collisionality of tungstenplasma streams are fundamental to an accurate description. Akinetic description is required in the early stages of plasmastream interaction. At higher velocities, this may also be-come important in aluminum and even lower atomic numbermaterials, and so is an essential part of the process. It may bepossible to include the effects of early time collisionless flowin the fluid model by means of a collision operator, as de-scribed in Refs. 38,39, although this is essentially an ap-proximation to the full kinetic approach. At later times, fol-lowing the onset of collisionality, both the fluid and kineticdescriptions show a similar formation mechanism, and thiscorrelates well to the experimental results.

On generators such as the Z machine, the distinction be-tween materials is likely to be less significant. The highercurrent level and faster rise rate leads to the ablation of aconsiderably greater mass of material from the wires than onMAGPIE. The plasma flow is therefore likely to be colli-sional from early time, and for 20 mm arrays the interactionat the array axis will be dominated by collisional flow. Theimportance of collisionless plasma flow may increase signifi-cantly if the plasma flow velocity is high. For lower mass20 mm diameter tungsten arrays this may indeed be the case40, and for 40 mm diameter arrays, the inferred velocity ofmaterial streaming to the array axis may be as high as 4105 ms1 41. Plasma may counterstream at the array axisfor a period of time, and therefore this behavior must beincluded in the modeling of these arrays.

V. TWISTED WIRE ARRAYS

In this modification, a 2.3 cm long, 16 mm diameter wirearray was twisted at an angle of/4, /2, or 3/4 by rota-tion of the top electrode relative to the bottom electrode.

FIG. 17. Filtered x-ray output from hydrodynamic model foraluminum black and tungsten dashed grey.


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Twisted wire arrays were considered as a method of mitigat-ing Rayleigh-Taylor RT instabilities during the implosionphase of an array, as was observed for uniform density gas-

puff experiments with an hourglass axial distribution 42.In this work twisted arrays are used to investigate propertiesof the precursor flow inside the array.

The smallest angle of rotation can be used to measure theazimuthally averaged radial distribution of the coronalplasma inside the array. The twisting effectively removes theazimuthal structures, formed by the collision of the streamsfrom different wires, from the end-on images shocks shownin Fig. 3. For twisted arrays the shocks are not seen due toaveraging of the gradients along the probing path of the laserbeam. This is shown for a 32 wire aluminum array in Fig. 10of Ref. 22.

For arrays with larger angles of twist, the axial magnetic

field Bz, generated inside the array is large enough to notice-ably affect the dynamics of the inward flow of the coronalplasma. The precursor plasma column on the array axis isstill formed, but this happens later and the delay increaseswith the twist angle. As this is increased, the distance rincreases, and hence the distance from the wires to the arrayaxis at the mid-plane of the array decreases Fig. 18. Thetime of flight for plasma ablated from the wires reduces ac-cordingly and it is expected that the precursor column shouldform at progressively earlier times as the twist angle is in-creased. Figure 19 shows the time of the precursor formationmeasured by radial streak photography in the middle of thearray for the different twist angles. The solid line representsthe predicted formation time from the reducing time of flightto the array axis, assuming fixed velocity, and is normalizedto the experimental point for zero twist. The formation timefor /4 is earlier than for an untwisted array, and very simi-lar to the flight time prediction. This suggests that small twistangles do not significantly disturb the plasma flow behaviorinside the array. At larger twist angles, however, the precur-sor column formation time increasingly deviates from thetime-of-flight predictions. Both 3/4 and /2 twist anglesdisplay a later column formation time than untwisted arrays,despite wires being substantially closer to the array axis atthe array midplane. This observed delay of the precursor

formation could occur due to deceleration of the coronal

plasma by the pressure of the axial magnetic field Bz gen-erated inside the array, although precise mechanisms are un-clear at present.

For arrays with a /2 twist, the axial magnetic field gen-erated in twisted wire arrays was measured using two mag-netic probes with opposite orientation of the turns, installedinside the array at the axial position at the end of the array,inside the hollow top electrode 43. The measured value ofBz at 150 ns was 1.5 T, which corresponds to a magneticfield of3 T in the middle of the array. It should be empha-sized that this estimate of the magnetic field in the middle ofthe array was obtained neglecting an effect of shielding ofthe magnetic field at the position of the probe by the elec-trode, and thus represents a lower estimate of the field insidethe array.

Calculated magnetostatic values ofBz for the twist angleof/2 suggest Bz5 T at time 150 ns assuming that thecurrent is flowing through or very close to the wires andneglecting attenuation of the field by the currents induced inthe electrodes. The fact that experimental and calculatedvalues are reasonably close may indicate that the calculationassumptions are correct, in that the current flows close to thewires. It is important to note that if current carrying plasmafrom adjacent wires merged, current would be able to flowdirectly axially and no Bz would be measured in the experi-ment. If plasma streams remain discrete for a significant pro-portion of the array radius, a considerable amount of thedrive current may be carried towards the array axis, and a B

zfield would still be measured. However, in untwisted alumi-num cylindrical arrays, streams are observed to interactshock structures on Fig. 3 approximately 1 mm from thewire position. The persistence of the axial magnetic fieldtherefore suggests that current primarily flows in regions nocloser to the axis than 1 mm inside the array radius.

VI. SCALING AND APPLICATIONS OF THE PRECURSOR

COLUMN

The precursor column demonstrates several interestingplasma physics processes occurring during its formation, in-

FIG. 18. Diagram of a 32 wire array with a /2 twist, showingthe reduction in radius along the z axis as a result of the twist.

FIG. 19. Variation of precursor formation time at the array halfheight with distance of wires from array axis. Data is shownwith error bars, and solid line represents predicted formation timefrom time-of-flight variation at fixed velocity.


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cluding transition from collisionless to collisional flow re-gimes, and the effect of radiation loss on the dynamics of theplasma. On the MAPGIE generator, precursor column densi-ties shortly after formation are of the order of10191020 ions/cm3 and temperatures of 80100 eV are in-ferred, which vary with array material. These values lie inwarm dense matter WDM regime. For the relatively longtime scales on which the precursor column has been ob-

served in this work 200 ns the constant or increasingdensity and decreasing temperature ensure the precursor col-umn remains close to this region. By studying the behaviorof the precursor column at different times during its lifetime,several different regions of parameter space may be acces-sible.

To investigate how the precursor column characteristicsscale with generator, we can attempt to extend previousanalyses to the Z Machine at Sandia National Laboratories20 MA, 100 ns rise time to see how this compares withpublished experimental data. Cuneo et al. 11 report thatprecursor structures with a FWHM diameter of 1.53 mmhave been observed on axis at times 0.5 stag where stag is

the stagnation time of the array on to the axis. Observationshave not been made at times earlier than this at present. Theprecursor remains at constant radius for 20 ns and thenexpands as the main array implosion begins at 0.75 stag, asdiscussed in Sec. III B. Analysis of time resolved x-ray pin-hole cameras XRPHC indicates precursor radiation bright-ness temperatures of 2530 eV at early time which rises to75 eV at the time that the main array mass implodes ontoit. The following discussion continues the pressure balanceapproach, which describes the precursor column on MAG-PIE well, for 20 mm diameter tungsten arrays on Z.

Section III C derived a relationship for the equilibriumradius of the precursor column immediately after formation

as a function of the generator current, the ablation velocityand the temperature see Eq. 4. This can be rearranged togive the temperature as a function of the radius at a givenpoint in the current drive

z + 1Tt = I2t R0/Va

0

tR0/VaI2tdt

VamionReq

2. 10

To determine the product z+1T at a given point in thecurrent drive, the equilibrium precursor radius, Req, and theablation velocity, Va, must be supplied. If the range of1.5 to 3 mm given above is taken, the temperature productcan be calculated for a range of ablation velocities as a func-tion of time.

At very early times this value will clearly be unphysical,since the precursor column will not have formed. The onsetof the formation process results from increasing radiationloss due to increasing on-axis density, which becomes com-parable and then exceeds the kinetic energy flux from incom-ing plasma streams allowing collapse of the density profile.As the radiation loss rate is proportional to ni

2, it is thereforethe on-axis ion density which is central to the timing of theprecursor column formation. The ion line density at the arrayaxis as a function of time and current is given by the mass

arriving at the axis derived from the rocket ablation modeldivided by the ion mass

nionaxist =0

4VaR0mion

0

tR0/Va

I2tdt. 11

In Al arrays on MAGPIE, ablated plasma streams are col-lisional from early times as they reach the axis, as is ex-pected to be the case for 20 mm W arrays on Z. A delay incolumn formation occurs for W arrays on MAGPIE due tothe lower ion number density ablated 1/mion and a periodof collisionless flow at the axis. To give an indication ofwhen formation of the precursor column is likely to occur on

Z, we will therefore examine the ion density in aluminum onMAGPIE at the time of formation observed in experiments.The first peak in the soft x-ray signature for Al occurs at anaverage time of 155 ns after current start see Fig. 5, andthis indicates when radiation-driven collapse and column for-mation has begun. The ion line density at the array axis atthis time is 61018 m1. If the column formation pro-cesses described in this work on MAGPIE occur in a similarfashion on Z, this represents a critical ion density, ni,crit, atwhich point the formation processes are occurring and theprecursor column will form. The time at which ni,crit isreached at the axis during an experiment therefore representsthe earliest time at which the precursor column is expected toform.

The product z+1T for a fixed column diameter of1.5 mm calculated from Eq. 11 is presented in Fig. 20 for arange of ablation velocities. These calculations assume thearray configuration is again over-massed and that the mainimplosion phase does not occur. The dashed vertical line isthe time at which the ion line density at the axis first exceedsthe critical value of 61018 m1. Values to the left of thisline i.e., earlier in time are prior to column formation andtherefore unphysical. The value ofZ+1T at ni,crit is greatestfor the largest ablation velocity, and this decreases with de-creasing velocity. Rapid cooling is apparent for all ablation

FIG. 20. Calculated variation of the product TZ+ 1 with timefor a precursor column of fixed diameter 1.5 mm for three abla-tion velocities. Dashed vertical lines indicate expected precursorcolumn formation time see text.


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velocities and the plots converge to similar values after

100 ns.The time at which the precursor column is expected toform varies considerably with ablation velocity. The forma-tion time for the lowest velocity 1.5105 ms1 gives 77 nsis 30 ns later than for the highest velocity 3105 ms1

gives 47 ns. This difference represents a third of the currentrise time on Z and is therefore highly significant in the un-derstanding of pre-implosion processes in these arrays. Itshould be noted that the line ofni,crit assumes collisional flowdominates interaction at the axis, and this would move laterin time for a given ablation velocity if collisionless flow isimportant. It should also be noted that the ni,crit line positionin time is relatively insensitive to changes in the current

shape for Z. The fast high current rise dominates the process,and changes in the peak current reached 1820 MA andthe inclusion of the experimental 1 MA foot pulse changesthis position by 2 ns. In addition to these features there isalso a prepulse current which rises to 45 kA over 300 ns.If the ablation velocity and plasma streaming behavior aresimilar during this period it is unlikely that the mass ablatedduring the prepulse will affect the time of formation of theprecursor column, however experiments to investigate thisearly time behavior are required to clarify this.

To estimate a temperature for the precursor column thecoronal equilibrium model used in the MAGPIE calculationsis again used here, and the resulting plots are given in Fig.21. The temperature of the precursor column at ni,axisni,crit varies significantly with the ablation velocity as-sumed. For the lowest velocity the temperature is 600 eVat formation 77 ns and drops to 200 eV by 130 ns. In-creasing this velocity by a factor of 2 gives a precursor col-umn which forms at 47 ns with a temperature of 900 eVand which again drops to 200 eV by 130 ns. The calcu-lated temperatures shortly after formation do not contradictexperimental observations on Z see Figs. 4 and 7 in 11.

The above estimates are obtained using a precursor col-umn diameter which is constant in time to allow a solution inthe equilibrium situation. The real variations in the precursor

column temperature after formation may be different due tochanges in the precursor diameter with time during the ex-periment, and this should be investigated to further this in-terpretation. To provide some indication of the variation ofthe calculated temperature with column diameter, Eq. 11can be evaluated for 1.5 mm and 3 mm diameter columnsusing a fixed ablation velocity of 1.5105 ms1 and againusing a CE ionization model to recover a temperature. Theseplots are given in Fig. 22. The change in precursor diameterdoes not affect the time at which ni,crit is reached during thecurrent drive, but the larger precursor is formed with a highertemperature. The pressure balance model used here dictatesthat the larger precursor is required to maintain a higher tem-

perature in order to balance plasma stream kinetic energy. Atformation the 3 mm diameter precursor is 950 eV, whilethe 1.5 mm diameter column is 600 eV, and even at latetimes the difference in temperature is 50 eV.

It is interesting to compare the time at which the criticalion density at axis is reached for these estimations with theexperimental times at which precursor structures have beenobserved for 20 mm diameter tungsten arrays on Z. The av-erage stagnation time for these arrays is 95.5 ns, and so thetiming of the earliest observation of the precursor structuresat the array axis corresponds to 48 ns on the plots in Figs.2022 in this paper. The time at which ni,crit is reached, andhence the time at which the precursor column should formand be observed dashed lines on plots, is calculated to belater than the experimental results above for both of the twolower velocities investigated. Only the highest velocity ofVa =310

5 ms1 yields a timing comparable to observa-tions. For comparison, one can also consider the time offlight TOF of ablated plasma from the array radius to theaxis for each of the ablation velocities investigated here. Forthe lowest velocity this TOF is 67 ns, and no ablated materialwill have reached the axis by the time of the experimentalobservation of a precursor structure at 48 ns 0.5stag. Simi-larly for Va =210

5 ms1 the TOF is 50 ns, and plasma hasbarely reached the axis by the time precursor structures are

FIG. 21. Calculated variation temperature with time for a pre-cursor column of fixed diameter 1.5 mm for three ablation veloci-ties, using a CE ionization model. Dashed vertical lines indicateexpected precursor column formation time see text.

FIG. 22. Calculated variation temperature with time for precur-sor column of 1.5 mm and 3 mm diameter using fixed ablationvelocity. Dashed vertical line indicates expected precursor columnformation time see text


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observed. The velocity of material streaming towards the ar-ray axis must therefore be greater than 2105 ms1 even ifthe precursor column forms immediately as streams collideat the axis. For the highest velocity, however, the TOF is33 ns which allows a brief period of increasing on-axis den-sity before ni,crit is reached, and the precursor column is ex-pected to form at 47 ns 0.5stag.

This high ablation velocity fits well with the formation

time of the precursor column from experiments, but is sig-nificantly higher than has been calculated in 111.40.3 ms1 and references therein. This would give ananomalously low mass ablation rate at the wires which isinconsistent with the fitting of implosion trajectories carriedout in 11. One explanation for this discrepancy could besuggested by comparison to experimental observations of theablation dynamics of wire arrays. It has been reported bymany authors e.g., 21,44,45 and references therein thatthe plasma ablated from wires and ejected towards the arrayaxis is periodically modulated in the axial directiontheablation wavelength. For tungsten, this wavelength is250 m. In addition, experiments demonstrate that gaps in

the wire cores are subsequently formed indicating that massablation is not uniform axially, and mass ablation is greaterat some points than at others. Some points along the wirelength may eject material with a high velocity, which wouldthen satisfy the early formation time of the precursor col-umn, but there must then also be regions where mass abla-tion is high to facilitate wire breakage and the start of theimplosion.

From the rocket model, the mass ablated at time t can bewritten as

m =1

Va 04R00

t

I2dt . 12For each wire in an array, the values in square brackets in

Eq. 13 are t

Documents

S. C. Bott et al- Dynamics of cylindrically converging precursor plasma flow in wire-array Z-pinch experiments