Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution. J.P. Chittenden Imperial College

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1 Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution J.P. Chittenden Imperial College In collaboration with S.V. Lebedev, S.N. Bland, F.N. Beg, J. Ruiz-Camacho, A.Ciardi, C.A. Jennings, A.R. Bell, and M.G. Haines from Imperial College, With additional experimental data from S.A. Pikuz, T.A. Shelkovenko, from P.N. Lebedev Physical Institute and D.A. Hammer from Cornell University With grateful thanks for funding from the AWE William Penney Fellowship scheme Sandia National Laboratories and the U.S. Department of Energy Dr. Jeremy P. Chittenden, William Penney Research Fellow, Plasma Physics Group, Blackett Laboratory, Imperial College of Science, Technology and Medicine Prince Consort Road, London, SW7 BZ, U.K. tel , fax , . j.chittenden@ic.ac.uk APS-DPP, Quebec

2 Assumption of rapid shell formation followed by D(r,z) MRT instability omits plasma formation effects and other important 3D phenomena If we were to assume that the initial flow of current causes rapid and uniform explosion of the wires then an almost uniform cylindrical shell of plasma results. THIS DOESN T HAPPEN The growth of the magneto-rayleigh-taylor instability is then responsible for shell broadening which determines the X-ray rise-time. THIS IS NOT THE ONLY EFFECT AND SOMETIMES ISN T IMPORTANT AT ALL. Radius in mm innermost bubble spikes thin shell D model Time in ns Rise-time ~ shell thickness / velocity THIS CANNOT EXPLAIN LOW WIRE NUMBER RESULTS APS-DPP, Quebec

3 Wire arrays cover a wide range of parameters but exhibit the same physical processes Owl II, xµm Al, 7mm φ SATURN x5µm Al, 7mm φ MAGPIE x5µm Al, mm φ MAGPIE x5µm Al, mm φ + x5µm Al, mm φ 5 Z Current in MA 5 SATURN SATURN Long Pulse Z, x7.5µm W, mm φ + x7.5µm W, mm φ A wide range of materials and diameters are used Owl II MAGPIE Time in ns Total currents vary considerably but currents per wire and inter-wire gaps are similar APS-DPP, Quebec

4 MAGPIE wire array experiments show intrinsically 3D phenomena with scales ranging from a few µm to several mm Side-on laser schlieren, r-z modulation (m= like instabilities in each wire?) End-on laser interferometer, r-θ modulation radial plasma streams Side-on X-pinch X-ray back-lighter reveals dense wire cores embedded within the coronas 79ns mm At late times, structure apparently resembles a global Rayleigh-Taylor instability For details on experiments see DO.7 MP. WO. Simultaneous laser schlieren shows relative size of coronas APS-DPP, Quebec

5 Talk Outline Philosophy Bench-mark D and 3D models in detail against MAGPIE wire array data and several single wire experiments. Use these models to understand behaviour of similar experiments at higher currents on SATURN, Z, X.. Cannot model whole problem (3D + global & fine scale structures) simultaneously. Therefore model different phases separately and attempt to link them Radius in mm Wire Initiation (Plasma Formation) Instability growth in each wire plasma Time in ns Coronal merger, mass injection and precursor formation Global instability development Nested array interaction Stagnation and X-ray generation Research Topics. D and D(r,z) cold-start single wire calculations :- formation of the core-corona structure, m= instability growth in individual wire plasmas.. D(r-θ) plane calculations:- how core-corona structure affects dynamics radial plasma streams, coronal merger, precursor. the physics of what controls the core ablation rate 3. A brief discussion of the physics of the precursor. D(r-θ) plane calculations of nested wire arrays :- momentum and current transfer during collision how these determine which mode of implosion results 5. 3D simulations of a single wire in an array :- origins of local and global perturbations differences in behaviour from single wires structure and trajectory of implosion APS-DPP, Quebec

6 Plasma formation in wires depends on complex EOS and transport coefficients ηj dt exceeds energy budget to heat, melt, vaporise and ionise all material in wires within a few ns. However this energy is not deposited uniformly, formation of a plasma corona greatly reduces energy transfer rate to cold, dense wire core, allowing it to survive until late times. -5 ρ solid / Pressure (MBar) ev 3 ev perfect gas ev.3 ev condensation. ev 3 ev degeneracy Resistivity in Ωm - -7 ρ solid /3 ρ solid Melting point Spitzer - like 3 Density (kg/m 3 ) Modified Thomas-Fermi Equations of State In condensed phase electron pressure is allowed to go negative, so that total pressure is zero. This is an oversimplification, but appears to work. Numerically such an EOS is a pain to use. However after a few ns, core expansion is sufficient for it to be approximated by a cold unionised gas. -. Temperature in ev Lee and More s transport model Modifications to transport remain important long after modifications to EOS, not least because Ohmic heating is found to be the dominant mechanism for energy transfer to the core. Considerable uncertainty remains over the resistivties around -ev [see GP. M.P. Desjarlais] APS-DPP, Quebec

7 D cold-start MHD simulations show formation of core-corona structure Consider a single 5µm Aluminium wire with ~ka/ns current Current Density (A/m ) Density (kg/m 3 ). E-3 E- 3.5x.x.5x.x 5.x 9 3 Radius in µm Temperature (ev) Total Pressure (Pa). T e T i Z * 3.x 7 Eqn. of State.5x 7.x 7 5.x Perfect Gas 3 Radius in µm ns Once vaporised core expands at roughly it s sound speed. Surface regions drop to low density and are readily ionised. Current gradually transfers from core to corona, which heats and expands. Core pressure << perfect gas value. Current Density (A/m ) Density (kg/m 3 ). E-3 E- 3.x.x.x.x.x.x 3 Radius in µm Temperature (ev) Total Pressure (Pa). 3.x 9.5x 9.x 9 5.x APS-DPP, Quebec T e T i Z *.5x 9 Eqn. of State Perfect Gas 3 Radius in µm 5ns Magnetic field becomes sufficient to pinch corona back onto core, triggering increased current flow and Ohmic heating core. Core continues to expand at roughly constant speed. Density is now low enough for pressure to be close to perfect gas value.

8 In D(r,z) cold-start simulations of a single wire, pinching of the corona excites the m= instability Z axis in mm 3 > Z axis in mm 3 Z axis in mm 3 Z axis in mm 3 Z axis in mm 3 Z axis in mm 3 R axis in mm R axis in mm R axis in mm R axis in mm R axis in mm APS-DPP, Quebec R axis in mm ns 5ns 3ns 35ns ns 5ns Short wavelengths at early times give way to longer wavelengths as plasma expands so that λ / radius roughly constant Current by-passes contorted path through flares and flows through narrow region just outside the core. Initially necks fail to penetrate core which remains virtually unperturbed. Since the core retains the majority of the mass, when the necks eventually penetrate to the axis, this represents a dramatic increase in total perturbation amplitude. Depletion of the core material in the region of penetration results in high temperatures and X-ray bright-spots

9 Experiment at 5ns Comparison to single wire data provides benchmark tests for D MHD code plus EOS and transport models therein Simulation at 5ns Experiment at 5ns Simulation at 5ns For example comparison to laser probing and X-pinch radiography of µm Al wires at Cornell [D. Kalantar and D. Hammer, Phys. Rev. Lett. 7, 3 (993)] allows simultaneous tests of wavelength and amplitude of m= in corona plus core expansion. Alternatively recent quantitative X-pinch radiography of low current Al wires at Cornell [S.A. Pikuz and T.A. Shelkovenko] provides more detailed test of core expansion Radius (mm) Corona Min. Corona Max. Core Min. Core Max. Corona Exp. Core Exp. Areal Density (µg/cm ) Time (ns) Radius (mm) APS-DPP, Quebec

10 3D behaviour of wires in arrays limits the application of D single wire calculations to scaling arguments M= instability in single wires Amplitude and wavelength increase as corona expands Necks penetrate cores forming X-ray bright-spots Growth dependent on current per wire Z axis in mm 3 R axis in mm ns 3ns ns x 5µm Al on SATURN Instability (or just modulation?) in wires in arrays ns Amplitude, wavelength and size in azimuthal direction are almost constant in time X-ray bright-spots not observed before global implosion initiates Growth is a weak function of current per wire Z axis in mm 3 R axis in mm x 5µm Al on SATURN ~9% of mass remains in core For low current per wire, instability doesn t penetrate core, perturbation amplitude remains small For higher current per wire (N < ), core is penetrated before array implodes, perturbation amplitude in each wire reaches %, perturbation for whole array ~ / N APS-DPP, Quebec

11 Y Axis in mm D(x,y) plane simulations show how core-corona structure radically alters implosion dynamics x5µm Aluminium on MAGPIE 5 ns Y Axis in mm ns Y Axis in mm 9 ns Y Axis in mm 5 ns Y Axis in mm X Axis in mm ns X Axis in mm Y Axis in mm X Axis in mm ns X Axis in mm X Axis in mm Use D cold-start to initialise X Axis in mm Low density corona accelerated by jxb and swept around wire cores forming radial plasma streams Dynamical balance between rate of material ejection from core and injection into interior of array Streams reach axis at ½ implosion time forming precursor Cores remain stationary until % of implosion time Majority of array mass on axis prior to implosion Dynamics and structures accurately reproduce experiments APS-DPP, Quebec

12 The same phenomena persist for higher wire numbers and larger, faster rising currents x7.5µm Aluminium on SATURN Low density parts of plasma streams merge early on Cores remain stationary and intact until % of implosion time During flight, cores merge to form azimuthally symmetric shell. Significant fraction of array mass on axis prior to implosion Radial mass profile agrees well with initial conditions assumed by Whitney and Thornhill [IEEE Transactions on Plasma Science p (99)]. Absence of axial dimension means short, sharp radiation pulse obtained. APS-DPP, Quebec

13 Use reduced zone around one wire for high resolution e.g. One of 3 5µm Al wires on MAGPIE In this case, jxb force redirects ablating material towards array axis without applying force directly to the core Close-up of core region Y Axis in mm Y Axis in mm Y Axis in mm X Axis in mm X Axis in mm X Axis in mm Log ρ(x,y) and log v(x,y) j z (x,y) Log ρ(x,y) and jxb(x,y) APS-DPP, Quebec

14 Ohmic heating of core material determines implosion trajectory Cool cores Warm cores Close cores j z j z Dynamical balance between core ablation and mass injection Residual cores remain stationary until % of implosion time. Mass averaged radius versus time similar to thin shell model Once T e > ev current transferred to cores, some jxb force applied directly cores rapidly heat and expand Trajectory similar to thin shell model Reduces injection of material and current between cores. Trajectory similar to thin shell model R R R T T j z T APS-DPP, Quebec

15 The precursor plasma is an apparently stable, uniform and long-lived, D plasma. Density in Kg/m 3 Z star carbon aluminium tungsten. stationary precursor.... flux through boundary. radially convergent stream Radius in mm Radius in mm Temperature in ev V r in ms - Gated soft X-ray images of precursor indicate that equilibrium radius is a strong function of material Can be modelled in high resolution with D MHD x -.x 5 -.5x 5 Radius in mm Radius in mm Precursor lifetime > ns Initial formation phase is collisionless. Once collisonal, converges to a two component equilibrium of high density stationary precursor and lower density convergent radial plasma stream. Pressure balanced by ρv of bombarding stream. Little or no current. Kinetic energy delivered (ρv 3 A) is the roughly balanced by radiation losses. Similar to a test developed to evaluate different artificial viscosity formulations in D hydrodynamics (W.F. Noh, J. Comp. Phys. 7, p7 (97). Density ratio between precursor and stream ~[(γ+)/(γ-)]. Data suggests for Al γ 5/3. For W precursor density much higher γ. Ideal test-bed for opacity measurements, X-ray laser experiments and benchmarking radiation hydrodynamics codes. APS-DPP, Quebec

16 There are at least 3 different theoretical modes of nested wire array dynamics Hydrodynamic Collision (or Shell on Shell) Mode Transparent Inner (or Current Transfer) mode Flux Compression (or Magnetic Buffer) mode 7 9 Time in ns 7 9 Time in ns Outer Inner 7 9 Time in ns APS-DPP, Quebec

17 Outer D(x,y) simulations reproduce collapse dynamics of nested arrays on MAGPIE ns 5 ns ns 3 ns..... ns Y Axis in cm... Y Axis in cm... Y Axis in cm... Y Axis in cm... Y Axis in cm... Inner..... X Axis in cm.. 9 ns..... X Axis in cm.. 5 ns..... X Axis in cm.. ns..... X Axis in cm.. 3 ns..... X Axis in cm.. ns Y Axis in cm... Y Axis in cm... Y Axis in cm... Y Axis in cm... Y Axis in cm X Axis in cm + x 5µm Aluminium (equal length arrays)..... X Axis in cm Inner wires heated by bombarding plasma streams from outer Small fraction of current flowing through inner array produces B θ between arrays Compression of this flux by implosion of outer produces sufficient current to drive inner ahead of outer. Radius in mm X Axis in cm Time in ns..... X Axis in cm Outer Sim. Inner Sim. Outer Expt. Inner Expt X Axis in cm APS-DPP, Quebec

18 Model inner and outer arrays on Z separately, first calculate radial plasma flux from outer array, then use this to bombard the inner array.. x7.5µm tungsten wires on a mm diameter ns Similar features to lower wire number cases, Y (or θ) Axis in mm -.. ns -.. ns ns X (or R) Axis in mm Dense wire cores retain most of the mass until implosions commences. Low density corona swept around cores forming radial plasma streams. At 75ns precursor stream extends down to mm and contains % of mass. In D, the remaining % is in a mm wide shell After this stage the plasma is largely homogeneous in the azimuthal direction. Use the flux through the LHS of the outer array simulation to provide RHS boundary conditions for simulation of an inner array wire. APS-DPP, Quebec

19 3 ns 9 ns ns ns 7ns D(x,y) simulations predict the implosion modes of nested arrays on Z Inner array of x.5 µm W wires Y (or θ) Axis in mm Y (or θ) Axis in mm Y (or θ) Axis in mm Y (or θ) Axis in mm Y (or θ) Axis in mm X (or R) Axis in mm X (or R) Axis in mm Plasma stream from outer of x7.5µm W Y (or θ) Axis in mm Y (or θ) Axis in mm Y (or θ) Axis in mm Y (or θ) Axis in mm Y (or θ) Axis in mm X (or R) Axis in mm X (or R) Axis in mm X (or R) Axis in mm Inner array wires see ns of low density coronal bombardment followed by main mass in mm wide shell. At first little expansion of inner wires outer material streams through, setting up bow shock. Later bombardment by denser main mass heats each wire with ~GW of kinetic flux. Inner wires expand rapidly allowing effective momentum transfer. Compression of magnetic flux carried by plasma stream effectively increases momentum transferred. Trajectory similar to hydrodynamic collision mode with reduced radiation at collision. Trajectories consistent with transparent inner mode require 3 wires in inner. APS-DPP, Quebec

20 5 ns Even with 3% amplitude perturbation on ρ(z) with.5mm wavelength, apparent modulation is much less than in experiment r z ns Side-on laser schlieren of Al arrays on MAGPIE show: modulation in corona from ~ns roughly constant amplitude (r + - r - ) and wavelength ns 3D MHD simulation shows: initial modulation amplitude retained in core & corona no apparent growth or change in modulation no apparent difference in cross-section for different axial positions Maybe this isn t an MHD instability at all? APS-DPP, Quebec

21 3D simulation of m= instability in ideal MHD equilibrium pinch: growth rate agrees well with analytic theory Similar results have been obtained for m= instabilities [S.G. Lucek, private communication] APS-DPP, Quebec

22 Modulating core resistivity versus z, gives results similar to experiment Z Axis in mm ns ns ns R Axis in cm Z Axis in mm R Axis in cm R Axis in cm Lower core resistivity in centre, higher core resistivity at ends modulated core heating and ablation. During implosion wire core breaks, current penetrates inside wire array, cold core regions left behind. Z Axis in mm APS-DPP, Quebec

23 Two and Three-dimensional Modelling of the Different Phases of Wire Array Z-pinch Evolution Conclusions D cold-start models illustrate important processes involved in plasma formation phase and provide model verification through comparison to single wire experiments. Absence of 3D effects, however, severely limits their ability to predict the behaviour of wires in an array. D(x,y) simulations show how the flow of material ablating from the core is redirected by j B forces forming the radial plasma stream and the precursor. Require better resistivity models to cover all array parameters. D(x,y) simulations of nested arrays model momentum transfer and magnetic flux compression during collision. All shots to date on Z have been hydrodynamic collision like, transparent inner mode requires fewer wires on the inner and larger (>.5mm) inter wire gaps. Preliminary 3D modelling suggests MAGPIE data can be explained in terms of wire breaking and not necessarily Rayleigh-Taylor. Can this model be extrapolated to Z? More experimental data is needed. APS-DPP, Quebec