The science of compressional heating on the LM26 magnetized target fusion experiment S. J. Howard , D. P. Brennan , K. Epp, P. Forysinski, D. Plant, M. Reynolds , A. Froese , N. Sirmas , D. Krotez , V. Suponitsky, R. Zindler, E. Love, C. Macdonald, N. Kumar, Z. Seifollahi Moghadam, K. Conquergood, A. Wong, W. Zawalski, B. Rablah, W. Kozicki, P. Carle , A. Rohollahi , C. Preston , A. M. D. Lee , J. Hobbis, L. Santos, X. Feng, M. Schellenberg-Beaver, R. Tingley, R. Underwood, J. Sanchez Rojo, J. Gorenstein, D. Froese, E. Cessford, J. Pratt, J. Crofts, J. Sardari, G. Faust, D. Ross, J. Wilkie, S. Bernard, S. Edwards, R. Oosterom, M. Yurkiv, J. Y. J. Cheng, C. Connor, S. Bolanos, C. Gutjahr, E. Chan, M. Greenwood, E. Ng, A. Massey, K. Chen, R. Svihra, A. Gromer, S. Lee, X. Zhu, L. Marshall, C. Eyrich, A. Mahoney, M. Davidson, H. Feng, A. Rudy, and M. Laberge General Fusion Inc., Richmond, British Columbia, Canada (Dated: June 22, 2026) The Lawson Machine 26 (LM26) operating at General Fusion has demonstrated compressional heatingofasphericaltokamakdeuteriumplasmaasitwascompressedbyanimplodingsolidlithium liner. Results from the first 11 compression experiments on LM26 are presented, the highest- performing of which show more than a three-fold increase in electron temperature, a ten-fold in- crease in density, and a ten-fold increase in poloidal field in the plasma driven by three-fold radial compression. The experimental device and its instrumentation are reviewed in detail, followed by direct observations from each of the key diagnostics for liner trajectory and plasma properties, measuringincreasesinmagneticfield,electrondensity,aswellasemissionofneutrons,X-rays,and visibleradiation. Observationsfromfast-cameraimagesduringcompressionprovidedetailedcontext for interpreting the spatial structure of plasma-wall interaction. Overviews of the various models developedandusedintheanalysisarepresented. Diagnosticdataareusedtoreconstructtheexper- imentalequilibriumstateincomputationalmodelingasafunctionoftime. Theresultsindicatethe reconstructions are sufficiently accurate to match the essential observations from the experiment, validating the models and building confidence in the stability and transport analyses that support the key conclusions. Trends across the full set of 11 compression shots are presented, and a more detailed examination of the high-performance shots are given individually. The key conclusions of theintegratedphysicsmodelspecificallyindicatethatcompressionalheatingwasachievedinthisset ofexperiments,asevidencedbythebalanceofheatingpowerfromcompression,Ohmicheatingfrom plasma current, and losses to the boundary necessary to match the experimental data. A majority oftheincreaseintemperatureisattributabletocompressionalheating. Anincreaseinneutronflux is also observed during compression. The results provide a basis for planned improvements to the LM26facilitythatwillenablethecompressionofmagnetizedplasmatoincreasinglyhigherdensities and temperatures. 1. INTRODUCTION 1–11. The initial proof-of-concept design of the experi- ment uses a cylindrical domain to hold a spherical toka- mak(ST)plasmawhichiscompressedtowardsastraight Magnetizedtargetfusion(MTF)isafusionpowercon- centralshaft,relyingonpoloidalfluxconservationwithin cept in which a fuel is confined by magnetic fields while thelithiumlinertocompresstheconfiningmagneticfield being compressed and heated inside an imploding con- of the plasma. Solid lithium is used due to its low mass ductive liner. This method is economically attractive density,highelectricalconductivity,mechanicalsoftness, because the energy used to bring the fuel up to fusion low atomic number, and its chemical getter ability that conditions is primarily supplied by the motion of the enables a low recycling-coefficient boundary. The trajec- walls,whichcanbeproducedinexpensively. Usingasolid tory of the lithium liner is designed to shape the plasma lithium liner in Lawson Machine 26 (LM26), we have boundary, control the speed of compression, and deter- made progress at demonstrating the viability of heating mine when the confining chamber is closed by making a deuterium plasma to fusion-relevant temperatures us- contact with the axial endplates. Future versions of the ing rapid compression as the primary means of heating. experimentwillincludeanupgradetothecenterconduc- Thispaperwillreportontheevidenceofcompressional torgeometry,providingadouble-nappedconetoincrease heating and observations of magnetically stable confine- the rate of geometric convergence and better maintain ment of the plasma, which has been gathered during the self-similarity during compression. sequence of plasma compression experiments conducted so far on LM26. At the time of this writing 11 plas- A suite of diagnostics is used to directly measure fun- mas have been successfully compressed by imploding a damentalplasmaquantities, suchasmagneticfield, tem- lithiumliner;thosediscussedbelowaredenotedasLMC- perature, and density. Those measurements are fitted to N (LawsonMachineCompression),withN rangingfrom two separate models to reconstruct the evolving equilib- 2 riumstate. TheBayesianPlasmaReconstruction(BPR) 1.2. General Fusion MTF experiments model [1] treats the discharge as sequential independent time slices, using time-dependent boundary conditions ExperimentsatGeneralFusion(GF)haveexploredthe reconstructed from magnetic data. The Integrated Sys- viability of compressing a plasma with a spherical toka- tem Model (ISM) [2] fits to time-dependent integrated mak magnetic configuration [1]. Because of their low simulations of the full compression. The observed in- aspect ratio, ST plasmas exhibit favorable confinement crease of electron temperatures during compression are and stability properties [19, 20]. They operate at high well explained in these fits by a combination of compres- toroidal beta β (the ratio of plasma pressure to toroidal t sionalandOhmicheating. Stabilityanalysesandexperi- magneticfieldpressure)andcansustainhighplasmacur- mentaldataindicatetheplasmasremainstableuntildeep rents due to their low aspect ratio and natural shaping. into compression. The results form a basis for under- The energy confinement time, τ , has a distinctive fea- E standingthekeyphysicsgoverningcompressionalheating ture in ST plasmas: it depends strongly on the toroidal processes and inform continuing research into compres- magnetic field and weakly on the plasma current [21]. sional heating using flux conserving metal liners. Because of these features, General Fusion decided to pursue the spherical tokamak magnetic configuration as a promising approach to fueling a compact and low-cost 1.1. Magnetized target fusion fusion pilot plant [22]. In a sequence of subscale exper- iments, ST plasmas were compressed using chemically- The MTF concept dates back to the LINUS project driven implosions of aluminum liners [1, 23–25]. These built at the Naval Research Laboratory in the 1970s [3]. PCS experiments provided an understanding of plasma LINUS was intended to compress plasma with a molten behaviorandconfinementscalinglawsduringrapidcom- sodium-potassium liner driven by pressurized gas, but pression. the testing that was done used water [4]. Due to a Concurrently,large-scaleplasmagenerationwastested shiftinfocustowardtokamaks,researchonplasmaheat- inthePlasmaInjector3experiment(PI3)[26]. Thegoal ing through magnetic compression subsequently became of PI3 was to demonstrate that large (0.6–0.7m major dominated by tokamak experiments, including ATC [5– radius) plasma targets that are suitable for compression 8], TOSCA [9], TFTR [10, 11], JET [12, 13], and to fusion temperatures could be generated with coaxial TUMAN-3M [14]. In particular, the ATC experiment helicity injection (CHI) [27]. CHI is a method of gen- was designed specifically for this purpose, producing sig- erating magnetized plasma at a location distant from nificantcompressionratios,andsuccessfullydemonstrat- the flux conserver where the plasma will settle and be ing much of the basic concept. However, with the many compressed. Such separation is required to protect the advancesinphysicsunderstandingandcomputationalca- plasmaformationcomponentsfromthelinercompression pability in the years since these experiments, it has be- system to enable reusability in a commercial reactor de- come evident that the conventional tokamak configura- sign. TheresultsfromPI3showedthathighperformance tion is not the optimal choice for magnetic compression OhmicallyheatedSTplasmascanhavegoodenergycon- in terms of efficiency, stability, and confinement. finementtimeandhighinitialtemperature,makingthem At the turn of the millennium, dedicated MTF ex- suitableasthetargetplasmainanMTFcompressionex- periments began to be designed and built. They took periment. advantage of new progress in plasma physics and con- Once large-scale plasma formation was demonstrated trol electronics to advance the concept. FRX-L [15], in PI3 and rapid plasma compression was demonstrated a collaboration between the U.S. Air Force Research inPCS,thenextstageintestingwastocombinethetwo Laboratory and Los Alamos National Laboratory, gen- experiments in Lawson Machine 26 (LM26). erated and translated field-reversed configuration (FRC) The rest of the paper describes LM26 and its first re- plasmoids suitable for MTF injection and compression sults. In Sec. 2 the experimental setup is described in schemes. The next machine, FRCHX [16], used an elec- detail, including the geometry, plasma injector, lithium tromagnetically driven flux-compression liner to rapidly lineranditsdynamicsduringcompression,andthediag- implode an FRC plasma. During the Plasma Compres- nostics suite. In Sec. 3 we highlight some of the key sion Science (PCS) campaign [1], General Fusion com- experimental observations stemming from direct mea- pressed spheromaks and ST plasma configurations using surements during compression experiments, including aluminum liners. Helion Energy collided FRC plasmas poloidal flux, density, X-ray emission and temperature and compressed them with pulsed magnetic fields [17]. increases, as well as phenomena such as X-ray crashes. MagLIF (Magnetized Liner Inertial Fusion) at Sandia In Sec. 4 we present the physics analysis and model- National Laboratories used the Z pulsed-power machine ing that lead to the main conclusion of compressional to implode a cylindrical liner around axially-magnetized heating, including the equilibrium reconstruction meth- laser-preheated deuterium fuel [18]. While diverse in de- ods, magnetohydrodynamic (MHD) stability analyses, sign, approach, and scale, these experiments shared the and quantitative analysis differentiating the compres- goalofreachingfusionconditionsthroughrapidcompres- sional heating from Ohmic heating, taking into account sion of a transient pre-magnetized plasma. transport losses. Finally, in Sec. 5 we summarize the re- 3 sults and review our conclusions from the experimental of the vessel to center the lithium liner relative to the outcomes. shaft. 2. THE LM26 EXPERIMENT 2.2. Plasma injector LM26 is a large-scale MTF machine built to demon- TheSTplasmasinLM26areproducedviaCHIusinga strate that compressional heating of an ST plasma can Marshall gun, repurposed from the earlier PI3 device. A achieve fusion-relevant conditions. A Marshall gun fulldescriptionoftheMarshallgundesign,operationand forms, by CHI, a magnetized deuterium plasma inside powersupplycircuitparameterscanbefoundinthepre- a 1.6 m diameter hollow lithium cylinder within a com- vious work [26]. Here, we will focus on the modifications positevacuumvessel. Asurrounding36-turntheta-pinch and upgrades made to implement the new configuration. coil generates a strong magnetic field to symmetrically During a compression shot in the LM26 experiment, the compressthelinerandtheplasmaitcontainsinlessthan liner impacts the endplates, loading the structure with a 4 ms. The compression of the liner occurs without dam- large axial shock. The gun was lengthened by 0.5 m to aging any major structures, allowing the device to be make room for a high-current sliding electrical contact, rapidly restored for further experiments. whichmechanicallydecouplestheMarshallgunfromthe compression region endplates. 2.1. Geometry 2.2.1. Power supplies A schematic of this first iteration of the LM26 exper- iment is shown in Fig. 1. The confinement region is The plasma formation system draws on several dedi- roughly cylindrical, bounded on the inside by an alu- catedcapacitorbanks. Thepowertogeneratetheplasma minum shaft (10.8 cm radius) running along the axis and drive the CHI process comes from a 2.7 mF, 25 kV, of the machine, on the outside by a replaceable lithium 0.8 MJ formation bank. The toroidal field is generated liner,andboundedateachendbyaluminumplates. The by current that runs along the central shaft and returns Marshall gun opening is a 23 cm concentric gap whose oncopperstrapsoutsidethevacuumvessel. Thecurrent outer surface is flush with the liner. The lithium liner is initially produced by a 30 mF, 10 kV, 1.5 MJ peaking is surrounded by a composite vacuum vessel with stain- bank and then resistive decay in the aluminum shaft is less steel flanges mounting to the Marshall gun on one compensated by a 48 mF, 11 kV, 2.9 MJ sustain bank. side and the removable diagnostic endplate on the other. Shaftcurrentcanalsobeincreasedrapidlywitha42mF, The compression coil is composed of an axial stack of 12 kV, 3.0 MJ ramp bank. It is subdivided into 24 mod- 36 single-turn aluminum-plate coils with G10 fiberglass ules which can be triggered in sequence to provide fine support components. It is mounted on a decoupled sup- control of the ramp rate. port structure that surrounds the vessel. Each turn The purpose of the shaft ramp bank is to balance the is designed to have low inductance and high mechani- toroidalfieldintheMarshallgunwiththatintheconfine- cal strength. To minimize the error fields produced by mentregionastheplasmaisbeingcompressed, untilthe the coil, there are current-focusing features at the input gun is closed off when the liner contacts the endplates. gaps of each turn and the gaps are distributed toroidally In an ideal compression scenario, the shaft ramp should around the machine. This design concept was first used track the inverse of the coaxial inductance of the plasma on the Super-Magnetized Ring Test (SMRT) device [28] confining volume to maintain the original current profile at General Fusion and has proven effective at generat- shape and prevent the plasma from being squeezed back inghighlysymmetric,fastrise-time,multi-Teslapoloidal into the gun [1]. However, we find that in the present compression fields. cylindrical(rectangularcrosssection)geometryofLM26, Thecompositevacuumvesselisbuiltaroundaprimary theplasmaistrappedwithintheconfinementregionbet- cylinder(20mmthick,1.8mID)ofmulti-layerfiberglass ter than in the PCS experiments, so less shaft ramp cur- and high-vacuum epoxy (EP42HT-2AO-1 Black) com- rent is required to obtain reasonably good results. As posite and is capable of high vacuum operation (10−8 such, we typically run with the shaft ramp significantly Torr range). It is significantly more shock resistant and below the inverse-inductance curve. cost effective than glass or ceramic alternatives of com- parablesize. Thevesselmustbenon-conductivetoallow thepoloidalcompressionfieldtopenetrateitwithoutat- 2.2.2. Control of plasma parameters tenuation. A set of axial copper straps are bonded to the outside of the composite vessel. They are evenly The plasma is well described by a Grad–Shafranov distributed to provide a return path for current in the equilibrium with a pre-compression total poloidal flux coaxialcircuitthatgeneratestoroidalmagneticfield. An of 100–170 mWb. The closed poloidal flux is due to the array of ceramic blocks are attached to the inner surface toroidal plasma current contained within the last closed 4 support column cathode liner trajectory grounded anode end coil etalp ssecca piezo gas puff valves sliding contact retem 1 liner drive coils solid lithium liner Penn t i r n a g p interferometers initial plasma magnetic probes 7 mera spectroscopy ca aluminum shaft gun t=7.2 ms 3 a flux er m coils t=6.2 ms c a t=5.5 ms t=4.8 ms t=3.5 ms composite shell axial copper strapping FIG. 1. Schematic of LM26 experiment, with liner trajectory (lower half-view) and poloidal flux (upper half-view) shown for the LMC-9 shot as an example. flux surface (LCFS). There is approximately 6–14 mWb coolingtoroomtemperature,thecastingisremovedfrom of open “buffer” flux between the outside of the LCFS the mold and cleaned. It is mounted in a large vertical and the inner surface of the liner. Adjusting the rate lathe, where it is formed to a rough size by rolling and of change of externally driven shaft current modifies the machined to final dimensions with a single point cutter, toroidalfieldattheedgeoftheplasma[26]. Thispermits leaving a clean, smooth plasma-facing surface. All these some control over the current density profile and is used operations are performed under argon to prevent rapid to balance against the resistive decay of plasma current. corrosion of the lithium. The completed liners have an Setting the shaft-current slope above or below the slope outer radius of 880 mm to match the dimensions of ce- of an unsustained shot causes an inductive decrease or ramic centering blocks in the vacuum vessel, and have increase of the edge plasma current, respectively. thicknesses ranging from 50–70 mm and lengths of 1– The density profile of the formed plasma can be var- 1.2 m. The liner mass has varied from 153 to 218 kg to ied from flat to peaked by adjusting plasma breakdown explore different collapse velocities and impact loads. conditions in the gun. A flat profile contains more to- tal thermal energy than a peaked profile with the same Transfer and positioning of the liner within the com- core density and temperature. A peaked profile has re- pressionchamberofLM26isconductedwithinlargedry- duced wall interaction. Experimental data for a typical argon filled polyethylene tubular bags, using a custom high-performance plasma discharge are shown in Fig. 2. multi-axis motorized transfer and positioning system to place the liner within the chamber onto the centering blocks. A large 2 m diameter gate valve uses a 1 in thick aluminum honeycomb panel as a sliding shutter that is 2.3. Lithium liner manufacture and transfer raised and lowered by electric winches. It serves as an air lock component needed for attaching and removing The lithium liners used in LM26 are manufactured at new bags during different stages of the lithium transfer General Fusion. Manufacturing begins with lithium in- and diagnostic shaft installation processes. The shutter gotsthataremeltedinasteelmold,whichiscoatedwith is removed before plasmaoperations begin. After thein- boron nitride to prevent adhesion. After solidifying and stallation of the new liner under argon gas, the machine 5 is pumped down to vacuum ahead of plasma formation. 2.4. Operation of compression campaigns Prior to each compression, the plasma performance must be optimized. A campaign of forming plasmas into 0.5 the static liner is executed to condition the machine and 0.4 improve the pre-compression plasma. Each plasma that 0.3 isformedcleansresidualcontaminantsfromthelinerand 0.2 other plasma facing surfaces. Periodically, we evapora- 0.1 tively deposit thin layers of lithium on the plasma facing 0.0 surfaces to further getter contaminants and reduce ion recycling [26]. Oncethemachineissufficientlyclean,thecompression shotisdesignedbyvaryingmachineparameterstoestab- lishabaselineofplasmabehaviorandoptimizethecom- pressibility of the plasma target. In addition to optimiz- ing parameters such as initial plasma temperatures and magnetic lifetime, a candidate plasma needs to remain quiescent during the 3 ms interval that compression will takeplace. Anexampleofanoptimizednon-compression plasma can be found in Fig. 2. 2.5. Compression dynamics of solid lithium liners Thelithiumlineriselectromagneticallyacceleratedra- diallyinwardsbyastackofcoilsinthemannerofatheta pinch. Thecompressioncoilassemblycontains36single- turn aluminum coils, which are driven in pairs by 18 iso- lated circuits. Each pair is located symmetrically across themid-planeandwiredinseriestoensurethefieldgen- erated is symmetric. Each isolated circuit is connected to a capacitor bank with a capacity of approximately 1 MJ (19.2 mF, up to 10.5 kV), for an overall capacity of 18MJacrossallcircuits. Whenthebanksaredischarged into the compression coils, this produces a field around the lithium liner that briefly reaches 4.5 T (80 atm). During liner implosion, a higher inward radial velocity is generated at the ends of the cylinder compared to the middle, which causes a converging crimp shape with a central pocket that contains the plasma during compres- sion. See the lower half of Fig. 1 for examples of liner shape during the compression. This shape is formed be- causethecoilstackislongerthanthelinerandhasagap in the middle. Additional shaping of the liner trajectory can be accomplished by connecting only select compres- sion coils to the power supplies. The thickness of the liner increases during compression due to radial conver- gence. Thisthickeningincreasesthestiffnessoftheliner, causing the liner to decelerate later in the compression. Once the liner contacts the shaft endplates at both endsitformsaclosedcoaxialcircuitfortheshaftcurrent that had been, up to that point, driven externally. Now flowing in a closed poloidal loop, this current passively rampsupasthefinalstageofcompressionproceeds,with total toroidal magnetic flux within that loop being held ]AM[ I p 1.0 0.8 0.6 0.4 0.2 0.0 0 5 10 15 20ms Time after formation [ms] Ishaft [MA] 4 3 2 1 0 ]3-mc 3101[ e n 0.4 0.3 0.2 0.1 0.0 ]T[ lop B 100 mA 100 nA 1 mA 1 nA 10 µA 10 pA 0.1 0.01 ]V[ mn 055 II iL 0.1 0.01 0.001 Li I 671 nm [V] 2.0 1.5 1.0 0.5 0.0 Btor [T] a) B6tor Shot 6530 LMC-9 Prototype Magnetic fields B6pol B1pol B13pol Plasma and Shaft Currents b) I shaft I p Interferometry c) IF2 IF3 IF5 IF6 AXUV Raw Signals P_MY5 P_MY11 d) P_NF Q_NF E2_NF S_NF R1_NF e) Lithium Filterscope Signals S4 S4 S3 S3 S2 S2 FIG. 2. Example data from non-compression ST plasma for- mationshot6530,whichistheprototypeforcompressionshot 6531 (LMC-9). a) Poloidal magnetic field at three positions (B1, B6, and B13 in Fig. 6) and toroidal field at B6 in red. b) Plasma current, I , and shaft current, I , determined p shaft by all edge poloidal and toroidal field measurements, respec- tively. c) Interferometer channels that measure line average electron density at four different radial locations. d) AXUV signals which are described in detail in Fig. 8. e) Filterscope signalsmeasuringLiIlightat671nmandLiIIlightat550nm, both on three chords shown in Fig. 6. See sections 2.6 and 3 for more details about plasma diagnostics used on LM26. nearly constant, but the inductance rapidly decreasing. The liner length is chosen based on the desired touch- down time and location, where shorter liners are used to delay the time of contact and the possibility of ejecta, and longer liners are used to close the coaxial current sooner. Comparisonsbetweenshotparametersandkeyperfor- mance metrics for LMC-2 through LMC-11 are shown in Table I. LMC-1 was a commissioning shot and is not included in this comparison. 6 TABLE I. Experimental shot parameters and performance for liner dynamics. Here C is the liner radial R compression ratio as defined in Sec. 2.5.3. LMC- Metric / Parameter 2 3 4 5 6 7 8 9 10 11 LinerThickness[mm] 69.3 69.5 63.2 72.0 62.2 54.3 70.6 63.1 65.2 54.6 LinerLength[mm] 1081 1033 1126 1218 1063 1060 1011 1012 1060 1013 LinerMass[kg] 204 184 193 218 182 162 195 176 186 153 CompressionCapEnergy[MJ] 12.7 12.0 9.8 12.7 11.3 14.8 12.8 14.9 13.1 15.9 PeakKineticEnergy†[MJ] 3.41 3.09 2.50 3.47 3.09 4.49 3.57 4.36 3.74 4.70 PeakLinerVelocity†[m/s] 170 165 147 167 177 242 188 223 200 255 CompressionStartTime[ms] 3.505 2.519 3.714 7.012 2.509 2.112 2.013 3.516 3.518 2.514 CR at1stTouchdown† 1.60 1.55 2.05 1.10 1.65 2.05 2.20 2.75 2.00 3.20 CR at2ndTouchdown† 1.60 1.70 2.20 1.45 2.05 2.90 2.30 2.90 2.00 3.30 NormalizedBpol Increase 2.5 4.0 2.7 1.0 6.0 7.0† 7.0 9.1 6.6 10.8 CR atMaxBpol † 1.82 2.25 1.95 1.0 2.35 2.6† 2.52 2.83 2.46 3.05 †Estimates taken from best-matching simulations reconstructed from experimental data. Liner velocity taken at plasma mid-plane. 2.5.1. Liner trajectory and imaging diagnostics FOVlensassembly(Fig.3). A15W,637nmmultimode fiber-coupled laser source is guided into the center shaft. Light emitted from the fiber is collimated and then re- Trackingtheevolutionofthelinershapeduringimplo- flected from a multi-faceted diamond-turned aluminum sion is achieved using a set of diagnostics that measure optic that creates four arcs, initially covering 60° to 80° liner velocity and position, which is required for plasma intoroidalrange,atfouraxialpositions. Laserlightscat- compression modeling and to assess the performance of tered from the lithium liner is viewed through the imag- the machine. ing assembly, coupled to a monochrome high-speed cam- PhotonDopplerVelocimetry(PDV)isusedtomeasure era via an imaging fiber bundle and fitted with a 636/10 the velocity of the inner surface of the lithium liner wall. nm bandpass filter to enhance contrast and block most A 12-channel PDV measurement system is implemented otherplasmalight. Astationarylandmarkinglaserbeam using1Wfiber-coupled1550nmlasers,13GHzphotode- inthefieldofviewistrackedtofacilitateimagestabiliza- tectors, and oscilloscopes sampling at 1GS/s. Optical tion. Asthelithiumlinermovesinward,thearcpositions collimators are located on the center shaft and directed aretrackedandprocessedtoreconstructthelinertrajec- radially outwards towards the inner surface of the liner. tory using methods as described in [29] and [30]. SLR is There is an axial array and a toroidal array, which are suitable for assessing the uniformity of liner collapse and both shown in Fig. 3. Raw PDV signals are transformed measurement of liner perturbations with toroidal mode to velocity-time spectrograms [29]. Using the measured numberequaltoorgreaterthann=6. SeeFig.5forout- initial liner radius, the velocity can be integrated to get put data from liner symmetry analysis using SLR data. the instantaneous position of the moving liner. The final trajectories reconstructed from PDV input data are in Fig. 4. Imagingoftheplasmaconfinementregionisperformed using in-vacuum lens assemblies coupled to color and Depending on the liner trajectory and surface charac- monochrome high-speed cameras. Experiments can in- teristics, impurities may be ejected by jetting when the clude up to three wide (approximately 90◦) field of view liner impacts the endplates. A narrow FOV in-vacuum (FOV)assemblies. ApairofcalibratedwideFOVassem- lens assembly is used to observe the liner where it con- blies coupled to color cameras is used to provide stereo- tacts the formation-side endplate. In some compression scopic reconstruction of plasma structures. See Fig. 22 experiments, light emitted by the plasma is used to il- for views of the plasma during LMC-9. These imaging luminate the FOV to observe the presence or absence of views are situated at the edge of the plasma cavity at ejectacreatedfromlinerimpact. Inotherexperiments,a radiithatarecutoffbythelineratcompressionratiosof laserscatteringsystemisimplementedtodetectejectaby approximately 2.5–3.5 depending on the view and liner projectingalaserlinefromtheshaftoutwardsparallelto shape, so this data is only available for earlier compres- theformation-sideendplatesurfaceandperpendicularto sion. the narrow FOV observation plane. A filter rejects most The Structured Light Reconstruction (SLR) system is of the visible plasma radiation enabling recording of the usedtomeasurethelinerinnersurfacepositionandshape scattered laser light to provide qualitative evaluation of with finer toroidal resolution than the PDV and consists ejecta. This system is called the Laser Scattering from of a projected laser pattern imaged by a calibrated wide Ejecta (LSE) diagnostic. 7 t=0.75 ms )m( R 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 Z (m) 1VDP SLR3 2 1 0 2VDP 3VDP 5VDP 6VDP that contains the exact machine, liner dimensions, and baseline lithium material properties. The nominal tra- t=1.50 ms jectoryisgenerallyinagreementwiththePDVandSLR measurements. Wherenecessary,thetrajectoryisrefined t=2.25 ms through a Bayesian reconstruction method. The refine- ment is achieved by generating a database that sweeps t=3.00 ms through a physically reasonable range of parameters in theJohnson–Cookconstitutivemodelforlithium. These SLR parametersaremeanttocoveruncertaintiesinthemetal- camera lurgy of the machined liners, specifically the coefficients view related to strain hardening, strain rate hardening, and the initial yield stress. Reconstruction of the liner trajectory includes a prediction for the distribution of the poloidal flux, ψ (R,Z,t), from the theta-pinch coils and the more FIG. 3. Fitted simulation trajectory for LMC-9 outlining in- 0 ner and outer surfaces of the cross section of the liner as it slowly varying bias coils. Both the nonuniform currents approachesthemachineaxis(blue),PDVintegratedposition inducedinthelinerandmachinebythesecoilsandadvec- data points sampled from different toroidal angles (green), tion as the liner moves are taken into account. However, SLR reconstructed liner position, averaged over toroidal an- theCOMSOLsimulationsdonotincludeplasmacurrent. gle range of approximately 70–90 degrees (red). Time labels This is justified by the fact that the plasma current and denote time since start of compression. induced image currents only weakly affect the trajectory oftheliner. Theeffectofplasmacurrentisaccountedfor in the next step of reconstruction, described in 4.2. 2.5.2. Liner trajectory modeling and liner reconstruction Reconstructionofthelinertrajectoryforuseinplasma 2.5.3. Liner implosion performance compression modeling is performed using a combination of experimental measurements and simulations. Two- Key liner performance metrics are tracked for com- dimensional axisymmetric simulations are conducted us- parison between shots based on experimental measure- ingCOMSOLMultiphysics,AnsysLS-DYNA,andacus- ments and supplemented by simulation results. Metrics tomized OpenFOAM solver. These models employ cou- include the maximum and time-resolved histories of the pled electromagnetic, structural mechanics, and thermal implosion velocity, as well as the timing, position, and solverstocapturethephysicsoftheimplodingliner. The velocity of the liner at plate touchdown. Some of these simulation domain includes the liner, compression coils, values can be found in Table I. Of particular interest is and cavity geometries. Simulated coils are connected to the trajectory at the mid-plane of the flux conserver, re- external circuits that replicate the experimental setup, ferred to as the equator, with results generally presented using measured voltages as inputs. Further details on in terms of the liner radial compression ratio at this lo- themodelingcanbefoundin[29],[31],and[32]. Beyond cation, C (t)≡R /R (t), where R (t) is the radius R eq,0 eq eq linerprofilereconstruction,thenumericaltoolsalsoserve ofthelinerattheequatorandR =R (0). Tobetter eq,0 eq tooptimizetheshotconditionsduringthecampaign,and compare different shots, the results are also mapped in evaluate the dynamic response of the overall assembly. terms of C , rather than absolute time, normalizing out R A sample trajectory obtained from simulation and variations in implosion velocity. compared with experimental measurements for LMC-9 The trajectories at the equator for shots LMC-6 to isshowninFig.3,whichincludeslinerpositionmeasure- LMC-11 are compared in Fig. 4, with the evolution of ments from PDV and SLR. Measurements from PDV, radial velocity, radius, and radial compression ratio up shown by the green symbols, are obtained by using the to C =4. R known initial liner radius and integrating the measured LMC-11wasthefastestshottodate,utilizingacapac- velocity. For axial positions where PDV channels have itor bank energy of 15.9MJ. For all cases, as shown in toroidal redundancy, the average radial value is plotted. Table I, approximately 25–30% of the energy supplied is The SLR data, indicated by the red markers, shows the transferred to the liner’s peak kinetic energy. For LMC- radiallinerpositionaveragedoveralltoroidalpositionsof 11, this leads to a peak radial velocity of approximately each laser arc at different compression times. Note that 255m/s, which was achieved by C =1.5. Higher veloc- R since the laser projection arcs are angled with respect to ities occur near the ends of the liner. machine axis they provide liner radius at different axial Asthelinermassisincreased,byincreasingthelength positions as the compression proceeds. or thickness, and the energy supply is reduced, the tra- Trajectories obtained from COMSOL are used for jectoriesbecomeslower. LMC-6isanexampleofaslower plasma reconstruction. Nominal trajectories are ob- compression, which reaches a lower peak radial velocity tainedbyapplyingthemeasuredvoltagesintothemodel of177m/sattheequatorandachievespeakcompression 8 0 50 − 100 − 150 − 200 − 250 − 0 1 2 3 4 5 )s/m( .leV laidaR (a) 0.8 0.6 0.4 0.2 0 1 2 3 4 5 )m( suidaR enalp-diM (b) 4 3 2 1 0 1 2 3 4 5 Time since start of compression (ms) oitaR noisserpmoC LMC-6 LMC-8 LMC-10 LMC-7 LMC-9 LMC-11 0.04 Reconstruction 0.03 Test(PDV) 0.02 0.01 0.00 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Compression Ratio (C ) R (c) FIG. 4. Comparison of reconstructed equatorial liner trajec- tories for shots LMC-6 through LMC-11, showing the evo- lution of (a) radial velocity, (b) radius, and (c) compres- sionratioasfunctionsoftimesincethestartofcompression. The reconstructed trajectories are obtained from simulations constrained by experimental PDV and SLR data. Dashed curvesintheradialvelocityplotaremeasurementsfromPDV, computed by taking the average PDV measurements within ±50 mm of the equator. after 5 ms. With increased mass and lower peak veloci- ties,theselinersexhibithigherlevelsofdecelerationprior to peak compression, eliminating impact with the shaft at this location, and not exceeding C ≈4. R Anotherkeytrajectorymetricisthetoroidalsymmetry of the liner during compression. Evidence of asymmetry ispresentinallshots,asobservedfromthepost-shotliner state. However, the final state provides limited informa- tion on the magnitude of asymmetry during implosion, because of the highly nonlinear nature of any buckling Rd xam SLR0 SLR1 SLR2 SLR3 max FIG. 5. Measurements of liner symmetry (dR = max |R−R| /R)forLMC-9,obtainedfordifferentSLRchords max (SLR0–3) and plotted against the radial compression ratio. behavior. Additionally, contact between the liner and the shaft at peak compression modifies the inner sur- face topology, making it difficult to estimate the level of asymmetry present during implosion based on the final state. The limited number of toroidally redundant PDV channelsalsolimitstheabilitytodrawconclusionsabout liner symmetry. Consequently, estimates on asymmetry are limited to using SLR data, which can be used as a representative estimate for the degree of symmetry due to its toroidal range spanning 70–90 degrees at varying axial position depending on compression ratio. A liner symmetry met- ric (dR =|R−R| /R) is defined as the maximum max max variation of the reconstructed radius from the mean, di- videdbythemeanradius. Becauseofthelimitedtoroidal range, and uncertainty in the SLR reconstruction asso- ciated with calibration and alignment, it is not expected that distortions with toroidal mode number below n=6 can be reliably detected using this method. The current calibration alignment procedure is expected to calibrate out decentering of the liner associated with the n = 1 mode. Due to resolution limitations and apparent width of the laser line signals it is also expected that high fre- quency modes will be detected to a reducing extent. SLR measurements are only available up to C ≈ 2.5 R due to geometrical constraints limiting the depth to which the SLR signal is distinguishable and viewable, varyinglinershapesobscuringtheviewatdifferentcom- pression ratios, and light from the plasma obscuring the SLR signal at different times. However, beyond C ≈ 2.25, close to the cutoff time of the diagnostic, R fewer points are distinguishable, and measurements are not reliable for estimating dR . max For the representative case of LMC-9, dR is plot- max ted in Fig. 5 for the different SLR chords, against com- pression ratio. Variation is due to the extraction uncer- tainty of the laser reflection lines from the image, and the varying toroidal range of discernible signal for ex- 9 traction. The toroidal extractable range tends to reduce andthedR tendstoincreaseascompressionratioin- max 0.8 creases,meaningthatdeviationsfromthetrendaremore notable for later stages of compression ratio. It can be 0.6 seen that dR remains below 5% up to a compression max AXUV Q ratio of 2.25, which is a general trend observed for other LMC shots as well. LMC-10wasdesignedtohavethelinerimpacttheend- AXUV S plates at an earlier compression ratio to test if there was ejecta from the liner created upon its impact with the 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 plates. Using the laser scattering system, the liner was observed to travel across the narrow FOV during com- pression with no signs of ejecta or dust created upon impact traveling ahead of the liner. 2.6. Overview of plasma diagnostics LM26isequippedwithasuiteofdiagnosticsproviding measurement of fundamental plasma parameters. The flux conserver has Mirnov coils installed at eight unique radial positions on the endplates and five unique axial positions on the center shaft, all distributed across multiple toroidal angles to measure both poloidal and toroidal magnetic fields. These are shown as red dots in Fig. 1. The sensors are installed inside metal wells with thin stainless steel covers, and a frequency and sensitivity calibration is performed after installation to compensate the attenuation and filtering effects of the surrounding metal [26]. Edge poloidal field data can be used to determine the total toroidal plasma current, and along with edge toroidal field data the profiles of current density and safety factor q can be reconstructed. Toroidal modes up to n=3 can be determined with the array, providing information about transitions between MHD stable and unstable behavior. Magnetic field rise during compression is shown in Fig. 11. To measure plasma density, a two-color HeNe/CO 2 heterodyne interferometer (IF) which measures plasma density along four chords is employed. As illustrated in Fig. 6, IF2/3/5/6 provide line average density measure- ment through the plasma. During compression the core of the plasma starts near IF2 and passes sequentially through each of the chords until it typically reaches the IF6 chord near the point of maximum compression. Due to access limitations, IF2 uses an internally mounted retroreflector located within the center electrode, as having a through-chord was not possible. The rest of the chords are through-beams with external optics. Multi-color X-ray detectors are used to infer electron temperature (T ) from the ratio of soft X-ray radiation e incidentonfilteredabsoluteextremeultraviolet(AXUV) diodes with different filter thickness and thus different cutoff energies in the soft X-ray (SXR) range, see Fig. 7. Fortwofiltersofthesamematerialviewingaplasmawith uniform T and n , the ratio is a monotonic function of e e )m( R 1.0 (a) B13 AXUV P B12 B1 0.4 B11 T1 AXUV E B10 B2 B9 0.2 LASER B3 AXUV R B4 B5 B6 B7 B8 0.0 Z (m) 0.8 0.6 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 )m( R 1.0 (b) S4 IF2 IF3 0.4 WV3 IF5 WV2 S3 0.2 WV1 S2 IF6 NV1 0.0 Z (m) FIG. 6. Location of diagnostics in LM26 overlaid on flux surfaces from an example uncompressed plasma. (a) Mag- neticprobesinred,Thomsonscatteringchordsingreen,and AXUV sight cones (P, Q, E, S, and R) in other colors. See Fig. 8 for more detail. Thomson laser is the solid green line, T1 viewing chord is the dotted green line, and scattering volume is highlighted as black box. (b) Filterscope, survey spectrometer, and ion Doppler chords in magenta (S2-S4), interferometerchordsinblue(IF2-IF6),andfastcameraview cones in other colors (wide view WV1–3, narrow view NV1). The mid-plane is at Z =2.37 m. T [1,26,33]. Sevenofthesedetectorarraysareinstalled e in a spanning set of roughly axial view cones. Each de- tector consists of a 2×2 array of AXUV diodes each of which can be configured with any of a set of different thickness of aluminum-coated Mylar (5.47, 11.0, 22.1, or 125 µm) or beryllium foil (21.6, 49.3, 110.0 µm) or with no filter to collect all light from visible to hard X-ray as a broadband detector. These different filter types will be denoted MY5, MY11, MY22, MY125, Be22, Be49, Be110, and NF (no filter) throughout the rest of this pa- per. More details on the AXUV measurement position canbeseeninFigs.6,8,andamorecompleteaccounting of operational assumption, analysis methods and obser- vations are described in Sec. 3.4, and a full description of error analysis is given in Appendix A. A Thomson scattering (TS) T measurement was e added for LMC-11. The geometry is shown in Fig. 9. A1064nm, 10nsbeamissentthroughthemachinefrom gun to recovery side using a repurposed interferometer through-chord. The laser operates in burst mode which can deliver four 1.9J pulses separated by 250µs each. 10 FIG. 7. Detail of AXUV filtered photodiode array assembly. In the 2×2 array, side-adjacent diode pairs can have views of the plasma that overlap between 71.9% and 83.4% at the plasma’smid-plane,whilediagonalpairshavebetween62.5% to75.4%overlappingareaatthemid-plane. Whengradients of plasma emissivity are small compared to the distance be- tween non-overlapping components of the view cones, (such as when the core of the plasma is passing through the view), theratiooffilteredsignalswithdifferentSXRenergycut-offs can provide 0D estimates of chord-averaged T . Ratios with e significant gradients can still be used to constrain thermal modelsoftheplasmaviatime-dependentreconstructionwith the Integrated System Model (ISM). The collection optics are mounted on a concrete pillar to provide mechanical isolation from machine motion. The 75mm diameter, f = 500mm, plano-convex lens images a3cmlengthofthe5mmdiameterlaserintoa3×2mm cross section, 30m long imaging fiber. The scattering volume is 1.9m from the lens and about 115mm off the cavity midplane towards the recovery flange. The scat- tering angle between the laser chord and view is about 10degrees,whichsignificantlynarrowsThomsonspectral broadening. For a 1keV plasma the broadening would extend to only 1040nm on the blue side of the laser line. The fiber carries the collected light to a UKAEA poly- chromator designed for low broadening measurements. Above about 350eV, the broadening is measurable in FIG. 8. Maps of individual AXUV view cones as they inter- three polychromator channels. The polychromator spec- sectthemid-planeoftheplasmaareshownfortheLMC-9(a) tral response was measured with a pulsed white light andLMC-11(b)configurations. Eachcampaignmadeaseries source, a monochromator, and an absolutely calibrated of improvements to the AXUV detector electronics and filter combinations. The filter types are labeled as NF (black) for detector. nofilter,withMylarfiltersMY5(blue),MY11(green),MY22 The LM26 spectroscopy system contains five survey (red),andMY125(lavender). StartingonLMC-10beryllium spectrometers and 20 filterscope channels. Survey spec- filerswereused(Be22,Be49,Be110). Thenumberinthelabel trometers are employed to measure the emission lines in indicatesthefilterthicknessinµm. Whentherearetwoofthe arangeof200–1000nmwithaspectralresolutionof0.3– same type of filter for gradient comparison (e.g. LMC-9 E1) 0.6nm/pixel and a temporal resolution of 0.5ms/frame. the filter labels are enumerated 1, 2. For context, the radial Filterscopes are used to measure the temporal evolution locations of the magnetic axis of the plasma are shown for ofemittedlightwithinasetofwavelengthsusingnarrow C R =1,2,3. On LMC-11 Thomson scattering was installed, band-pass filters in front of avalanche photodiodes. The laser path shown in yellow with scattering volume in black. wavelengths measured include 523.2–523.8 nm (no asso- ciated lines), 548.5nm (LiII) 610.3nm (LiI), 656.1nm 11 3.1. Poloidal field increase during compression The poloidal field B that confines the plasma pol greatlyincreasesduringcompressionasthenear-constant amount of poloidal flux is compressed into a smaller and smaller cross sectional area. Poloidal flux conservation within the plasma during compression can be a direct indicator of plasma core resistivity, and indirectly an in- dicator of temperature. For a given resistivity model, a relationship between the evolution of T (t) and the FIG.9. Thomsonscatteringlaserandviewline,withscatter- e poloidalfluxψ(t)canbeobtained[1]ifitisassumedthat ingvolumeattheintersectionhighlightedinblack. Linerand thecurrentprofileintheplasmaremainsinvariantunder poloidalfluxcontoursshownforcompressedplasmainLMC- 11 at the time of the last Thomson pulse (t=5.211ms). compression. As our plasma is well explained by resis- tive MHD, significant (worst-case) cooling of the plasma always shows up immediately as a rapid resistive down- turn in ψ(t), which is observed as a drop in B (t) at pol (Dα), and 670.1nm (LiI). The location of the three fil- a fixed central point towards which the plasma is be- terscope chords (S2, S3, S4) can be seen on Fig. 6. ing compressed. The B6 Mirnov probes in the LM26 shaft, as shown in Fig. 6, are located at this central Two chords of ion Doppler spectroscopy (IDS) data point. Whereas with the ideal, perfectly flux conserv- are collected and used to measure the ion temperature ing case, the poloidal magnetic field will keep increas- of the plasma by analyzing CV impurity line radia- ing, in the case of constant current profile this will be tion at λ = 227 nm. Each system makes use of a B (t)/B (0)∼ C2(t), as long as plasma continues to high-resolutionspectrometer(HoribaIHR500)witha16- pol pol R behotenoughtohavearesistivedecaytimemuchlonger channelHamamatsuphotomultipliertubelineararrayas than the remainder of the compression time. a detector, each channel is digitized at 10MHz sampling rate. Doppler shift and broadening of the line can then WhathasbeenobservedintheLM26experimentsisa bedeterminedasafunctionoftimeinaprocessingcode. general trend in improvement in poloidal flux conserva- tion, as measured by the poloidal field rise relative to its In addition to IDS, a deuterium ion temperature mea- starting value. This trend in improvement follows from surement is derived from neutron counting: a measured achieving better initial plasma conditions and running neutronyield,combinedwiththeinterferometricelectron at higher liner compression velocities, as highlighted in density measurements, is inverted through the Bosch– Fig. 10. The poloidal field behavior is shown in detail in Hale reactivity [34] of the D(d,n)3He fusion reaction to Fig. 11. LMC-11 and LMC-9 had the highest poloidal extract T [35]. Neutrons are counted with eight liquid i field increases of any Li liner compressions to date and scintillators (Eljen EJ-301 and EJ-309) coupled to pho- maintained their flux to a larger C value before a final tomultiplier tubes, with active volumes from 830 cm3 to R thermal crash. LMC-10 was similar to LMC-8, which 3375 cm3 placed at various distances from the plasma had lower compression velocities. LMC-7 has some in- to ensure a large dynamic range of pulse rates. Each dications from AXUV and camera data that the plasma scintillatorishousedinasteelenclosurewithleadplates survivedtosignificantC values. However,shaftMirnov R thatsuppresselectronicnoiseandpartiallyshieldagainst data was not successfully recorded on LMC-7 due to a theambientgammaflux;theresidualgammacomponent higher-than-normal transient electrical surge on the sig- is rejected by pulse-shape discrimination (PSD) on the nal cables during the compression shot. digitized waveforms. On compression shots, all detected Theobservationthatmostofthecompressionshotsex- pulses occurring more than 1 ms after formation have ceeded the ideal trend with B (t)/B (0) > C2(t), is their PSD identification manually reviewed to ensure no pol pol R explainableintermsofthecurrentprofilebecomingmore noise spikes, gamma pulses with large noise fluctuations, hollow during the shot, whereby the radial distribution pile-up pulses, or other pulse-shape altering features are of flux is more concentrated toward the edge, giving a counted as neutrons. higher magnetic field strength than if the original profile had been maintained. This interpretation is supported byBPRandISManalysesthatindicateadecreaseinin- ternalinductancel duringthecompression,seeFig.24c. i 3. EXPERIMENTAL OBSERVATIONS 3.2. Density increase during compression Inthefollowingsections,wediscussthekeyexperimen- talobservationsandresultsobtainedduringcompression, Thelineaveragedensityisobservedtoincreaseduring includingincreasesinpoloidalfield,density,andtemper- compression, see Fig. 12. For the IF6 chord, inboard of ature. themagneticaxisatthestartofcompression,thelineav- 12 FIG.10. TrendofincreaseofpeakB withincreasingpeak pol liner velocity at the mid-plane. The case of no B increase pol is the y=1 dashed line. FIG. 12. Vibration-compensated line average density of each IF chord vs compression ratio C compared across relevant R LM26 compression campaigns. The solid curves show exper- imental interferometer signals, while the dashed curves show the corresponding volumetric compression model n (t) = e FIG.11. Thepoloidalfieldincreasefromthemagneticprobes n e (t = t comp )C V (t) ∼ n e (C R = 1)C R 3, which serves as a closest to the center of the shaft (B6 or B5, see Fig. 6) for simple guide to detect if the plasma seen by that IF chord all compression shots on LM26, as a function of compres- has its density falling significantly below, or rising above the sion ratio. For LMC-6 to LMC-10 the curves shown are the constant-inventory curve. Most, but not all shots begin with toroidalaverageofthemultipleMirnovsignalsattheB6po- a drop in density below this simple model, followed by a rise sition. Thefieldrises,drivenbycompressionwhileconserving neartheendwhichcouldbeduetoimpurityinflux. C V isde- poloidal flux until near a maximum compression ratio. The terminedfromanestimateoftheplasmavolumeasafunction curveforLMC-7isbasedonanISMfittotheendplatemag- oftimeduringthecompression,usingaTaylorstatemagnetic netic probes because the shaft probe data was not available. model calculated as part of liner trajectory reconstruction. TheriseabovetheidealfluxconservingC2 trendisduetoa R broadeningoftheinternalplasmacurrentprofileduringcom- pression, seen in reconstruction in terms of the decrease of internal inductance l shown in Fig. 24c. i 13 erage density increases as the plasma is compressed and 16 LMC-8 moved onto the chord. The general behavior of density (a) 14 evolution during compression can be explained in terms of some variable amount of particle loss from the edge 12 region in addition to nearly particle-conserving scaling with inverse volume near the core, followed late in com- 10 pressionbywhatislikelyanetparticleinventoryincrease frominfluxofimpuritiesfromtheboundaryneartheend 8 phase of compression. 6 The full set of data from interferometry provides in- 4 formation on the electron density profile and total elec- tron inventory. The profile is obtained by fitting data 2 from multiple interferometers making use of flux surface geometry reconstructed from Mirnov data (described in 0 T IF2 IF3 IF5 0 1 2 3 4 5 6 Sec. 4.2.1). We illustrate this reconstruction in Fig. 13, which shows the flux surface resolved electron inventory 12 versus time, as well as the total inventory contained LMC-9 (b) withintheLCFS,forLMC-8(upperpanel),LMC-9(mid- 10 dlepanel),andLMC-11(lowerpanel). Thincurvesshow totalinventorywithinfluxsurfacesidentifiedbyconstant toroidalfluxΦ,withspacing10%ofthetoroidalfluxini- 8 tially contained within the LCFS. The envelope of these curves, the heavy curve, represents the total inventory 6 within the LCFS. Dots indicate times when curves ter- minatebecausetheirfluxsurfacefallsoutsidetheLCFS, 4 e.g., due to “flux soak”, the resistive diffusion of mag- netic flux into the liner and walls. Compression trigger time, 2.0ms for LMC-8, 3.5ms for LMC-9, and 2.5ms 2 forLMC-11,isindicatedbytheverticalsolidlinelabeled “T”. The times when interferometers are removed from 0 T IF2 IF3 IF5 0 1 2 3 4 5 6 7 thedensityfitareindicatedbyverticaldashedlines(IF2, IF3, and IF5). LMC-11 does not have IF5 because that 12 port was repurposed for the TS diagnostic. As the liner LMC-11 (c) movesinwarditsequentiallyblocksthesetofinterferom- 10 eterchordsandsoweexcludeinterferometerdataafterits cutoff time from the fitting process to prevent the result becoming unreliable on account of this. Considering the 8 caseofLMC-8,wehavefourchords,spanningtheplasma from core to edge. The figure shows that for LMC-8 the 6 inventory settles down solidly after the CHI formation process, becoming quite constant after t=1ms (t is the 4 timeafterplasmaformation). Adropininventorybegins abruptly at t = 1.8ms, about 0.2ms before compression 2 is triggered. Determination of the inventory distribution within the plasma is naturally of limited spatial resolu- tion with only four interferometers, but the data does 0 T IF2 IF3 0 1 2 3 4 5 6 indicate that inventory loss occurs from the edge, and not globally. After this drop the inventory is again quite stable, until a second inventory drop starts gradually at FIG.13. Fluxsurfaceresolvedelectroninventoryversustime aboutt=4ms. Withonlythreeinterferometersthespa- for LMC-8 (upper panel), LMC-9 (middle panel), and LMC- tial resolution is reduced, but we can notice that losing 11(lowerpanel). Theverticalsolidlinelabeled“T”indicates the IF3 interferometer does not cause any discontinuity the compression trigger time, and the vertical dashed lines in the result. Inventory starts to rise after t = 5ms, (IF2, IF3, and, except in LMC-11, IF5) indicate the times probablyduetoaninfluxofimpuritiesattheedgeofthe when interferometers are removed from the density fit. plasma. Atthistimewehavereliabledatafromonlytwo interferometers (IF5 and IF6) and profile reconstruction becomes moreuncertain. After theIF5 cutoff we areleft with only the IF6 interferometer and profile reconstruc- 14 tion is no longer possible. In this final period we simply assume the profile shape is no longer changing and hold theprofileshapedeterminedjustbeforeIF5iscutoff. In the case of LMC-9 (middle panel in Fig. 13), the behav- ior during compression is similar, with an inventory loss shortly after compression is triggered and again later in compression. In the case of LMC-11 (lower panel) there is a similar inventory loss event observed shortly after compression is triggered (there are too few interferome- ters to indicate density profile after the IF3 cutoff). The observation of this drop in electron inventory oc- curring at near compression trigger time seems to be mostly coincidental in nature since it is not limited to just compression shots and has a moderate degree of re- peatability to it. Non-compression and LMC shots with similar system control parameters will tend to have an inventory drop event at around this time, and it even sometimes happens just before the compression is trig- gered, such as in LMC-8. So the logical conclusion is thatthisisanaturalintrinsicbehavioroftheplasmaand has little to do with the start of compression. The den- sity profile reconstruction indicates that the inventory is lost from the edge-most flux surfaces, and not from the plasma core. Further information about this plasma re- organization event can be determined from the AXUV data and analysis of the crash inversion radius in section 3.3. The increase in plasma density during compression (Fig. 12) is approximately a factor 10 before interpreta- tion of interferometric measurements is complicated by the increase in inventory attributed to impurities enter- ing the LCFS. For example, the core density determined by interferometric reconstruction increases by a factor of {10,8,10}inLMC-8,LMC-9,andLMC-11,fromdensity measured at 5.60ms, 6.55ms, and 5.36ms, respectively. 3.3. AXUV-crash inversion radius When looking at the set of non-filtered AXUV chan- nels,whichspantheradialextentoftheinboardplasma, the effective total emissivity profile can be determined. In Fig. 14, there are radial profiles obtained at three different times in the plasma evolution, using a smooth- ing spline applied to the set of NF (no filter) AXUV sig- nals on the LMC-9 prototype shot 6530, which used all the same control settings as LMC-9 (6531), except that 6530 did not have the compression driver banks or shaft ramp circuit triggered. The radial extent of each posi- tion of AXUV measurement is included in the graph as FIG. 14. Observation of X-ray emissivity profile during a triosofthreelinkedpoints,thecenterofwhichislocated crash event. at the center of the mid-plane footprint of the view cone and is given a 10% uncertainty, while the edge points of the view cone footprint are given 20% uncertainties profiles follow the same method but from using AXUV for the smoothing spline fit. The data from the AXUV NF measurements taken at 2.0 and 3.24ms respectively. NF measurements at t=3.34ms during the peak of the These sample times are shown as vertical lines in the X-ray crash are shown in red, which then generates the middle graph of the raw signals for shot 6530. For the solidorangesmoothingsplineprofile. Thegreenandblue first1msafterplasmaformationthetotalemissivitypro- 15 file is somewhat hollow, but then by 2ms has become centrally peaked and stays peaked for the remainder of the discharge except when the periodic AXUV crashes are happening (see Fig. 2 for indications of periodic be- havior on multiple diagnostics). During the crash event regions near the core decrease in X-ray brightness, while regions at larger minor radius than the inversion radius will temporarily increase in brightness. For this first crash beginning at t = 3.25ms, the inversion radius is R =0.25±0.03m, which corresponds to a normalized inv minor radius of r/a = 0.66. This can be compared to availableBPRdataforshot6530(seeSec.4.2.2formeth- ods) where the reconstructed safety factor q(R) value at t = 3ms is found to be q(0.25m) = 3.3+1.0, while after −0.8 the crash at t=4ms it has dropped a moderate amount to q(0.25m) = 2.7+0.7, so it is plausible that this inver- −0.6 sion radius might be near the q = 3 surface. This MHD event is most likely related to the loss of electron inven- tory from the outer edge of the plasma just after com- pression begins on LMC-9 (see Fig. 13) and other shots. It also coincides with an n = 0 increase of all B sig- pol nalscorrespondingtoasmallnetincreaseintotalplasma 900 current I , see Fig. 2. p 800 The nature of this plasma reorganization is of con- siderable interest as it may constrain the initial condi- 700 tions of the plasma just prior to the start of the com- 600 pression. Notably, the reproducibility of the plasma for- mation method was sufficiently precise that when shot 500 6531 was initiated it returned to a similar state, with 400 the same first crash possibly delayed by only 300µs, but was then triggered to begin with the action of the com- 300 pression field directly modifying the plasma boundary 200 through the lobes of flux fringing around the ends of the liner (see Fig. 23). In this case there was still an in- 100 crease in brightness in the outer region of the plasma, 0 followed by a drop, however the measurement closest to the core (AXUV Q) only increases in brightness in this early compression phase before significant motion of the Li liner has happened, possibly due to the direct mag- netic compression effects. For shot 6531 there was no data recorded from the AXUV P array due to rapid me- chanical damage to the diode electronics from the liner impact shock. 3.4. Increase of X-ray emission and T e Plasma electron temperature estimates on LM26 are inferred from measuring the plasma’s soft X-ray spec- trumwithacoarselyresolvedbasissetofdifferenttrans- mission filters, detecting the photons that have enough energy to pass through the filter with absolutely cali- brated X-ray photodiodes (AXUV), using a generaliza- tion of previously existing methods [26]. The tempera- tureestimateiscalculatedfromalookuptableoffiltered AXUV diode ratios as a function of a 0D value for T . e The ratio-T tables are constructed using a simple ra- e diation model where emissivity as a function of photon ] Ve[ eT 10 5 0 1.0 1.5 2.0 2.5 3.0 3.5 CR Bpol(t)/Bpol(CR=1) LMC-11 (Shot 7567) Electron Temperature Summary A IS X M U m V o T d e e l for LM P C o - w 11 e r c l o a r w e T fi e t v T s e( C C R R ) = 2 1 0 C R 1 . 5 3 TS T po h s o i m tio s n o n fl u S x c c a o tt o e d ri i n n g a t T e e ( y_ ) (b) TS: 718 eV +/- 80 eV (0.18) ISM: 690 eV R1: 667 eV - + 1 1 3 3 0 6 e e V V E1 E2 S R2 E1(early) (0.22) Q (0.39) (0.30) P 137 eV 118 eV Normalized poloidal field at B6 f =33o f =213o f =123o f =303o ISM normalized Bpol at B6 FIG.15. ElectrontemperatureriseforLMC-9(a)andLMC- 11 (b) as a function of C , with AXUV ratio measurements R of T compared to ISM best-fit model. Thomson scattering e measurementsofT areincludedforLMC-11with(ψ¯)values e of the TS measurement position shown in point labels. For mostoftheAXUVdatapointsthevaluesaretakenwhenthe magnetic axis passes the average center of that view cone. Severalexceptionsforthisexist,theE1(peak),E1(early)and R1 and R2 data taken at alternative times where valid data exists when not viewing through the core. See main text for further discussion. AXUV T data with (◀▶) markers have e been calibrated for diode variability, data with (◦) markers use the manufacturer’s responsivity value. Also shown are B measurements at B6, and the corresponding synthetic pol signal in solid black from the best-fit ISM simulation, the ISM core T values are shown as a dashed black curve. The e sharp drop in temperature near C = 2.57–2.66 is required R inbothmodelstoobtainamagneticfieldthatrollsoffatthe observedpeak. Thereisalsoapowerlawfitshownindashed red. 16 energy E is assumed to be proportional to exp(−E/T ), Off-core measurements can be made relatively insen- e andthisismultipliedbythetransmissioncurveT (E)for sitive to gradient effects by using a diagonally sym- k thek-thfilterandthenintegratedtogiveadimensionless metric filter set of two thickness values. For exam- synthetic diode signal. Physical scalar coefficients in the ple, on the LMC-9 campaign the E1 AXUV array emissivity involving plasma density will cancel out when was set up with aluminized-Mylar filters arranged as taking the ratio of diodes looking at a spatially homo- {{11µm,22µm},{22µm,11µm}}, which allows for di- geneous plasma, and so a table of dimensionless diode rect measurement of the radial variation of the MY11 ratios as a function of T is sufficient. signals, and possible toroidal variation of the MY22 sig- e nals. To get an improved-accuracy ratio for temperature The filtered diodes are mounted in-vacuum within estimation with the E1 module it is possible to take the a closely packed 2 × 2 rectangular array (12.2mm × average of both MY11 signals, which will have a concen- 11mm), witheachdiodehavingacollectionregion5mm tricviewtotheaverageofbothMY22signals,producing in diameter. This set of four filtered diodes views the a ratio that is less sensitive to plasma gradients. See plasma through an aperture in the aluminum inner cav- Fig. 8 for the orientation of diode view cones and filter ity wall plate, with a distance to the aperture ranging selections used for LMC-9. between 482mm and 765mm, depending on which port the diode package is mounted. This arrangement pro- In the final phase of compression, after the magnetic duces a tight cluster of mostly-overlapping view cones field has begun to rapidly drop and non-axisymmetric for each of the four diodes. Depending on the geomet- modes are larger in amplitude, the ratios of the three ric optics of each AXUV port, side-adjacent diode pairs distinct pairs (MY11/MY5, MY22/MY5, MY22/MY11) can have views with between 71.9% and 83.4% of areal will very strongly disagree in estimating T e from the dis- overlap at the plasma’s mid-plane, while diagonal pairs parate ratios—in particular for the R1 and R2 arrays have between 62.5% to 75.4% overlapping area at the at smallest radius (see Fig. 16 for analysis of effect of mid-plane. As such, taking ratios of diode signals (thick Ar on the R2 data). This occurs before the magnetic filter/thinfilter)approximatelycancelsalldependenceon axis has passed through the center of their views, and density and overall brightness, due to mostly observing thelengthscaleofdisplacementbetweennon-overlapping the same cone of plasma with the pairs of diodes being components(30mm)becomesasignificantfractionofthe compared. However, the accuracy of this approximation plasmaemissivitygradient(∼170mmfortotalemissivity willdependonthelengthscaleofthegradientinplasma across the minor radius and ∼70mm for the core region emissivity that is being observed by this 2×2 cluster of hot enough to generate photons that pass the 13µm fil- views, as well as the accuracy of the filter thickness used ter). During this time, toroidal variations in emissivity and overall responsivity of the photodiodes being well due to observed n = 1 modes could also contribute to characterized. The most accurate measurement will oc- ratioerrors. ThethreeestimatesofT e fromtheR2array cur when the magnetic axis of an axisymmetric plasma takenatthepeakofthepoloidalmagneticfield, disagree with a centrally peaked temperature profile is passing stronglywitheachother,andthemostplausibleinterpre- through the central view of a filtered diode array, then tation is that the true core temperature at that moment the measurement is taking place at the “top” of the hill is on the lower end of those values, more likely agreeing where there is zero average gradient, and so ratios be- with the ISM thermal model that matches the resistive tween diode signals will represent a measurement of a downturn of the B pol signals. common core temperature. In addition to gradient effects, another factor which As much as possible, we focus our analysis on the T must be carefully considered is the fact that certain im- e valuesmeasuredthroughthecoreoftheplasmatoreduce purities can distort the relationship between filtered ra- sensitivitytogradients. However,thereareindicationsof tios and electron temperature. Although higher energy temperature rise even at points away from the magnetic photonsgettingthroughthefiltersaremostlycontinuum axis; these are also presented here as further evidence of radiation from recombination and bremsstrahlung emis- global heating of the plasma, observed as the sampling sion,itispossiblethatlineemissionfromcertainimpuri- location is shifting relative to the plasma core. A sum- tiescouldbesufficientlybrightthatthefilteredratioswill mary of electron temperature data from AXUV and TS start to deviate from what is expected with the simple is shown in Fig. 15, where a clear trend of core electron radiation model lookup curve. temperature rise is observable during compression up to To address this, the total emission from various po- C =2.5 in both the LMC-9 and LMC-11 data. LMC-6 tential impurities that pass through each filter can be R andLMC-7alsoshowedpreliminaryevidenceofheating, modeled from a set of FLYCHK [36] simulations. This but these shots only had a single AXUV ratio pair at a is done over a range of simulated T values and impurity e single location to provide the peak measurement. Sig- concentrations to determine a trend for the error that nificant effort was made on later shots to collect higher arises from interpreting measured ratios with the simple qualitydataatmorelocationswithredundancytoshowa radiation model. As described in Appendix A, this can consistentrepeatablesetofresults. Analysistointerpret provide error bounds on T estimates for specified upper e thisincrease interms ofcompressionalheating versusan limits on the concentrations of different plausible impu- increase in Ohmic heating is described in Sec. 4.5. rities that might be present in the LM26 system. 17 2000 1000 0 6.7 6.8 6.9 7.0 7.1 time (ms) )Ve( T derrefni e (a) T inferred from AXUV ratios e Best fit Te(t) at [Ar] = 100% R2 11/5 at [H]=100% R2 22/11 at [H]=100% R2 22/5 at [H]=100% 1.00 0.75 0.50 0.25 0.00 6.7 6.8 6.9 7.0 7.1 time (ms) oitar VUXA (b) Comparison to Ar model 11/5 (observed) 11/5 (Ar best-fit) 22/11 (observed) 22/11 (Ar best-fit) 22/5 (observed) 22/5 (Ar best-fit) 1.00 0.75 0.50 0.25 0.00 200 400 600 800 1000 T (eV) e oitar VUXA (c) Ratio to T lookup curves e 11/5 (H) 11/5 (Ar) 22/11 (H) 22/11 (Ar) 22/5 (H) 22/5 (Ar) 10000 5000 0 3 4 5 6 7 8 time (ms), extended ).u.a( ytisnetnI TABLEII.ElectrontemperatureinferredfromfilteredAXUV ratiosduringcompression,measuredatthestartofcompres- sion(att )andatpeakT (att )beforetheplasmaap- initial e final pearstocrash. TimesinmillisecondsandT ineV.Theseesti- e matesusealookuptablebasedonthesimpleradiationmodel. The error bars account for statistical variation in equivalent measurementsanduncertaintyinfilterthickness. Forsimplic- itytheyaresymmetricaveragesoftheasymmetricupperand lower uncertainties. Effects of possible variations in plasma composition and gradient effects, as well as the presence of electrical pickup effects have been used as criteria to exclude data from this analysis. LMC-8 data is absent because the AXUVelectronicsweredamagedbycompressionshock. Only shotswithpeaklinervelocitygreaterthan175m/sareshown. Shot t T (t ) t T (t ) initial e initial final e final LMC-6 2.35 220±151 5.70 634±127 LMC-7 2.09 266±144 4.72 688±138 LMC-9 3.50 239±57 6.38 546±100 LMC-10 3.45 226±51 6.51 429± 85 LMC-11 2.49 210±53 5.47 667±134 ThemostpertinentcaseforLMC-9isthatofargonim- purities possibly entering the plasma in the final phase of compression, which could explain much of the dis- agreement in the R2 AXUV ratios (see Fig. 16). The best-match model, from FLYCHK data using 100% Ar plasma composition, gives a lower T estimate similar e to the ISM result. Visible spectral lines associated with Ar impurities are observed to increase at late time dur- ing LMC-9 in the survey spectroscopy data. This model doesnotperfectlymatchtheobservedR2ratios,soother effects due to plasma gradients, uncorrected differences in diode responsivity, or filter imperfections are likely to be contributing factors. The overall result across multiple LM26 campaigns is thatelectrontemperaturewillriseduringcompressionof (d) Evolution of possible Ar lines this class of Ohmically heated ST plasmas if the peak liner velocity is sufficiently fast (greater than 175 m/s) Ar I (772.4 nm) andtheplasmaremainsMHDstable. Theseoveralltem- Ar II (624.3 nm) peratureincreasesareshowninTableIIforLMC-6,7,9, 10, and 11 for which there was sufficient AXUV data to determine electron temperature. Based on the resistive evolution of the poloidal magnetic flux, we can also con- clude that LMC-8 must have had a similar temperature rise as was observed on LMC-10, given the strong simi- larityinmagneticevolutionandlinervelocityonthetwo shots. FIG. 16. Possible contribution to AXUV R2 ratio discrep- 3.5. Thomson scattering measurements ancy observed on LMC-9 due to the ingress of Ar, late in compression. Spectral line intensity is shown from spectrom- For LMC-11, T was measured by Thomson scatter- eter 2 positioned at the R = 125mm port. The best-fit Ar e model for T at late time (red) shown in (a) ranges from 240 ing at a single location and at four points in time during e to 400 eV, which is on the low end of the outputs of the H- compression. The laser trigger was timed such that the only radiation model, and gives an improved match to the fourth pulse occurred about 100µs before liner cutoff of threedivergentratiosin(b)withthedashedcurvesfromthe the laser on the recovery side. Raw polychromator data Ar model (c), and (d) shows Ar data. for the fourth pulse during LMC-11 is shown in Fig. 17 alongside a vacuum (no-plasma) test shot. For LMC-11, 18 50 CChhaannnneell 11 ((ssttrraayy lliigghhtt)) 7567 raw 7567 fit 25 7562 raw 0 80 CChhaannnneell 22 40 0 CChhaannnneell 33 160 80 0 CChhaannnneell 44 100 50 0 0 100 200 300 400 500 time (ns) ).bra( esnopser lartceps rotamorhcylop (a) (b) FIG. 18. Thomson scattering measurements of (a) electron temperature and (b) relative density as a function of time after formation, taken during compression of LMC-11. The sametemperaturedataareshownversuscompressionratioin Fig. 15. The temperature at the fourth pulse is 718±80 eV. Ch 3/2 Ch 4/2 FIG. 17. Raw and Gaussian-fit polychromator data for laser Ch 4/3 pulse4forLMC-11(7567)andano-plasmavacuumtestshot (7562). the oscilloscope’s channel gain was set to a conservative level to avoid saturation during compression and so dig- itization noise was apparent on channels 2–4. Stray light at 1064nm was measured by the dedicated polychromatorspectralchannel1. Straylightwasaneg- ligiblesourceoferrorandwasatasimilarlevelduringthe compression shot as it was when shooting into vacuum (Fig. 17). The LMC-11 Thomson T and relative n measure- e e ments are shown in Fig. 18. The relative residuals indi- cated a good fit, which can also be seen from the agree- ment of the polychromator signal ratios in Fig. 19. The calculated uncertainty in T takes into account e background plasma shot noise, Thomson shot noise, in- trinsic electrical noise, and the scattering angle. The scattering angle was estimated to be 9.9±0.5deg and, accordingtoasensitivityanalysis,contributesa10%un- certainty to T . More details on the error analysis can e be found in Appendix B. 3.6. Neutron and gamma rates during compression In this section we discuss the observations of neutrons andgammasfromtheLMCshotswithdeuteriumplasma. ItisworthnotingthattestdischargesofLM26performed with hydrogen instead of deuterium do not generate any scintillator pulses, indicating that the observed gammas and neutrons are a consequence of D-D fusion. oitar lennahc rotamorhcylop FIG. 19. Expected polychromator channel signal ratios for a rangeoftemperatures. Verticaldashedlinesarethemeasure- ments for LMC-11 and their implied temperature. Pulses 1 and2onlyshowthechannel3/2ratiobecausechannel4was in the noise. The neutrons counted on the liquid scintillators are converted into a plasma neutron yield (neutrons emitted per unit time) using a per-detector detection probabil- ity. The probability is the product of an inverse-square geometric factor and an intrinsic efficiency. The geomet- ric factor scales as 1/d2, where d is the distance from s s the plasma center to the detector; across the array d s ranges from 4.48 m to 10.48 m. Some detectors were moved closer to the plasma between LMCs to improve sensitivitytolower-yielddischarges. Thedetectorintrin- sic efficiency ϵMCNP, computed from MCNP simulations s performed by the UKAEA [35], is 0.565 for the smaller EJ-309 detectors, 0.612 for the EJ-301 detectors, and 0.633 for the single larger EJ-309 detector. The total efficiency of detecting neutrons (ϵ ) is thus estimated det as: ϵ = (cid:88) ϵMCNP A s , (1) det s 4πd2 s s 19 where s represents each of the eight detectors, and A is s the cross sectional area of the scintillator. ϵ ranges det from 2.2 × 10−4 for LMC-7, LMC-8, and LMC-9, to 2.0×10−4 forLMC-11. Theseshotsareselectedbecause theyshowthegreatestevidenceofincreaseinyieldpost- compression. a) LMC-7 (shot 4985) The effect of lab geometry on neutron scattering and 100 1010 efficiency of measurement are not considered in this 80 109 model other than two bounding assumptions for the at- tenuation of the neutron signal by the lithium liner: one 60 108 conservative case (“no Li”) in which the liner causes no 40 107 attenuation, and one (“max Li”) in which the signal is attenuated by e−dLi/λ, where λ ≈ 10 cm is the mean 20 106 free path of 2.45 MeV D-D neutrons in lithium and d Li 0 105 is the lithium path length along the direct line of sight b) LMC-8 (shot 5805) to the scintillator. In LMC-9, for example, ϵ det drops 1010 from 2.2×10−4 before the liner begins to enter the lines 80 109 of sight, to 0.6×10−4 at 7ms in the case of maximum 60 attenuation. 108 Becauseindividualneutrondetectionsarediscreteand 40 107 theirratevariesbyordersofmagnitudeacrossadischarge (frequent during the early plasma but sparse just before 20 106 compression and after termination of plasma), the rate 0 105 is reconstructed without fixed-width time bins, which c) LMC-9 (shot 6531) would otherwise wash out the early structure or leave 1010 the late-time signal dominated by empty bins. Instead 20 109 we use an adaptive estimate in which the local averag- ing timescale is set by the data themselves: each de- 15 108 tected neutron in the aggregate across all scintillators, 10 107 contributesaGaussiankernelwhosewidthh isthetime i interval to its k=7th nearest-neighboring detection, so 5 106 the estimate is sharp where detections are dense and 0 105 smoothly broadened where they are sparse [37]. The de- d) LMC-11 (shot 7567) tected count rate (λˆ(t)) is: 50 1010 λˆ(t)= (cid:88) h−1ϕ (cid:0) (t−t )/h (cid:1) , (2) 40 109 i i i 30 108 i 20 107 where the t are the detection times and ϕ is the unit i Gaussian distribution. The statistical uncertainty at 10 106 each time is set by the effective number of detections 0 105 contributing locally (n ), which is calculated: 1 2 3 4 5 6 7 eff time [ms] (cid:0)(cid:80) h−1ϕ (cid:0) (t−t )/h (cid:1)(cid:1)2 n = i i i i . (3) eff (cid:80) (cid:0) h−1ϕ (cid:0) (t−t )/h (cid:1)(cid:1)2 i i i i Fromthis,a90%confidencebandisobtainedfromPois- son counting statistics. Dividing λˆ(t) by ϵ converts det the detected rate into the plasma neutron yield; the two bounding liner-attenuation assumptions then translate into a corresponding band on the yield that widens as the liner enters the lines of sight. Theneutrondiagnosticindicatesthattheyieldishigh early in the discharge, falls more or less quickly depend- ing on the shot, and then in the best performing shots, exhibits a modest increase during compression. Fig. 20 shows the pulse time data for LMC-7, LMC-8, LMC-9, andLMC-11,starting1msafterplasmaformation,when nib sm 2.0 rep tnuoc esluP Neutron yield (s ) 1 indeterminate count neutron yield gamma count no liner attenuation neutron count 90% confidence level manual pulse curation starts compression start peak magnetic field first liner touchdown maximum liner compression FIG. 20. Neutron yield and scintillator signals (stacked his- togramsofcountsfromallscintillators)asafunctionoftime for LM26 compression shots LMC-7 (a), LMC-8 (b), LMC-9 (c),andLMC-11(d). Dataafterthefirst1msisshown,when STmagneticequilibriumhasbecomefullyestablished. Pulses whichhaveeitherasecondarypulseorlargenoisefluctuation on the tail preventing the normal PSD algorithm from work- ing, but are obviously real particle-interaction based pulses, are labeled indeterminate. accuratepulseidentificationbecomesfeasible. Inallfour shots an increase in neutron and gamma yield is evident during compression. It is possible to obtain a continuous determination of T (t) from the yield curve obtained from scintillator i 20 counts via (2), and the assumption that the ions have TABLEIII.Iontemperatureinferredfromneutrondatadur- had sufficient time to thermalize to a Maxwellian dis- ingcompression,assumingisothermalcompression. Theanal- tribution, with additional knowledge of the deuterium ysis window is t < t < t (times in ms), the number density profile. The result is plotted as the blue band in min max of neutron events in the window is N, and the inferred ion Fig. 21 for LMC-8, LMC-9, and LMC-11 in comparison temperature is T (in eV), with error bars given by Poisson i to ion Doppler data. It is again shown in comparison to statistics. Analysis is performed with worst case Li attenua- a simple model in Fig. 26 for LMC-9 and LMC-11. The tion (max Li) and ignoring Li attenuation (no Li). method, similar to that used in [1, 26, 35], is as follows. Shot t t N T (max Li) T (no Li) min max i i To obtain a T (t) curve, the continuous yield data in- LMC-7 4.0 5.3 21 391±16 378±15 i cluding liner attenuation (from Fig. 20) is converted to LMC-8 4.0 5.8 21 382±15 368±14 a temperature by adjusting the core T in a D-D fusion LMC-9 4.0 7.0 2 278±32 269±31 i forward model to match the yield at each time in the LMC-11 4.0 5.6 15 411±20 397±19 reconstruction. The neutron yield is calculated by inte- gratingtheBosch–Halereactivity[34]overthevolumeof theplasma,withprofilesofdensityn e (ψ¯)takenfromthe simplify the analysis, we assume the conservative case of interferometer fit, and with an assumed linear T i profile isothermalcompressionwiththesamelinearT i (ψ¯)profile of the form T i ∝ 1−ψ¯. A temperature confidence band as before. In this case the increase of neutron yield with is found by adjusting the core T i to reproduce the yield time is due to the increase of density. The density in the values at the upper and lower limits of the yield confi- core of the plasma is assumed to increase inversely with dence band. In the yield calculation, the plasma density the volume of a small flux surface of constant toroidal profile is not corrected for the possibility of impurity in- flux, with density referenced to the interferometer den- flux, which may underestimate T i at later times. For sity at the time compression is triggered. In this way we the fuel dilution factor f D = n d /n e we use a constant avoid mistaking the impurity influx at the edge for an value f D =0.7. This value is arbitrary but, in our opin- increase in deuterium density in the core. The varying ion, reasonable to represent the average behavior during attenuationofemittedneutronsbythecollapsinglithium earlycompression. Inorderforthisvaluetobeconsistent liner coming between the plasma and the scintillators is with the Z eff and Z avg values (also arbitrary but reason- takenintoaccount. Theprobabilitydistributionfortem- able) assumed for the ISM simulations, there should be perature is calculated as P(T )∝ (cid:81) P(k |λ ) where i t,s t,s t,s afractionofprotonsintheplasma, ascouldbeexpected the product is over bin times t and scintillators s, k is t,s to be generated from residual water, hydrocarbons, or the observed count, λ is the expected count rate given t,s hydroxides in the machine. thetemperatureT andtheplasmaconfigurationattime i The resulting T (t) confidence band is plotted in t, and P(k|λ)=λke−λ/k! is the Poisson distribution. In i Fig. 21. It is worth noting that the confidence band for this calculation the bin width is 0.1ms. For error bars temperature is much tighter than that for yield because we give the range of T i where P(T i ) exceeds e−1 of its ofthefastdependenceofreactivityontemperature. The peak value. Expectation values for T i from this analysis figure shows that the ion temperature is high immedi- are shown in Table III and also represented graphically ately after plasma formation. This is probably due to in Fig. 26 together with with the continuous T i (t) curve. dissipation of magnetic energy by the initial reorganiza- The scintillator signals also provide evidence of both tion of helicity after injection, and likely also has a con- prompt and delayed gamma production, relative to neu- tribution from the stagnation of the fast MHD flow into tron production. The prompt gammas are evident be- thechamber. Thefallofiontemperatureafterformation cause neutrons and gammas both rise when the emission may be due to e-i collisions (see Section 4.4), which will rate increases, both at formation and during compres- beespeciallyeffectivebeforetheelectronsbecomeOhmi- sion. Promptgammasaretobeexpectedasabyproduct callyheatedaftertheformationofclosedfluxsurfaces. It ofD-Dfusion: gammasfromreactionswhenenergeticfu- isalsopossiblethattheionsareinitiallynotthermalized sionprotons,3H,and3Hehitthewall,andcapturegam- (non-Maxwellian distribution) and that at least part of mas emitted when fusion neutrons thermalize and are the apparent temperature drop is due to reduction of an absorbed in the machine and surrounding material (neu- enhanced reactivity by thermalization of the ion veloc- tronswillthermalizein20–50µsandthenbecapturedin ity distribution, as discussed in [35]. Here we continue 100–500µs). The evidence for delayed gamma produc- with the working assumption of a Maxwellian distribu- tion is strongest in LMC-9, because no clear neutrons, tion. For reference, the ion collision time in LMC-11 at but many gammas, are detected in the interval between the start of compression is τ i ≈0.3ms. 1ms and 6ms, before neutrons associated with compres- An alternative, simpler, method of estimating T dur- sionappear. Duringthisintervalthegammayielddecays i ing compression is possible directly from the scintillator withahalf-lifeoft =1.5±0.2ms. Thesedelayedgam- 1/2 neutron histogram. Table III shows the result of this mas may result from a confined fraction of charged fu- method to estimate the ion temperature that would pro- sionproductsleavingtheplasmaandimpactingthewall. duce the observed neutron signal from the compressed There may also be a gamma contribution from decay of plasma, using the histogram data shown in Fig. 20. To neutron-activated isotopes or isomers, a mechanism that 21 atures from neutron data (blue bands) and curves from TABLE IV. Relative enhancement of curated scintillator two cases (χˆ =4m2/s, χˆ =2m2/s) of the simple zero- counts (neutrons, gammas, and indeterminate pulses) dur- i i dimensional transport-only compression model used to ing compression. The pre-compression decay is fit to an exponential ∝ e−t/τ over 1ms < t < t + 1.5ms (af- estimate the level of ion heat transport required to ex- comp plain the evolution of ion temperature indicated by neu- ter formation and before the compression starts, see Ta- ble I), where τ is the fitted decay time constant (half-life tron yield, as will be described in Sec. 4.4. Synthetic d t = τ ln2), and is extrapolated across the compression IDS model data are shown by the black and red dot- 1/2 d window t +1.5ms < t < t +5ms. N is the mea- dashed curves assume a fixed temperature profile shape comp comp sured count of all types of radiation in the window, N exp the (T i ∝ 1−ψ¯) with core T i determined by neutron mea- count predicted by the extrapolated decay, and R=N/N exp surement(thevalueofthedot-dashedcurveindicatesthe the enhancement factor. Uncertainties on τ and N exp are T (ψ¯) at the minimum value of ψ¯ along the IDS view). i propagated from thePoisson-weighted fit covariance; the un- This assumes the hottest part of the chord is also the certainty on R additionally includes the Poisson error on N. brightest by a large enough factor that it dominates the Shot τ (ms) N N R d exp spectral line. This simple model does not take into ac- LMC-7 0.93±0.05 142 24.6±4.4 5.8±1.1 counttheexpectedabundanceoftheCVionizationstate LMC-8 1.16±0.11 249 59.8±12.8 4.2±0.9 asafunctionoflocalT ,anddisagreementwithmeasured LMC-9 2.42±0.36 43 37.5±9.2 1.1±0.3 e IDS data would most likely be explainable that the ac- LMC-11 1.47±0.31 132 20.2±9.7 6.5±3.2 tual T profile is spatially broader than the linear model i assumed for this analysis. probablyaccountsforthelongtailofgammaemissionob- In LMC-8 the IDS T i is nearly constant during most servedevenafterthedischargeisover. Butthebrightness of compression, only rising late in time, while LMC-9 ofthefastergammadecaybeforecompressionisdifficult and LMC-11 begin to rise gradually almost immediately to explain this way. after the compression start time (labeled “T” for trig- The pre-compression scintillator signal observed in ger time in Fig. 21). The initial rise is explainable in LMC-7andLMC-8isdifferentfromthatinLMC-9,with terms of some amount of modest compressional heating more gamma counts, a faster decay (t ≃ 0.6-0.9ms), compounded with the IDS chords measuring deeper into 1/2 and neutrons observed as well as gammas (N /N ≃ theplasmaasthecompressionmovestheplasmaradially γ n 6–11). inward towards the chords. Overall,thegeneralpropertyoftheradiationfieldhav- InLMC-9(middlepanelofFig.21)theCViontemper- ingmoregammasthandirectneutrons,yetbeingroughly ature shows a near continuous rise to a maximum value proportional, and having zero gammas emitted when near the end of compression. forming hydrogen-only plasmas, implies that both PSD- InLMC-11(lowerpanelofFig.21),thereareusefulsig- identified neutrons and the prompt and delayed gammas nals from two IDS channels, J and M, with views along serve as an indication of the D-D fusion rate on LM26 the S2 and S3 diagnostic chords (see Fig. 6) which inter- deuterium plasmas. The rise of the more abundant and cepttheequatorialplaneat0.266mand0.236m,respec- easily detected gammas during compression on high per- tively. The IDS T reaches a maximum in both channels i forming shots (Fig. 20) provides some degree of support- aroundt≈4.9msbeforefallingatlatetime. Atthetime iveevidencethatthemeasuredincreaseinPSD-identified of this maximum, according to BPR, the magnetic axis neutroncountsisindeedfromanenhancedfusionratein is at R≃0.314m, still outside the radius of the channel the plasma during compression. Table IV shows the rel- views. The magnetic axis is estimated to cross the views ative increase during the compression window compared at 5.11ms and 5.36ms. At the time of the IDS T peak i to the window immediately preceding. LMC-11 is con- there is no corresponding peak on the neutron T and i sistent with showing the largest relative increase of all the neutron T is lower than the IDS T . Thus the IDS i i LMCs. T peak may indicate a preferential heating of CV over i D. Overall, the trends observed with neutron-inferred T i 3.7. Ion Doppler C V temperature are more consistent with the case χˆ = 4m2/s than the i moreoptimisticcaseof2m2/s,whencomparedtothe0D TheIDSdiagnostic[1,26]givesaCViontemperature transport model, Model A. evolution shown in Fig. 21 (solid black and red traces) Thefinalphaseofcompressioncontainsanarrowspike forcompressionshotsLMC-8,LMC-9,andLMC-11. The in ion Doppler T on LMC-8 and LMC-9, starting at i IDS data for the carbon ion temperature is found from 5.2ms and 6.2ms, respectively, reaching nearly 1keV, a Gaussian fit to the λ = 227nm spectral line shape, however this spike up is absent from LMC-11 data. The with instrumental broadening subtracted in quadrature spike is too fast to be compressional heating. We at- before applying the Doppler broadening formula for T . tributethisspikeinionDopplerT toheatingofthecar- i i Raw signal of the CV line is well above the noise floor bon ions by magnetic reconnection, presaged by MHD throughout compression. activity observed in the Mirnov probe signals as shown, For comparison, we also show deuterium ion temper- for example, in Fig. 25. The neutron data shows no 22 1.2 simultaneous increase in deuterium temperature. Such LMC-8 (a) preferential heating of carbon ions relative to deuterium 1 ionswhenreconnectionoccurshasbeenobserved,forex- ample, in MST [38, 39], with deuterium temperature re- maining more stable the higher the density. 0.8 0.6 3.8. Observations with fast-cameras 0.4 For most LMC campaigns we had one to three high- speed RGB color video cameras viewing the plasma be- havior through wide view internal fish-eye optics that 0.2 could record 1 frame per 25µs in the compression cav- ity (up to two camera views), and when present an ad- T IF2 IF3 IF5 0 0 1 2 3 4 5 6 ditional camera that viewed forward through the Mar- shall gun section that could record 1 frame per 3.5µs for 1 LMC-9 studying the gas breakdown and CHI bubble-out pro- 0.9 (b) cess [1, 40] within the gun section. Frames from two of these cameras at three times during LMC-9 are shown 0.8 in Fig. 22. These high speed videos can provide global 0.7 information about the plasma discharge that line aver- 0.6 aged or point-wise individual diagnostic signals can not directlyprovide. Thevisiblelightismostlyemittedfrom 0.5 theplasmaneartheedgewherelowionizationstatescan 0.4 exist. This can indicate the location, degree of intensity, andatomicnumberofimpuritiesinvolvedinplasma-wall 0.3 interaction. Noticeable on most frames are red filaments 0.2 on the liner inner surface, which are best explained as 0.1 high mode number electron density fluctuations at the LCFS, which have radial displacements into the ambient T IF2 IF3 IF5 0 0 1 2 3 4 5 6 7 fieldofneutralDemittednearlyuniformlyfromthewall, lighting up with red Dα emission where there are both 1.2 LMC-11 significantfreeelectronsandneutraldeuterium. Outside (c) the LCFS the electron density is almost zero in compar- 1 ison and so appears dark, while deep inside the plasma there are too few neutrals to observe with the camera. 0.8 4. PHYSICS ANALYSIS AND MODELING 0.6 4.1. Plasma behavior during compression 0.4 The plasma geometry changes dramatically during 0.2 compression,asevidencedbythelinershapeandpoloidal magnetic flux function ψ(R,Z), reconstructed at three 0 T IF2 IF3 times during LMC-9 shown in Fig. 23. Reconstructions 0 1 2 3 4 5 6 combine data on the liner location, magnetic fields, den- sity and temperature to find fits to the equilibrium state FIG. 21. Ion temperature obtained by neutron yield (blue as discussed below. Starting with a liner inner radius of confidence band) and by IDS (black curve, and red curve in 80 cm, a plasma major radius of ∼47 cm, and a minor LMC-11) for LMC-8 (upper panel), LMC-9 (middle panel), radiusof∼33cm,theplasmaissqueezedtoafinalminor andLMC-11(lowerpanel). Bluesolid anddashedcurvesare radius of roughly 8.6 cm and a major radius of ∼20 cm from Model A (Sec. 4.4). Dot-dashed curve(s) indicates IDS at full compression, representing a ∼50-fold reduction in T i trend based on neutron T i . The vertical solid line labeled plasma volume. Geometric and plasma equilibrium pa- “T” indicates the compression trigger time, and the vertical rameters from an ISM fit to LMC-9 are shown as curves dashedlines(IF2,IF3,and,exceptinLMC-11,IF5)indicate inFig.24asafunctionoftimeafterformation,wherethe thetimeswheninterferometersareremovedfromthedensity compressiontriggerat3.5msisindicatedbythevertical fit. gray lines. 23 FIG.22. Fastcolor-cameraobservationsofvisiblelightemissionfromneartheedgeoftheplasmaduringtheLMC-9compres- sion. Magneticfilamentscanbeseen,aswellastheonsetoftoroidal-modebucklingoftheliner. Thegreenfilamentintheleft frameisfromLiIIionsemittedfromthecenterelectrode. RedlightistypicallyfromtheDαlineorfromneutralLiat671nm. The poloidal extent of camera 3 (WV2) and camera 7 (WV3) view angles are indicated in Fig. 1 as yellow cones. The safety factor profile q is characterized by q in section with δ ∼ 0.1 to a more D-shaped plasma with min Fig. 24a and q in Fig. 24b. The safety factor profile is δ ∼ 0.3. The early-compression increase in triangular- 95 monotonic throughout, so q = q and both are well ity at 3.7 ms is driven by the flux from the compression 0 min preservedwith1.3<q <1.8formostofcompression. coils pushing around the ends of the liner, while the sec- min Incontrast,thenearedgevaluesdecreaseasfluxsurfaces ondincreaseat5.5msoccursasthelinercompressesthe diffuse into the edge during compression. Initially, the plasma against the shaft (Figs. 1, 23). Elongation and plasma is diverted and q ∼ 5. When the compression aspect ratio steadily increase from initial values of 1.8 95 coilsareenergized,thecompressionfieldsqueezesaround and 1.4 respectively to 3.4 and 2.5 by the end of com- the liner and separates the plasma from the gun flux, pression. The fixed radius of the shaft is the primary which leads to a brief increase in q . The plasma then drivertoincreasingA,whiletheincreaseinκreflectsthe 95 becomeslimitedandasthepoloidalfluxdiffusesintothe cylindrical compression produced by the liner trajectory. liner, the edge q falls rapidly. Once q drops to less Though toroidal β increases, as shown in Fig. 24f, the 95 than 3, n = 1 magnetic instabilities appear (see Fig. 15 normalized β slightly decreases through compression as for the occurrence of magnetic instability on LMC-9 and shown in Fig. 24e, and remains well below the Troyon LMC-11). limit of 4l , heuristically indicative of β limits in toka- i maks. This suggests any observed MHD instabilities are The plasma current I is shown in Fig. 24c. It starts p likely to be current driven. at 500 kA, half the shaft current, and slowly decreases by 20% until compression starts. Then it grows to 1.5 MA and becomes larger than the shaft current. The aspect ratio A ≡ R/a, elongation κ, compression ratio R /R, and triangularity δ are shown in Fig. 24d. Tri- 0 angularity increases from the initially rectangular cross 24 results in time-dependent magnetic boundary conditions 1 that are reconstructed using the PsiBC code. PsiBC takes as input the COMSOL liner reconstruction result: 0.8 time-dependent geometry and poloidal field ψ (R,Z,t) 0 duetocompressioncoilandbiascoilcurrentsandthecur- 0.6 rents they induce in the metal. Then, using the Mirnov 0.4 data from experiment, PsiBC reconstructs the field gen- eratedbyplasmacurrent,includingtheeffectofskincur- 0.2 rentsinducedinthemetalbyplasmacurrent,byfittinga (a) 3.700ms GSequilibriumateachtimepoint. Betweentimepoints, 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 thediffusionofthemagneticfieldintothemetalissimu- lated while holding the plasma current density constant. And, as the liner moves, the magnetic flux in the liner is advected. Inthismanner,PsiBCproducesthetotaltime- dependent field ψ(R,Z,t) = ψ (R,Z,t) + δψ(R,Z,t), 0 where δψ(R,Z,t) is the additional poloidal flux due to plasmacurrentandeddycurrentsinducedinthemetalby theplasma. Thistotalfieldissavedforuseasaboundary condition in subsequent plasma reconstruction using the Bayesian Plasma Reconstruction (BPR) code [1], which isdescribedinthenextsection. Amoredetaileddescrip- tion of the PsiBC code is given in Appendix C. We find that useful preliminary interpretation of the plasma data can be done using the magnetic flux sur- facegeometryproducedbyPsiBC,despitethesimplicity of its plasma model. Interferometer data, for example, can be fit to characterize the density profile and electron inventory, as was shown in Fig. 13. 4.2.2. Bayesian plasma reconstruction More complete and flexible reconstruction of the plasma state is performed with the BPR code. It em- ploys Bayesian inference to match the magnetic probes totabulatedplasmaequilibria[1,26]. Forselectedpoints in time throughout the shot, we precalculate a multi- FIG. 23. Example geometry during a PsiBC reconstruction dimensional lookup table of GS equilibria based on the offluxsoakduringcompressionofLMC-9,correspondingap- changing wall geometry and poloidal flux boundary con- proximately to the times of the images shown in Fig. 22. ditions calculated by PsiBC at the surface of the flux conserver. Each equilibrium in the table is weighted by its fit quality, defined as the normalized likelihood 4.2. Equilibrium reconstruction w = exp(−χ2/2), where χ2 is the sum of squared nor- malized residuals between the forward-modeled mag- We reconstruct the evolving plasma equilibrium state netic fields and Mirnov probe measurements. That is, in the experiment by two separate methods: recon- χ2 = (cid:80) ((B −B )/σ)2. These weights are nor- probes m s structions of sequential time slices, and fitting to time- malized to sum to unity across the table and used to dependent integrated simulations of the full compres- compute posterior statistics for each output parameter, sion. Both methods rely on the fact that the velocity such as poloidal flux or q profile. of the liner is much slower than the Alv´en velocity, so The GS equilibria are calculated with the Flagships the plasma is always in quasi-equilibrium and, therefore, code[42],assumingnotoroidalcurrentoutsidetheLCFS. governed by the Grad–Shafranov (GS) equation [41]. The fitting space is regularly sampled over six parame- ters, which typically include the plasma current, current density profile, and β . The plasma current profile is pol 4.2.1. Diffusion of poloidal flux constrained to be of the form [43] f(ψ¯)=exp (cid:2) aψ¯+bψ¯2−(y+a+b)ψ¯n(cid:3) , (4) The poloidal flux sourced by plasma current diffuses intothelinerandothermetalduringthedischarge. This where the free parameters are a, b, y, and n. The be- 25 FIG. 24. Geometric and plasma equilibrium parameters as a function of time after formation in LMC-9. Compression starts at 3.5 ms, indicated by vertical lines. Curves are sourced from a time-dependent ISM fit. Error bars are sourced from BPR reconstruction and show 1-σ range. (a) Safety factor minimum, q in black, and on axis q in blue. (b) Safety factor at min 0 95% flux surface, q . The ISM model does not include the Marshall gun, so the two models differ until the liner closes off 95 the gun. (c) Plasma current, I , and shaft current, I . (d) Elongation, κ, aspect ratio, A ≡ R/a, and compression ratio, p shaft C ≡ R /R(t), where R is the initial major radial location of the outboard mid-plane and R(t) is its value as a function R 0 0 of time. The elongation for the BPR model is initially higher because the plasma extends to the X-point in the gun. The triangularity, δ, is in blue with its axis on the right. (e) Internal inductance, l , and the Troyon metric β ≡ β /(I /aB), i N t p whereI istheplasmacurrentshownontherightaxisinblue, B isthetoroidalmagneticfield, aistheminorradius. (f)The p poloidal β ≡µ P/B2 , with P the pressure on axis, and the toroidal β ≡µ P/B2. p 0 pol t 0 haviourofthecurrentdensitynearthecoreisdetermined only the BPR model includes the gun region. The BPR byaandb,thecurrentdensityattheLCFSisdetermined model includes the gun flux, so the plasma becomes di- by y, and the internal inductance of the plasma is deter- verted and extends into the gun to an X-point, which mined by n. results in greater elongation and a smaller q . This dif- 95 ference between the BPR and ISM models becomes less While BPR can produce detailed multi-dimensional significant as the liner closes off the gun region during probability density functions, Fig. 24 shows the error compression. for LMC-9 presented more simply. The error bars for each output parameter span the 16th–84th percentile range of the posterior distribution, where each equilib- rium is weighted by its normalized fit quality such that 4.2.3. Time-dependent reconstruction all weights sum to unity. The time-dependent ISM fit curves in the plot are described in the next section, but Bayesian methods offer a rigorous probabilistic ap- itisclearthatinmostcasestheyareenclosedwithinthe proach to reconstructing instants in time independently. error region calculated by BPR. The two exceptions are This can be highly advantageous for understanding how thesafetyfactoratthe95%fluxsurface, q inFig.24b, the accuracy of the fitting evolves in time, and what is 95 and the elongation, κ in Fig. 24d, which differ because driving uncertainty, such as mode activity. In steady 26 state scenarios, such approaches can be sufficient. How- tentiallycarryingtheplasmaacrossastabilityboundary. ever, in the dynamic environment of an MTF compres- The key driving factors governing the stability of the sion, the inclusion of time-dependent physics to the re- plasma are the cross-sectional shape, safety factor pro- constructionprocesscanfacilitateunderstandingfroman file,currentdensityprofile,temperatureprofile,andden- entirely different perspective. The combination of these sity profile. These are all affected by operating param- twoapproachesisoftennecessarytofullyunderstandthe eters and internal plasma processes. Operating parame- experimental outcome. ters, suchasthelinertrajectory, shaftcurrentovertime, The Integrated System Model (ISM) is a comprehen- plasma formation power, and timings of the gas puff, sivecodeframeworkdevelopedbyGFthatprovidestime- plasma formation, and start of compression, influence dependentreconstructionsandpredictionsforthebehav- the plasma shape, initial conditions, and boundary con- ior of magnetized target fusion systems [2, 44, 45]. The ditions. Plasma processes, such as resistive transport, plasma component of the framework, ISM-plasma [45], flux loss into the boundary, and impurity transport, de- is a 1.5D plasma transport code based on the assump- termine the evolution of the profiles. Plasma stability tionthattheevolutionofamagneticallyconfinedplasma can be maintained well into compression with carefully can be represented as a sequence of 2D axisymmetric selected operating parameters. Grad–Shafranov equilibria linked through 1D transport The stability of equilibria reconstructed by ISM from processes (resistive and thermal diffusion, etc.). ISM in- experimentaldataareanalyzedwiththeResistiveDCON cludes plasma evolution with resistive diffusion of mag- code (RDCON) [51–54], and the NIMROD extended neticfieldsintothesurroundingstructure,self-consistent MHD code [55, 56]. RDCON calculates both the ideal motionofthemetalliner,andmechanismsfordrivingthe MHD perturbed potential energy, δW, and the resis- liner. Forreconstructionsoftheplasma,thelinermotion tive MHD stability based on the method of matched is specified from experimental data via the liner trajec- asymptotic expansions in full toroidal geometry. In this tory reconstruction process (Sec. 2.5.2). Synthetic diag- analysis, the linear form of NIMROD is used to evolve nostic outputs from ISM allow direct comparison with MHD perturbations over an equilibrium state, including plasma experimental data, such as magnetic probes, in- resistivity and viscosity throughout the plasma. Both terferometers, and AXUV filter ratios, which are used to codes implement a small vacuum region surrounding the constrainthethermalconfinementintheplasmaviaaχ2 plasma. approach. If the δW calculated in RDCON of any mode is neg- In the studies described here, ISM-plasma uses a two- ative, this represents a lower energy state with the per- temperaturemodel, withelectronsattemperatureT e (ψ) turbations, and the equilibrium is ideal unstable. If the and all ion species assumed to be in thermal equilibrium equilibriumisidealstable,itcanstillbeunstabletoslow- at temperature T i (ψ). Both temperatures are assumed growing resistive modes. The free energy available to a to be 10eV at the LCFS, as modifying this value did non-ideal (i.e. resistive) mode given some resistivity in a not have a significant effect on the results. Thermal ex- surface layer is described by the local stability index ∆′, change between electrons and ions is included. Thermal calculated from the resonant response at each layer. transportismodeledbysimplifiedperpendicularthermal In general, at each surface in the plasma where the diffusivitiesforelectronsandions,χˆ andχˆ respectively. e i safety factor matches the ratio of integer poloidal and Forthepurposesofreconstruction,theseareeffectivedif- toroidal mode numbers, q =m/n, a resonance occurs in fusivitieswhichlumptogetherallthermaltransportpro- the (m,n) component of the MHD mode. Such surfaces cesses including radiation. Magnetic flux and particles will form resonant layers in the plasma, with highly lo- aregraduallylostfromtheedgeoftheplasmadomainas calizedcurrentssensitivetotheresistivityoftheplasma. the magnetic field resistively diffuses across the plasma The resistivity allows for magnetic reconnection and in- or soaks into the liner. Particles are otherwise conserved terchange in the layer, allowing the global mode to grow inthismodel. Furtherdetailsofthemodelaredescribed at a rate determined by the physics in the layers. De- in Appendix D. pending on the growth rate, structure and location of the instability, it could be observed in experiment, with consequences ranging from disruptive to benign. 4.3. MHD stability Overall, analyses have been conducted on every com- pression shot, and indicate that the plasma profiles at In all axisymmetric toroidally-confined plasmas, as thestartofcompression,aswellashowquicklytheshaft higher performance plasma states are reached, instabil- current is ramped during the early compression before ities can appear suddenly and lead to disruptive events trapping the toroidal flux, are critical to subsequent sta- wheremostoftheconfinedenergyisrapidlylost[46–50]. bility. The results from a stability analysis of the ISM Disruptive events have been observed in previous com- reconstructions of LMC-9 are summarized in Fig. 25. pression experiments at GF [1], evidently caused by in- The safety factor q at the LCFS drops significantly a stabilities driven by magnetohydrodynamic (MHD) lim- during LMC shots from initial values q ∼ 20 to q < 4 a a its. As the plasma is compressed, the geometry of the near the end of compression as the plasma begins to ter- flux conserving walls and plasma properties change, po- minate. Flux loss to the boundary reduces the edge q 27 edutilpma edoM 0.35 0.30 0.25 0.20 0.15 0.10 0.05 stable 0.00 6.0 6.2 6.4 6.6 6.8 7.0 Time (ms) )1-s(γetar htworg edoM 107 106 105 104 103 102 edutilpma edoM 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 2 Time (ms) 4 6 7 )1-s(γetar htworg edoM 107 (a) L 0 is t´he plasma inductance at the start of compression (L = R−1dRdZ). With insufficient ramping, the Ideal LCFS qprofileintheedgeregioncanfallrapidly,causingglobal RDCON 106 MHD instability. With sufficient shaft current ramping, NIMROD theq profileandstabilitycanbemaintained,thoughitis Mode amp not necessarily advantageous to track the ideal ramping 105 rate exactly. In practice, lagging the ideal rate induces edge current in the plasma, which heats the edge plasma 104 viaOhmicheating,possiblyimprovingperformance. The shotspresentedherealllagtheidealratetosomedegree. Beforecompression,theplasmasaretypicallyfoundto 103 bestabletodisruptiveidealandresistiveMHDinstabili- tiesinexperimentandanalyses. Insomecases,earlyvery 102 stable slow-growing resistive modes (γ ≲2×103s−1) are found in stability analyses that are either too slow to be ob- served on the timescale of the experiment or are, in fact, (b) stable. In Fig. 25, one early resistive instability is cal- culated to be weakly unstable by RDCON, with growth rate∼4×102s−1,whileNIMRODfindsasimilarsteady low growth rate. This growth rate is too low to manifest on the experimental timescale, where 104s−1 or above is aconcern, andwouldlikelynonlinearlysaturateatsmall size. Becauseq ∼1.85inthereconstruction,asshown min inFig.24,theq =2surfaceisrelativelyclosetothemag- netic axis in low shear, with variable contribution to the stability as the equilibrium changes. The q = 3 surface is more steady, and is a stabilizing influence. The com- bination causes an intermittent instability for RDCON in Fig. 25. NIMROD’s model includes viscosity and is more accurate for these plasmas. However, there are not significant signals above the noise level on the magnetic probes that would suggest growing instabilities, as evi- denced by the relatively flat n=1 mode amplitude until ∼6ms. FIG. 25. The resistive MHD growth rates of the n=1 mode inLMC-9fromRDCON(magenta),fromNIMROD(orange), Shortly after compression starts, the flux from the 10%oftheinverseAlfv´entimewhenidealinstabilityisfound compression coils pushes through the gaps at the axial (red), and the experimentally determined mode amplitude ends of the liner, which pushes the flux surfaces radially frommagneticprobemeasurements90◦ apart(cyan),forthe inwardinbothregions. Thiseffectcanbeseenonthetop full discharge (a) and deep in compression (b). Dashed lines plot of Fig. 23 and it typically has the effect of increas- connectsequentialunstablepoints,withgapswheretheyare ingthetriangularityjustasthelinerhasbeguntomove, found stable. Very early the mode is marginal with one low whenC ∼1,asshowninFig.24. Thetriangularitydur- growth rate found by RDCON and a steady low growth rate R ing compression falls in the range 0.1 < δ < 0.4, which found by NIMROD, but evidently too low and short lived to tendstoimprovestability, ascanbeseenbythestabiliz- manifest. Through most of compression the plasma remains stable. Deepincompressiontheplasmaisfoundunstablejust ing influence on NIMROD in Fig. 25. Although, trian- as an instability grows and the shot terminates. gularity can be destabilizing if it increases further [57]. An interesting aspect of these shots is that they are con- ducted in positive triangularity, whereas previous stud- aftercompressionstarts,whilethecoreplasmaq remains ies [53, 54], and the target configuration for upcoming relatively steady, slowly decreasing, indicating good core experimentsinLM26areinnegativetriangularity. How- confinement. Initiating the plasma with higher edge q ever, even in these negative triangularity cases, deep in and ramping the shaft current during compression both compression the triangularity can reverse to positive as counter the effect of flux loss to the boundary. As com- the plasma is compressed against the center shaft. pression begins, the liner must first traverse the radial Laterincompression,astheedgeshearinq isreduced, gap to the endplates to make contact and complete the resistive instabilities are found unstable in some cases, poloidal circuit that traps the toroidal flux. During this with growth rates ≳ 104s−1. Above this growth rate traversal, the shaft current is ramped to mimic the trap- the mode can e-fold more than ten times in a millisec- ping of toroidal flux. The ideal ramping scenario would ond, and therefore may become an observed instability track the inverse inductance I (t) ∝ L /L(t) where as its growth is fast enough to surpass the linear regime shaft 0 28 and transition to a nonlinear state within the timescale in experiment and stability analyses, q can strongly min of the experiment. Even if the mode is marginally sta- affect the resistive stability, especially when the central ble, or too slow-growing to be observed, such marginal value is near a m/n resonance. A point of observation modes are highly susceptible to error fields, which can in experiment is that there is no evidence of character- drive an instability in a manner nearly indistinguishable istic transient instability as q = 2 is crossed in either min fromspontaneousinstabilitiesinpractice[58]. InFig.25, LMC-8 or LMC-9. This suggests 1.7<q <2 early in min late in compression, the plasma is found ideal unstable these shots, as per χ2 of the reconstructions. Thus, the by RDCON as indicated by the red dots. These markers case shown in Fig. 24 and 25 is chosen taking both the are placed at 10% of the inverse Alfv´en time 0.1/τ to ISM and BPR fits into account, with q ∼1.85. A min indicateanupperboundforwhatisexpectedoftheideal growthratesofthemode. Intwoofthesecasesat6.2ms and 6.45ms, the δW briefly turns slightly negative as As discussed above, stability analyses indicate insta- theedgeq oftheplasmaapproaches3andthen2respec- bility late in compression as q edge falls to q < 4, and a tively. The resistive stability growth rates from RDCON idealinstabilityisfoundshouldq cross2,asistypicalin a andNIMRODareinitiallyingoodagreement,increasing tokamak discharges [59]. In the experiment, MHD activ- as the q =3 edge is crossed, though the RDCON growth ity is often observed to begin growing at an earlier time rates are significantly below the NIMROD result as the than the stability analyses indicate. One aspect of the edge q approaches 2. In these final moments, the RD- onsetofinstabilityisthattheyarefirstdetectedatnearly CON assumption of localized current layers at the q =2 thesamecompressionratioC ∼1.8inshotswherethey R surfacearelessaccuratethantheNIMRODmodel,which areobserved,exceptforLMC-11wherethemodeappears makesnosuchassumption,asthetemperaturedropsand at C ∼ 2.2. This fact suggests a mechanism based on R the resistive layers become too wide. plasma geometry, although other interpretations can not Approaching7ms,theq =2surfaceleavestheplasma, be ruled out. Buckling of the liner during compression, and RDCON finds ideal instability as the plasma termi- as discussed in Sec. 3.8 could lead to magnetic pertur- nates. The red dots in Fig. 25 are only markers of the bations as the current flowing through the liner becomes upper bounds expected of ideal modes, while without displaced. As discussed above, error field penetration resonant surfaces any unstable mode is driven by non- can drive instabilities that can be difficult to distinguish resonant mechanisms internally, and may be related to fromspontaneousinstabilities[58]. Thisisespeciallytrue theexternalkink[59]. Ifanidealgrowthratewerecalcu- near marginality for the mode, where a finite frequency latedforthesetimes,itmayinfactbesignificantlylower to the mode can appear, which translates into a propa- thanthemarkers. NIMRODfindsmostofthetimesfrom gating mode stemming from a static error field. Indeed 6.75ms to 7ms as stable, except for a slow growing case relatederrorfieldinteractionsinvolvingfinitefrequencies at 6.9ms. NIMROD’s model includes resistivity, viscos- can persist into the nonlinear regime with large islands ity and adiabatic pressure, which may be responsible for [61]. the discrepancy. In any case, at the mode amplitudes observed in experiment, these instabilities are nonlinear andthereforethelinearpictureisonlypartiallyrelevant, Thestabilityanalysesoffertheabilitytoexploremod- though their evolution may include linear drive [56, 60]. ifications of the experimental conditions which can im- Though the total plasma current is well constrained, prove stability, by building understanding of the driving it remains a challenge to reconstruct the detailed struc- mechanisms involved. Shortly after formation, as q min ture of the current profile, in particular the initial hol- decreases, assuming a fixed profile shape, the locations lowness at the start of compression. Hollowness in the of the q = 2 and 3 surfaces, along with all others, are initial current profile is primarily driven by the CHI for- moved outwards toward the edge. As the surfaces move mation process, where current is driven on outer flux outward, the shear in q increases and the distance to surfaces and relaxes minor-radially inward. How hollow the conducting wall decreases, and in a simplified view the current profile is can directly affect stability because of Furth, Rutherford and Selberg [62], the plasma be- it determines q min as well as the current gradient across comes less easily driven unstable. Also, the shaft ramp surfaces, two important factors to stability. current, which primarily supports the near edge current Note the BPR reconstructions of LMC-9 indicate a profile, is more effective for near edge surfaces. How- higher initial q ≳2 than the best fit from ISM, where ever,thetemperaturegenerallydecreasesduringthispre- min q ∼ 1.5 initially. In fact, reconstructions from either compression period, having been driven up by the initial min BPR or ISM with a q value in close agreement with formationprocess,andnowreliantsolelyonOhmicheat- min the other would show little difference in χ2 from their ing to counter transport losses. As such, stability con- best fits, as the minima in χ2 are somewhat shallow. cernsmustbebalancedagainsttheadvantagesofstarting Thisfactalsomanifestsasasignificantvariabilityinq compression with sufficient plasma performance. Over- min as a function of time reconstructed with BPR, while the all,thegoalistoachieveacompressedplasmastatethat ISMvaluesareconstrainedtovarysmoothlybythetime- achieves high confinement and remains stable through dependentphysicsoftheintegratedsimulation. However, compression. 29 4.4. Simple models for ion temperature e-i collisions. In the case of LMC-9 we use an ad hoc T (t) during the pre-compression phase to illustrate, in e As shown in Secs. 3.4–3.7, the ion temperature does Fig. 26, a possible scenario in which χˆ i is constant and not heat as much as electron temperature during com- the initial ion cooling is dominated by e-i collisions. The pression. Our initial interpretation of the ion tempera- model curves are calculated using for the effective ther- ture evolution is based on the preliminary plasma recon- mal diffusivity the value χˆ i ≃ 4m2/s. The initial T i is struction provided by PsiBC. We start by using a simple chosensothecurvepassesthroughthecompressedT i de- zero-dimensional (0D) compressional heating model (de- termined by the isothermal analysis of Table III. For the scribed in Ref. [1]), which we call “Model A”: pre-compression geometry, the chosen value of χˆ i corre- sponds to an ion energy confinement time τ ≃ 5ms. ˆ E,i T (t) (cid:20) V(0) (cid:21)2/3 (cid:18) t (cid:19) Notethatwhencomparingtomoresophisticatedmodels T i (0) = V(t) exp −χˆ i k(t′)2dt′ (5) thatproperlyaccountfordensityprofileandfluxsurface i 0 geometry, τ is a more appropriate quantity than χˆ . E,i i where χˆ is an adjustable constant and k(t) = 2.4/a(t) Moresophisticated1Dmodelingoftheelectronandion i with a the minor radius, where 2.4 the first zero of the temperature evolution using ISM-plasma is described in Bessel function J [1]. For the volume V(t) we use the Sec. 4.5. The levels of ion heat transport inferred from 0 reconstructed volume of a small flux surface with con- the experiment using ISM-plasma are similar to those stanttoroidalflux. Thebasicassumptionunderlyingthis appearing in the simple models described above, when model is that all ion thermal losses can be lumped into compared on the basis of energy confinement time. an energy confinement time τ that scales during com- E,i pression as a2. The constant χˆ can be interpreted as i an effective thermal diffusivity, and serves as a way to 4.5. Compressional heating and confinement characterize τ . In Fig. 21 we compare this model to E,i reconstructedT andfindthatavalueχˆ ≃4m2/sissuf- Time-dependentreconstructionisperformedusingthe i i ficient to prevent significant compressional ion heating, ISM code described in Sec. 4.2.3. The ISM reconstruc- while χˆ ≃ 2m2/s would result in a significant temper- tionsofLMC-9andLMC-11areconstrainedbymagnetic i ature rise. This simple model does not distinguish colli- field probes, interferometers, AXUV filter ratios, Thom- sional electron-ion energy exchange (Equation 19) from son scattering (for LMC-11 only), and inferred ion tem- cooling of the ions due to actual ion heat transport. The perature. In addition to providing direct magnetic infor- collisional energy exchange depends on both the colli- mation, the time evolution of the magnetic field probes sion frequency and the temperature difference between yieldsinformationaboutelectrontemperatureduetothe electrons and ions. Therefore, collisional cooling of ions resistive evolution of the plasma. The time-dependent should diminish and even become a heating term for the approach of ISM reconstruction leverages this relation- ions as the electron temperature rises to exceed the ion ship,alongwithothertime-dependencieslikeelectron-ion temperatureduringcompressionasshownbytheAXUV heat exchange, leading to a higher degree of consistency measurements. The absence of a significant ion temper- across measurement domains. atureincreasethereforeindicatesthationheattransport Another benefit of the time-dependent reconstruction is a dominant loss mechanism during the compression approach is that translation of the compressing plasma phase of the experiment. across AXUV view cones reveals information about the We next apply a more detailed (but still 0D) model gradient of electron temperature. A two-parameter χˆ e that separately accounts for electron-ion energy ex- profile was therefore used in the ISM reconstruction to change, which we call “Model B”: improve agreement with observed AXUV data, T˙ =− 2V˙ T −χˆ k2T +ν¯i|e(T −T ) (6) χˆ e (ψ¯)=χˆa e xis+(χˆL e CFS−χˆa e xis)ψ¯2, (7) i 3V i i i ϵ e i where χˆaxis and χˆLCFS are the effective electron thermal e e Fig. 26 compares the neutron T data from LMC-9 and diffusivitiesatthemagneticaxisandLCFS,respectively. i LMC-11 to this model. For the rate-of-compression Here ψ¯ is, as usual, defined to be linear in ψ and having factor V˙/V we again use the volume V(t) of a small values0atthemagneticaxisand1attheLCFS.Theion flux surface with constant toroidal flux. The e-i colli- thermal diffusivity was assumed to be uniform, χˆaxis = i sional term involves rate ν¯i|e, defined in Appendix D, χˆLCFS, as there is insufficient data to constrain the ion ϵ i depending on n and T . For n (t) we use the pre- temperature profile. e e e compression core density from interferometry, scaling it It is important to note that the χˆ parameter in ISM i during compression inversely as the volume V(t). For simulations can have a rather different value than in the T we specify T (t) = (238eV)C0.971 for LMC-9 and 0D simulations of Sec. 4.4. This is because in ISM the e e R T (t)=(210eV)C1.53forLMC-11,onthebasisofAXUV density profile is taken into account, whereas in the 0D e R data (Fig. 15). Early in the discharge, the decay of neu- simulations the density profile is implicitly incorporated tron yield allows us, in principal, to estimate a lower into χˆ . Also, the ISM utilizes full flux surface geometry i bound for T by attributing the decay to ion cooling via whereas the 0D models assume cylindrical flux surfaces. e 30 TABLE V. Summary of thermal confinement parameters for the ISM reconstructions of LMC-9 and LMC-11. Effective thermal diffusivities χˆ are given in m2/s. Thermal confine- ment times within the 50% poloidal flux surface and the LCFS, τ and τ respectively, are taken at the start E,50 E,100 ofcompressionandgiveninms. Forreconstructionpurposes, these fitted values of χˆ lump together all possible thermal transportprocesses,includingradiation,andthereforedonot represent the true diffusivity in the plasma. Shot Channel χˆaxis χˆLCFS τ τ E,50 E,100 Electrons 9.0 27 3.2 2.4 LMC-9 Ions 12 12 4.3 4.3 Combined – – 3.6 3.0 Electrons 6.5 24 4.3 2.6 LMC-11 Ions 10 10 5.0 5.3 Combined – – 4.6 3.5 The relevant quantity for comparison is instead the en- ergy confinement time, τ . E,i Thebestfitdiffusivitiesandresultingthermalconfine- ment times are summarized in Table V. The evolution of the corresponding power balance terms and temper- atures are shown in Fig. 27 and Fig. 28, respectively. The power balance of LMC-11 is expanded in terms of 1DpowerdensityprofilesinFig.29,sampledatt=5ms. TheseISMresultsassumeaneffectiveionchargeZ =2 eff and average ion charge Z =1.2. avg In order to match the final rapid turnover of the mea- sured magnetic field, it is necessary to increase χˆ by a e large factor, resulting in a rapid decrease in temperature which accelerates the resistive decay of the plasma. For LMC-9 this change occurs at 6.75ms, a timing which is consistent with the MHD stability analysis discussed in Sec. 4.3. Before compression begins, the electron temperature profiles are approaching a steady state with transport losses being largely balanced by Ohmic heating. The onset of compression leads to a subsequent rise in elec- trontemperature,duetocompressionalandOhmicheat- ing. Ohmic heating increases during compression as the FIG. 26. Ion temperature from neutron yield data (blue plasma current density grows due to flux compression. curve and confidence band) for shot LMC-9 (upper panel) Transport of heat becomes more effective as the plasma and LMC-11 (lower panel). The magenta box indicates the becomes smaller. T fromtheisothermalanalysisofscintillatordata(TableIII, i For MTF, the important power balance is that of the max Li case). Also shown are model results for T (black i curve) using T from experiment (red curve) in the simple coreplasma,takenheretobethecentral50%ofpoloidal e zero-dimensional Model B, given by equation (6), that sep- flux and represented in Fig. 27 by the solid curves. The arately accounts for e-i collisions, ion heat transport (χˆ = powerbalanceoftheplasmaasawholeisrepresentedby i 4m2/s), and compressional heating. Electron temperature is thedashedcurves. Thereasonthecoreplasmaisrelevant based on the AXUV data during compression. Compression for MTF is that compressional heating, being propor- startsattimeindicatedbyverticalline. InLMC-9anad hoc tionaltopressure, predominantlyoccursinthecore, and early-time T (t) is imposed to illustrate the possible initial e itistherethefusionyieldwillbegenerated. Inaddition, fast cooling of ions by e-i collisions. In LMC-11 a constant inthepresentexperimentstheOhmicheatingintheedge early-time T is assumed. e plasmaisenhancedbythelagofshaftcurrent, discussed inSec.4.3,generatingheatthattransportsquicklytothe wall. The 1D distribution of these power balance terms is exemplified by Fig. 29. Focusingonthecoreplasma,thesimulationshowsthat compressional heating is fully exceeding Ohmic heating 31       @:>UHZRSQRUWFHO( /0&,60SRZHUEDODQFH 3 3 FRPS 3 3 L H H   3 3 FRPS 3 3 L H H   D             W>PV@ @:>UHZRSQR, 3 3 FRPS 3 L L L  3 3 FRPS 3 L L L  E       @:>UHZRSQRUWFHO( /0&,60SRZHUEDODQFH 3 3 FRPS 3 3 L H H   3 3 FRPS 3 3 L H H   F            W>PV@ @:>UHZRSQR,               W>PV@ 3 3 FRPS 3 L L L  3 3 FRPS 3 L L L  G FIG.27. PowertermsfromtheISMreconstructionsof(a,b) LMC-9 and (c, d) LMC-11, separated by (a, c) electron and (b,d)ionchannels. PowerduetothermaltransportlossesPχ (blue),compressionalheatingPcomp (orange),Ohmicheating PΩ (green), and electron-ion exchange Pi (red) are shown. Toillustratethevariationofpowertermsacrossthecoreand edge, each term is evaluated within the 50% poloidal flux surface (solid lines) as well as the LCFS (dashed lines), de- noted by subscripts 50 and 100 respectively. Compression power data after temperature crashes is not reliable due to stepchangesinχˆ andisthereforeomitted. Compressionbe- e gins at t=3.5ms for LMC-9 and 2.5ms for LMC-11 @9H>HUXWDUHSPH7 /0&,60WHPSHUDWXUHHYROXWLRQ 7 H 7 L D            W>PV@ @9H>HUXWDUHSPH7 /0&,60WHPSHUDWXUHHYROXWLRQ 7 H 7 L E FIG.28. TemperatureevolutionfromtheISMreconstructions of (a) LMC-9 and (b) LMC-11. Temperatures are shown for electrons(solidlines)andions(dashedlines)atpositionspro- gressing from the magnetic axis (brightest color) to ψ¯=0.75 (darkestcolor)inuniformincrements. Bothtemperaturesare fixedto10eVattheLCFS.Compressionbeginsatt=3.5ms for LMC-9 and 2.5ms for LMC-11.            @ P:>\WLVQHGUHZR3  /0&,60SRZHUSURILOHVDWW PV î T T FRPS T T L FIG. 29. Power density profiles taken from the ISM recon- structionofLMC-11att=5ms. Profilesareshownforther- mal transport losses qχ (blue), compressional heating qcomp (orange), Ohmic heating qΩ (green), and electron-ion ex- change qi (red). Electron and ion terms are denoted by solid and dashed lines, respectively. 32 at5msforLMC-9and3.5msforLMC-11. Inthecaseof ing in LM26 compression has a large contribution from LMC-11, compressional heating dominates Ohmic heat- its lower current density (H ∼0.5). ing by a factor of 2 at 5ms. Ion compressional heating WenoteherethatthePCSobservationofenhancedion exceeds ion losses in both cases, yielding peaks in ion energy confinement (τ ≈ 1.6ms, χˆ ≈ 0.52m2/s) in a E,i i temperature consistent with those in Table III. timeintervalfollowingplasmaformation[1]hasalsobeen Even for the power balance of the plasma as a whole, observed in the neutron data for PI3 [35] and in early compressionalheatingbecomessimilartoOhmicheating shotsinLM26. Thisenhancedconfinementbehaviordoes during compression in the case of LMC-11. notoccurinthelaterLM26compressionshotsconsidered in the detailed transport analysis. Theinferredtransportlevelinthepresentexperiments 4.6. Discussion is similar to neoclassical ion heat diffusivities predicted for these plasma parameters. Comparable transport lev- els have also been observed in NSTX experiments with In these LM26 experiments, the energy confinement similar magnetic field (B ≈ 0.4T) and aspect ratio time is lower than observed in PI3 [26]. The energy [63]. Gyrokinetic simulations of the similar PI3 machine confinement time for the combined electron and ion en- showed that ion heat transport can be predominantly ergy is τ ≃ 3.5ms from Table V. This value is below E driven by ITG turbulence [64]. In NSTX, ITG modes the range of values τ ≃ 4–12ms observed in PI3 with E werelargelysuppressedduetostrongE×Bsheargener- fresh lithium coating. A lower τ in LM26 is to be ex- E ated by neutral beam injection, resulting in ion thermal pected because it is a smaller plasma (scaling the LM26 transport close to neoclassical levels. In contrast, PI3 energyconfinementtimetoPI3majorradiusasR2 gives (andLM26)hasnosignificantexternalmomentuminjec- τ ≃6ms, in the range of PI3 observed values, but E,scaled tion source to stabilize ITG modes, and turbulent con- at the low end). We believe a more stringent compari- tributions to the transport may therefore still contribute son would be confounded by various effects that we now significantly to observed ion heat losses. ITG turbulence discuss. The LM26 plasma is more elongated, and in- would also contribute to electron heat transport. Dis- creasing elongation typically improves τ , which would E tinguishing between neoclassical and turbulence-driven reduce the effect of the geometry change. The observed transport mechanism is beyond the scope of the present lower τ in LM26 may also be partly explained by im- E work. purities, whichmightbemoreprevalentduetotheshort Assuming the transport approximately follows gyro- period of time between machine assembly and the com- Bohm scaling, χ ∼ρ ρ v ∼ρ2v /a, where ρ = pression experiment, a few weeks, compared to the ex- i,GB ∗ i th,i i th,i √ ∗ ρ /a is the normalized ion gyroradius, ρ = m T /eB tended periods of time spent under vacuum in PI3 cam- i i i i is the thermal ion gyroradius, a is the minor radius, and paigns, many months. In addition, the τ E in PI3 is from (cid:112) v = T /m istheionthermalspeed. Intermsofthe measurements made 5ms after plasma formation, and th,i i i magneticfieldB,gyro-Bohmscalingmeansthetransport the plasmas in LM26 are compressed earlier than that. scales as χ ∼ ρ2v /a ∝ 1/B2. Assuming this scaling Finally, and possibly most importantly, the τ value for i i th,i E remainsapproximatelyvalid,a50%increaseinmagnetic LM26isaneffectivevalue,referencedtopre-compression fieldstrengthBwouldreducethetransportbymorethan geometry,describingtransportfortheplasmabeforeand a factor of two. Experimentally, the energy confinement during compression, and thus is not expected to fully time in STs is observed to improve roughly linearly with correspond to the non-compressing PI3 plasmas. B [21, 63]. Even in this less optimistic scenario, increas- InthePCSexperiments[1],theionheattransportdur- ing B could be an effective way to improve the outcome ingcompressionwassimilartowhatwehaveobservedin from weak ion heating to strong compressional ion heat- LM26, with χˆ ≈ 5m2/s for the effective thermal dif- fusivity in a 0 i D model, giving τ ≃ 122µs when re- ing, in the LM26 machine. E,i ferred to pre-compression geometry. The compression time was τ ≃ 140µs, giving τ /τ ≃ 0.87. In LMC- C E,i C 11 the energy confinement time is significantly higher, 5. CONCLUSIONS τ ≃ 5.3ms (Table V), and the compression time is E,i τ = R /v = 3.6ms, giving τ /τ ≃ 1.5. Go- Inthisstudy, wepresentedaseriesofexperimentalre- C eq,0 E,i C ing from PCS to LM26 the ratio τ /τ increases by sults from the LM26 device at General Fusion and ex- E,i C a factor of 1.7, consistent with the modeled ion temper- plore relevant analyses and modeling to draw conclu- ature increasing in LM26 and not in PCS. The reason sions about future directions for MTF research. The theτ /τ ratiowasnotincreasedlinearlywithmachine initial pre-compression state of the CHI-formed ST deu- E,i C sizeisthattheexplosivelydrivencompressionvelocityin terium plasmas has sufficiently good confinement prop- PCS is much faster than the coil-driven compression ve- erties to remain in a nearly-axisymmetric magnetic field locityinLM26. TheelectronenergyconfinementinPCS configurationuntilreachingradialcompressionfactorsof was worse than in LM26, in terms of χˆ . This can be at- C >2.8. Significantelectronheatingisaccomplishedby e R tributed to the very high current density in PCS (Hugill compression, due to the liner remaining sufficiently sym- numberH ∼0.04). Sotheimprovementofelectronheat- metricdeepintocompression,inadditiontocompression- 33 enhanced Ohmic heating from increasing plasma current the early slow phase of compression, followed by an ion density. As seen in Table II, the experiments achieve a reheating in the final faster phase of compression. Our three-fold increase in electron temperature in LMC-11, best-fitting models show that contributions from com- with the majority of this increase attributable to com- pressional heating and thermal transfer from the hotter pressional heating (as shown in Sec. 4.5). electrons exceed overall ion cooling from transport. This Plasmadensityalsoincreasessignificantlyduringcom- results in a net positive slope of T i (t) at deep compres- pression (Fig. 12, 13) with trends that typically show a sion on the best performing shots (see section 4.5). We monotonically decaying particle inventory until the very believe the observed level of ion heating is due to ion end of compression. At that point, rapid additions to heat transport being too high for the overall size of the the electron population are best explained by ingress plasma and the existing speed of compression. Trans- of higher Z impurities from the wall that ionize and port modeling and experimental evidence from the liter- cool the edge plasma by radiation. These impurities are ature (see section 4.6) indicate that ion transport could mostlikelyvaporizedLiDfromthesurfaceofthelithium be sufficiently reduced to result in significant compres- linerandpossiblytrappedargon(Ar)releasedfromvoids sional heating of ions with the existing achieved liner ve- withintheliner. Weconcludethattheimpuritiesarepri- locities if the plasma magnetic field strength is increased marily in gaseous form because the LSE diagnostic used by 50%. Shaping can also have a significant impact on onLMC-10detectednoparticulateejectawheretheliner performance. Negative triangularity is known to have first contacted the wall, nor was any indicated by spec- both a positive impact on stability and offer stronger troscopy data, AXUV error analysis, and analysis of the compressionalheatingbyenhancingtheself-similarityof color fast camera data. the equilibrium states during compression [54]. These improvements will be implemented in the upcoming up- Magnetic stability is maintained quite well until a ra- grade to the LM26 machine. dial compression of C ∼ 1.8 (C = 2.2 in LMC-11), R R after which a weak n = 1 mode begins to grow and is likely to cause a moderate increase in energy loss rate due to transport. The appearance of this mode tends ACKNOWLEDGMENTS to occur slightly before MHD instability is identified in reconstructedequilibria. However,evensmalldifferences The authors would like to thank Martin Cox, Tony inreconstructionscanresultinsignificantchangesinthe Donne,KurtSchoenberg,BrianLloyd,andNedSauthoff predicted timing of the instability. One possible cause of fortheirinsightfuldiscussionandconstructivecomments. themodeisthatlinerbuckling,asobservedwiththeSLR The authors also thank UKAEA for their support with andplasmafastcameras,growssufficientlytobegincou- Thomson scattering diagnostics. pling to the plasma edge and drive instability. Another possible cause is that as edge q falls, spontaneous insta- bility becomes more likely, where even small differences Appendices inreconstructionscanresultinsignificantchangesinthe predictedtimingoftheinstability. Inanycase,thissmall amplitude rotating n = 1 mode of the plasma poloidal Appendix A SOURCES OF ERROR IN AXUV T e magnetic field does not lead to disruption immediately, CALCULATIONS AND MODELING and the plasma continues to survive significant further compression. The AXUV diagnostic arrays use the Opto Diode This general observation of stability is further sup- model AXUV20HS1 silicon photodiode which has a ported by suppression of AXUV crashes after the first 1 nearly flat responsivity above 20 eV photon energies. msoflinercompression,asevidencedbycomparingcom- The manufacturer provides an absolute calibration for pression shots to their preceding prototype shots. In the responsivity (R) in A/W as a function of photon wave- prototypeshots,non-disruptiveX-raycrasheswithn=0 length (5.2 to 1100 nm), which shows an average value magneticstructuretypicallyoccurwithanintervalof2–3 of R = 0.261A/W for photon energy E in the range of ms. Each crash transiently increases the plasma current, 20eV < E < 235eV, which we use for converting raw and at least one would have fallen within the compres- signal in amperes to total power incident on the diode sion window. However, once rapid compression has be- in watts. These photodiodes continue to have similar gun, the next expected AXUV crash is suppressed, or responsivity up to E ∼ 10keV. However, the manufac- at least delayed, due to plasma conditions shifting away turer does not specify an expected diode-to-diode varia- from what would have caused a crash to happen. Stabil- tion in responsivity. Therefore, starting with the LMC-9 ity analysis supports this conclusion as well, with ideal campaign, we have implemented our own process to de- disruptive instabilities delayed until the end of compres- termine a relative calibration correction within the set sion. of 4 photodiodes of a single AXUV diagnostic module. Thedatacollectedbythescintillatorarrayandtheion Errorsinoverallresponsivityofthewholediagnostic, in- Doppler spectrometers are consistent with the interpre- cluding electronics, can then be compensated for, which tationthationtransportcausesT tofallbelowT during is important since systematic errors in individual diode i e 34 responsivities are magnified in the error of the ratio of from the simple radiation model as used in our primary those signals (5% errors in diode R can cause a maxi- calculations: T (T ) = r−1 (r (T )), where e,apparent e ab,H ab,mix e mum of 11% error in their ratio). The calibration setup r (T )istheratiocurveforthesimpleradiationmodel ab,H e uses a 40 kV, 80 µA X-ray source, and relies on attenua- and r (T ) is the ratio curve calculated for the mix- ab,mix e tion of soft X-ray components through 20 cm of air such ture. that only a beam of hard X-rays illuminates the array of For hydrogen plasmas with a single impurity, the con- 4 photodiodes. This enables doing the calibration with centration at which the impurity causes |T − e,apparent the filters in place just before installation on LM26. As T |/T >0.1isshowninFig.30fortheMylarfiltersand e e suchitmeasuresrelativevariationinthedioderesponsiv- in Fig. 32 for beryllium filters. Emissions from light ele- ityaboveE >5keV. Wefindthatmostofthediodesare ments such as lithium and oxygen have limited effect on within 3% of each other, however some have been found measuredratios,especiallyathighertemperatures. Even to deviate from neighbors by as much as 25%, which if small traces of iron and tungsten can cause the apparent leftuncompensatedforwouldleadtosignificanterrorsin temperaturetobelowerthanthetruetemperature,with T estimation. Assuch,wehaveaugmentedthereported e the effect more pronounced at higher temperatures. error bars for AXUV data before LMC-9 to represent However, at some temperatures, certain impurities this increase in measurement uncertainty. In the future, skewtheapparenttemperaturehigherthanthetruetem- this X-ray calibration setup could also better character- perature. Of these, aluminum and argon deserve partic- izethefiltertransmissioncurvesthroughouttheVUVto ular consideration due to their prominence in machine soft X-ray range, which could reduce the present ∼20% construction and operations. RGA studies before com- uncertaintyinT associatedwithfilterthickness. Forthe e pression shots estimate a background argon density of results reported here, we are relying on established cal- 1014m−3, which at our measured temperatures would culated transmission functions [65] for our calculations. haveaneffectwithinourexistingerrorbars, butpockets Theeffectoffilterthicknessuncertaintywasdirectlycal- of argon released from the liner as it deforms or alu- culated for LMC-11 results and retroactively applied as minum scattered during liner touchdown have the po- a minimum of 20% uncertainty in T for previous shots. e tential to impact results. Figures 31 and 33 show the Reported error bars in Table II and Fig. 15 will exceed worst-case impact of each species on the measured tem- this minimum if there is a larger disagreement in T val- e perature, when using the Mylar or beryllium filter sets, uescalculatedfromalternateratiosatthesameposition, which are qualitatively similar for both types of filter. in which case the error bars represent the spread in dif- One remaining practical consideration that is worth ferent calculated values. discussingasapotentialerrorsource,andafactorinfilter Theoretical modeling from FLYCHK [36] data was selection in an MTF-relevant application is the compari- used to estimate the direction and magnitude of par- son of the visible-light rejection properties of aluminized ticular species of impurities that may impact the T e Mylar and beryllium filters, as well as their mechanical estimate derived from the X-ray diagnostics using the properties. Thevapor-depositedcoatingofaluminumon simple radiation model. The simple radiation model is the Mylar substrate has the potential to develop small basedonapproximatingtheemissionspectrumusingthe pinholes which can pass near-UV and visible light with bremsstrahlung spectral shape ∼exp(−E/T ), where E e wavelengthλ>250nm. Al-layerpinholescanbepresent is the photon energy. fromthefabricationprocess,orcanformduringexposure Synthetic AXUV ratios for a given element are deter- to plasma. As a simple, preliminary measure to protect mined as follows: given ion fraction concentrations c (cid:80) k againstintroducingvisiblelightcontamination,aquality (including deuterium concentration to make c =1), k k controlstephasbeenconductedpriortoeachinstallation we determine the ion dilution factor of AXUV filter-diode assemblies on LM26 and just after n c c f = i = i = i (8) disassembly after a compression shot, in which a high- i n e (cid:80) j c j Z¯ j Z avg brightnesswhiteLEDflashlightisapplieddirectlyonthe filtered aperture while a digital multimeter measures the notingthatmeanchargestateZ¯ mayvarywithtemper- forward-biased resistance across each AXUV diode. A i ature. Total power seen by an AXUV diode, assuming new diode typically has a resistance value (when in com- fixed density and temperature, is plete darkness) of R ≈ 1.700MΩ, with measurement d ˆ resolution of 1kΩ, using the multimeter applying a for- ∞ P (T )=C (cid:88) f (T )n2 T (E)p (E,T )dE (9) ward polarity test voltage of V test = +0.5VDC. Diode- k e i e e k i e to-diodevariabilityforR isaround1%buthasarepeat- i 0 d able measurement value for each diode. Because photo- where T (E) is the transmission curve for filter k, and voltaic current is in the negative direction, absorption k p (E,T ) is the emission spectrum for element i down- of light causes the measured forward-biased current to i e loaded from FLYCHK, normalized by the n n used in decrease, this reduction of current at I(V ) is shown i e test theFLYCHKcalculation. TheprefactorC disappearsin on the multimeter as an increase in measured resistance. the ratio r (T ) = P (T )/P (T ). The apparent T is For an AXUV diode without any filter, the measured re- ab e a e b e e determined by inverting the derived ratio with the curve sistance will increase to R ≈ 4MΩ when the flashlight d 35 is applied. This corresponds to a photovoltaic current levelonanyparticularfilter, thickorthin, duetoforma- of I = 0.169µA when compared to the nominal dark tion of new pinholes. pv resistance. A filtered photodiode assembly is considered Such cases can be detected within the redundant ar- to fail this light-leak flashlight test if the measured resis- ray of measurements at similar radius, and indicated tanceincreasesby1kΩormore,inwhichcasethefilteris retroactively by failing the flashlight leak test during removedandreplacedwithanewfilterofthesamethick- post-compression disassembly. Due to the fact that high ness. Thephotovoltaiccurrentrequiredtocausethisnull energy photons are only emitted from a smaller region upper bound for filter acceptance is near the core of the plasma, visible light contamina- tion is easily noticed in the raw signals by the presence I =(0.5/1.7−0.5/1.701)×10−6A=1.73×10−10A null of significant (> 10%) signals during early compression when equivalent neighboring signals are below the mini- For a filter that passes this test, the fraction of incident mumdetectionfloor,orwhenratiosofthick/thinfiltered visible light transmitted by the filter will be signals exceed unity. All identified cases of measurable f <I /I =0.00102, visible-light leakage have been excluded from considera- vis null pv tion in AXUV T estimates in this study. e When considering the overall effect of visible light leak- An advantage of beryllium filters is their robust abil- age on T measurement error, it is important to note ity to completely block all near-visible light, eliminating e that visible light makes up only a small fraction of to- the possibility of pinhole formation during plasma ex- tal EM emissions from the plasma. The upper limit of posure. This was recognized as a way to produce more the fraction of near-visible emissions (1eV<E <5eV), reliablemeasurements,requiringlessdata-qualitycontrol emerges from the optimistic case of a pure H plasma, effortandsoberylliumfilterswereinstalledinoneAXUV which at T = 150eV was calculated to be 2.35% of to- package on the LMC-10 campaign to compare to the Al- e tal light from the plasma. This visible fraction decreases Mylar filter performance and also to confirm that the to 0.75% for pure H at T = 510eV, which is the upper more brittle beryllium foils did not fracture from the in- e limit of our FLYCHK scan. The final fractional error tensemechanicalshockfromthehigh-speedlinerimpact. also needs to take into account the fact that the higher The first test was successful and so the use of beryllium energy photons are also being filtered. This error can be filters was expanded to three AXUV modules on LMC- defined as 11. Overall, we find that the beryllium and properly- curated aluminized Mylar filters give similar results on f P ϵ (k)= vis vis (10) the LM26 system. Both give similar T values, which vis (f P +P (T )) e vis vis k e are further validated with the Thomson scattering mea- surementsobtainedonLMC-11. Bothtypesoffiltersare using (9) for an individual diode signal collecting light intrinsically susceptible to plasma-gradient effects if not through an Al-Mylar filter that has passed this visible- properlyarranged,aswelltotheeffectsofpossiblehigh-Z lightleaktest. Themaximumestimateforpossibleerror impurities, these being the dominant error contributions forthistermcomesfromthe150eVpure-Hextremecase: to either AXUV measurement method. ϵ (MY 5µm)<1.23% vis ϵ (MY 11µm)<12.5% vis Appendix B THOMSON SCATTERING T ϵ (MY 22µm)<69.6%. e vis CALCULATION AND ERROR ANALYSIS As an example of more realistic impurity composition, (1% Li, 0.1% O) can be added to the H simulation and Electron temperature T from Thomson scattering is e the visible light fraction drops to 1.0% at T = 150eV calculated by finding the best fit between noise-weighted e falling further to 0.5% at 510eV. For this example polychromator channel measurements and synthetic sig- case, the visible light fractional errors would have upper nals from a forward model. The required scaling of the bounds of: signal can also provide the relative change in electron density n between laser pulses. e ϵ∗ vis (MY 5µm)<0.18% AclosedformestimateofT e uncertaintyduetophoton ϵ∗ (MY 11µm)<1.14% shot noise and intrinsic detector electrical noise is calcu- vis ϵ∗ (MY 22µm)<9.1%. lated as described in [66]. The background shot noise vis and intrinsic electrical noise is the standard deviation of Foranyparticularplasmacompositionthisisasystem- a channel’s trace before the laser pulse. aticeffectthatwillpreferentiallyincreaseratiosinvolving Thomson scattering shot noise is conservatively esti- the thicker filters, and so will generally cause an larger mated from the variation of the channel 1 stray light over-estimate of T with those ratios. The effect of this pulse height in a series of 20 pulses. For an average sig- e error, however, will also decrease as true T increases. nalof10mV,thevariationwas2mVandatleasthalfof e Additionally, it is also possible that during plasma oper- this variation could be accounted for by the background ations, we could have f increase above the initial null andintrinsicnoisemeasurementbeforethepulse. There- vis 36 0 10 1 10 2 10 3 10 4 10 5 10 6 10 2 3 10 10 noitartnecnoc (a) MY22/MY11 0 10 1 10 2 10 3 10 4 10 5 10 6 10 2 3 10 10 noitartnecnoc (b) MY22/MY5 0 10 1 10 2 10 3 10 4 10 5 10 6 10 2 3 10 10 T (eV) e noitartnecnoc Critical impurity concentration for ±10% apparent-T bias e 3 10 2 10 2 3 10 10 C O Al Ar Fe Mo W +10% -10% (c) MY11/MY5 FIG. 30. Concentrations of impurities in an otherwise pure hydrogen plasma at which there is a significant deviation of 10% between the true temperature and the temperature estimated from AXUV ratios assuming the simple radia- tion model. The Li spectral shape is extremely similar to bremsstrahlungforthetransmittedregion,sothereisnosuch concentration. For elements with a solid line, at concentra- tions above that line there is potential for the temperature obtained using the simple radiation model to overestimate the true temperature. )Ve( T tnerappA e (a) MY22/MY11 3 10 2 10 2 3 10 10 )Ve( T tnerappA e (b) MY22/MY5 3 10 2 10 2 3 10 10 T (eV) e )Ve( T tnerappA e Apparent vs True T e at 100% concentration model Li C O Al Ar Fe Mo W H (c) MY11/MY5 FIG. 31. Electron temperature T derived from applying a e simple-radiation-model-based ratio-T lookup curve to ratios e forward-modeledfrompure(100%concentration)plasmasof various elements. This bounds the possible effect of each im- purity species on AXUV T calculations. e 37 0 10 2 10 4 10 6 10 2 3 10 10 noitartnecnoc (a) BE110.0/BE49.3 0 10 2 10 4 10 6 10 2 3 10 10 noitartnecnoc (b) BE110.0/BE21.6 0 10 2 10 4 10 6 10 2 3 10 10 T (eV) e noitartnecnoc Critical impurity concentration for ±10% apparent-T bias e 3 10 2 10 2 3 10 10 Al Ar Fe Mo W +10% -10% (c) BE49.3/BE21.6 FIG.32. Concentrationsofimpuritiesthatwouldgiverisetoa 10%errorinestimatedT valuewhenusingberylliumfilters, e repeating the analysis method shown in Fig. 30. Carbon, oxygen and lithium were included in the analysis but do not appearinthegraphbecausethereisnoconcentrationofthese low-Z impurities that would give rise to a 10% error in T e output, as can be seen in Fig. 33. )Ve( T tnerappA e (a) BE110.0/BE49.3 3 10 2 10 2 3 10 10 )Ve( T tnerappA e (b) BE110.0/BE21.6 3 10 2 10 2 3 10 10 T (eV) e )Ve( T tnerappA e Apparent vs True T e at 100% concentration model Li C O Al Ar Fe Mo W H (c) BE49.3/BE21.6 FIG. 33. T lookup curves similar to Fig. 31 but with beryl- e lium filters, this shows a reduction in error for low-Z impuri- ties when using beryllium filters rather than aluminized My- lar, however high-Z impurities have line radiation at higher photon energy which can cause either overestimates (for Ar andAl)orunderestimates(forFe,W,Mo)toasimilardegree for beryllium filters as the Mylar filters. 38 Here ψ¯ is, as usual, defined to be linear in ψ and hav- TABLE VI. Reference point coordinates and uncertainties ing values 0 at the magnetic axis and 1 at the LCFS. used to define the Thomson measurement scattering angle. The shaft current F (t) is set to the value measured by The z coordinate is along the axis of the machine, with the 0 toroidal-field Mirnov coils. The derivative of F with re- equator at z=2367mm. spect to ψ is Reference point x [mm] y [mm] z [mm] View port 108±2 63±2 3499±2 F′ =λψ¯′ f′(ψ¯) (13) Cavity wall spot 283±10 164±10 1743±2 ψ ψ Beam exit port 266±4 154±4 3615±2 Beam entry port 0±8 0±8 −1640±20 where ψ¯ ψ ′ ≡ dψ¯/dψ. The scaling parameter, λ, with units of inverse length, is adjusted during Picard iter- ation to achieve the solution ψ(R,Z) satisfying the GS fore, we take the Thomson shot noise to be 10% for a 10 equation with specified boundary conditions and desired mV signal. Behaving as a Poisson process, the relative plasma current. The one-parameter shape function used √ Thomsonshotnoisethenscalesas1/ N whereN isthe in PsiBC is number of Thomson photons. The scattering angle is a significant source of error. f′(ψ¯)=−1−aψ¯+(1+a)ψ¯2 (14) Sensitivity analysis shows that at a 10 degree scattering We have edge value f′(1) = 0 to avoid discontinuity in angle, for every 0.5 degrees in error there is a 10% er- J at the edge of the plasma that would interfere with ror in T . The scattering angle was estimated from a 3D φ e stable solution of the GS equation. Positive values of modelofthemachinetakingintoaccountpotentialdevi- the shape parameter, a, correspond to a hollow plasma ations between the model and reality, and the alignment current profile, negative values to peaked. In order to uncertaintyofthelaserandtheviewvectors. Fourrefer- keep the toroidal current density J > 0 in the plasma encepointsweredefinedtocalculatethescatteringangle. φ we need to restrict a to satisfy a > −2. The integral of These point are listed in Table VI. The cavity wall spot the shape function, with edge value f(1)=0, is reference point is created by backfilling the fiber with white light and projecting the light through the aligned collection lens onto the formation cavity wall inside the f(ψ¯)=1−ψ¯− a(cid:0) ψ¯2−1 (cid:1) + 1+a(cid:0) ψ¯3−1 (cid:1) (15) 2 3 vacuum. The resulting spot can then be imaged with a camera through a nearby window and compared to the Theboundaryconditionψ =ψ +δψfortheGSequation 0 3D model to determine the coordinates of the spot using uses ψ from COMSOL and δψ from the previous time 0 convenient port features on the cavity wall. Solving for stepofPsiBC.AmarchingalgorithmidentifiestheLCFS the angle and propagating the error gives 9.9±0.5deg. at each Picard iteration, so that ψ¯(R,Z) can be calcu- Thiscontributesa10%uncertaintyinT e whichisadded lated within the LCFS. During the iteration the scale in quadrature to the uncertainty generated from the T e factor λ and the solution δψ are rescaled to match the fit. Mirnov data of the equatorial probe, while a is held con- stant. This makes the GS solution a function of one variable, a. This function is wrapped in a 1D optimizer Appendix C DESCRIPTION OF PSIBC thatvariesatofittheMirnovprobesalongtheshaftand (early in compression) on the endplates. Mirnov probes Here we provide a description of the method used in on the endplates are progressively excluded from the fit the PsiBC code to reconstruct the soaked flux boundary as the liner moves inward. After a is determined, the condition from Mirnov data. A finite difference method next step is to update the flux soak, which is done by is used to solve both the GS equation and the flux diffu- solving the magnetic diffusion equation for poloidal flux, sion equation. A Cartesian grid in (R,Z) is used, with a typical resolution ∆R = ∆Z = 5mm. Points on the ∂(δψ) η = ∆∗(δψ) (16) grid are identified as being in metal or plasma (vacuum) ∂t µ 0 regions based on the machine geometry and shape of the liner, comingfromthe COMSOLlinerreconstruction, at insidethemetal(machinecomponentsandliner), withη the time t under consideration. In the vacuum region the resistivity, and, simultaneously, solving PsiBC solves the GS equation for the poloidal flux field ∆∗ψ =−µ RJ (17) ψ(R,Z), namely 0 φ ∆∗ψ =−FF′ (11) outsidethemetal,withJ φ (R,Z)theplasmacurrentden- ψ sity from the preceding GS solve. This system of equa- Pressure is neglected. The poloidal current F(ψ) = tionsissolvedbyaseriesofimplicitbackwardEulertime RB (R,Z)isafluxfunctiongivenintermsoftheprofile steps. After this is done, the soaked magnetic flux is ad- φ function f (described below) by vectedbyinterpolatingδψfromtheCartesiangridtothe Lagrangianmeshoftheliner,movingthelinertoitsnew F(ψ)=F +λf(ψ¯) (12) position, and interpolating back onto the Cartesian grid. 0 39 By repeating this cycle of GS solve, flux soak update, The electron and ion pressure profiles are evolved ac- and advection, PsiBC produces a time series of plasma cording to the following equations parameters during the discharge, including ψ boundary condition, informed by Mirnov data. 3/2 ∂ (cid:104) p (V′)5/3 (cid:105) + 1 ∂P e =Q −Q , (22) Reconstruction of the density profile from PsiBC out- (V′)5/3∂t e ρ V′ ∂ρ Ω i ρ ρ put is performed as a post-processing step. The method 3/2 ∂ (cid:104) (cid:105) 1 ∂P used is an unconstrained linear fit to a set of polynomial (V′)5/3∂t p i (V ρ ′)5/3 + V′ ∂ρ i =Q i . (23) shapes chosen based on the number of interferometers, ρ ρ both unconstrained sparse polynomials in ψ¯ and sparse Here p = n T is the pressure of species s. Densities polynomials constrained to be low density at ψ¯=1, and s s s arerelatedbyn =Z n ,wheretheaverageioncharge e avg i taking the best fit that satisfies the positivity constraint Z is a parameter in the code. The coordinate ρ labels (n (ψ¯)≥0). avg e thefluxsurfacesandV′dρisthevolumebetweensurface ρ ρ and surface ρ+dρ. In ISM-plasma, ρ is determined by the enclosed mass (and vice versa). Since particles are Appendix D DESCRIPTION OF ISM conserved,thischoiceoffluxlabelmeansdensityisdeter- minedsimplyastheratioofthefixedmasstothevarying This section describes the method used for time- volume between flux surfaces. The first term on the left- dependent simulations in the Integrated System Model hand sides of (22) and (23), the pressure time derivative (ISM) [2, 44, 45]. The ISM-plasma model, being a 1.5D term,iswritteninaformsuitableforaLagrangiantreat- plasma transport code, is divided into an equilibrium ment of compressional heating, keeping pV5/3 constant solver and transport model. The former is implemented in the ideal adiabatic case. The second term represents as a direct equilibrium q-solver wherein the safety factor thedivergenceoftheheatflux. Ontheright-handsideof and entropy profiles are used to determine the plasma (22), Q = ⟨ηJ2⟩ is the volumetric flux surface average Ω equilibrium. These profiles evolve as determined by 1D oftheelectronheatingbyOhmicdissipationofmagnetic flux surface averaged transport processes. The poloidal energy, calculated using resistivity η = η with η (T ) ∥ ∥ e flux along the boundary of the plasma domain is calcu- the Spitzer parallel resistivity with enhancement factors latedself-consistentlybysolvingthediffusionandadvec- for impurities and neoclassical effects [67]. ISM results tion of magnetic flux within the liner and center con- presented in the present paper assume an effective ion ductor. For reconstructions, the external toroidal field charge Z =2 and average ion charge Z =1.2. eff avg function F 0 (t) is prescribed from an average of toroidal The resistive diffusion and dissipation of toroidal and field probes located on the center conductor. poloidal fluxes in the plasma evolve according to The evolution of thermal energy in ISM-plasma is de- ˆ terminedbyabalancebetweenthermaltransport,Ohmic ∂Φ η 1 = ∇F ·dS, (24) heating, and compressional heating. Heat flow between ∂t 2πµ R2 0 fl u u si x ng surfacesforspeciessisgivenbypowerP s ,calculated ∂ ∂ ψ t =− µ η (cid:16) FF ψ ′ + 2 µ π 0V L ′p′ ψ (cid:17) , (25) ˆ 0 3 dT P =− n (ψ)χˆ s ∇ψ·dS. (18) where Φ is the enclosed toroidal flux and V′ denotes the s 2 s s dψ derivative of enclosed volume with respect t L o inductance L with Here s=e,i indicate electron and ion transport, respec- ˆ tively, and ψ is the poloidal flux (but can be any flux 1 L= dRdZ. (26) surface label). In general, χˆ s can be a function of ψ. R The collisional energy transfer Q between electrons i and ions is given by: Equations (22) to (25) are discretized into N flux sur- faces, resulting in a system of ordinary differential equa- 3m Q = en ν (T −T ) (19) tions which are advanced in time using a second order i m i e e e i Runge–Kutta scheme. 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