Modelling of runaway electron dynamics during argon-induced disruptions in ASDEX Upgrade and JET
K. Insulander Björk, O. Vallhagen, G. Papp, C. Reux, O. Embreus, E. Rachlew, T. Fülöp, ASDEX Upgrade Team, JET contributors, EUROfusion MST1 Team
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Modelling of runaway electron dynamics duringargon-induced disruptions in ASDEX Upgrade andJET
K. Insulander Björk , O. Vallhagen , G. Papp , C. Reux ,O. Embreus , E. Rachlew , T. Fülöp , the ASDEXUpgrade Team ‡ , JET contributors § and the EUROfusionMST1 Team k Chalmers University of Technology, Gothenburg, 412 96, Sweden Max Planck Institute for Plasma Physics, D-85748 Garching, Germany CEA, IRFM, F-13108 Saint-Paul-lez-Durance, FranceE-mail: [email protected]
Abstract.
Disruptions in tokamak plasmas may lead to the generation ofrunaway electrons that have the potential to damage plasma-facing components.Improved understanding of the runaway generation process requires interpretativemodelling of experiments. In this work we simulate eight discharges in theASDEX Upgrade and JET tokamaks, where argon gas was injected to triggerthe disruption. We use a fluid modelling framework with the capability to modelthe generation of runaway electrons through the hot-tail, Dreicer and avalanchemechanisms, as well as runaway electron losses. Using experimentally based initialvalues of plasma current and electron temperature and density, we can reproducethe plasma current evolution using realistic assumptions about temperatureevolution and assimilation of the injected argon in the plasma. The assumptionsand results are similar for the modelled discharges in ASDEX Upgrade and JET,indicating that the implemented models are applicable to machines of varying size,which is important for the modelling of future, larger machines. For the modelleddischarges in ASDEX Upgrade, where the initial temperature was comparativelyhigh, we had to assume that a large fraction of the hot-tail runaway electronswere lost in order to reproduce the measured current evolution.
Keywords : Runaway electrons, tokamaks, fluid modelling, ASDEX Upgrade, JET
Submitted to:
Plasma Physics and Controlled Fusion ‡ See the author list of “H. Meyer et al , 2019 Nucl. Fusion § See the author list of “E. Joffrin et al , 2019 Nucl. Fusion k See the author list of “B. Labit et al , 2019 Nucl. Fusion unaway dynamics in argon-induced disruptions
1. Introduction
Runaway electrons (RE) may cause severe damageto plasma-facing components in tokamaks [1], wherethey may occur as a consequence of disruptions. Toavoid this, different schemes are being developed tomitigate, limit or entirely avoid the formation of REs.Massive material injections (MMI) are proposed to thisend, which may be realized through a gas injection(massive gas injection - MGI) or the injection of solidpellets, which can be shattered when entering thevacuum chamber (shattered pellet injection - SPI)[2, 3]. In medium-sized tokamaks, the potential ofthese measures have been demonstrated [4–13], butin the much larger devices envisaged for fusion energygeneration, the plasma conditions will be significantlydifferent (higher temperatures and densities, largerplasma currents), and whether the proposed measuresfor RE mitigation or avoidance will be effective also inthese devices can currently only be assessed throughplasma physics modelling.Several theoretical models for the physics ofrunaway electrons in tokamak disruptions have beendeveloped and implemented in computational tools[14]. To assess to which extent these models canbe applied to make useful predictions for the plasmabehaviour during disruptions, they must be validatedagainst existing experimental data. In currenttokamaks, disruptions are often deliberately triggeredby MGI, leading to a thermal quench (TQ) oftenfollowed by a formation of REs and a rapid decay ofthe ohmic plasma current (a current quench - CQ).The experimental data collected during such dischargesconstitutes a valuable dataset for validation of modelsfor the formation of REs in the presence of impurities(often in the form of massive amounts of noble gasses).In this work, we use such experimental data fromthe tokamaks ASDEX Upgrade (AUG) and JointEuropean Torus (JET) to investigate the applicabilityof the models, possible modifications needed, andqualitative differences between the mechanisms behindRE formation in these two differently sized tokamaks.The main computational tool used in the presentpaper is called go , a fluid code that describes the radialdynamics of the current density and the electric field,in the presence of impurities. The models implementedin this tool are briefly described in section 2, and moredetails are given in Refs. [15–18]. In the version of go that is used in the paper, hot-tail, Dreicer and avalanche RE generation models are implemented, aswill be further described in section 2.REs are defined as electrons having a momentumlarger than the critical momentum p c (or, analogously,a velocity larger than the critical velocity v c ), abovewhich the accelerating force exerted by the electricfield (induced in the disruption) is larger than thefriction force due to collisions, i.e. electrons abovethis threshold are accelerated until their momentum islimited by radiative energy losses. When the collisiontime at the critical momentum is longer than theduration of the TQ, hot-tail is the dominant primarygeneration mechanism [19]. In future fusion devices,this is expected to be the case [20], so significant hot-tail generation is expected in e.g. ITER [21] or SPARC[22]. For this reason, it is important to properlyunderstand and being able to model the interplayof the various runaway generation mechanisms inexperimental scenarios.Kinetic simulations of AUG discharges, using thetool code [23], were recently presented by InsulanderBjörk et al. [24]. code solves the linearized Fokker-Planck equation including radiation reactions andavalanche source, as well as the electric field evolution[25]. Kinetic simulations of JET discharges with code were attempted, but became prohibitively slow due tothe high electric fields induced. Also, code lacks radialdiffusion of electric field and current, mechanismswhich have been shown to be important for modellingof disruptions in large machines such as ITER [18], soin this sense, the two codes are complementary.In this paper, we begin by comparing simulationsof AUG discharges with go and code and verifythat these tools make qualitatively similar predictions,despite the different models used. In agreementwith recent results based on coupled fluid and kineticsimulations with go and code [26], we find thattaking into account all the hot-tail electrons in AUGoverestimates the final runaway current. To be able tomatch the experimentally obtained current evolution,we vary the loss-fraction of the hot-tail RE. Such lossesare expected to occur due to the breakup of magneticflux surfaces that accompanies the TQ.Furthermore, we find that the self-consistentcalculation of the temperatures of ions and electronsusing time-dependent energy transport equations oftenresults in a temperature evolution which does notagree with measurement data. The main reason whythe energy transport equations fall short of accurately unaway dynamics in argon-induced disruptions
2. Fluid modelling with go go [15–18] simulates the radial dynamics of the of thecurrent, temperature and electric field. Models for thepreviously mentioned RE generation mechanisms areimplemented in the code, as well as RE generationthrough Compton scattering and tritium decay, albeitthe two latter mechanisms are not relevant for thesimulations presented here.The hot-tail RE generation is a result of a rapidcooling of the plasma, during which there is notenough time for the fastest electrons (the "hot tail"of the electron velocity distribution) to thermalize,thereby ending up above the critical momentum. Theanalytical model used in go for hot-tail generation isderived in Ref. [27]. dn h RE dt = − du c dt u c H ( − du c /dt )( u c − τ ) / Z ∞ u c e − u u du ( u − τ ) / , (1)where u c = ( v c /v T + 3 τ ) / , τ = ν R t n ( t ) /n dt , n is the electron density, v T is the thermal electronspeed, v c is the critical velocity, subscript denotesan initial value, H is the Heaviside function, ν = ne ln Λ / (4 πǫ m e v T ) , ln Λ is the Coulomb logarithm, ǫ is the vacuum permittivity and m e the electronrest mass. This expression takes into account thedirectivity of the electric field, in contrast to thesimplified expression used in Refs. [21, 28] that is basedon counting the number of electrons with v > v c .The Dreicer
RE generation is instead a conse-quence of momentum space diffusion of electrons tomomenta above the critical threshold, due to small an-gle collisions [29]. For modelling of Dreicer RE genera-tion, a neural network was used, trained on the outputof kinetic simulations of plasmas with impurities, whichwere performed with code [30].RE generation through these two mechanismsis referred to as primary generation . REs thusgenerated may transfer part of their momentum to bulkelectrons, knocking them above the critical momentumand thereby creating new REs at an exponentiallygrowing rate. This a secondary generation mechanism,referred to as avalanche [31, 32]. The avalanchegeneration is modelled by a semi-analytical formula[33], that has been carefully benchmarked againstkinetic code simulations of disruptions in impurity-containing plasmas. go has the ability to calculate the temperatures offree electrons, bulk ions and impurity ions usingtime-dependent equations for the energy transport ofthe respective species in a one-dimensional cylindricalmodel [16]. The temperature evolution in this work iscalculated by ∂ ( nT ) ∂t = 3 n r ∂∂r (cid:18) χr ∂T∂r (cid:19) + σ ( T, Z eff ) E − X i,k n e n ik L ik ( T, n e ) − P Br − P ion , (2)where n is the total density of all species (electronsand ions), σ is the conductivity, Z eff is the effectiveion charge, n ki is the density of the i th charge state( i = 0 , , ..., Z − ) of the ion species k (e.g. deuterium,argon), P Br and P ion are the Bremsstrahlung andthe ionization energy losses, respectively. The lineradiation rates L ik ( T, n e ) are extracted from ADAS[34]. In the simulations we use a constant heat diffusioncoefficient χ = 1 m / s , and it was confirmed that theresults are insensitive to this choice.In the simulations presented here the temperatureis assumed to be equal for all species, since this iscomputationally less expensive. This simplificationaffects the heat capacity with a factor of at mosttwo [18], which affects the self-consistent temperaturecalculations used in the initial simulations described insection 4.1.1. For reasons to be discussed shortly, theinitial part of the temperature evolution, where most ofthe thermal energy is lost, is modelled by a prescribedexponential temperature drop, and therefore thissimplification is not expected to have an importantimpact. The injected argon is instead assumed to beheated through instantaneous heat exchange with theparticles present before the injection.During the initial phase of the simulationswhen the temperature decays rapidly (TQ), themagnetic flux surfaces are broken up due to magneto-hydrodynamic (MHD) instabilities [35]. The rapidradial transport due to these effects is the main drivingforce behind the initial temperature change, and theenergy transport equations are not valid. Hence, inour simulations the on-axis temperature T oa is initiallyprescribed to follow an exponential decay [21, 27]: T oa = T end + ( T initial − T end ) · e − tt TQ , (3)where the initial temperature T initial is deduced fromexperimental data, and the final temperature T end and thermal quench time t TQ may be treated asfree parameters which can be adjusted to yield an I p evolution similar to the measured data. Anotherreason for using this prescribed temperature evolutionduring the TQ is to be consistent with the assumptions unaway dynamics in argon-induced disruptions T end =1 eV, until T oa reached a pre-set value T switch .It has been estimated [36] that transport due tostochastic flux surfaces could drive a temperature dropdown to ∼ eV more effectively than radiation,which provides a physically motivated value for T switch . Then the simulation was restarted usingthe time-dependent energy transport equation (2) fortemperature determination, i.e. the value T end = 1 eVis not reached. After the switch to using the energytransport equations the temperature is calculatedlocally in each radial point by go .In many cases, this strategy resulted in a rapidincrease in the calculated on-axis temperature up toa few 100 eV after the switch. Re-heating has beensporadically observed in natural disruptions [37], buthas not been possible to investigate experimentallyin MGI-induced disruptions. On ASDEX Upgrade,the electron cyclotron emission temperature diagnostic(see section 3.1.1) is in density cut-off after the MGI,therefore any potential re-heating can not be directlymeasured. However, it is likely that the losses inthe energy transport equation are underestimated,for example due to radiation from wall impurities orremaining transport losses that are not included, orunderestimated, in the present model. The radiationlosses with argon impurities present follow a non-monotonic behaviour as a function of temperature,with maxima at ∼ eV and ∼ eV. Even a smallunderestimate of the radiative losses can make theohmic heating overcome the cooling in the ∼ eVrange, and make the temperature evolve towardsthe ∼ eV range where the ohmic heating is lessefficient. However, if the calculated losses are largeenough, the temperature will instead move towardsthe ∼ eV range, where it remains until the ohmicheating has decayed. When the simulations result insuch a temperature evolution also the experimental I p evolution was reproduced fairly well.There is close connection between the current de-cay rate and the temperature through the conductivity.In cases where re-heating occured in the simulations,the experimentally observed current decay rate was not reproduced sufficiently well, thus we used a differentstrategy for the temperature determination. In thesecases, equation (3) was used for the entire simulation,and the temperature T end was adjusted, together withother free parameters, to match the measured I p evo-lution as described in section 4.1.2. The values of T end which could match the I p evolution were approximately20 eV, i.e. T end is the free electron temperature duringmost of the CQ. go calculates the ionization states of the argon and theresulting free electron density from the time-dependentrate equations d n ik d t = n e (cid:2) I i − k n i − k − ( I ik + R ik ) n ik + R i +1 k n i +1 k (cid:3) , (4)using as input the free electron density prior tothe injection, the injected amount of argon and theinitial density profile. I ik denotes the electron impactionization rate and R ik the radiative recombination ratefor the i th charge state of species k , respectively. Theionization and recombination rates are extracted fromADAS [34]. The injected argon is assumed to distributeaccording to the same density profile as the initialdeuterium density.In the current simulations, the quantity ofassimilated argon is given in terms of the ratio r Ar / D ofthe argon density n Ar to the initial deuterium density,assumed to be equal to the initial free electron density n e0 . Lacking reliable experimental data on the rateof assimilation of argon into the plasma, the argonis assumed to be present at the very start of thesimulation. The temperature of the argon is calculatedassuming heat exchange as described in section 2.1,and the associated density of free electrons and of eachionization state is calculated by go . The effect ofmodelling the Ar density as exponentially increasingafter the beginning of the simulation was investigated,but this introduced another free parameter (the timerate of Ar assimilation) which affected the simulationresults in a similar manner as assuming a constant r Ar / D , so the simpler approximation was preferred. In the disruptions modelled in this paper, the measuredplasma current I p typically displays a small peakcoinciding with the beginning of the TQ, whichindicates an MHD event during which the currentdensity is redistributed radially due to a breakup ofmagnetic flux surfaces [35]. Towards the end of theTQ, the magnetic flux surfaces re-form and the currentcan be confined until it is dissipated resistively.The rapid relaxation of the current profile due tonon-axisymmetric MHD is not modelled by go , and unaway dynamics in argon-induced disruptions I p does not display this peak. Afterthe peak, I p starts to decay rapidly (the CQ) due tothe increase in plasma resistivity associated with thedecreasing temperature ( σ k ∝ T / e ), which is modelledby go . Due to this temperature dependence, the I p evolution is strongly affected by T e evolution, andhence by the parameters describing it: T initial , T end and t TQ . As previously stated, the exact value of T end does not affect the results significantly if we switch tothe time-dependent energy transport equation before T end is reached. However, the value becomes importantif equation (3) is used throughout the simulation, asdiscussed in sections 4.2 and 4.1.2. In these cases, T end rather plays the role of the equilibrium temperaturethat marks the end of the TQ, but not necessarily thefinal temperature reached after the CQ when no ohmiccurrent remains causing resistive heating of the plasma. go has the ability to model elongated plasmas [38],however, this capacity was not used in the presentedsimulations. The modelled discharges featuredslightly elongated plasmas, κ ≈ ¶ . Inclusion of the elongation inthe simulations did not result in any qualitativechanges in the simulation results, and thus it wasdecided to reduce the parameter space by omittingthis parameter. This simple geometry also facilitatescomparison with previous kinetic simulations whichwere zero-dimensional in real space.
3. Experimental data
In order to assess the applicability of the go model fortokamaks with different parameter ranges, we modeldischarges in AUG and JET, and compare our resultsto experimental data. Table 1 shows some basic datatypical for the discharges simulated for the respectivetokamaks, and in the following sections, more specificdata for each simulated discharge are presented. ASDEX Upgrade is a medium sized tokamak, and wemodel four discharges which were specifically designedfor the study of runaway electron dynamics [5, 12].The runaway discharges studied in this paper are near-circular (elongation κ ≈ . ), inner wall limited, havecore ECRH (Electron Cyclotron Resonance Heating)and low pre-disruption density ( n e0 ≈ · m − ).The discharges are terminated using argon MGI froman in-vessel valve. A 30 ms time period, starting ¶ EFIT/ELON at the argon valve trigger, was simulated to ensurethat the entire current quench was covered in allthe simulated cases. The plasma position remainedradially and vertically stable in the experiments duringthe simulated time window.Four discharges with different plasma parametersand amounts of injected argon were selected formodelling and an overview of the basic parametersfor these discharges is presented in table 2. Thesedischarges were selected from the set of 11 dischargesmodelled earlier in Ref. [24].
The temperature and its radialdistribution in the plasma is measured using electroncyclotron emission (ECE). For the present simulations,we only use the initial temperature profile. Theinitial on-axis temperature T initial used in equation(3) is determined as the average temperature over thecircular area within 1 cm of the plasma axis and thetime interval between 1 ms and 1.5 ms after the argoninjection valve trigger. The time interval was selectedto exclude both the beginning of the TQ and anyinitial temperature variations due to the shut off ofthe heating system shortly before the disruption. The free electron density is measuredby CO interferometry, which yields the line integratedfree electron density along the line of sight of theinterferometer. The density profile is fitted by thetool augped , which fits a modified hyperbolic tangentfunction [39] to the radial density data points obtainedwith interferometry. The average measured freeelectron density during the first 1.5 ms after the argonvalve trigger is used as the initial on-axis free electrondensity n e0 , since after these 1.5 ms, the argon injectioncauses the measured density to start to increase. Theinitial density profile is scaled with n e0 to obtain theinitial density in each radial data point.In each discharge, argon gas was injected into thevacuum vessel to trigger the disruption [5, 40]. Theinjected number of argon atoms N Ar is calculated fromthe pressure in the MGI reservoir holding the argongas before injection, the reservoir volume (0.1 l) andthe gas temperature (300 K). This quantity is listedin table 2 for the respective discharges. The amountof argon which finally assimilates in the plasma as afunction of time is difficult to assess. In previous work[24], the Ar assimilation fraction f Ar was assumed tobe the same for all discharges, and was assessed bymatching the current decay rate during the RE plateauphase. It was found that a reasonable estimate wasthat by the end of the TQ, 20% of the injected argonwas assimilated in the plasma volume V p , as defined bythe minor and major radii listed in table 1. We hencemake the same assumption for the present simulations, unaway dynamics in argon-induced disruptions ∗ At the time of the modelleddisruptions, the minor radius was 0.8 m.Tokamak AUG JETMajor radius R [m] 1.65 3.0Minor radius a [m] 0.50 1 ∗ Initial plasma current I p0 [MA] 0.76 1.3 - 1.4Initial on-axis free electron temperature T initial [keV] 5.3 - 7.2 1.6 - 2.0Initial on-axis free electron density n e [ m − ] 2.4 - 3.1 4.5 - 8.7Toroidal magnetic field on axis B [T] 2.5 3Table 2: Basic parameters for the four simulated discharges in AUG, and the notation for these parameters usedin this paper. Further descriptions of the origins of these parameters are found in sections 3.1.1 and 3.1.2 below.Discharge number 33108 34183 35649 35650Injected number of Ar atoms N Ar [ ] 175 74 94 96Initial free electron temperature on axis T initial [keV] 7.2 5.5 6.2 5.3Initial on-axis free electron density n e0 [ m − ] 3.1 2.8 2.6 2.4 Parameters used in code and initial go simulations Thermal quench time parameter t TQ [ms] 0.152 0.198 0.178 0.177Argon-to-deuterium density ratio r Ar / D n Ar as N Ar /V p · f Ar = N Ar /V p · . . The ratio r Ar / D is calculated as n Ar /n e0 . JET is significantly larger than ASDEX Upgrade, asshown in table 1. Four discharges in which runawayelectrons were formed were chosen for modelling,representing a set of varying plasma parameters. Inparticular, the size of the Ar injection varies by morethan an order of magnitude. The basic parameters forthese discharges is presented in table 3.
The free electron temperatureis measured by high resolution Thomson scattering(HRTS) diagnostic which yields both a temperatureprofile and an sufficiently reliable value of the on-axis temperature before the Ar injection ( T initial usedin equation (3)). The initial temperature profile issmoothed using the rloess algorithm in matlab [41]to remove signal noise. Equation (3) was used for theentire simulation, and the TQ time parameter t TQ andthe final temperature T end were varied along with theother free parameters (see section 4.1.2) in order to finda set of parameters which reproduced the measured I p evolution. The free electron density n e is mea-sured by several different diagnostics. Interferometry yields the line integrated free electron density along theline of sight of the interferometer. In these simulations,we used the fast interferometer signal + . The shape ofthe initial free electron density profile was retrievedfrom the HRTS data at the last available data pointfor this diagnostic before the Ar injection, which wasat most 130 ms before the injection. The profile wassmoothed to remove signal noise using rloess [41] justas for the temperature profile, and then scaled so thatthe integral of the profile yielded the same initial lineintegrated density as the interferometry based densitydata.The final number of injected argon atoms iscalculated from the volume of the reservoir holdingthe argon before the injection (DMV3, with volume0.35 L), the gas vessel pressure before and after theinjection ∗ , and the assumption that the gas is heldat room temperature (300 K). The fraction of thisargon which actually assimilates in the plasma isdifficult to assess experimentally, and the gas alsoleaves the injection reservoir during a period of sometens of ms. Hence, the ratio r Ar / D was chosenso that the maximum free electron density regardedas a free parameter, constrained by the conditionthat the calculated maximum line integrated free + KG4C/LDE3 signal, compensated for fringe jumps and scaledto give a pre-disruption density in agreement with the DF/G1C-LD 4. Results go and code results for AUG Four discharges that were previously modelled using code [24] were modelled with go . First, we aimed toreproduce the previous code simulations to verify thatthe models, although very different, give qualitativelysimilar results. Then, we adjusted the models toimprove the matching between the simulated andmeasured I p evolution. code In the first set of simulations, the four AUGdischarges were simulated using the same modellingparameters t TQ and r Ar / D as earlier [24] to verifythat the modelled quantities show a qualitativelysimilar behaviour for the two different codes. Forthese simulations, the modelling strategy was to useequation (3) initially, and then switch to the time-dependent energy transport equation after the on-axis T e had dropped to below a pre-defined temperature T switch . This method most closely resembles thestrategy used in Ref. [24]. However, note thatin the simulations presented in Ref. [24], afterthe temperature fell below T switch , the temperaturewas determined assuming equilibrium between Ohmicheating and line radiation losses, rather than the time-dependent energy transport equation as was used here(see equation (2)). Furthermore, the densities ofthe various ionization states were calculated assumingan equilibrium between ionization and recombination,instead of the time-dependent rate equations as is donehere (see equation (4)). Due to these differences, inaddition to the differences caused by the fluid vs kineticapproach and the omission of the radial dynamics inRef. [24], the results in the two cases are not expectedto agree quantitatively, even if the input parameters are the same. In our simulations, we set T switch =6 eV, which is similar to the temperature equilibriumvalue found in several of the simulations in Ref. [24].Furthermore, we set T end = 1 eV, and note that thevalue of T end has a minor impact on the exponentialfunction, as long as it is lower than T switch , sincethe exponential function is abandoned before T end isreached.The simulation results are similar for all fourmodelled discharges, and plots of the most importantsimulation output is shown in figures 1 and 2 forAUG discharge code and go simulations, and differs from the measured evolutionin a similar manner. In figure 2, the RE generationrates G m = 1 /n e · d n RE ,m / d t are displayed separatelyfor each generation mechanism m . The Dreicer REgeneration rate was calculated using a neural networkdescribed in Ref. [30], and the avalanche growth ratewas calculated using the semi-analytical formula givenin Ref. [33]. We note that also the generation ratesare qualitatively similar in both cases, in that theDreicer RE seed generation is small compared withthe hot-tail RE seed and that avalanche generationcontributes significantly towards the end of the CQ,but the timing of the Dreicer and hot-tail generationpeaks differs, as does the relative importance of hot-tail and avalanche generation. The absolute valuesof the RE generation rates differ between the code and the go simulations, partly because the valuesobtained with code are representative of our estimatesof the on-axis conditions, whereas the values plotted forthe go simulations are the generation rates averagedover the plasma volume. Another reason for thedifferent absolute values is that REs, by necessity,are differently defined in the different simulations; inthe code simulations they are defined by having amomentum p fulfilling p/m e c > . and in the go simulations by p > p c .In order to display the qualitative similaritybetween the code and go results, relevant parametersfor comparison of the go and code results are shown in unaway dynamics in argon-induced disruptions I p spike to the point where d I p / d t < kA/ms,and the post-CQ total plasma current as the measuredor calculated value of I p at this time point. Fromthe listed parameters, we can conclude that while thesimulation results are not identical for the two codes,which would not be expected, they are qualitativelysimilar in that for all simulated discharges, both codes • predict a partial CQ (i.e. neither full conversionnor complete CQ), • overestimate the total RE generation, leadingto an underestimation of the CQ time and anoverestimation of the post-CQ I p • overestimate the I p decay rate during the CQ and • predict a small Dreicer RE seed generation( G Dreicer ), relative to the hot-tail RE seed.The post-CQ total plasma current calculated by code for go gives qualitatively similarresults to code , but also identified a need to adjust ourmodels to obtain better matching between calculatedand measured I p evolution. I p evolution Asdiscussed in Ref. [24], the (almost) consequentoverestimation of the post-CQ total plasma current islikely due the fact that losses of the RE seed populationcaused by the stochastization of the magnetic fluxsurfaces during the TQ are not modelled by code (and also not in the initial go simulations). In code ,the 2D momentum distribution of the entire electronpopulation is modelled as a continuum, i.e. no explicitdistinction is made between bulk electrons and RE.In contrast, go models the REs as a distinct electronpopulation, and also models each RE generation modeseparately. This makes it possible to model theloss of REs generated by the hot-tail mechanism bymultiplying the number of hot-tail REs generated ineach time step by a damping factor f d . For clarity, wediscuss the hot-tail seed survival fraction f HT = 1 − f d in the following. The loss of hot-tail generated REsis expected to be significant since they are generatedduring the TQ when confinement is impaired due tothe magnetic flux surfaces being broken up as noted insection 2.1. The losses of REs generated through theDreicer and avalanche mechanisms are expected to beless important since they are predominantly generatedduring the later phase of the disruption when the fluxsurfaces have healed again.The four AUG discharges were simulated withhot-tail losses, but the overprediction of the post-CQ I p remained also when assuming loss of all hot-tail (a)(b)Figure 1: Total and RE currents calculated by (a) code and (b) go . In both panels, the experimentallymeasured total plasma current is plotted for reference.In (b) the temperature was calculated using equation(3) until T oa reached T switch = 6 eV when equation(2) was invoked. The vertical lines mark the end ofthe CQ for the measured and calculated I p evolutionrespectively, see table 4.RE. When the hot-tail seed was assumed lost, theDreicer generation increased instead, still resulting ina high total RE generation. In these cases, it was alsofound that the I p evolution depended strongly on thetemperature at which the switch was made from usingequation (3) to invoking the energy transport equation,while the model is only predictive of the temperatureevolution during the CQ if it is insensitive to T switch inthe physically motivated 10-100 eV range. Moreover,the calculated temperature in many cases increasedto about 100 eV after the switch, which contradictsmeasurement data, as discussed in section 2.1. Wetherefore turned to a different modelling strategy, usingequation (3) for the entire simulation.As mentioned in section 2.3, the I p evolution issensitive to the temperature evolution, determined bythe parameters T initial , T end and t TQ . T initial can bemeasured and is thus taken from experimental data,but t TQ and T end are regarded as free parameters.The hot-tail RE generation is exponentially sensitiveto t TQ [27], with a quick TQ (small t TQ ) resulting ina large hot-tail RE generation early in the disruption unaway dynamics in argon-induced disruptions go and code simulations. G total and G Dreicer are the totaland Dreicer RE generation rates, respectively.Discharge number 33108 34183 35649 35650Measured CQ time [ms] 3.8 4.4 4.7 4.2Measured post-CQ I p [MA] 0.24 0.24 0.21 0.22Code used code go code go code go code go Calculated CQ time [ms] 1.9 1.1 1.8 1.4 1.5 1.3 2.0 1.2Calculated post-CQ I p [MA] 0.22 0.62 0.38 0.60 0.41 0.61 0.32 0.60max( G total )/max( G Dreicer ) 9 · · · · · · · · (a)(b)Figure 2: (a) Total RE generation rate calculated by code , complemented with the Dreicer and avalanchegeneration rates calculated with separate scripts asdescribed in [24]. (b) Total, Dreicer, avalanche andhot-tail RE generation rates calculated with go . Notethat in both panels, the Dreicer generation rate isscaled to be visible in the respective plots. The scalingfactors are given directly in the plots.and consequently a higher post-CQ I p . An increased t TQ results in a smaller RE generation, see figure3(a). The hot-tail seed loss parameter can obviouslybe adjusted to counteract the effect of a small t TQ , buta small t TQ results not only in increased hot-tail REgeneration, but also increased Dreicer RE generation,so the effect of a small t TQ can not be completelycancelled by assuming that the hot-tail seed is lost. As shown in figure 3(b), variations of f HT below 1%have a minor impact on the results for the typical sizeof the hot-tail seed in these simulations, because atthis point, Dreicer generation becomes the dominantRE generation mechanism.The parameter T end affects the I p decay rateduring the CQ. Since the duration of the TQ ( ∼ ms)is much shorter than the duration of the CQ ( ∼ ms), T e ≈ T end for most of the CQ and the plasma resistivity σ k ∝ T / e determines the I p decay rate, see figure 3(c).We can therefore infer T end by choosing it to match theinitial I p decay rate.The amount of Ar assimilated in the plasma,quantified by the parameter r Ar / D , has a direct effecton I p , a higher r Ar / D leading to a lower post-CQ I p ,see figure 3(d), until the post-CQ I p approaches zero.The search of the parameter space was doneiteratively, starting from the set of parameters usedin the initial simulations. The values of r Ar / D usedin the initial simulations were kept the same in thesesimulations. T end was then chosen to match the I p decay rate during the CQ, defined as ∆ I p / ∆ t duringhalf of the CQ (while I p decreased through 25% to 75%of the CQ). Then the hot-tail seed loss was adjustedto match the measured post-CQ I p . For discharges I p remained toohigh even when all hot-tail RE seed was removed, so t TQ was increased, keeping the hot-tail seed loss at100%, until the the measured post-CQ I p was matched.Since t TQ also affects the I p decay rate to a smallextent, the process had to be iterated a few times.The conditions for matching were that ∆ I p / ∆ t shouldbe within 10% of the measured value, and that thepost-CQ I p should be within 10 kA from the measuredvalue. The chosen values of the simulation parametersare listed in table 5.Again using discharge I p evolution and the RE generation rates areshown in figure 4. The corresponding figures forthe other three modelled discharges are similar. Thetransition between the CQ and the RE plateau phase unaway dynamics in argon-induced disruptions I p evolution to the simulations parametres (a) t TQ (b) f HT , (c) T end and (d) r Ar / D . (a)(b)Figure 4: (a) Current evolution and (b) RE generationrates for AUG discharge T fin = 20 eV and a hot-tail RE seed loss of99.9%. The other parameters are t T Q = 0.35 ms and r Ar / D = 0.64.is less marked in these simulations than when thetime-dependent energy transport equations were used,since the TQ ends less abruptly when prescribed byequation (3), as seen when comparing figure 4(a) withfigure 1(b). The exponentially decaying temperaturedoes not represent the physical T e evolution exactly,but is a better approximation than the significantre-heating predicted when switching to the time-dependent energy transport equations. The REgeneration rates shown in figure 4(b) differ from thoseshown in figure 2(b) mainly in the almost completelysuppressed hot-tail generation. When the hot-tailgeneration is suppressed, the electric field is allowedto build up, resulting in larger Dreicer generation. go Simulations of the JET disruptions were first at-tempted with an initial exponentially decaying tem-perature and a switch to time-dependent energy trans-port calculations, but with similar complications as forthe AUG discharges, i.e. for discharges I p evolution. For JET dis-charges unaway dynamics in argon-induced disruptions T switch = 100 eV down to approx-imately 20 eV where it remained for a ms after contin-uing down to the final equilibrium temperature of 1 eV.This is shown for t TQ = 0.175 ms and r Ar / D = 0.25. The resulting I p evolution and RE generation rates are shown in figure6, and, as shown, the I p evolution was reproduced withthis temperature evolution. It was concluded that,again, the failure to reproduce the I p evolution in theother discharges was mainly a consequence of a failureto predict the temperature evolution. Hence, simu-lations were performed with a prescribed exponentialtemperature drop down to a temperature of approxi-mately 20 eV, which made it possible to reproduce the I p evolution also for the other JET discharges listed intable 3.As stated in section 3.2.2, the fraction of theinjected Ar atoms that assimilate in the plasma, f Ar ,is constrained by the condition that the maximumcalculated line integrated free electron density, which isvery sensitive to the assimilated Ar density, should notdeviate by more than 10% from the measured value.Since the calculated free electron density is not verysensitive to the other simulation parameters, the Arassimilation fraction, and thereby the ratio r Ar / D , waschosen first.The other parameters were found iteratively, asfor the AUG discharges. Once the assimilationfraction f Ar , and hence the values of r Ar / D ( r Ar / D = f Ar N Ar / ( V p n e0 ) = 0 . N Ar / ( V p n e0 )), were chosen, T end was chosen to match the I p decay rate, and the valueof t TQ was chosen to match the measured post-CQ I p . In all cases, since Dreicer was the dominantRE generation mechainsm, the I p evolution couldbe modelled without assuming hot-tail losses, so thehot-tail seed loss was neglected. The conditions for (a)(b)Figure 6: (a) Current evolution and (b) RE generationrates for JET discharge ∼ ∆ I p / ∆ t (for definition see section 4.1.2) shouldbe within 10% of the measured value, and that thepost-CQ I p should be within 10 kA from the measuredvalue.The I p evolution shown in figure 7 shows a lessabrupt transition from the CQ to the RE plateau phasethan the measured data, just as for the AUG case infigure 4(a). This is, once again, due to the smoothend of the TQ given by the exponential approximation.The comparatively small hot-tail peak is visible tothe left in figure 7(b), whereas the Dreicer generationcontinues for several ms. This differs significantly fromthe generation rates shown in figure 6(b), where theDreicer peak is almost as narrow as the hot-tail peak.The calculated line-integrated free electron den-sity is shown together with a measurement-based es-timate of the same quantity in figure 7(c). The esti-mate is based on the line integrated density measuredby interferometry (KG1 diagnostic, corrected for fringejumps). The much faster increase of the calculateddensity immediately at the start of the simulation isdue to the assumption that all Ar enters the plasma in-stantaneously, which is obviously not correct, but hasa minor impact on the calculated plasma current. unaway dynamics in argon-induced disruptions I p evolution. For the JET cases, the hot-tailsurvival fraction is unconstrained, since the I p evolution could be reproduced without assuming hot-tail losses.Argon-to-deuterium Thermal quench Hot-tail Finaldensity ratio time parameter survival fraction temperature r Ar / D t TQ f HT T fin ms % eV AUG discharges ≤ ≤ JET discharges 5. Discussion and conclusions The most important parameters for the simulateddischarges in JET and AUG are listed in table 1and the simulation parameters in table 5. We notethat the initial free electron densities and also theexternally applied magnetic field are similar for the twosets of discharges, whereas the machine size and theinitial plasma current are significantly larger in JET.It should be noted that the higher current in JET isa direct consequence of the larger machine size - thecurrent densities are similar (1.5 – 1.8 MA/m for AUGand 1.9 – 2.0 MA/m for JET). In AUG, the initialtemperature and the injected Ar density ( r Ar / D · n e )are larger than in the JET discharges.The simulation parameters needed to reproducethe I p evolution when using the same modellingstrategy are similar, except that the hot-tail seedsurvival fraction, had to be chosen very small toreproduce the AUG discharges, whereas its value wasunimportant for modelling of the JET discharges.Our simulations indicate that the hot-tail REgeneration is much smaller in JET as compared withAUG. This may be understood by the fact that at afixed TQ time, the hot-tail seed increases exponentiallywith initial temperature due to the longer slowing-down time of the hot-tail electrons. According to thesimulations, the TQ times in the two devices appear tobe similar, although this property would be expectedto scale with machine size [1]. This may be becausethe higher initial temperature in the AUG dischargesis compensated by a relatively higher density ofimpurities (due to the smaller plasma volume), whichcan then radiate the heat more efficiently. An important remaining problem to be solvedbefore reliable predictive simulations can be madeis the self-consistent modelling of the temperatureevolution. Although we found that a time-dependentenergy transport model including ohmic heating andimpurity radiation losses allowed for reproduction ofthe plasma current evolution for some JET discharges,this model was prone to predict a re-heating of theplasma after a forced temperature drop in otherdischarges in a way that contradicts the experimentaldata. This suggests that some additional lossesare present, such as radiation from wall impuritiesand transport losses remaining during the CQ. Theimportance of transport losses is expected to behigher in a smaller machine, which might explainwhy the prediction of an experimentally excluded re-heating was less common for JET than for AUG.Time dependent radial profiles of the magnetic fieldfluctuations are difficult to obtain from experiment,although their amplitude during the current quenchhas often been found to be fairly small [42]. Numericalmodelling of the transport due to magnetic fluctuations[43] would therefore require exploration of a fairly largeadditional parameter space and the effect of this on theresults presented remains an open question.The main conclusion of the presented set ofsimulations is that the RE generation models in go are able to reproduce experimentally measuredrunaway currents for both AUG and JET using similarassumptions and modelling strategies. This increasesour confidence that the RE models adequately describethe runaway physics in tokamaks of different sizes. Therelative simplicity of the code makes it possible torun a large number of simulations and qualitatively EFERENCES Acknowledgements The authors are grateful to L Hesslow, M Hoppe, ISvenningsson, I Pusztai, A Boboc and G Pautasso forfruitful discussions. This work has been carried outwithin the framework of the EUROfusion Consortiumand has received funding from the Euratom researchand training programme 2014 - 2018 and 2019 - 2020 under grant agreement No 633053 andfrom the European Research Council (ERC) underthe European Union’s Horizon 2020 research andinnovation programme under grant agreement No647121. The views and opinions expressed hereindo not necessarily reflect those of the EuropeanCommission. The work was also supported by theSwedish Research Council (Dnr. 2018-03911) andthe EUROfusion - Theory and Advanced SimulationCoordination (E-TASC). 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