Validity of models for Dreicer generation of runaway electrons in dynamic scenarios
S. Olasz, O. Embreus, M. Hoppe, M. Aradi, D. Por, T. Jonsson, D. Yadikin, G.I. Pokol, EU-IM Team
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Validity of models for Dreicer generation ofrunaway electrons in dynamic scenarios
S. Olasz , , O. Embreus , M. Hoppe , M. Aradi , D. Por , T.Jonsson , D. Yadikin , G.I. Pokol , , EU-IM Team ∗ Institute of Nuclear Techniques, Budapest University of Technology andEconomics, M˝uegyetem rkp. 3, Budapest 1111, Hungary Fusion Plasma Physics Department, Centre For Energy Research, Budapest,Hungary Department of Physics, Chalmers University of Technology,SE-41296 G¨oteborg,Sweden Barcelona Supercomputing Center (BSC), Barcelona, Spain KTH Royal Institute of Technology, Stockholm, Sweden Department of Space, Earth and Environment, Chalmers University ofTechnology,SE-41296 G¨oteborg, Sweden ∗ https://users.euro-fusion.org/eu-imE-mail: [email protected] December 2020
Abstract.
Runaway electron modelling efforts are motivated by the risk theseenergetic particles pose to large fusion devices. The sophisticated kinetic models cancapture most features of the runaway electron generation but have high computationalcosts which can be avoided by using computationally cheaper reduced kinetic codes.In this paper, we compare the reduced kinetic and kinetic models to determine whenthe former solvers, based on analytical calculations assuming quasi–stationarity, canbe used. The Dreicer generation rate is calculated by two different solvers in parallelin a workflow developed in the European Integrated Modelling framework, and thisis complemented by calculations of a third code that is not yet integrated into theframework.
Runaway Fluid , a reduced kinetic code,
NORSE , a kinetic code usingnon-linear collision operator, and
DREAM , a linearized Fokker–Planck solver, areused to investigate the effect of a dynamic change in the electric field for differentplasma scenarios spanning across the whole tokamak–relevant range. We find thaton time scales shorter than or comparable to the electron collision time at the criticalvelocity for runaway electron generation kinetic effects not captured by reduced kineticmodels play an important role. This characteristic time scale is easy to calculate andcan reliably be used to determine whether there is a need for kinetic modelling, orcheaper reduced kinetic codes are expected to deliver sufficiently accurate results. Thiscriterion can be automated, and thus it can be of great benefit for the comprehensiveself–consistent modelling frameworks that are attempting to simulate complex eventssuch as tokamak start–up or disruptions. alidity of models for Dreicer generation of runaway electrons in dynamic scenarios
1. Introduction
Correct understanding and accurate simulations of runaway electron generation are ofgreat importance as they pose a severe risk to the upcoming ITER experiment [1, 2].Runaway electrons can appear when large electric fields are generated in variousevents, such as disruptions or the tart–up of a tokamak discharge [3, 4, 5]. A largefraction of the pre–disruption current can be converted into runaway current on a fewmillisecond timescale and on large devices such as JET the runaway current can reachthe magnitudes of megaamps [6, 7].Runaway electrons can be generated in plasmas because of the reduced friction forceexperienced by fast particles compared to the thermal population. An applied electricfield can create a net accelerating region, called runaway region, in momentum spacewhere the particles are accelerated to high energies until synchrotron radiation losses canbalance the acceleration of the electric field. A primary runaway electron generationmechanism, named Dreicer generation, is the diffusion of particles in velocity spaceinto the runaway region due to collisions [8]. The expression was generalized to therelativistic case by J. Connor and R. Hastie [9]. Both formulas describe the generationin steady–state cases and are used in the present paper as a basis for comparison.The steady–state generation rates are often used in numerical models to calculatethe runaway electron generation – these models are often referred to as fluid orreduced kinetic codes. One of these codes is
Runaway Fluid ‡ , which is used inthe European Integrated Modeling (EU-IM) framework [10], and is also integratedinto the ITER Analysis and Modelling Suite (IMAS) [11]. Runaway Fluid usesthe analytical formulas by Connor and Hastie [9] for the Dreicer generation rate toestimate the runaway electron density. The European Transport Simulator workflowwith the
Runaway Fluid has been benchmarked against the GO code [10]. GO [12]is a transport solver for cylindrical geometry for the study of runaway electrons indisruptions which has a similar implementation of generation rates as the RunawayFluid code. Reduced kinetic models have also been used in other comprehensivesimulation tools, such as JOREK [13] and ASTRA-STRAHL [14]. The common usageof these models in integrated modelling tools motivates the study of the range ofapplicability of this approach.Kinetic solvers give a wider range of opportunities in runaway modelling. TheNORSE code [15] § solves the kinetic equation with a non-linear collision operator.Therefore, it is suitable for slide–away scenarios, as it takes runaway–runawayinteraction into account. The DREAM [16] k code, on the other hand, solves the bounceaveraged Fokker–Planck equation with a linearized collision operator, while analyticallysolving the background plasma evolution n a manner similar to GO . There are severalother kinetic solvers commonly used for the study of runaway electrons, including, ‡ https://github.com/osrep/Runafluid § https://github.com/hoppe93/NORSE k https://github.com/chalmersplasmatheory/DREAM alidity of models for Dreicer generation of runaway electrons in dynamic scenarios CODE [17, 18],
KORC [19, 20] and
LUKE [21, 22], all considering the problem todifferent degrees of complexity. The present paper shows results from
NORSE and
DREAM .These kinetic solvers are becoming more and more accurate in simulating theevolution of the runaway electron distribution functions, but the price of the accuracyis paid in computation time. Therefore, self–consistent simulations tend to use fluid–like models, which provide satisfying results in most cases, but the underlying steady–state assumptions fail in highly dynamic scenarios. In the present paper, we study thebehavior of the time evolution of the primary generation rate with various models inscenarios with an instantaneously introduced electric field. For this purpose,
RunawayFluid and
NORSE are integrated into a workflow called the Runaway Electron TestWorkflow in the EU-IM framework [10, 23]. The workflow is used to study the effectof electric field on the time evolution of the primary runaway generation for differentelectron densities and temperatures with main focus on specifying the conditions inwhich it is safe to use the fluid-like approach. A case of rapidly evolving temperature,referred to as hot-tail generation, is not addressed, as it has been considered in otherpapers recently [24, 25, 26].Four cases with differing electron temperature and density values have been selectedfor the current study in a way to span the whole tokamak-relevant range of plasmaparameters. Consequently, the chosen cases are relevant to extreme cases of plasmaoperation where runaway electron generation might be expected. The focus of ourstudy is on the range of applicability of the analytical formulas and linear collisionoperator for these cases. The Dreicer generation rate and runaway electron density arecalculated. We find that a kinetic effect can lead to a peak in Dreicer generation rateon short timescales. The time over which this peak occurs can be characterized by theelectron-electron collision time at the critical velocity for runaway electron generation.The other primary generation rates, such as hot-tail generation [27], and the secondaryor avalanche generation [28] is not considered in the present paper but their possibleeffects are discussed in the conclusions. The modelling tool used for the simulations isdescribed in Section 2, the results for the various cases are presented in Section 3 andwe conclude in Section 4.
2. Model description
The present work is carried out in the EUROfusion Integrated Modeling framework (EU-IM), which was developed to enable coupling different codes simulating different physicalphenomena in complex workflows. This is achieved by defining a standardized datastructure and providing standardized access to the named data structure, so the differentcodes can exchange input and output data with ease [23]. The framework also allows forexperimental data to be imported to the data structure. The integration of physics codesinto workflows is managed by the Kepler graphical workflow engine ¶ . The European ¶ https://kepler-project.org alidity of models for Dreicer generation of runaway electrons in dynamic scenarios Transport Simulator (ETS) workflow aims to simulate tokamak discharges and hasbeen proposed for self–consistent simulation of tokamak disruptions [10]. Each modulein ETS used for simulating different physical aspects of discharges have multiple codesintegrated and it can be chosen which of these codes is used during a simulation.The purpose of
Runaway Indicator , the simplest runaway model in ETS is toindicate when the physical parameters during a simulation are suitable for runawayelectron generation. It calculates the critical electric field for given plasma parametersand indicates when an electric field larger than the critical field is present. It alsoestimates the primary runaway electron generation rate with a simple formula and givesa warning if it is larger than a preset value [10] + .During a simulation, Runaway Fluid can give a conservative estimate of therunaway electron population. It calculates the primary and secondary (avalanche)generation of runaway electrons. The analytical formulas derived by J. Connor andR. Hastie [9] are implemented for primary generation calculation including the mostgeneral formula (63) in the paper and it was used for the simulations presented in thecurrent paper. The avalanche generation rate is calculated using the formula derivedby M.N. Rosenbluth and S.V. Putvinski [28]. Both generation mechanisms can beaugmented by the use of a number of different correction factors. Using these generationrates,
Runaway Fluid estimates the runaway electron density and current [10]. In thepresent work, no correction factor is used and the secondary generation is not calculatedas we are focusing on the behavior of Dreicer generation for various plasma scenarios.
NORSE is a more advanced tool that solves the kinetic equation to calculate theelectron distribution function [15]. It uses a non-linear collision operator which makesit suitable for taking runaway-runaway interaction into account. It is also capable ofsimulating slide-away scenarios. The
NORSE code is written in Matlab [15]. Theinterfacing of the code to the EU-IM framework was done using Python ∗ . A MatlabEngine is used for the communication between the different programming languageswhich allows for Python codes to execute Matlab commands. The Python interface is,in turn, integrated into the Kepler workflow as an actor, and it provides the inputto the NORSE code and writes the output to the required data structure. Thedefault
NORSE output contains the electron distribution function, the correspondingmomentum grid, the runaway density and current. Additional information, such asplasma parameters for the simulation, the value of the critical field, runaway electrondensity, current and generation rate is output as an option from
NORSE in HDF5format.These codes have been interfaced in the EU-IM framework to create so-calledactors in Kepler workflows. The dedicated runaway electron workflow, the RunawayElectron Test Workflow is shown in Figure 1. The boxes in the Figure are compositeactors consisting of an embedded workflow themselves, which perform different actionsduring simulations. The input parameters can be given under the shot parameters. + https://github.com/osrep ∗ https://github.com/osrep alidity of models for Dreicer generation of runaway electrons in dynamic scenarios Figure 1.
The dedicated runaway electron workflow in the EU-IM framework.
The shotnumber , runnumber and machine parameters determine which data will beused from the database, while the run out parameter determines what run number theoutput will be stored in. Simulation parameters such as time step , starting time ,and number of iterations can also be given. As Runaway Fluid and
NORSE output are of the same data type, they have to be differentiated by a parameter calledoccurrence number. The local occurrence Runaway fluid is used for this purpose for
Runaway Fluid , while the occurrence number for
NORSE can be given at the actorlevel.The workflow starts at the
Initialize composite actor where the input data isread from the database as specified by the simulation parameter settings. The mainpart of the workflow is the time loop, where the different physics codes are integrated.This is controlled by the
LoopOrganizer which stops the loop after a preset number ofiterations. The post-processing of the simulation results is done in the
Postprocessor actor. The wrapper composite actors contain the physics codes, as shown for the
Runaway Fluid in Figure 2. First, the input data for the
Runaway Fluid actor isextracted from the bundle and delivered to the code. On the other side, the outputdistribution is passed onto the next workflow level, and any possible run-time messages alidity of models for Dreicer generation of runaway electrons in dynamic scenarios Figure 2.
The content of the runaway-fluid-wrapper composite actor. are output by the
Message composer composite actor. The output from the differentphysical actors is merged in another composite actor and the bundle is given back tothe loop organizer. The main advantage of this method is that the different codes canbe run parallel with identical input and their output can be easily compared.The
DREAM kinetic solver has not yet been integrated into the EU-IM framework,as some parts of it are still under development [16] ♯ . However, we use it here for studyingthe Dreicer generation rate obtained with a linearized test-particle collision operator.The DREAM code is specifically designed for the self–consistent study of runawaygeneration during tokamak disruptions and to solve a bounce-averaged Fokker–Planckequation, along with equations for background plasma parameters such as the electricfield, electron temperature and ion densities. In this work, our main pieces of interestare transient effects on the Dreicer runaway generation rate and therefore we use thecode to only solve for the electron distribution function in a homogeneous plasma withprescribed parameters, i.e. equivalent to
NORSE but with a linearized test-particlecollision operator.
3. Simulation results
In this section, the simulation results are presented. Firstly, the physical parametersused for the different scenarios are shown. The evolution of the distribution functionfrom
NORSE calculations is presented and the calculated generation rate from thedifferent models is given. The evolution of the generation rate based on the distributionfunction and its effects on the runaway electron density are discussed. Lastly, thetimescales characterizing the kinetic effects are introduced. ♯ https://github.com/chalmersplasmatheory/DREAM alidity of models for Dreicer generation of runaway electrons in dynamic scenarios Table 1.
The electrons density and temperature values for the various scenariosstudied.
Low densitydischarge (LD) Start–upphase (SU) End ofdisruption (ED) Start ofdisruption (SD)Density [m − ] 5 · · Temperature [eV] 10000 300 300 10000
The simulations were carried out for the four different combinations of density andtemperature values given in Table 1 using the Runaway Electron Test Workflow.The different values were chosen to represent extreme cases of electron densitiesand temperature still relevant for tokamak operations. A low density and lowtemperature scenario is relevant for the start–up (SU) phase of tokamak discharges,where large electric fields are induced, which are capable of creating a runaway electronpopulation [4]. During the flat top phase of a plasma discharge, a starting disruption(SD) might create large enough electric field to generate runaway electrons despitehigh electron density initially at high temperature, while low temperature and highdensity case is relevant for the end of a disruption (ED). Finally, the high temperaturelow density case is relevant for low density discharges (LD) where runaway electrongeneration is studied [5].For all the scenarios all the plasma parameters were kept constant in time, anda step in the electric field time evolution is introduced to observe the response ofthe system. For each case, the electric field value was chosen to have a moderatebut observable runaway electron generation and to avoid the slide-away effect, wherecalculations of
Runaway Fluid would be completely invalid. The magnetic fieldstrength was set to be zero to omit the synchrotron radiation losses on the distributionfunction by
NORSE and
DREAM . Toroidicity effects were also omitted as
NORSE cannot take them into account. The physical parameters for the different scenarios arelisted in Table 2.
Table 2.
The various plasma parameters used for the simulations. The collision timeat the critical velocity was found to be the relevant time scale.
Low densitydischarge (LD) Start–upphase (SU) End ofdisruption (ED) Start ofdisruption (SD)Electric field [V/m] 2 . · − . · − . · − Critical field [V/m] 5 . · − . · − . · − . · − Normalized electric field [-] 5.55 71 52.5 5Coulomb logarithm [-] 19.9 16.3 13.7 17.2Collision time atcritical velocity [s] 2 . · − . · − . · − . · − alidity of models for Dreicer generation of runaway electrons in dynamic scenarios NORSE , is shown in Figure 3 for start–up scenario. The green dashedline shows the boundary of the runaway region calculated by
NORSE based on anestimate by Smith et al. [29]. In the first two plots, the shifting of initial Maxwelliandistribution into a Spitzer–like distribution dominates as the electric field is applied.During this time, the distribution function is still mostly isotropic as it shifts slightlytowards the direction of the accelerating electric field. In later stages of the simulation,shown on the bottom two panels, we can see the formation of the high energy tailconsisting of runaway electrons. a)b)c)d)
Figure 3.
The 2D distribution function throughout the simulation for the start–upcase. The runaway boundary is marked with a dashed line. alidity of models for Dreicer generation of runaway electrons in dynamic scenarios a) b)c) d) Figure 4.
The Dreicer generation rate calculated with
NORSE , Runaway Fluid and
DREAM in the four different cases. The collision time at the critical velocity isindicated on the plots with a vertical dashed line. The dot–dashed lines in the start–up(SU) case on plot (b) indicate the times where the distribution function is shown inFigure 3.
The Dreicer generation rates from
NORSE and
Runaway Fluid are shown in Figure 4for all four scenarios. The plots look qualitatively similar, but the time scales are widelyvarying among the figures. A large peak in the generation rate from
NORSE can beseen in all the four cases. The peak might is caused by the slight but rapid shift anddistortion of the whole Maxwellian into a Spitzer-like distribution, see Figure 3, as theelectric field is introduced. The overlap of the bulk population with the runaway regiongives rise to the large but temporary peak in generation rate, which smooths out asrunaway electrons are accelerated to higher energies. The generation later relaxes to theanalytical generation rate value as the Maxwellian distribution develops a large runawayelectron tail due to Dreicer generation. In the later stages, the runaway generationmechanism is the diffusion of electrons in the velocity space through the critical velocityboundary, as expected, and the kinetic generation rate closely follows the analyticalcalculations in the quasi–stationary limit [8, 9].The same scenarios have also been run with the kinetic solver
DREAM , which usesa test particle collision operator to calculate the Dreicer generation rate. The runawayboundary for the plotted generation rate was defined at | p | = p c = 1 / q E/E c −
1, atwhich point collisional friction balances electric field acceleration for an electron with alidity of models for Dreicer generation of runaway electrons in dynamic scenarios
DREAM generation rate closely follows the analytical value in the later stages. Thepeak is also reproduced with
DREAM although it has a slightly higher maximum than
NORSE , most likely due to the differing runaway region definitions. In cases LD andSD a significant runaway electron fraction is generated. In Figure 4 plots (a) and (d),
DREAM slightly overestimates the generation rate even for later times, which is dueto this decrease in bulk electron density. The fall after the peak does not reach theanalytical value, and also a steady–state is not reached in the simulated time intervals.The interaction of runaway electrons with the bulk population could compensate for theincreased runaway electron generation due to the reduced bulk density, but in DREAMthis is not calculated leading to the increased generation rate. In these two cases,the
NORSE generation rate undershoots the analytical value and approaches it frombelow. NORSE takes the runaway-bulk interaction into account which explains thelower generation rate compared to DREAM.It can be concluded that there is a good qualitative agreement between
NORSE and
DREAM , and the time scale of the evolution of the initial peak is similar. Amore thorough comparison of the two codes is out of scope for the current paper, as theobserved differences do not affect the conclusions.
The runaway electron densities were also plotted as functions of time in all cases, asshown in Figure 5. The generation rate peak observed in the kinetic calculations causes aconstant shift in the runaway density between
NORSE and
Runaway Fluid for casesSU and ED (see plots (b) and (c)) where no undershoot was observed. In LD and SDcases (plots (a) and (d)), where the generation rate temporarily fell below the analyticalgeneration rate before recovering to the asymptotic value, the runaway electron densityfrom
NORSE is slightly lower than the density from
Runaway Fluid on long timescales but the difference is not significant in any of the cases. The relative difference issmaller for longer time scales in all scenarios.We note that the avalanche generation was not considered in the current simulation,which could cause an exponential divergence of the runaway electron density from thetwo codes in case of prescribed plasma parameters. On the other hand, in simulationswith self–evolving plasma parameters one would expect the electric field induced by therunaway current to decrease the differences between the results, but a quantitativestudy of this effect requires self–consistent evolution of plasma parameters with acomprehensive model.
The characteristic time scale governing the Dreicer generation rate is related to theelectron–electron collision time at the critical velocity. The critical velocity of runaway alidity of models for Dreicer generation of runaway electrons in dynamic scenarios a)c) b)d) Figure 5.
The runaway electron density as a function of time calculated by
NORSE and
Runaway Fluid in the four scenarios. electron generation only depends on the electric field normalized to the critical field as v c = c q EE c , (1)where E c is the critical electric field. The velocity dependent electron-electron collisiontime at this velocity can be calculated as τ ee = 4 πε m e v c e n e ln Λ (2)where m e is the mass of the electron, e is the electron charge, n e is the elementarydensity and ln Λ is the Coulomb logarithm [30]. This quantity depends on three plasmaparameters, namely the electron density, temperature and the electric field. The densitydependence appears explicitly and through the Coulomb logarithm in the denominatoras well as through the critical field in the critical velocity. The temperature dependencecomes solely from the Coulomb logarithm and hence it has weaker effect than the densityor electric field. The strength of the electric field relative to the critical electric field willinfluence the collision time through the critical velocity. The explicit dependence of thecollision time on these physical parameters is τ ee = 4 πε m e c e n e ln Λ( EE c ) / . (3)The critical field is dependent on density and temperature as [9] E c = n e e ln Λ4 πε m e c , (4) alidity of models for Dreicer generation of runaway electrons in dynamic scenarios
10 20 30 40 50 60 70
Normalized electric field − − − C o lli s i o n t i m e [ s ] · m − ,
300 eV5 · m − , eV10 m − ,
300 eV10 m − , eV Figure 6.
The dependence of the collision time at the critical velocity on thenormalized electric field. where the temperature dependence is again hidden in the Coulomb logarithm andsubstitution to the collision time yields τ ee ( n, T, E ) = s m e πε n / e (ln Λ) / E / , (5)where the explicit dependence on the plasma parameters can be seen.The relaxation of the generation rate peak happens on different absolute time scalesfor the different scenarios. The simulation time interval was chosen to achieve thesteady–state solution from NORSE in all cases. The time to converge to the steady–state solutions is governed by the distortion of the distribution function at the boundaryof the runaway region, which is driven by the collisions of the electrons. The time scaleof the relaxation was found to be related to the collision time at the critical velocityof runaway electrons, and it was indicated in Figure 4 with a vertical red line. It canbe seen that the peak of the generation rate is already relaxing at the critical velocitycollision time. On longer time scales the kinetic and
Runaway Fluid generation ratesshow good agreement with differences only caused by the varying definitions of therunaway region.The collision time as a function of the normalized electric field is shown in Figure6. Cases with identical densities are indicated with the same colors. The low densityscenarios are plotted in red while the high density in blue. Dashed lines are used forlow temperatures, while solid lines for high temperatures. The temperature has onlya small effect on the collision time and the main dependence is on density and electricfield.In Figure 7 the collision time is plotted as a function of density and electric fieldwith constant temperature. The temperature was chosen to be 10 keV, although it alidity of models for Dreicer generation of runaway electrons in dynamic scenarios Figure 7.
The dependence of the collision time at the critical velocity on the densityand the electric field. does not significantly affect the dependence only the magnitude: the magnitude of thecollision time is changed by a mere factor of about 1 .
4. Conclusion
We have investigated three different models for Dreicer generation: a reduced kinetic, alinear kinetic and a non-linear kinetic. We have studied four different scenarios meantto represent the full experimentally relevant range.
Runaway Fluid uses analyticalformulas to calculate the runaway generation rates.
NORSE and
DREAM were used
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Acknowledgments
This work has been carried out within the framework of the EUROfusion Consortiumand has received funding from the Euratom research and training programme 2014–2018and 2019–2020 under grant agreement No 633053. The views and opinions expressedherein do not necessarily reflect those of the European Commission. G. I. Pokol andS. Olasz acknowledge the support of the National Research, Development and InnovationOffice (NKFIH) Grant FK132134.
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