- Date:22 Nov 2020
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Kinetic modelling of runaway electron dynamics 2, In this contribution we discuss improvements to the model which enable us to study. the effect of hot tail runaway generation on the electron distribution Section 2 which. can be the dominant mechanism in rapidly cooling plasmas as well as an improved model. for the knock on collisions leading to avalanche multiplication of the runaway population. Section 4 This model takes the energy dependence of the runaway distribution into. account We also discuss the implementation of a full linearized collision operator and. demonstrate its conservation properties Section 3, The improvements described in this contribution enable the detailed study of. runaway processes in dynamic situations such as disruptions and the conservative. collision operator makes self consistent calculations of the runaway population and. current evolution in such scenarios feasible 5,2 Time dependent plasma parameters. To be able to investigate the behavior of the electron population in dynamic scenarios. such as disruptions or sawtooth crashes it is necessary to follow the distribution function. as the plasma parameters change To this end CODE has been modified to handle. time dependent background plasma parameters Since the kinetic equation is treated in. linearized form the actual temperature and density of the distribution are determined. by the background Maxwellian used in the formulation of the collision operator This. allows for a scheme where the kinetic equation is normalized to a reference temperature. T and number density n so that the discretized equation can be expressed on a fixed. reference grid in momentum space Throughout this paper we will use a tilde to denote. a reference quantity By changing the properties of the Maxwellian equilibrium around. which the collision operator is linearized plasma parameter evolution can be modelled. on the reference grid without the need for repeated interpolation of the distribution. function to new grids, Analogously to Ref 2 the kinetic equation in 2D momentum space for the electron. distribution function f experiencing a parallel electric field E and collisions can be. expressed as, Hereqwe have introduced a convenient normalized momentum y v v e where.

v e 2T m is the reference electron thermal speed and the cosine of the pitch angle. yk y Using m3 v e3 3 2 n we have also defined the distribution function F f. normalized so that F y 0 1 for a Maxwellian with T T and n n time t ee t. and electric field E eE mv e ee as well as the normalized operators C C ee. and S S ee with ee 16 e4 n ln 3m2 v e3 the reference electron thermal collision. time e m and v the charge rest mass and speed of the electron c the speed of light. and the relativistic mass factor C is the Fokker Planck collision operator and S. an operator describing close large angle Coulomb collisions These operators will be. Kinetic modelling of runaway electron dynamics 3, discussed more thoroughly in Sections 3 and 4 respectively for now we just state the. new formulation of the collision operator employed in Ref 2. cC v e3 y 2 y2 2 F 1 2 2,y x y v e 2x, Here a bar denotes a quantity normalized to its reference value i e v e ve v e. x y v v e is the normalized speed cC 3 ee 4 c Zeff v e2 4 x2 2 Zeff. is the effective ion charge x v e and x v e v e2 v e 1 xd d x v e 2x2. are the error and Chandrasekhar functions respectively and v e c is assumed to be. a small parameter, Changes to the plasma temperature manifest as shifts in the relative magnitude. of the various terms in Eq 2 through and the quantities with a bar as well as a. change in the overall magnitude of the operator whereas changes in density only have. the latter effect In both cases the distribution is effectively colliding with and relaxing. towards a Maxwellian different from the one native to the reference momentum grid. Heat or particles are introduced to or removed from the bulk of the distribution when. using this scheme as all changes to plasma parameters are described by changes to. the Maxwellian This provides a powerful way of simulating rapid cooling for instance. associated with a tokamak disruption,2 1 Hot tail runaway electron generation. If the time scale of the initial thermal quench in a disruption event is short enough. comparable to the collision time the tail of the initial Maxwellian electron distribution. will not have time to equilibrate as the plasma cools The particles in this supra thermal. tail may constitute a powerful source of runaway electrons should a sufficiently strong. electric field develop before they have time to reconnect with the bulk electrons This. process is known as hot tail generation and can be the dominant source of runaways. under certain conditions 6 7 and has previously been investigated analytically or using. computationally expensive Monte Carlo simulations 7 8 9 Using CODE to model a. temperature drop we may study a wider range of scenarios and verify the validity of. the analytical models, Figure 1a compares the runaway density evolution computed with CODE to.

analytical formulas derived in Ref 9 for a typical hot tail scenario The calculations. followed the prescribed temperature evolution shown in Fig 1b and the avalanche source. was excluded The collision operator used in Ref 9 is the non relativistic limit of. Eq 2 with c 0 the distribution is isotropic since there is no electric field see. below CODE results using both this operator and the full Eq 2 are plotted in. Fig 1a with the latter producing 50 more runaways in total This difference. can likely be explained by the relatively high initial temperature 3 keV in the scenario. considered in which case the non relativistic operator is not strictly valid for the highest. energy particles The analytical formulas use different definitions for nr the CODE. Kinetic modelling of runaway electron dynamics 4,10 3 3 1 2. 10 4 2 0 8,CODE rel with E,10 5 CODE non rel,0 6 0 8 1 1 2 0 0 4 0 8 1 2. Figure 1 a Hot tail runaway density obtained using CODE solid with black. and without red yellow an electric field included during the temperature drop and. several analytical formulas dashed for the temperature and E field evolution in b. ED is the Dreicer field and the cited equation numbers refer to Ref 9. results are expected to agree with Eq 19 in Ref 9 and good agreement is indeed. seen for the saturated values in the figure, The evolution of the temperature and electric field are shown in Fig 1b These are. the same as those used in Fig 5 of Ref 9 as are all other parameters The analytical. formulas are derived in the absence of an electric field only an exponential drop in. the bulk temperature is assumed The q electric field shown in Fig 1b is only used to. define a runaway region y yc 1 E Ec 1 with Ec the critical electric field for. runaway generation so that the runaway fraction can be calculated In other words. it is assumed that the electric field does not have time to influence the distribution. significantly during the temperature drop A CODE calculation where the electric. field evolution is properly included in the kinetic equation is also shown in Fig 1a. solid black showing increased runaway production by less than a factor of 2 For the. parameters used the above assumption can thus be considered reasonable. 3 Conservative linearized Fokker Planck collision operator. Treating the runaway electrons as a small perturbation to a Maxwellian distribution. function the Fokker Planck operator for electron electron collisions can be linearized. and written as C f C l f f C tp f1 fM C fp fM f1 where fM denotes a. Maxwellian and f1 f fM the perturbation to it The so called test particle term. C tp describes the perturbation colliding with the bulk of the plasma whereas the field. particle term C fp describes the reaction of the bulk to the perturbation The full. linearized operator C l conserves particles momentum and energy The field particle. term mainly affects the bulk of the plasma and is therefore commonly neglected when. studying runaway electron kinetics however the test particle term alone only ensures. the conservation of particles not momentum or energy. Under certain circumstances it is necessary to use a fully conservative treatment. also for the runaway problem in particular when considering processes where the. conductivity of the plasma is important In the study of runaway dynamics during. Kinetic modelling of runaway electron dynamics 5,Kinetic energy Ek Ek 0. p arb units,0 1 l n 5 1019 m 3,Cnon rel cons,0 50 t0 150 200 250 300 0 50 t0 150 200 250 300.

time ee t time ee t, Figure 2 a Parallel momentum and b energy moments of the distribution function. in CODE using different collision operators Initially E 50 V m and Zeff 1 were. used but for t t0 the electric field was turned off and the ion charge set to Zeff 0. to avoid momentum transfer to the background ions Using two Legendre modes for. the field particle term was sufficient to achieve good conservation. a tokamak disruption using a self consistent treatment of the electrical field accurate. plasma current evolution is essential and the full linearized collision operator must be. used A linearized operator valid for arbitrary particle energy has been formulated. 10 11 The collision operator originally implemented in CODE is the result of an. asymptotic matching between the highly relativistic limit of the test particle term of. that operator with the usual non relativistic test particle operator 12 and is given in. Eq 2 The relativistic field particle term is significantly more complicated however. and its use would be computationally expensive Here we instead implemented the. non relativistic field particle term as formulated in Ref 13 As will be shown this. operator together with the non relativistic limit of Eq 2 accurately reproduces the. Spitzer conductivity for temperatures where the bulk is non relativistic Using the. normalization in Section 2 the operator is,3 2 e v e x 4 2. where G and H are the Rosenbluth potentials obtained from the distribution using. v e2 v2 H 4 F v e2 v2 G 2H 4, The system of equations composed of Eqs 3 4 together with the non relativistic limits. of Eqs 1 2 y x and 0 is discretized see Ref 2 and solved using an efficient. method described in Ref 14 The inclusion of the field particle term introduces a full. block for each Legendre mode into the normally sparse matrix describing the system. however since only a few modes are required to accurately describe the Rosenbluth. potentials the additional computational cost is modest. The conservation properties of the full non relativistic collision operator as well as. the relativistic test particle operator in Eq 2 are shown in Fig 2 An electric field was. initially used to supply some momentum and energy to the distribution As expected. the full operator conserves energy and momentum in a pure electron plasma Zeff 0. Kinetic modelling of runaway electron dynamics 6,Cnon rel E Ec. 0 10 20 30 0 10 20 30,time ee t time ee t, Figure 3 a Conductivity normalized to the Spitzer value and b runaway density.

for different collision operators and E field strengths considering only Dreicer runaway. generation The parameters T 1 keV n 5 1019 m 3 and Zeff 1 were used. after the electric field is turned off at t t0 100 collision times whereas the operator in. Eq 2 does not The electric field continuously does work on the distribution a large. part of which heats the bulk electron population but the linearization of the collision. operator breaks down if the distribution deviates too far from the equilibrium solution. As long as a non vanishing electric field is used together with an energy conserving. collision operator an adaptive sink term removing excess heat from the bulk of the. distribution must be included in Eq 1 to guarantee a stable solution Physically this. accounts for loss processes that are not properly modelled such as line radiation and. radial heat transport The magnitude of the black line in Fig 2b therefore reflects the. energy content of the runaway population not of the total distribution The sink term. is not included for t t0 since E 0 and the energy conservation observed is due to. the properties of the collision operator itself, Figure 3 demonstrates that CODE reproduces the expected Spitzer conductivity. S for moderate electric field strengths if the conservative collision operator is used and. the initial Maxwellian adapts to the applied electric field on a time scale of roughly 10. collision times For field strength significantly larger than Ec the conductivity starts to. deviate from S as a runaway tail begins to form Fig 3b in this regime the analytical. calculation is no longer valid Using the collision operator in Eq 2 consistently leads. to a lower conductivity by about a factor of 2 as expected The runaway growth is. also affected with the conserving operator leading to a larger runaway growth rate. 4 Improved operator for knock on collision, The Fokker Planck collision operators discussed in Section 3 accurately describe grazing. collisions small angle deflections which make up the absolute majority of particle. interactions in the plasmas we consider Large angle collisions are usually neglected as. their cross section is significantly smaller but in the presence of runaway electrons they. can play an important role in the momentum space dynamics as an existing runaway. Kinetic modelling of runaway electron dynamics A Stahl 1 O Embr eus E Hirvijoki G Papp2 M Landreman3 I Pusztai1 and T F ul op 1 1 Department of Applied Physics Chalmers University of Technology G oteborg Sweden 2 Max Planck Institute for Plasma Physics Garching Germany 3 University of Maryland College Park MD USA E mail stahla chalmers se

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