Theory of Runaway Electrons in ITER: Equations, Important Parameters, and Implications for Mitigation Allen H

Theory of runaway electrons in ITER: Equations, important parameters, and implications for mitigation Allen H. Boozer

Citation: Physics of Plasmas (1994-present) 22, 032504 (2015); doi: 10.1063/1.4913582 View online: http://dx.doi.org/10.1063/1.4913582 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/22/3?ver=pdfcov Published by the AIP Publishing

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Theory of runaway electrons in ITER: Equations, important parameters, and implications for mitigation Allen H. Boozera) Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA (Received 17 December 2014; accepted 13 February 2015; published online 10 March 2015) The plasma current in ITER cannot be allowed to transfer from thermal to relativistic electron carriers. The potential for damage is too great. Before the final design is chosen for the mitigation system to prevent such a transfer, it is important that the parameters that control the physics be understood. Equations that determine these parameters and their characteristic values are derived. The mitigation benefits of the injection of impurities with the highest possible atomic number Z and the slowing plasma cooling during halo current mitigation to տ40 ms in ITER are discussed. The highest possible Z increases the poloidal flux consumption required for each e-fold in the number of relativistic electrons and reduces the number of high energy seed electrons from which exponentia- tion builds. Slow cooling of the plasma during halo current mitigation also reduces the electron seed. Existing experiments could test physics elements required for mitigation but cannot carry out an integrated demonstration. ITER itself cannot carry out an integrated demonstration without excessive danger of damage unless the probability of successful mitigation is extremely high. The probability of success depends on the reliability of the theory. Equations required for a reliable Monte Carlo simulation are derived. VC 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4913582]

I. INTRODUCTION the current of relativistic electrons if the transfer does occur: Any phenomenon that causes a dissipation of the plasma current in ITER on a time scale faster than tens of seconds • The local loop voltage V‘. can result in a large fraction of the plasma current being The local loop voltage4 is the slippage of the poloidal carried by relativistic runaway electrons. The effect and its magnetic flux outside of a surface that contains a given toroi- importance to ITER were pointed out by Fleischmann and 1 2 dal flux, V‘ ð@wp=@tÞ , and is equal to the average value co-workers in 1993. Rosenbluth and Putvinski published Ð wt of g~j d~‘ per toroidal circuit, where d~‘ is the differential more detailed calculations in 1997. distance along a magnetic field line. A successful ITER program implies a proof-of-principle As discussed in Sec. II, Minimum loop voltage for run- demonstration that a tokamak can be routinely operated aways, the drag force on background electrons exceeds the above a plasma current, Ip տ 5 MA, at which damage can become severe from the formation of a large runaway current acceleration of the parallel electric field in ITER for electrons of all energies when V‘Շ3nb Volts, where nb is the of relativistic electrons. For example, the transfer of the cur- 20 3 rent to relativistic electrons generally results in a significant background electron density in units of 10 =m . In many change in the current profile, which can drive resistive tear- ITER plasma scenarios, nb 1. When the loop voltage is ing modes with the resistivity determined by the parameters larger, it is potentially possible for the parallel electric field of the cold background plasma, not the relativistic electrons,3 to accelerate to arbitrarily high energies electrons that have 2 2 2 Sec. VIII. The loss of magnetic surfaces through tearing an initial kinetic energy ðc 1Þmec with 3nbc =ðc 1Þ modes could result in a sudden release of the relativistic Volts > V‘. Thesepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are called runaway electrons. The Lorentz electrons from the plasma to the chamber walls. factor is c 1= 1 v2=c2 with v the electron speed and c The required runaway avoidance and mitigation techni- the speed of light. ques cannot be developed empirically while operating at a • The poloidal flux change w required for an e-fold in high current. The danger to ITER is too great. Theory is efold the strength of the relativistic-electron current. required. This paper derives required equations, determines the critical parameters, and discusses the implications for the As discussed in Sec. III, Avalanche effect, the dominant mitigation of runaway electrons on ITER. effect of Coulomb scattering for most plasma phenomena is given by the cumulative effect of many small changes. A. Critical parameters Although the probability is smaller by the Coulomb loga- Four parameters determine whether a current transfer rithm lnK, an energetic electron in a single collision can to relativistic electrons will occur and the magnitude of elevate a cold electron to a sufficiently high energy to run away. The effect is an exponential increase in the number of a)[email protected] runaway electrons, which is called a runaway avalanche.

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When the loop voltage is significantly greater than the • The time sR required for the runaway distribution to reach voltage required for runaway, an initial current carried by a steady-state form. runaway electrons will e-fold each time the poloidal flux out- Section VII, Power balance for runaways shows that after side of a magnetic surface that contains a given toroidal flux a relaxation time s ¼ 2ðc 1Þs ln K the distribution func- changes by w տ w . Where the required poloidal flux R r c efold c tion for runaways assumes a steady-state form though expo- change for an e-fold under idealized assumptions, such as no nentially increasing in magnitude. The effective kinetic scattering,1,2 is w 2:3V s for ITER. c energy required for runaway is K ¼ðc 1Þm c2. Avalanche The magnitude of w is inversely proportional to the r r e efold theory is simplified when the runaway distribution assumes a minimum effective energy for runaway, which is sensitive to steady state form early in the process. Approximations that the pitch-angle distribution of newly energized electrons, were made by Rosenbluth and Putvinski2 are not valid for which is itself dependent on the pitch-angle distribution of times short compared to s . the runaway distribution as a whole. The pitch-angle distri- R 2 bution used by Rosenbluth and Putvinski and by later • The time sT required for the transfer of the plasma current authors is based on the assumption that the runaway elec- to relativistic electrons. trons are moving in perfect alignment with the magnetic The time to transfer the current, which is given by field. Section III derives the pitch and energy distribution of sT ¼ðrswe-foldÞ=V‘, is in large part determined by the resis- newly energized electrons for an arbitrary pitch and energy tivity of the cold background plasma, which determines the distribution of the runaways and explains why the loop voltage V‘. That is, until a large fraction of the total Rosenbluth-Putvinski approximations may not have plasma current is carried by runaway electrons, the develop- adequate accuracy. ment of the runaway current is a passive process, which is • The ratio ers of the initial number of electrons above the determined by the properties of the cold background plasma. critical energy for runaway, called seed electrons, to the The transfer time sT need not be long compared to the time number of relativistic electrons required to carry the full sR required for the runaway distribution to reach a steady- plasma current. state form.

As discussed in Sec. IV, Control of avalanche seed, the C. Effects modifying critical parameters obvious source of seed electrons is the tail of the pre-thermal-quench Maxwellian. This source, as well as seed Strategies to avoid the transfer of plasma current to rela- electrons from electron current drive, can be reduced to tivistic electrons must be based on modifications to the criti- insignificance if the cooling is sufficiently slow տ40 ms. cal parameters. Several effects either can or potentially could Other sources for the required electron seed are found to be produce such modifications: far from obvious. • Pitch-angle scattering

• The initial poloidal magnetic flux w0 between the mag- The addition of a species with a high atomic-number Z netic axis and the chamber walls. to the background plasma has important beneficial effects by The poloidal flux consumption required to transfer the both increasing Kr, the kinetic energy required for runaway, and wefold, the poloidal flux change required per e-fold. plasma current to relativistic electrons is rswefold. Assuming the chamber walls are perfectly conducting on the This is discussed in Sec. V, Kinetic equation and Coulomb scattering. time scale of the current transfer, the remaining poloidal flux

inside the plasma chamber is wr w0 rswefold. The rela- • Direct losses of energetic electrons tivistic current is I ¼ w =L , where L is the inductance of r r r r When high energy electrons have a characteristic loss the plasma current in the profile that it assumes after the time s that is shorter than w =V , the avalanche mecha- transfer. Appendix A, Magnetic flux changes during run- ‘ efold ‘ nism becomes sub-dominant and the transfer of current to away acceleration, shows that L l R. Generally it is r 0 relativistic electrons cannot occur. There are two processes thought that up to about 3 MA of runaway current could be that could cause a rapid loss: (1) The break up of magnetic safely handled on ITER, which corresponds to w 20 V s. r surfaces can allow high energy electrons to rapidly leave the plasma by moving along the magnetic field lines.5 This B. Important time scales effect is discussed below under mitigation strategies. (2) Studies of the development of a runaway current must Pitch-angle scattering can place electrons into trapped consider three time scales: particle trajectories, which can be poorly confined in a non- axisymmetric plasma, Sec. V. • The characteristic time sc required for a high energy electron to slow to a thermal energy. • Synchrotron radiation As discussed in Sec. II, the time it takes a relativistic Highly relativistic electrons can lose power more rapidly 2 electron with kinetic energy ðc 1Þmec to slow to thermal through synchrotron radiation than from Coulomb drag on energy of the background plasma is ss ¼ðc 1Þsc, where the background plasma. Section VI, Synchrotron radiation, sc 22:7ms=nb. Methods of reducing the number of seed derives the effective collision operator for electrons due to electrons must operate on this time scale to be effective. radiative losses. The derivation has subtleties that have not

been adequately discussed in the literature. Generally, syn- the change in poloidal flux required for an e-fold in the num- chrotron radiation should not greatly modify the transfer of ber of runaways; and (3) reduces the radiative loss for a current to runaway electrons in ITER though it can be im- given electron drag. Deeply trapped electrons in weakly ion- portant as a diagnostic tool.6,28 ized high-Z atoms contribute to runaway drag but have a negligible radiative power loss. • Microturbulence A potential disadvantage of high-Z impurities is that the Microturbulence could in principle enhance pitch-angle average energy of the runaway electrons is increased, but scattering and energy loss—both are beneficial. Nevertheless, this disadvantage is of little importance if the runaway cur- as discussed in Sec. IX, Microturbulence, existing theories do rent is reduced exponentially in magnitude. not give cause for optimism that this will occur. Since the • Ensure the magnetic surfaces are destroyed whenever the effects of microturbulence make mitigation easier if they arise, ignoring the effects of microturbulence in a theory of loop voltage is large. mitigation is a reasonable conservative assumption. Magnetic surfaces may be destroyed during rapid thermal quenches. But even a narrow annulus of magnetic surfa- D. Mitigation strategies ces can confine relativistic electrons in ITER due to their Not only will any naturally arising process that causes a small gyroradius. It is neither obvious that all magnetic sur- rapid cooling of the plasma tend to cause a transfer of the face are destroyed nor that they remain destroyed. It may be plasma current to relativistic electrons but also the halo- possible to design tokamaks, so the induction current in the current mitigation system will drive this transfer. For halo- walls that occurs when the plasma current rapidly drops current mitigation, the plasma current should be ramped would ensure the required breakup of surfaces, Sec. X. This 7,8 down significantly faster than the wall time, which in ITER was not done for ITER, and the only theoretical studies Շ have used highly simplified models. with w0 100V s implies w0=V‘ 150 ms and a loop voltage V‘ տ 700 V. This voltage is far above the critical voltage for runaway unless the background density is increased 100’s of times, which may not be possible and pro- II. MINIMUM LOOP VOLTAGE FOR RUNAWAYS duces problems of its own. The parallel electric field, E , must be sufficiently large The operational lifetime of ITER could be set by the jj for runaway electrons to persist. The lowest value for this E effectiveness of the runaway-mitigation system. Several jj is called the critical electric field, E , which is calculated strategies for improving the runaway-mitigation systems are c assuming that the only non-ideal effect on high-energy elec- possible and two could probably be implemented on ITER. trons is the Coulomb drag of the cold background electrons. • The use of slow plasma cooling, 40 ms, for halo Ec is not an electric field; eEc is the minimum drag force mitigation. exerted on an electron by the background electrons. The expression for E is given in Eq. (12) of Connor and Hastie9 Slow cooling gives adequate time for the high-energy c electrons to slow, which allows a Maxwellian distribution to 3 3 4pnbe ln K nbe ln K be maintained as the plasma cools and eliminates the large Ec ¼ ¼ 0:075nb (1) m c2 4p2m c2 number of high energy electrons present during lower hybrid e 0 e current drive. Rapid cooling produces a non-Maxwellian tail in Gaussian and in Standard International units, where lnK is that serves as a runaway seed. the Coulomb logarithm. The number density of background The plasma cooling time should not be confused with 20 3 electrons nb has units of 10 /m when the critical electric the time scale for the plasma current decay. The rate of cur- field has units of Volts per meter. A typical density for pro- rent decay is determined by the electron temperature—not jected ITER operations is nb 1. The drag force on an elec- the rate of cooling—through the plasma resistivity. It may be tron that has a velocity v is beneficial to preemptively cool plasmas that are in high danger of disruption to avoid the rapid plasma cooling of a natu- c2 c2 f ¼ eE ¼ eE : (2) ral thermal quench. drag v2 c c2 1 c Existing experiments could test and demonstrate some of the benefits of slow cooling, but the time required for Only relativistic electrons experience the minimum force. magnetic field penetration through the surrounding walls and In addition to drag, Coulomb collisions with the back- plasma densities differ between existing experiments from ground plasma cause pitch angle scattering, which will be those in ITER in a way that complicates a comprehensive discussed in Sec. V. Section V will also give the relativisti- test. cally correct collision operator for high-energy electrons colliding with a cold background plasma from which the • The use of higher atomic number, Z, impurities for run- c2=v2 factor in the drag force is derived. away mitigation, the higher the Z the better. No electrons can runaway unless the component of the The enhanced pitch-angle scattering of the high-energy electric field along the magnetic field lines satisfies Ejj > Ec. electrons: (1) increases the energy required for runaway, As demonstrated in Appendix A, this is equivalent to the which reduces the number of seed electrons; (2) increases requirement that the local loop voltage V‘ satisfies

R element in electron runaway in the presence of a large V‘ Vc 2pREc 3nb Volts; (3) RITER change in the poloidal magnetic flux Dwp. The importance of a single scattering event converting a cold into a runaway where R is an average major radius of a magnetic surface electron to ITER was pointed out by Fleischmann and and RITER is the major radius of ITER, 6.2 m. co-workers,1 but the effect is usually called the Rosenbluth 4 The local loop voltage is the rate of change of the avalanche due to the more detailed treatment of Rosenbluth poloidal magnetic flux wp outside of a surface that contains a and Putvinski.2 As will be discussed in this section, this kind given toroidal flux, wt of scattering can cause the number of runaways to increase as expðjDw j=w Þ, where w տ w 2:3V sina @w p efold efold c V p : (4) tokamak the size of ITER. The exponential increase is called ‘ @t wt the runaway avalanche. The source function for newly energized electrons When magnetic surfaces do not exist, the relevant loop volt- Sðp; kÞ is defined so, if that were the only term, the kinetic age is an average over the volume covered by a single mag- equation for energetic electrons would be @f =@t ¼ S. The netic field line, Eq. (A12). source function Sðp; kÞ for a newly energized electron of The quantity of poloidal flux enclosed by the magnetic momentum p and pitch k is the result of a collision of an axis and available to produce runaways depends on the initial electron of negligible energy with an energetic electron that current profile and whether the flux outside the conducting had momentum pe and pitch ke before the collision. That is, structures that surround the plasma can penetrate those struc- ð tures on the time required to exacerbate the runaway. 2 Nevertheless as discussed in Appendix A, the available flux SðÞpe; ke; p; k fpðÞe; ke 2ppedpedk SpðÞ; k ¼ : (8) is approximately l0RIp ¼ 117 V s when the plasma current 2pp2 is 15 MA in ITER. Consequently, runaways cannot be produced in ITER when the current changes on a time scale The function Sðpe; ke; p; kÞ gives rate of change in the num- longer than l0RIp=Vc 40 s=nb. Stated differently, the pos- berofelectronswithinthe differential distance dpdk of the sibility of runaway production must be considered when the momentum space position ðp; kÞ due to collisions with current in ITER changes rapidly compared to a 40 s time electrons that were within the differential distance dpedke scale. of the momentum space position ðpe; keÞ before the The background electrons drag down the kinetic energy collision. of relativistic electrons, which means c –1 1, on the time The momentum p, the pitch k cos #, and the gyrophase scale up are spherical momentum coordinates. They are related to Cartesian momentum coordinates by px ¼ p sin # cos up; s ¼ðc 1Þs ; (5) s c py ¼ p sin # sin up,andpz ¼ p cos #. The phase angle up, which advances at the rapid gyrofrequency, will be averaged where over. After averaging, a momentum-space position is specified m c 22:7ms by ðp; kÞ, and the momentum-space p volume element is s e : (6) c eE n c b d3p ¼ 2pp2dpdk: (9) When the loop voltage V is larger than V , the drag on pffiffiffiffiffiffiffiffiffiffiffiffiffi ‘ c By the definition of spherical coordinates, sin # 1 k2 background electrons, Eq. (2), assures that electrons with ¼ 2 2 2 is always positive. kinetic energy less that ðcc 1Þmec with ðcc 1Þ=cc ¼ The function Sðpe; ke; p; kÞ is determined by Vc=V‘ cannot runaway. For V ‘ V c S0ðpe; ke; p; kÞ, which gives the rate at which an electron Vc with momentum and pitch ðpe; keÞ takes an electron with cc 1 ¼ : (7) 2V‘ negligible kinetic energy to a momentum and pitch ðp; kÞ in a single collision. While doing this, energy and momentum conservation implies the phase-space position of the ener- III. AVALANCHE EFFECT getic electron is changed to ðpn; knÞ, where pn and kn are The standard Coulomb collision operator for electrons functions of ðpe; ke; p; kÞ,so gives the diffusive effect of a large number small changes in S¼ð1 dðp peÞdðk keÞ the momentum of an electron due scattering from the rest of d p p d k k p ; k ; p; k : (10) the plasma. The cross section for electron-electron scatter- þ ð nÞ ð nÞÞS0ð e e Þ ing, the Møller cross section10,11 when quantum and relativ- Energy conservation implies c ðpnÞ¼1 þ c ðpeÞcðpÞ istic effects are included, is singular for small changes in the n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie 2 electron momentum. The resolution of this singularity gives with the Lorentz factor written as cðpÞ¼ 1 þðp=mecÞ . the Coulomb logarithm that appears in Ec. Momentum conservation implies knpn ¼ kepe kp. These Finite changes in the momentum of an electron due to are solvable equations for ðpn; knÞ as functions of Møllor scattering are subdominant by a factor of 1=lnK in ðpe; ke; p; kÞ. Momentum conservation in the two directions most collisional processes in plasmas but are a critical orthogonal to the magnetic field is irrelevant due to the

average over the rapid gyromotion. S0ðpe; ke; p; kÞ is deter- that the binding energies are small compared to the energy mined by two functions, required for runaway.

drMðÞpe; p S ðÞp ; k ; p; k ¼ n v PðÞp ; k ; p; k ; (11) A. Approximate S of Rosenbluth-Putvinski 0 e e b e dp e e Rosenbluth and Putvinski2 gave an approximate expres-

where nb is the number of cold background electrons. sion for S, which is the standard expression in the literature The first function that appears in S0ðpe; ke; p; kÞ is the on the avalanche effect. They made three approximations: rate per energetic electron at which collisions occur that (1) the fractional transfer of kinetic energy is small, 1, result in the energization of an electron to a momentum in (2) the energetic electron is relativistic, ce –1 1, and (3) the range dp about p. This rate is given by vedrM=dp, where the momentum of energetic electron is well aligned with the magnetic field, k 1. ve ¼ pe=ceme is the speed of the energetic electron and drM j ej! is the differential Møllor cross section in the form given by When the fractional energy transfer is small, ! 0, the Ashkin et al.11 change in the energy and momentum of the energetic electron 2 is negligible, so S!S0. In addition, the x term dominates 2 2 ce drM. That and the second approximation, ce –1 1, imply drMðÞce; ¼ 2pro 2 ðÞce 1 ðÞce þ 1 ! eEcc dKs 2 v dr ¼ ; (18) c 1 e M 2n ln K K2 x2 3x þ e ðÞ1 þ x d; (12) b s ce 2 where Ks ¼ðc 1Þmec is the kinetic energy of the newly 1 energized electron. In the relativistic limit c –1 1, both x ; (13) e ðÞ1 vedrM=dp and P are independent of the momentum of the c 1 energetic electron, pe The third approximation, jkej!1, : (14) which is equivalent to the limit k2 ! 0, implies ce 1 Pðpe; ke; p; kÞ¼dðk k1Þ, a delta function about k1, which becomes independent of c in the limit c 1 1. The classical radius of an electron, ro, is defined by e e 2 2 2 2 The three Rosenbluth-Putvinski approximations make e =ð4p0roÞ¼mec . That is, 2pr0 ¼ eEc=ð2mec nb ln KÞ. The Lorentz factors are c for the newly energized electron the source function Sðp; kÞ proportional to the number density of runaway electrons, independent of the details of after the collision and ce for the energetic electron before the collision. The fractional transfer of kinetic energy is . the distribution function f ðp; kÞ, which simplifies calculations. The number of runaway electrons n increases as The second function that appears in S0ðpe; ke; p; kÞ is Ð r S2pp2dpdk,or the probability Pðp; k; pe; keÞ that the newly energized electron will have a pitch in the range dk after the collision given d ln nr ecEc 1 the momentum p and pitch k of the energetic electron ¼ ; (19) e e dt 2lnK K before the collision and the momentum p of the energized r Ðelectron after the collision. Since P is a probability, where K is the effective kinetic energy required for 1 r 1 Pdk ¼ 1. Section III B shows that four-momentum con- runaway. servation implies 1. Effective energy for runaway 1 PðÞ¼pe; ke; p; k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (15) 2 2 The primary subtlety in Eq. (19) for runaway exponen- p k2 ðÞk k1 tiation is the appropriate value for the effective kinetic 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy for runaway Kr ¼ðcr 1Þmec . The electric-field ðÞce þ 1 ðÞc 1 acceleration exceeds Coulomb drag for non-relativistic elec- k1ðÞp; pe; ke ke; (16) ðÞce 1 ðÞc þ 1 trons with a velocity v perfectly aligned with the magnetic 2 1 field when ðv=cÞ > Ec=Ejj. Nevertheless, Jayakumar et al. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 used v =c ¼ 2Ec=Ejj as the non-relativistic runaway condi- 2ðÞce c 2 2 k2ðÞp; pe; ke 1 ke ; (17) tion as did Rosenbluth and Putvinski in their Eq. (6). As rec- ðÞce 1 ðÞc þ 1 ognized by Jayakumar et al., the precise determination of the effect K for use in Eq. (19) has subtleties and is changed by when k k k . Otherwise, P p ; k ; p; k 0. r j 1j 2 ð e e Þ¼ pitch-angle scattering. Here a formula equivalent to Eq. (7) When a significant fraction of the background electrons of Rosenbluth and Putvinski2 will be used when pitch-angle are bound in partially ionized ions, the binding energy and scattering is ignored the momentum of the bound electrons must be taken into ÀÁ account unless either their binding energy is small compared d ln nr eEjj Ec V‘ Vc to kinetic energies required for runaway or the binding ¼ ¼ ; (20) dt 2mec ln K w energy is large compared to the energy of the energetic elec- c trons. A Thomas-Fermi model of ions would give a reasona- mec R wc 4pR ln K 2:32 V s: (21) ble approximation, but here it will be assumed for simplicity e RITER

Energy conservation, Sec. VII A, implies that, even 2. Limitations of the Rosenbluth-Putvinski though exponentially increasing in magnitude, the distribu- approximation tion function for the energetic electrons relaxes to a steady- Although the assumptions that give the Rosenbluth- state form with an average Lorentz factor Putvinski approximation to S appear to hold in a number of c 1 2lnK (22) situations, the approximation may not give an accurate expression for the poloidal flux change required to transfer 2 when jjj enrc, a result given by Rosenbluth and Putvinski. the plasma current into a relativistic runaway current. This expression for c is consistent with the first two of the As discussed in Sec. VII, a long time may be required 2 Rosenbluth-Putvinski assumptions, ðc 1Þmec Kr for the runaway distribution to reach its steady, though and c 1 1. Under many conditions the average pitch exponentially increasing, form. During this period, the angle of the runaways is small, jd#j1=ð2lnKÞ, Eq. (84), Rosenbluth-Putvinski assumptions are not generally valid, which is consistent with the third Rosenbluth-Putvinski and a large error can result in the poloidal flux change assumption. required to transfer the current from near-thermal to runaway When c 1 Kr, negligible energy is required to cre- electrons. During the early phases of the avalanche, a large ate a new runaway, but significant energy is required to fractional change in the energy of the energetic electrons accelerate the newly created runaway electron from cr to c. may be required to produce additional runaways by the ava- The required poloidal flux change wc to obtain an e-fold is lanche mechanism and the typical runaway electron may not given by flux change required to bring the energy of a newly be strongly relativistic. energized electron to the average energy, c 1 2lnK,of Under many conditions the average pitch angle of the the runaway distribution, which can be calculated using the runaways is small, jd#j1=ð2lnKÞ, Eq. (84), as assumed conservation of toroidal canonical momentum by Rosenbluth and Putvinski. However, a newly energized

Pu ¼ cRmevu þ ewp=2p. electron, when non-relativistic, moves almost perpendicular When the loop voltage is large compared to the critical to the momentum of the energetic electron that produced it, voltage, V‘ Vc, Eq. (20) predicts the density of runaway which means almost perpendicular to the magnetic field electrons nr increases from its initial, or seed, value ns as lines. Such electrons gain energy only slowly from the paral- ! lel electric field and are prone to magnetic trapping, which jDwpj prevents gaining energy from the parallel electric field. The nr ¼ ns exp ; (23) Rosenbluth-Putvinski approximation is that Pðp ; k ; p; kÞ wc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e e dðk k1Þ with k1 ðc 1Þ=ðc þ 1Þ. When the newly energized electron is not relativistic, its pitch is jkj¼v=2c. where jDwpj is the change in the poloidal magnetic flux outside a surface that encloses a given amount of toroidal mag- The spread in pitch angle of subrelativistic newly energized netic flux. electrons about k ¼ k1 is given by k2 d#e, where d#e is The transfer from an Ohmic current carried by thermal the spread in pitch of the energetic electrons. Especially in electrons to runaway current is on a time-scale set by the the early stages of runaway, d#e need not be small. The effective kinetic energy of runaway K is determined by the Ohmic loop voltage. When jDwpjreq is the change in the r poloidal flux required to obtain necessary number of e-folds, behavior of newly energized electrons, and the required flux the required time for the transfer is change to transfer the current is inversely proportional to Kr.

jDwpjreq B. Scattering probability Pðpe; ke; p; kÞ sT ¼ : (24) V‘ Electron-electron scattering is energy and momentum The electric field drops to critical field for producing conserving. When two electrons collide, one with a far larger runaways when the full plasma current is carried by kinetic energy than the other, energy and momentum conser- runaways. vation determine the probability Pðpe; ke; p; kÞ that the elec- Although c 1 2lnK, the maximum kinetic energy tron that is energized will have a pitch in the range dk after 2 the collision given the momentum pe and pitch ke of the ðcmax 1Þmec that an electron can reach is much higher. Using results for Appendix A, one can show c ¼ ejDw j= energetic electron before the collision and the momentum p max p of the energized electron after the collision. ð2pRmecÞ¼cjDwpj=wc. Any uncertainty in the effective energy for runaway K r 1. Scattering angle translates into an uncertainty in the average kinetic energy of 2 runaway electrons ðc 1Þmec , Eq. (82), and hence the An important element in the derivation of P is the angle poloidal flux change wefold required for an e-fold in the hd between the velocity vector of secondary electron and the number of runaway electrons. An uncertainty in wefold velocity vector of the high energy electron that was the other implies an uncertainty in the poloidal flux jDwpjreq that must participant in the collision. be consumed to transfer the current from thermal to relativis- The properties of the Lorentz transformation imply the tic electrons, and therefore the magnitude of the current of angle hd is uniquely determined by the kinetic energy, 2 relativistic electrons. The calculation of of Kr requires an Ks ¼ðc 1Þmec , imparted to the secondary electron. This accurate source function for newly energized electrons. section will use these properties to show the secondary

electrons move on the surface of a cone oriented around the the direction of the velocity of the newly energized electron velocity direction of the high energy electron with the open- in the lab frame. ing angle of the cone hd given by Eq. (31) In the lab frame in whichP the newly energized electron l When the fractional energy transfer is small and the had been stationary, q ¼ lKlqf with the sign of vf in secondary electron is sub-relativistic, c –1 1, Eq. (31) Lorentz transformation matrix, Eq. (25), changed. The time- implies the secondary electron moves in an essentially 0 like component q ¼ cmec is given by the Lorentz orthogonal direction to the direction of the high energy elec- transformation tron. When the secondary electron is itself strongly relativistic, it moves in essentially the same direction as the high v2 q0 ¼ 1 f cos h c2m c (29) energy electron. c2 c f e In relativity theory, the four momentum of an electron is l p ¼ðc;~vÞcme and pl ¼ðc;~vÞcme. The index l, which which can be used to determine cos hc in terms of c, cf, and runs from zero to three, numbers the components. The time- vf =c. When this relation is used in the Lorentz transformed x^ 1 like component of a four vector changes sign when the index component in the lab frame, qx q , the result is l is changed from a superscript to a subscript. The dot prod- v v uct of any two four vectors—one must be superscript and the f f 2 c 1 P qx ¼ cos hc cf mec ¼ mec: (30) l c c vf =c other in subscript form—is a Lorentz invariant, so lp pl ¼ðc2 v2Þc2m2 ¼m2c2, which implies c2 ¼ 1= e e The form of the Lorentz factor implies square of the three ð1 v2=c2Þ, the well known expression for the relativistic momentum is q2 c2 1 m c 2. The angle h between Lorentz factor. ¼ð Þð e Þ d the momentum of newly energized electron and the momen- When the frame of reference is changed so the new 2 2 2 tum of the energetic electron is defined by cos hd qx=q . frameismovingwithavelocityvf x^ relative to the origi- nal frame, the four momentum, or any other four vector, in Using Eqs. (26) and (30), new frame p is related to the four vector in the old frame P f 2 ce þ 1 c 1 l cos hd ¼ : (31) by pf ¼ lKlp , where the Lorentz transformation matrix c 1 c þ 1 is e 0 1 vf B cf cf 00C 2. Pitch distribution B c C B vf C B C This section will derive the pitch distribution of elec- K ¼ B cf cf 00C (25) l B c C trons that are energized by the collisions of an energetic elec- @ A 0010 tron that is moving in the x^ direction. When #e is the pitch 0001 angle of the fast electron relative to the magnetic field and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ue the gyrophase, the magnetic field direction is and c ¼ 1= 1 v2=c2. f f ^ When an energetic electron with four momentum b ¼ x^cos #e y^sin #e cos ue þ z^sin #e sin ue: (32) l p ¼ðc; vex^Þceme strikes a background electron with four l The newly energized, or secondary, electron that results momentum q ¼ðc;~0Þme, the center-of-momentum frame is 1 1 from the collision is moving in the direction defined so pf þ qf ¼ 0. That is, the x^ component of the sum of the two momenta vanishes. The Lorentz transformed 1 1 s^ ¼ x^cos hd þ y^sin hd cos ud þ z^sin hd sin ud; (33) momenta are pf ¼ðve vf Þcf cme and qf ¼vf cf me,so ðve vf Þc vf ¼ 0. The speed and the Lorentz factor of the where hd is the angle between its momentum and that of the center-of-momentum frame are determined by the Lorentz energetic electron. factor, ce, of the energetic electron The pitch of the newly energized secondary electron is k s^ b^,so v 2 c 1 f ¼ e ; (26) c ce þ 1 k ¼ cos hd cos #e þ sin hd sin #e cos ua; (34) rffiffiffiffiffiffiffiffiffiffiffiffiffi where u ðu þ u Þ is an arbitrary angle since it ce þ 1 a e d cf ¼ : (27) depends on the arbitrary pitch angles. Equation (31) gives 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos hd; sin hd ¼þ 1 cos hd, and ke cos #e. The pitch A bared ql denotes the post-collision four momentum of angle of the secondary electron is then the newly energized electron. In the center-of-momentum frame k ¼ k1 þ k2 cos ua; (35) v v where k1 is given by Eq. (16), k2 is given by Eq. (17), and u ql ¼ 1; f cos h x^; f sin h y^; 0 c m c; (28) a f c c c c f e has an equal probability of having any value in the range 2p ua > 0. where hc is the angle through which the electron is scattered The probability that a newly energized electron has a in that frame. Note hc is distinct from the angle hd that gives pitch in a certain range is proportional to the range of ua

over which it can have this value of k, and dua=dk for producing a dangerous runaway current is about twenty 21 ¼1=ðk2 sin uaÞ. The pitch k has a maximum and a mini- three, which means about Ns > e seed electrons are 53 Ðmum value since it must lie in the range jk k1jk2. Since required. The total number of electrons in ITER is Nb e , kmax ðdu =dkÞdk ¼p, the probability that a new energized so at 10 MA a dangerous runaway requires kmin a electron will have a pitch k is given by Eq. (15). Ns ns ers տ e32: (37) IV. CONTROL OF AVALANCHE SEED Nb nb

The avalanche mechanism can amplify the number of run- B. Hot-Maxwellian as avalanche seed away electrons by expðjDwpj=wcÞ. Nevertheless, a certain number of seed runaways are required to initiate an avalanche. The most obvious source of seed electrons for an ava- This section considers the number of seed runaways that are lanche is the remaining tail of the high temperature, Th, required and how the high energy electrons that were part of a Maxwellian distribution of the pre-thermal-quench plasma. Maxwellian distribution before a thermal quench can be pre- When the cooling time scool > sc 23ms=nb vented from forming that seed. When the thermal quench is on 2 mec scool a time scale longer than 40 ms, it will be shown that pre-ther- ns nb exp : (38) mal-quench energetic electrons do not provide a seed. A simi- Th sc lar cooling time is required to prevent mildly relativistic For a typical ITER plasma, Th ¼ 20 keV, the ratio electrons that result from lower hybrid current drive from pro- 2 mec =Th 25, so the Maxwellian tail is not an adequate viding a seed that is so strong that little avalanching is seed when scool տ 40ms=nb. This assumes the cooled temper- required. Studies of the generation of seed electrons by faster ature of the background electrons satisfies Tc < Kr=rs, cooling have been reported by Smith and collaborators.12–14 where Kr is the required kinetic energy for runaway, which Other sources of seed electrons are problematic. The only 2 must satisfy Kr > ðmec =2ÞðVc=V‘Þ for V‘ Vc. The intrinsically radioactive component in a fusion plasma is trit- required cooled temperature is typically 100’s of eV. ium, and the maximum energy of its beta-decay electrons is Halo current mitigation should use slow cooling. The 18.6 keV. Alpha particles can have energies up to 3.5 MeV, time sc is sufficiently short compared to the wall time in but the resulting speed of a cold electron that is struck by an ITER that it should be possible to both achieve both halo alpha particle is less than twice the speed of the alpha particle, current mitigation, which is generally based on reducing the which means the maximum kinetic energy is 4me=ma times decay time of the plasma current below the wall time, and the energy of the alpha particle. That is, less than 1.9 keV. the elimination of the Maxwellian tail as a significant source When the critical energy for runaway is greater than 18.6 keV, of the electron runaway seed. neither provides a seed. High energy electrons from the region Radiative cooling times scale scales inversely with the outside the plasma cannot cross the magnetic field between the electron density times the density of the radiating impurity, walls and the plasma through their gyromotion when their 3 so a strong increase in nb, which accompanies a rapid radia- Lorentz factor is less than about 10 . Gamma rays from decays tive collapse of the temperature, favors a strong runaway. in the wall could produce runaways through Compton scattering though the cross section is small. V. KINETIC EQUATION AND COULOMB SCATTERING Under the assumption that background-electron drag is the only impediment to the transfer of the current from near- This section will consider the kinetic equation that thermal to runaway electrons, the magnetic axis in ITER determines the distribution function, f, for high energy electrons when two effects are present: electric field acceleration encloses sufficient toroidal flux for l0RITERIp=wc 50 along the magnetic field lines and Coulomb collisions with a e-folds in the number of runaway electrons when Ip ¼ 15 MA. If all the e-folds were used to produce the runaway cur- cold background plasma. The Coulomb collision operator rent, the runaway current would of negligible magnitude. will be given in its relativistically correct form with both the Even at 15 MA, there can be not more than about forty drag and the pitch-angle scattering terms. e-folds and still have a dangerous runaway current. It will be found that when the atomic number of the ions Z satisfies ð1 þ ZÞEc=Ejj 1 the critical energy for runaway A. Required number of runaways is greatly increased, Eq. (44), as is the required change in the poloidal flux to produce an e-fold in the number of runaway Number of relativistic electrons required to carry a electrons, Eq. (45). current Ip is It is the atomic number of the ions Z, not their charge state, that enters Coulomb scattering when the energy of the 2pR 44 Ip R Nr ¼ Ip e : (36) electrons being scattered is far above the binding energy of ec 15MA RITER the most tightly bound electrons in the ions. The background In forty-four e-folds, one runaway electron could be multi- electron density nb includes the bound as well as the free plied into a sufficient number of runaway electrons to carry electrons. Nevertheless, the Coulomb logarithm, lnK, which the entire current in ITER. Even at 15 MA, the number of appears in Ec, is subtle when a significant fraction of the possible e-folds is less that forty-four for a dangerous run- electrons are bound in high-Z impurity ions. The quantity K away current, and at 10 MA the maximum number of e-folds is the ratio of the largest distance over which the scattered

particle feels the charge of the scatterer divided by the clos- isotropic pitch distribution. The parallel current requires a est approach of the scattering particle to the scatterer, which deviation a from isotropy. When the current carriers are rel- can be due to either classical or quantum effects.4 When the ativistic electrons, scattering particles are free electrons and ionized ions, the largest distance is the Debye length. When the scatterer is ei- jjj ¼ aenrc: (42) ther a bound electron or a less than fully ionized ion, the The deviation from isotropy is given by the ratio of the largest distance is related to the size of the electron cloud of electric-ﬁeld acceleration to the pitch-angle scattering, c=c , the partially ionized ion. Rosenbluth and Putvinski2 took k or bound electrons as having half the effect of free electrons.

Figures showing more complete calculations of the effect of a cEjj=ð1 þ ZÞEc (43) bound electrons on the scattering have been presented by Aleynikov et al.15 for relativistic electrons, which means with a kinetic energy 2 2 ðc 1Þmec mec . A. Electric field acceleration Runaway is only possible when electrons at a kinetic energy c 1 m c2 obtain more power from the electric Letting ^z be the direction of the magnetic field, the term ð Þ e field, E j , where j is their current density, than they lose to in the kinetic equation that gives the effect of the parallel jj jj jj drag on background electrons. The drag is en cE when electric field on the electron distribution function is r c տ c 1 1. The implication is that the required c factor for @f @f 1 @f runaway is eE z^ eE z^ p^ z^ #^ ; jj ¼ jj þ @~p @p p @# 2 ! Ec @f 1 k2 @f cr 1 ðÞ1 þ Z : (44) ¼ eE k þ ; Ejj jj @p p @k ÀÁ !!The poloidal flux change required to increase the number of k @ p2f @ 1 k2 ¼ eE þ f ; (39) runaways by a factor of e is predicted by Eq. (19) to be jj p2 @p @k p inversely proportional to cr 1, so rffiffiffi ^ where z^ p^ ¼ cos # and z^ # ¼sin #. The pitch angle of p ðÞ1 þ Z Ec wefold ¼ wc; (45) the electron relative to the magnetic field is #, the pitch 3 Ejj k cos #, and the magnitude of the electron momentum p ¼j~pj¼cmev. where wc is the requiredp fluxffiffiffiffiffiffiffiffi change without pitch-angle scattering, Eq. (21). The p=3 in the equation for wefold is B. Coulomb collisions implied by Eq. (18) of Rosenbluth and Putvinski2 when 1 þ Z 1. The collision operator of high energy electrons colliding Pitch-angle scattering can cause relativistic electrons to with cold background electrons and ions can be written in become mirror trapped in the magnetic field variation when spherical momentum space coordinates as Շ their c ð1 þ ZÞEc=Ejj before they can have a significant 2 acceleration. In non-axisymmetric systems, trapped electrons CcðÞf 1 @ 2 c ¼ 2 p 2 f can drift across the magnetic surfaces at a typical speed eEc p @p v dðq=RÞv where q ¼ cm v=B is their gyroradius, v their veloc- m c2 @ @f e e ðÞ2 ity, and d is the non-axisymmetry in the magnetic field. The þðÞ1 þ Z 2 1 k : (40) 2vp @k @k time they stay trapped is of order the pitch-angle scattering This operator is given in Secs. II and III of Karney and time sk. The drift distance between collisions in ITER is then Fisch16 and is based on the relativistic collision operator of ð20c=nbÞd=ð1 þ ZÞ in meters. Beliaev and Budker.17 The injection of very high-Z impurities may be critical for achieving adequate runaway electron mitigation through C. Pitch-angle scattering and required poloidal flux the increase they cause in (1) the critical energy for runaway, (2) the poloidal flux change required for an avalanche e-fold, When c –1 1, the importance of pitch-angle scattering and (3) the scattering into poorly confined trapped-electron is reduced by a factor 1=c compared to the term in the kinetic trajectories. equation that gives the effect of the electric field acceleration. Pitch-angle scattering dominates Ejj acceleration when D. Lifshitz and Pitaevskiii anisotropy Ec c < ck ðÞ1 þ Z (41) The runaway anisotropy can be obtained using an Ejj a electrical conductivity calculation of Lifshitz and Pitaevskiii. 18 assuming ck 1. Lifshitz and Pitaevskii derive the collision frequency for At any c for which pitch-angle scattering dominates Ejj the scattering of electrons on ions of charge Z and number acceleration, the electron distribution function has a nearly density nZ

4pZ2e4 ln K where p is the momentum, k cos # is the pitch and # the ¼ n : (46) ei vp2 Z pitch angle relative to the magnetic ﬁeld direction of the electron. The characteristic time for synchrotron radiation is Assuming an exponential distribution of relativistic electrons ss. The radiated power, in energy, which deﬁnes a runaway temperature Tr, the aver- 2 2 age relativistic c factor is c ¼ 3Tr=mec 1. The conductiv- 1 p? Ps ¼ ; (52) ity that they derive is se me

3 cmec nr is implied by the first term in Cs, Eq. (62), where rr ¼ 2 ; (47) 2 2 2 3pZe ln K ne p? ¼ð1 k Þp . Andersson et al.22 gave a related expression for the where nr=ne is the fraction of the electrons that lie in the rela- effect of synchrotron radiation on the electron kinetic equa- tivistic distribution and ZnZ ¼ ne. tion, which has been used as a basis for a number of papers The electron current along the magnetic field is including.21 The result by Andersson et al. was not originally jjj ¼ aenrc, where a is the anisotropy of the relativistic strictly consistent with the required form, Sec. VI A, of a ve- electrons. Since jjj ¼ rrEjj, the anisotropy is locity space divergence. 2 The momentum-drag term in Cs is a factor of c larger rrEc Ejj ¼ ; (48) than the term which exerts a drag on the pitch. a en c E r c Consequently, only the momentum-drag term is important 4 c Ejj when c–1 1. ¼ (49) 3 Z Ec The drag force of synchrotron radiation is only important when it exceeds the Coulomb collisional drag on the when a 1. background electrons. The ratio of the magnitude of the synchrotron radiation force to the magnitude of the collisional VI. SYNCHROTRON RADIATION drag force is given by The high-energy electrons that carry a runaway current Ps=c 2 2 not only lose energy through drag on background electrons asc ðÞ1 k ; (53) but also through synchrotron radiation, which arises from the eEc electron acceleration associated with cyclotron motion. 2 2 2 xc0 1 2 B Synchrotron radiation enters the kinetic equation for the high as ¼ 2 9:7 10 ; (54) 3 x ln K nb energy electrons in the form of a collision operator, Eq. (50), p0 which will be derived in this section. where xc0=xp0 is the ratio of the cyclotron to the plasma fre- Hazeltine and Mahajan19 derived the effective collision quency of the background electrons, B is in Tesla, and nb is operator for synchrotron radiation for sub-relativistic elec- the number density of background electrons is in units of trons ðc 1Þ1 based on a calculation of the radiative 1020/m3. Even when synchrotron drag exceeds Coulomb reaction given by Rohrlich.20 However, the power loss 2 drag, the effects can be small when Ejj Ec. through synchrotron radiation, Ps, which increases as c , Synchrotron radiation is of greatest importance when generally competes with Coulomb drag as an energy loss c –1 1, and simplified derivations are possible in this mechanism only when c 1 1. ð Þ limit. Section VI B 1 uses expressions from Landau and Rohrlich’s expression for the four force of the radiative Lifshitz in The Classical Theory of Fields23 in the c –1 1 reaction20 can be used to obtain the synchrotron collision op- limit. Equation (52) for the radiated power can be obtained erator for electrons for an arbitrary Lorentz factor c. The col- from their Eqs. (73.7) and (74.2), and the force on a charged lision operator that represents the effects of synchrotron particle due to synchrotron radiation ~f is given in their Sec. radiation is derived in Sec. VI A in terms of the force ~f s s 76. Section VI B 2 derives the force ~f from the radiative exerted on electrons by the synchrotron radiation. Section s power loss using the properties of the Lorentz transformation VI C uses Rohrlich’s expression for the four force of the in the large c limit. radiative reaction to obtain ~f and the collision operator that s The published literature is confusing on the synchrotron represents the effects of synchrotron radiation. The correct radiation of an electron aligned with a curved magnetic field form for this operator has been given in spherical momentum coordinates by Stahl et al.21 line. No matter how well aligned an electron may be with a ! curved magnetic field, it will radiate due to its perpendicular ðÞ2 motion. A non-zero perpendicular velocity ~v? is required by 1 @ 2 cp 1 k CsðÞf ¼ p f the relativistic equation of motion, which is cm d~v=dt p2 @p s e s ! ~ ^ ^ ~ ¼e~v B. Writing ~v ¼ vjjb þ~v?, where b B=B, then in 2 @ kðÞ1 k the limit ~v =v 0, the acceleration term is f : (50) j ? jjj! @k css ^ 2 ^ ~ ^ d~v=dt ¼ðdvjj=dtÞb þ vjj~j, where ~j b rb is the field-line 3 3 4p0ðÞmec curvature with R 1=j~jj the radius of curvature; b^ ~j ¼ 0, s ; (51) s 4 2 ^ ^ 2 e B which implied by b b ¼ 1. In the j~v?=vjjj!0 limit,

B. Highly relativistic derivation A. Form of synchrotron collision operator The expression for the synchrotron collision operator Since synchrotron radiation does not change the number can be derived simply in the highly relativistic limit using ei- density of electrons in ordinary space but does change their ther expressions from Landau and Lifshitz in The Classical density in momentum space, the electron kinetic equation Theory of Fields23 or from properties of the Lorentz transfor- when only synchrotron radiation is retained has the form mation in terms of the radiated power. Both derivations are @f given in this section. ¼ C ðÞf ; (56) @t s 1. Landau and Lifshitz expressions @ C ðÞf ¼ F~ : (57) s @~p s Landau and Lifshitz in Sec. 76 of The Classical Theory of Fields23 give the force exerted by synchrotron radiation The effective collision operator for synchrotron radiation on an individual, highly relativistic electron is a momentum-space divergence of a momentum-space ~v ﬂux F~s. ~ f s ¼Ps 2 : (63) The drag force on electrons produced by synchrotron c radiation can be obtained by multiplying Eq. (56) by the mo- ~ Using this expression for f s, Eqs. (59) and (60) give Eq. (50) mentum ~p and integrating over all of momentum space, for the synchrotron collision operator in the highly relativis- ð ð tic limit. The pitch term in the collision operator is zero @ 3 3 ~pfd p ¼ ~pCsðÞf d p because p^ #^ ¼ 0. Note, p^ ~p=j~pj¼~v=j~vj since ~p ¼ cm ~v. @t e ð The result is Eq. (50). 3 ¼ F~sd p; (58) The last term in Eq. (76.3) of The Classical Theory of Fields has a misprint and should read ð2e4=3m2c5Þ Ð Ð i ‘ km with the use of the identity ~xr~ ~vd3x ¼ ~vd3x, which is u ðFk‘u ÞðF umÞ. In the second paragraph below Eq. (76.3) easily derived in ordinary space, but of course applies in mo- is a clause: those terms in the space components of the four- mentum space as well. vector (76.3) increase most rapidly which come from the The collision operator that gives the effect of synchro- third derivatives of the components of the four-velocity. This clause should read that it is the term proportional to the four- tron radiation on the electron distribution function is more 3 transparent in spherical momentum coordinates. The opera- velocity cubed, which increases as c , that increases most rapidly. This is the term in Eq. (76.3) that has the misprint. tor ð@=@~pÞF~s, where F~s ¼Fsp^, can be written in spherical momentum coordinates using the standard expression for Despite these misprints, their expression for the force on a the divergence in spherical coordinates single electron in the strongly relativistic regime is correct. In particular, the force is opposite to the total velocity and @ 1 @ ÀÁ not just its perpendicular component. F~ ¼ p2p^ F~ @~p s p2 @p s 1 @ ÀÁ 2. Synchrotron force-power relation þ sin ##^ F~ : (59) p sin # @# s A Lorentz transformation into the frame of reference in which the electron is moving with velocity ~v from a frame in The distribution function f ð~x;~p; tÞ for an individual elec- which it is instantaneously at rest demonstrates that as c !1 tron is proportional to a delta function in momentum space, the synchrotron radiation force (1) is aligned with the total ve- so under the standard assumption that the electrons radiate locity of the electron and (2) is uniquely determined by the incoherently Eq. (58) implies synchrotron power loss Ps. The relativistic equation of motion is dpl=ds ¼ Kl, F~ ¼ ~f f ð~x;~p; tÞ; (60) s s where ds ¼ dt=c is time interval in the frame of rest of the ~ particle and Kl is the four force. The ordinary force ~f on a where f s is the force on an individual electron due to synchrotron radiation. particle is given by the three space-like components of the 1;2;3 The energy of an electron, including its rest mass four force K , divided by c. TheP power transferred to the 2 0=c l 2 energy, is cmec ,soEq.(50) implies the power loss through particle is given by cK . Since P lplp ¼ðmecÞ , the l synchrotron radiation is four force must obey the constraint lplK ¼ 0.

Let Kl be the four force on an electron in the frame in 2e4B2 k : (68) which it is at rest and Kl be the four force in the frame in s 2 3 34ðÞp0 mec which it is moving. TheP relation between the four force in l ~ the two frames is K ¼ lKlK . The convention of Kl of Note that B is the magnetic field in which the particle is Eq. (25) implies the velocity difference between the moving, not the radiation field. For strongly relativistic elec- 2 two frames is ~vf ¼~v with the x^ axis, which has index 1, trons, the c term in the four forces is dominant and gives the aligned with the velocity ~v. The Lorentz transformation same result as used in Sec. VI B. implies K0 ¼ cðK0 þK1v=cÞ; K1 ¼ cðK0v=c þK1Þ¼K0; The three spatial components of the four force in the K2 ¼K2, and K3 ¼K3. Since the usual three-space force, ~f , laboratory frame are related to the ordinary three force by ~ ~ ~ ~ ~ is the space-like components of the four-force divided by c, f s ¼ K=c, so using the identities B? ¼ B ~v B=v and the space-like force components that are perpendicular to the k ¼ð~v B~Þ=ðvBÞ, particle velocity scale as 1=c compared either to the space- 1 ~f B2 ~v B~B~ like component aligned with the velocity, K =c, which is s ¼c2 ?~v þ ? ; (69) 0 0 2 2 equal to K =c, or to theP power transfer cK =c ¼Ps. ks B B l The constraint p K ¼ 0, which in the frame in 2 l l ðÞ~ ~ ÀÁðÞ~ 0 0 2 ~v B B 2 ~v B which the electron is at rest is mecK ¼ 0, implies K ¼ 0. ~ ~ ¼c v þ 2 þ c 1 2 v; (70) Unlike K0, the space-like components of the four force K1;2;3 B ðÞvB need not vanish. B~ ¼c2ðÞ1 k2 ~v kk~v v : (71) B C. General derivation ~ Equation (4) in Rohrlich’s paper20 gives the four forces As c ! 1, the force is f s ¼ks~v?, which agrees with due to radiation for a particle moving in given electric and Ref. 19. The two important components of the synchrotron magnetic fields. In the rest frame of the particle, force for scattering are 2 4 XÀÁ 2 2 p? l 2 e la l 0a ~f p^ ¼kc ðÞ1 k v ¼k ; (72) K ¼ F Fa0 d F Fa0 ; (64) s 2 2 2 0 mev 3 4p0mec a ~f #^ ¼kv cos # sin #: (73) where the electromagnetic field tensor is s ^ ~ ~ ^ 0 1 Since ~v p^ ¼ v; ~v # ¼ 0; B p^ ¼ kB, and B # ¼sin #B. 0 Ex=cEy=cEz=c These equations together with Eqs. (59) and (60) give B C B C Eq. (50) for the synchrotron collision operator. B Ex=c 0 Bz By C l B C F ¼ B C: (65) B C @ Ey=c Bz 0 Bx A VII. POWER BALANCE FOR RUNAWAYS E =cB B 0 2 z y x The energy equation is the ðc 1Þmec moment of the kinetic equation, which is calculated by multiplying the ki- Writing the four force as Kl ¼ðK0; K~Þ, one finds K0 ¼ 0, netic equation for the high energy electrons, but @f @f eE z^ ¼ C ðÞf þ C ðÞf þ S; (74) 4 @t jj @~p c s ~ 2 e ~ðÞr ~ðÞr K¼ 2 2 E B ; (66) 3 4p0mec 2 by the kinetic energy of a runaway electron ðc 1Þmec and integrating over d3p ¼ 2pp2dpdk. The average relativistic c where the superscript (r) has been added to the electric and factor, or equivalently average energy, is magnetic fields in which the particle is moving to indicate Ð they are to be determined in the rest frame of the particle. ðÞc 1 fd3p c 1 þ Ð : (75) In the lab frame, the particle has a velocity vx^ and is fd3p moving in a magnetic field B~ with the electric field ~ zero, E ¼ 0. A Lorentz transformation of the electric field, TheÐ number of runaway electrons per unit volume is 3 which follows from a Lorentz transformation of the electro- nr ¼ fd p. ðrÞ ðrÞ magnetic field tensor, gives Ex ¼ 0; Ey ¼cvBz, and The power balance equation will be derived using the ðrÞ Ez ¼ cvBy. The transformation of the magnetic field gives simplified source function S of Rosenbluth and Putvinski, ðrÞ ðrÞ ðrÞ Bx ¼ Bx; By ¼ cBy, and Bz ¼ cBz. Therefore, the four Sec. III A, of the avalanche mechanism. Their S adds par- forces in the rest frame of a particle that is moving with a ve- ticles but otherwise plays a negligible role in power balance locity vx^ are because the critical relativistic factor for runaway, cr, satisfies cr 1 c 1. The avalanche source function is the B2 þ B2 B B B B only term in the kinetic equation that does not conserve Kl ¼ k v 0; c2 y z ; c x y ; c x z ; (67) s B2 B2 B2 particles.

The term inÐ the kinetic equation involving the parallel Equation (82) has Eq. (22) as a special case, c 1 2lnK, 2 3 electric field, ðc 1Þmec eEjjz^ ð@f =@~pÞd p, can be when cr 1 ¼ Ec=Ejj and a 1. 2 analyzed using Eq. (39) and the identity @c=@p ¼ v=mec , With the inclusion of pitch-angle scattering, the run- 2 whichqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi follows from writing the Lorentz factor as away condition becomes cr 1 ¼ð1 þ ZÞðEc=EjjÞ , Eq. 2 (44). Equation (82) implies c 1 2 1 Z E =E ln K. c ¼ 1 þðp=m cÞ . The resulting expression is E j where ¼ ð þ Þð c jjÞ a e Ð jj jj The implication is that pitch-angle scattering is not important 3 the parallel current jjj e kvfd p. for the typical runaway electron. To prove this first assume 2 The equation @c=@p ¼ v=mec can be also used to sim- Ð the a 1; the result is c ð1 þ ZÞEc=Ejj, so pitch-angle plify the term ðc 1Þm c2C d3p. The resulting expression Ð e c scattering is not important. Then assume a cEjj= 2 3 Շ is eEc ðc =vÞfd p. When c 1 1, this expression sim- ð1 þ ZÞEc 1. The equation c 1 ¼ 2ð1 þ ZÞðEc=EjjÞa ln K plifies to Ececnr. implies 1 ¼ 2lnK, which is a contradiction. Consequently when ð1 þ ZÞEc=Ejj > 1, pitch-angle scattering is negligible A. Without synchrotron losses for the typical runaway electron, a 1, and Ignoring the power loss from synchrotron radiation, the 2 ðÞ1 þ Z Ec ðc 1Þmec moment of Eq. (74) implies c 1 2lnK ; (83) Ejj dc dn m c2n þ ðÞc 1 m c2 r ¼ E j E ecn : (76) dt e r e dt jj jj c r where Ejj is assumed to satisfy ð1 þ ZÞEc Ejj Ec.

The rate of change of the number of runaways, dnr=dt,is B. Synchrotron radiation when Z 1 given by Eq. (19). Using sc ðeEc=mecÞ, Eq. (6) Section VII A showed that even when ð1 þ ZÞEc=

dc c 1 Ejj jjj Ejj 1, the typical value for the runaway kinetic energy sc þ ¼ 1: (77) 2 ðc 1Þmec is sufﬁciently high that pitch-angle scattering dt 2ðÞcr 1 ln K Ec enrc has a small effect for most runaway electrons. The effects of

When Ejj is time independent, Eq. (77) implies c relaxes synchrotron emission are then limited because with only to a steady state value on the time scale sR Coulomb effects giving pitch-angle scattering the width in pitch angle can narrow to d# ck=c ¼ð1 þ ZÞðEc=EjjÞ=c. cðtÞ1 ¼ðcð1Þ 1Þð1 et=sR Þ; (78) Using Eq. (83)

s 2 c 1 s ln K; (79) 1 R ð r Þ c d# : (84) 2lnK cðÞ1 1 Ejj jjj ¼ 1 : (80) This expression for d# follows from properties of pitch 2ðÞcr 1 ln K Ec enrc 24 angle scattering. A particle that initially has a pitch k0 can

Equation (19) for dnr/dt can be rewritten using Eq. (80) as have a range of values of pitch ks after a time s. Sampling ÀÁ from a binomial distribution and assuming the collision frequency times s is small, k ¼ð1 k Þð1 sÞ d ln nr e Ejj jjj=enrc Ec qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 0 ¼ : (81) 2 dt mec cðÞ1 1 6 ð1 k0Þs, where 6 is a random sign. The largest magnitude of ks is jksj¼1 s=2. The rate at which the runaways exponentiate is not affected Assuming synchrotron radiation is more important than by whether t is long or short compared to s , provided R drag, the ratio the power gained from the electric field, Ejjenrc c t 1, but the average kinetic energy of the runaways 2 2 ð Þ to that lost by synchrotron radiation Psnr ¼ðEcenrcÞasc d# , increases linearly in time, cðtÞ1 ¼ðt=srÞðcð1Þ 1Þ which implies runaway is only possible when when t sR. To simplify notation, c will denote cð1Þ. 1=3 The necessary condition for runaway production, Ejj 2 > ðÞ1 þ Z as : (85) Ejj > Ecenrc=jjj, only approximates the sufficient condition Ec when the runaway electrons are moving in close alignment with the magnetic field, so jjj enrc. Pitch-angle scattering tends to isotropize the electrons in pitch, which can reduce VIII. OHM’S LAW FOR TEARING MODES the current density of runaway electrons from its maximum Even when the current is carried by runaways, the devia- value enrc. tion in the plasma response from an ideal Ohm’s law during In general, the current density of runaway electrons is a resistive instability is determined by the Ohm’s law of the jjj ¼ aenrc, where a is the anisotropy of the runaway elec- thermal electrons.3 Since runaway electrons are moving at trons, a 1. The critical electric field for runaway is the velocity of light, the only way to change the runaway Ejj > Ec=a. When Ejj Ec=a, Eq. (80) implies current is to change the number of runaways, which is over a time scale scEc=ðEjj EcÞ. There is a long lag time between c 1 E ¼ 2 ln K jj : (82) a change in the electric field and the response of the runaway c 1 a E ~ ~ ~ r c current. Thermal electrons respond as gjth ¼ E þ~v B on

the time scale of their collisions. So deviations from an ideal their kinetic energy Ks. Knowledge of this angular distribu- response, such as the opening of islands, are determined by tion is critical for determining the effective energy required the thermal electrons. for runaway since trapped electrons cannot obtain significant energy from the parallel electric field. IX. MICROTURBULENCE The poloidal flux change required for a current transfer can be increased by reducing the number of seed electrons. The microinstability that is thought to be of greatest The required flux change is discussed in Sec. IV as are the potential importance to the runaway issue on ITER is an benefits of slowing the plasma cooling to a time scale of instability of the whistler waves.25,26 This instability requires 40 ms, when possible, to eliminate the high energy elec- a strong anisotropy of the perpendicular to parallel electron trons that survive from the pre-thermal-quench Maxwellian momentum in a beam jp =p j1, and is stabilized quasi- ? jj or from current drive. Preemptive cooling of disruption linearly by spreading p . The p spreading has two effects: ? ? prone plasmas is the only obvious method of avoiding the (1) The power loss from synchrotron radiation, which rapidly cooling of a naturally arising thermal quench. A depends quadratically on p is enhanced.27 (2) If pitch-angle ? 40 ms cooling time should be adequate to prevent the damag- scattering were large enough to reverse the direction of elec- ing halo currents that arise when loss of axisymmetric con- trons along the magnetic field, which means p reversal, then jj trol results in the plasma being pushed into the wall, so the the effect on runaways is similar to pitch-angle scattering halo-current mitigation system should be made consistent when c < ð1 þ ZÞE =E . Unfortunately, quasi-linear stabili- c jj with this time scale. zation appears to occur well before the electron momentum In addition to slowing on background electrons, the along the magnetic field is reversed.26 pitch angle of high energy electrons is scattered through collisions with the background plasma at a rate proportional X. DISCUSSION OF PHYSICS AND MITIGATION to 1 þ Z, where Z is the atomic number of the background The achievement of the ITER mission will require the ions. Because of the high energy of runaway electrons, successful avoidance of large currents of relativistic elec- whether the ions are fully ionized or not makes only a mod- trons. The potential for transferring the plasma current from est difference, 2, in the scattering. As discussed in Sec. V, thermal to relativistic electrons exists whenever the poloidal pitch angle scattering produces a major modification of the Շ magnetic flux content of an ITER plasma changes on a time runaway process when the parallel electric field Ejj ð1 þ ZÞ scale faster than tens of seconds. The danger to the ITER Ec or equivalently when the loop voltage V‘Շð1 þ ZÞVc. The device posed by a large current of relativistic electrons primary changes are: (1) The critical kinetic energy for run- 2 requires a focus on avoidance rather than on the control of away, ðcr 1Þmec , increases from cr 1 Ec=Ejj to 2 such currents if they occur. cr 1 ð1 þ ZÞðEc=EjjÞ . (2) The poloidal flux change The poloidal flux must change faster than a critical value required for an e-fold in the number of relativistic electrons

for a significant number of relativistic electrons to persist in increases from wc 2:3V stowefold ð1 þ ZÞ an ITER plasma. This time scale is discussed in Sec. II and ðEc=EjjÞwc. (3) When the plasma is strongly non- can be expressed as a critical loop voltage, Eq. (3), which is axisymmetric, a sufficient number of high energy electrons Vc 3nb Volts when pitch angle scattering is ignored; nb is may drift out of the plasma as trapped particles to prevent an the background electron density in units of 1020/m3. In many avalanche. ITER operational scenarios, nb 1. The loop voltage times Synchrotron radiation, Sec. VI, drains energy from rela- the time gives the change in the poloidal magnetic flux. tivistic electrons, but its importance to the process of the When the loop voltage is large compared to the critical transfer of current from thermal to relativistic electrons loop voltage, the number of electrons above a certain energy, appears marginal except when the background plasma den- 27 called the effective runaway energy Kr, will increase expo- sity is low and the magnetic field strength is high. nentially and naturally give a distribution of relativistic elec- Nevertheless, the derivation of the collision operator that 2 trons with an average kinetic energy ðc 1Þmec . Ignoring gives the effect of synchrotron radiation on the electron dis- pitch-angle scattering, c 1 2lnK with lnK the Coulomb tribution function is derived in some detail. Contrary to some logarithm. Given an initial number density of electrons with expectations, the force exerted by synchrotron radiation is an energy above Kr, called seed electrons, the number of rel- not aligned with the electron velocity perpendicular to the ativistic electrons will be nr ¼ ns expðjDwpj=wefoldÞ, where magnetic field~v? but with the full velocity~v, and this affects jDwpj is the change in the poloidal flux that has occurred and the form of the collision operator. wefold is the flux change required for an e-fold. Ignoring The energy moment of the kinetic equation is used in pitch-angle scattering, wefold ¼ wc 2:3V s. This Sec. VII to find the average kinetic energy of the relativistic 2 increase in the number of relativistic electrons continues electrons that carry a runaway current, ðc 1Þmec . Without until either all of the available poloidal flux has disappeared pitch angle scattering c 1 2lnK, where lnK is the Շ or all of the plasma current is carried by relativistic electrons. Coulomb logarithm. When Ejj ð1 þ ZÞEc the average The time scale for the current transfer, jDwpj=V‘, is deter- energy of runaways is increased to c 1 2ð1 þ ZÞ mined by the Ohmic loop voltage, V‘ of the background ðEc=EjjÞ ln K. This section also shows that it is possible for plasma. This current transfer process, called the electron ava- synchrotron radiation to prevent runaway in a high-Z plasma, lanche, is discussed in Sec. III and the angular distribution of but the range of Ejj=Ec over which this can occurs appears the newly energized electrons is derived as a function of very limited in ITER scenarios.

Sections VIII and IX mention results that have already the opening of the magnetic surfaces but also makes their appeared in the literature: When tearing modes arise, the force larger. break up of magnetic surfaces occurs on the resistive time The danger is too great to use an empirical validation of scale of the cold background plasma.3 It appears that micro- the runaway mitigation system on ITER itself. turbulence does not have a major role in the phenomenon of Computational simulations are required, and this paper gives current transfer from thermal to runaway electrons. the physics that could be used to relatively easily and quickly Anomalies have been seen in experiments involving run- carry out such simulations. aways,28 which could make avoidance of a current transfer much easier. More analysis is required to know whether ACKNOWLEDGMENTS these empirical anomalies might be due to other effects, such This material is based upon work supported by the U.S. as pitch angle scattering by impurities or the time scale of Department of Energy, Office of Science, Office of Fusion the experiments. Monte Carlo simulations of the kinetic Energy Sciences under Award No. De-FG02-03ER54696. equation for high energy electrons could relatively simply The author would like to thank Igor Kaganovich and Edward address much of the uncertainty. Startsev for useful discussions and Per Helander for making Appendix A gives the response of the trajectory and the a number of comments and suggestions. energy of an electron to a slippage of the poloidal relative to the toroidal magnetic flux. The rate of this slippage is the APPENDIX A: MAGNETIC FLUX CHANGES DURING loop voltage. RUNAWAYACCELERATION A final decision will need to be made on the mitigation system that will be installed on ITER in just a few years. It is Relativistic electrons have a small gyroradius qe critical that simulations address the effectiveness of pro- compared to characteristic plasma dimensions in ITER; posed mitigation strategies for avoiding strong currents of qe ¼ cmc=eB 1:7mmðc=BÞ, where B is in Tesla and the 2 relativistic electrons. Fortunately, these could be done rela- electron kinetic energy is ðc 1Þmec . Consequently, the tively easily and quickly by representing the collision opera- electron motion can be followed using the relativistic drift 24 29 tors by their Monte Carlo equivalents. Simulations of what Hamiltonian, dpu=dt ¼@H=@u; dph=dt ¼@H=@h. The happens if the transfer of current from thermal to relativistic relativistic canonical momenta of the electron drift electrons actually occurs are far more difficult. The current Hamiltonian are pu ¼ðl0G=2pBÞcmevjj þ ewp=2p, and carried by the relativistic electrons must be known at each ph ¼ðl0I=2pBÞcmevjj ewt=2p. The toroidal current I and point in the plasma. Fortunately, constraints on the relativis- the toroidal magnetic flux wt lie in the region enclosed by a tic current density make a point by point calculation of the magnetic surface. The poloidal current G and the poloidal relativistic current simpler than might be expected, magnetic flux wp lie outside the same magnetic surface. Appendix B. When magnetic surfaces exist, the poloidal h and the toroidal The issue of current transfer from thermal to relativistic u angles can be chosen so the magnetic field can be written 4 electrons is central to the success of the ITER program but in two forms far less central to the success of magnetic fusion energy. For ~ ~ ~ ~ ~ example, little change would occur in stellarator reactor 2pB ¼ ru rwpðwt; tÞþrwt rh; (A1) designs if the constraint were imposed that the net plasma ¼ l Gðw ; tÞr~u þ l Iðw ; tÞr~h current must be smaller than 5 MA. 0 t 0 t ~ Even tokamaks, such as ITER, operating well above 5 þ bðwt; h; u; tÞrwt: (A2) MA might be designed to prevent a current transfer from = h thermal to runaway electrons. The poloidal field due to the For passing electrons, one can average dph dt over and dp =dt over u. The averaged p and p are independent net plasma current penetrates the walls surrounding the u h u of time. The conservation of p and p in time-independent plasma on a time scale of 100’s of milliseconds in ITER. h u magnetic fields can be obtained from the expression for the When the current in the plasma collapses on a faster time guiding-center velocity ~v ¼ðv =BÞrð~ A~þ q B~Þ, where scale, a toroidal current equal to the plasma current must be gc jj jj q is the gyroradius calculated using the component of the induced in the walls. If the wall structures were designed to jj velocity that is parallel to the magnetic field.4 Equation (A1) minimize the forces on the walls due to this current, a mini- implies 2pA~ ¼ w r~h w r~u,soA~ A~þ q B~ can be mization of j~j B~j=j~j B~j, a sudden drop in the plasma cur- t p jj written as rent would cause a destruction of the magnetic surfaces in the plasma and prevent a transfer of the remaining plasma ~ ~ ~ 2pðeÞA ¼ phrh puru: (A3) current from thermal to relativistic electrons.7,8 During nor- mal ITER operations, the net toroidal current flowing in the Passing particle trajectories lies in the surfaces of the effec- ~ ~ walls is small, and the effect on the magnetic surfaces would tive magnetic field B~ rA when such surfaces exist. be negligible. Since the natural path of currents through the Since the surfaces of B~ are perturbed from those of B~ only wall is known, small error-field effects could be compen- by terms of order gyroradius to system size, ph and pu vary sated with error-field control coils, but this does not appear by terms only of that order. The same result does not hold to be required. The usual design of allowing these induced for trapped particles because B~ is singular at turning points, currents to flow toroidally not only eliminates their drive for the points where vjj ¼ 0.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.129.120.3 On: Tue, 19 May 2015 20:38:02 032504-16 Allen H. Boozer Phys. Plasmas 22, 032504 (2015) Since ph and pu are conserved @A~ u ¼V‘r~ þ ~u B~þ r~s; (A11) @t ~x 2p d2pp =e l G dcmev dw h ¼ 0 jj p ¼ 0; (A4) dt eB dt dt where the loop voltage is V‘ @wpðwt; h; u; tÞ=@t, the veloc- d2pp =e l I dcmev dw ity of a ðwt; h; uÞ point through space is ~u, and s is a well h ¼ 0 jj þ t ¼ 0: (A5) dt eB dt dt behaved function of position and time. Since E~ ¼@A~=@t r~U, the average loop voltage in a Ignoring the terms that become negligible as the parallel region covered by a magnetic ﬁeld line is gyroradius qjj goes to zero relative to the system size, which ð 2p E~ B~ means terms proportional to cvjj instead of its time deriva- Ð V ‘ ¼ dwtdhdu: (A12) tive, these two equations imply dwtdhdu B~ r~u

dwp G When magnetic surfaces exist the integrals are only over an ¼ V‘; (A6) dt G þ iI inﬁnitesimal range of wt, and the averaged loop voltage can be written V‘ðwt; tÞ. Equations (A1) and (A2) then imply dwt I 2 ¼ V‘; (A7) 1=ð2pB~ r~uÞ¼l ðG þ iIÞ=ð2pBÞ ,so dt G þ iI 0 þ dcmevjj eB Ejj dhdu ¼ V‘; (A8) V‘ ¼ l0ðÞG þ iI 2 : (A13) dt l0ðÞG þ iI B ðÞ2p

where the rotational transform i ð@wp=@wtÞt ¼ 1=q and With the standardÞ Ohm’s law and the resistivity g constant 2 the loop voltage V‘ ð@w =@tÞ : The total time derivatives within a surface, ðE =BÞdhdu=ð2pÞ ¼ ghjneti=hBi, where p wt jj mean along the trajectory of an electron. jnet is the net current density, which has the property that jnet/ The form of the poloidal flux wp is non-intuitive to B is constant over the surface, so the averages over jnet and B many. Its form can be clarified by considering cylindrical can have any desired form. Consequently, one can write coordinates ðr; h; z ¼ RuÞ, which means the cylinder is peri- ~ hBi odic in the z direction with a period 2pR and ru ¼ z^=R. hE i¼ V : (A14) jj l ðÞG þ iI ‘ Equation (A1) implies @wpðr; tÞ=@r ¼ 2pRBh. When the 0 poloidal flux is held fixed on a cylindrical shell at r ¼ b, let The implication is that the rate of increase of the electron Ðwpðb; tÞ¼0. The poloidal flux is then wpðr; tÞ¼2pR r momentum, Eq. (A8), is due to the surface-averaged parallel b Bhdr. When the current density is constant for r < a and electric field as one would expect. zero for a < r < b, then for r < a ! b 1 r2 APPENDIX B: SIMULATION OF RUNAWAY CURRENT w ¼l R ln þ 1 I ; (A9) p 0 a 2 a2 p The evolution of the runaway current can be followed using Monte Carlo methods. A collision operator can be con- 24 where Ip is the current in the plasma. The poloidal flux verted into a Monte Carlo operator. This Appendix shows enclosed by the magnetic axis, r ¼ 0, is negative wpð0; tÞ¼ that the current obtained from a Monte Carlo calculation can l0Rðlnðb=aÞþ1=2ÞIp while the loop voltage, which is due have far less statistical noise than might be expected because to the plasma resistivity, is positive. Consequently, the poloi- of physics constraints that imply spatially averaged quanti- dal flux at the axis rises towards the value of zero, which is ties accurately define the local current density. ~ ~ its value at r ¼ b throughout the evolution. The toroidal flux The current density of runaway electrons is j ¼ KnB=l0 2 ~ ~ in the cylindrical model is wt ¼ pBur . þ dj, where dj is proportional to and given by the pressure The enclosed poloidal flux depends on the current pro- tensor of the runaway electrons. The net parallel current of 2 2 file. When the initial current density jðrÞ/1 r =a , where the runaways, which is given by Kn, must be constant along 2 2 2 2 a is the plasma radius, the current IðrÞ¼Iað2 r =a Þr =a , each magnetic field line. That is, Kn ¼ l0hjjj=Bi, which the poloidal flux enclosed by the magnetic axis means an average along the magnetic field line. By explicitly evaluating the field-line average, the noise in computing the 3 b runaway current density can be greatly reduced. The net par- w ¼ l R þ ln I : (A10) p 0 4 a a allel current density of the runaways is approximately a hun- dred times larger than their perpendicular current density. In standard tokamaks, I/G 1, and the inward pinching The electric field exerts a far smaller force on runaway in toroidal flux of an accelerating electron, Eq. (A7), is small. electrons than does the magnetic field, so force balance 2 In a cylindrical model I=G ¼ðBhrÞ=ðBuRÞ¼ðr=RÞ iðrÞ, across the magnetic field for runaway electrons is given by where the rotational transform i ¼ 1=q ¼ RBh=aBu. 4 $ Equation (A1) can be generalized to represent and arbi- r~ P ¼ ~j B~; (B1) trary magnetic field for which B~ r~u is non-zero by letting $ wp be a function of ðwt; h; u; tÞ. The Appendix to Ref. 4 where P is the pressure tensor of the runaway electrons and shows the time dependence of the vector potential is ~j is their current density.

2 The divergence of the runaway current obeys j? cmenrc =aB qr ¼ ; (B11) j en c a @n jj r r~ ~j ¼ e r ; (B2) @t where a is the plasma radius and the gyroradius of a runaway 3 electron is q cc=xce ¼ 1:7 10 c=B, where qr is in where @nr=@t is the rate runaways are created per unit vol- r ume, which is approximately four orders of magnitude meters and B is in Tesla. For a typical ITER runaway elec- 3 smaller than the characteristic value of the divergence of the tron, c 20, B ¼ 5 T, and a ¼ 2m,sojj?=jjjj6:8 10 . runaway current jr~ ~jjenrc=R, where R is the spatial scale of the plasma. Therefore, one can assume

1 ~ ~ R. Jayakumar, H. H. Fleischmann, and S. J. Zweben, Phys. Lett. A 172, rj ¼ 0: (B3) 447 (1993). 2M. N. Rosenbluth and S. V. Putvinski, Nucl. Fusion 37, 1355 (1997). Equations (B1) and (B3) imply the runaway current den- 3P. Helander, D. Grasso, R. J. Hastie, and A. Perona, Phys. Plasmas 14, sity can be accurately given as 122102 (2007). 4A. H. Boozer, Rev. Mod. Phys. 76, 1071 (2004). 5G. Papp, M. Drevlak, T. Ful€ op,€ and G. I. Pokol, Plasma Phys. Controlled ~ jjj ~ ~ j ¼ B þ j?; (B4) Fusion 54, 125008 (2012). B 6M. Landreman, A. Stahl, and T. Ful€ op,€ Comput. Phys. Commun. 185, 847 $ (2014). B~ r~ P 7 ~j ¼ ; (B5) A. H. Boozer, Plasma Phys. Controlled Fusion 53, 084002 (2011). ? B2 8H. M. Smith, A. H. Boozer, and P. Helander, Phys. Plasmas 20, 072505 (2013). j 9 B~ r~ jj ¼r~ ~j : (B6) J. W. Connor and R. J. Hastie, Nuclear Fusion 15, 415 (1975). B ? 10C. Møller, Ann. Phys. 406, 531 (1932). 11A. Ashkin, L. A. Page, and W. M. Woodward, Phys. Rev. 94, 357 (1954). ~ ~ 12 € € An average of rj? over the volume of an infinitely H. M. Smith, P. Helander, L.-G. Eriksson, and T. T. Fulop, Phys. Plasmas long magnetic flux tube must vanish when the current den- 12, 122505 (2005). 13H. M. Smith and E. Verwichte, Phys. Plasmas 15, 072502 (2008). sity along the tube remains finite, so 14H. M. Smith, T. Feher, T. Ful€ op,€ K. Gal, and E. Verwichte, Plasma Phys. ð Controlled Fusion 51, 124008 (2009). L 15 ~ ~ d‘ P. Aleynikov, K. Alynikova, B. Breizman, G. Huijsmans, S. Konovalov, rj? S. Putvinski, and V. Zhogolev, “Kinetic modeling of runaway electrons ~ ~ 0 B hrj?i lim ð ¼ 0; (B7) and their mitigation in ITER,” paper presented at the International Atomic L!1 L d‘ Energy Agency 25th Fusion Energy Conference, Saint Petersburg, Russia, 0 B 13–18 October 2014, Paper No. TH/P3-38. 16C. F. F. Karney and N. J. Fisch, Phys. Fluids 28, 116 (1985). 17S. T. Beliaev and G. I. Budker, Sov. Phys. Dokl. 1, 218 (1956). where d‘ is the differential distance along a magnetic field 18 line. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, translated by J. B. Sykes and R. N. Franklin (Pergamon Press, New York, 1981), p. 209. The current density of the runaways parallel to the mag- 19R. D. Hazeltine and S. M. Mahajan, Phys. Rev. E 70, 046407 (2004). netic field is, therefore, given by 20F. Rohrlich, Phys. Lett. A 283, 276 (2001). 21A. Stahl, E. Hirvijoki, J. Decker, O. Embreus, and T. Fulop,€ “Effective j critical electric field for runaway electron generation,” Phys. Rev. Lett. (to l jj ¼ K þ K ; (B8) 0 B n ps be published). 22F. Andersson, P. Helander, and L.-G. Ericksson, Phys. Plasmas 8, 5221 jjj (2001). K l ; (B9) 23 n 0 B L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 3rd re- vised English ed., translated by Morton Hamermesh (Pergamon Press, @ 1 New York, 1971). ~ ~ 24 Kps ¼ rj?: (B10) A. H. Boozer and G. Kuo-Petravic, Phys. Fluids 24, 851 (1981). @‘ B 25B. Breizman, Reviews of Plasma Physics, Vol. 15 (Consultants Bureau, New York, 1990). By far the largest term in the runaway current density is 26G. Pokol, T. Fulop, and M. Lisak, Phys. Plasmas Controlled Fusion 50, the net current Kn. The Pfirsch-Schluter€ current, which is 045003 (2008). 27T. Fulop, H. M. Smith, and G. Pokol, Phys. Plasmas 16, 022502 (2009). given by Kps, and the perpendicular current density are far 28 smaller. R. S. Granetz, B. Esposito, J. H. Kim, R. Koslowski, M. Lehnen, J. R. Martin-Solis, C. Paz-Soldan, T. Rhee, J. C. Wesley, L. Zeng, and the The ratio of the perpendicular current to the net ITPA MHD Group, Phys. Plasmas 21, 072506 (2014). current is 29A. H. Boozer, Phys. Plasmas 3, 3297 (1996).