Production and Damping of Runaway Electrons in a Tokamak
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1 International Sherwood Fusion Theory Conference Madison, WI, April 4-6, 2016 Production and Damping of Runaway Electrons in a Tokamak Boris Breizman 1) and Pavel Aleynikov 2) 1) Institute for Fusion Studies, Austin, TX, USA 2) Max-Planck-Institut für Plasmaphysik, Greifswald, Germany Work supported by the U.S. DOE Contract No.DEFG02-04ER54742 and by ITER contracts ITER-CT , ITER/CT/15/
2 Outline q Thermal quench and primary runaway production q Runaway avalanche and runaway sustainment q Damping of runaway current q Micro-instabilities q Summary Page 2
3 Runaway production q Electrons cool down, plasma becomes more resistive, and loop voltage increases during thermal quench. q Strong toroidal electric field produces relativistic runaway electrons. E c = 4πn e e3 Λ mc 2 = n e,20 q Friction force decreases with energy for nonrelativistic electrons and asymptotes to ee c at relativistic energies. q Critical loop voltage in ITER is about 1V. Page 3
4 Thermal quench model q Maxwellian pre-quench electrons (n 0, T 0 ) with Spitzer correction (j 0 ). q Pellet delivers cold ions ( n cold > n 0 after ionization). q Kinetic treatment of the hot electrons: the timescales for collisions are ν cold cold ν hot cold ν hot hot q The current density is constant during TQ; the electric field evolves accordingly: F, m -3 1x x x x x x x x x x10 20 Initial hot and cold populations Ar ee force E, kev T 0 j 0 = ev! F( t,e)dpsin(θ)dθ + σ cold E( t) q σ cold is determined by the hot population energy release and lineradiation Page 4
5 2-d evolution of the hot electron distribution Conservation of the total current precludes complete slowing down of the hot population The surviving hot electrons form a runaway beam Page 5a
6 Hot electron survival in the limit of σ cold =0 Phase I q Electric field rises in step with the loss of the hot population. q The remaining hot electrons form a beam; total current is carried by the hot population Phase II q Electric field decays slowly as the surviving electrons accelerate E/E c ; W kin, kev E; Ε kin, kev Ι I ΙΙ II n hot, m -3 F, m Time variable, s ~100μs E, kev Page 6
7 Seed reduction by bulk conductivity q If σ cold =0, all the post-tq current is carried by the hot electron beam (no need for the avalanche). q Finite σ cold reduces the electric field and makes it more difficult for the hot electrons to survive. Lower temperature More surviving electrons E/E c n hot, m -3 j cold, MA/m eV, 25eV, 37eV, 60eV Time, t/τ Page 7
8 Self-consistent thermal quench Thermal quench scenario: 1 Hot electrons heat the cold bulk via Coulomb collisions 2 The bulk overtakes a fraction of the current 3 Bulk conductivity drops due to radiative losses.the hot population decreases in the meantime There are two possible outcomes: 1. Prompt conversion regime: purple & blue (low energy REs carry the total current at low electric field). 2. Seed for avalanche regime: green & red (Ohmic current -> high electric field -> high energy seed REs for avalanche). W kin, kev T e, ev E n hot, m -3 j cold, MA/m Time variable, s Page 8
9 Scan over initial plasma parameters Contour plot of the surviving hot electron population (normalized to j 0 /ec) The prompt conversion area represents 100% of sub-mev RE current q The electric field is supercritical (E/E c ~4 8) in the prompt conversion regime (due to relatively slow transformation of the 100keV RE into 1MeV RE current ) q The seed density has a non-monotonic dependence on pre-quench temperature with a maximum at T 0 ~4keV Page 9
10 Summary of runaway seed modeling in DIII-D and ITER DIII-D: q RE seed peaks toward the core, because T 0 < 5 kev q RE-free disruption for n Ar < m -3 q Non-uniformity of the plasma allows the post-quench current to be carried by two distinct runaway populations (sub-mev and ultra-relativistic). q Prompt conversion of the total current into RE current for n A r > m -3 q High current density (~3 MA/m 2 ) facilitates Dreicer generation ITER-relevant conditions: q Non-monotonic RE profile expected, because T 0 > 5 kev q RE-free disruption for n Ar < m -3 q Prompt conversion of the total current into RE current for n Ar > m -3 q Negligibly weak Dreicer mechanism Page 10
11 Avalanche mechanism q Seed runaway electrons produce secondary electrons via large-angle (Möller) collisions with the bulk. q The source of secondary electrons is proportional to the density of existing runaways. q The runaway population grows exponentially and saturates when it takes large part of the total current q Points of interest: q Critical electric field for avalanche onset q Avalanche growth rate q Runaway distribution function C. Møller, Ann. Phys. (Leipzig) 14, 531 (1932) Yu. A. Sokolov, JETP Letters 29, 218 (1979) M. N. Rosenbluth and S. V. Putvinski, Nucl. Fusion 37, 1355 (1997). Page 11
12 Runaway kinetics and separation of time scales q Kinetic equation: F t + ee 1 p 2 p p2 cosθf ĈF = mc τ 1 p 2 ˆRF = mc 1 τ rad m 2 c 2 p 2 1 psinθ Small-angle collisions: θ sin2 θf = ĈF + ˆRF + ŜF (Z +1) mc p ( 2 + m 2 c 2 p p2 + m 2 c 2 )F + 2sinθ p 3 Synchrotron radiation reaction: p p3 m 2 c 2 + p 2 sin 2 θf + 1 psinθ Knock- on collisions (Möller source): ŜF θ θ sinθ θ F pcosθ sin 2 θ m 2 c 2 + p F 2 Time scales: τ m 2 c 3 4πn e e 4 Λ τ rad 3m3 c 5 2e 4 B 2 q Möller source ( ŜF) is weaker than electron drag (by Coulomb logarithm). q Two-step description of the runaway avalanche: 1 Examine sustainment of the runaways in the absence of knock-on collisions Page 12 2 Use the distribution function of the sustained runaways to predict their multiplication or loss due to knock-on collisions.
13 Source of knock-on electrons The simplified source [Nucl. Fusion 37, 1355(1997)] implies ultra-relativistic primary electrons without angular spread. The needs to relax these constraints: runaway scattering on high-z impurities energy limitation by synchrotron radiation moderate energy of primary electrons at high electric field Exact source : S = n cold c γ γ 0 δ cosθ cosθ p dσ dδ dσ / dδ = Møller cross-section The gyro-averaged δ-function is δ cosθ cosθ p = 1 π p p 0 pp p!p 0! γ 1 pp 0 γ +1 γ 0 +1 γ Red curve - shape of the simplified source. Color-coded exact source for primary electrons with γ 0 5 and λ 0 = 0.2. Page 13
14 Near-threshold ordering Page 14 q The time-scale for pitch-angle equilibration is much shorter than the momentum evolution time-scale for moderate values of the electric field. q Relatively long synchrotron time-scale ( τ rad >> 1 ). Ε (Z +1) F + p 2 q The lowest order kinetic equation is:. F = G( t; p) exp[ Acosθ ] 2sinh A q Lowest order solution:. A( p) 2Ε p 2 (Z +1) p F p 3 sinθ θ = 0 q Integration of the kinetic equation over all pitch-angles eliminates the lowest order terms and gives a continuity equation in momentum: G t + p U(p)G = 0 1 U(p) A p ( ) 1 tanh( A( p) ) A p 2 +1 Ε 1 1 p + Z +1 p Eτ rad p A p ( ) 1 tanh( A( p) )
15 Momentum space attractor 1 U(p) A p ( ) 1 tanh( A( p) ) Ε 1 1 p + Z +1 p Eτ rad p A p ( ) 1 tanh( A( p) ) 0.4 Peaking of the distribution function around p max Flow velocity, U(p) p min p max Electron momentum, p The roots p min and p max merge at a certain electric field E=E 0. This is the minimal electric field required for runaway sustainment p, (mc) λ Page 15
16 Avalanche growth rates comparison 2 Avalanche growth rate, 1/n dn/ds Normalized growth rate ( Γ ) Rosenbluth-Putvinski Truncated model (Martin-Solis) Kinetic near-threshold solution Rosenbluth-Putvinski: Γ = 1 τ lnλ π E 1 3( Z + 5) E c Normalized Electric inductive field, E field (E/E c ) Page 16 PRL, 114,155001(2015).
17 Marginal stability model for runaway current damping q Characteristic times of runaway electron stopping and their production via avalanche mechanism are much shorter than the current decay time-scale. q The local inductive electric field must be very close to the avalanche threshold to maintain runaway current at any flux surface q The threshold electric field is determined by plasma density 1 r r r E r = 4π j c 2 t Ohm s law : j > 0 E = E c (n) j = 0 E < E c (n) Page 17 Nucl. Fusion 54, (2014)
18 Dynamics of the runaway current profile q Critical field: q Initial current profile: E c = 4πn ee 3 Λ E mc r 2 a 2 j = j 0 1 r a q Time-scale of current decay: t decay = π j 0 a2 c 2 E 0 q The current density decreases and the current channel narrows over time: dρ dτ = 1 ρ 2 + 2ρ 2 ln ρ 4ρ ( 1 ρ τ )ln ρ ; t π j 0a 2 c 2 E 0 τ; r / a ρ Page 18
19 Critical field model and prior work Total current Radius of the current channel Dimensionless time Normalized current density Instantaneous profile Initial profile Normalized radius 0.9 Page 19 Damping of relativistic electron beams by synchrotron radiation F. Andersson, P. Helander, and L.-G. Eriksson, Phys. Plasmas 8, 5221 (2001) Normalized electric field Instantaneous electric field Normalized radius Critical field for avalanche
20 Microinstabilities q Can microinstabilities enhance runaway scattering? Fan instability observed experimentally: V.V. Alikaev et al., Sov. J. Plasma Phys. 1, 303 (1975). Modes of interest: q Electron cyclotron (upper hybrid) waves: ω = ω c 2 + ω p 2 q Magnetized plasma waves: ω = ω p q Whistlers: ω = ω c k kc 2 ω p 2 1+ k 2 c 2 ω c 2 /ω p 4 k c k 2 c 2 + ω p 2 q Waves can be excited excited via anomalous Doppler resonance ( ω + ω c /γ = k V ) and Cherenkov resonance ( ω = k V ). Page 20
21 Collisional damping for resonant modes 1 Normalized collisional damping rate Γ/ν for anomalous Doppler resonance ω p /ω c =0.35 γ=20 γ=30 q Collisional damping and instability threshold have been overestimated in prior work by other authors q Collisional damping exhibits significant frequency dependence q Whistler mode has the lowest damping rate Normalized frequency ω/ω c Damping rates for waves driven via anomalous Doppler resonance Page 21
22 Kinetic drive Γ = R(ω;γ ) 2πmcγ p 2 F b θ dθ ω c n ω p 2 R(ω;γ ) n b n π k 1+ ie 4 γ ω 2 + e c 3 γ 2 1 R c ω c 2 ( 1 e 2 ) ω + ω ω 2 2 pω c ω 2 2 ( ω c ) 2 ie ω c ω 2 p ω ω c 2 ω 2 2 ω c 2 ( ) 2 + ωe ( ) γ 2 1 γ 2 k! c ω c γ Dots magnetized wave Solid whistlers Blue no FLR Red full Bessel q The drive for whistlers from a narrow electron beam is weaker than the drive for the magnetized plasma branch Γ/ω c q Finite Larmor radius effect suppresses magnetized plasma waves q Whistlers have the lowest collisional damping ω/ω c Page 22
23 Ray tracing shows internal reflection and linear transformations of wave packets Wave packet transformation Red curve - whistler wave Green curve - magnetized plasma wave Poloidal motion of the wave packet Page 23
24 Statistical analysis Wave trajectories diverge after multiple reflections and due to small fluctuations of plasma parameters This randomness calls for statistical analysis: 1. Launch many waves at a reference temperature 2. Calculate drive and damping separately and find trajectories with maximum amplification factor 3. Scale damping with temperature 4. Find minimal temperature for instability to appear. Page 24
25 Simulation results for ITER parameters n e ~ m 3 T j re ~ 1.0MA / m 2 e ~ 22eV E re ~ 15MeV Fast analysis of several million wave packets determines instability onset Kinetic drive and damping Total amplification factor The local growth rate at the plasma core becomes positive at T e ~ 12eV Ray-tracing analysis requires higher temperature for the instability onset. Page 25 Nucl. Fusion 55, (2015).
26 Summary q Self-consistent kinetic modeling of primary runaway formation during thermal quench (prompt conversion of the plasma current into runaway current is advantageous for plasma position control). q Kinetic near-threshold theory for runaway sustainment and runaway avalanche in presence of synchrotron losses (enhanced critical electric field found for avalanche onset). q Marginal stability scenario for runaway-dominated current quench (runaway avalanche threshold determines the current decay timescale). q Revised thresholds of runaway-driven micro-instabilities (instability window quantified for ITER-relevant parameters). Page 26
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