J. Plasma Fusion Res. SERIES, Vol.2 (1999) 35-38

Enhancement of Runaway Avalanche by Lower Hybrid Waves
CHAN Vincent S., PARKS Paul B., CHIU Shiu-Chur and HARVEY Robert W.2
General Atomics, San Diego, Califurnia, U.S.A.
tsunrise R&M, Inc., San Diego, Califurnia, U.S.A.
2CompX, Del Mar, Califurnia, u.S.A.
(Received: 1 March 1999 / Accepted: 15 April 1999)

Abstract
The runaway avalanche process is different from the Dreicer runaway mechanism by the muchsmaller threshold electric field. The "knock-on" source produced by large angle Coulomb scattering is a
key ingredient for secondary energetic electron generation which reduces the electric field requirement.
The lower hybrid waves, by effectively accelerating the secondary electrons, can alter the growth rate andmagnitude of the avalanche process. This paper presents the first demonstration by numerical simulation
ofthe Fokker-Planck equation, including large angle Coulomb scattering, that significant enhancement of
the runaway production rate is possible.

Keywords:
runawav electrons. lower hybrid waves

1. lntroduction

reproduced the runaway avalanche growth rate [6].

The "knock-on" source is a key ingredient in recent
studies of secondary electron production by runaway
avalanche during tokamak disruptions [] and disruption
mitigation using giant pellet injection [2]. The avalanche
depends on the existence of some trace amount of very

(LH) waves can affect the runaway avalanche process.
Even when the minimum phase velocity alkx of thelower hybrid spectrum is significantly larger than theelectron thermal velocity (hence is not expected toThe present study considers how the lower hybridenergetic electrons (MeV and above) in the plasma.
These electrons can knock out bulk electrons to supercritical energies (r.e., above the energy where theelectric field acceleration balances the collisional drag)through large angle collisions. They become runaways
and can positively feedback on the knock out process
resulting in an avalanche effect. This process is distinct
from the process discussed by Dreicer [3] by the muchsmaller threshold electric field E" (smaller than theDreicer field Ee Ay v?tc2) and the relatively long growthtime for the knock-out runaways to build up. A comprehensive, relativistic theory of knock-on avalanche has

produce a large current according to quasilinear theory),
the waves can still effectively generate an initial highenergy tail with much higher density than any
considered in the disruption scenario. One would then
expect the growth of the run-away avalanche current tobe much more expedient, and should increase toobservable magnitude within the duration of present day

tokamak operation. Furthermore, extending the orbit
analysis to this case, the avalanche growth rate is shown

to be also significantly enhanced by the LH waves. In
the following, a model describing the characteristics ofthe runaway avalanche current, and the dependence of
its amplitude and growth rate on wave and electric field

been developed, and has been validated by Monte-Carlosimulations [4] and Fokker-Planck calculations [5]. A
momentum orbit analysis has also analyticallyC o rre spondin

I autho r' se -parameters is presented. This model is compared against@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Researchmail : c hanv @ gav. gat. c om
35

Chan V.S. et al., Enhancement of Runaway Avalanche by Lower Hybrid Waves

with t = fn"4tteamln(A)l(mc)311, p = wllw, n, is theinstan-taneous runaway density, and n" is the bulk

TFokker-Planck simulations which clearly demonstrate
the existence of an enhancement effect.
This result suggests that runaway avalanche physics
can be validated in existing tokamaks with limited voltseconds through the use of LH waves. It also offers anew explanation for the LH "spectral gap" problem, anissue having to do with the discrepancy between theneed of low alkx velocity to justify the prediction ofquasilinear theory and the low nn= k$la values of thedensity.

The Fokker-Planck equation with the source S has
been solved analytically [4] for the runaway avalanche

growth rate Tft, and validated against Monte-Carlo andFokker-Planck calculations. A slightly modified
expression obtained from fitting against a Fokker-Planckcode is also available [5].

spectra launched in some ex periments, which has notbeen resolved with complete satisfaction [7].

avalanche, the momentum orbit analysis turns out to be

2. Enhanced Runaway Gurrent by "KnockOn" Avalanche and LH Waves

very elucidating. Reference 6 developed the trajectoriesof the secondary electrons which initially lie on the"knock-on" source orbit (dash-dot line in Fig. 1) wrro =In studying the effect of LH waves on runaway(l + w2s)1t2 - l, in the presence of an electric field and

The Landau-Fokker-Planck collision operator treats

only the dom-inant small-angle Coulomb scattering

small angle Coulomb collisions. They are topologicallyseparated by two separatrices (dashed lines in Fig. 1)

contribution. To study "knock-on" effects one has tomodify the Fokker-Plank equation by adding a source
term ,S to account for the electrons knocked out from thebulk to large energies through large-angle scatterings.
Furthermore, at least a weakly relativistic treatment(justified for Tlmc2 << l) is required. The following
assump-tions are used in the derivation of the source. Ina two particle collision process, the slower particle fromthe bulk is taken to have zero initial momentum, while
the high energy (runaway) particle has finite momentum
only parallel to the electric field, 1.e.. fii = ri initially.
The high energy particle is scattered to momentum il,,knocking out the idle electron to momentum il. ttre
relationship of the initial and final momenta can bederived using conservation of momentum and energy.

defined by the equations of motion. The intersection
between the initial "knock-on" orbit and the negative
slope separatrix demarcates the minimum momentum(wrr.in) above which the secondary electrons will
runaway and contribute to the avalanche. It is clear fromthis picture that the LH waves can effectively modify

runaway avalanche if and only if the upper LH
resonance boundary (dash-dot-dot line in Fig. 2), w =(nl1^1" wll- l)t'', intersects the "knock-on" source orbit

trajectory above w1;-in as shown. The process can be
modified in two ways.
First, the seed population of the runaway electrons
can be significantly enhanced. This can be estimatedfrom previous studies of Dreicer runaway enhancementby LH waves. In Ref. [9], a minimum alkl = Vrln wEls
imposed and an explicit dependence of the modified

This relationship is geometrically expressed by theequation of an ellipse\,lt I-t)=o'(1)x55

where y2 = | + w2, w = ulc.In the limit Ti -+ *, Eq. (1)es0

becomes parabolic: wn= w2Ll2. Using the conservation

2-15of particles in the laboratory and CM frame, the source
S is related to the Mgller cross-section [8] which
accounts for large-angle scattering. The complete formis given in Ref. [5]. It is useful for analytic evaluation ofthe growth rate to change variables from (w11, wr) to (lr,p) so in the limit Ti -->

2.35

L

*g=n'ln-(A)2fi220t.361 trt - trto@)l w-2

dl ,1.4

1.5

t.6

wll

l'"

a,lr1t.^; 'Q)

Fig. 1 Momentum orbit trajectories of secondary
electrons for z = 4, E/E" = 2.
36

Chan V.S. et al., Enhancement of Runaway Avalanche by Lower Hybrid Waves

Dreicer runaway flux l-p on the wave power and3. Fokker-Planck Simulations

spectrum was derived

The CQL3D Fokker-Planck code [10] is a general
purpose code for studying the inter-actions of charged( ' 2-'l'"\fo-exp \-._rd 2'.

-*

(3)

l'particles via Coulomb collisions and under auxiliary
heating conditions to which a "knock-on" source has
been added. It solves the bounce-averaged FokkerPlanck equation in velocity space. For the present study,
we will neglect particle trapping and focus on themagnetic axis of a tokamak. The LH waves are modeledby a quasi-linear diffusion operator in momentum spacewith a2 = E(l + D)/ED, Es- 2tte3n"ln(zl)/I, and D6 is
the LH quasilinear diffusion coefficient normalized by
the collisional diffusion coefficient. The dependence on
D6 is in good agreement with numerical Fokker-Planck

calculations. Second, the avalanche growth rate can also
be changed. From the momentum space orbit analysis,
the growth rate ya* JSd3w * l/w;1r1n in the absence of#1,,=# [u'#]LH waves. In their presence, "knock-out" electrons withmomentum as low as the minimum phase velocity(6)

For simplicity, we choose a constant Dor.
For this study, we wish to compare the results withthose of previous study [5], hence the plasma parameters
are chosen to be the same: ne = I x 10la cm-3, Z" - 100
eV. Also. is set to be l. and it is useful to remember E= 0.102 npYlm.

(momentum) of the LH spectrum, V.in = c//l;1qs1 coo
contribute to the growth rate, depending on the applied
power of the LH waves. The maximum enhancement
factor for Tru over the no LH case is thus lrl;1-tnr?11max.
To describe the temporal evolution of the runaway
density (and equivalently the runaway current "/ra), thetwo effects are combined in a model to sive

Figure 2 presents the runaway current fromto explore any variation for E >> E", we numericallysimulation under different conditions. The lowest two
curves are with Ohmic field alone with ElE" = 1.5,
without and with the "knock-on" sources, both are verysmall compared with the Ohmic current and notobservable within this time duration. The middle two
curves are with lower hybrid turned on at DorlD"o1 = 5,
D"o;1 is the Coulomb diffusion coefficient, E/E"- O, andwith S on and off. Even though the source exists below
ymin, there is no mechanism to accelerate these particles
to the resonant region and beyond, and no avalanchetakes place. The resulting cunent is as predicted by thequasilinear theory of lower hybrid current drive. The
second from the top curve is with E/E" = 1.5, LH

solve the Fokker-Planck equation with the "knock-on"

applied but S = 0. The current approaches saturation, isdn.
t=Trun,+I
D,(4)

ot

Assuming that T,a and l-p are time independent, thesolution at large time isr"n,u=jexp{7,"r}, anddln(J..)())
d, =y,".

ata

The separation of time scales between the saturation ofthe Dreicer mnaway tail, which takes several collisiontimes, and the avalanche growth time which is much
longer, justifies the assumption that I-p and y,^ is time
independent. To evaluate the accuracy ofthis model andsignificantly larger than with LH only and represents thesource.

LH enhanced Dreicer runaway current [9]. The top
curve is with E, LH, and S all turned on. It shows an
exponential growth to orders of magnitude larger than10{the quasilinear prediction within an interval of two-4- -1--'-

seconds.

To compare with the proposed model, Eq. (5) theminimum phase velocit! vain of the LH spectrum isa-.---.-t-.-.-.-r-.r-ll-lr.E| + -LH+EISt00
Eo3r!

| -; -E+S| - -x-.E| '.+..1HI

varied while keeping E and Dot fixed, and S turned on as
in the top curve of Fig. 2. The relative magnitudes of thecurrent support the picture that lowering vmin results in alarger seed population for the avalanche. Furthermore,
the magnitude of the current is found to be self-similarby the factor exp{-v2^1"/2} according to Eq. (4). This isIIIf-.--LH+s |___t-7c, - ' - -:-L------'--"'-4'---.
" " - -' - - "' - 4 - - - -',-'-:.:-:-:':'4,x"

0.01

c0,0001
0.0

0.5

1.5

1.0

2,02.5t (s)convincing evidence that the amplitude of the runaway
curent is determined by the LH enhanced Dreicer flux.

Fig. 2 Runaway currents under different conditions.
37

Chan V.S. et al., Enhancement of Runaway Avalanche by Lower Hybrid Waves
1000

large angle Coulomb scatterings can accelerate thermal
electrons through the gap to high energy. The combinedinteraction of the source, a small Ohmic electric field,
and the lower hybrid waves is capable of significantly
enhancing the noninductive current, beyond thequasilinear prediction.

100Ei10E!tI\=-=t---:----*-----

Acknowledgments
We gratefully acknowledge discussions with M.N.
Rosenbluth and Y.R. Lin-Liu. This is a report of work
sponsored by the U.S. Department of Energy underGrant No. DE-FG03-95ER54309.

0.1

1.0

0.0

1.5

r (s)

Fig. 3 Dependence of runaway current groMh rate on
LH power.

References

tll R. Jayakumar, H.H. Fleischmann and S. Zweben,
To determine whether 7,u depends on the applied
LH power, DotlD"ot is varied. Three values, 1, 5, and 10Phys. Lett. 1^172,447 (1993).

t2l M.N. Rosenbluth, in Plasma Physics andControlled Nuclear Fusion Research (Proc. l6th
Int. Conf. MontrEal, 1996) (IAEA, Vienna), Vol.2,

are plotted in Fig. 3 with the top curve corresponding to
the highest Dor value. The growth rate shows a large
initial increase with DorlD"on and saturation as this value
approaches 10. The computed enhancement of thegrowth rate at high LH power compared with no LH
waves is approximately 102. Both results are consistent
with the model put forth using the orbit analysis.

p.979.

t3l H. Dreicer, Phys. Review ll7,329 (1960).
[4] M.N. Rosenbluth and S.V. Putvinski, Nucl. Fusion
37, r3ss (1997).

t5l S.C. Chiu, M.N. Rosenbluth, R.W. Harvey andV.S. Chan, Nuclear Fusion 3E, 17l I (1998).

4. Discussion[6] P.B. Parks, M.N. Rosenbluth and S.V. Putvinski,
Phys. Plasmas 6,2523 (1999).
[7] F. Santini, Proc. 4th Int. Symp. "Heating in
Toroidal Plasmas," Rome, Mar 2l-28, 1984, Vol. 2,p.1303.
t8l W. Heitler, The Quantum Theory of Radiation,Oxford University Press, Oxford, England (1994).
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This study presents the first demonstration thatelectron runaway avalanche can be significantly
enhanced by LH waves. A model of the runaway
current, validated by Fokker-Planck calculations,correctly predicts the dependence of the amplitude andgrowth rate on wave parameters. A corollary result isthat the non-inductive current driven by LH waves inthe presence of a small Ohmic electric field can be increased by large angle Coulomb scatterings. This offers
a new explanation to the LH "spectral gap" issue. Mostof the previous explanations had to do with filling thespectral gap with additional waves. In contrast, the newexplanation invokes the novel yet classical idea thaton Advances in Simulation and Modeling ofThermonuclear Plasmas, Montr6al, 1992 (IAEA,Vienna, 1993), p. 489-526.

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