Download Suppression of runaway electron avalanches by radial

Suppression of runaway electron avalanches by radial diffusion
P. Helander, L.-G. Eriksson, and F. Andersson
Citation: Phys. Plasmas 7, 4106 (2000); doi: 10.1063/1.1289892
View online: http://dx.doi.org/10.1063/1.1289892
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Published by the American Institute of Physics.
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PHYSICS OF PLASMAS
VOLUME 7, NUMBER 10
OCTOBER 2000
Suppression of runaway electron avalanches by radial diffusion
P. Helander
EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, United Kingdom
L.-G. Eriksson
Association EURATOM-CEA sur la Fusion, CEA Cadarache, F-13108 St. Paul lez Durance, France
F. Andersson
Department of Electromagnetics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
共Received 6 April 2000; accepted 29 June 2000兲
The kinetic theory of runaway electron avalanches caused by close Coulomb collisions is extended
to account for radial diffusion. This is found to slow down the growth of avalanches. An
approximate analytical formula for the growth rate is derived and is verified by a three-dimensional
Monte Carlo code constructed for this purpose. As the poloidal magnetic flux that is available to
induce an electric field in a tokamak is limited, avalanches can be prevented altogether by
sufficiently strong radial diffusion. The requisite magnetic fluctuation level is sensitive to the mode
structure and the speed of the disruption. It is estimated to be ␦ B/B⬃10⫺3 for parameters typical
of large tokamaks. 关S1070-664X共00兲03010-X兴
I. INTRODUCTION
II. SECONDARY RUNAWAY GENERATION
The generation of runaway electrons in disruptions is a
common feature in present-day tokamaks. The runaways are
generated in the thermal quench of the plasma, when the
plasma temperature drops to the region of 10 eV.1 The resistivity then becomes very large and the loop voltage therefore
rises rapidly—typically to a few hundred volts in the center
of the Joint European Torus 共JET兲.2 A large fraction 共typically around a half in many tokamaks兲 of the pre-disruption
plasma current can then be converted to a runaway current,
and the loop voltage drops to nearly zero. For future larger
experiments, runaway electrons can be a severe problem
since their loss to the first wall may cause severe localized
surface damage. It is thus desirable to avoid runaway production in disruptions as far as possible. Magnetic fluctuations may be helpful in this respect since they increase the
loss rate of runaway electrons, as has been reported from
experiments in Ref. 3.
In the present paper we investigate to what extent the
presence of radial diffusion caused by magnetic fluctuations
can be expected to prevent the production of runaways in
large tokamaks. Our paper is organized as follows. In Sec. II
we briefly review the physics of runaway production, focusing on so-called ‘‘secondary’’ generation, which is thought
to be the dominant mechanism in tokamaks with large
plasma current. In the following section the influence of radial diffusion on runaway production is calculated analytically. The result is then compared with simulations done
with a three-dimensional, time-dependent Monte Carlo code
constructed for this purpose, as described in Sec. IV. Our
conclusions are summarized in Sec. V, where we estimate
the magnetic fluctuation level required to suppress avalanches.
The generation of runaway electrons has been studied
theoretically for over four decades. Early work focused on
the formation of fast electrons through the combined effects
of a constant electric field and velocity–space diffusion as
described by the Fokker–Planck equation.4–7 This is sometimes referred to as the primary generation of runaways.
More recently it has been realized that in a tokamak with
sufficiently large plasma current, close Coulomb collisions
共which are not described by the usual Fokker–Planck equation兲 between thermal electrons and existing runaways can
lead to exponential multiplication of the latter—a so-called
runaway avalanche.8–10 The point is that a single such collision can kick the thermal electron above the critical energy
for runaway acceleration by the electric field. The total number of e-foldings caused by this ‘‘secondary generation’’
mechanism in a disruption can be estimated as10
1070-664X/2000/7(10)/4106/6/$17.00
冕␥
r dt⬃
I
,
I A ln ⌳
共1兲
where I is the plasma current and I A ⫽4 ␲ m e c/ ␮ 0 e⫽0.017
MA is the Alfvén current. Consequently, in small tokamaks
this mechanism is not effective, but becomes the dominant
mechanism in a next-step device. It has been observed on a
small scale in nondisrupting TEXTOR 共Tokamak Experiment for Technology Oriented Research兲 plasmas.11
Mathematically, secondary runaway production is governed by the kinetic equation,
冉
冊
⳵ f eE 储 ␰ ⳵ f 2␭ ⳵ f
⫽C 共 f 兲 ⫹S,
⫺
⫺
⳵ t m ec ⳵ p
p ⳵␭
共2兲
for the electron distribution f . 10 The velocity–space coordinates are the relativistic momentum p, which has been normalized to m e c 共with m e the rest mass兲, and ␭⫽p⬜2 / p 2 b
⫽(1⫺ ␰ 2 )/b, where ␰ ⫽ p 储 /p and b⫽B/B max is the magnetic
4106
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Phys. Plasmas, Vol. 7, No. 10, October 2000
Suppression of runaway electron avalanches by . . .
field normalized to its maximum on the flux surface in question. Coulomb collisions are described by the right-hand side
of the equation, where C( f ) denotes the Fokker–Planck collision operator. For superthermal electrons it has the form7
C共 f 兲⫽
1
␶p
2
冋
册
⳵
共 1⫹ p 2 兲 f ⫹ 共 1⫹Z eff兲 冑1⫹ p ⫺2 L共 f 兲 ,
⳵p
共3兲
where
Z eff
is
the
effective
ion
charge,
␶
⫽4 ␲ ⑀ 20 m 2e c 3 /n e e 4 ln ⌳ is the collision time for relativistic
electrons, and
1 ⳵
⳵
L⫽
共 1⫺ ␰ 2 兲
2 ⳵␰
⳵␰
共4兲
is the Lorentz scattering operator. Relativistic effects associated with the thermal electron distribution have been neglected, but are retained for the fast electrons. S is a source
term of fast electrons produced by close collisions between
existing runaways and slow electrons. In the limit where the
former are strongly relativistic and move nearly parallel to
the magnetic field while the momentum imparted to the slow
electrons is relatively small, S can be calculated by retaining
only the leading term in the quantum-mechanical Mo” ller
scattering formula.12 This approximation,10 whose accuracy
was confirmed in Ref. 13, gives
S 共 p,␭ 兲 ⫽
␭ 2⫽
nr
2 ␲ ␶ ln ⌳
2b ⫺1
冑1⫹p 2 ⫹1
冑b ⫺1 ⫺␭
␦ 共 ␭⫺␭ 2 兲 ⳵
p2
冉
1
⳵ p 1⫺ 冑1⫹ p 2
冊
,
共5兲
.
Note that this source is proportional to the density of existing
runaways n r and to the collision frequency for close collisions 1/( ␶ ln ⌳) rather than the usual Fokker–Planck collision frequency 1/␶ .
Rosenbluth and Putvinski10 solved the orbit-average of
Eq. 共2兲 analytically in several limits and constructed an interpolation formula,
␥ r␶ ⯝
E⫺1
ln ⌳
冉
冑
␲␸
3 共 Z eff⫹5 兲
1
4 ␲ 共 Z eff⫹1 兲 2
⫻ 1⫺ ⫹
E 3 ␸ 共 Z eff⫹5 兲共 E 2 ⫹4/␸ 2 ⫺1 兲
冊
⫺1/2
,
共6兲
for the runaway production rate ␥ r ⫽d ln nr /dt, whose correctness they confirmed numerically. Here E⫽ 兩 E 储 兩 /E c is the
normalized electric field, ␸ ⫽1⫺1.46⑀ 1/2⫹1.72⑀ describes
the effects of finite toroidicity, ⑀ ⫽r/R is the inverse aspect
ratio, and E c ⫽m e c/e ␶ is the critical electric field below
which no runaway is possible.7 Thus, we have approximately
␥ r␶ ⬃
E⫺1
.
2 ln ⌳
共7兲
The estimate 共1兲 is obtained by integrating this growth rate
over time, assuming E 储 ⰇE c , using E 储 ⯝L(dI/dt)/(2 ␲ R),
and approximating the plasma inductance by L⯝ ␮ 0 R.
In a tokamak disruption, the electric field usually exceeds the critical one, so that EⰇ1. The energy threshold for
4107
electron runaway is then low, and electrons with pⲏE ⫺1/2
run away. Most secondary electrons are thus generated with
relatively low energy (pⰆ1) and with velocity vectors
nearly perpendicular to the magnetic field (␭⯝1); see Eq.
共5兲. However, as in Ref. 10, we assume that they are superthermal, thus taking (T e /m e c 2 ) 1/2ⰆE ⫺1/2Ⰶ1, where T e is
the bulk electron temperature.
III. RADIAL DIFFUSION
In the analysis reviewed above it was assumed that there
is no loss of runaway electrons. In practice, these particles
undergo radial transport due to magnetic fluctuations and are
thus imperfectly confined. This reduces the growth rate of
runaway avalanches since the runaways leaving the plasma
do not produce any new ones in close collisions with thermal
electrons. In this section, we calculate the reduction of the
avalanche growth rate caused by radial diffusion described
by the addition of a term,
⳵f
1 ⳵
rD ,
r ⳵r
⳵r
共8兲
on the right-hand side of Eq. 共2兲, where D is the diffusion
coefficient. The effect of diffusion on primary runaway generation has previously been investigated by Catto et al.,14 but
this problem does not appear to have been analyzed in the
context of secondary runaway production other than in the
numerical study of Ref. 15.
The addition of the diffusion term 共8兲 adds another dimension to the kinetic equation, which then becomes very
difficult to solve analytically. 关Already without diffusion the
kinetic equation 共2兲 poses a formidable mathematical problem, which was, however, successfully solved in Ref. 10.兴
However, an approximate analytical solution can nevertheless be found by noting that there is a separation of time
scales in the problem. The acceleration to relativistic speed
of an electron that has recently been knocked above the runaway threshold by a close collision occurs on a time scale
␶ acc⫽ ␶ /E, but the time scale for growth of the runaway
population is much longer: as we have already noted, it is
␥ r⫺1 ⬃(2 ln ⌳)␶acc . Therefore we expect that finite radial diffusion can interrupt the avalanche even if the diffusion is not
strong enough to alter the velocity–space kinetics of the runaway generation process.
The momentum space can thus be divided into a lowenergy region (p⬍ p ) where the avalanche mechanism op*
erates undisturbed, and a high-energy region (p⬎p ) where
*
the accelerating runaways undergo radial diffusion.16 The
flux of runaways from the former to the latter region is given
by the Rosenbluth–Putvinski formula 共6兲. In the simple, but
unrealistic, case where the diffusion coefficient is independent of energy, integrating the kinetic equation 共2兲 over the
high-energy region then gives the following simple equation
for the runaway density:
⳵nr
⳵nr
1 ⳵
⫽ ␥ rn r⫹
rD
,
⳵t
r ⳵r
⳵r
where the growth rate ␥ r is given by Eq. 共6兲. 共The source
term S can be ignored for p⬎ p .) If, for simplicity, the
*
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4108
Phys. Plasmas, Vol. 7, No. 10, October 2000
Helander, Eriksson, and Andersson
plasma has a circular cross section, this equation is easily
solved by separation of variables. Keeping only the 共Bessel兲
eigenfunction corresponding to the lowest eigenvalue,
n r 共 r,t 兲 ⬀J 0 共 kr 兲 ,
冉 冊
a 2
.
2.4
共9兲
The diffusion coefficient D of fast electrons is thought to
be governed by magnetic turbulence in the plasma,17 but its
precise value is not known. In the simplest picture, the electrons follow stochastic magnetic field lines and thus diffuse
radially out of the plasma with a diffusion coefficient of the
order
D RR ⫽ ␲ q v 储 R
冉 冊
␦B
B
L共 f 兲 ⯝
冕
⬁
共10兲
which was first derived by Rechester and Rosenbluth.18 Here
␦ B is the magnetic fluctuation level, and the magnetic field is
assumed to be fully stochastic. However, runaway confinement is known to be much better than that predicted by
Rechester–Rosenbluth diffusion.17 It was first pointed out by
Mynick and Krommes19 that the finite Larmor radius and
magnetic drift velocity of the fast electrons can considerably
reduce the diffusion coefficient. Some of their results were
later disputed by Myra and Catto20 共and subsequently corrected in a joint paper21兲, who found that the poloidal component of the drift does not, in fact, reduce the diffusion
coefficient in broad-band magnetic turbulence. Their analysis
shows that the effect of magnetic drift on diffusion is generally very complicated and depends on details of the turbulence such as its poloidal localization and radial correlation
length. In general, drifts tend to be important when the orbit
width ⌬ e exceeds the mode width ⌬ m . The details can vary
substantially, however, and the diffusion coefficient may decrease slowly 共algebraically兲 or rapidly 共exponentially兲 with
⌬ e /⌬ m depending on the poloidal structure of the turbulence. In any case, since the orbit width for a relativistic
electron increases linearly with p,
共11兲
the confinement improves progressively for fast electrons as
they are accelerated by the electric field.22 Here q is the
tokamak safety factor and B T the magnetic field strength in
Teslas.
The situation is thus considerably more complicated than
indicated by the simple criterion 共9兲, and we need to calculate how a velocity-dependent diffusion process affects a
runaway avalanche. To this end, we again integrate the kinetic equation 共2兲 in the high-energy region, p⬎p , but this
*
time only take the integral over perpendicular momentum.
Since most runaways have nearly parallel momenta, p⯝p 储 ,
the pitch-angle scattering operator 共4兲,
p 2储
⳵
⳵f
p⬜
⫹O 共 f 兲 ,
2 p⬜ ⳵ p⬜ ⳵ p⬜
0
C 共 f 兲 d 2 p⬜ ⫽
冉
冊
1⫹Z eff F
1 ⳵
,
共 1⫹ p ⫺2 兲 F⫹O
␶ ⳵p
␶
p2
for p⯝ p 储 Ⰷ p⬜ , where F is the distribution function in parallel momentum,
F⫽
冕
⬁
0
qm e c qp
⌬ e⯝
⫽ 1.7 mm,
eB
BT
冊
if ⳵ f / ⳵ p 储 ⬃ f /p 储 and ⳵ f / ⳵ p⬜ ⬃ f /p⬜ with p⬜ Ⰶ p 储 . The first
term, which is the largest, is therefore annihilated by integration over perpendicular momentum, and the integral of the
collision operator 共3兲 becomes
2
,
冊 冉
is approximately
k⫽2.4/a,
where a is the minor radius, one obtains the confinement
time ␶ e ⫽1/k 2 D. If this is shorter than the inverse avalanche
growth rate ␥ r⫺1 no avalanche occurs. Thus, in this simple
case the avalanche is expected to be interrupted if
D⬎D crit⫽ ␥ r
冉
p ⳵
p 储 ⳵ p⬜2 ⳵ f
p储 ⳵ f
L共 f 兲 ⫽
⫺
⫺
2 ⳵ p 储 p⬜ ⳵ p⬜ p ⳵ p 储 p⬜ ⳵ p⬜
f 2 ␲ p⬜ d p⬜ .
Thus we obtain the following equation for strongly relativistic runaway electrons:
␶
⳵F
⳵F ␶ ⳵
⳵F
⫹ 共 E⫺1 兲
⫽
rD 共 p 兲
.
⳵t
⳵p r ⳵r
⳵r
共12兲
The second term in Eq. 共12兲 contains the difference between
acceleration by the electric field and collisional drag. As before, the population of high-energy electrons governed by
Eq. 共12兲 is fed from below 共in p兲 by the avalanche mechanism, which operates in the region p⬍p according to Eq.
*
共6兲 and thus provides a boundary condition on F(p ),
共 E⫺1 兲 F 共 p ,r,t 兲 ⫽ ␥ r ␶ n r ⫽ ␥ r ␶
*
冕
*
⬁
p
共13兲
F 共 p 兲 d p.
*
Note the unconventional form of this condition: the boundary data depend on the integral of the solution. In applying
the strongly relativistic approximation 共12兲 to all electrons in
the high-energy region p⬎p , we have assumed that p
*
*
Ⰷ1.
It is not difficult to find separable solutions to Eq. 共12兲,
冋
F 共 p,r,t 兲 ⫽J 0 共 kr 兲 exp ␥ t⫺
␶
E⫺1
冕
p
p
册
共 ␥ ⫹k 2 D 共 p ⬘ 兲兲 dp ⬘ .
*
共14兲
In general, the full solution must be sought as a superposition
of several of these eigensolutions, but in the present case the
boundary condition 共13兲 is satisfied by a single eigensolution
if the growth rate ␥ is chosen so that
E⫺1⫽ ␥ r ␶
冕
⬁
p
*
冋
d p exp ⫺
␶
E⫺1
冕
p
p
册
„␥ ⫹k 2 D 共 p ⬘ 兲 …dp ⬘ .
*
共15兲
More generally, Eqs. 共12兲 and 共13兲 are satisfied by any sum
of such solutions corresponding to different Bessel eigenvalues k. Again, for our purposes it is sufficient to keep only the
term corresponding to the lowest k since this term has the
highest growth rate. Therefore, the relevant solution is given
by Eq. 共14兲 with k⫽2.4/a.
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Phys. Plasmas, Vol. 7, No. 10, October 2000
Suppression of runaway electron avalanches by . . .
As already mentioned,16 the parameter p should satisfy
*
k D ␶ p /(E⫺1)Ⰶ1. In this case, the range of both integrals
*
in Eq. 共15兲 may be extended down to p⫽0 since the contribution from the interval 0⬍ p⬍ p only makes a small con*
tribution. Thus, although we earlier assumed p Ⰷ1, we may
*
still take p ⫽0 in Eq. 共15兲, which determines the avalanche
*
growth rate ␥ if the runaway diffusion coefficient D(p) is
known. If the latter is independent of p, we simply obtain
4109
IV. MONTE CARLO SIMULATION
2
␥ ⫽ ␥ r ⫺k 2 D,
共16兲
which reproduces the criterion 共9兲. As already remarked, in
practice D(p) decreases with increasing p; in fact, the integral
冕
⬁
0
共17兲
D 共 p 兲 dp
In order to verify the ideas described above and to fully
simulate the avalanche process, we have constructed a Monte
Carlo code, ARENA 共Avalanche of Runaway Electrons, Numerical Analysis code兲. This code numerically solves the
orbit-average of Eq. 共2兲 along the lines described in Ref. 10,
but with the additional diffusion term 共8兲.
The Monte Carlo solution of this equation involves following a large number of Monte Carlo particles in phase
space (r,p,␭); the phase space positions of these particles
are changed periodically, at time intervals ⌬t, by applying
Monte Carlo operators representing collisions, the accelerating electric field, and the diffusion term 共8兲. The Monte
Carlo operator for the latter is given by
⌬r⫽
usually converges fairly rapidly. In this case, Eq. 共15兲 can be
simplified further by noting that, because of Eq. 共7兲,
␥␶
1
ⱗ
Ⰶ1,
E⫺1 2 ln ⌳
so that the first term of the integral within the exponent in
Eq. 共15兲 is small over the convergence region of the term
involving D(p ⬘ ). The integration of the latter can thus be
extended to infinity, which gives us a completely explicit
formula for the growth rate,
冉
␥ ⫽ ␥ r exp ⫺
k 2␶
E⫺1
冕
⬁
0
冊
D 共 p 兲 dp .
共18兲
This simple formula is the main result of the present paper.
Note that although the avalanche growth rate is reduced by
the presence of radial diffusion, it cannot become negative if
the integral 共17兲 converges. The physical reason for this is
that the confinement time of the fast electrons approaches
infinity sufficiently quickly as they are accelerated by the
electric field. To understand this in a simple way, consider
the simple time-dependent diffusion equation,
⳵nr 1 ⳵
⳵nr
⫽
D共 t 兲
,
⳵t
r ⳵r
⳵r
which models the transport of electrons whose diffusion coefficient changes with time 共because of acceleration in our
problem兲. By introducing the new time variable,
t ⬘⫽
冕
t
0
D 共 t 兲 dt,
where ␨ is a random number with zero mean and unit variance. Monte Carlo operators for the orbit-averaged effects of
Coulomb collisions and the electric field are given in Refs.
10 and 23. It should be noted here that a Monte Carlo particle given by (r,p,␭) represents a number of real particles
uniformly distributed in time along the unperturbed guidingcenter orbit given by the three invariants (r,p,␭).
Thus, the procedure for obtaining the numerical solution
is as follows: at the beginning of the calculation a large number of Monte Carlo particles are initialized with a Maxwellian velocity distribution and with a radial density profile
which can be freely specified; after each time step, ⌬t, the
particle positions in phase space are updated with the aid of
the Monte Carlo operators; in addition, new runaway particles are added in accordance with the source term 共5兲; diagnostic output 共runaway growth rate, averaged energy, etc.兲
are produced after suitably spaced time intervals 共i.e., some
fraction of a collision time兲.
The ARENA code has been benchmarked carefully 共without diffusion兲 against the Rosenbluth–Putvinski formula 共6兲,
which has been found to be accurate in all physically relevant cases. Adding radial diffusion, we have used the code
to verify the approximate analysis given in Sec. III. If the
diffusion coefficient is taken to be independent of momentum, the growth rate is found to drop linearly with increasing
diffusion, in practically exact agreement with Eq. 共16兲; see
Fig. 1. In the more realistic case where the diffusion coefficient does depend on momentum, we have verified Eq. 共18兲
by running the code with a radial diffusion coefficient given
by
2
D 共 p 兲 ⫽D 0 e ⫺(p/⌬p) ,
this equation is transformed into an ordinary diffusion equation. It is immediately clear that if the integral defining t ⬘
converges as t→⬁, a fraction
冉 冕
exp ⫺k 2
⳵D
⌬t⫹ ␨ 冑2D⌬t,
⳵r
⬁
0
D 共 t 兲 dt
冊
of the electrons will always stay in the system, regardless of
the absolute value of the diffusion coefficient. The avalanche
growth rate 共18兲 contains exactly this reduction factor.
for various values of the width ⌬ p. The runaway electrons
were initialized with the 共almost flat兲 radial density profile
n r (r)⫽n r0 关 1⫺0.9(r/a) 2 兴 0.1, and the ratio between the accelerating field and the critical one was E⫽ 兩 E 储 /E c 兩 ⫽100.
The result of the simulations is shown in Fig. 2, where the
growth rate obtained from the simulations and the one from
Eq. 共18兲 are plotted as functions of ⌬p. As can be seen, the
agreement is good, which indicates that Eq. 共18兲 is a convenient and accurate estimate of the reduction of the runaway
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4110
Phys. Plasmas, Vol. 7, No. 10, October 2000
FIG. 1. The normalized avalanche growth rate ␥␶ versus the diffusion coefficient in the case when the latter is assumed to be independent of momentum. The growth rate decreases linearly with D and vanishes when D
⫽D crit , as predicted by Eq. 共16兲.
growth rate caused by momentum-dependent diffusion. If
⌬p becomes very large the agreement is worse since the
integral 共17兲 then converges slowly. In such cases the more
accurate expression 共15兲 should be used to calculate the
growth rate.
V. CONCLUSIONS
Our analytical calculation and numerical simulations
demonstrate that radial diffusion slows down runaway avalanches, so that the growth rate ␥ r calculated in Ref. 10 is
reduced according to Eq. 共15兲. If the integral 共17兲 converges
quickly enough, this equation is reduced to the more explicit
equation 共18兲. At first sight, this expression suggests that
introducing a large diffusive loss of fast electrons does not
Helander, Eriksson, and Andersson
prevent an avalanche from occurring, since the growth rate
共18兲 cannot become negative. However, in practice there are
at least two reasons why sufficiently rapid diffusion should
still be able to effectively prevent avalanches. First, the runaway energy is not unlimited, as assumed in Sec. III. Several
physical processes 共like synchrotron radiation or cyclotron
instability兲 limit the energy and hence the confinement time
of runaways, so that avalanches may be completely stabilized. This may, however, require a very large diffusion coefficient. Second, and perhaps more importantly, for practical purposes it is sufficient to reduce the growth rate to a
level where there are not enough volt-seconds of induced
electric field to achieve significant avalanche growth. Thus,
although complete avalanche stabilization by radial diffusion
may be difficult, it is nevertheless possible to reduce the
avalanche size to a manageable, or even negligible, level.
In order to estimate the magnetic fluctuation level required for this, we use the approximate expression 共1兲 for the
total number of avalanche e-foldings, modified by the reduction factor from Eq. 共18兲,
冕␥
dt⬃
冉
I
k 2␶
exp ⫺
I A ln ⌳
E⫺1
冕
⬁
0
冊
D共 p 兲d p .
共19兲
We assume that the diffusion coefficient D(p) is equal to the
Rechester–Rosenbluth value 共10兲 for small p and falls off at
higher energies when the orbit width 共11兲 becomes comparable to the mode width of the magnetic turbulence.20 The
mode width is, of course, quite uncertain; it has recently
been estimated to be 0.5–4 cm in TEXTOR-94.24 This determines the value of the D(p) integral, which we write as
冕
⬁
0
D 共 p 兲 d p⫽p crit␲ qcR
冉 冊
␦B
B
2
,
where p crit characterizes the speed of convergence. Requiring
that the number of e-foldings 共19兲 should not be larger than
some number N defining the maximum tolerable avalanche
size implies the following lower limit on the magnetic fluctuation level:
␦B
B
ⲏ
冑
a
2.4
冉
冊
E⫺1
I
ln
.
␲ qcR ␶ p crit NI A ln ⌳
This limit is practically independent of the argument within
the logarithm, and becomes for typical large-tokamak parameters about 10⫺3 . 共For JET, E 储 ⱗ104 V/m, R⫽3 m, a⫽1 m,
n e ⫽2⫻1019m3 , and p crit⬎1.) This greatly exceeds the fluctuation level in quiescent plasmas but is not unrealistic in a
disruption, where ␦ B r / ␦ B ␪ ⯝0.04 has been reported.25
Thus, it appears possible that the naturally occurring, or any
externally induced, magnetic fluctuations could significantly
reduce the size of secondary runaway avalanches.
ACKNOWLEDGMENTS
FIG. 2. The normalized avalanche growth rate ␥␶ vs ⌬p for the diffusion
2
coefficient D(p)⫽D 0 e ⫺(p/⌬p) , with D 0 ⫽R 2 / ␶ , a/R⫽0.31, and Z eff⫽1.5.
The solid line shows the result from the numerical simulation, and the dotted
line the analytical approximation 共18兲.
Two of the authors 共F.A. and P.H.兲 are grateful for the
hospitality of CEA Cadarache, where some of this work was
carried out. It was funded by the U.K. Department of Trade
and Industry, and by Euratom under association contracts
with Sweden, France, and United Kingdom.
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