pfi Vo ?. 2, 2 DRFC/CAD
EUR-CEA-FC-1415
Stochastic Instability of Relativistic Runaway Electrons due to Lower Hybrid Waves J. M. RAX, L. LAURENT, D. MORgAU
Février 1991
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4
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STOCHASTIC INSTABILITY OF RELATIVISTIC RUNAWAY ELECTRONS DUE TO LOWER HYBRID WAVES
J-M. Rax*, L. Laurent, and D. Moreau Association EURATOM-CEA sur la Fusion Contrôlée 13108 Saint-Paul-lez-Durance, FRANCE
Abstract Runaway elections and lower hybrid waves in tokamak discharges are coupled because of the relativistic decrease of the cyclotron frequency. Two interaction regimes must be considered. For the low energy pan of the relativistic runaway spectrum, neighboring low harmonic nonlinear cyclotron resonances may overlap. For the high energy pan, the influence of the higher order resonances are more conveniently studied with the help of a map describing nearly perpendicular Landau coupling. In both cases it is shown that the stochasticity threshold can be reached in typical discharges, leading to quasilinear diffusion.
PACS NUMBERS:
52.55.Pi
*Present adress: Plasma Physics Laboratory, Princeton University Princeton NJ 08540 USA
52.60,+h
41.70.+L
1. Ordering and modeling Lower hybrid (LH) current drive is now widely used to generate the toroidal current in tokamak [1, 2] and to control the associated profile [3] through a Landau parallel resonance with suprathermal electrons. During the transient phase of tokamak discharges large values of the electric field in low density plama result in the production of runaway electrons (RE) [4]. Such fast particles can also interact with LH waves because of the relativistic decrease of the electron cyclotron frequency. Cyclotron and perpendicular Landau resonances occur between LH waves and RE, ultimately leading to stochasticity when appropriate criteria are fullfUled. This letter considers this direct interaction. Indirect interaction due to the LH current profile modification inducing perturbed RE orbits is not considered here. The notations will be as follows, -e and m are the electron charge and mass, c is the velocity of light, p the ratio of the particle velocity to c, p the momentum and the index » and J. indicate parallel and perpendicular directions relative to the magnetic field. A complete model of the RE dynamics in wave driven tokamaks is beyond the scope of this letter whose main goal is to point out the existence of new non linear resonances and to calculate the associated stochasticity thresholds. The relativistic RE are a population within the energy range (y is the ratio of the relativistic energy to the rest energy): O(l) 0(10), N11 = 2, N^O(IO),
5N11TN11SO(I)
(2)
under current drive conditions. The LH spectral width 8N1I induced by toroidal broadening is of the order of one or larger [5, 6]. This ordering allows a Landau coupling to the parallel velocity of the suprathermal electrons [1,2]. The large value of the perpendicular index is responsible for a coupling to the thermal ions because their perpendicular velocity can match the the perpendicular phase velocity of the wave [7, 8,9]. On the other hand the perpendicular degree of freedoom of the thermal electron population is not coupled to the LH wave, because the (non relativistic) electron cyclotron frequency Q is larger than ft), the LH wave frequency, and the thermal Larmor radius is smaller than the perpendicular wavelength. But these latter conclusions are to be revisited if one considers a relativistic
population: the cyclotron frequency is lowered by a factor y. We shall show that the resulting increase of the Larmor radius and decrease of the relativistic cyclotron frequency down to Q/y will allow a set of unsuspected non linear resonant coupling. The RE free dynamics is governed by the balance between the loop voltage acceleration and the synchrotron slowing down (we consider an ideal axisymmetric tokamak and neglect the coupling with the ripple [10] ). The time scale associated with these two processes is of the order of few seconds. In the following we shall deal with oscillatory processes taking place on far much shorter time scale. Thus, we will consider a fixed RE energy spectrum; the influence of cyclotron losses is briefly considered in the conclusion. A small finite value of the perpendicular kinetic energy of RE: 0(10-2) < Px < 0(1),
(3)
as been observed experimentally [11,12, 13], its origin is not yet completely elucidated. While this energy is negligible compared to the parallel one, it nevertheless allows a coupling with the LH wave with the following values of the wave-particle coupling parameters: (4)
This RE-LH coupling under typical current drive conditions indicates that the perpendicular wavelength can be of the order of the the Larmor radius and even smaller; and that the wave frequency can be of the order of the relativistic cyclotron frequency and even larger. Under these conditions, we shall see that, cyclotron harmonic interaction and nearly perpendicular Landau interaction between the wave and the particle occur. The length scales of the wave particle interaction problem are smaller than the macroscopic gradient scales of the tokamak configiration, thus an homogeneous model is sufficient to study this interaction. We consider a homogeneous static magnetic field directed along the z axis, and for each index of the spectrum an electrostatic waves propagating in the x, z plane. The unperturbed motion of the RE is the combination of the usual relativistic cyclotron motion around the z axis with a constant velocity along this direction.
2. Low energy relativistic cyclotron interaction To analyse the influence of the previous wave on this motion we introduce the Hamiltonian: mc2V 1+P2An2C2 + —- sinj; sinfj;c [N1XH-NnZ-Ct] + $}, O)Nx \ '
(5) 2
and define a set of action variables of the unperturbed motion, ç = p^mc, n = p^/2Qm, which are canonically conjugated to the wave rest frame drift a = mc(z - ct/N|j) 4- (j>mc2/a)N|| and to the gyroangle O, the associated Hamiltonian becomes: l=+oo H = m c 2 l + C2 + 2£ty/mc2 -
N
+— J,(rj)sin a + IQ coN,1 4-J l \mc2 /
H
(6)
- - l=-oo
J] are the Bessel function of integer index resulting from the harmonic cyclotron expansion and T| is defined in equation (4). This Hamiltonian enables us to identify a set of resonances. We consider the unperturbed trajectory, along which, a perturbation analysis fails to converge (because of small denominators) when the argument of the 1 sinusoidal wave is stationnary: 1— -0 + P8N11(O = O. iy
H
H
(7)
Under typical current drive conditions (NN * 2, §N|/N,| » O[l], $\\ *1) this equation indicates that resonances are roughly given by C = Y = - IQAu ( these are anomalous Doppler resonances, PH — 1 [4] gives normal resonances 7 = lQ/3co that can be investigated in the same way). Thus we are lead to consider two energy regimes. When ytû/£ï =1 low order Doppler cyclotron resonances 1 = -1, 1 = -2 are effective, when fù/Q. >l higher order resonances are to be considered. In the first case the small Larmor radius is valid and in the second one because of the large value of the y factor it does remain valid and the interaction becomes a quasi perpendicular Landau resonance considered in the next section. To study the low energy, y = O(10), case we perform a second canonical transform of the unperturbed Hamiltonian near the I1*1 resonance : C1 = ^, fi' = Ji - mc2lC/coN||, 6' = 6, a1 = a + mc219/(uN|| and use the small Larmor radius expansion of the Bessel functions: (8,
We recognize a non linear oscillator with angle a' and action C', the width of the trapped domain is:
Thus for each NU we have a set of islands in phase space corresponding to the various values of 1 allowed by the resonance condition. Using dy/dC = PU »1 at constant |i and differentiating the equation (7) we obtain the distance between two non linear resonances separated by Sl and SNg, SC(Sl. SN||) * I51Q/Q + 8N||Yl, ( y * C ). |8l| is of the order of unity as well as |SN|||, for example if we consider the distance between the upper and lower pan of the LH spectrum associated with neighboring harmonics ( N||=3,1 =-2 with Nj = 2,1 =-l)we obtain SC »0 wich indicate a very strong interaction. More generally the coupling strength between two non linear resonances is given by the Chirikov parameter [ 14,15]: S =
AC1O, N n ) + AC1O', N1Ii) « IE ^ " --1^—It-- 4 ~A/o5TyP.L51 + 5N|, 2SC'(r-l,N'rN|,) Q \2B 0 c IHJ -' "Q'
.
(10)
Where we have introduce B0 the value of the static magnetic field. When this stochasticity parameter is of the order of unity the two resonances overlapp and the motion becomes stochastic. With typical parameters: E,, = 7. KV/cm, NH = 2., û) = !Ore GHz, Nx = 20., B0 = 3. T, 7 = 20 and Px = 2. 10"1 we see that the stochasticity parameter S approach unity and thus the low energy part of the RE spectrum is unstable with respect to the stochastic instability induced by the low (anomalous) harmonic cyclotron (non linear) resonances of the LH wave. Under experimental conditions, because the wave is not plane, and the unperturbed drift trajectory not a straight line additionnai decorrelation mechanism ate to be considered and we expect that this instability will thus be enhanced.
3. High energy quasi perpendicular Landau interaction Now let us consider the other regime (y = Q[IOO]) when the wave length is smaller than the Larmor radius and when the cyclotron frequency is below the LH wave one. Rather than using the canonical Hamiltonian formalism we shall calculate the exange of energy mc^y and paralell momentum mci; between the wave and the particle directly from the Lorentz force. 3t
=
nï [EA«»( t )+ E1Pi 1 COSG)[N1P1J sin( t) + (N n P n -I)I]
BT =m^Eiic°s(^NiP4sin1 the interaction will be localized near these stationary points (large number of wavelength in one larmor radius and the particle experiences many oscillations at the stationary points). Thus in this high energy regime the exchange of energy and momentum can be modeled as a set of localized kicks. We integrate the differential equation (11) with the stationary-phase method: we expand, up to second order in t around t~, the trajectory inside the argument of the cosine of the wave. .The corresponding energy and momentum increments are :
'*™'
(W)
We have introduced w(y,^)= TJ "V 1-u2 and used the equations (12, 13). We can easily build an invariant I of this impact dynamic: I=N||Y~ £, C+ ^ - In" = O. The upper and lower boundaries in the previous integral can be extended toward infinity because the oscillatory character of the integrand far from s = O cancel out the positive and negative large s values contribution to the sum. P
n+r Yn= rc«[± 1. This criterion is a rough estimate but it as been confirmed by numerical study of the map. This map display invariant Jines coresponding to the zero of K : W-UT) arccos(u)= rt/4 + im (zero of the cosine) and invariant points corresponding to the values of y,
1, can be reached in typical current drive experiments. The quasilinear diffusion coefficients will be calculated elsewhere, nevertheless the amount of LH power going into this fast population is expected to be negligible. This study was motivated by the unexplained behaviour of RE in Tore Supra during LH operation. It is not yet confirmed if this new mechanism is relevant to this set of experiments, but it appears that with this new mechanism LH waves under proper conditions can be used to disrupt RE beams as this energetic population is to be controlled for safer tokamak operations. Aknowledgements: We whish to thank G. Martin and M. Chatelier for fruitfull discussions on the TS runaway experimental program. One of us (JMR) would like to aknowledge PPPL for its hospitality and N. Fisch for simulating discussions.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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