ФИЗИКА ПЛАЗМЫ, 2014, том 40, № 12, с. 1050-1056
ТОКАМАКИ
УДК 533.9
INFLUENCE OF PLASMA PARAMETERS ON THE ABSORPTION COEFFICIENT OF ALPHA PARTICLES TO LOWER HYBRID WAVES
IN TOKAMAK
© 2014 г. J. Wang, X. Zhang, L. Yu, and X. Zhao
Department of Physics, East China University of Science and Technology, P.O. Box 385, Shanghai 200237, People's Republic of China e-mail: zhangxm@ecust.edu.cn Поступила в редакцию 08.11.2013 г. Окончательный вариант получен 14.02.2014 г.
In Tokamaks, fusion generated a particles may absorb Lower Hybrid wave energy, thus reducing the Lower Hybrid current drive efficiency. The absorption coefficient ya of Lower Hybrid waves due to a particles changing with some typical parameters is calculated in this paper. Results show that, the ya increases with the parallel refraction index ny while decreases with the frequency of LH waves ю over a wide range. Higher background plasma temperature and toroidal magnetic field will increase the absorption. ya increases with ne when ne < 8 x 1919 m-3 while decreases with ne when ne becomes larger, and there is a peak value of ya when ne« 8 x 1019 m-1 for ITER-like scenario. The influence of spectral broadening in parametric decay instabilities (PDIs) on the absorption coefficient is evaluated. The ya with n| | being 2.5 is almost two times larger than that with n | | being 2.0, and the case of 2.9 is even lager, which will obviously increase the absorption of LH power by alpha particles. DOI: 10.7868/S0367292114100084
1. INTRODUCTION
In the past decades, lower hybrid waves have been successfully used for auxiliary heating and current drive in experimental tokamaks [1—4]. It has been proved that Lower Hybrid Current Drive (LHCD) is one of the leading candidates for off-axis current profile control in order to sustain steady-state scenarios in future reactors [5, 6]. However, the lower hybrid waves can efficiently interact with fast ions in fusion plasmas [7—13]. In particular, the fusion-generated 3.5 MeV alpha particles might absorb a large fraction of the lower-hybrid wave power, thus reducing the current drive efficiency [2, 3, 14—17]. The experiments also presented the evidence of the interaction of LH waves with ICRH minority ions of few MeV of energy [18, 19].
Fortunately, the damping LH waves on alpha particles could be neglected by a proper choice of the LH frequency [2, 15, 20]. In fact, from these preliminary numerical studies, the fraction of LH power absorbed by alpha particles depends sensitively not only on the frequency of the LH waves but also on various parameters of plasma and LH waves, i.e. the refractive index of the LH wave vector along the magnetic field «lj, the background plasma temperature spatial profile Te toroidal magnetic field B^ and the density profile of plasma background ions n. For these reasons, this pa-
per is aimed at calculating the absorption coefficient with respect to all these parameters.
Previous studies have shown that the spectral broadening in PDIs is weakly sensitive to changes of the peak of the launched antenna spectrum and the operating magnetic field. Instead, the LH spectral broadening depends mainly on the changes in the electron temperature profile of the SOL and the lower LH pump electric field [21]. A smaller spectral broadening is expected in case of higher electron temperatures and lower LH pump electric field, which may provide an essential way of decreasing the absorption of alpha particles on lower hybrid waves.
The paper is organized as follows. In section 2, we present the models employed to calculate the absorption coefficient ya. The numerical results are presented in section 3. We will evaluate the influence of parametric decay instabilities (PDIs) on the absorption coefficient in section 4. Finally, we summarize the main results of the paper and give some discussion in section 5.
2. THE CALCULATING MODEL OF THE ABSORPTION COEFFICIENT
An evolution equation for the LH waves power flowing along the ray path is integrated simultaneously with the ray equations and has the form [5]
dP}
LH
dt
= -2 (Ye + ) Plh,
(1)
(2)
Where
4 2
Do = P4HL + P2n, + Po .
Po = S||[(n,| - s,) - sxy]
P2 = (6|| + s, )( n2 - s, ) + sX P4 = s,.
(4)
(5)
With
N.
spec 2
s, = 1 - E
j = : N
spec
EE
№
p,j
22
, № - №c <
j = 1 c'j
Nsp
spec 2
s = 1 v №p ' j
s|| = 1 - E ~
j=1 №
(6)
Ns
spec 2
№p ;№c
E
"p J^cj
equation, and the hot plasma dielectric tensor can be written as [5, 22]
where PLH is the LH waves power, ye and ya are the damping rates on electrons and alpha particles. In this paper, we are mainly concentrated on the calculation of Ya with respect to various plasma and LH waves parameters.
The plasma dispersion relation can be written as [22]
D(x, k, ® - iy) = D0(x, k, ® - iy) +
+ iIm[D(x, k, ® - iy)] = 0, where D0 is the real parts of D and Im[D(x, k, ® — /y)] is the imaginary parts of the dispersion relation that results from the damping mechanisms. Expanding D for small y and Im[D(x, k, ® — /y)] with D0 = 0, yields [5, 22, 23]
y = Im [ D(x, k, ® - iy ) ]
la - r. • VJ/
d Do
5®
The expression for the real parts of D, D0 is written
as:
K = I + № j2np± J ¿p E
№ 0 -œ m = -a
where
№ - ^||V|| - m№c
(7)
S =
mJm\2 .v UmJmJm
J2
vLW—m x
-i vLUm
JmJm
V||
V, U( J m ) -iV, WJmJm
i V|| UJmJm
V|| WJm
(8)
with
U=-+-f
W = M№ f - Mm — Í fpL - fp,
dpN p,vôpy dp,
(9)
where p and pL are the particle momentum components, respectively, parallel and perpendicular to the magnetic field, M is the relativistic mass (M = ym0,
y being the relativistic factor). The Jm and Jm are the modified Bessel function of the first kind and its derivative, with argument x = k1v1/®c, a.
In the quasi-linear theory, the damping phenomena corresponds to the anti-Hermitian part of the dielectric tensor [24—26]. The resonance between LH waves and alpha particles appears when ® — k| |V| | — m®c, a = 0, occurring when the denominator in equation (10) tends to zero. One can rewrite this denominator as follows:
1
№ - k||V|| - m—c
=P
1
v№ - k|| V|| - m№c>ay
(10)
22 = 1 №(№c,j - № )
In equation (5) and (6), Sy, s± and sxy are the elements of plasma dielectric tensor. « and nL are the wave refraction index parallel and perpendicular to the local magnetic field respectively. Nspec is the total number of ion species, ® the frequency of the electromagnetic wave and ®c, j and ®p, j are respectively the plasma and cyclotron frequencies for species j. Here the alpha particles' contribution to the real part of D is neglected since we consider them as a small minority.
The expression for the imaginary parts of D is derived from the hot plasma dielectric tensor, which is derived from a linearized treatment of the Vlasov
- in8(® - k||V||- m®c>a),
where 8 is the Dirac function and P denotes the principal value.
After some basic assumptions and simplification [5], one can get
22
Im(D(x, k, ® - iy)) = - (nL«||) + (S|| - nL) x x (2s_l - n2 - nj- 2Sxy) x
ff_! 2 (11)
X
œ
30
œ
V
,
sxy
Therefore the damping on alpha particles can be thus written as:
Y = --- (nJn | | ) 2 + ( E II - n J-) ( 2 E J_ --- n 2 - n2 - 2 Exy) y
Ya *
dD
dœ
, p±max \
x , -2nm*Zn J ip ^
/ r c 12 (12)
JL-
Plm
n^ 1 - 1 ma
where n is the wave refraction index, and c is the speed of light in vacuum. Za, ma and na are the alpha charge, mass and density respectively. pimax = maV0 and Pimm = mac/nlmax, is the maximum and minimum of the perpendicular momentum respectively, at which the resonance occurs. nlmax is the maximum of local n±, and V0 is the alpha birth velocity, which is 1.3 x 107 m/s. D is the LH dispersion relation, andf is the alpha perpendicular distribution function.
Alpha particles are created continuously by the D-T fusion reaction at a 'source rate' S with energy approximately 3.5 MeV, i.e. velocity 1.3 x 107 m/s and their birth distribution is isotropic in velocity direction. So we treat the alpha particles as a 'beam' of energetic ions and assume f as the slowing down distribution [27]:
/l =
Se0ma mi
(
nZeffZyin A
1
, 1 + F3/ Vj
(13)
F < Fo ; /l = 0, F > Fo
with
Vcrit _ ( 2 Wa, crit/ma) _
= 3 zeff (n/2) [ Te/(me mt )] ,
where Fcrit is the beam-ion 'critical' velocity at which the contributions of background ions and electrons to the slowing down rate are exactly equal. mi is the mass of background ion particles and e is the absolute electronic charge. lnA is the Coulomb logarithm. Other notations have the normal meanings. Note that we use the equation (13) as the distribution function with a
normalization such that
p±m
P±m
2 n/LpLdpL = 1 .
It is worth mentioning that the LH wave absorption by alpha particles is estimated in linear approximation without taking into account the evaluation of the alpha particle distribution function induced by the interaction with LH waves. However, it is still useful for plasma experiments to calculate the absorption rate ya with respect to various parameters of plasma and LH waves numerically.
3. NUMERAL RESULTS
As has been mentioned, the absorption coefficient Ya depends on various parameters of fusion reactors and the LH waves, which are «||, œ, Te, n, B9 respectively. All parameters chosen below should satisfy the accessibility conditions and the landau damping condition, namely n± should be real and n± > 23 respectively.
The results of calculation of the absorption coefficient Ya versus n|| for the parameters of ITER-like scenario and the perpendicular refraction n± versus n are shown in fig. 1.
The upper and lower limits of n can be easily estimated by recalling that the parallel refractive index of the injected waves must satisfy the accessibility conditions. The changing of ya with respect to n are shown in fig. 1a with B9 = 5T, œ = 3.7 GHz, Te = T, = 5 keV, 10 keV, 20 keV; ne = nt = 7.0 x 1019 m-3; 50% deuterium and 50% tritium mixture, and the initial value of n is 2.0. The density of alpha particles na is 3.0 x 1016 m-3 at
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