Proceedings of ITC/ISHW2007

Study of poloidal electric field generation
by ECH in a helical plasma
I.Higashi, S.Murakami
Department of Engineering, Kyoto University, Kyoto 6060-8501, Japan
Poloidal electric field generated by ECH is investigated in a helical plasma. A linearized Fokker-Planck
equation is solved by the adjoint method assuming a helically symmetic configuration for simplicity. It is foundthat the generated electric potential in the helical plasma is about 20% larger than that in the tokamak plasma.
This indicates that about two times greater poloidal electic field is generated in a l=2 helical plasma.
Keywords: poloidal electric field, ECH, helical plasma

1 Introduction
Electron cyclotron heating (ECH) accelerates electrons perpendicularly and generates trapped particles,
which tend to localize at the resonance region. Those
resonant trapped particles would enhance a inhomogeneous electrostatic potential on a flux surface andgenerate a poloidal electric field resulting in the large
radial transport due to the radial drift by E × Bdrift.
M.Taguchi(1992)[1] evaluated the poloidal electric field generated by ECH in a tokamak plasma bysolving an adjoint equation to the linearized FokkerPlanck equation with a quasi-linear diffusion term.
Because of the deeper magnetic ripple by the helicalcoils the larger poloidal electric field would be generated in helical plasma than that in tokamaks.
In this paper, the poloidal electric field generated by ECH is investigated in a helical plasma in thecollisionless regime. Applying the same method by
Taguchi, the poloidal electric field is calculated assuming helically symmetric configuration and compared with one in a tokamak plasma.

2Basic equation

We consider a toroidal plasma, where the magnetic
field is expressed in the Booer coordinates (ψ, θ, ζ),
where ψ, θ, ζ are the toroidal flux, the poloidal angle and the torodial angle, respectively. When the RF
power is assumed to be weak, the gyrophase-averaged
distribution function for the electrons is slightly distorted from the Maxwell distribution function fe0 =ne0 (me /2πT e )3/2 exp(−me v2 /2T e ), where me and T e
are the electron rest mass and electron temperature.
The distorted part fe1 is determined by the lineraizedauthor’s e-mail: higashi@p-grp.nucleng.kyoto-u.ac.jp

drift kinetic equation:∂ fe0∂W∂ fe0+Qrf ( fe0 ) −∂t

vk b · ∇ fe1 − Ce ( fe1 ) = evk E · b(1)where the energy W = 12 mv2 , the magnetic momentµ = v2⊥ /2B and σ = vk /|vk | as independent variables in velocity space. b = B/B ,vk = v · b, v⊥ =(v2 −v2k )1/2 , Ce is the linearized Fokker-Plack collision
operator, and Qrf and E = −∇φ are the velocity-spacediffusion and the poloidal electric field due to ECH,respectively. To calculate the electrostatic potential φ
we introduce the adjoint equation
vk b · ∇ f˜m,n + Ce ( f˜m,n ) = −νee fe0 ei(mθ+nζ)√g(2)where νee = (4πne0 e4 ln Λ)/m2e v3e , ve = (2T e /me )1/2 ,ln Λ is the Coulomb logarithm, θ is the poloidal angle√
and 1/ g is Jacobian. We multiply (2) by fe1 / fe0 ,
integrating over velocity space and averaging over theflux surface. Then, the electrostatic potential φ can beexpressed as*+ne0 e 1 i(mθ+nζ)φ =√ eTeg*Z ˜ "# +fm,n1∂ fe0−
Qrf ( fe0 ) −dv
νeefe0∂t*+ne1+ √ ei(mθ+nζ)(3)gwhereZ
ne1 ≡fe1 dv(4)Here, hAi is the flux surface average of a quantity A:
IIdθdζ
dθdζ
hAi =
√ A/
√gg

Proceedings of ITC/ISHW2007

3

And we have used the relation*Z+*Z+f vk b · ∇gdv = −gvk b · ∇ f dv

In the collisionless regime we expand f˜m,n as f˜m,n =(0)(1)(0)
f˜m,n+ f˜m,n+ · · · . Then the distribution functions f˜m,n(1)and f˜m,n satisfy the equations

and the adjoint property of the collision operator,Z

fe1
Ce ( f˜m,n )dv =fe0Z(0)vk b · ∇ f˜m,n=0f˜m,nCe ( fe1 )dv
fe0νee fe0(1)(0)vk b · f˜m,n+ Ce ( f˜m,n) = − √ ei(mθ+nζ)
gA similar relation can also be derived for ions. It becomes+*+*ni0 eZi 1 i(mθ+nζ)
ni1= − √ ei(mθ+nζ)(5)√ eTigg
Zi is the charge number of ion and T i is the ion temperature.
We expand the electrostatic potential in a Fourier
series:∞Xφ=φm,n e−i(mθ+nζ)(6)m,n=−∞The left hand side of (3) becomes*+ne0 e 1
LHS =√ φm,nTeg
ne0 e2πH √=φm,nTegdθdζ(7)

Using charge neutrality, the second term in the righthand side of (3) becomes+ *+*Zi ni1ne1 i(mθ+nζ)= √ ei(mθ+nζ)√ egg*+ni0 eZi 1 i(mθ+nζ)= −Ziφ√ eTigne0 eZi2πH √φm,n (8)=−TigdθdζTransposing the second term in the right hand sideand arranging the equation give us
eφm,nFm,nνrf=−Teνee 1 + Zi T e /T i
RFm,n =(9)ihf˜m,n∂ fe0fe0 Qrf ( fe0 ) − ∂t dv
DREW
T e Qrf dv

H √1ne0 T e*Z+WQrf dv

λ = 2µ/v2 . Here the bounce average hAib is definedbyhAiI ZZ√√hAib = gdθdζA/gdθdζ
(11)(0 ≤ λ < λc )(λc < λ ≤ λmax )where λc = 1/Bmax λmax = 1/Bmin . In order to solve(15), we consider the following eigenvalue equation:" DE dG #d1/2λ (1 − λB)b dλdλ*+B
1+κG = 0(16)2 (1 − λB)1/2 b

subject to the boundary conditions

where(13)where x = v/ve . For simplicity we use this approximate collision term here. Then the solubility condition can be written as(0) E ∂ f˜m,n1 ∂  D
1/22(1 + Zi ) 3
λ (1 − λB) b∂λ x ∂λ  (0) +*fe0 ∂  f˜m,n B−
(1 − λB)1/2 b x2 ∂x  fe0 *+B
1=−
√ ei(mθ+nζ) fe0 (15)1/2g(1 − λB)b(0 ≤ λ < λc , λc < λ ≤ λmax )2π(12)(0)This function f˜m,nis determined by the solubility condition for (13). The linearized Fokker-Planck collision operation Ce is approximated well for v > (2 −3)v3 by!
νee vk ∂∂fCe ( f ) ≈ (1 + Zi ) 3µvk∂µx B ∂µ!
νee ∂ f− fe0 2(14)x ∂x fe0gdθdζ(10)νrf ≡Solution of the adjoint equation

G(λc − 0) = G(λc + 0)dG(λc − 0) dG(λc + 0)=dλdλG(0) = 1,dG(λmax )is f initedλProceedings of ITC/ISHW2007
The eigenfunctions Gn with eigenvalues κn satisfy theorthogonality condition+Z λmax *BGnGm dλ = 0 (κn , κm )10(1 − λB) 2 b

This resonance condition becomes a semicircle in velocity space

Using this, we can express the solution of (16) in theformwhere u0 = (ω − lωc )/kk ve is the normalized parallel
velocity of the resonant electrons and S = lωc ve /kk c2
relates the strength of the relativistic correction.

(0)
f˜m,n=∞Xx3
Gn (λ)S n fe0(1 + Zi )κn + 3
n=14.1bNote that the eigenfunction G0 = 1, so that S 0 = 0.
Substituting the solution (17) into (9), we obtainH √gdθdζ1DREFm,n =W2πT e Qrf dv+*Z∞XSn
3x Gn Qrf dv(18)×(1 + Zi )κn + 3n=1since*Z∂ fe0x Gndv∂t
*Z λmax
B!1/2
11 − h cos θ01−xdθ00)0(1−cosθ1−hhθ1(24)
!−1/2
Z θ00
2
1 − h cos θ1β=1−xdθ00(1−)
1 − hhθ1(25)!−1/2
Z θ0021dα
11 − h cos θ=−
1−xdθ0dx
2 θ10 (1 − h )1 − h(26)Z θ0α=++GnG0 dλ = 0 (n ≥ 1)

10(1 − λB) 2

Helical symmetric configuration

We assume a helical symmetric plasma with magneticNζ)).
configuration B = Bh (1 − h cos θ0 ) (θ0 = M(θ + M
Using x = λBh (1 − h ) we rewrite (16) and boundary
condition .
!
d1d2
dα+α
G + βκG = 0 (23)xα 2 G + xdx
dx
2dx

3

∝

(22)

(17)

where κ0 (= 0) < κ1 < κ2 < · · · and
R λmax
B1 i(mθ+nζ√Gedλn10(1−λB) 2 gbSn =R λmax
B2Gdλ1
n0(1−λB) 2!2!2vk1v⊥
1 − 4u0 S−+=ve 2Sve
(2S )2b2(19)4

1G → const × [1 − κ(1 − x)]
2
1G → 1 − κxx→02

Evaluation of φm,nThe quasi-linear diffusion term Qrf for electroncyclotron damping is written as!2(l−1)1 ∂ v⊥
QRF =Dv⊥v⊥ ∂v⊥ ve#!
lωc∂ fe0× δ ω−− kk vk
(20)
γ∂v⊥
1where γ = [a − (v/c)2 ]− 2 , c is the speed of lightin vacuum and l is the harmonic number; ω and ωcare the frequency of the injected wave and the nonrelativistic electron-cyclotron frequency respectively.
Throughout this paper we consider only the X-modewave. Then D is taken to be independent of the velocity. Moreover, for simplicity, we approximate therelativistic resonance condition as!
lωcv2
ω−− kk vk ≈ ω − lωc 1 − 2 − kk vk = 0γ2c(21)x → 1 (27)(28)

We evaluate the eigenfunctions Gn and eigenvalues κnby a relaxation method. The eigenvalues κn are given
in table 1.
\n0.1
0.3

1
2.886
2.377

2
8.400
4.983

3
15417
12.837

4
26.64320.314

Table 1 eigenvaluesκn540.19631.568

Proceedings of ITC/ISHW2007

4.2

Results

We evaluated the poloidal electric field in a helicalsymmetric plasma and the results are compared withtokamak ones by Taguchi[1]. The poloidal electricfield is represented as Eq.(9) and is analayzed in ahelical plasma.
Figure 1 shows Fm values in a tokamak and ahelical plasma changing the magnetic field ripple ;0.1, 0.3 and the normalized parallel velocity; 0, 0.5, 1.0. We can see that the Fm increases as the  becomeslarger in both helical and tokamak plasma cases. And
the differences of Fm is larger in the larger  cases.
The value of φh1 /φt1 ∼ 1.25 − 2 in the case of Fm with
 = 0.3. The values of Fm with various  of a helicalplasma are shown in figure 2.

5Conclusion

We have studied the poloidal electric field generated by ECH in a helical plasma. The linearized
Fokker-Planck equation has been solved by the adjoint method assuming a helically symmetic configuration for simplicity. We have found that the generated electric potential of the helical plasma is about
20% larger than that of the tokamak configuration.
This indicates that the two times greater poloidal electic field is generated in a l=2 helical plasma.

Fig. 2 Plots of Fm for helical plasma with  = 0.05, 0.1,0.15, 0.2
Fig. 1 Comparisons of Fm in the tokamak and helicalplasma[1] M.Taguchi, J.Plasma Physics. 47(1992), part 2, 261[2] S.Murakami, et al, Nucl. Fusion 40(2000) 693