Ray tracing is based on WKB approximation: if the wavelength is short compared to the scale length of tokamaks, the wave quantities such as electric field is taken to be wary as
( ) exp



 x x
    z
 



Here the x axis is taken to be along the direction of the inhomogeneity. The wave propagates with fixed k y
, k z
, and ω and with k x
(x) determined through the local dispersion relation, the parameters of which are functions of x.
E
0 exp( i
  relation.
The direction of energy flow is given by the group velocity . g
  
/
 k
The vector k is determined by a dispersion relation of the form: D ( , , )

0 and the fact that along the ray path:
The ray tracing equations can be given by:
/
 
D /
 k dk dt
 
D /

)( / )

0
/ ( / ) /( /

)
/ ( / ) /( D /

)

Along the ray path, based on the requirement of conservation of energy flux and weak attenuation approximation, the evolution of the electrostatic wave power is given by 𝑑𝑃 𝑑𝑡
=−2 𝛾 𝑡
Where 𝛾 𝑡
P, 𝛾 𝑡
= 𝛾 𝑒
+ 𝛾 𝑖
+ 𝛾 𝑐 is calculated by writing:
Expending D for small 𝛾 𝑡 and 𝐷 (𝛼) with 𝐷
0
(𝒙, 𝒌, 𝜔 )=0, yields:
The expressions for 𝐷
(𝛼) are given by [*]
[*]

The TORLH[J. C. Wright, 2009] is a full wave electromagnetic code, which solves a wave equation of the form: c
2
ω 2
𝛻 × 𝛻 × E = SE
⊥
+ iD(b ×E
⊥
) + PE
∥
+
4πi
ω
( J e
2
+ J
ANT
) ,
4πi
ω
J e
2 c
2
= −
ω 2
𝛻
⊥
× λ
0
𝛻
⊥
× E
⊥ c
2
+ i
ω 2
{𝛻
⊥
× [ξ
0 b ∙ 𝛻 E
∥ b + b b ⋅ 𝛻 ξ
0
λ
0
=
1 ω
2 pe
2 Ω
2 ce v
2 te c 2
[−x e
Z x e
] , ξ
0
=
1 ω
2 pe
2 ωΩ ce v
2 te c 2
[x 2 e
Z ′ x e
] b ∙ 𝛻
⊥
× E
⊥
} , where S, D, and P, are the Stix components of the dielectric tensor, expanded in the LH range of frequencies ( Ω 𝑐𝑖
≪ 𝜔 ≪ Ω 𝑐𝑒
)
A semi‐spectral ansatz is used in TORLH to represent the electric field in terms of poloidal (m) and toroidal (n) Fourier modes:
E x = E m,n
(Ψ)e imθ+inϕ m,n
The radial dependence of the coefficients in the above equation is expressed in terms of cubic
Hermite interpolating polynomials. In the full wave simulations carried out for EAST we have taken 2047 poloidal modes ( −1023 < 𝑚 < +1023) and 480 radial elements in order to resolve the short wavelength electrostatic LH wave ( 𝜆
⊥
∼ 1 − 3𝑚𝑚) .

Quasilinear calculations[*] have shown that damping of LH waves begins at a phase velocity of 𝜔 𝑘
∥ 𝑣 𝑡𝑒
= 2.5
, which can be restated as 𝑛
∥
= 5.7 𝑇 𝑒
(𝑘𝑒𝑉) . Current drivie efficiency scales[#] as 𝑛 𝑒
/𝑛
∥
, thus the accessibility limit sets the maximum current drive effciency. If the electron temperature is not sufficiently high, it requires high 𝑛
∥ sufficient Landau damping. The difference between the launched 𝑛
∥ to have
,which tends to be just above the accessibility limit, and the slower wave velocity (require higher 𝑛
∥
) at which damping occurs is known as the spectral gap.
[*]P.T. Bonoli, IEEE Trans. Plasma Sci. 12,95(1984)
[#]N.J. Fisch, PRL, 41,873(1978)

Note that in toroidal geometry the poloidal mode number of LH wave is not a conserved quantity and can vary considerably 𝑚 𝑘
∥
= 𝒌 ∙ 𝒃 = [ 𝑟
𝑩 𝜽
+ ( n
R
) 𝑩 𝝓
]/ 𝑩
Toroidal effects move the value of 𝑘
∥ up and down during propagation: A rise helping absorption, and a fall retarding absorption.
The rise in 𝑘
∥ is a stronger effect as density, poloidal field inside the plasma increase and as aspect ratio (R/a) decreases [*]
Motivations of the following works:
The spectral gap for LHW in EAST is expected to be very large, and the LHW is in a low single pass absorption regime, which poses a challenge for the simulation of
LHCD in EAST device with large aspect ratio.
[*] K. Kupfer and D. Moreau, \Wave chaos and the dependence of LHCD eciency on temperature," Nucl. Fusion 32 1845 (1992).
H. Takahashi, et al., \Accessibility for lower hybrid waves in PBX{M," in 1993 Conference on Controlled Fusion and Plasma
Physics (Proc. Conf. Lisbon, 1994) Vol 17C, Part III, European Physical Society, Geneva (1993) 901.
F. Paoletti, et al., \LHCD Accessibility Study with Reconstructed Magnetic Equilibria in PBX-M," Nucl. Fusion 34 771-776 (1994).

Study of the Spectral gap problem in EAST LHCD using ray-tracing (GENRAY)
LHW(2.45GHz, 𝑛
∥
~2.1
) power deposition profile modeled by the GENRAY with the parameters: Te
=1.5keV, 𝑛 𝑒
= 1.0 × 10 19 𝑚 −3
LHW undergoes multiple passes before toroidal effects give rise to sufficient upshift in the parallel wavenumber
Simulation results shows that the spectral gap in EAST LHCD is large and the Landau damping relies very much on the up-shift in the parallel wavenumber due to the toroidal effects, since the power is not able to be absorbed on-axis where the temperature is higher but the poloidal field ~0.

Comparisons between the GENRAY and the TORLH simulation for the LHW in EAST for the Maxwellian case
Maxwellian electron distribution function assumed for both the GENRAY and the TORLH simulation:
Spectrum of Re(E) versus m
~ r/a
0.10
0.30
0.50
0.70
0.90
1.00
1000 500 0 500
Poloidal mode number - m
Poloidal spectrum modeled by TORLH
1000 𝑛
∥
Parameters:
= 2.1, 𝑇 𝑒 𝑛 𝑒
= 1.5𝑘𝑒𝑉,
= 1.0 × 10 19
The power deposition related to the up-shift of 𝑛
∥ due to the toroidal effect calculated by the GENRAY and the TORLH agree well with each other. Besides, the full wave effects play another important role in the 𝑛
∥ up-shifting.

Comparisons between the GENRAY and the TORLH with different temperature
1.5 keV 5.0 keV
电子温度越高，低杂波 spectral gap 越小，吸收加强，朗道阻尼对 𝑘
∥
上移的依赖性越小，采用
不同的最大极向模数的全波结果就越接近，全波结果和射线追踪结果也越接近。

Comparisons between the GENRAY and the TORLH simulation for the LHW in EAST for the Non-Maxwellian case
Using the same Non-Maxwellian electron distribution function provided by the iteration between GENRAY and
CQL3D for both the GENRAY and the TORLH simulation:
Maxwellian results of wave electric field modeled by TORLH:
Quasi-linear results of wave electric field modeled by TORLH
For the non-Maxwellian case, the two deposition profiles look similar to each other except for a slight difference between the positions of the peaks, which indicates that the discrepancy of the two codes is diminished as the results of the iteration between the GENRAY and the CQL3D being used.

• The spectral gap for LHCD in EAST has been studied and proved to be very large as expected. The electron Landau damping of the LHW relies very much on the upshift in the parallel wavenumber, and the LHW is in a low single pass absorption regime, which poses a challenge for the simulation of LHCD in EAST.
• The GENRAY and TORLH agree well with each other on the calculation of the upshift of 𝑛
∥ due to the toroidal effect. The full wave effects also play important role in up-shifting in the parallel wavenumber with sufficient maximum poloidal mode numbers. But the discrepancy of the two codes is diminished in the quasi-linear case as the damping becomes stronger.
• Simulations using Genray-CQL3D and TORLH models for LHCD in EAST would be an important addition to the validation studies of those models being used in low single pass absorption scenarios of tokamaks with large aspect ratio.