D.2 Lower Hybrid Current Drive

Analytic expression for the properties of LH waves can be obtained in the cold plasma description if further approximations are made. Then, an analytical expression can be obtained for the LH diffusion coefficient.

D.2.1 Electrostatic Dispersion Relation

We consider the wave equation (D.4)

(D.31)

The electric field can be separated into its longitudinal and transverse components with respect to the normalized wave vector N_b:

(D.32)

so that the wave equation (D.31) becomes

(D.33)

Electrostatic waves, and also electromagnetic waves approaching resonances, such as LHW, can satisfy the condition

(D.34)

and therefore (D.33) reduces to

(D.35)

Dot-multiplying (D.35) by _b = N_b∕N_b leads to the electrostatic dispersion relation

(D.36)

and the transverse electric field is given by

(D.37)

We note from (D.37) and (D.34) that |E_bT |≪|E_bL|. The electric field is quasi longitudinal, which justify the term electrostatic approximation given to the condition (D.34).

D.2.2 Cold Plasma Limit

Using (D.6), the electrostatic dispersion relation (D.36) in the cold plasma limit is then given by

(D.38)

which gives an expression for N_b⊥ as a function of N_b∥ and ω_b:

(D.39)

D.2.3 Lower Hybrid Waves

We consider the lower-hybrid range of frequency in a electron-ion plasma, where the following ordering applies

(D.40)

Assuming in addition that

(D.41)

leads to approximate expressions of the dielectric tensor components given by:

K⊥	≃ 1 + -
K∥	≃-	(D.42)
Kxy	≃	(D.43)

where K_⊥ can be rewritten as

K⊥	=	(D.44)
	=	(D.45)
	=	(D.46)

where

(D.47)

is the lower hybrid frequency.

The perpendicular index of refraction becomes

(D.48)

We recognize that N_b⊥ →∞ as the the lower-hybrid frequency approaches the wave frequency. However, in most LHCD scenarios, wave frequency are sensibly higher than the lower-hybrid frequency in order to avoid conversion to IBW. Typically, in Alcator C-Mod, we have

(D.49)

In that case, and as shown in D.2.7, the contribution of Φ_bT to the power flow can be neglected. The cold plasma description therefore remains valid.

D.2.4 Polarization

The LH Waves of concern for LHCD are quasi-electrostatic, and therefore the electric field is quasi-longitudinal (E_b ∥ N_b) and we have

(D.50)

for any component i, so that the polarization elements become

eb,+	≡ = = e+iα	(D.51)
eb,-	≡ = = e-iα
eb,∥	≡ =

D.2.5 Determination of Θ_k^b,LH

Lower Hybrid Current Drive (LHCD) results from momentum exchange from the LH wave to the plasma through Landau damping (harmonic n = 0). In this case, the coefficient (4.308) becomes

Θkb,0	= eb,+e-iαJ -1 + eb,-e+iαJ 1 + eb,∥J0	(D.52)
	= J 1 + eb,∥J0	(D.53)

Using (D.51), we see that the perpendicular components cancel and we are left with

(D.54)

The argument of the Bessel function in (D.54) is

(D.55)

where an expression for N_⊥ is given by (D.48). The argument of the Bessel function becomes

(D.56)

and we see that for ω_pe ~ ω_ce and ω_b ~ 2ω_LH, we get

(D.57)

Given that most electrons concerned with LHCD have p_⊥~ p_Te, and that for LHCD we have typically N_b∥~ 2, we see that it is reasonable to take the limit

(D.58)

as long as the plasma is not too relativistic (β_Te ≪ 1). This limit is consistent with the validity condition of the cold plasma description. In this limit, valid for most LHCD scenarios, we have

(D.59)

D.2.6 Determination of Φ_bP ^LH

The vector Φ_b^LH describes the relation between the energy flux and the electric field. Although LHW are almost electrostatic, the dominant contribution to the power flux is the wave Poynting flux Φ_bP ^LH, assuming that the wave frequency remains sensibly higher than the LH frequency. The contribution of the Kinetic power flux is calculated in D.2.7. In most LHCD scenarios, this contribution is not more than a few percents of the total flux and can therefore be neglected. We have (D.28).

(D.60)

LH waves are quasi-electrostatic, so that the longitudinal and transverse components are given by (D.37)

ebL	≃ 1	(D.61)
ebT	= K ⋅ ebL	(D.62)

so that the first term in (D.60) can be neglected, and

ΦbP LH	≃ Re
	= Re	(D.63)

Note from (D.63) and (D.36) that the Poynting flux is in the direction perpendicular to the wave vector:

(D.64)

We finally find, using (D.6),

(D.65)

D.2.7 Determination of Φ_bT ^LH

In the analysis above the kinetic part of the power flux, due to the coherent motion of charge carriers, has been neglected. This approximation must be justified by comparing the Poynting and the kinetic fluxes.

The normalized expression for the kinetic power flux associated with an electromagnetic wave in a kinetic plasma is given by (4.286)

(D.66)

Electrostatic Waves

In the electrostatic approximation, the electric field is quasi-longitudinal (D.37)

(D.67)

and (D.66) becomes

(D.68)

In the frame of the wave vector _b, (D.68) can then be expressed as

(D.69)

where

(D.70)

is the longitudinal (electrostatic) component of the dielectric tensor.

Lower Hybrid Waves

An expression of the electrostatic dispersion relation for LH waves that includes the first-order thermal corrections is given by [?]:

(D.71)

where the finite Larmor radius (FRL) effects are scaled by the parameters

(D.72)

with v_T ² = T∕m.

The k-dependence of the dispersion relation is found in the thermal correction terms so that

(D.73)

Inside the plasma, we have k_b∥² ≪ k_b⊥² and therefore k_b⊥²∕k_b² ≃ 1. We finally get the following expression for the kinetic flux associated with quasi-electrostatic LH waves:

(D.74)

The kinetic flux is oriented in the direction of the wave vector, and the incident flux on the flux surface

(D.75)

This incident kinetic flux must be compared to the incident Poynting flux taken in the cold plasma limit from (D.65)

(D.76)

The ratio of the incident kinetic power flux to the Poynting flux is given by

(D.77)

Using (D.39) we have

(D.78)

so that

(D.79)

where β_Te² = T_e∕mc².

It can be seen from (D.79) that the kinetic part of the power flux becomes significant only as the wave approaches the lower-hybrid resonance very closely. In a typical LHCD context (for instance Alcator C-Mod, [?]), we have

(D.80)

so that the kinetic part of the power flux is not more than a few percents.

D.2.8 LH Diffusion Coefficient

General expression in small FLR limit and ES approximation

The normalized bounce-averaged diffusion coefficient for the Fokker-Planck equation is given by (4.331)

DbLH(0)(p,ξ 0)	= ΨθbDb,0LH,θb ×
	HHT δ2	(D.81)

with

Db,0LH,θb	= Pb,inc	(D.82)
N∥resθbLH	=	(D.83)

Within the electrostatic approximation and in the small FLR limit, we have obtained the following expressions for the LHW properties (D.59), (D.39), (D.65)

Θk,θbb,LH	=	(D.84)
Nb⊥	= Nb∥	(D.85)
ΦbLH	= -	(D.86)

so that

	=	(D.87)
	≃	(D.88)
	=	(D.89)

and where K_⊥ and K_∥ are given by (D.44) and (D.42) respectively

Simplified expression for LHCD

We consider the limit of a large aspect ratio tokamak with circular flux-surfaces (limit of a cylindrical plasma). In that case, r_θb∕R_p → 0 and the effects of magnetic trapping disppear. We can use the following asymptotic expressions

Ψθb	→ 1	(D.90)
λ	→ 1	(D.91)
	→ 1	(D.92)
	→ 1	(D.93)

Because of cylindrical symmetry, the dependence θ_b on disappears.

The QL diffusion coefficient for LHCD (D.81) can therefore be written as

(D.94)

with

Db,0LH	= finc,bl+1∕2P b,inc	(D.95)
N∥resLH	=	(D.96)

and where we used N_b ≃ N_b⊥ and assume a square power spectrum as in (4.351).

In order to compare with LHCD operators found in the litterature, we redefine the LH constant factor such that

(D.97)

with

Db,0,newLH	= Db,0LH	(D.98)
	= finc,bl+1∕2P b,inc	(D.99)

A familiar expression for the LH QL operator is obtained in cylindrical geometry. From (4.239) and (4.242-4.245) we see that for LHCD, the QL operator can be rewritten as

QLH(f)	= D∥∥RF
	= ∑ bDb,0,newLHHH	(D.100)

Although it is better to keep the exact formula (D.100), the factor v_Te∕ has often been neglected in the litterature, which is an acceptable approximation when D_b,0,new^LH ≫ ν_ep_Te² and ΔN_b∥≪ N_{b∥ min}. In this case, considering only one ray b, the LH Operator reduces to

(D.101)

with

v1	=	(D.102)
v2	=	(D.103)
D0,newLH	= Db,0,newLH	(D.104)
	= fincl+1∕2P inc	(D.105)

where is an averaged value of which can be taken to be and then

(D.106)