8 LH-wave

8.1 Dispersion relation

The dielectric tensor↔ǫ can be very complex depending on the situation and

the phenomena that are investigated. One can add a tremendous amount ofphysics in it e.g. relative effects, collisions, warm or hot plasma effects andanisotropy. In case of LH waves, the cold plasma approximation (vph ≫ vth)is enough to get a reasonable accuracy in the dispersion relation except nearthe resonance where hot plasma effects are important. The warm plasmaeffect on the dispersion relation is described in next section. Using thisapproximation of the dielectric tensor

↔ǫ and if the coordinates axes are

chosen so that the magnetic field is along the z-axis and the wave propagatesin the x-z plane, the wave equation can be expressed in a matrix form

S − N2‖ iD N⊥N‖

iD S − N2 0N⊥N‖ 0 P − N2

⊥

·

Ex

Ey

Ez

= 0, (2)

where,

S = 1 −ω2

pe

ω2 − ω2ce

−ω2

pi

ω2 − ω2ci

(3)

iD = iω2

piωci

ω(ω2 − ω2ci)

− iω2

piωce

ω(ω2 − ω2ce)

(4)

P = 1 −ω2

pe

ω2−

ω2pi

ω2(5)

Where, ωpe is electron plasma frequency, ωpi is ion plasma frequency, ωce

is electron cyclotron frequency, and ωci is ion cyclotron frequency. And,the notation Nx

∼= N⊥, Nz∼= N‖ is adopted. The subscripts parallel and

perpendicular refer to the direction of the external magnetic field B0. Inorder to have non trivial solutions the determinant of the multiplying matrixhas to be zero. This condition gives the dispersion relation

D(N, ω) = AN4⊥ + BN2

⊥ + C = 0 (6)

where,

A = S (7)

B = (N2‖ − S)(S + P ) + D2 (8)

C = P[

(N2‖ − S)2 − D2

]

. (9)

An approximation form of the dispersion relation, known as the ‘ electro-static approximation,’ is used frequently in lower-hybrid theories. The elec-trostatic approximation is given by,

SN2⊥ + PN2

‖ = 0. (10)

131

8.2 Wave propagation and accessibility

The perpendicular refractive index N⊥ can be solved from Eq. (6)

N2⊥ =

−B ±√

B2 − 4AC

2A, (11)

where the plus sign corresponds the slow wave and the minus sign is for thefast wave. In the case of LH grill the sign of the N⊥ must be chosen so thatthe energy of the wave goes radially outward, and if imaginary, is damped.

There exits a wave resonance (N⊥ → ∞) when the denominator ofEq. (11) goes to zero. Equating Eq. (3) to zero and solving it for the fre-quency gives, in the limit of ωci ≪ ω ≪ ωce, the resonance frequency

ωLH = ωpi

(

1 +ω2

pe

ω2ce

)−1/2

(12)

where, ωLH is the lower hybrid resonance frequency. In the early daysof LH heating the power was proposed to be absorbed by this resonancebut later due to the accessibility conditions and strong Landau damping itwas abandoned. Since then also other heating schemes e.g. stochastic ionheating have been tried but the most reliable and reproducible absorptionmechanism has proven to be the electron Landau damping.

LH wave also exhibits a cut-off (N⊥ → 0) when the nominator of Eq. (11)goes to zero. For the slow wave this can happen only when C → 0 that is

C = P ((N‖ − S)2 − D2) = 0 (13)

The condition (N2‖ − S)2 = D2 produces the cut-offs of the fast wave

NFC‖ =

√S + D, (14)

and the condition P = 0 gives the LH-wave (slow-wave) cut-off. Again, inthe limit ωci ≪ ω ≪ ωce, the LH cut-off condition can be solved to give thecut-off density

nc =ǫ0me

e2ω2 ∝ ω2. (15)

The cut-off density is an important parameter for the coupling becausethe wave can not propagate below it. Below the cut-off density the wave isevanescent and it can only tunnel into the higher densities. The LH waveis expected to reflect almost totally if the distance between the cut-off layerand the grill mouth is too large compared to the wavelength. Notice thatwe have 6-cm wavelength for KSTAR 5.0-GHz LHCD system.

When the lower hybrid resonance does not exist in the plasma, thatis, ω > ωLH , the condition for wave penetration to the maximum densitywithout mode conversion to the fast wave is

N‖ crit =ωpe

ωce+ S1/2. (16)

This is well-known accessibility condition. Above equation is obtained withthe approximation of S in Eq. (3) and iD in Eq. (4) in the limit of ωci ≪ω ≪ ωce. This critical value may be called linear turning point as shown infigures in next chapter.

132

8.3 Phase Velocity and Group Velocity

In the valid limit N2‖ ≫ 1, the use of Eq. (12) gives a simplified equation of

Eq. (11)

N2⊥

N2‖

=mi

me· ω2

LH

ω2 − ω2LH

. (17)

This equation states that a wave with a certain N‖ has also a certain N⊥.Eq. (17) can be solved for the wave frequency as a function of the wavenumber. And it gives the group velocity of the wave

vg‖ =∂ω

∂k‖=

ω

k‖

ω2 − ω2LH

ω2, (18)

vg⊥ =∂ω

∂k⊥=

ω

k⊥

ω2LH − ω2

ω2. (19)

The wave frequency ω is usually larger than the lower hybrid resonance fre-quency ωLH implying that the perpendicular phase velocity vp⊥ = ω/k⊥ isnegative with respect to the group velocity since the perpendicular group ve-locity in Eq. (19) must be positive. The relation between the phase velocityand the group velocity gives the interesting phenomenon

vg‖vg⊥

= −k⊥k‖

. (20)

This suggests the phase velocity and the group velocity are at right angles inthe cold plasma approximation. Another interesting thing is that the higherplasma density results in the smaller angle of the propagation cones to thetoroidal direction. Because the N‖ is determined from the grill structureand N⊥ is increased as the plasma density increases. One should note thatthe wave vector ~k and the propagation direction are at right angles.

8.4 Parametric study of the 5.0-GHz LH-wave propagationin the KSTAR tokamak

In this section, we calculate the parametric dependence of the wave prop-agations in the KSTAR tokamaks. The main equilibrium parameters aresummarized in Table 1. In this table, R0 is the major radius, a is theplasma minor radius, A = R0/a is the aspect ratio, κ is the ellipticity, δ isthe triangularity, Rgr is defined as the grill position of the LH antenna, q(a)is the safety factor at the edge. The q factor is defined as

q(r) =RBφ(r)

2π

∫

ds1

R2Bθ(r), (21)

Let us now specialize the simple circular plasma model in toroidal geom-etry with local toroidal coordinates (r, θ, φ). r is the radius measured fromthe magnetic axis of the torus, θ is the poloidal angle, and φ is the toroidalangle rotated with respect to vertical coordinate. We neglect the ellipticity

133

Table 1: The main equilibrium parameters of the KSTAR tokamak

Parameter Value

Ip (MA) 2.0

BT (T) 3.5

ne0 (m−3) 1.0 × 1020

Te0 (keV) 10 ∼ 20

R0 (m) 1.8

a (m) 0.5

A 3.6

κ 2.0

δ 0.8

q(a) 3 - 10

Rgr (m) 2.3

and the triangularity in subsequent calculations. The magnetic field in thiscircular plasma is given as below

Br = 0 (22)

Bθ =√

B2R + B2

Z =µ0Ip

2πr(1 − (1 − (r/a)2)q(a)) (23)

Bφ = R0BT /(R0 + r cos θ) (24)

B2 = B2r + B2

θ + B2φ. (25)

Here, the pitch angle, p, between magnetic field lines and the toroidal di-rection will be needed in our analysis. The variation along the midplane of

the pitch angle, p = arctanBpol/Bφ, where Bpol =√

B2r + B2

θ , is plotted

in Fig. (6) for the KSTAR. And, the electron temperature and the densityprofiles are modelled to be parabolic-like as below

ne(r) = ne(0)(1 − r2/a2

)α(26)

Te(r) = Te(0)(1 − r2/a2

)β(27)

Ti(r) = Ti(0)(1 − r2/a2

)β. (28)

For both α and β less than 1, we have broad density and temperatureprofiles. If they are higher than 1, we get more peaked squared parabola.

Since the plasma frequencies ωpe and ωpi are functions of the densityand the cyclotron frequencies ωce and ωci are functions of the magneticfield, those frequencies are given as functions of the radial coordinate of theplasma. Therefore, we get the perpendicular refractive index N⊥ and the

134

critical parallel refractive index N‖crit as a function of the radial coordinateof the plasma.

For the KSTAR tokamak with the central density ne(0) = 1 × 1020 m−3

and the toroidal magnetic field at the plasma center B0 = 3.5 T and the LHfrequency of 5.0 GHz, the critical parallel refractive index is 2.27. Withoutapproximation of S and iD, the N‖crit is solved from equating B2−4AC = 0and it becomes 2.18 for the same parameter as above. Fig. 7 shows N‖crit

as a function of radial coordinate for central densities ne(0) = 0.2, 0.5, 1.0 inunit of 1020 m−3 with the broad profile (α = 1).

Figure 8 shows N⊥ as a function of radial coordinate for various N‖ withthe central density ne(0) = 1 × 1020 m−3. Each N‖ values in ascendingcorresponds to the phase differences, 60 ◦, 90 ◦, 120 ◦, and 150 ◦ betweenadjacent waveguides of the grill. One may find that there exists evanescentzone due to low edge plasma density. In addition, we find that the wavewith the launched N‖ value less than N‖crit cannot penetrate into the centerand the mode conversion from the slow wave to the fast wave. In this figure,the solid line corresponds to the slow wave and the dotted line to the fastwave. If the central density decreases, the wave can penetrate into the centerbecause N‖crit decreases as the plasma density decreases (see Eq. (16)).

Fig. 9 shows N⊥ for various central densities with N‖ = 2.14. But, thereexists the longer evanescent zone for the lower central density.

In Figs. 8 and 9, N‖ values are maintained with constant value as thewave propagates into the plasma. However, it actually varies downward orupward in the tokamak which has toroidal geometry. The wavelength mustbecome shorter in regions of a smaller major radius in order to accommodatethe same number of wave periods within a shorter toroidal circumference.The toroidal mode number n

grφ , imposed by the grill located at Rgr, is

related to the toroidal component, Ngrφ , of the refractive index vector at

the grill, through Ngrφ = cn

grφ /(ωRgr). The constancy of the mode number

(nφ = ngrφ ) then requires the toroidal refractive index, Nφ = cnφ/(ωR), to

be inversely proportional to the major radius, i.e.

N‖ = Nφ =Rgr

RN

grφ . (29)

This is a most basic toroidal effect, and will be called a “wedge effect”.Fig. 10 shows that N‖ is gradually increased as the wave propagates intothe plasma. The two lines of Fig. 11 show the re-plots of N⊥ in the caseof N

grφ = 2.14 in Fig. 8 with the wedge effect and without wedge effect,

respectively. Interesting thing is that the wedge effect increases the N‖value so that the wave can penetrate into the center.

8.4.1 Spectral gap and N‖ shifting

There is an aspect of wave damping mechanism that has not been fully un-derstood. The lower-hybrid waves in the current drive regime are theoreti-cally expected to damp through Landau damping by resonantly interactingwith electrons that are moving at speeds near the wave phase speed paral-lel to the magnetic field. The spectrum of waves launched into a tokamakplasma by an antenna has, however, a phase speed often much greater than

135

the thermal speed of electrons, and there are few electrons that are reso-nant with the waves. This gap between the parallel phase speed of launchedwaves and electron thermal speed is commonly known as the ‘spectral gap.’Upshifting of N‖ can fill this gap, causing the waves to damp. Although adirect experimental confirmation of N‖ upshifting is difficult, it has, never-theless, become widely accepted as an explanation for how the lower-hybridwaves damp in spite of the spectral gap. The spectral gap can be largeor small depending upon the wave phase speed and electron temperature.There exists upper and lower bounds of N‖ shifting during the wave prop-agation in a tokamak plasma. The main reason of N‖ shifting comes fromthe toroidal effect.

The wavenumbers conjugate to the spatial coordinates (r, θ, φ) are givenas

~k = (kr, mθ/r, nφ/R). (30)

Where, the toroidal mode number nφ is a constant of motion due to thetoroidal symmetry. The toroidal effects comes from the variation in mθ andmagnetic shear. By definition, the parallel wavenumber k‖ = ωN‖/c alongto the magnetic field is

k‖ =~k · ~B

| ~B|. (31)

The magnetic field is given by Eqs. (22)-(24). The perpendicular wavenum-ber k⊥ to the magnetic field is given by

k2⊥ = |~k|2 − k2

‖. (32)

Using Eq. (30) and the magnetic field components gives

k2‖ =

mθBθ/r + nφBφ/R

B2(33)

k⊥ = k2r +

mθBφ/r − nφBθ/R

B2. (34)

Substituting mθBθ/r from Eq. (33) into Eq. (34) we obtain an equation fork‖

(k‖√

1 − γ2 − kφ)2 = γ2(k2⊥ − k2

r) (35)

where γ = Bθ/B. The perpendicular wave vector ~k⊥ is a function of k‖through the local dispersion relation. From Eq. (35), noting that k2

r ≥ 0, weobtain the expression

(k‖√

1 − γ2 − kφ)2 ≤ γ2(k2⊥). (36)

In the electrostatic limit (k2⊥ = −(P/S)k2

‖) from Eq. (10), Eq. (36) breaksinto the following two inequalities:

N‖ =k‖c

ω≤

( c

ω

) kφ√

1 − γ2 −√

−P/Sγ= N‖, up (37)

136

N‖ =k‖c

ω≥

( c

ω

) kφ√

1 − γ2 +√

−P/Sγ= N‖, down. (38)

The right-hand side in Eq. (37) corresponds to the extreme upshift, andthe right-hand side in Eq. (38) corresponds to the extreme downshift. Theyvaries as function of the pitch angle and the dielectric tensor elements, S andP. The solution of k‖ becomes infinite when the denominator of Eqs. (37)and (38) vanishes. If the denominator of the upshifting becomes very smallunder some conditions, the upper bound increases rapidly.

The admissible range of N‖ is defined by the lowest upper bound, thehighest lower bound, the fast wave cutoff, and the mode conversion to thefast wave. The variation of the admissible range of N‖ as a function of theposition defines a ‘wave domain’ (WD).

Waves with a high N‖ value will damp strongly through electron Landaudamping. The condition that the wave phase speed be a certain multiple,λ, of the electron thermal speed, ve, can be expressed as,

Ndmp‖ =

c

λve≈ 5.33√

Te. [in unit of keV] (39)

For the phase speed equal to the three times the thermal speed (λ = 3), thedamping is strong. The damping is exponentially weaker at a higher phasespeed.

For the KSTAR tokamak plasma, the N‖ shifting is investigated in thefollowing figures for various plasma conditions: the central density ne(0), thecentral temperature Te(0), the plasma current Ip, the safety factor q, andthe α = 1 or 2. Fig. 12 shows the fast wave cutoff (FC) by Eq. (14) and theupshift and downshift of N‖ (Eqs. (37) and (38)) as a function of the radialposition in the mid-plane for the central density. The solid line correspondsto ne(0) = 0.5×1020 m−3 and the dotted line to ne(0) = 1.0×1020 m−3. Theother plasma parameters, Ip = 2MA, q = 3, α = 1, and N

grphi = 2.14. Note

that this plot includes the wedge effect. From this figure, it is shown thatthe higher central density plasma gives more upshift in N‖. The Landaudamping zones are over plotted for various central electron temperature inFig. 13. In this figure, the solid line in Fig. 12 is used for the N‖ shifting.The damping zone (DZ) is defined as the overlap region between the wavedomain region and the Landau damping region in Fig. 13. As the centralelectron temperature decreases, the damping zone goes to the upper regionof the wave domain and hence the narrower region for the damping.

The N‖ shifting is also investigated for the plasma current variations.With the plasma conditions of ne(0) = 1 × 1020 m−3, α = 1, q = 3, andN

grφ = 2.14, the N‖ shifting is plotted in Fig. 14. In this figure, the more

upper shifting happens for the higher plasma current.For α = 2, we get a peaked profile of the electron density. We compared

the N‖ shifting of the broad profile with that of the peaked profile in Fig. 15.The figure shows that the broad profile gives more upshifting with few changeof the downshifting. Here, we used ne(0) = 1 × 1020 m−3, Te(0) = 10 keV,and Ip = 2 MA.

The dependency of the N‖ shifting on the launched parallel refractive

index Ngrφ is shown in Fig. 16. When we get the higher launched value of

Ngrφ at the grill, the overall shifting is shifted up. In addition, the range

137

between the upper limit of the upshift and the lower limit of the downshiftincreases as shown in Fig. 17. The upper limits and the lower limits are alsoindicated inside the brace for each case of the launched N

grφ values. These

values are obtained for ne(0) = 1 × 1020 m−3, Te(0) = 10 keV, and Ip = 2MA.

138

Figure 6: The magnetic pitch angle of KSTAR plasma in mid-plane. Ip =2 MA and B0 = 3.5 T

Figure 7: The critical N‖ value vs radial position in mid-plane for variouscentral density. Broad density profile (α = 1) is used in this plot.

139

Figure 8: The perpendicular refractive index vs radial position in mid-planefor various N

grφ . ne(0) = 1.0 × 1020 m−3 and α = 1.

Figure 9: The perpendicular refractive index vs radial position in mid-planefor various central density. N

grφ = 2.14 and α = 1.

140

Figure 10: The variation of Nφ vs radial position.

Figure 11: N2⊥ vs radial position in mid-plane with constant Nφ = N

grφ =

2.14 (solid line) and with increasing Nφ due to wedge effect.

141

Figure 12: The up-shift and down-shift in N‖ and the fast wave cut-off (FC)for two central densities and fixed Te(0) = 20 keV. The “WD” is defined asthe region bounded by up and down shifts and FC. Here, N

grφ = 2.14.

Figure 13: The wave domain and damping zone in KSTAR plasma forne(0) = 1 × 1020 m−3 with broad profile. The dashed lines are the sig-nificant Landau damping for various central temperatures. Here, N

grφ =

2.14.

142

Figure 14: The up-shift and down-shift in N‖ vs radial position in mid-plane

for the plasma current. Here, Ngrφ = 2.14.

Figure 15: The up-shift and down-shift in N‖ vs radial position in mid-plane

for broad and peaked profiles. Here, Ngrφ = 2.14.

143

Figure 16: The up-shift and down-shift in N‖ vs radial position in mid-plane

for Ngrφ .

1.5 2.0 2.5 3.0 3.5

1

2

3

4

5

6

7

8

9

10

11

(3.57, 2.61)(2.86, 2.09)

(2.14, 1.56)(1.43, 1.04)

(2.14, 6.24)

(3.57, 10.39)(2.86, 8.32)

(1.43, 4.16)

Lower limit of downshift

Upper limit of upshift

N||

N gr

Figure 17: The upper limits of up-shift and the lower limits of down-shiftvs N

grφ , which are results from Fig. 16.

144

8.5 Dispersion relation with thermal correction

The local dispersion relation can be written

D(~r,~k, ω) = |~k~k − k2 + (ω2/c2)↔K (~r,~k, ω)| = 0 (40)

if the electromagnetic portions of Maxwell’s equations are retained or

D(~r,~k, ω) = ~k·↔K (~r,~k, ω) · ~k = 0 (41)

in the electrostatic approximation assuming N‖ = k‖c/ω ≫ 1, hence there-

fore ∇ × δ ~E = ~k × δ ~E ≃ 0. Here↔K is the hot plasm dielectric tensor (see

section 3. Much of the physics of the propagation is found by a “warm-

plasma” expansion of↔K (~r,~k, ω) in which first-order temperature effects for

ions and electrons are retained. After such an expansion (see section 3.1and 3.2, Eq. (40) becomes

D(~r,~k, ω) = k4⊥K⊥ (42)

+ k2⊥

(

[k2‖ − (ω2/c2)K⊥](K‖ + K⊥) + (ω2/c2)(K2

xy + 2KxyK2))

+ K‖(

[k2‖ − (ω2/c2)K⊥]2 − (ω4/c4)K2

xy

)

= 0,

and

D(~r,~k, ω) = k2⊥K⊥ + k2

‖K‖ = 0, (43)

where

K⊥ = S − αk2⊥; α = 3

ω2pi

ω2

V 2T i

ω2+

3

4

V 2Te

ω2ce

, (44)

K‖ = P

(

1 − k2⊥V 2

Te

ω2ce

+ 3k‖V

2Te

ω2

)

, (45)

Kxy = D

(

1 − 3

2

k2⊥V 2

Te

ω2ce

)

, (46)

K2 =ω2

pe

ωωce

k2‖V

2Te

ω2, (47)

V 2T i, e =

κTi, e

mi, e. (48)

The thermal corrections are also valid if k2⊥V 2

T i/ω2, k2⊥V 2

Te/ω2ce, k2

‖V2Te/ω2

are all much less than unity, and above equations can be rewritten

K⊥ = S − αk2⊥; α = 3

ω2pi

ω2

V 2T i

ω2+

3

4

V 2Te

ω2ce

, (49)

K‖ = P, (50)

Kxy = D, (51)

K2 = 0, (52)

V 2T i, e =

κTi, e

mi, e. (53)

145

8.6 Wave absorption

An estimate of the wave absorption can be found only adding imaginaryparts to the dispersion relation according to the imaginary parts of theplasma dispersion or Z function of Fried and Conte contained in the expres-

sions for↔K (~r,~k, ω). The asymptotic expansion of the Z function is useful

for finding the good approximation of the damping term

Z(x) =1√π

∫ ∞

−∞

exp(−t2)

t − xdt (54)

≃ i√

π exp(−x2) − (1

x+

1

2x3+

3

4x5+ . . .).

It is important to note the appearance of the imaginary term in Eq. (55),arising from the pole contribution at t = x. This resonant part will giverise to a collisionless (or Laundau) damping of the Lower-hybrid wave. Thedamping terms for the electrons and ions (de, di) to be added to the electro-static equation Eq. (43).

D = ℜ(D) + iℑ(D) = Dr + iDi = k2⊥K⊥ + k2

‖K‖ + i(de + di) (55)

The decrease in wave power P due to electron Landau damping and ionLandau damping is given by

P = P0 exp

(

−2

∫

ℑ(k⊥)dr

)

. (56)

The expansion of Eq. (55) about the real term of k⊥ to the first order of theimaginary part of k⊥ gives the expression of the ℑ(k⊥).

ℑ(k⊥) =de + di

(∂D/∂k⊥)k⊥=k⊥, r

=de + di

[2k⊥(∂D/∂k2⊥)]k⊥=k⊥, r

. (57)

With the help of good approximations of

λe, i =k2⊥V 2

Te, i

ω2ce, i

≪ 1, (58)

χi =ω√

2k⊥VT i

≫ 1, (59)

ξe =ω√

2k‖VTe

≫ 1, (60)

the damping terms de and di are expressed simply as

de =

√2ω2

pe

V 2Te

ω√2k‖VTe

exp

(

− ω2

2k2‖V

2Te

)

, (61)

di =

√2ω2

pi

V 2T i

ω√2k⊥VT i

exp

(

− ω2

2k2⊥V 2

T i

)

. (62)

One may note that λe, i is the argument of the modified Bessel function andχi and ξe are the arguments of the Z function. With Eqs. (59)-(61), Eq. (57)are rewritten as

ℑ(k⊥)

ℜ(k⊥)=

√π

∂D/∂k2⊥

(

F (ξe)ω2

pe

ω2

k2‖

k2⊥

+ F (χi)ω2

pi

ω2

)

. (63)

Where, the function F (x) = x3 exp(−x2).

146