Plasma Sheath Lab - POSTECHpsl.postech.ac.kr › material › lecture › wavesinplasmas.pdf · 2004-09-24 · 1 Dispersion Relation in a Cold Uniform Plasmas As long as Te = Ti =

Lecture note

Plasma Sheath Lab†

Theory of Waves in Plasmas

(PSL-LN-01)

September 13, 2004

Y. S. Baea and W. Namkung

Department of Physics, POSTECH

†homepage:http://psl.postech.ac.kraDepartment of Physics, Pohang University of Science and TechnologySan 31, Hyoja-dong, Nam-gu, Pohang 790-784, Korea.e-mail: [email protected]

Contents

1 Dispersion Relation in a Cold Uniform Plasmas 11.1 Resonances (N → ∞) . . . . . . . . . . . . . . . . . . . . . . 51.2 Cut-offs (N = 0) . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 CMA diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 EC-wave propagation using Mathematica 82.1 O-X propagation for 2nd harmonic resonance for KSTAR

tokamak with low density . . . . . . . . . . . . . . . . . . . . 82.2 O-X propagation for 2nd harmonic resonance for KSTAR

tokamak with high density . . . . . . . . . . . . . . . . . . . . 152.3 O-X-B heating . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Dispersion Relations in a Hot Plasma 323.1 Electromagnetic Dispersion Relation . . . . . . . . . . . . . . 323.2 Electrostatic Dispersion Relation . . . . . . . . . . . . . . . . 50

3.2.1 Electrostatic Modes in Hot Plasma . . . . . . . . . . . 55

4 Dispersion plots of electron modes using Mathematica 644.1 Electron modes . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Electron Bernstein (EB) modes . . . . . . . . . . . . . . . . . 69

5 Landau Damping 74

6 ECR Heating [or Damping] Rates 83

6.1 Fund. Harm. Damping Rate - classical approach(det↔M) . . . 83

6.1.1 The Dielectric Tensor for ω ≫ ωpi, Ωi and ω ∼ |Ωe| . . 836.1.2 Damping Rates near the ECR Region . . . . . . . . . 86

6.2 Damping Rates Using Quasi-linear Theory . . . . . . . . . . . 976.2.1 Higher Harmonics (n ≥ 2) . . . . . . . . . . . . . . . . 1036.2.2 Fundamental Harmonic (n = 1) . . . . . . . . . . . . . 1086.2.3 O-mode & X-mode Heating . . . . . . . . . . . . . . . 115

7 Calculation of ECR optical depth using Mathematica 121

8 LH-wave 1318.1 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . 1318.2 Wave propagation and accessibility . . . . . . . . . . . . . . . 1328.3 Phase Velocity and Group Velocity . . . . . . . . . . . . . . . 1338.4 Parametric study of the 5.0-GHz LH-wave propagation in the

KSTAR tokamak . . . . . . . . . . . . . . . . . . . . . . . . . 1338.4.1 Spectral gap and N‖ shifting . . . . . . . . . . . . . . 135

8.5 Dispersion relation with thermal correction . . . . . . . . . . 1458.6 Wave absorption . . . . . . . . . . . . . . . . . . . . . . . . . 146

9 Ray Tracing in Inhomogeneous Media 1479.1 Electric and Magnetic fields of E-M waves in a Tokamak with

a cold Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

i

9.2 Phase velocity and group velocity of EM waves in a tokamakwith a cold plasma . . . . . . . . . . . . . . . . . . . . . . . . 153

9.3 Raytracing of EC-wave in KSTAR tokamak . . . . . . . . . . 156

A Calculation of Sx 177

B The reason of validity of cold plasmadielectric tensor in the calculationof harmonic damping rates(n ≥ 2) 179

C Quasi-linear Theory 180

ii

List of Figures

1 CMA diagram for a two-component plasma. The ion-to-electron mass ratio is chosen to be 2.5. Bounding surfaces ap-pear as lines in this two-dimensional parameter space. Crosssections of wave-normal surfaces are sketched and labeled foreach region. For these sketches the direction of the magneticfield is vertical. The small mass ratio can be misleading here:the L = 0 line intersects P = 0 at Ωi/ω = 1 − (Zme/mi).From T. Stix’s book (AIP, 1992). . . . . . . . . . . . . . . . . 7

2 Contour I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Contour II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 Contour III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 Real and imaginary parts of the frequency as a function of

wave number for a stationary one-component plasma in ther-mal equilibrium. The frequency is given in units of ωp, whilethe wave number is expressed in units of the Debye wave num-ber (kD). The dotted curves represent approximate formulasderived in this section. . . . . . . . . . . . . . . . . . . . . . . 79

6 The magnetic pitch angle of KSTAR plasma in mid-plane. Ip

= 2 MA and B0 = 3.5 T . . . . . . . . . . . . . . . . . . . . . 1397 The critical N‖ value vs radial position in mid-plane for var-

ious central density. Broad density profile (α = 1) is used inthis plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8 The perpendicular refractive index vs radial position in mid-plane for various N

grφ . ne(0) = 1.0 × 1020 m−3 and α = 1. . . 140

9 The perpendicular refractive index vs radial position in mid-plane for various central density. N

grφ = 2.14 and α = 1. . . . 140

10 The variation of Nφ vs radial position. . . . . . . . . . . . . . 141

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. 14112 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 as the region bounded by up and down shiftsand FC. Here, N

grφ = 2.14. . . . . . . . . . . . . . . . . . . . 142

13 The wave domain and damping zone in KSTAR plasma forne(0) = 1×1020 m−3 with broad profile. The dashed lines arethe significant Landau damping for various central tempera-tures. Here, N

grφ = 2.14. . . . . . . . . . . . . . . . . . . . . . 142

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

plane for the plasma current. Here, Ngrφ = 2.14. . . . . . . . 143


plane for broad and peaked profiles. Here, Ngrφ = 2.14. . . . . 143


plane for Ngrφ . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

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

grφ , which are results from Fig. 16. . . . . . . . . . . . . . 144

18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

iii

21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16426 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16527 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16931 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17032 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17335 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17436 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17537 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17638 Velocity distribution for “bump-on-tail” instability. Real part

of unstable frequencies are such that v = ω0(k)/k lies inregion where vdf0/dv is positive (opposite sense to Landaudamping). Quasi-linear diffusion due to these modes tends toflatten out the bump. . . . . . . . . . . . . . . . . . . . . . . 181

iv

1 Dispersion Relation in a Cold Uniform Plasmas

As long as Te = Ti = 0, the waves described can easily be generalized toan arbitrary number of charged particle species and an arbitrary angle ofpropagation θ relative to the magnetic field. Waves that depend on finiteT , such as ion acoustic waves, are not included in this treatment.The fourth Maxwell equation:

∇× ~B = µ0(~j + ǫ0 ~E)

where ~j is the plasma current due to the motion of the various chargedparticle species s, with density ns, charge qs, and velocity vs:

~j =∑

s

nsqsvs

Considering the plasma to be a dielectric with internal currents ~j,

∇× ~B = µ0~D

where~D = ǫ0 ~E +

i

ω~j

Here we have assumed an exp(-iωt) dependence for all plasma motions.

A conductive tensor↔σ (because of the magnetic field B0z);

~j =↔σ · ~E

Thus,

~D = ǫ0(↔1 +

i

ǫω

↔σ ) · ~E =

↔ǫ · ~E

The effective dielectric tensor of the plasma:

↔ǫ = ǫ0(

↔1 +

i

ǫω

↔σ )

where↔1 is the unit tensor.

To evaluate↔σ , we use the “linearized fluid equation” of motion for species

s, neglecting the collision and pressure terms:

ms∂ ~vs

∂t= qs( ~E + ~vs × ~B0)

Defining the cyclotron and plasma frequencies for each species as

Ωs ≡ |qsB0

ms| ωps ≡

n0q2s

ǫ0ms

Note that Ωs > 0 hereafter.

1

We can separate “linearized fluid equation” into x, y, and z componentsand solve for vs, obtaining

vxs =iqs

msω

[Ex ± i(Ωs/ω)Ey]

1 − (Ωs/ω)2

vys =iqs

msω

[Ey ± i(Ωs/ω)Ex]

1 − (Ωs/ω)2

vzs =iqs

msωEz

where ± stands for the sign of qs. The plasma current is

~j =∑

s

n0sqs ~vs

so that

i

ǫ0ωjx =

∑

i

in0s

ǫ0ω

iq2s

msω

[Ex ± i(Ωs/ω)Ey]

1 − (Ωs/ω)2

=∑

i

ω2ps

ω2

[Ex ± i(Ωs/ω)Ey]

1 − (Ωs/ω)2

Using the identities

1

1 − (Ωs/ω)2=

1

2[

ω

ω ∓ Ωs+

ω

ω ± Ωs]

± Ωs/ω

1 − (Ωs/ω)2=

1

2[

ω

ω ∓ Ωs− ω

ω ± Ωs],

1

ǫ0ωjx = −1

2

∑

s

ω2ps

ω2[(

ω

ω ∓ Ωs+

ω

ω ± Ωs)Ex + (

ω

ω ∓ Ωs− ω

ω ± Ωs)iEy]

Similarly, the y and z components are

1

ǫ0ωjy = −1

2

∑

s

ω2ps

ω2[(

ω

ω ± Ωs− ω

ω ∓ Ωs)iEx + (

ω

ω ∓ Ωs+

ω

ω ± Ωs)Ey]

1

ǫ0ωjz = −

∑

s

ω2ps

ω2Ez

These give

1

ǫ0Dx = Ex −

1

2

∑

s

[ω2

ps

ω2(

ω

ω ∓ Ωs+

ω

ω ± Ωs)Ex +

ω2ps

ω2(

ω

ω ∓ Ωs− ω

ω ± Ωs)iEy]

Similarly with the y and z components, we obtain

ǫ−10 Dx = SEx − iDEy

ǫ−10 Dy = iDEx + iSEy

ǫ−10 Dz = PEz

2

Where

R ≡ 1 −∑

s

ω2ps

ω2(

ω

ω ± Ωs)

L ≡ 1 −∑

s

ω2ps

ω2(

ω

ω ∓ Ωs)

S ≡ 1

2(R + L) D ≡ 1

2(R − L)

P ≡ 1 −∑

s

ω2ps

ω2.

Or,

S = 1 −∑

s

ω2ps

ω2 − Ω2s

D =∑

s

ω2ps

ω

±Ωs

ω2 − Ω2s

P = 1 −∑

s

ω2ps

ω2.

From ~D =↔ǫ · ~E,

↔ǫ = ǫ0

S −iD 0iD S 00 0 P

≡ ǫ0↔K

The wave equation by taking the curl of the equation

∇× ~E = − ~B and substituting ∇× ~B = µ0↔ǫ · ~E:

∇×∇× ~E = −µ0ǫ0(↔K · ~E) = − 1

c2

↔K · ~E

Assuming an exp(i~k ·~r) spatial dependence of ~E and defining a vector indexof refraction

~N =c

ω~k,

the wave equation becomes

~N × ( ~N × ~E)+↔K · ~E = 0

The uniform plasma is isotropic in the x-y plane (i.e. ky = 0).

If θ is the angle between ~k and ~B0 we then have

Nx = n sin θ Nz = n cos θ Ny = 0

Using the elements of↔K,

↔M · ~E ≡

S − N2 cos2 θ −iD N2 sin θ cos θiD S − N2 0

N2 sin θ cos θ 0 P − N2 sin2 θ

Ex

Ey

Ez

= 0

From this it is clear that the Ex, Ey components are coupled to Ez only ifone deviates from the principal angles θ = 0, 90.

3

The above equation is a set of three simultaneous, homogeneous equations;

the condition for the existence of a solution is that the determinant of↔M

vanish: ‖↔M ‖ = 0.

That is,

A′N4 − B′N2 + C ′ = 0. “Cold Plasma Dispersion Relation”

Where

A′ = S sin2 θ + P cos2 θ,

B′ = RL sin2 θ + PS(1 + cos2 θ),

C ′ = PRL

We have used the identity S2 − D2 = RL.

The solution of dispersion relation:

N2 =B′ ± F

2A′ ,

withF 2 = (RL − PS)2 sin4 θ + 4P 2D2 cos2 θ.

Alternately, using the notation of ~N = N⊥x+N‖z = N sin θx+N cos θz,the dispersion relation can be rewritten by

AN4⊥ + BN2

⊥ + C = 0

Where

A = S,

B = −(S + P )(S − N2‖ ) + D2,

C = P [(S − N2‖ )2 − D2]

The solution of N2⊥:

N2⊥ =

−B ± (B2 − 4AC)1/2

2A

• For Electron Cyclotron Wave (EC-wave)

1. Low Field Side (LFS) launch:

a. (+) sign : O-mode (or Fast Wave)

b. (– ) sign : X-mode (or Slow Wave)

2. High Field Side (HFS) launch:

a. (+) sign : X-mode (or Slow Wave)

b. (– ) sign : O-mode (or Fast Wave)

4

• For Lower Hybrid Wave (LH-wave)

a. (+) sign : Slow Wave

b. (– ) sign : Fast Wave

The dispersion relation was put into another form by Astrom and Allis:

Expanding in minors of the second column of↔M , we then obtain

(iD)2(P−N2 sin2 θ)+(S−N2)×[(S−N2 cos2 θ)(P−N2 sin2 θ)−N4 sin2 θ cos2 θ] = 0

By replacing cos2 θ by 1 − sin2 θ, we can solve for sin2 θ, obtaining

sin2 θ =−P (N4 − 2SN2 + RL)

N4(S − P ) + N2(PS − RL)

We have used the identity S2 − D2 = RL, too. Similarly,

cos2 θ =SN4 − (PS + PL)N2 + PRL

N4(S − P ) + N2(PS − RL)

Dividing the last two equations, we obtain

tan2 θ =P (N4 − 2SN2 + RL)

SN4 − (PS + RL)N2 + PRL

Since 2S = R + L, the numerator and denominator can be factored to

tan2 θ =P (N2 − R)(N2 − L)

(SN2 − RL)(N2 − P )

• When θ = 0,

P = 0 (Langmuir wave)N2 = R (R-wave)N2 = L (L-wave)

• When θ = 90,

N2 = RL/S (extraordinary wave)N2 = P (ordinary wave)

1.1 Resonances (N → ∞)

We then havetan2 θres = −P/S

θres is the resonance cone angle.This shows that the resonance frequencies depend on angle θ.

• If θ = 0,

P = 0 : Plasma resonance

S = ∞

R = ∞ Electron Cyclotron ResonanceL = ∞ Ion Cyclotron Resonance

• If θ = 90,

P = ∞ : No occurrence for finite ωp and ωS = 0 : Upper Hybrid Resonance (ωUH frequency) and Lower HybridResonance (ωLH frequency)

5

1.2 Cut-offs (N = 0)

Let N = 0 in ‖↔M ‖ = 0 and again using S2 − D2 = RL,

PRL = 0 indepedent of θ

• R = 0 (ωR cutoff frequency)

• L = 0 (ωL cutoff frequency)

• P = 0 (resonance for longitudinal wave, a cutoff for transverse waves):this degeneracy is due to our neglect of thermal motions.

1.3 Polarization

From wave equation,

iDEx + (S − N2)Ey = 0

Thus the polarization in the plane perpendicular to B0 is given by

iEx

Ey=

N2 − S

D

a. At resonance (N2 = ∞), “Linearly Polarized”

b. At cutoff (N2 = 0; R = 0 or L = 0; thus S = ±D),

iEx

Ey= − S

D= ∓1 : “Circularly Polarized”

c. At θ = 0 (N2 = R or N2 = L)

• For N2 = R

iEx

Ey=

R − S

D=

R − 1/2(R + L)

1/2(R − L)= 1 : a right-hand circular polarization

• For N2 = L

iEx

Ey=

L − S

D=

L − 1/2(R + L)

1/2(R − L)= −1 : a left-hand circular polarization

1.4 CMA diagram

The information contained in the cold dispersion relation is summarized inthe Clemmow-Mullaly-Allis (CMA) diagram as seen in Fig. 1. One furtherresult, not in the diagram, can be obtained easily from this formulation.

6

Figure 1: CMA diagram for a two-component plasma. The ion-to-electronmass ratio is chosen to be 2.5. Bounding surfaces appear as lines in thistwo-dimensional parameter space. Cross sections of wave-normal surfacesare sketched and labeled for each region. For these sketches the direction ofthe magnetic field is vertical. The small mass ratio can be misleading here:the L = 0 line intersects P = 0 at Ωi/ω = 1 − (Zme/mi). From T. Stix’sbook (AIP, 1992).

7

2 EC-wave propagation using Mathematica

2.1 O-X propagation for 2nd harmonic resonance for KSTARtokamak with low density

8

Propagation of 84-GHz Microwave in

KSTAR tokamak for Second Harmonic

Resonance with low plasma density

Electron density in unit of 10^20 m^-3

Electron temperuture in unit of keV

Toroidal magnetic field in unit of Tesla

All frequencies in unit of GHz

KSTAR major radius: 1.8 m

KSTAR plasma minor radius: 0.5 m

KSTAR toroidal magnetic field, B0: 1.5 T

KSTAR ECH system frequency: 84 GHz

Clear "Global` "

Off General::spell ;

Off General::spell1 ;

a 0.5;

R0 1.8;

f 84.0;

bz0 1.5;

te0 10.;

ne1 0.;

Nh 2;

Nnu 1.;

Tnu 1;

Az 1;

massr 2000. Az;

sc 3 10^8;

mc2 511.0;

OX-propagation-2ndHarm-lowDensity.nb 1

9

ne ne0 1 rho^2 ^Nnu ne1;

te te0 1 rho^2 ^Tnu;

ve sc Sqrt 2 te mc2 ;

bz bz0 1 a R0 rho ;

fce 28.0 bz;

fci fce massr;

fpe 90.0 Sqrt ne ;

fpi fpe Sqrt massr ;

w 2.0 Pi f;

wce 2.0 Pi fce;

wpe 2.0 Pi fpe;

wci 2.0 Pi fci;

wpi 2.0 Pi fpi;

SS = 1 - 2pe / ( 2- 2

ce)

DD = (- 2pe /( 2- 2

ce)) ( ce / )

PP = 1 - 2pe / 2

If define

q = 2pe / 2

u = 2ce / 2

q wpe^2 w^2;

u wce^2 w^2;

SS 1 q 1 u ;

DD q 1 u Sqrt u ;

PP 1 q;

AA SS;

BB SS PP SS Npar^2 DD^2;

CC PP SS Npar^2 ^2 DD^2 ;

Disc BB BB 4 AA CC;

Nppsq BB Sqrt Disc 2.0 AA ;

Npnsq BB Sqrt Disc 2.0 AA ;


10

+ sign : O-mode , - sign : X-mode

Density plot (low density)

NePlot Plot 10 ne . ne0 0.8, rho, 1, 1 ,

PlotStyle Thickness 0.005 , GrayLevel 0.5

1 0.5 0.5 1

2

4

6

8

Graphics

BTPlot Plot bz, rho, 1, 1 , PlotStyle Thickness 0.005 , GrayLevel 0.5

1 0.5 0.5 1

1.2

1.4

1.6

1.8

Graphics

Cutoff and Resonances

O-mode cutoff: q = 1 ( P = 0)

X-mode cutoff: q = (1- u )(1-N 2 ) ( (S-N 2 )2 - D2= 0 )

Upper Hybrid Resonance: q = 1 - u ( 2= 2pe+ 2

ce )

Electron Cyclotron Resonance: u = 1 ( = N ce )

O-X conversion: q = 1 + u [(1-N 2 )/(2 N )]2 ( B2 - 4 AC = 0)

Clear Npar ;


11

ocut 1;

xcut 1 Sqrt u 1 Npar^2 ;

uhr 1 u ;

ecr Nh Sqrt u 1;

oxc 1 u 1 Npar^2 2 Npar ^2;

O-Mode Cutoff position (for maximum density of 0.8 x 10^20 m^-3)

solocut Solve q ocut . ne0 0.8, rho

rho 0. 0.298142 , rho 0. 0.298142

No O-mode cutoff!

X-mode Cutoff position for N = 0.5

solxcut Solve q xcut . ne0 0.8, Npar 0.5 , rho

rho 3.70834 , rho 3.4765 , rho 0.846925 , rho 0.723424

xcutrho1 rho . solxcut 4 ;


Upper Hybrid Resonance position

soluhr Solve q uhr . ne0 0.8, rho

rho 4.06125 , rho 2.97094 , rho 0.535528 , rho 0.367715

uhrrho1 rho . soluhr 4 ;


ECR position

solecr Solve ecr 0, rho

rho 7.2 , rho 0.

ecrrho rho . solecr 2 ;

O-X Conversion [No O-X conversion]

[1] For fixed centeral density, ne0 =0.8x 1020m

3 and fixed N = 0.5

soloxc1 Solve q oxc . ne0 0.8, Npar 0.5 , rho

rho 3.63975 0.38162 , rho 3.63975 0.38162 ,

rho 0.0397483 0.482289 , rho 0.0397483 0.482289


12

No solution!

[2] For fixed centeral density, ne0 =0.8 x 1020m

3 and fixed N = 0.7

soloxc2 Solve q oxc . ne0 0.8, Npar 0.7 , rho

rho 3.60977 0.188404 , rho 3.60977 0.188404 ,

rho 0.00976686 0.352002 , rho 0.00976686 0.352002

No solution!

[3] For fixed = 0.2 and fixed N = 0.5

soloxc3 Solve q oxc . rho 0.2, Npar 0.5 , ne0

ne0 1.02193

oxcne0 ne0 . soloxc3 1 ;

[4] For fixed = 0.2 and fixed ne0 =1.0 x 1020m

3

soloxc4 Solve q oxc . rho 0.2, ne0 1 , Npar

Npar 1.88051 , Npar 0.531771 , Npar 0.531771 , Npar 1.88051

No solution!

Thus, for fixed N of 0.5, the maximum density > 1.0 x 1020m 3 is

required to have O-X mode conversion at = 0.2.

For the maximum density of 1.0 x 1020m 3 , the parallel refractive

index, N > 0.5 to have O-X mode conversion at = 0.2.

O-X Propagation Plot for N|| = 0.5

OX_temp1 Plot Nppsq . ne0 0.8, Npar 0.5 , Npnsq . ne0 0.8, Npar 0.5 ,

rho, 1, 1 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1 ,

Thickness 0.008 , RGBColor 1, 0, 0 , PlotRange 0, 5

Omode1 Plot Nppsq . ne0 0.8, Npar 0.5 ,

rho, 1, 1 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1

Xmode1 Plot Npnsq . ne0 0.8, Npar 0.5 ,

rho, xcutrho1, 1 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0


rho, uhrrho2, uhrrho1 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0


13


rho, 1, xcutrho2 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0

OXWavePlot Show Omode1, Xmode1, Xmode2, Xmode3 , PlotRange 0, 5

Show NePlot, BTPlot, OXWavePlot ,

Graphics Dashing 0.01, 0.01 , Line xcutrho1, 0 , xcutrho1, 10 ,

Dashing 0.01, 0.01 , Line xcutrho2, 0 , xcutrho2, 10 ,

Dashing 0.01, 0.01 , Line uhrrho1, 0 , uhrrho1, 10 ,

Thickness 0.015 , GrayLevel 0.5 , Line ecrrho, 0 , ecrrho, 10 ,

Graphics Text "N 0.5", 0.2, 6 , Text "O", 0.6, 0.5 ,

Text "UHR", uhrrho1 0.1, 4 , Text "Ne", 0.25, 8. , Text "BT", 0.6, 2.2 ,

Text "X", 0.9, 0.2 , Text "X cutoff", xcutrho1, 2 , Text "X", 0.42, 3 ,

Frame True, FrameLabel " ", "N2 , Ne 1019 m 3 , BT T " ,

PlotRange 1, 1 , 0, 10

0.75 0.5 0.25 0 0.25 0.5 0.75 1

2

4

6

8

10

N2

,e

N0

19

1m

3,

BT

T

N 0.5

O

UHR

Ne

BT

X

X cutoff

X

Graphics


14

2.2 O-X propagation for 2nd harmonic resonance for KSTARtokamak with high density

15


KSTAR tokamak for Second Harmonic

Resonance with high plasma density









Clear "Global` "



a 0.5;

R0 1.8;

f 84.0;

bz0 1.5;

te0 10.;

ne1 0.;

Nh 2;

Nnu 1.;

Tnu 1;

Az 1;

massr 2000. Az;

sc 3 10^8;

mc2 511.0;

OX-propagation-2ndHarm-highDensity.nb 1

16




bz bz0 1 a R0 rho ;

fce 28.0 bz;

fci fce massr;

fpe 90.0 Sqrt ne ;


w 2.0 Pi f;

wce 2.0 Pi fce;

wpe 2.0 Pi fpe;

wci 2.0 Pi fci;

wpi 2.0 Pi fpi;

SS = 1 - 2pe / ( 2- 2

ce)

DD = (- 2pe /( 2- 2

ce)) ( ce / )

PP = 1 - 2pe / 2

If define

q = 2pe / 2

u = 2ce / 2

q wpe^2 w^2;

u wce^2 w^2;

SS 1 q 1 u ;

DD q 1 u Sqrt u ;

PP 1 q;

AA SS;



Disc BB BB 4 AA CC;




17


Density plot (low density)

NePlot

Plot 2 ne . ne0 1, rho, 1, 1 , PlotStyle Thickness 0.005 , GrayLevel 0.5

1 0.5 0.5 1

0.5

1

1.5

2

Graphics

BTPlot Plot bz, rho, 1, 1 , PlotStyle Thickness 0.005 , GrayLevel 0.5

1 0.5 0.5 1

1.2

1.4

1.6

1.8

Graphics





ce )

Electron Cyclotron Resonance: u = 1 ( = N ce )


Clear Npar ;


18

ocut 1;


uhr 1 u ;

ecr Nh Sqrt u 1;


O-Mode Cutoff position (for maximum density of 0.8 x 10^20 m^-3)

solocut Solve q ocut . ne0 1, rho

rho 0.359011 , rho 0.359011

ocutrho1 rho . solocut 2 ;



solxcut Solve q xcut . ne0 1, Npar 0.5 , rho

rho 3.68869 , rho 3.50128 , rho 0.88288 , rho 0.784159




soluhr Solve q uhr . ne0 1, rho

rho 4.01963 , rho 3.04426 , rho 0.677797 , rho 0.541688



ECR position

solecr Solve ecr 0, rho

rho 7.2 , rho 0.


O-X Conversion

[1] For fixed centeral density, ne0 =1x 1020m

3 and fixed N = 0.5

soloxc1 Solve q oxc . ne0 1, Npar 0.5 , rho

rho 3.63314 0.345317 ,

rho 3.63314 0.345317 , rho 0.0523865 , rho 0.11867


19

oxcrho1 rho . soloxc1 4 ;



index, N > 0.5 to have O-X mode conversion at ~ 0.1.

O-X Propagating for N|| = 0.5 (O-X conversion occurrence)

OXconv_temp2 Plot Nppsq . ne0 1, Npar 0.5 , Npnsq . ne0 1, Npar 0.5 ,



Omode21 Plot Nppsq . ne0 1, Npar 0.5 ,

rho, oxcrho1, 1 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1

Omode22 Plot Nppsq . ne0 1, Npar 0.5 ,

rho, oxcrho2, 1 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1

Xmode21 Plot Npnsq . ne0 1, Npar 0.5 ,



rho, oxcrho1, uhrrho1 0.013 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0


rho, oxcrho2, uhrrho2 0.02 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0


rho, 1, xcutrho2 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0

OXWavePlot2 Show Omode21, Omode22, Xmode21, Xmode22, Xmode23, Xmode24


20

Show NePlot, BTPlot, OXWavePlot2 ,


Dashing 0.01, 0.01 , Line xcutrho1, 0 , xcutrho1, 3 ,

Dashing 0.01, 0.01 , Line oxcrho2, 0 , oxcrho2, 3 ,





Graphics Text "N 0.5", 0.2, 2.7 , Text "O X Conv", oxcrho1 0.1, 1 ,

Text "Ne", 0.25, 2. , Text "BT", 0.75, 2 ,

Text "O X Conv", oxcrho2 0.1, 1 , Text "O", 0.6, 0.3 ,

Text "UHR", uhrrho1 0.1, 2 , Text "X", 0.9, 0.2 ,

Frame True, FrameLabel " ", "N2 , Ne 0.5 x 1020 m 3 , BT T " ,

PlotRange 1, 1 , 0, 3

0.75 0.5 0.25 0 0.25 0.5 0.75 1

0.5

1

1.5

2

2.5

3

N2

,e

N5.

0x

01

02

m3

,B

TT

N 0.5

O X Conv

NeBT

O X Conv

O

UHR

X

Graphics


21

2.3 O-X-B heating

22


KSTAR tokamak for Fundamental Harmonic

Resonance with high density plasma









Clear "Global` "



a 0.5;

R0 1.8;

f 84.0;

bz0 3.5;

te0 10.;

ne1 0.;

Nh 1;

Nnu 1.;

Tnu 1;

Az 1;

massr 2000. Az;

sc 3 10^8;

mc2 511.0;

OXB.nb 1

23




bz bz0 1 a R0 rho ;

fce 28.0 bz;

fci fce massr;

fpe 90.0 Sqrt ne ;


w 2.0 Pi f;

wce 2.0 Pi fce;

wpe 2.0 Pi fpe;

wci 2.0 Pi fci;

wpi 2.0 Pi fpi;

SS = 1 - 2pe / ( 2- 2

ce)

DD = (- 2pe /( 2- 2

ce)) ( ce / )

PP = 1 - 2pe / 2

If define

q = 2pe / 2

u = 2ce / 2

q wpe^2 w^2;

u wce^2 w^2;

SS 1 q 1 u ;

DD q 1 u Sqrt u ;

PP 1 q;

AA SS;



Disc BB BB 4 AA CC;



OXB.nb 2

24


NePlot

Plot 100 ne . ne0 1, rho, 1, 1 , PlotStyle Thickness 0.005 , GrayLevel 0.5

1 0.5 0.5 1

20

40

60

80

100

Graphics

BTPlot Plot 10 bz, rho, 1, 1 , PlotStyle Thickness 0.005 , GrayLevel 0.5

1 0.5 0.5 1

35

40

45

Graphics





ce )

Electron Cyclotron Resonance: u = 1 ( = ce )


Clear Npar ;

OXB.nb 3

25

ocut 1;


uhr 1 u ;

ecr Sqrt u 1;


O-Mode Cutoff position

solocut Solve q ocut . ne0 1, rho

rho 0.359011 , rho 0.359011





rho 3.79521 , rho 3.34727 , rho 1.22572 , rho 0.972991

xcutrho rho . solxcut 4 ;


soluhr Solve q uhr . ne0 1, rho

rho 4.47818 , rho 1.82883 0.848917 ,

rho 1.82883 0.848917 , rho 0.935832

uhrrho rho . soluhr 4 ;

ECR position

solecr Solve ecr 0 . ne0 1, rho

rho 7.8 , rho 0.6


O-X Conversion

[1] For fixed centeral density, ne0 =1x 1020 m 3 and fixed N = 0.5


rho 3.75345 0.755581 , rho 3.75345 0.755581 ,

rho 0.153449 0.672412 , rho 0.153449 0.672412

OXB.nb 4

26

No solution!

[2] For fixed centeral density, ne0 =1 x 1020 m 3 and fixed N = 0.8


rho 3.61662 0.243947 ,

rho 3.61662 0.243947 , rho 0.244533 , rho 0.277768



[3] For fixed = 0.2 and fixed N = 0.5

soloxc3 Solve q oxc . rho 0.2, Npar 0.5 , ne0

ne0 1.53094

oxcne0 ne0 . soloxc3 1 ;

[4] For fixed = 0.2 and fixed ne0 =1 x 1020 m 3

soloxc4 Solve q oxc . rho 0.2, ne0 1 , Npar

Npar 1.32994 , Npar 0.751912 , Npar 0.751912 , Npar 1.32994

oxcnpar Npar . soloxc4 3 ;

Thus, for fixed N of 0.5, the maximum density > 1.53 x 1020m 3 is

required to have O-X mode conversion at = 0.2.


index, N > 0.75 to have O-X mode conversion at = 0.2. However, for

large N , there



rho 3.70122 , rho 3.48506 , rho 1.10163 , rho 0.986692


Graphics`Graphics`

$TextStyle FontFamily "Times", FontSize 14 ;

OXB.nb 5

27

Electron Bernstein ModeQ,be

be

1

pe2

e2

whereQ,be

be

1

Q2 12

1 3 be

Q2 12 Q2 22

1 3 5 be2

Q2 12 Q2 22 Q2 32

with

Qe

, bek 2 Ve

2

kperp2 10. w 3. ^2 Npb2;

be kperp2 ve^2 2. wce^2 10^18 ;

Qe w wce;

XX wpe^2 wce^2;

alphaovbe 1 Qe^2 1 3 be Qe^2 1 Qe^2 2^2 ;

solNpb Solve alphaovbe 1 XX, Npb2

Npb2 23.1843 4. 0.734694 1. 0.277778 rho 2

1. 0.734694 1. 0.277778 rho 2 1.

1. 0.734694 1. 0.277778 rho 2

1.18568

1. 0.277778 rho 2 0. ne0 1. 1. rho21.

1. 0.277778 rho 2 1. 1. rho2

NEBW Npb2 . solNpb 1

23.1843 4. 0.734694 1. 0.277778 rho 2

1. 0.734694 1. 0.277778 rho 2 1.

1. 0.734694 1. 0.277778 rho 2

1.18568

1. 0.277778 rho 2 0. ne0 1. 1. rho21.

1. 0.277778 rho 2 1. 1. rho2

OXB.nb 6

28

EBWPlot Plot NEBW . ne0 1, rho, ecrrho, uhrrho 0.015 ,

PlotStyle Thickness 0.008 , PlotRange Automatic

0.65 0.7 0.75 0.8 0.85 0.9

70

80

90

100

Graphics

X-EBW Heating Scheme

Omode1 Plot 10 Nppsq . ne0 1, Npar 0.5 ,

rho, ecrrho, 1 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1

Omode2 Plot 10 Npnsq . ne0 1, Npar 0.5 ,

rho, ecrrho, ocutrho2 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1


rho, 1, ocutrho2 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1

Xmode1 Plot 10 Npnsq . ne0 1, Npar 0.5 ,

rho, xcutrho, 1 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0


rho, ecrrho, uhrrho 0.014 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0

Xmode3 Plot 10 Nppsq . ne0 1, Npar 0.5 ,

rho, 1, ecrrho , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0

OXWavePlot Show Omode1, Omode2, Omode3, Xmode1, Xmode2, Xmode3

OXB.nb 7

29

Show NePlot, BTPlot, OXWavePlot, EBWPlot ,

Graphics Dashing 0.01, 0.01 , Line ocutrho1, 0 , ocutrho1, 110 ,

Dashing 0.01, 0.01 , Line ocutrho2, 0 , ocutrho2, 110 ,

Dashing 0.01, 0.01 , Line xcutrho, 0 , xcutrho, 110 ,

Dashing 0.01, 0.01 , Line uhrrho, 0 , uhrrho, 110 ,


Graphics Text "N 0.5", 0.2, 80 , Text "O", 0.75, 8 ,

Text "UHR", uhrrho 0.09, 50 , Text "Ne", 0.55, 80 ,

Text "BT", 0.6, 37 , Text "X", 0.75, 22 , Text "EBW", 0.8, 80 ,

Frame True, FrameLabel " ", "N2 , Ne 1018 m 3 , BT kG " ,

PlotRange 1, 1 , 0, 110

0.75 0.5 0.25 0 0.25 0.5 0.75 1

20

40

60

80

100

N2

,e

N0

18

1m

3,

BT

Gk N 0.5

O

UHR

Ne

BT

X

EBW

Graphics

O-X-EBW Heating Scheme

OXconv_temp2

Plot 10 Nppsq . ne0 1, Npar 0.8 , 10 Npnsq . ne0 1, Npar 0.8 ,



Omode21 Plot 10 Nppsq . ne0 1, Npar 0.8 ,

rho, ecrrho, 1 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1


rho, ecrrho, oxcrho2 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1


rho, 1, oxcrho1 , PlotStyle Thickness 0.008 , RGBColor 0, 0, 1



OXB.nb 8

30


rho, ecrrho, uhrrho 0.016 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0


rho, oxcrho2, ecrrho , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0


rho, 1, oxcrho1 , PlotStyle Thickness 0.008 , RGBColor 1, 0, 0

OXWavePlot2 Show Omode21, Omode22, Omode23, Xmode21, Xmode22, Xmode23, Xmode24

Show NePlot, BTPlot, OXWavePlot2, EBWPlot ,




Dashing 0.01, 0.01 , Line uhrrho, 0 , uhrrho, 110 ,


Graphics Text "N 0.8", 0.15, 80 , Text "O X Conv", oxcrho1, 50 ,

Text "Ne", 0.55, 80 , Text "BT", 0.6, 37 ,

Text "O X Conv", oxcrho2, 50 , Text "O", 0.75, 6 ,

Text "UHR", uhrrho 0.09, 50 , Text "X", 0.75, 20 , Text "EBW", 0.8, 80 ,

Frame True, FrameLabel " ", "N2 , Ne 1018 m 3 , BT kG " ,

PlotRange 1, 1 , 0, 110

0.75 0.5 0.25 0 0.25 0.5 0.75 1

20

40

60

80

100

N2

,e

N0

18

1m

3,

BT

Gk N 0.8

O X Conv

Ne

BT

O X Conv

O

UHR

X

EBW

Graphics

OXB.nb 9

31

3 Dispersion Relations in a Hot Plasma

3.1 Electromagnetic Dispersion Relation

Vlasov Equation for a Collisionless Plasmas

∂fs

∂t(~r,~v, t) + ~v · ~∇rfs(~r,~v, t) +

(qs

ms

~E +qs

ms~vs × ~B

)

~∇vfs = 0

Maxwell’s equations

~∇ · ~E =1

ǫ 0

∑

s

qs

∫

fs d3v

1

µ0

~∇× ~B = ǫ0∂ ~E

∂t+

∑

s

qs

∫

~vfs d3v

~∇× ~E = −∂ ~B

∂t~∇ · ~B = 0

Let

fs(~r,~v, t) = fso(~r,~v) + fs1(~r,~v, t)

~B = ~B0(~r) + ~B1

~E = ~E0 + ~E1 = 0 + ~E1

and, fs1, ~B1, ~E1 are dependent of ei(~k·~r−ωt).

1© Zeroth order

∂fs0

∂t= 0

~v · ~∇rfs0 +

(qs

ms~v × ~B0

)

· ~∇vfs = 0

~∇ · ~E0 =1

ǫ0

∑

s

qs

∫

fs0 d3v

1

µ0

~∇× ~B =∑

s

qs

∫

~vfs0 d3v

2© First order

∂fs1

∂t+ ~vs · ~∇rfs1 +

(qs

ms~vs × ~B0

)

· ~∇vfs1

︸︷︷︸

= − qs

ms

(

~E1 + ~vs × ~B1

)

· ~∇vfs0

︸︷︷︸

dfs1

dtS(~r,~v, t)

32

i~k · ~E1 = 1ǫ0

∑

s qs

∫fs0 d3v

1µ0

~k × ~B1 = −ω(

ǫ0 ~E1 + iω

∑

s qs

∫~vfs1 d3v

)

~B1 = 1ω~k × ~E

Let ~E1 + iǫ0ω

∑

s qs

∫~vfs1 d3v =

↔K · ~E1

∴1

µ0

~k × ~B1 = −ωǫ0

↔K × ~E1 = −ω

↔ǫ · ~E1

=1

µ0ω~k × (~k × ~E1)

∴ ~k × (~k × ~E1) + µ0ǫ0ω2

↔K · ~E1 = 0

⇒ ~k(~k · ~E1) − k2 ~E1 + µ0ω2ǫ0

↔K · ~E1 = (~k~k + µ0ǫ0ω

2↔K −k2−→1 ) = 0

Thus Det (~k~k + µ0ǫ0ω2

↔K −k2

↔1 ) = 0

Fs1 =

∫ t

−∞S dt′ = − qs

ms

∫ t

−∞dt′[ ~E1(~r

′(t′), t′)+~vs(t′)× ~B1(~r

′(t′), t′)] · ~∇v′fs0

when

S = − qs

ms[ ~E1(~r

′(t′), t′) + ~vs(t′) × ~B1(~r

′(t′), t′)] · ~∇v′fs0

= − qs

ms[ ~E1(~r

′(t′), t′) + ~v(t′) × 1

ω(~k × ~E1)] · ~∇v′fs0

Since ~v × (~k × ~E1) = (~v · ~E1)~k − (~k · ~v) ~E1

S = − qs

ms

[(

1 −~k · ~v′(t′)

ω

)

~E1(~r′, t′) +

1

ω(~v′(t′) · ~E1(~r

′, t′))~k

]

· ~∇v′fs0

33

Thus

f1(~r,~v, t) = − q

m

∫ t

−∞

[(

1 −~k · ~v′(t′)

ω

)

~E1(~r′, t′) +

1

ω(~v′(t′) · ~E1(~r

′, t′))~k

]

·~∇v′f0 dt′

Where, I dropped the sub-index s

It is assumed that

~E1(~r′, t′) = ~E exp[i(~k · ~r′ − ωt′)]

f0(~r,~v) = f(v⊥, vz)

v2⊥ = v2

x + v2y

d~r′

dt= ~v′,

d~v′

dt=

q

m~v′ × ~B0

⊙Particle motion in a uniform field

∂~v′

∂t′=

q

m( ~E + ~v′ × ~B0)

~v′ = ~v′(t′)

~E = 0

~B0 = B0z

Let q ~B0

m = ~Ω (sign contained)

∂~v′

∂t′= ~v′ × ~Ω = v′yΩx − v′xΩy + 0z

⇒∂ ~v′‖∂t′

= 0

∴ v′‖ = v′z(t′) = const = vz(t), t

′ < t

dv′xdt′

= v′yΩ

dv′ydt′

= −v′xΩ

34

(1) + i(2) =∂v′x∂t′

+ i∂v′y∂t′

= v′yΩ − iv′xΩ′

⇒ d

dt′(v′x + iv′y) = −iΩ(v′x + iv′y)

Let v′x + iv′y = v+′

⇒ d

dt′v+′

= −iΩv+′

(d

dt′v+′

+ iΩv+′= 0) × eiΩt′

⇒ d

dt′(v+′

eiΩt′) = 0

∫ t′

t

d

dt′′(v+′

eiΩt′′)dt′′ = v+′(t′)eiΩt′ − v+′

(t)eiΩt = 0

⇒ v+′(t′) = v+′

(t′)eiΩ(t−t′)

= v⊥eiαeiΩ(t−t′)

= v⊥ei(α+Ω(t−t′))

= v⊥ei(α−Ω(t′−t))

∴ v′x(t′) = Re[v+′(t′)] = v⊥ cos(α − Ω(t′ − t))

v′y(t′) = Im[v+′

(t′)] = v⊥ sin(α − Ω(t′ − t))

v′z(t′) = v+′

(t) = vz(t)

Note Ω contains sign

dx′(t′)dt′

= v′x(t′) = v⊥ cos(α − Ω(t′ − t))

dy′(t′)dt′

= v′y(t′) = v⊥ sin(α − Ω(t′ − t))

dz′(t′)dt′

= v′z(t′) = vz

⇒∫ t′

t

x′(t′)dt”

dt” = x′(t′) − x′(t) =v⊥Ω

[sin(α − Ω(t′ − t′)) − sin(α − Ω(t′ − t))]

=v⊥Ω

[sin(α − sin(α − Ω(t′ − t))]

∴ x′(t′) = x +v⊥Ω

[sin(α) − sin(α) cos(Ω(t′ − t)) + cos(α) sin Ω(t′ − t)]]

= x +v⊥Ω

[sin(α)(1 − cos(Ω(t′ − t))) + cos(α) sin Ω(t′ − t)]]

= x +1

Ω[vx0 sinΩ(t′ − t) + vy0(1 − cos Ω(t′ − t)]

35

Similarly

y′(t′) = y − 1

Ω[vx0 cos Ω(t′ − t) − vy0(1 − sinΩ(t′ − t)]

= y +1

Ω[−vx0 cos Ω(t′ − t) + vy0(1 − sinΩ(t′ − t)]

z′(t′) = z + vz(t′ − t)

A. Calculation of

[(1 −~k · ~v′(t′)

ω) ~E1(~r′, t

′) +1

ω(~v′(t′) · ( ~E1(~r′, t

′))~k]

Since ~E1 = ~E1ei(~k·~r′−ωt′) = ~E1e

i(~k·~r′−ω(t′−t))e−iωt

~k′ · ~r′ = ~k · ~r +kx

Ω[vx0 sinΩ(t′ − t) − vy0(cos Ω(t′ − t) − 1)]

+ky

Ω[vx0(cos Ω(t′ − t)) − vy0 sinΩ(t′ − t)] + kzvz(t

′ − t)

= ~k · ~r +1

Ω(kxvx0 + kyvy0) sinΩ(t′ − t)

− 1

Ω(kxvy0 − kyvx0) cos Ω(t′ − t)

+1

Ω(kxvy0 − kyvx0) + kzvz(t

′ − t)

= ~k · ~r +k⊥v⊥

Ωsin(Ω(t′ − t) − α) +

k⊥v⊥Ω

sinα + kzvz(t′ − t)

~E1ei(~k·~r′−ωt′) = ~E1e

i(~k·~r−ωt)∑

n

∑

m

Jm(k⊥v⊥

Ω)Jn(

k⊥v⊥Ω

) × exp[in(Ω(t′ − t) − α)]

× exp[imα] exp[i(kzvz − ω)(t′ − t)]

where, we used eia sin x =∑

Jm(a)eimx

36

[(

1 −~k · ~v′

ω

)

+1

ω

(

~v′ · ~E1

)

~k

]

· ~∇v′f0

=∂f0

∂v′z

[(

1 − kxv′xω

−kyv

′y

ω− kzv

′z

ω

)

Ez +(v′xEx + v′yEy + v′zEz

) kz

ω

]

+∂f0

∂v′⊥

[(

1 − kxv′xω

−kyv

′y

ω− kzv

′z

ω

) (

Exv′xv′⊥

+ Ey

v′yv′⊥

)

+(v′xEx + v′yEy + v′zEz

)(

kx

ω

v′xv′⊥

+ky

ω

v′yv′⊥

)

v′z

]

Ez

=

[∂f0

∂vz

kzv⊥ω

+∂f0

∂v⊥

(

1 − kzvz

ω

)] [

Exv′xv⊥

+ Ey

v′yv⊥

]

+

[∂f0

∂vz

(

1 − kxv′xω

− kyvy

ω

)

+∂f0

∂v⊥

(kx

ω

v′xv⊥

+ky

ω

v′yv⊥

)

v′z

]

Ez

v′xv⊥

= cos(α − Ω

(t′ − t

)),

v′yv⊥

= sin(α − Ω

(t′ − t

))

=

[∂f0

∂vz

kzv⊥ω

+∂f0

∂v⊥

(

1 − kzvz

ω

)][Ex cos

(α − Ω(t′ − t)

)+ Ey sin

(α − Ω(t′ − t)

)]

+

[∂f0

∂vz

(

1 − k⊥v⊥ω

cos(α − Ω(t′ − t)

))

+∂f0

∂v⊥

k⊥vz

ωcos

(α − Ω(t′ − t)

)]

Ez

= C

∴ f1 =q

m

∑

n

∑

m

∫ t

−∞Jm

(k⊥v⊥

Ω

)

Jn

(k⊥v⊥

Ω

)

e−i(n−m)αe−inΩ(t′−t)ei(kzvz−ω)(t′−t)

×C dt′

We dropped time dependence of ei(~k·~r−ωt).

Let t′ − t = τ , dt′ = dτ ,∫ t−∞ dt′ =

∫ 0−∞ dτ .

Thus,

f1 = − q

m

∑

n

∑

m

e−i(n−m)α

∫ 0

−∞dτ Jm

(k⊥v⊥

Ω

)

Jn

(k⊥v⊥

Ω

)

einΩτei(kzvz−ω)τ × C

A.1) For the Ex component

cos(α − Ωτ) =1

2

(

ei(α−Ωτ) + e−i(α−Ωτ))

1

2

∫ 0

−∞dτ einΩτei(kzvz−ω)τ

(eiαe−iΩτ + e−iαeiΩτ

)

=1

2

[ieiα

ω − kzvz − (n − 1)Ω+

ie−iα

ω − kzvz − (n + 1)Ω

]

37

∴ f1 = − iq

mω

∑

n

∑

m

1

2Jm

(k⊥v⊥

Ω

)

Jn

(k⊥v⊥

Ω

)

×[

e−i[(n−1)−m]α

ω − kzvz − (n − 1)Ω+

e−i[(n+1)−m]α

ω − kzvz − (n + 1)Ω

]

UEx

n − 1 → n ⇒ Jn → Jn+1

n + 1 → n ⇒ Jn → Jn−1

∑

n

1

2Jn

(k⊥v⊥

Ω

) [

e−i[(n−1)−m]α

ω − kzvz − (n − 1)Ω+

e−i[(n+1)−m]α

ω − kzvz − (n + 1)Ω

]

=∑

n

e−i(n−m)α

ω − kzvz − nΩ

(Jn+1 + Jn−1

2

)

=∑

n

ei(m−n)α

ω − kzvz − nΩ

n

λJn(λ)

where we used nJn

λ = 12(Jn+1 + Jn−1).

Thus,

f1 = − iq

mω

∑

n

∑

m

ei(m−n)α

ω − kzvz − nΩ

n

λJm(λ) Jn(λ) U Ex

where λ = k⊥v⊥Ω , U = ∂f0

∂vzkzv⊥ + ∂f0

∂v⊥(ω − kzvz).

A.2) For the Ey component

sin(α − Ωτ) =1

2i

(

ei(α−Ωτ) − e−i(α−Ωτ))

1

2i

∫ 0


(

ei(α−Ωτ) − e−i(α−Ωτ))

=1

2i

[ieiα

ω − kzvz − (n − 1)Ω− ie−iα

ω − kzvz − (n + 1)Ω

]

similarly,

f1 = − iq

mω

∑

n

∑

m

ei(m−n)α

ω − kzvz − nΩJm(λ)

(1

i

)(−J ′

n(λ))

U Ey

= − iq

mω

∑

n

∑

m

Jm(λ)ei(m−n)α

ω − kzvz − nΩ

(iJ ′

n(λ))

U Ey

where we used 2J ′n = Jn−1 − Jn+1.

38

A.3) For the Ez component

∑

n

∑

m

JmJne−i(n−m)α

∫ 0


(

1 − k⊥v⊥ω

cos(α − Ωτ)

)

=∑

n

∑

n

iJmJnei(m−n)α

ω − kzvz − nΩ− k⊥v⊥

ω

iJmei(m−n)α

ω − kzvz − nΩ

nJn

λ

=∑

n

∑

m

1

ω

iJmJnei(m−n)α

ω − kzvz − nΩ(ω − nΩ)

∑

n

∑

m

JmJne−i(n−m)α

∫ 0

−∞dτeinτei(kzvz−ω)τ kzvz

ωcos(α − Ωτ)

=∑

n

∑

m

(1

ω

) (

iJmJne−i(n−m)α

ω − kzvz − nΩ

)

n

λik⊥vz

=∑

n

∑

m

1

ω

iJmJne−i(n−m)α

ω − kzvz − nΩ

nΩ

v⊥vz

Thus,

f1 = − iq

mω

∑

n

∑

m

iJm(λ)e−i(n−m)α

ω − kzvz − nΩ

[

(ω − nΩ)∂f0

∂vz+

nΩ

v⊥vz

∂f0

∂v⊥

]

×EzJn(λ)

Putting together as components gives the perturbed distribution func-tion f1

∴ f1 =iq

mω

∑

n

∑

m

iJm(λ)e−i(n−m)α

ω − kzvz − nΩ

[

−Exn

λUJn − iEyUJ ′

n − EzWJn

]

where

λ =k⊥v⊥

Ω, Jn = Jn(λ), J ′

n =d

dxJn(λ)

U = (ω − kzvz)∂f0

∂v⊥+ kzv⊥

∂f0

∂vz

W =nΩ

v⊥vz

∂f0

∂v⊥+ (ω − nΩ)

∂f0

∂vz

B. Calculation of∫

vf1d3v

Let α = φ

39

B.1) v⊥ cos φx

∫

dvv⊥ cos φf1x =

∫ ∞

∞dvz

∫ ∞

0v⊥dv⊥

∫ 2π

0dφv⊥ cos φf1x

=iq

mω

∑

n

∑

n

∫ ∞

∞dvz

∫ ∞

0v2⊥dv⊥

∫ 2π

0cos φei(m−n)φdφ

×[

−Exn

λUJn − iEyUJ ′

n − EzWJn

] Jm(λ)

ω − kzvz − nΩx

=iq

mω

∑

n

∑

n

∫ ∞

∞dvz

∫ ∞

0v2⊥dv⊥ · 2π

δm,n+1 + δm,n−1

2

×[

−Exn

λUJn − iEyUJ ′

n − EzWJn

] Jm(λ)

ω − kzvz − nΩx

=iq

mω

∑

n

∫ ∞

∞dvz

∫ ∞

02πv2

⊥dv⊥[

−Exn

λUJn − iEyUJ ′

n − EzWJn

]

× 1

ω − kzvz − nΩ

(Jn+1 + Jn−1

2

)

x

=iq

mω

∑

n

∫ ∞

∞dvz

∫ ∞

02πv⊥dv⊥

1

ω − kzvz − nΩv⊥

(n

λJn

)

×[

−Exn

λUJn − iEyUJ ′

n − EzWJn

]

x

=−iq

mω

∑

n

∫

d3v1

ω − kzvz − nΩ

×[

v⊥

(nJn

λ

)2

UEx + iv⊥n

λJnJ ′

nUEy + v⊥n

λJ2

nEy

]

x

B.2) v⊥ sinφy

∫ 2π

0sinφei(m−n)φdφ = 2π

1

2i(−δm,n+1 + δm,n−1)

∫

d3vv⊥ sinφf1y

=iq

mω

∑

n

∫

d3v1

ω − kzvz − nΩv⊥

1

iJ ′

n

×[

−Exn

λUJm − iEyUJ ′

n − EzWJn

]

y

= − iq

mω

∑

n

∫

d3v1

ω − kzvz − nΩ

×[

−iv⊥Un

λJnJ ′

nEx + vperpU(J ′n)2Ey + iv⊥WJnJ ′

nEz

]

y

B.3) vz z∫ 2π

0ei(m−n)φdφ = 2πδn,m

40

∴

∫

d3vvzf1z

=iq

mω

∑

n

∫

d3v1

ω − kzvz − nΩvz · Jn

[

−Exn

λUJn − iEyUJ ′

n − EzWJn

]

z

=iq

mω

∑

n

∫

d3v1

ω − kzvz − nΩ

[

vzn

λJ2

nEx + ivzJnJ ′nUEy + vzWJ2

nEz

]

z

Thus, the dielectric tensor,↔K

↔K · ~E =

↔1 · ~E +

i

ǫ0ω

∑

s

qs

∫

~vfs1d3v

=⇒↔1 +

∑

s

ω2ps

ω2

1

ns

∞∑

n=−∞

∫

d3v

↔S

ω − kzvz − nΩ=

↔K

where∫

d3v =∫ ∞−∞ dvz

∫ ∞0 2πv⊥dv⊥

↔S=

v⊥(nJn

λ )2U iv⊥nλJnJ ′

nU v⊥W nλJ2

n

−iv⊥U nλJnJ ′

n v⊥U(J ′n)2 −iv⊥WJnJ ′

n

vznλIJ2

n ivzJnJ ′nU vzWJ2

n

and,

ω2ps =

nsq2s

msǫ0

U = (ω − kzvz)∂fs0

∂v⊥+ kzv⊥

∂fs0

∂vz

W =nΩs

v⊥vz

∂fs0

∂v⊥+ (ω − nΩs)

∂fs0

∂vz

Ωs =qsB0

ms

λs =k⊥v⊥Ωs

41

C. For an isotropic Maxwellian plasma

fs0 = nsfs0 = ns(a√

π)−3e−v2/a2

where the thermal velocity a =√

2Ts/ms. and ns is the numberdensity of the species s.** Fried-Conte function

Z(z) =1√π

∫ ∞

−∞

e−x2

x − zdx

and

Z ′(z) = −2[1 + zZ(z)]

(1)

∫ ∞

0e−a2x2

xJ2n(px)dx =

1

2a2e−

p2

2a2 In(p2

2a2)

(2)

∫ ∞

0e−a2x2

x2J ′n(px)Jn(px)dx =

p

4a4e−

p2

2a2 [I ′n(p2

2a2) − In(

p2

2n2)]

(3)

∫ ∞

0e−

x2

2b x3[J ′n(x)]2dx = be−b[n2In(b) − 2b2(I ′n(b) − In(b))]

where In is the modified Bessel function of order n ;I ′n denotes the derivative of In with respect to its argument.

(

⋆

∫ ∞

0tJν(pt)Jν(qt)e

−a2t2dt =1

2a2exp(−p2 + q2

4a2)Iν(

pq

2a2

)

when p=q,

(1)

∫ ∞

0tJ2

n(pt)e−a2t2dt =1

2a2e−

p2

2a2 In(p2

2a2)

d

dp(1) ⇒

∫ ∞

02t2Jn(pt)J ′

n(pt)e−a2t2dt =1

2a2(− p

a2)e−

p2

2a2 In(p2

2a2) +

1

2a2e−

p2

2a2 (p

a2)I ′n(

p2

2a2)

⇒ (2)

∫ ∞

0t2Jn(pt)J ′

n(pt)e−a2t2dt =p

4a4e−

p2

2a2 [I ′n(p2

2a2) − In(

p2

2a2)]

(3)

∫ ∞

0e−

x2

2b x3[J ′n(x)]2dx = be−b[n2In(b) − 2b2(I ′n(b) − In(b))]

42

Let p = k⊥Ω = λ

v⊥and a2 → 1

a2 = m2T , x = v⊥, b =

k2⊥T

mΩ2

(1)m

2πT

∫ ∞

02πv⊥J2

n(k⊥v⊥

Ω)e−

mv2⊥

2T = e−bIn(b)

where b =k2⊥T

mΩ2

(2)m

2πT

∫ ∞

02πv2

⊥Jn(k⊥v⊥

Ω)J ′

n(k⊥v⊥

Ω)e−

mv2⊥

2T =m

2πT

∫ ∞

02πv2

⊥Jn(λ)J ′n(λ)e−

mv2⊥

2T

=k⊥T

mΩe−b[I ′n(b) − In(b)]

(3)m

2πT

∫ ∞

02πv3

⊥[J ′n(λ)]2e−

mv2⊥

2T =1

2

(2T

m

)

e−b

[n2

bIn(b) + 2bIn(b) − 2bI ′n(b)

]

(* Note∑

n J2n = 1;

∑

n e−bIn(b) = 1 → ∑

n In(b) = eb )


∂v⊥+ kzv⊥

∂fs0

∂vz

fs0 = ns(a√

π)−3

e−v2⊥+v2

z

a2

→ ∂fs0

∂v⊥= −msns

2πT

2

a3√

πv⊥e−

v2⊥+v2

z

a2 = v⊥A

→ ∂fs0

∂vz= −msns

2πT

2

a3√

πvze

− v2⊥+v2

z

a2 = vzA


∂v⊥+ kzv⊥

∂fs0

∂vz

= ω∂fs0

∂v⊥− kzvzv⊥A + kzvzv⊥A

= ω∂fs0

∂v⊥= ωv⊥A

W =nΩ

v⊥v⊥vzA + (ω − nΩ)vzA

= nΩvzA + (ω − nΩ)vzA

= ωvzA

where

A = −msns

2πT

2

a3√

πvze

− v2⊥+v2

z

a2

D. Integration over velocity space

43

D.1 Sxx component

∫ ∞

−∞dvz

∫ ∞

02πv⊥dv⊥

v⊥n2

λ2 J2n

ω − kzvz − nΩωv⊥A

• Integration over v⊥

∫ ∞

02πv⊥dv⊥v⊥

n2

λ2J2

nωv⊥A

= nsn2Ω2

k2⊥

ω(− 2

a3)e−

v2z

a2m

2πT

∫ ∞

02πdv⊥v⊥J2

ne−v2⊥

a2

=n2Ω2

k2⊥

ω(− 2

a3)e−

v2z

a2 e−bIn(b)

• Integration over vz

1√π

∫ ∞

−∞dvz

e−v2z

a2

ω − kzvz − nΩ= − 1√

π

∫ ∞

−∞dvz

e−v2z

a2

kzvz − (ω − nΩ)

= − 1√π

1

kz

∫ ∞

−∞dx

e−x2

x − ω−nΩkza

= − 1

kzZn(ξn)

where ξn = ω−nΩkza and x = vz

a

∴

∫

d3vSxx

ω − kzvz − nΩ= ns

n2Ω2

k2⊥

2ω

a3kze−bIn(b)Zn(ξn)

= nsω

akz

n2Ω2

k2⊥

2

a2e−bIn(b)Zn(ξn)

= nsω

akz

n2Ω2m

k2⊥T

e−bIn(b)Zn(ξn)

= nsω

akz

n2

be−bIn(b)Zn(ξn)

Thus

Kxx = 1 +ω2

ps

ω2

ω

kza

∑

n

n2

be−bIn(b)Zn(ξn)

D.2 Sxy component

∫

d3viv⊥

nλJnJ

′

nU

ω − kzvz − nΩ

44


∫ ∞

02πv⊥dv⊥v⊥

n

λJnJ

′

nωv⊥A = nsnΩω

k⊥

∫ ∞

02πv2

⊥dv⊥JnJ′

nωA

= nsnΩω

k⊥

m

2πT(− 2

a3)e−

v2z

a2

∫ ∞

02πv2

⊥dv⊥JnJ′

nωe−v2⊥

a2

= nsnΩω

k⊥(− 2

a3)e−

v2z

a2k⊥T

mΩe−b[I

′

n(b) − In(b)]

= ns2T

m(nω

a3)e−b[I

′

n(b) − In(b)]e−v2z

a2

= ns−nω

ae−b[I

′

n(b) − In(b)]e−v2z

a2


1√π

∫ ∞

−∞dvz

e−v2z

a2

ω − kzvz − nΩ= −Zn(ξ)

kz

∴

∫

d3vSxy

ω − kzvz − nΩ

= insω

kzane−b[I

′

n(b) − In(b)]Zn(ξn)

Thus

Kxy = −Kyx = iω2

ps

ω2

−ω

kza

∑

n

ne−b[I′

n(b) − In(b)]Zn(ξn)

D.3 Sxz component

∫

d3vv⊥W n

λJ2n


∫

d3vv⊥ωvz

nλJ2

nA

ω − kzvz − nΩ

=

∫

dvzωvz

ω − kzvz − nΩ

∫

2πv⊥dv⊥v⊥n

λJ2

nA


2

a3e−

v2z

a2nΩ

k⊥

(

−msns

2πT

) ∫ ∞

02πv⊥dv⊥J2

ne−v2⊥

a2

= −2ns

a3e−

v2z

a2nΩ

k⊥e−bIn(b)

45


1√π

∫ ∞

−∞dvz

vz

ω − kzvz − nΩe−

v2z

a2 = − 1

kz

1√π

∫ ∞

−∞

vze− v2

za2

vz − ω−nΩkz

dvz

= − 1

kz

a√π

∫ ∞

−∞

xe−x2

x − ζndx

= − a

kz

[

1√π

∫ ∞

−∞e−x2

dx +1√π

∫ ∞

−∞

ζne−x2

x − ζndx

]

= − a

kz[1 + ζnZn(ζn)] =

a

2kzZ ′

n

∴

∫

d3vv⊥W n

λJ2n

ω − kzvz − nΩ= −2ωns

a3

nΩ

k⊥

a

2hze−bInZ ′

n

= −ωns

kza

nΩ

k⊥

1

ae−bInZ ′

n

= −ωns

kza

nΩ

k⊥

√m

2Te−bInZ ′

n

= −ωns

kza

(

± n√2b

)

e−bInZ ′n

(+sign : ion−sign : electron

)

Thus

Kxz = −ω2

p

ω2

ω

kza

∑

n

(

± n√2b

)

e−bInZ ′n

sinceSzx = vz

n

λUJ2

n =n

λωv⊥vzJ

2nA = Sxz

Kzx = Kxz = −ω2

p

ω2

ω

kza

∑

n

(

± n√2b

)

e−bInZ ′n

D.4 Syy component

∫

d3vv⊥U [J ′

n]2


∫

d3vωv2

⊥[J ′n]2A

ω − kzvz − nΩ


2ns

a3

(

− m

2πT

) ∫

2πv⊥dv⊥v2⊥(J ′

n)2e−v2⊥

a2 =

(

−2ns

a3

)1

2a2e−b

[n2

bIn(b) − 2b(I ′n − In)

]


1√π

∫ ∞

−∞dvz

e−v2z

a2

ω − kzvz − nΩ= − 1

kzZn(ζn)

46

∴

∫

d3vv⊥U [J ′

n]2


1

kz

2ωns

a3

1

2a2e−b

[n2

bIn − 2b(I ′n − In)

]

Zn

=ωns

kzae−b

[n2


]

Zn

Thus

Kyy = 1 +ω2

ps

ω2

ω

kza

∑

n

e−b

[n2


]

Zn

D.5 Syz component

Syz = −iv⊥WJnJ ′n = −iωv⊥vzAJnJ ′

n

Szy = +ivzUJnJ ′n = +iωv⊥vzAJnJ ′

n = −Syz

∫

d3viωv⊥vzAJnJ ′

n

ω − kzvz − nΩ


2

a3

(

−msns

2πT

) ∫

2πv⊥dv⊥v⊥JnJ ′ne−

v2⊥

a2 =

(

−2ns

a3

)k⊥T

mΩe−b(I ′n − In)


1√π

∫ ∞

−∞dvz

vze− v2

za2


a

2kzZ ′

n ⇐= we already calculated.

∴

∫

d3v−iωv⊥vzInI ′nω − kzvz − nΩ

= i2ωns

a2

k⊥T

mΩe−b(I ′n − In)

a

2kzZ ′

n

= iωns

kza

√m

2T

k⊥T

mΩe−b(I ′n − In)Z ′

n

= iωns

kza

(

±√

b

2

)

k⊥T

mΩe−b(I ′n − In)Z ′

n

Thus,

Kyz = −Kzy = iω2

ps

ω2

ω

kza

∑

n

(

±√

b

2

)

e−b(I ′n − In)Z ′m

47

D.6 Szz component

∫

d3vvzWJ2

n


∫

d3vωv2

zJ2nA

ω − kzvz − nΩ


(

−2ns

a3

)m

2πT

∫ ∞

0dv⊥2πJ2

ne−v2⊥

a2 = −2ns

a3e−bIn(b)


1√π

∫ ∞

−∞dvz

v2ze− v2

z

a2

ω − kzvz − nΩ= − 1

kz√

π

∫ ∞

−∞dvz

v2ze

− v2z

a2

vz − (ω−nΩ)kz

= − a2

kz√

π

∫ ∞

−∞dx

x2e−x2

x − ζn

∫ ∞

−∞dx

x2e−x2

x − ζn=

∫ ∞

∞dx

x2 − ζ2n + ζ2

n

x − ζne−x2

dx

=

∫ ∞

−∞(x + ζn)e−x2

dx + ζ2n

∫ ∞

−∞

e−x2

x − ζndx

= ζn

√π +

√πζ2

nZn = ζn

√π(1 + ζnZn)

∴ − a2

kz√

π

∫ ∞

−∞dx

x2e−x2

x − ζn= −a2

kzζn(1 + ζnZn)

=a2

2kzζnZ ′

n

∴

∫

d3vvzWI2

n

ω − kzvz − nΩ= −2ωns

a3

a2

2kzζne−bInZ ′

n

= −ωns

kzae−bInζnZ ′

n

Thus,

Kzz = 1 −ω2

ps

ω2

ω

kza

∑

n

e−bInζnZ ′n

Therefore, the dielectric tensor,

48

↔K=

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

where

Kxx = 1 +∑

s

ω2ps

ω2

ω

kza

∑

n

n2

be−bInZn

Kyy = 1 +∑

s

ω2ps

ω2

ω

kza

∑

n

n2

be−bInZn − 2be−b(I ′n − In)Zn

Kzz = 1 −∑

s

ω2ps

ω2

ω

kza

∑

n

e−bInζnZ ′n

Kxy = −Kyx = i∑

s

ω2ps

ω2

ω

kza

∑

n

ne−b(I ′n − In)Zn

Kxz = Kzx = −∑

s

ω2ps

ω2

ω

kza

∑

n

(

± n√2b

)

e−bInZ ′n

Kyz = −Kzy = i∑

s

ω2ps

ω2

ω

kza

∑

n

(

± b√2

)

e−b(I ′n − In)Z ′n

where,

a =

√

2T

m

b =k2⊥T

mΩ2is the argument of In

ζn =ω − nΩ

kzais the argument of Zn

Ω = qsB0

m =⇒ sign contained, Z ′ denotes the derivative of Zn with respectto its argument, and the summation run for all integer n.

49

3.2 Electrostatic Dispersion Relation

K(~k, ω) → ǫ = 1 +∑

s

ω2ps

k2

∑

n

∫

d3vJ2

n

ω − k‖v‖ − nΩs[nΩs

v⊥

∂fs0

∂v⊥+ k‖

∂fs0

∂v‖]

[Harris Dispersion Relation]

• Derivation:

Maxwell’s equations for Electrostatic:

~E1 = −∇φ1 = −i~kφ1

∇× ~E1 ≃ 0 ⇒ B1 = 0

∇ · ~E1 =1

ǫ0

∑

s

qs

∫

fs1d3v ⇒ i~k · ~E1 =

1

ǫ

∑

s

qs

∫

fs1d3v

First-order Vlasov equation :

∂fs1

∂t+ ~v · ∇fs1 + (

qs

ms~v × ~B0) · ∇vfs1 = − qs

ms

~E1 · ∇vfs0

⇒ dfs1

dt= S(~r, v, t) = − qs

ms

~E1 · ∇vfs0

∴ fs1 = − qs

ms

∫ t

−∞~E1(~r′, t) · ∇v′fs0dt′ (2.2.1)

~E1 has fourier component of ei(~k·~r′−ωt′)

From section 3.1,

~k · ~r′ = ~k · ~r +k⊥v⊥

Ωsin(Ω(t′ − t) − α) +

k⊥v⊥Ω

sinα + kzvz(t′ − t)

Then,

~E1 = ~E1(~r)ei(~k·~r′−ωt′) = ~E1(~r)e

i(~k·~r−ωt′)∑

n

∑

m

Jn(k⊥v⊥

Ω)Jm(

k⊥v⊥Ω

)

× exp[in(Ω(t′ − t) − α)] exp[imα] exp[i(kzvz − ω)(t′ − t)]

= ~E1(~r)T (~r,~v, t′).

where T (~r,~v, t′) = ei(~k·~r−ωt)∑

n

∑

m JnJmein(Ω(t′−t)−α)eimαei(kzvz−ω)(t′−t)

and we used eiα sin x =∑∞

m=−∞ Jm(a)eimx

The integrand of Eq.(2.2.1) is

~E1(~r′, t′) · ∇v′fs0 = T ~E1(~r) · ∇v′fs0

= T [Ez∂fs0

∂v′z+ (Ex

v′xv′⊥

+ Ey

v′yv′⊥

)∂fs0

∂v′⊥]

= TEz∂fs0

∂v′z+ [Ex cos(α − Ω(t′ − t)) + Ey sin(α − Ω(t′ − t))]

∂fs0

∂v′⊥

50

We dropped sub index 1.Note that

v′x = v′⊥ cos(α − Ω(t′ − t)) = v⊥ cos(α − Ω(t′ − t))

v′y = v′⊥ sin(α − Ω(t′ − t)) = v⊥ sin(α − Ω(t′ − t))

Let t′ − t = τ, dt′ = dτ,∫ t−∞ dt′ =

∫ 0−∞ dτ

Thus Eq.(2.2.1) becomes

fs1 = − qs

ms

∑

n

∑

m

ei(m−n)αJn(k⊥v⊥

Ω)Jm(

k⊥v⊥Ω

)

×∫ 0

−∞dτeinΩτei(kzvz−ω)τEz

∂fs0

∂vz+ [Ex cos(α − Ωτ) + Ey sin(α − Ωτ)]

∂fs0

∂v⊥

Integration for Ex component gives

∫ 0

−∞dτeinΩτei(kzvz−ω)τ cos(α − Ωτ) =

1

2[

ieiα

ω − kzvz − (n − 1)Ω+

ie−iα

ω − kzvz − (n + 1)Ω]

Integration for Ey component gives

∫ 0

−∞dτeinΩτei(kzvz−ω)τ sin(α − Ωτ) =

1

2i[

ieiα

ω − kzvz − (n − 1)Ω− ie−iα

ω − kzvz − (n + 1)Ω]

Integration for Ez component gives

∫ 0

−∞dτeinΩτei(kzvz−ω)τ =

1

i

−1

ω − kzvz − nΩ(2.2.2)

∑

n

1

2Jn

(k⊥v⊥Ωs

) [

ei(m−(n−1))α

ω − kzvz − (n − 1)Ωs+

ei(m−(n−1))α

ω − kzvz − (n + 1)Ωs

]

(n − 1 → n ⇒ Jn → Jn+1)

(n + 1 → n ⇒ Jn → Jn−1)

=∑

n

1

2

ei(m−(n−1))α

ω − kzvz − nΩs(Jn+1 + Jn−1)

=∑

n

ei(m−(n−1))α

ω − kzvz − nΩs

nΩ

k⊥v⊥Jn(

k⊥v⊥Ωs

) (2.2.3)

Where, we used

Jn

x=

1

2(Jn+1(x) + Jn−1(x))

and

∑

n

1

2iJn

(k⊥v⊥Ωs

) [

ei(m−(n−1))α

ω − kzvz − (n − 1)Ωs− ei(m−(n+1))α

ω − kzvz − (n + 1)Ωs

]

51

=∑

n

1

2i

ei(m−n)α

ω − kzvz − nΩs(Jn+1 + Jn−1)

=∑

n

(1

i

) ei(m−(n−1))α

ω − kzvz − nΩs(−J ′

n) (2.2.4)

Where, we used

2J ′n = Jn−1 − Jn+1

Thus, putting the result of integration, Eqs. (2.2.2)-(2.2.4)

fs1 = − iqs

ms

∑

n

∑

m

Jm

(k⊥v⊥Ωs

) ei(m−n)α


×[ExJnnΩs

k⊥v⊥

∂fs0

∂v⊥+ EyJ

′n

∂fs0

∂v⊥+ EzJn

∂fs0

∂vz]

The electrostatic dispersion relation comes from

i~k · ~E1 =1

ǫ0

∑

s

qs

∫

fs1d3v ⇒ k2φ =

1

ǫ0

∑

s

qs

∫

fs1d3v

⇒ k2φ − 1

ǫ

∑

s

qs

∫

fs1d3v = ǫk2φ = 0

∫

fs1d3v = ?

∫

fs1d3v =

∫ ∞

−∞dvz

∫ ∞

0v⊥dv⊥

∫ 2π

0dαfs1

Since fs1 has the dependence of ei(m−n)α, the integration of dα gives2πδmn.

∫

fs1d3v = − iqs

ms

∑

n

Jn

∫ ∞

−∞dvz

1


∫ ∞

02πv⊥dv⊥

[Ex

nΩs

k⊥v⊥Jn

∂fs0

∂v⊥

+EyJ′n

∂fs0

∂v⊥+ EzJ

′n

∂fs0

∂vz

]

But, Ex = −ikxφ, Ey = −ikyφ, Ez = −ikzφ (2.2.5)

Assume ~k = kx~x + kz~z = k⊥~x + kz~z (ky = 0) (2.2.6)

Now, we define the integration in velocity space as

52

∫

d3v =

∫ ∞

−∞dvz

∫ ∞

02πv⊥dv⊥ (2.2.7)

Then, from Eqs.(2.2.5)-(2.2.7)

∫

fs1d3v = −qsφ

ms

∑

n

∫

d3vJ2

n


[nΩs

v⊥

∂fs0

∂v⊥+ kz

∂fs0

∂vz

]

ǫk2φ = k2φ − 1

ǫ0

∑

s

qs

∫

d3vfs1

= k2φ +∑

s

qsφ

msǫ0

∑

n

∫

d3vJ2

n


[nΩs

v⊥

∂fs0

∂v⊥+ kz

∂fs0

∂vz

]

= k2φ +∑

s

ω2psφ

∑

n

∫

d3vJ2

n


[nΩs

v⊥

∂fs0

∂v⊥+ kz

∂fs0

∂vz

]

(Note : f0 is the normalized distribution function)

∴ ǫ = 1 +∑

s

ω2ps

k2

∑

n

∫

d3vJ2

n

(k⊥v⊥Ωs

)


(

nΩs

v⊥

∂f0

∂v⊥+ kz

∂f0

∂vz

)

(2.2.8)

The E.S. Dispersion Relation for M-B distribution

f0 = (a√

π)−3e−v2

a2

(

a =

√

2Ts

ms

)

∗∫ ∞

∞e−s2x2

xJ2n(px)dx =

1

2s2e−

p2

2a2 In(p2

2s2)

∂f0

∂v⊥= ∂

∂v⊥

[

(a√

π)−3e−(v2⊥

a2 +v2z

a2 )

]

= (a√

π)−3(−2v⊥

a2

)e−

v2

a2 = − 2a5π

√πe−

v2

a2

∂f0

∂vz= − 2

a5π√

πvze

− v2

a2

53

∫

d3vJ2

n(k⊥v⊥Ωs

)


[nΩs

v⊥

(

− 2

a5π√

πv⊥e−v2

a2

)

+ kz

(

− 2

a5π√

πvze

− v2

a2

) ]

= − 2

a5π√

π

∫ ∞

−∞dvz

∫ ∞

02πv⊥dv⊥

J2n(k⊥v⊥)

ω − kzvz − nΩs(nΩs + kzvz)e

− v2

a2

=4

a5√

π

∫ ∞

−∞dvz

e−v2z

a2

ω − kzvz − nΩs(nΩs + kzvz)

[

1

2a2e

−a2k2⊥

2Ω2s In

(a2k2

⊥2Ω2

s

) ]

= − 2

a3√

πe−a2k2

⊥2Ω2

s In

(a2k2

⊥2Ω2

s

) ∫ ∞

−∞dvz

nΩs + kzvz

ω − kzvz − nΩse−

v2z

a2

The first term of the integrand:

∫ ∞

−∞dvz

e−v2z

a2

ω − kzvz − nΩs= nΩs

(

− 1

kz

) ∫ ∞

−∞dvz

e−v2z

a2

vz − ω−nΩs

kz

=nΩs

kz

∫ ∞

−∞dx

e−x2

x − ω−nΩs

kza

= −nΩs

kz

√πZn(ζn)

where Zn(ζn) is the dispersion function and its argument ζn = ω−nΩs

kza .

The second term of the integrand:

∫ ∞

−∞dvz

kzvz

ω − kzvz − nΩze−

v2z

a2 = −∫ ∞

−∞dvz

vz

vz − ω−nΩs

kz

e−v2z

a2

= −a

∫ ∞

−∞dx

xe−x2

x − ζn

= −a

∫ ∞

∞dx

(x − ζn + ζn)

x − ζne−x2

= −a

[∫ ∞

−∞dxe−x2

+ ζn

∫ ∞

−∞dx

e−x2

x − ζn

]

= −a(√

π +√

πζnZn(ζn))

Then,∫

d3vJ2

n


[

nΩs

v⊥

∂f0

∂v⊥+ kz

∂f0

∂vz

]

= − 2

a3√

πe−a2k2

⊥2Ω2

s In

(a2k⊥2Ω2

s

) [

−nΩs

kz

√πZn(ζn) − a

√π − a

√πζZn(ζn)

]

=2

a2e−a2k2

⊥2Ω2

s In

(a2k⊥2Ω2

s

) [

1 +

(nΩs

kza+

ω − nΩs

kza

)

Zn(ζn)

]

=2

a2e−a2k2

⊥2Ω2

s In

(a2k⊥2Ω2

s

) [

1 +ω

kzaZn(ζn)

]

54

Thus, the dielectric constant ǫ is

ǫ = 1 +∑

s

ω2ps

k2

2

a2

∑

n

e−a2k2

⊥2Ω2

s In

(a2k⊥2Ω2

s

) [

1 +ω

kzaZn(ζn)

]

= 1 +∑

s

1

k2λ2Ds

∑

n

e−bIn(b)

[

1 +ω

kzaZn(ζn)

]

(2.2.9)

Where

λ−2Ds

=neq

2s

ǫ0Ts=

2ω2ps

a2: Debye Length

b =a2k2

⊥2Ω2

s

= k2⊥

(Ts

msΩ2s

)

= k2⊥ρ2

ζn =ω − nΩs

kza

a =

√

2Ts

ms

∴ The dispersion relation:

ǫ = 0 = 1 +∑

s

1

k2λ2Ds

∑

n

e−bIn(b)

[

1 +ω

kzaZn(ζn)

]

(2.2.10)

3.2.1 Electrostatic Modes in Hot Plasma

A. Electron Modes (kz 6= 0, ω ≫ ωpi, Ωi, low temperature)

ω ≫ ωpi, Ωi

For B0 = 3T

ne = 1.0 × 1014cm−3

Ωe = 84GHz

ωpe = 90GHz

ωpi = 2.1GHz

Ωi = 46MHz

For low temperature, we do Taylor expansionFor large argument

Z(x) −−−−−→x ≫ 1 i

√πe−x2 − 1

x

(

1 +1

2x2+ · · ·

)

From Eq.(2.2.9)

ǫ(~k, ω) = 1 +∑

s

2ω2ps

k2V 2s

∑

n

e−bIn(b)

[

1 +ω

kzVsZn(ζn)

]

(2.2.1.1)

55

where

Vs =

√

2Ts

ms

Assume that ǫ(~k, ω) = 1 + χe(~k, ω) + χi(~k, ω)

χe(~k, ω) =2ω2

pe

k2V 2e

∞∑

n=−∞e−beIn(be)

[

1 +ω

kzVeZn(ζne)

]

(2.2.1.2)

χi(~k, ω) =2ω2

pi

k2v2i

∞∑

n=−∞e−biIn(bi)

[

1 +ω

kzviZn(ζni)

]

(2.2.1.3)

where be =k2⊥V 2

e

2Ω2e

, bi =k2⊥v2

i

2Ω2i

ζne = ω+nΩe

kzVe, ζni =

ω − nΩi

kzvi

Ωe =∣∣∣qeB0

me

∣∣∣ , Ωi =

qiB0

mi> 0

χe(~k, ω) =2ω2

pe

k2v2e

e−be

I1(be)

[

2 +ω

kzVe(Z1(ζ1e) + Z−1(ζ−1e))

]

+I0(be)

[

1 +ω

kzv3Z0(ζ0e)

]

+

∞∑

n=2

In(be)

[

2 +ω

kzVe(Zn(ζne) + Z−n(ζ−ne))

]

(2.2.1.4)

Here we used In(be) = I−n(be)

Since ζne is large, we use the asymptotic expansion of the dispersionfunction.

Z1(ζ1e) + Z−1(ζ−1e) ≃ i√

π(

e−ζ21e + e−ζ2

−1e

)

− 1

ζ1e− 1

2ζ31e

− 1

ζ−1e− 1

2ζ3−1e

= i√

π

[

exp

(

−(ω + Ωe)2

k2zv

2e

)

+ exp

(

−(ω − Ωe)2

k2zv

2e

)]

−(

kzVe

ω + Ωe+

kzVe

ω − Ωe+

1

2

k3zv

3e

(ω + Ωe)3+

1

2

k3zv

3e

(ω − Ωe)3

)

= i√

π

[

exp

(

−(ω + Ωe)2

k2zv

2e

)

+ exp

(

−(ω − Ωe)2

k2zv

2e

)]

−kzVe2ω

ω2 − Ω2− 1

2k3

zv3e

2ω(ω2 + 3Ω2)

(ω2 − Ω2)3

2 +ω

kzVe(Z1(ζ1e) + Z−1(ζ1e))

= i√

πω

kzVe

[

e−ζ21e + e−ζ2

−1e

]

+ 2 − 2ω2

ω2 − Ω2e

− 1

2k2

zV2e

2ω2(ω2 + 3Ω2e)

(ω2 − Ω2e)

3

= i√

πω

kzVe

[

e− (ω+Ωe)2

k2zVe + e

− (ω+−Ωe)2

k2zVe

]

− 2Ω2

ω2 − Ω2e

− k2zV

2e

ω2(ω2 + 3Ω2e)

(ω2 − Ω2e)

3(2.2.1.5)

56

Zn(ζne) + Z−n(ζ−ne) ≃ i√

π(

e−ζ2ne + e−ζ2

−ne

)

− 1

ζne− 1

ζ−ne

= i√

π(

e−ζ2ne + e−ζ2

−ne

)

− kzVe2ω

ω2 − n2Ω2e

≃ −2kzVeω

ω2 − n2Ω2e

⇒ 2 +ω

kzVe(Zn(ζne) + Z−n(ζ−ne)) = 2 − ω

kzVe2kzVe

ω

ω2 − n2Ω2e

=−2n2Ω2

e

ω2 − n2Ω2e

(2.2.1.6)

Z0(ζ0e) ≃ i√

πe−ζ20e − 1

ζ0e− 1

2ζ30e

= i√

πe1 ω2

k2zV 2

e − kzVe

ω− 1

2

k3zV

3e

ω3

⇒ 1 +ω

kzVeZ0(ζ0e) ≃ i

√π

ω

kzVee− ω2

k2zV 2

e + 1 − ω

kzVe

kzVe

ω− 1

2

ω

kzVe

k3zV

3e

ω3

= i√

πω

kzVee− ω2

k2zV 2

e − k2zV

2e e

2ω2(2.2.1.7)

Substitution of Eqs. (2.2.1.5)-(2.2.1.7) into Eq. (2.2.1.4)Then,

Re(χe) =2ω2

pe

k2V 2e

e−be

I1(be)

[

− 2Ω2e

ω2 − Ω2e

− k2zV

2e

ω2(ω2 + 3Ω2e)

(ω2 − Ω2e)

3

]

+ I0(be)

[

−k2zV

2e

2ω2

]

+∞∑

n=2

In(be)

[ −2n2Ω2

ω2 − n2Ω2e

]

For low temperature, i.e., be ≪ 1

e−be ≃ 1 − be

I1(be) ≃ b

2I0(be) = 1

Re(χe) =2ω2

pe

k2V 2e

(1 − be)

[−be

2(

2Ω2e

ω2 − Ω2e

+ k2zV

2e

ω2(ω2 + 3Ω2e)

(ω2 − Ω2e)

3 ) − k2eV

2e

2ω2

]

−2ω2

pe

k2V 2e

∞∑

n=2

e−beIn(be)2n2Ω2

e

ω2 − n2Ω2e

57

We take first-order term of be. Therefore ǫR,e is

Re(χe) =2ω2

pe

k2V 2e

(

−be

2

) [2Ω2

e

ω2 − Ω2e

+ k2zV

2e

ω2(ω2 + 3Ω2e)

(ω2 − Ω2e)

3

]

−2ω2

pc

k2V 2e

k2zV

2e

2ω2

+2ω2

pe

k2V 2e

(be)k2

zV2e

2ω2−

2ω2pe

k2V 2e

∞∑

n=2

e−beIn(be)2n2Ω2

e

ω2 − n2Ω2e

=ωpe2

k2V 2e

k2⊥V 2

e

2Ω2e

2Ω2e

ω2 − Ω2e

− ωpe2

k2V 2e

k2⊥V 2

e

2Ω2e

k2zV

2e

ω2(ω2 + 3Ω2e)

(ω2 − Ω2e)

3 − 2ωpe2

k2V 2e

k2zV

2e

2ω2e

+2ωpe2

k2V 2e

k2⊥V 2

e

2Ω2e

k2zV

2e

2ω2e

− 2ωpe2

k2V 2e

∞∑

n=2

e−beIn(be)2n2Ω2

e

ω2 − n2Ω2e

= − ωpe2

ω2 − Ω2e

k2⊥

k2− ωpe2

2Ω2e

k2⊥

k2

k2zV

2e

ω2

ω4(ω2 + 3Ω2e)

(ω2 − Ω2e)

3 − ωpe2

ω2e

k2z

k2

+ωpe2

2Ω2e

k2⊥

k2

k2zV

2e

ω2− 2ωpe2

k2V 2e

∞∑

n=2

e−beIn(be)2n2Ω2

e

ω2 − n2Ω2e

= −ωpe2

ω2e

k2z

k2− ωpe2

ω2 − Ω2e

k2⊥

k2+ ǫte (2.2.1.8)

where

ǫte =ωpe2k2

z

2ω2k2

[k2⊥V 2

e

Ω2e

− k2⊥V 2

e

Ω2e

ω4(ω2 + 3Ω2e)

(ω2 − Ω2e)

3

]

− 1

k2λ2De

∞∑

n=2

e−beIn(be)2n2Ω2

e

ω2 − n2Ω2e

Im(χe) =2ω2

pe

k2V 2e

e−be

I1(be)

[

e− (ω+Ωe)2

kz2V 2

e + e− (ω−Ωe)2

kz2V 2

e

]

+ I0(be)e− ω2

kz2V 2

e

≃√

π2ω2

pe

k2V 2e

ω

kzVe(1 − be)

be

2

[

e− (ω+Ωe)2

k2zV 2

e + e− (ω−Ωe)2

k2zV 2

e

]

+ e− ω2

k2zV 2

e

≃√

π2ω2

pe

k2V 2e

ω

kzVe

be

2

[

e− (ω+Ωe)2

k2zV 2

e + e− (ω−Ωe)2

k2zV 2

e

]

+√

π2ω2

pe

k2V 2e

ω

kzVee− ω2

k2zV 2

e −√

π2ω2

pe

k2V 2e

ω

kzVebee

− ω2

k2zV 2

e

=√

π

2ω2pe

2Ω2e

k2⊥

k2

ω

kzVe

[

e− (ω+Ωe)2

k2zV 2

e + e− (ω−Ωe)2

k2zV 2

e

]

+1

k2λ2De

ω

kzVe

[

1 − k2⊥V 2

e

2Ω2e

]

e− ω2

k2zV 2

e

(2.2.1.9)

where λ−2De =

2ω2pe

V 2e

58

Similarly, we get χi(~k, ω) easily.

ωpe −→ ωpi

Ωe −→ Ωi

Ve −→ Vi =

√

2Ti

mi

λ−2De −→ λ−2

Di =2ω2

pi

V 2i

be −→ bi =k2⊥V 2

i

2Ω2i

Re(

χi(~k, ω))

= −ω2

pi

ω2

k2z

k2−

ω2pi

ω2 − Ω2i

k2⊥

k2+ ǫti (2.2.1.10)

where

ǫti =ω2

pik2z

2ω2k2

[k2⊥V 2

i

Ω2i

− k2⊥V 2

i

Ω2i

ω4(ω2 + 3Ω2i )

(ω2 − Ω2i )

3

]

− 1

k2λ2Di

∞∑

n=2

e−biIn(bi)2n2Ω2

i

ω2 − n2Ω2i

(2.2.1.11)

Im(χi) =√

π

[

ω2pi

2Ω2i

k2⊥

k2

ω

kzVi

e− (ω+Ωi)

2

k2zV 2

i + e− (ω−Ωi)

2

k2zV 2

i

+1

k2λ2Di

ω

kzVi

(

1 − k2⊥V 2

i

2Ω2i

)

e− ω2

k2zV 2

i

]

(2.2.1.12)

For getting electron modes with kz 6= 0, one may neglect χi(~k, ω)

59

∴ ǫ(~k, ω) = 1 + χe(~k, ω) = ǫR,e + iǫI,e

ǫR,e = 1 −ω2

pe

ω2

k2z

k2−

ω2pe

ω2 − Ω2e

+k2⊥

k2+ ǫte (2.2.1.13)

ǫI,e =√

π

[

ω2pe

2Ω2e

k2⊥

k2

ω

kzVe

(

e− (ω+Ωe)2

k2zV 2

e + e− (ω−Ωe)2

k2zV 2

e

)

+1

k2λ2De

ω

kzVe

(

1 − k2⊥V 2

e

2Ω2e

)

e− ω2

k2zV 2

e

]

(2.2.1.14)

ǫte =ω2

pek2z

2ω2k2

[k2⊥V 2

e

Ω2e

− k2⊥V 2

e

Ω2e

ω4(ω2 + 3Ω2e)

(ω2 − Ω2e)

3

]

− 1

k2λ2De

∞∑

n=2

e−beIn(be)2n2Ω2

e

ω2 − n2Ω2e

(2.2.1.15)

ǫR,e = ǫR,e(~k, ω) = 0 gives

1 −ω2

pe

ω2

k2z

k2−

ω2pe

ω2 − Ω2e

k2⊥

k2= 0 (2.2.1.16)

⇒ k2ω2(ω2 − Ω2e) − k2

zω2pe(ω

2 − Ω2e) − k2

⊥ω2peω

2 = 0

⇒ k2ω4 − k2Ω2eω

2 − k2zω

2peω

2 − k2⊥ω2

peω2 + k2

zω2peΩ

2e = 0

⇒ k2ω4 − (k2Ω2e + (k2

z + k2⊥)ω2

pe)ω2 + k2

zω2peΩ

2e = 0

⇒ k2ω4 − k2(Ω2e + ω2

pe)ω2 + k2

zω2peΩ

2e = 0

⇒ k2ω4 − k2ω2UHω2 + k2

zω2peΩ

2e = 0

where, ω2UH = ω2

pe + Ω2e

∴ ω2 =k2ω2

UH ±√

k4ω4UH − 4k2k2

zω2peΩ

2e

2k2

=1

2

[

ω2UH ± (ω2

UH − 4Ω2eω

2pek

2z/k2)1/2

]

(2.2.1.17)

For k⊥ → ∞, ω2 = 12(ω2

UH ± ω2UH) = ω2

UH (Upper Hybrid Resonance)

• Plots of Eq. (2.2.1.17)

The plots of Eq. (2.2.1.17) are inserted using Mathematica program.

• Lists of plots

1. Plot 3D: ω vs. (kx, kz)

60

2. Contour plots of (1)

3. Plot 2D: ω vs. (kx, kz)

B. Electron Bernstein Waves (kz → 0, ω = |nΩe|)In the limit kz → 0(kz ≪ k⊥), ǫ = 0 gives electron Bernstein modes atω ≃ |nΩe|.

ǫ(~k, ω) = 1 +∑

s

2ω2ps

k2V 2s

∑

n

e−bIn(b)[1 +ω

kzVsZn(ζn)]

= 1 +2ω2

pe

k2V 2e

∞∑

n=−∞e−beIn(be)[1 +

ω

kzVeZn(ζne)]

where ζne = ω+nΩe

kzVeand Ωe = | qeB0

me| > 0

In the limit kz → 0, the damping terms (imaginary parts) disappearexcept precisely at ω = |nΩe|.

i) For n = 0

ǫ(~k, ω) ≃ 1 +2ω2

pe

k2V 2e

e−beI0(be)

[

1 +ω

kzVe

(

i√

πe− ω2

k2zv2

e − kzVe

ω

)]

≃ 1 +2ω2

pe

k2v2e

e−beI0(be)

[

1 − ω

kzVe

kzVe

ω

]

= 1 + 0

No contribution in summation

ii) For n 6= 0

ǫ(~k, ω) = 1 +2ω2

pe

k2V 2e

∞∑

n=1

e−beIn(be)[2 +ω

kzVe(Zn(ζne) + Zn(ζne))]

2 +ω

kzVe(Zn(ζne + Z−n(ζ−ne))

≃ 2 +ω

kzVe[i√

πe−ζ2ne − 1

ζne+ i

√πe−ζ2

−ne − 1

ζ−ne]

≃ 2 +ω

kzVe(i√

π)(e−ζ2ne + e−ζ2

−ne) − ω

kzVe

[kzVe

ω + nΩe+

kzVe

ω − nΩe

]

≃ 2 − ω

[1

ω + nΩe+

1

ω − nΩe

]

= 2 − ω2ω

ω2 − n2Ω2e

=−2n2Ω2

e

ω2 − n2Ω2e

61

Here, the imaginary terms is neglected if one may preserve ω → ω +iν, ν > 0. And, we just took the first term in its asymptotic expansion,Zn(ζne) ≃ −1

ζ .

Thus, for kz → 0

ǫ(~k, ω) = 1 −2ω2

pe

k2V 2e

∞∑

n=1

e−beIn(be)2n2Ω2

e

ω2 − n2Ω2e

= 1 −2ω2

pe

k2⊥V 2

e

∞∑

n=1

e−beIn(be)2n2Ω2

e

ω2 − n2Ω2e

Since, be =k2⊥V 2

e

2Ω2e

,

ǫ(~k, ω) = 1 −ω2

pe

Ω2e

2

be

∞∑

n=1

e−beIn(be)n2

(ω/Ωe)2 − n2

= 1 −ω2

pe

Ω2e

α(Q, be)

be= 0 (2.2.1.18)

This is a “Dispersion Relation for E-Bernstein Waves”.

Where,

α(Q, be) = 2∞∑

n=1

e−beIn(be)n2

Q2 − n2(2.2.1.19)

and Q =ω

Ωe(1)

The solution of Eq (2.2.1.18) gives electron Bernstein waves.The function α(Q, be) can be expressed an expansion in ascending power ofbe,

α(Q, be) =be

Q2 − 12+

1 · 3b2e

(Q2 − 12)(Q2 − 22)+

1 · 3 · 5b3e

(Q2 − 12)(Q2 − 22)(Q2 − 32)+ . . .

This expression shows that resonance at the nth cyclotron harmonic ap-pear only when terms up to at least bn−1

e are pertained in the dispersionrelation.

The characteristics of α(Q, be) is shown in page 295-300 of T.H. Stix‘sbook in detail.

It is convenient to put Eq(2.2.1.18) into the from

62

Ω2e

ω2pe

=α(Q, be)

be(2.2.1.20)

The plots of Eq.(2.2.1.20): Q vs.√

be for several values ofω2

pe

Ω2e

Since Q = ωΩe

and be =k2⊥V 2

e

2Ω2e

= k2⊥ρ2

e (ρ2e = V 2

e

2Ω2e)

√

be = k⊥ρe

Q vs.√

be = k⊥ρe ⇒ (ω

Ωe) vs. k⊥ρe

Eq. (2.2.1.20) becomes

α(Q, be) = be1

ω2pe/Ω2

e

→ α(Q, be)

be=

1

ω2pe/Ω2

e

=1

X

Left-hand Side :

α(Q, be)

be=

1

Q2 − 12+

1 · 3 · be

(Q2 − 12)(Q2 − 22)+

1 · 3 · 5 · b2e

(Q2 − 12)(Q2 − 22)(Q2 − 32)+ · · ·

Then, for some harmonics,

(1) n = 2 :α(Q, be)

be=

1

Q2 − 12+

1 · 3 · be

(Q2 − 12)(Q2 − 22)=

1

X

(2) n = 3 :α(Q, be)

be=

1

Q2 − 12+

1 · 3 · be

(Q2 − 12)(Q2 − 22)

+1 · 3 · 5 · b2

e

(Q2 − 12)(Q2 − 22)(Q2 − 32)

=1

X

(3) n = 4 :α(Q, be)

be=

1

Q2 − 12+

1 · 3 · be

(Q2 − 12)(Q2 − 22)

+1 · 3 · 5 · b2

e

(Q2 − 12)(Q2 − 22)(Q2 − 32)

+1 · 3 · 5 · 7 · b3

e

(Q2 − 12)(Q2 − 22)(Q2 − 32)(Q2 − 42)

=1

X

63

4 Dispersion plots of electron modes using Math-ematica

4.1 Electron modes

64

Electron Modes

( vs. kx with kz 0, >>

pi, ci )$TextStyle FontFamily "Times", FontSize 16 ;

Graphics`Graphics3D`;

Graphics`Graphics`;

wuh2 wpe^2 wce^2;

disc Sqrt wuh2^2 4.0 wce^2 wpe^2 kz^2 kx^2 kz^2 ;

w1 wpe_, wce_ Sqrt 0.5 wuh2 disc ;

w2 wpe_, wce_ Sqrt 0.5 wuh2 disc ;

kxw1d wpe_, wce_, kz_ Sqrt 0.5 wuh2 disc ;

kxw2d wpe_, wce_, kz_ Sqrt 0.5 wuh2 disc ;

plot1 Plot Evaluate Table kxw1d 1, 1.2, kz , kz, 0.2, 1.0, 0.2 , kx, 5, 5

plot2 Plot Evaluate Table kxw2d 1, 1.2, kz , kz, 0.2, 1.0, 0.2 , kx, 5, 5

wuh Sqrt 1 1.2^2

1.56205

electron_mode.nb 1

65

show1 Show plot1, plot2 , Graphics Text "kz 0.2", 1, 0.1 ,

Text "kz 1.0", 1.2, 1.35 , Text "kz 1.0", 2, 0.45 ,

Text " uh", 0.3, 1.6 , Text " ce", 0.3, 1.18 , Text " pe", 0.3, 1.02 ,

PlotRange 5, 5 , 0, 1.7 , FrameLabel "kx", " " , Frame True

4 2 0 2 4

kx

0.25

0.5

0.75

1

1.25

1.5

kz 0.2

kz 1.0

kz 1.0

uh

ce

pe

Graphics

Trievelpiece-Gould Mode ( vs. kz)

kzw1d wpe_, wce_, kx_ Sqrt 0.5 wuh2 disc ;

kzw2d wpe_, wce_, kx_ Sqrt 0.5 wuh2 disc ;

plot3 Plot Evaluate Table kzw1d 1, 1.2, kx , kx, 0.2, 1.0, 0.2 , kz, 5, 5

plot4 Plot Evaluate Table kzw2d 1, 1.2, kx , kx, 0.2, 1.0, 0.2 , kz, 5, 5

electron_mode.nb 2

66

show2 Show plot3, plot4 , Graphics Text "kx 0.2", 0.5, 1.25 ,

Text "kx 0.2", 1, 1.0 , Text "kx 1.0", 1.5, 1.53 , Text "kx 1.0", 2, 0.7 ,

Text " uh", 0.3, 1.6 , Text " ce", 0.3, 1.18 , Text " pe", 0.3, 1.0 ,

PlotRange 5, 5 , 0, 1.7 , FrameLabel "kz", " " , Frame True

4 2 0 2 4

kz

0.25

0.5

0.75

1

1.25

1.5

kx 0.2

kx 0.2

kx 1.0

kx 1.0

uh

ce

pe

Graphics

plot5 Plot3D w1 1, 1.2 , kz, 5, 5 , kx, 5, 5 , PlotPoints 160, Mesh False,

ViewPoint 2.298, 2.915, 3.351 , AxesLabel "kx", "kz", " 1"

plot6 Plot3D w2 1, 1.2 , kz, 5, 5 , kx, 5, 5 , PlotPoints 160, Mesh False,

ViewPoint 2.298, 2.915, 3.351 , AxesLabel "kx", "kz", " 2"

show3 Show plot5, plot6 , ViewPoint 2.051, 3., 0.8 ,

AxesLabel "kx", "kz", " "

52.5

02.5

5kx

52.5 0 2.5 5

kz

0

0.5

1

1.5

52.5

02.5

5kx

52.5 0 2.5 5

kz

Graphics3D

electron_mode.nb 3

67

plot7 ContourPlot w1 1, 1.2 , kz, 5, 5 ,

kx, 5, 5 , ContourLines True, FrameLabel "kx", "kz"

4 2 0 2 4

kx

4

2

0

2

4z

k

ContourGraphics

plot8 ContourPlot w2 1, 1.2 , kz, 5, 5 ,

kx, 5, 5 , ContourLines True, FrameLabel "kx", "kz"

4 2 0 2 4

kx

4

2

0

2

4

zk

ContourGraphics

electron_mode.nb 4

68

4.2 Electron Bernstein (EB) modes

69

Electron Bernstein Modes

(kz 0.0, |n ce|)

2pe

2ce x

be k

Q ce

Clear "Global` "



be kpR^2;

alp 2

n 1

M

Exp be BesselI n, be n^2 Q^2 n^2 ;

x1 1.0;

x2 3.0;

x3 5.0;

x4 8.0;

x5 10^10;

equations M_, x_ alp be 1.0 x

2n 1

M be BesselI n,be n2

Q2 n2

kpR21.

x

equations 2, x

2kpR2 BesselI 1,kpR2

1 Q24 kpR2 BesselI 2,kpR2

4 Q2

kpR21.

x

electron_Bern_mode.nb 1

70

solM2x1 Solve equations 2, x1 , Q ;




















QM2x11 Q . solM2x1 2 ;


























71











QM5x12 Q . solM5x1 10 ;

QM5x22 Q . solM5x2 10 ;

QM5x32 Q . solM5x3 10 ;

QM5x42 Q . solM5x4 10 ;

QM5x52 Q . solM5x5 10 ;

Graphics`Graphics`;

plot1 Plot QM2x11, QM2x21, QM2x31, QM2x41, QM2x51 , kpR, 0, 5 , PlotRange All








plot9 Plot 2, 3, 4, 5 , kpR, 0, 5 , PlotStyle Dashing 0.03, 0.03



72

show1 Show plot1, plot3, plot5, plot7, plot9,

Graphics Text "X 1", 0.5, 1.2 , Text "3", 0.55, 1.5 , Text " ", 2, 1.45 ,

Text " ", 2.2, 2.6 , Text " ", 2.6, 3.7 , Text " ", 2.8, 4.8 ,

PlotRange 0, 3 , All , Frame True, FrameLabel "k ", " ce"

0.5 1 1.5 2 2.5 3

k

1

2

3

4

5

ec

X 1

3

Graphics


73

5 Landau Damping

In the first order, we denote the perturbation in f(~r,~v, t) by f1(~r,~v, t):

f(~r,~v, t) = f0(~v) + f1(~r,~v, t)

The first-order Vlasov equation for electron is

∂f1

∂t+ ~v · ~∇f1 −

e

m~E1 ·

∂f0

∂~v= 0

Where we let ~B0 = ~E0 = 0and we assumed the ions are massive and fixedand that the waves are plane waves in the x direction

f1 ∝ ei(kx−ωt)

Then the first-order Vlasov equation becomes

−iωf1 + ikvxf1 =e

mEx

∂f0

∂vx

∴ f1 =ieEx

m

∂f0/∂vx

ω − kvx

Poisson’s equation

ǫ0~∇ · ~E1 = ikǫ0Ex = −en1 = −e

∫ ∫ ∫

f1d3v

ikǫ0Ex = −e

∫ ∫ ∫ieEx

m

∂f0/∂vx

ω − kvxd3v

→ 1 =−e2

kmǫ0

∫ ∫ ∫∂f0/∂vx

ω − kvxd3v

If we replace f0 by a normalized function f0 ;that is, f0 = n0f0

1 = −ω2

p

k

∫ ∞

−∞dvz

∫ ∞

−∞dvy

∫ ∞

−∞

∂f0(vx, vy, vz)/∂vx

ω − kvxdvx

For a one-dimensional Maxwellian distribution

1 = −ω2

p

k

∫ ∞

−∞

∂f0/∂vx

ω − kvxdvx

=ω2

p

k2

∫ ∞

−∞

∂f0/∂vx

vx − ω/kdvx

74

Dropping the subindex x,

1 =ω2

p

k2

∫ ∞

−∞

∂f0/∂v

v − ω/kdv : dispersion relation

singularity at v = ω/kNo problem, because in practice ω is almost never real.The integral must be treated as a contour integral in the complex v plane.

• For an unstable wave, with Im(ω) > 0

Figure 2: Contour I

• For a damped wave, with Im(ω) < 0

Figure 3: Contour II

∫

C1

Gdv +

∫

C2

Gdv = 2πiR(ω/k)

Where G is the integrand, C1 is the path along the real axis, C2 is thesemicircle at infinity, and R(ω/k) is the residue at ω/k.This Works If The Integral over C2 Vanishes. Unfortunately, this does nothappen for a Maxwellian Distribution, which contains the factor

exp(−v2/v2th)

75

This factor becomes large for v → ±i∞, and the contribution from C2 can-not be neglected.

But, for the case of large phase velocity and weak damping (small imag-inary Im(ω), the contour integral is possible as shown in Fig. 5.

Figure 4: Contour III

• An approximate dispersion relation for the case of large phase velocityand weak damping.

For Re(vφ) ≫ 1 and Im(vφ) ≪ 1, the contour in Fig. 5 is used.

Then the dispersion relation becomes

1 =ω2

p

k2

[

P

∫ ∞

−∞

∂f0/∂v

v − (ω/k)dv + iπ

∂f0

∂v|v=ω

k

]

Where P stands for the Cauchy principal value.

1) The evaluation of P∫ ∞−∞

∂f0/∂vv−(ω/k)dv : stop just before encountering

the pole

∫ ∞

−∞

∂f0/∂v

v − (ω/k)dv =

[f0

v−vφ

]∞

−∞−

∫ ∞

−∞

−f0

(v − vφ)2dv

=

∫f0

(v − vφ)2dv (∵ f0 << 1 for large in vφ)

= (v − vφ)−2

Since vφ >> v

(v − vφ)−2 = v−2φ

(

1 − v

vφ

)−2

= v−2φ

(

1 +2v

vφ+

3v2

v2φ

+4v3

v3φ

+ · · ·)

The odd terms vanish upon taking the average,

(v − vφ)−2 ≃ v−2φ (1 +

3v2

v2φ

)

1

2mv2 =

1

2kTe

76

Thus, the dispersion relation becomes

1 =ω2

p

k2

1

v2p

(1 + 31

v2p

kTe

m)

=ω2

p

k2

k2

ω2(1 + 3

k2

ω2

kTe

m)

∴ ω2 = ω2p +

ω2p

ω2

3kTe

mk2

We assumed Im(ω/k) << 1If the thermal correction is small, ω2 ≈ ω2

p

∴ ω2 = ω2p +

3kTe

mk2

2) The evaluation of the imaginary termNeglect the thermal correction to the real part of ω

ω2 ≃ ω2p

1 =ω2

p

k2

[

P

∫ ∞

−∞

∂f0/∂v

v − (ω/k)dv + iπ

∂f0

∂v|v=ω

k

]

=ω2

p

k2

1

v2φ

+ω2

p

k2iπ

∂f0

∂v|v=ω

k

=ω2

p

ω+ iπ

ω2p

k2

∂f0

∂v|v=ω

k

ω2

(

1 − iπω2

p

k2

∂f0

∂v|v=ω

k

)

= ω2p

∴ ω = ωp

(

1 − iπω2

p

k2

∂f0

∂v|v=ω

k

)−1/2

≃ ωp

(

1 + iπ

2

ω2p

k2

∂f0

∂v|v=ω

k

)

f0 =1√πvth

exp(− v2

v2th

) : one − dimensional Maxwellian v2th =

2kTe

m

∂f0

∂v= (πv2

th)−1/2(−2v

v2th

) exp(− v2

v2th

)

= − 2v√πv3

th

exp(− v2

v2th

)

∂f0

∂v|v=ω

k= − 2ω

k√πv3

th

exp(−ω2/k2

v2th

)

77

Im(ω) = −π

2

ω3p

k2

2√π

ω

k

1

v3th

exp(−ω2/k2

v2th

)

ω2 = ω2p +

3kTe

mk2

keep thermal correction term in the exponent.

Im(ω) = −π

2

ω3p

k2

2√π

ω

k

1

v3th

exp(−ω2

p/k2

v2th

) exp(−3

2)

= −√

πωp(ωp

kvth)3 exp(−

ω2p/k2

v2th

) exp(−3

2)

∴ Im(ω

ωp) = −0.22

√π(

ωp

kvth)3 exp(− 1

2k2λ2D

)

Where λ2D =

v2th

2ω2p

If Im(ω) < 0, collisionless damping of plasma waves: “Landau damp-ing”This is the analytical result.

• The contour integral by numerical approach was presented(J.D. Jackson, Plasma Phys. 1(1960) pp. 5)→ Fried and Conte have provided tables for the case when f0 is aMaxwellian.

The below figure shows the analytical results and numerical results.

78

Figure 5: Real and imaginary parts of the frequency as a function of wavenumber for a stationary one-component plasma in thermal equilibrium. Thefrequency is given in units of ωp, while the wave number is expressed in unitsof the Debye wave number (kD). The dotted curves represent approximateformulas derived in this section.

79

• From the Harris Dispersion Relation

The electrostatic dispersion relation in hot plasma

1 +∑

s

1

k2λ2D

∑

n

e−bIn(b)[1 +ω

kzvthZn(ζn)] = 0

where λ2D =

v2th

2ω2p

, b =v2thk2

⊥2Ω2

, ζn =ω − nΩ

kzvth

vth =

√

2kTe

m

Zn(ζn) is “Fried-Conte” function or “Dispersion function”Zn(ζn) can be evaluated numerically.

For Landau damping in unmagnetized plasmas (B0 = 0)

Ω → 0, k⊥ → 0, (n → 0)

kz → k

∴ 1 +1

k2λ2D

[ 1 +ω

kvthZ(

ω

kvth) ] = 0

⊙ Power Series of Z(ζ)

for ζ << 1,

Z(ζ) = iπ1/2e−ζ2 − 2ζ[1 − 2ζ2/3 + 4ζ4/15 − 8ζ6/105 + · · ·]

= iπ1/2e−ζ2 − ζ∞∑

n=0

(−ζ2)nπ1/2/(n + 1/2)!

⊙ Asympotic Expansion

for ζ >> 1,

Z(ζ) ≃ iπ1/2σe−ζ2 − 1

3

[

1 +1

2ζ2+

3

4ζ4+ · · ·

]

= iπ1/2σe−ζ2 −∞∑

n=0

ζ−(2n+1)(n − 1

2)!/π1/2

Where σ =

0 y > 01 y = 0 ζ = x + iy2 y < 0

80

Assuming vφ = ωk >> vth

Z(ω

kvth) ≃ iπ1/2σe

− ω2

k2v2th − kvth

ω− k3v3

th

2ω3− 3k5v5

th

4ω5

A. Real term in dispersion relation

1 +1

k2λ2D

[

1 +ω

kvth

(

−kvth

ω− k3v3

th

2ω3− 3k5v5

th

4ω5

)]

= 0

→ 1 +1

k2λ2D

[

1 − 1 − k2v2th

2ω2− 3k4v4

th

4ω4

]

= 0

→ 1 +1

k2λ2D

[(

−k2v2th

2ω2

) (

1 +3k2v2

th

2ω2

)]

= 0

→ 1 +2ω2

p

k2v2th

(

−k2v2th

2ω2

) (

1 +3k2v2

th

2ω2

)

= 0

→ 1 −ω2

p

ω2

(

1 +3k2v2

th

2ω2

)

= 0

∴ ω2 = ω2p +

ω2p

ω2

3k2v2th

2

= ω2p +

ω2p

ω2

(3

2

2kTe

mk2

)

= ω2p +

ω2p

ω23kTe

m k2

B. Imaginary term in dispersion relation

0 = 1 +1

k2λ2D

[

1 +ω

kvth(iπ1/2σe

− ω2

k2v2th − kvth

ω− k3v3

th

2ω3− 3k5v5

th

4ω5)

]

0 = 1 +1

k2λ2D

[

1 +ω

kvthiπ1/2σe

− ω2

k2v2th − 1 − k2v2

th

2ω2

]

= 1 +2ω2

p

k2v2th

(ω

kvth

)

iπ1/2σe− ω2

k2v2th −

2ω2p

k2v2th

k2v2th

2ω2

ω2

ω2p

= 1 − 2ω3

k3v3th

iπ1/2σe− ω2

k2v2th

∴ω

ωp=

[

1 − 2(ω

kvth)3iπ1/2σe

− ω2

k2v2th

]1/2

≃ 1 −(

ω

kvth

)3

iπ1/2σe− ω2

k2v2th

≃ 1 −(

ωp

kvth

)3

iπ1/2σe− ω2

p

k2v2th e−

32

81

∴ Im( ωωp

) = −0.22√

πσ(ωp

kvth)3e

− 1

2k2λ2D

82

6 ECR Heating [or Damping] Rates

6.1 Fund. Harm. Damping Rate - classical approach(det↔M)

(↔M= ~k~k +

ω2

c2

↔ǫ −k21 = 0)

6.1.1 The Dielectric Tensor for ω ≫ ωpi, Ωi and ω ∼ |Ωe|

Talking first order in the temperature from the Hot Plasma Dispersion Re-lation

b =k2⊥T

mΩ2≪ 1

Identities

In = I−n

In(x) =∞∑

s=0

1

s!(s + n)!(x

2)2s+n

In(x) = I−n(x)

I0(b) =∞∑

s=0

1

s!(s)!(b

2)2s

= 1 +b2

4+

1

4

b4

16+ · · · ≃ 1 +

b2

4

I1(b) =∞∑

s=0

1

s!(s + 1)!(b

2)2s

=b

2+

1

2

b3

8+ · · · ≃ b

2

∑

n

⇒ n = −1 & n = 0 & n = 1,

∑

s

ω2ps

ω2≃

ω2pe

ω2(only electron, ωpe ≫ ωpi)

83

ǫxx = 1 +ω2

pe

ω2

ω

kza

1 − b

b[I1Z1 + I−1Z−1] = 1 +

ω2pe

ω2

ω

kza

1 − b

bI1(Z1 + Z−1)

ǫyy = 1 +ω2

pe

ω2

ω

kza

1 − b

bI1(Z1 + Z−1) −

ω2pe

ω2

ω

kza2b(1 − b)(I ′0 − I0)Z0

ǫzz = 1 −ω2

pe

ω2

ω

kza(1 − b)[I0ζ0Z

′0 + I1(ζ1Z

′1 + ζ−1Z

′−1)]

ǫxy = −ǫyx = iω2

pe

ω2

ω

kza(1 − b)[(I ′1 − I1)Z1 + (I−1 − I ′−1)Z−1]

= iω2

pe

ω2

ω

kza(1 − b)(I ′1 − I1)(Z1 − Z−1)

ǫxz = ǫzx = −ω2

pe

ω2

ω

kza

((1 − b)√

2b

)

I1(Z′1 + Z ′

−1)

ǫyz = −ǫzy = iω2

pe

ω2

ω

kza

(√

b

2

)

(1 − b)[(I ′1 − I1)(Z′1 − Z ′

−1) + (I ′0 − I0)Z′0]

Since

I0(b) = 1 +b2

4+ · · · &I1(b) =

b

2+ · · ·

I ′0(b) − I0(b) =

(b

2− 1 − b2

4

)

,

and the large argument expansion of Zn(ζ), we may neglect the last termsin ǫyy and ǫyz. Then ǫxx = ǫyy

Note that

ζn =ω − nΩs

kzve

For electrons, n = −1 is the resonant term.

ζ = ζ−1 =ω + Ωs

kzve,

ω − nΩs

kzve≫ 1 for n 6= −1 (Ωe < 0)

Z(x)−→x≫1 i

√πe−x2 − 1

x(1 +

1

2x2+ · · · )

: non-resonant term (large argument expansion)

84

ǫxx = 1 +ω2

pe

ω2

ω

kza

1 − b

b· b

2(Z1 + Z−1)

= 1 +ω2

pe

2ω2

ω

kza(1 − b)(Z1 + Z−1)

= 1 +ω2

pe

2ω2

ω

kzveZ(ζ) +

ω2pe

2ω2

ω

kzve

(

− 1

ζ1

)

= 1 −ω2

pe

2ω2

ω

kzve· kzve

ω + |Ωe|+

ω2pe

2ω2

ω

kzveZ(ζ)

= 1 −ω2

pe

2ω(ω + |Ωe|)+

ω2pe

2ω2

ω

kzveZ(ζ) = ǫyy

where a = ve ≡√

2Te/me, Z(ζ) = Z−1(ζ−1) : “resonant term”

|Ωe| =

∣∣∣∣

eB

me

∣∣∣∣

ζ−1 = ζ =ω + Ωe

kzve=

ω − |Ωe|kzve

ǫzz = 1 −ω2

pe

ω2

ω

kza(1 − b)[I0ζ0Z

′0 + I1(ζ1Z

′1 + ζ−1Z

′−1)]

≃ 1 −ω2

pe

ω2

ω

kzaI0ζ0Z

′0 −

ω2pe

ω2

ω

kzaI1( ζ1Z

′1

︸︷︷︸

≃ζ11

ζ21

= 1ζ1

≪1

+ ζ−1Z′−1)

≃ 1 −ω2

pe

ω2

ω

kza· 1 · ζ0

(1

ζ20

)

−ω2

pe

ω2

ω

kza· b

2· ζZ ′(ζ)

= 1 −ω2

pe

ω2−

ω2pe

ω2

ω

kzabζZ ′(ζ)

ǫxy = −ǫyx = +iω2

pe

ω2

ω

kza(1 − b)(I ′1 − I1)(Z1 − Z−1)

≃ +iω2

pe

ω2

ω

kza(I ′1 − I1)(Z1 − Z−1)

≃ +iω2

pe

ω2

ω

kza

(1

2− b

2

)

(Z1 − Z−1)

≃ +iω2

pe

ω2

ω

kza(Z1 − Z−1)

≃ +iω2

pe

ω2

ω

kza

(

− 1

ζ1

)

− iω2

pe

ω2

ω

kzaZ(ζ)

= −iω2

pe

2ω(ω + |Ωe|)− i

ω2pe

ω2

ω

kzaZ(ζ)

85


pe

ω2

ω

kza

((1 − b)√

2b

)

I1(Z′1 + Z ′

−1)

≃ −ω2

pe

ω2

ω

kza

(1 − b)√2b

b

2(Z ′

1 + Z ′−1)

≃ −ω2

pe

ω2

ω

kza

√

b

2(Z ′

1 + Z ′−1)

(

∵ Z ′1 ∼ 1

ζ21

)

≃ −ω2

pe

ω2

ω

kzve

√

b

2Z ′(ζ)

ǫyz = −ǫzy ≃ +iω2

pe

ω2

ω

kza

√

b

2(1 − b)(I ′1 − I1)(Z

′1 − Z ′

−1)

≃ +iω2

pe

ω2

ω

kza

√

b

2(I ′1 − I1)(Z

′1 − Z ′

−1)

≃ +iω2

pe

ω2

ω

kza

√

b

2

1

2(1 − b)(Z ′

1 − Z ′−1)

≃ +iω2

pe

ω2

ω

kza

√

b

2Z ′(ζ)

= iǫxz

Thus, finally we obtain

ǫxx = ǫyy = 1 −ω2

pe

2ω(ω + |Ωe|)+

ω2pe

2ω2

ω

kzveZ(ζ)

ǫzz = 1 −ω2

pe

ω2−

ω2pe

ω2

ω

kzabζZ ′(ζ)

ǫxy = −ǫyx = −iω2

pe

2ω(ω + |Ωe|)− i

ω2pe

ω2

ω

kzaZ(ζ)


pe

ω2

ω

kzve

√

b

2Z ′(ζ)

ǫyz = −ǫzy = iǫxz

where ve =√

2Te/me, b = be = k⊥TmeΩ2

e,

|Ωe| = | eBme|, ζ = ω−|Ωe|

k‖ve

6.1.2 Damping Rates near the ECR Region

M = Det↔M= Det(~k~k + µ0ǫ0ω

2 ↔ǫ −k2 ↔

1 ) = 0

⇒ det( ~N ~N+↔ǫ −N2 ↔

1 ) = 0

where N = kc/ω

In tokamak

Bz(x) = BT R0(1 − x

R)

86

The argument of the Fried-Conte function

ζ =ω − Ωe(x)

k‖ve=

√me(ω − Ωe(x))

k‖√

2Te

since Ωe(x) = eBz(x)me

= eBT

me(1 − x

R) = Ωe − ΩexR

when ω = Ωe

ω − Ωe(x) = ΩexR

∴ ζ =Ωe

xR

k‖√

2

√me

Te=

xΩe√2ω

c N‖

1

R

√me

Te

=x√2

Ωe

Ωe

c N‖

1

R

√me

Te=

x√2

1

N‖R

√

mec2

Te

=x√2∆

where ∆ = RN‖√

Te/mec2, c is the speed of light.

“the half width of resonance zone”

Let σ = ωk‖ve

= ωN‖ω

c

√2Teme

= 1√2

1

N‖

√Te

mec2

= R√2∆

Now the dielectric tensor components can be written as

ǫxx = ǫyy = 1 − ωpe2

2ω(ω + Ωe)+

ωpe2

2ω2σZ(ζ)

ǫzz = 1 −ω2

pe

ω2− ωpe

2

2ω2σbζZ ′(ζ)

since b =R2

⊥Te

meΩ2e

=ω2N2

⊥Te

mec2Ω2e

1

ω2σb =

1

ω2

σ2

σb =

1

ω2

1

σ

1

2N2‖

Te

mec2

ω2N2⊥Te

mec2Ω2e

=1

2Ω2e

N2⊥

N2‖

1

σ

⇒ 1

ω2σ2b =

1

2Ω2e

N2⊥

N2‖

⇒ σ√

b

ω=

1√2

1

Ωe

N⊥N‖

87

∴ ǫzz = 1 −ω2

pe

ω2−

ω2pe

4Ω2e

N2⊥

N2‖

ζ

σZ ′(ζ)

ǫxy = −ǫyx = −iω2

pe

2ω(ω + Ωe)− i

ω2pe

2ω2σZ(ζ)


pe

2ω2σ

√

b

2Z ′(ζ) = −

ω2pe

2ω

σ√

b

ω

1√2Z ′(ζ)

= −ω2

pe

2ω

1√2

1

Ωe

N⊥N‖

1√2Z ′(ζ) = −

ω2pe

4ωΩe

N⊥N‖

Z ′(ζ)

= −ω2

pe

4ωΩ2e

N⊥N‖

Z ′(ζ)

ǫyz = −ǫzy = iǫxz

We define α =ω2

pe

Ω2e

and F = 14σ

ω2pe

Ω2e

ζN2

‖

Z ′(ζ)

ǫxx = 1 − α

4+

α

2σZ

ǫxy = −i(α

4+

α

2σZ

)

ǫxz = − α

4N‖Z ′N⊥

ǫzz = 1 − α − FN2⊥

Then, the dispersion relation becomes

↔M=

∣∣∣∣∣∣∣

N2⊥ + ǫxx − (N2

⊥ + N2‖ ) ǫxy N⊥N‖ + ǫxz

−ǫxy ǫxx − (N2⊥ + N2

‖ ) iǫxz

N⊥N‖ + ǫxz −iǫxz N2‖ + ǫzz − (N2

⊥ + N2‖ )

∣∣∣∣∣∣∣

= 0

⇒ [N2⊥ + ǫxx − N2

⊥ − N2‖ ][(ǫxx − N2

⊥ − N2‖ )(N2

‖ + ǫzz − N2⊥ − N2

‖ ) − ǫ2xz]

−ǫxy[−ǫxy(N2‖ + ǫzz − N2

⊥ − N2‖ ) − iǫxz(N⊥N‖ + ǫxz)]

+(N⊥N‖ + ǫxz)[iǫxyǫxz − (ǫxx − N2⊥ − N2

‖ )(N⊥N‖ + ǫxz)] = 0

88

Left-hand side:

(ǫxx − N2‖ )[(ǫxx − N2

⊥ − N2‖ )(ǫzz − N2

⊥) − ǫ2xz] + ǫxy[ǫxy(ǫzz − N2⊥) + iǫxz(N⊥N‖ + ǫxz)]

+(N⊥N‖ + ǫxz)[iǫxyǫxz − (ǫxx − N2⊥ − N2

‖ )(N⊥N‖ + ǫxz)]

= (ǫxx − N2‖ )[N4

⊥ − (ǫxx − N2‖ + ǫzz)N

2⊥ + (ǫxx − N2

‖ )ǫzz − ǫ2xz]

+ǫxy[−ǫxyN2⊥ + ǫxyǫzz + iǫxzN⊥N‖ + iǫ2xz]

+(N⊥N‖ + ǫxz)[iǫxyǫxz + ǫxzN2⊥ + N3

⊥N‖ − (ǫxx − N2‖ )N⊥N‖ − ǫxz(ǫxx − N2

‖ )]

= (ǫxx − N2‖ )N4

⊥ − (ǫxx − N2‖ )(ǫxx − N2

‖ )N2⊥ + (ǫxx − n2

||)2ǫzz − ǫ2xz(ǫxx − N2

‖ )

−ǫ2xyN2⊥ + ǫ2xyǫzz + iǫxyǫxzN‖N⊥ + iǫxyǫ

2xz + iǫxyǫxzN⊥N‖ + iǫxyǫ

2xz + ǫxzN

3⊥N‖ + ǫ2xzN

2⊥

+N4⊥N2

‖ + ǫxzN3⊥N‖ − (ǫxz + N⊥N‖)(ǫxx + ǫ2xzN

2⊥)N⊥N‖

−ǫxz(ǫxz + N⊥N‖)(ǫxx − N2‖ )

= ǫxxN4⊥ + (ǫxx − N2

‖ )(ǫzz − N2⊥) − (ǫxx − N2

‖ )(ǫzzN2⊥ + ǫ2xz + N2

⊥N2‖ + ǫxzN‖N⊥ + ǫ2xz

+ǫxzN⊥N‖) − ǫ2xy(N⊥ − ǫzz) + 2iǫxyǫxzN⊥N‖ + 2iǫxyǫ2xz + 2ǫxzN

3⊥N‖ + ǫ2xzN

2⊥

= ǫxxN4⊥ + (ǫxx − N2

‖ )(ǫzz − N2⊥) − (ǫxx − N2

‖ )(ǫzzN2⊥ + 2ǫ2xz + 2ǫxzN‖N⊥ + N2

⊥N2‖ )

+ǫ2xy(ǫzz − N⊥) + 2iǫxyǫxzN⊥N‖ + 2iǫxyǫ2xz + 2ǫxzN

3⊥N‖ + ǫ2xzN

2⊥

=(

1 − α

4+

α

2σZ

)

N4⊥ +

(

1 − α

4+

α

2σZN2

‖

) (1 − α − FN2

⊥ − n⊥2)

−(

1 − α

4+

α

2σZ − N2

‖

)[

(1 − α − FN2⊥)N2

⊥ +α2

8N2‖Z ′2N2

⊥ − α

2N‖Z ′N‖N

2⊥ + N⊥N2

‖

]

−α2

16

(1 + 4σZ + 4σ2Z2

)(1 − α − FN2

⊥ − N2⊥) − 2

α

4(1 + 2σZ)

α

4N‖Z ′N2

⊥N‖

+2α

4(1 + 2σZ)

α2

16N2‖Z ′2N2

⊥ − 2α

4N‖Z ′N‖N

4⊥ +

α2

16N2‖Z ′2N4

⊥

89

(1) N4⊥ Coefficient

[

1 − α

4+

α

2σZ +

(

1 − α

4+

α

2σZ − N2

‖

)

F − α

2+

α2

16N2‖Z ′2

]

= σα

2Z + 1 − α

4+

(

1 − α

4− N2

‖

)

F +α

2ZσF − α

2Z ′ +

α2

16N2‖Z ′2

σF = α4

ζN2

‖

Z ′ and Z ′ = −2(1 + ζZ) ⇒ ζZ = −12Z ′ − 1

∴α2 ZσF = α

2 Z α4

ζN2

‖

Z ′

= α2

8N2‖

Z ′ (−12Z ′ − 1

)= − α2

16N2‖

Z ′2 − α2Z′

8N2‖

= σα

2Z + 1 − α

4+

(

1 − α

4− N2

‖

)

F − α2

16N2‖Z ′2 − α2

8N2‖Z ′ − α

2Z ′ +

α2

16N2‖Z ′

= σα

2Z + 1 − α

4+

(

1 − α

4− N2

‖

)

F − α

2

(

1 +α

4N2‖

)

Z ′ = A

(2) N2⊥ Coefficient

−(

1 − α

4+ α2σZ − N2

‖

)2(F + 1) −

(

1 − α

4+

α

2σZ − N2

‖

)[

(1 − α) +α2

8N2‖Z ′2 − α

2Z ′ + N2

‖

]

+α2

16[1 + 4σZ(1 + σZ)](F + 1) − α2

8(1 + 2σZ)Z ′ +

α3

32N2‖(1 + 2σZ)Z ′2

=

[

−(

1 − N2‖ − α

4

)

− ασZ(

1 − N2‖ − α

4

)

− α2

4σ2Z2

]

(F + 1)

−(

1 − α

4− N2

‖

)[

(1 − α) +α2

8N2‖Z ′2 − α

2Z ′ + N2

‖

]

− α

2σZ

[

(1 − α) +α2

8N2‖Z ′2 − α

2Z ′ + N2

‖

]

+α2

16[1 + 4σZ(1 + σZ)] (F + 1) − α2

8Z ′ − α2

4σZZ ′ +

α3

32N2‖Z ′2 +

α3

16N2‖σZZ ′

= −(1 − N2‖ )

(

1 − N2‖ − α

2

)

F − α2

16F −

(

1 − N2‖ − α

4

)2− ασ ZF

(

1 − N2‖ − α

4

)

−ασZ(

1 − N2‖ − α

4

)

− α2

4σ2Z2F − α2

4σ2Z2 −

(

1 − α

4− N2

‖

)

(1 − α)

− α2

8N2‖

(

1 − α

4− N2

‖

)

Z ′2 +α

2

(

1 − α

4− N2

‖

)

Z ′ −(

1 − α

4− N2

‖

)

(1 − α)

−α

2σZ(1 − α) − α3

16N2‖σZZ ′2 +

α2

4σZZ ′ − α

2N2

‖σZ +α2

16F +

α2

4σZF (1 + σZ)

+α2

16+

α2

4σZ(1 + σZ) − α2

8Z ′ − α2

4σZZ ′ +

α3

32N2‖Z ′2 +

α3

16N2‖σZZ ′2

90

But, since α2 σZF = − α2

16N2‖

Z ′2 − α2

8N2‖

Z ′

−ασZF(

1 − N2‖ − α

4

)

− α2

4 σ2Z2F + α2

4 σZF (1 + σZ)

=

(

α2

8N2‖

Z ′2 + α2

4N2‖

Z ′) (

1 − N2‖ − α

4

)

+ α2

4 σ2Z2F

=

(

α2

8N2‖

Z ′2 + α2

4N2‖

Z ′) (

1 − N2‖ − α

4

)

+ α2 σZ

(

− α2

16N2‖

Z ′2 − α2

8N2‖

Z ′)

= α2

8N2‖

Z ′2 + α2

4N2‖

Z ′ − α2

8 Z ′2 − α2

4 Z ′ − α3

32N2‖

Z ′2 − α3

16N2‖

Z ′ − α3

32N2‖

Z ′2 − α3

16N2‖

Z ′

= α2

8N2‖

Z ′2 + α2

4N2‖

Z ′ − α2

8 Z ′2 − α2

4 Z ′ − α3

16N2‖

Z ′2 − α3

8N2‖

Z ′

= −(1 − N2‖ )

(

1 − N2‖ − α

2

)

F − (1 − N2‖ )2 +

α

2(1 − N2

‖ ) − ασZ(

1 − N2‖ − α

4

)

−(

1 − α

4− N2

‖

)

(1 − α) − α2

8N2‖

(

1 − α

4− N2

‖

)

Z ′2 +α

2

(

1 − N2‖ − α

4

)

Z ′

−(

1 − α

4− N2

‖

)

N2‖ − α

2σZ(1 − α) − α

2N2

‖σZ +α2

4σZ − α2

8Z ′ +

α3

32N2‖Z ′2 +

α2

8N2‖Z ′2

+α2

4N2‖Z ′ − α2

8Z ′2 − α2

4Z ′ − α3

16N2‖Z ′2 − α3

8N2‖Z ′ = B

For the terms which do not contain Z, Z ′, andF

−(1 − N2‖ )

2+

α

2(1 − N2

‖ ) − (1 − α

4− N2

‖ )(1 − α) − (1 − α

4− N2

‖ )N2‖

= −1 + 2N2‖ − N4

‖ +α

2− α

2N2

‖ − 1 + α +α

4− α2

4+ N2

‖ − αN2‖ − N2

‖ +α

4N2

‖ + N4‖

= −2 + 2N2‖ − 5

4αN2

‖ +7

4α − α2

4

= −2 +7

4α − α2

4+ (2 − 5

4α)N2

‖

For the coefficients of Z

−σα(1 − N2‖ − α

4) − α

2σ(1 − α) − α

2N2

‖σ +α2

4σ

= −σα + σαN2‖ +

α2

4σ − α

2σ +

α2

2σ − α

2N2

‖σ +α2

4σ

= −3

2σα +

1

2σαN2

‖ + σα2

= σα

2(N2

‖ − 3 + 2α)

91

For the coefficients of Z ′

α2

4N2‖− α2

4− α3

8N2‖

+α

2(1 − N2

‖ − α

4) − α2

8

=α2

4N2‖− α3

8N2‖

+α

2(1 − N2

‖ ) − α2

4− α2

8− α2

8

=α

2[1 − α − N2

‖ +α

2N2‖(1 − α

2)]

Thus, the coefficient of N2⊥ , B is

B = σα

2(N2

‖ − 3 + 2α)Z − 2 +7

4α − α2

4+ (2 − 5

4α)N2

‖

+α

2[1 − α − N2

‖ +α

2N2‖(1 − α

2)]Z ′ − (1 − N2

‖ )(1 − N2‖ − α

2)F

(3) Constant term

(1 − α

4+

α

2σZ − N2

‖ )2(1 − α) − α

16(1 + 4σZ + 4σ2Z2)(1 − α)

= (1 − α

4− N2

‖ )2(1 − α) + ασ(1 − α

4− N2

‖ )(1 − α)Z +α2

4σ2Z2(1 − α)

−(α2

16+

α2

4σZ +

α2

4σ2Z2)(1 − α)

= (1 − α

4− N2

‖ )2(1 − α) − α2

16(1 − α) + ασZ(1 − α)(1 − α

4− N2

‖ − α

4)

= σα

2(1 − α)(2 − α − 2N2

‖ )Z + (1 − N2‖ )(1 − N2

‖ − α

2)(1 − α)

= C

Therefore, M = AN4⊥ + BN2

⊥ + C = 0Where

A = σα

2Z + 1 − α

4− α

2(1 +

α

4N2‖)Z ′ + (1 − α

4− N2

‖ )F

B = σα

2(N2

‖ − 3 + 2α)Z − 2 +7

4α − α2

4+ (2 − 5

4α)N2

‖

+α

2[1 − α − N2

‖ +α

2N2‖(1 − α

2)]Z ′ − (1 − N2

‖ )(1 − N2‖ − α

2)F

C = σα

2(1 − α)(2 − α − 2N2

‖ )Z + (1 − N2‖ )(1 − N2

‖ − α

2)(1 − α)

Since F ∼ 1σ and σ ≫ 1

We may neglect the terms explicitly including F in A, B, and C.

<A. I. Akhiezer>

M = AN4⊥ + BN2

⊥ + C ⇒ Regroup about Z, Z’ and constants

92

M = σα

2ZN4

⊥ + (1 − α

4)N4

⊥ − α

2(1 +

α

4N2‖)Z ′N4

⊥

+σα

2(N2

‖ − 3 + 2α)ZN2⊥ + [−2 +

7

4α − α2

4+ (2 − 5

4α)N2

‖ ]N2⊥

+α

2[1 − α − N2

‖ +α

2N2‖(1 − α

2)]Z ′N2

⊥

+σα

2(1 − α)(2 − α − 2N2

‖ )Z + (1 − N2‖ )(1 − N2

‖ − α

2)(1 − α)

= [σα

2N4

⊥ + σα

2(N2

‖ − 3 + 2α)N2⊥ + σ

α

2(1 − α)(2 − α − 2N2

‖ )]Z

+(1 − α

4)N4

⊥ + [−2 +7

4α − α2

4+ (2 − 5

4α)N2

‖ ]N2⊥ + (1 − N2

‖ )(1 − N2‖ − α

2)(1 − α)

+[−α

2(1 +

α

4N2‖)N4

⊥ +α

2(1 − α − N2

‖ +α

2N2‖(1 − α

2))N2

⊥]Z ′

= σα

2[N4

⊥ − (3 − N2‖ − 2α)N2

⊥ + (1 − α)(2 − α − 2N2‖ )]Z

+(1 − α

4)N4

⊥ − [2 − 7

4α +

α2

4− (2 − 5

4α)N2

‖ ]N2⊥ + (1 − N2

‖ )(1 − α)(1 − N2‖ − α

2)

+[−α

2(1 +

α

4N2‖)N4

⊥ +α

2(1 − α − N2

‖ +α

2N2‖(1 − α

2))N2

⊥]Z ′

= σM0 + M1 + Z ′M2 = 0

where,

M0 =α

2[N4

⊥ − (3 − N2‖ − 2α)N2

⊥ + (1 − α)(2 − α − 2N2‖ )]Z

M1 = (1 − α

4)N4

⊥ − [2 − 7

4α +

α2

4− (2 − 5

4α)N2

‖ ]N2⊥ + (1 − N2

‖ )(1 − α)(1 − N2‖ − α

2)

M2 = −α

2(1 +

α

4N2‖)N4

⊥ +α

2(1 − α − N2

‖ +α

2N2‖(1 − α

2))N2

⊥

From the equation including σ,

we take Taylor expansion at N⊥0 of (σM0)N⊥0= 0

σM0 = 0 → M0 = 0

∵ M = σM0 + M1 + M2Z′ = 0

σ >> 1, → σM0 >> M1 + M2Z′

︸︷︷︸

weak-damping

→ M ≈ σM0 = 0)

Thus,

93

N2⊥0 =

1

2[(3 − N2

‖ − 2α) ±√

(3 − N2‖ − 2α)

2 − 4(1 − α)(2 − α − 2N2‖ )]

but, (3 − N2‖ − 2α)

2 − 4(1 − α)(2 − α − 2N2‖ )

= 9 + N4‖ + 4α2 − 6N2

‖ − 12α + 4αN2‖ − 8 + 4α + 8N2

‖ + 8α − 4α2 − 8αN2‖

= 1 + N4‖ + 2N2

‖ − 4αN2‖

= (1 + N2‖ )2 − 4αN2

‖

∴ N2⊥0 =

1

2

[

(3 − N2‖ − 2α) ±

√

(1 + N2‖ )2 − 4αN2

‖

]

(+) sign : X-mode like(– ) sign : O-mode like

Next order solution

σM0 = −(M1 + Z ′M2)

(σM0)N⊥0+ (

∂σM0

∂N⊥)N⊥=N⊥0

δN⊥ = −(M1 + Z ′M2)N⊥=N⊥0

∵ δN⊥ = −M1 + Z ′M2

(∂σM0∂N⊥

)|N⊥=N⊥0

⇒ “the change of refractive index of wave near the electroncyclotron resonance zone when the wave propagate”

Using the identity of the Fried-Conte function

Z′(x) = −2[1 + xZ(x)], and

∂σM0

∂N⊥|N⊥0

= σα

2[4N3

⊥0 − 2(3 − N2‖ − 2α)N⊥0]Z

= σα

2N⊥0[4N2

⊥0 − 6 + 2N2‖ + 4α]Z

= σαN⊥0[2N2⊥0 − 3 + 2N2

‖ + 2α]Z

[M1 + Z ′M2] = M1 − 2[1 + ζZ]M2 = M1 − 2M2 − 2ζM2Z.

∴ δN⊥ = − M1 − 2M2 − 2ζM2Z

σαN⊥0(2N2⊥0 − 3 + N‖ + 2α)

94

Since Z(ζ) is complex,

δN⊥ = −(M1 − 2M2 − 2ζM2Z)(ReZ − iImZ)

σαN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2

= − [M1 − 2M2 − 2ζM2(ReZ + iImZ)](ReZ − iImZ)

σαN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2

= −2ζM2|Z|2 − (M1 − 2M2)ReZ + i(M1 − 2M2)ImZ

σαN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2

⇒ Re δN⊥ = − 2ζM2|Z|2 − (M1 − 2M2)Re Z

σαN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2

Im δN⊥ = − (M1 − 2M2)Im Z

σαN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2

But, σ =R√2∆

, ζ =x√2∆

Z(ζ) =1√π

∫ ∞

−∞

e−t2

t − ζdt

When ζ is real,

Z(ζ) = iπ12 e−ζ2 − 2ζY (ζ) where, Y (ζ) =

1

ζe−ζ2

∫ ζ

0et2dt

Then, Im Z(ζ) =√

πe−ζ2=

√πe−

x2

2∆2

Thus,

Re δN⊥ = −2 x√

2∆M2|Z|2 − Λ1(Re Z)

R√2∆

αN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2

=

√2

R∆ − (

√2 x

∆)M2|Z|2 − Λ1(Re Z)

αN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2

Im δN⊥ = − Λ1√

πe−x2

2∆2

R√2∆

αN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2

=

√2π

α

∆Λ1e−x2

2∆2

RN⊥0(2N2⊥0 − 3 + N2

‖ + 2α)|Z|2 , ∆ = RN‖

√

Te

mec2

:“Damping Rate of Waves near the ECR zone”

95

Where,

Λ1 = (M1 − 2M2)N⊥0

= (1 − α

4)N4

⊥0 − [2 − 7

4α +

α2

4− (2 − 5

4α)N2

‖ ] N2⊥0 + (1 − N2

‖ )(1 − α)(1 − N2‖ − α

2)

+α(1 +α

4N2‖)N4

⊥0 − α[1 − α − N2‖ +

α

2N2‖(1 − α

2)]N2

⊥0

= (1 +3α

4+

α2

4N2‖))N4

⊥0 − [2 − 3α

4− 3α2

4+

α

4N2

‖ − 2N2‖ +

α2

2N2‖(1 − α

2)] N2

⊥0

+(1 − N2‖ )(1 − α)(1 − N2

‖ − α

2)

96

6.2 Damping Rates Using Quasi-linear Theory

(ref. Owen C. Eldridge and Won Namkung, ORNL/TM-6052)

The Fokker-Plank form from Quasi-linear theory (see Appendix 3.)

∂f

∂t= π(

e

2mω)2

∑

modes

∞∑

n=−∞ 1

v⊥

∂

∂v⊥[nΩ|A|2δ(ω − nΩ − kzvz)

× (nΩ

v⊥

∂f

∂v⊥+ kz

∂f

∂vz)] +

∂

∂vz[kz|A|2δ(ω − nΩ − kzvz)

× (nΩ

v⊥

∂f

∂v⊥+ kz

∂f

∂vz)]

with

A = v⊥E−eiθJn+1 + v⊥E+e−iθJn−1 + 2vzEzJn.

Jn = Jn(k⊥v⊥

Ω)

E+ = Ex + iEy, E− = Ex − iEy

Bz = BT R0/(R0 + x) = BT (R − x)/R

where BT is the toroidal magnetic field at cyclotron resonance and

nΩ(x) = ω(1 − x

R)

∗ ∑

modes is the summation over all possible perturbed modes,

and∑

n is the summation over all harmonics.

For Maxwellian Distribution,

f = ne(x)[m

2πT (x)]3/2 exp(−mv2

2T)

= ne(x)[m

2πT (x)]3/2 exp(−mv2

⊥2T

− mv2z

2T)

⇒ ∂f

∂v⊥= [

m

2πT]3/2ne(−

mv⊥T

)e−mv2

2T

∂f

∂vz= [

m

2πT]3/2ne(−

mvz

T)e−

mv2

2T

Where ne(x) is the plasma density.

⇒ nΩ

v⊥

∂f

∂v⊥+ kz

∂f

∂vz= [

m

2πT]3/2ne(−

m

T)[nΩ + kzvz]e

−mv2

2T

When, this term is integrated over vz, nΩ + kzvz = ω.Then, above term becomes

nΩ

v⊥

∂f

∂v⊥+ kz

∂f

∂vz= −[

m

2πT]3/2 nemω

Te−

mv2

2T

97

And, the argument of delta function is

δ(ω − nΩ − kzvz) = δ(ω − ω(1 − x

R) − kzvz)

= (ωx

R− kzvz)

Thus,

∂f

∂t= −π

e2ne(x)

4m2ω2

mω

T (x)[

m

2πT (x)]3/2

∑

modes

∞∑

n=−∞

1

v⊥

∂

∂v⊥

×[

nΩ(x)|A|2δ(ωx

R− kzvz) exp(−mv2

2T)

]

+∂

∂vz

[

kz|A|2δ(ωx


2T)

]

Since, ω2p = e2ne(x)

mǫ0

πe2ne(x)

4mωT (x)= π

ǫ0ω2p

4ω

1

T

∴∂f

∂t= −π

ǫ0ω2p

4ω

1

T[

m

2πT]3/2

∑

modes

∞∑

n=−∞

1

v⊥

∂

∂v⊥

×[

nΩ(x)|A|2δ(ωx


2T)

]

+∂

∂vz

[

kz|A|2δ(ωx


2T)

]

Where,A = v⊥E−eiαJn+1 + v⊥E+e−iαJn−1 + 2vzEzJn

For the small argument, the Bessel functions are expanded, but n + 1 orderBessel function is smaller than the other two and is neglected.

Jn(x) ∼ (1

2x)n/Γ(n + 1) =

xn

2nn!(n 6= −1,−2,−3, · · · )

For n = −1,−2,−3, · · · (electrons)

Using Jn(x) = J−M (x) = (−1)MJM (x)

Jn+1 = J−M+1(x)

= J−(M−1)(x)

= (−1)M−1JM−1(x)

And if ~k = (kx, 0, kz), θ = 0.Then,

A = v⊥E− (k⊥v⊥|Ω| )n−1

2n−1(n − 1)!(−1)n−1 + 2vzEz

(k⊥v⊥|Ω| )n

2nn!(−1)n (n = 1, 2, 3, · · · )

= E− kn−1⊥ vn

⊥(2|Ω|)n−1(n − 1)!

(−1)n−1 + Ez2kn

⊥vn⊥vz

(2|Ω|)nn!(−1)n

=kn−1

x vn⊥

(2|Ω|)n−1 (n − 1)!

(

E− +kxvzEz

n|Ω|

)

98

Thus,

|A|2 =(k2

x)n−1v2n⊥

(4Ω2)n−1[(n − 1)!]2∣∣E− +

kxvzEz

n|Ω|∣∣2

The perpendicular and parallel heating rates per unit volume are found byintegrating over the velocities,

d2W⊥dtdV

=

∫ ∞

−∞dvz

∫ ∞

0

1

2mv2

⊥∂f

∂t2πv⊥dv⊥ = mπ

∫ ∫

v3⊥

∂f

∂tdv⊥dvz

non-relativistic energy

d2W‖dtdV

=

∫ ∞

−∞dvz

∫ ∞

0

1

2mv2

‖∂f

∂t2πv⊥dv⊥ = mπ

∫ ∫

v2‖v⊥

∂f

∂tdv⊥dvz

For an energy flux ~S in a plane plasma, one has

~∇ · ~S =∂Sx

∂x= − d2W

dtdV= Im(−2kx)Sx

1

Sx

∂Sx

∂x= − 1

Sx

d2W

dtdV= Im(−2kx)

The total energy absorbed in the resonant surface.

Wabs = W0(1 − e∫ ∞−∞ dxIm(−2kx))

= W0(1 − e−η), where η =

∫ ∞

−∞dxIm(2kx)

1) Integration of d2W⊥dtdv ,

d2W⊥dtdV

= −mπω2

p

16ω

1

T (x)

[ m

2πT (x)

] 32

∫ ∞

−∞dvz

∫ ∞

0v3⊥

1

v⊥

∂

∂v⊥× [n|Ω||A|2δ(ωx

R− kzvz)e

−mv2

2T ]1

+∂

∂vz[kz|A|2δ(ωx

R− kzvz)e

−mv2

2T ]2

dv⊥

1)

∫ ∞

−∞dvz

∫ ∞

0v3⊥

∂

∂vz[2] dv⊥ =

∫ ∞

0v3⊥

∫ ∞=vz

−∞=vz

d[2] dv⊥

=

∫ ∞

0v3⊥

(

kz|A|2δ(ωx

R− kzvz)e

−mv2

2T

)

dv⊥ = 0

2)∂

∂v⊥

[nΩ2|A|2δ(ωx

R− kzvz)e

−mv2

2T

]= n|Ω| (k2

x)n−1

(4Ω2)n−1[(n − 1)! ]2δ(

ωx

R− kzvz)

×[

2n(v⊥)2n−1 − m(v⊥)2n+1

T

]

e−mv2

2T

∣∣E− +

kxvzEz

n|Ω|∣∣2

99

= n|Ω| (k2x)n−1

(4Ω2)n−1[(n − 1)! ]2δ(

ωx

R− kzvz)2n(v⊥)2n−1

(

1 − m

2nTv2⊥

)

e−mv2

2T

× |E− +kxvzEz

n|Ω|∣∣2

Then, the integration gives

∫ ∞

0

[∫ ∞

−∞dvzδ(

ωx

R− kzvz)

∣∣E− +

kxvzEz

n|Ω|∣∣2 e−

mv2z

2T

]

× n|Ω| (k2x)n−1

(4Ω2)n−1[(n − 1)! ]22n

(v2n+1⊥ − m

2nTv2n+3⊥

)e−

mv2⊥

2T dv⊥

=2n2|Ω|

kz

(k2x)n−1

(4Ω2)n−1[(n − 1)! ]2

∣∣∣∣E− +

kxωxEz

kzRn|Ω|

∣∣∣∣

2

exp(− m

2T

ω2x2

k2zR

2

)

×∫ ∞

0

(v2n+1⊥ a

− m

2nTv2n+3⊥

b

)e−

mv2⊥

2T dv⊥3

3) Calculation of 3

a)

∫ ∞

0v2n+1⊥ e−

mv2⊥

2T dv⊥

(∫ ∞

0e−ttzdt = Z!

)

Let,mv2

⊥2T

= t , v⊥ =(2T

mt)1/2

, dv⊥ =1

2(2T

mt)−1/2(

2T

m)dt

⇒∫ ∞

0

(2T

mt)n+ 1

21

2(2T

mt)−1/2(

2T

m)e−tdt

=1

2

(2T

m

)n+1 ∫ ∞

0tne−tdt =

1

2

(2T

m

)n+1

n!

b)m

2nT

∫ ∞

0v2n+3⊥ e−

mv2⊥

2T dv⊥ =m

2nT

1

2

(2T

m

)n+2 ∫ ∞

0tn+1e−tdt

=m

4nT

(2T

m

)n+2

(n + 1)!

100

(a)-(b)

=1

2

(2T

m

)n+1

n! − m

4nT

(2T

m

)n+2

(n + 1)!

=1

2

(2T

m

)n+1

n! − 1

2n

m

2T

(2T

m

)n+2

(n + 1)!

=1

2

(2T

m

)n+1

n! − 1

2n

(2T

m

)n+1

(n + 1)!

=1

2

(2T

m

)n+1

n!

(

1 − n + 1

n

)

=1

2

(2T

m

)n+1

(n)!

(−1

n

)

= −1

2

(2T

m

)n+1

(n − 1)!

Thus,

d2W⊥dtdV

= −mπ

kz

ǫ0ω2p

4ω

1

T (x)

[m

2πT (x)

] 32

2n2|Ω| (k2x)n−1

(4Ω2)n−1[(n − 1)! ]2

×(−1

2

)(2T

m

)n+1(n − 1)!

∣∣∣∣E− +

kxvzEz

n|Ω|

∣∣∣∣

2

exp(− m

2T

ω2x2

k2zR

2

)

since, kx = ωc Nx, kz = ω

c Nz

¦(k2

x)n−1

(4Ω2)n−1=

(1

4Ω2

ω2

c2N2

x

)n−1

≃(

N2x

4Ω2

n2ω2

c2

)n−1

=

(n2N2

x

4c2

)n−1

¦ − m

2T

ω2x2

k2zR

2= − m

2T

ω2x2

R2

c2

ω2N2z

= −1

2

mc2

T

1

N2z R2

x2 = − x2

2∆2

where, ∆2 = N2z R2

(T

mc2

)

¦ − kxωxEz

kzRn|Ω| =NxωxEz

NzRn|Ω| ≃Nx

Nz

x

REZ

∴d2W⊥dtdv

=mπ

kzT

ǫ0ω2p

4ω

( m

2πT

) 32n2|Ω|

(n2n2

x

4c2

)n−11

(n − 1)!

(2T

m

)n+1

×∣∣∣∣E− +

nx

nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

e−x2

2∆2

2) Integration ofd2V‖

dtdV

101

d2V‖dtdV

=mπ

∫ ∞

−∞dvz

∫ ∞

∞dv⊥v2

zv⊥∂f

∂t= −mπ

ǫ0ω2p

4ω

1

T

( m

2πT

) 32

×∫ ∞

−∞dvz

∫ ∞

−∞dv⊥v2

zv⊥

[1

v⊥

∂

∂v⊥n|Ω||A|2δ

(ωx

R− Rzvz

)

e−mv2

2T

+∂

∂vz

kz|A|2δ(ωx

R− kzvz

)

e−mv2

2T

]

(a) The first term of the integrand

∫ ∞

−∞dvz

∫ ∞

0dv⊥v2

z

[

n|Ω|(

n2n2x

4c2

)n−11

[(n − 1)!]2δ(ωx

R− Rzvz

)

×2n(v⊥)2n−1(

1 − m

2nTv2⊥)

× e−mv2

2T

]

×∣∣∣∣E− +

nx

nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

=1

kz

(ωx

kzR

) ∣∣∣∣E− +

nx

nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

e−x2

2∆2

×∫ ∞

0n|Ω|

(n2n2

x

4c2

)n−11

[(n − 1)!]2× 2n(v⊥)2n−1

(

1 − m

2nTv2⊥)

e−mv2

2T dv⊥

But,

∫ ∞

0v2n−1⊥ e−

mv2⊥

2T dv⊥ =1

2

(2T

m

)n

(n − 1)!

∫ ∞

0v2n+1⊥ e−

mv2⊥

2T dv⊥ =1

2

(2T

m

)n+1

n!

=1

kz

(ωx

kzR

) ∣∣∣∣E− +

nx

nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

e−x2

2∆2 2n2|Ω|(

n2n2x

4c2

)n−11

(n − 1)!2

×(

1

2

(2T

m

)n

(n − 1)! − m

2nT

1

2

(2T

m

)n+1

n!

)

= 0

(b) The second term of the integrand

102

∫ ∞

−∞dvz

∫ ∞

−∞dv⊥v2

zv⊥∂

∂vzkz|A|2δ

(ωx

R− kzvz

)

e−mv2

2T

=

∫ ∞

−∞dv⊥v⊥

∣∣∣∣v2zkz|A|2δ

(ωx

R− kzvz

)

e−mv2

2T

∣∣∣∣

∞

−∞−

∫ ∞

−∞dvz2vzkz|A|2δ

(ωx

R− kzvz

)

e−mv2

2T

=

∫ ∞

−∞dv⊥v⊥e−

mv2

2T ×[

(−2)ωx

kzR|A|2e−

x2

2∆2

]

= − 2

(ωx

kzR

)

e−x2

2∆2

(n2n2

x

4c2

)n−11

[(n − 1)!]2

∣∣∣∣E− +

nx

nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

×∫ ∞

0v2n+1⊥ e−

mv2⊥

2T dv⊥︸︷︷︸

12(

2Tm )

n+1n!

= − 2ωx

kzR

(n2N2

x

4c2

)n−11

[(n − 1)!]21

2

(2T

m

)n+1

n(n − 1)! ×∣∣∣∣E− +

Nx

Nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

e−x2

2∆2

= − ωx

kzR

(n2N2

x

4c2

)n−1n

(n − 1)!

(2T

m

)n+1 ∣∣∣∣E− +

Nx

Nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

e−x2

2∆2

∴d2W‖dtdV

= −mπǫ0ω

2p

4ω

1

T

[ m

2πT

] 32

(

− ωx

kzR

) (n2N2

x

4c2

)n−1n

(n − 1)

(2T

m

)n+1 ∣∣∣∣E− +

Nx

Nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

e−x2

2∆2

=mπ

T

ǫ0ω2p

4ω

[ m

2πT

] 32 ωx

kzR

n

(n − 1)!

(n2N2

x

4c2

)n−1 (2T

m

)n+1 ∣∣∣∣E− +

Nx

Nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

e−x2

2∆2

Thus,

d2W

dtdV=

d2W⊥dtdV

+d2W‖dtdV

=d2W⊥dtdV

[

1 +x

R

]

=mπ

T

ǫ0ω2p

4ω

[ m

2πT

] 32 n2|Ω|

(n − 1)!

(n2N2

x

4c2

)n−1 (2T

m

)n+1

×∣∣∣∣E− +

Nx

Nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2 [

1 +x

R

]

e−x2

2∆2

6.2.1 Higher Harmonics (n ≥ 2)

Let us calculate

∣∣∣E− + Nx

Nz

xR

ωn|Ω|Ez

∣∣∣

2

The nonzero components of the dielectric tensor in a cold electron plasma

103

are

ǫxx = ǫyy = 1 −ω2

pe

ω2 − Ω2e

= S

ǫxy = −ǫyx = iω2

pe

ω2 − Ω2e

Ωe

ω= −iD

and

ǫzz = 1 −ω2

pe

ω= P

From the dispersion relation

~N × ( ~N × ~E)+↔ǫ · ~E = 0 ( ~N =

~kc

ω)

−→

S − N2z −iD NxNz

iD S − N2 DNxNz D P − N2

x

Ex

Ey

Ez

=

000

(S − N2z )Ex − iDEy + NxNzEz = 0 (1)

iDEx + (S − N2)Ey = 0 (2)

NxnzEx + (P − N2x)Ez = 0 (3)

From equation (2)

i(iDEx) + i(S − N2)Ey = 0

−DEx + (S − N2)iEy = 0

DEx − (S − N2)iEy = 0 −→ Ex =S − N2

DiEy

From equation (3)

Ez =−NxNzEx

P − N2x

= − NxNz

P − N2x

S − N2

DiEy

E− +Nx

Nz

x

R

ω

n|Ω|Ez = Ex − iEy +Nx

Nz

x

R

ω

n|Ω|Ez

=S − N2

DiEy − iEy −

Nx

Nz

x

R

ω

n|Ω|NxNz

P − N2x

S − D2

DiEy

=

(S − N2

D− 1

)

iEy −x

RN2

x

ω

n|Ω|(P − N2x)

S − N2

DiEy

=

(S − D − N2

D− x

RN2

x

ω

n|Ω|(P − N2x)

S − N2

D

)

iEy

⋍S − D − N2

DiEy

104

∴

∣∣∣∣E− +

Nx

Nz

x

R

ω

n|Ω|Ez

∣∣∣∣

2

=(S − D − N2)2

D2|Ey|2

But,

Sx =1

4µ0Re

(

~E × ~B∗ +1

2~E∗ · ∂

↔ǫ

∂ ~N· ~E

)

x

→ Ref : T.H.Stix, page 74 (Eqs.(18) & (19))

≃ 1

4µ0cNx|Ey|2

D2(P − N2x)2 + (S − N2)2PN2

z

D2(P − N2x)2

Detailed calculation steps are seen in Appendix 1.

∗ For fundamental harmonic heating,↔ǫ is the dielectric tensor in hot plas-

mas. But for higher harmonic heating,↔ǫ is the cold plasma dielectric tensor.

(see Appendix 2)

1

Sx

∂Sx

∂x= − 1

Sx

d2W

dtdV= − 1

Sx

(

1 +x

R

) d2W⊥dtdV

= −4µ0c1

Nx

1

|Ey|2D2(P − N2

x)2

D2(P − N2x)2 + (S − N2)2PN2

z

mπ

kzT

ǫ0ω2p

4ω

[ m

2πT

] 32

×n2|Ω|(

n2N2x

4c2

)n−11

(n − 1)!

(2T

m

)n

|Ey|2(S − D − N2)2

D2e−

x2

2∆2

(

1 +x

R

)

= −4

c

mπ

kzT

ω2p

4ω

[ m

2πT

] 32 n2|Ω|

Nx

(n2N2

x

4c2

)n−11

(n − 1)!

(2T

m

)n

× (S − D − N2)2(P − N2x)2

D2(P − N2x)2 + (S − N2)2PN2

z

e−x2

2∆2

(

1 +x

R

)

= Im(−2kx)

Let η(n) =∫ ∞−∞ Im(2kx)dx : Optical depth

105

∴ η(n) =4

c

mπ

kzT

ω2p

4ω

[ m

2πT

] 32 n2|Ω|

Nx

(n2N2

x

4c2

)n−11

(n − 1)!

(2T

m

)n+1

×[

(S − D − N2)2(P − N2x)2

D2(P − N2x)2 + (S − N2)2PN2

z

] ∫ ∞

−∞e−

x2

2∆2

(

1 +x

R

)

dx

︸︷︷︸√

2∆2π=∆√

2π=NzR√

2πT

mc2

=4

c

mπ

kzT

ω2p

4ω

[ m

2πT

] 32 n2|Ω|

Nx

(n2N2

x

4c2

)n−11

(n − 1)!

(2T

m

)n+1

NzR

√

2πT

mc2

×[

(S − D − N2)2(P − N2x)2

D2(P − N2x)2 + (S − N2)2PN2

z

]

=4

c

mπ

kzTc

ω2p

4ω2

[ m

2πT

] 32

√

2πT

mc2ω

n2|Ω|N2

x

Nx

(n2N2

x

4c2

)n−11

(n − 1)!

(2T

m

)n

NzR

×[

(S − D − N2)2(P − N2x)2

D2(P − N2x)2 + (S − N2)2PN2

z

]

= πm

kzTcα

m

2πT

√m

2πT

√

2πT

m

1

cω

n2|Ω|N2

x

Nx

(n2N2

x

4c2

)n−2 (n2N2

x

4c2

)

× 1

(n − 1)!

(2T

m

)n−2 (2T

m

)2 (2T

m

)kzc

ωR

[(S − D − N2)2(P − N2

x)2

D2(P − N2x)2 + (S − N2)2PN2

z

]

= πα1

kzc2

1

2πn3ω

n|Ω|c2

Nx1

(n − 1)!

(n2N2

xT

2mc2

)n−2 (2T

m

)kzc

ωR

×[

(S − D − N2)2(P − N2x)2

D2(P − N2x)2 + (S − N2)2PN2

z

]

= 2παT

mc2

n3

(n − 1)!

(n2N2

xT

2mc2

)n−2R

λNx

[(S − D − N2)2(P − N2

x)2

D2(P − N2x)2 + (S − N2)2PN2

z

]

Thus the fractional absorbed energy by particles by only ray passing throughthe resonance region is given by

Wabs = fW0 (W0 is initial energy) = W0 − W0e−η

∴ f = 1 − e−η

= 1 − exp

[

−2παR

λ

T

mc2

(n2N2

xT

2mc2

)n−2

Nx

[(S − D − N2)2(P − N2

x)2

D2(P − N2x)2 + (S − N2)2PN2

z

]]

Where, α =ω2

p

ω2 , n is the harmonic number (n ≥ 2), T is the electrontemperature, R is the scale length of the tokamak, and λ is the wavelength

106

in free space. S, P , D is evaluated at ω = n|Ω|. The perpendicular indexof refraction is

N2x =

−B ± (B2 − 4AC)12

2A

(+ : O-mode

− : X-mode

)

with

A = S

B = −(S + P )(S − N2z ) + D2

C = P [(S − N2z )2 − D2]

107

6.2.2 Fundamental Harmonic (n = 1)

• The perpendicular heating rate per unit volume

d2W⊥dtdV

=mπ

KzT

ǫoω2p

4ω[

m

2πT]32 n2|Ω|(n

2N2x

4c2)n−1

1

(n − 1)!(2T

m)n+1 × |E− +

Nx

Nz

x

R

ω

n|Ω|Ez|2ex2

2∆2

For n = 1, (∆ =√

k2zR2Tmω2 )

d2W⊥dtdV

=mπ

kzT

ǫoω2p

4ω[

m

2πT]32 |Ω|(2T

m)2 × |E− +

Nx

Nz

x

R

ω

|Ω|Ez|2e−x2

2∆2

=mπ

kzT

ǫo

4ω[

m

2πT]32 (

2T

m)2ω2

p|Ω||E− +Nx

Nz

x

R

ω

|Ω|Ez|2e−x2

2∆2

= ǫo

√

m2π2

k2zT

2

1

16ω2(

m

2πT)3

16T 4

m4ω2

p|Ω||E− +Nx

Nz

x

R

ω

|Ω|Ez|2e−x2

2∆2

= ǫo

√

mω2

k2zω

4T(1

4)

1√2π

ω2pΩ|E− +

Nx

Nz

x

R

ω

|Ω|Ez|2e−x2

2∆2

=ǫoω

2p

2

|Ω|ω2

R

∆

e−x2

2∆2

√2π

|E− +Nx

Nz

x

R

ω

|Ω|Ez|2

• The parallel heating rate per unit volume

d2W‖dtdV

=mπ

kzT

ǫoω2p

4ω[

m

2πT]32

n2|Ω|(n − 1)!

(n2N2

x

4c2)n−1(

2T

m)n+1|E− +

Nx

Nz

x

R

ω

n|Ω|Ez|2 ×x

Re−

x2

2∆2

For n = 1, (∆ =√

k2zR2Tmω2 )

similarly

d2W‖dtdV

=ǫoω

2p

2

|Ω|ω2

x

∆

e−x2

2∆2

√2π

|E− +Nx

Nz

x

R

ω

|Ω|Ez|2

So, |E− +Nx

Nz

x

R

ω

|Ω|Ez|2 =?

From the Hot Plasma dispersion relation,

ǫxx − N2z ǫxy ǫxz + NxNz

−ǫxy ǫxx − N2 ǫyz

ǫxz + NxNz −ǫyz ǫzz − N2x

Ex

Ey

Ez

= 0

108

(ǫxx − N2z )Ex + ǫxyEy + (ǫxz + NxNz)Ez = 0

−ǫxyEx + (ǫxx − N2)Ey + ǫyzEz = 0

(ǫxz + NxNz)Ex − ǫyzEy + (ǫzz − N2x)Ez = 0

⇒ Ez = −(ǫxz + NxNz)E− + iNzNzEy

ǫzz − N2x

E− = Ex − iEy

= −i(ǫxx − iǫxy − N2

z )Ey − i(ǫxz + NxNz)Ez

ǫxx − N2z

Note ǫyz = ǫzy = iǫxz for the first-order approximation

E− +Nx

Nz

x

R

ω

|Ω|Ez = −i(ǫxx − iǫxy − N2

z )Ey − i(ǫxz + NxNz)Ez

ǫxx − N2z

−Nx

Nz

x

R

(ǫxz + NxNz)E− + iNxNzEy

ǫzz − N2x

= −iEy

ǫxx − iǫxy − N2

z + N2x

xR

ǫxx−N2z

ǫzz−N2x

ǫxx − N2z

−(ǫxz + NxNz)Ez

ǫxx − N2z

− Nx

Nz

x

R

ǫxz + NxNz

ǫzz − N2x

E−

* E− is smaller than Ey, Ez, by a factor of (∆R )

ǫxx − iǫxy ≃ 1 −ω2

p

2ω2(

ω

ω + Ω− R√

2∆Z) −

ω2p

2ω2(

ω

ω + Ω+

R√2∆

Z)

≃ 1 −ω2

p

ω2

ω

ω + Ω∼ order of 1

ǫxx ∼ order ofR

∆

E− +Nx

Nz

x

REz ≃ −iEy

(1

ǫxx − N2z

) (

ǫxx − iǫxy − N2z + N2

x

x

R

ǫxx − N2z

ǫzz − N2x

)

+ǫxz + NxNz

ǫxx − N2z

(ǫxz + NxNz)E− + iNxNzEy

ǫzz − N2x

≃ −iEy1

ǫxx − N2z

[

ǫxx − iǫxy − N2z − NxNz(ǫxz + NxNz)

ǫzz − N2x

+ N2x

x

R

ǫxx − N2z

ǫzz − N2x

]

109

ǫxx = 1 − α4 + α

2R√2∆

Z

ǫxy = − iα2 (1

2 + R√2∆

Z) = − iα4 − iαR

2√

2∆Z

ǫzz = 1 − α(1 + α4

N2x

N2z

xRZ ′)

= 1 − α[1 + α4

N2x

N2z

xR(−2)(1 + x√

2∆Z)]

= 1 − α + α2

4N2

x

N2z

xR(2 + 2x

2√

2∆Z)

ǫxz = −α4

Nx

NzZ ′ = α

4Nx

Nz(2)(1 + x√

2∆Z)

Where α =ω2

p

Ω2 ≃ ω2p

ω2

E− +Nx

Nz

x

REz ≃ −iEy(

1

ǫxx − N2z

)[ǫxx − iǫxy − N2z +

−NxNzǫxz + N2x

xRǫxx − N2

xN2z (1 + x

R)

ǫzz − N2x

]

−NxNzǫxz + N2x

x

Rǫxx = N2

x

x

R(1 − α

4+

α

2

R√2∆

Z) − NxNzα

2

Nx

Nz(1 +

x√2∆

Z)

= N2x

x

R− α

4N2

x

x

R− N2

x

α

2≃ −N2

x

α

2

ǫxx − iǫxy − N2z = 1 − α

4+

α

2

R√2∆

Z + i(iα

4+

iαR

2√

2∆Z) − N2

z

= 1 − α

4− α

4− N2

z

= 1 − α

2N2

z

ǫzz − N2x = 1 − α +

α2

4

N2x

N2z

x

R(2 +

2x√2∆

Z) − N2x

≃ 1 − α − N2x

ǫxx − N2z = 1 − α

4+

α

2

R√2∆

Z − N2z

≃ 1

2√

2

αR

∆Z (∵ 1 − α

4− N2

z ≪ αR

2√

2∆Z)

∴ E− +x

R

Nx

NzEz = −iEy

2√

2∆

αRZ

(1 − α2 − N2

z )(1 − α − N2x) − α

2 N2x − N2

xN2z

1 − α − N2x

= −iEy2√

2∆

αRZ

(1 − α2 − N2

z )(1 − α) − N2x + α

2 N2x + N2

xN2z − α

2 N2x − N2

xN2z

1 − α − N2x

= −iEy2√

2∆

αRZ

(1 − α2 − N2

z )(1 − α) − N2x

1 − α − N2x

∣∣∣∣E− +

x

R

Nx

NzEz

∣∣∣∣

2

=8∆2

α2R2

((1 − α

2 − N2z )(1 − α) − N2

x

1 − α − N2x

)2 |Ey|2|z|2

110

Thus, the perpendicular energy absorption

d2W⊥dtdV

≃ǫ0ω

2p

2

|Ω|ω2

R

∆

1√2π

8∆2

α2R2

((1 − α

2 − N2z )(1 − α) − N2

x

1 − α − N2x

)2 |Ey|2|Z|2 e−

x2

2∆2

≃ ǫ02√

2ω

πα

∆

R|Ey|2

((1 − α

2 − N2z )(1 − α) − N2

x

1 − α − N2x

)2

(√

πe−

x2

2∆2

|Z|2 )

Through Poynting’s theorem

~∇ · ~S +d2W⊥dtdV

= 0

~S ∼ e2i(kxx−ωt)

∂Sx

∂x= 2ikxSx = −d2W⊥

dtdV

⇒ 2ikx = − 1

Sx

d2W⊥dtdV

⇒ Real

since kx = kR + ikI = Re(kx) + iIm(kx)

⇒ 2Im(kx) =1

Sx

d2W⊥dtdV

But,

Sx =1

4µ0Re( ~E × ~B∗ +

1

2~E∗ · ∂

∂ ~N

↔ǫ · ~E)x

where↔ǫ is the dielectric tensor in Hot Plasma.

Sx =1

4µ0Re[(EyB

∗z − EzB

∗y) +

1

2(E∗

xXxx + E∗yXyx + E∗

zXzx)]

↔X=

∂

∂~n

↔ǫ · ~E = x

∂

∂Nx[xU+yV +zW ]+y

∂

∂Ny[xU+yV +zW ]+z

∂

∂Nz[xU+yV +zW ]

where

U = ǫxxEx + ǫxyEy + ǫxzEz

V = ǫyxEx + ǫyyEy + ǫyzEz

W = ǫzxEx + ǫzyEy + ǫzzEz

since Ny = 0

↔X= x

∂

∂Nx[xU + yV + zW ] + z

∂

∂Nz[xU + yV + zW ]

For Sx, we just calculate below components:

111

Xxx =∂

∂NxU

Xyx = 0

Xzx =∂

∂NxU

U = ǫxxEx + ǫxyEy + ǫxzEz

= (1 − α

4+

α

2

R√2∆

Z)Ex − i(α

4+

αR

2√

2∆Z)Ey +

α

4

Nx

Nz2(1 +

x√2∆

Z)Ez

= (1 − α

4)Ex − i

α

REy +

α

R2√

2∆Z(Ex − iEy) +α

2

Nx

Nz(1 +

x√2∆

Z)Ez

(put, Ex − iEy = E−)

= (1 − α

4)Ex − i

α

REy +

αR

2√

2∆Z(E− +

x

R

Nx

NzEz) +

α

2

Nx

NzEz

= (1 − α

4)Ex − i

α

REy +

α

2

Nx

NzEz +

αR

2√

2∆Z(−iEy)

2√

2∆

αR× (1 − α

2 − N2z )(1 − α) − N2

x

1 − α − N2x

=

[

(1 − α

4)Ex − i

α

REy +

α

2

Nx

NzEz

]

− iEy

[(1 − α

2 − N2z )(1 − α) − N2

x

1 − α − N2x

]

But,

Ez = −(ǫxz + NxNz)E− + iNxNzEy

ǫzz − N2z

≃ − iNxNzEy

1 − α − N2x

∴α

2

Nx

NzEz = −iEy

αN2x

2(1 − α − N2x)

and,E− = Ex − iEy ≃ 0

∴ Ex ≃ iEy

The first term in the square bracket in above equation becomes

(1 − α

4)Ex − i

α

4Ey +

α

2

Nx

NzEz = (1 − α

4)(iEy) − iEy

α

4− iEy

αN2x

2(1 − α − N2x)

= iEy(1 − α

4− α

4− αN2

x

2(1 − α − N2x)

)

= iEy

[

1 − α

2− αN2

x

2(1 − α − N2x)

]

Thus,

U = (+iEy)

[

1 − α

2− αN2

x

2(1 − α − N2x)

−(1 − α

2 − N2z

)(1 − α) − N2

x

1 − α − N2x

]

112

Xxx =∂U

∂Nx= iEy

[

−α

2

2Nx(1 − α − N2x) − N2

x(−2Nx)

(1 − α − N2x)2

−−2Nx(1 − α − N2x) −

[(1 − α

2 − N2z

)(1 − α) − N2

x

](−2Nx)

(1 − α − N2x)2

]

= iEy

[

−α

2

2Nx(1 − α)

(1 − α − N2x)2

+−2Nx(1 − α) − 2Nx

(1 − α

2 − N2z

)(1 − α)

(1 − α − N2x)2

]

= iEyNx

[

−α(1 − α) + 2(1 − α) − 2(1 − α

2 − N2z

)(1 − α)

(1 − α − N2x)2

]

= iEyNx2(1 − α)N2

z

(1 − α − N2x)2

Xzx =∂U

∂Nz= iEy

(2Nz(1 − α)

1 − α − N2x

)

E∗xXxx ≃ (iEy)

∗Xxx

= −iE∗y(iEy)Nx

2(1 − α)N2z

(1 − α − N2x)2

= |Ey|2Nx2(1 − α)N2

z

(1 − α − N2x)2

E∗zXzx ≃ −NxNz(iEy)

∗

1 − α − N2x

· iEy2Nz(1 − α)

1 − α − N2x

= −NxNz(−iE∗

y)

1 − α − N2x

· iEy2Nz(1 − α)

1 − α − N2x

= −|Ey|2Nx2N2

z (1 − α)

(1 − α − N2x)2

∴ E∗xXxx + E∗

zXzx = 0

And,

EyB∗z − EzB

∗y

=1

c[Ey(NxE∗

y) − Ez(NzE∗x − NxE∗

z )]

=1

cNx|Ey|2 −

1

c(−iEy)

NxNz

1 − α − N2x

(

Nz(−iE∗y) + Nx

−iE∗yNxNz

1 − α − N2x

)

=1

cNx|Ey|2 +

1

c(iEy)(−iE∗

y)NxNz

1 − α − N2x

(

Nz +N2

xNz

1 − α − N2x

)

=Nx

c|Ey|2

[

1 +Nz

1 − α − N2x

(

Nz +N2

xNz

1 − α − N2x

)]

=Nx

c|Ey|2

[

1 +Nz

1 − α − N2x

(1 − α)Nz − N2xNz + N2

xNz

1 − α − N2x

]

=Nx

c|Ey|2

[

1 +(1 − α)N2

z

(1 − α − N2x)2

]

113

Thus,

Sx =1

4µ0cNx|Ey|2

[

1 +(1 − α)N2

z

(1 − α − N2x)

]

2 Im(kx) =1

Sx

d2W⊥dt dV

= 4µ0c1

Nx|Ey|2

[(1 − α

2 − N2z

)(1 − α) − N2

x

]2

(1 − α − N2x)2 + (1 − α)N2

z

×ǫ02√

2ω

πα

√2∆

R|Ey|2 ·

√π

e−x2/2∆2

|Z|2

= −8ω

cπ

√2∆

R

1

αNxIm

(1

Z

) [(1 − α

2 − N2z

)(1 − α) − N2

x

]2

(1 − α − N2x)2 + (1 − α)N2

z

where

Im

(1

Z

)

=

√πe−x2/2∆2

|Z|2 .

η =

∫ ∞

−∞Im(2kx) dx

f = 1 − e−η =Wabs

W0: Fraction of absorbed energy to the input wave energy.

Calculation of

∫ ∞

−∞Im

(1

Z

)

dx =

∫ ∞

−∞

√πe−x2/2∆2

∣∣∣Z

(x√2∆

)∣∣∣

2 dx

since x√2∆

is real,

Z

(x√2∆

)

= i√

πe−x2/2∆2 − 2x√2∆

Y

(x√2∆

)

Im

(1

Z

)

= Im

[ReZ − iImZ

|Z|2]

= − ImZ

|Z|2

=−√

πe−x2/2∆2

|Z|2

For x√2∆

≫ 1

|Z|2 ≃ 1(

x√2∆

) from Asymptotic Expansion

Thus,

Im

(1

Z

)

= −√

π

2∆2x2e−x2/2∆2

114

∫ ∞

−∞Im

(1

Z

)

dx = −√

π

2∆2

∫ ∞

−∞x2e−x2/2∆2

dx

= −√

π

2∆2

1

2

√π

(1

2∆2

)3

= −π∆√2

Thus,

η = −8ω

cπ

√2∆

R

1

αNx

(

−π∆√2

) [(1 − α

2 − N2z

)(1 − α) − N2

x

]2

(1 − α − N2x)2 + (1 − α)N2

z

=8

cπ2π

c

λ· π∆2

RαNx

[(1 − α

2 − N2z

)(1 − α) − N2

x

]2

(1 − α − N2x)2 + (1 − α)N2

z(

∆2 =k2

zR2T

mω2

)

=8

cπ2π

c

λ· π

RαNx

k2zR

2T

mω2

[(1 − α

2 − N2z

)(1 − α) − N2

x

]2

(1 − α − N2x)2 + (1 − α)N2

z

=16π

α· R

λ

N2z

Nx

T

mc2

[(1 − α

2 − N2z

)(1 − α) − N2

x

]2

(1 − α − N2x)2 + (1 − α)N2

z

6.2.3 O-mode & X-mode Heating

A. Fundamental Harmonic (n = 1)

η =16π

α

R

λ

N2z

Nx

T

mc2

[(1 − α

2 − N2z

)(1 − α) − N2

x

]2

(1 − α − N2x)2 + (1 − α)N2

z

where N2x = 1

2

(

3 − N2z − 2α ±

√

(1 + N2z )2 − 4αN2

z

)

from section 6.1

a© X-mode

N2x =

1

2

(

3 − N2z − 2α +

√

(1 + N2z )2 − 4αN2

z

)

b© O-mode

N2x =

1

2

(

3 − N2z − 2α −

√

(1 + N2z )2 − 4αN2

z

)

a. “Near-normal incidence” (Nz is small)

N2x =

1

2

(

3 − N2z − 2α ±

√

1 + 2N2z + N4

z − 4αN2z

)

≃ 1

2

(

3 − N2z − 2α ±

√

1 + (2 − 4α)N2z

)

≃ 1

2

(

3 − N2z − 2α ±

(

1 + (1 − 2α)N2z − 1

8(1 − 2α)2N4

z

))

≃ 1

2

(3 − N2

z − 2α ±(1 + (1 − 2α)N2

z

))

115

i) X-mode

N2x ≃ 1

2

[3 − N2

z − 2α +(1 + (1 − 2α)N2

z

)]

=1

2

[3 − N2

z − 2α + 1 + N2z − 2αN2

z

]

=1

2

[4 − 2α

(1 + N2

z

)]

= 2 − α(1 + N2

z

)

≃ 2 − α

ηX =16π

α

R

λ

N2z√

2 − α

T

mc2

[(1 − α

2 − N2z

)(1 − α) − 2 + α

]2

[1 − α − 2 + α]2 + (1 − α)N2z

≃ 16π

α

R

λ

N2z√

2 − α

T

mc2

14 [(2 − α)(1 − α) − 2(2 − α)]2

1

= 4πR

λ

T

mc2· N2

z

(2 − α)3/2(1 + α)2

α

ii) O-mode

N2x ≃ 1

2

[3 − N2

z − 2α −(1 + (1 − 2α)N2

z

)]

=1

2

[3 − N2

z − 2α − 1 − (1 − 2α)N2z

]

=1

2

[2 − 2α − 2N2

z + 2αN2z

]

= (1 − α) − N2z (1 − α)

= (1 − α)(1 − N2z )

≃ 1 − α

ηO =16π

α

R

λ

N2z√

1 − α

T

mc2

[(1 − α

2 − N2z

)(1 − α) − 1 + α

]2

[1 − α − 1 + α]2 + (1 − α)N2z

≃ 16π

α

R

λ

N2z√

1 − α

T

mc2

14 [(2 − α − 2N2

z )(1 − α) − 2(1 − α)]2

(1 − α)N2z

=16π

α

R

λ

N2z√

1 − α

T

mc2

1

4

[(1 − α)(2 − α − 2N2z − 2)]2

(1 − α)N2z

=4π

α

R

λ

N2z√

1 − α

T

mc2

(1 − α)2(α + 2N2z )2

(1 − α)N2z

≃ 4πR

λ

T

mc2α(1 − α)1/2

116

or,

N2x ≃ (1 − α)(1 − N2

z )

N2x =

√1 − α

√

1 − N2z

and,1

Nx=

1√

(1 − α)(1 − N2z )

=1

√

1 − α − (1 − α)N2z

≃ 1√1 − α

We use√

1 − α√

1 − N2z (keep Nz term) for N2

x . For 1/Nx, we letNz → 0.

Therefore,[(

1 − α2 − N2

z

)(1 − α) − (1 − α)(1 − N2

z )]2

[1 − α − (1 − α)(1 − N2z )]2 + (1 − α)N2

z

=14

[(2 − α − 2N2

z )(1 − α) − 2(1 − α)(1 − N2z )

]2

(1 − α)N2z ((1 − α)N2

z + 1)

=1

4

[(1 − α)(2 − α − 2N2z − 2 + 2N2

z )]2

(1 − α)N2z ((1 − α)N2

z + 1)

=1

4

[(1 − α)(2 − α − 2N2z − 2 + 2N2

z )]2

(1 − α)N2z (1 + (1 − α)N2

z )

=1

4

(1 − α)2(−α)2

(1 − α)N2z (1 + N2

z (1 − α))

Thus,

ηO ≃ 4πR

λ

T

mc2

α(1 − α)1/2

1 + Nz2(1 − α)

b. “Normal-incidence” (Nz = 0)

i) O-mode

ηO = 4πR

λ

T

mc2

α(1 − α)1/2

1 + Nz2(1 − α)

∴ ηO(90) ≃ 4πR

λ

T

mc2α(1 − α)1/2

ii) X-mode

ηX = 4πR

λ

T

mc2Nz

2 (2 − α)3/2(1 + α)2

α∴ ηX(90) = 0

⇒ “Expanding to the next-order in the temperature is necessary”

ηX(90) ≃ π

2

R

λ

T 2

m2c4α(2 − α)3/2

117

B. Second Harmonic (n = 2)

η = 2πR

λ

T

mc2(Nx

2n2T

2mc2)n−2 n3

(n − 1)!αNx

(S − D − N2)2(P − N2x)2

D2(P − N2x)2 + PN2

z (S − N2)2

η(n = 2) = 2πR

λ

T

mc28αNx

(S − D − N2)2(P − N2x)2

D2(P − N2x)2 + PN2

z (S − N2)2

where

S = 1 − ωp2

ω2 − Ω2= 1 − ωp

2

ω2(1 − Ω2

ω2 )= 1 − α

1 − 14

= 1 − 4

3α

P = 1 − ωp2

ω2= 1 − α

D = − ωp2

ω2 − Ω2

Ω

ω= −4

3α

1

2= −2

3α

Nx2 =

1

2A[−B ±

√

B2 − 4AC]

A = S

B = −(S + P )(S − Nz2) + D2

C = P [(S − Nz2)2 − D2]

(+) sign : O-mode(– ) sign : X-mode

B2 − 4AC = [(S + P )(S − Nz2) − D2]2 − 4SP [(S − Nz

2)2 − D2]

= [(S − P )(S − Nz2) − D2]2

• O-mode:

Nx2 =

1

2S[(S + P )(S − Nz

2) − D2 + (S − P )(S − Nz2) − D2]

=1

2S[(S − Nz

2)(S + P + S − P ) − 2D2]

=1

S[S(S − Nz

2) − D2]

• X-mode:

Nx2 =

1

2S[(S + P )(S − Nz

2) − D2 − (S − P )(S − Nz2) + D2]

=1

S(S − Nz

2)P

♦ Normal incidence (Nz = 0)

118

• O-mode

N2x =

1

S(S2 − D2) =

1

S(S + D)(S − D)

= 31

3 − 4α(1 − 4

3α − 2

3α)(1 − 4

3α +

2

3α)

=1

3 − 4α(1 − 2α)(3 − 2α)

Nx =1

(3 − 4α)1/2(1 − 2α)1/2(3 − 2α)1/2

ηO(90, n = 2) = 2πR

λ

T

mc28αNx

(S − D − N2x)2

D2

S − D − N2x = (1 − 4

3α +

2

3α) − 1

3 − 4α(1 − 2α)(3 − 2α)

= (1 − 2

3α) − 1

3 − 4α(1 − 2α)(3 − 2α)

=1

3(3 − 2α) − (1 − 2α)(3 − 2α)

3 − 4α

=(3 − 2α)(3 − 4α)) − 3(1 − 2α)(3 − 2α)

3(3 − 4α)

=(3 − 2α)(3 − 4α − 3 + 6α)

3(3 − 4α)

=2

3α

3 − 2α

3 − 4α

ηO(90, n = 2) = 2πR

λ

T

mc28α

(1 − 2α)1/2(3 − 2α)5/2

(3 − 4α)5/2(−23α)2

4

9α

= 16πR

λ

T

mc2α

(1 − 2α)1/2(3 − 2α)5/2

(3 − 4α)5/2

• X-mode

N2x =

1

S(S − N2

z )P = P = 1 − α

Nx = (1 − α)1/2

ηX(90, n = 2) = 2πR

λ

T

mc28αNx

(S − D − N2x)2

D2

S − D − N2x = S − D − P

= 1 − 4

3α +

2

3α − 1 + α

=α

3

119

ηX(90, n = 2) = 16πR

λ

T

mc2α(1 − α)1/2 (α

3 )2

(−23α)2

= 4πR

λ

T

mc2α(1 − α)1/2

120

7 Calculation of ECR optical depth using Mathe-matica

121

Calculation of Optical Depth

for EC-wave Heating

Graphics`Arrow`;

Graphics`Colors`;

Graphics`FilledPlot`;

Graphics`Graphics`




R0 180;

a 50;

f 84;

w 2 Pi f;

fpe 90 Sqrt Ne0 1 rho^2 ;

fce n_ f n 1 a R0 rho ;

alpha fpe f ^2;

alphan n_ fce n f ^2;

lambda 0.3;

mc2 511;

Te0 10;

For Fundamental Harmonic

Nx1stX 0.5 3 Nz^2 2 alpha Sqrt 1 Nz^2 ^2 4 alpha Nz^2 ;

Nx1stO 0.5 3 Nz^2 2 alpha Sqrt 1 Nz^2 ^2 4 alpha Nz^2 ;

(+) sign: X-mode like, (--) sign: O-mode like

(Opposite sense to the signs of the solutions of

the cold plasma dispersion relation)etaFundO 16 Pi alpha R0 lambda Nz^2 Sqrt Nx1stO

Te0 mc2 1 alpha 2 Nz^2 1 alpha Nx1stO ^2

1 alpha Nx1stO ^2 1 alpha Nz^2 ;

etaFundX 16 Pi alpha R0 lambda Nz^2 Sqrt Nx1stX Te0 mc2

1 alpha 2 Nz^2 1 alpha Nx1stX ^2

1 alpha Nx1stX ^2 1 alpha Nz^2 ;

FundOmode1 Plot Log 10, etaFundO . Nz 0.1, rho 0 ,

Ne0, 0.0, 2 , PlotStyle Dashing 0.01, 0.01 , RGBColor 1, 0, 0

ECR_OptDepth.nb 1

122


Ne0, 0.0, 2 , PlotStyle Dashing 0.01, 0.01



Show FundOmode1, FundOmode2, FundOmode3, Graphics

Text "Nz 0.1", 0.2, 1.7 , Text "0.5", 0.4, 1.4 , Text "0.7", 0.6, 1 ,

Text "Fundamental Harmonic O mode ", 0.5, 2.6 , PlotRange 0, 1 , 2, 3 ,

FrameLabel "Electron Density at Resonance 1020m 3 ", "Log10 " , Frame True

0.2 0.4 0.6 0.8 1

Electron Density at Resonance 1020m 3

1

0

1

2

3

go

L0

1

Nz 0.10.5

0.7

Fundamental Harmonic O mode

Graphics

FundXmode1 Plot Log 10, etaFundX . Nz 0.1, rho 0 ,






ECR_OptDepth.nb 2

123

Show FundXmode1, FundXmode2, FundXmode3, Graphics

Text "Nz 0.1", 0.25, 0.9 , Text "0.5", 0.6, 2 , Text "0.7", 0.8, 3.1 ,

Text "Fundamental Harmonic X mode ", 1, 7 , PlotRange 0, 2 , 4, 8 ,


0.25 0.5 0.75 1 1.25 1.5 1.75 2


2

0

2

4

6

8

go

L0

1

Nz 0.1

0.5

0.7

Fundamental Harmonic X mode

Graphics

FundOmode4 Plot Log 10, etaFundO . Ne0 0.8, rho 0 ,

Nz, 0., 1 , PlotStyle RGBColor 1, 0, 0

FundXmode4 Plot Log 10, etaFundX . Ne0 0.8, rho 0 ,


ECR_OptDepth.nb 3

124

Show FundOmode4, FundXmode4,

Graphics Text "O mode", 0.5, 1.2 , Text "X mode", 0.6, 2.2 ,

Text "Fundamental Harmonic", 0.3, 3.5 , Text "Ne0 8 x 1019 m 3", 0.3, 3 ,

PlotRange 0, 1 , 1, 4 , FrameLabel "Nz", "Log10 " , Frame True

0.2 0.4 0.6 0.8 1

Nz

0

1

2

3

4

go

L0

1

O mode

X mode

Fundamental Harmonic

Ne0 8 x 1019 m 3

Graphics

For Higher Harmonics

Second Harmonic

SS n_ 1 alpha 1 alphan n ;

PP 1 alpha;

DD n_ alpha Sqrt alphan n 1 alphan n ;

AA SS n ;

BB SS n PP SS n Nz^2 DD n ^2;

CC PP SS n Nz^2 ^2 DD n ^2 ;

Disc Sqrt BB^2 4 AA CC ;

NxO BB Disc 2 AA ;

NxX BB Disc 2 AA ;

NO2 Nz^2 NxO;

NX2 Nz^2 NxX;

ECR_OptDepth.nb 4

125

etaHarmO n_ 2 Pi R0 lambda Te0 mc2 NxO n^2 Te0 2 mc2 ^ n 2

n^3 Factorial n 1 alpha Sqrt NxO SS n DD n NO2 ^2 PP NxO ^2

DD n ^2 PP NxO ^2 PP Nz^2 SS n NO2 ^2 ;

etaHarmX n_ 2 Pi R0 lambda Te0 mc2 NxX n^2 Te0 2 mc2 ^ n 2

n^3 Factorial n 1 alpha Sqrt NxX SS n DD n NX2 ^2 PP NxX ^2

DD n ^2 PP NxX ^2 PP Nz^2 SS n NX2 ^2 ;

SecondOmode1 Plot Log 10, etaHarmO 2 . Nz 0.1, rho 0 ,






Show SecondOmode1, SecondOmode2, SecondOmode3, Graphics

Text "Nz 0.1", 0.7, 0.8 , Text "0.5", 0.4, 0.6 , Text "0.7", 0.4, 0.8 ,

Text "2nd Harmonic O mode ", 0.5, 1.6 , PlotRange 0, 1 , 4, 2 ,


0.2 0.4 0.6 0.8 1


3

2

1

0

1

2

go

L0

1 Nz 0.1

0.5

0.7

2nd Harmonic O mode

Graphics

SecondXmode1 Plot Log 10, etaHarmX 2 . Nz 0.1, rho 0 ,






ECR_OptDepth.nb 5

126

Show SecondXmode1, SecondXmode2, SecondXmode3, Graphics

Text "Nz 0.1", 0.4, 2.8 , Text "0.5", 0.3, 1.6 , Text "0.7", 0.2, 1.5 ,

Text "2nd Harmonic X mode ", 0.25, 3.5 , PlotRange 0, 0.5 , 3, 4 ,


0.1 0.2 0.3 0.4 0.5


2

1

0

1

2

3

4g

oL

01

Nz 0.1

0.50.7

2nd Harmonic X mode

Graphics

SecondOmode4 Plot Log 10, etaHarmO 2 . Ne0 0.8, rho 0 ,


SecondXmode4 Plot Log 10, etaHarmX 2 . Ne0 0.4, rho 0 ,

Nz, 0.0, 1 , PlotStyle RGBColor 0, 0, 1

ECR_OptDepth.nb 6

127

Show SecondOmode4, Graphics Text "O mode", 0.5, 1.5 ,

Text "Second Harmonic", 0.35, 1.5 , Text "Ne0 8 x 1019 m 3", 0.35, 0.8 ,

PlotRange 0, 0.7 , 6, 2 , FrameLabel "Nz", "Log10 " , Frame True

0.1 0.2 0.3 0.4 0.5 0.6

Nz

5

4

3

2

1

0

1

2

go

L0

1

O mode

Second Harmonic

Ne0 8 x 1019 m 3

Graphics

Thrid Harmonic

ThirdOmode1 Plot Log 10, etaHarmO 3 . Nz 0.1, rho 0 ,






ECR_OptDepth.nb 7

128

Show ThirdOmode1, ThirdOmode2, ThirdOmode3, Graphics

Text "Nz 0.1", 0.7, 0.8 , Text "0.5", 0.65, 1.8 , Text "0.7", 0.3, 2 ,

Text "3rd Harmonic O mode ", 0.5, 2.5 , PlotRange 0, 1 , 4, 3 ,


0.2 0.4 0.6 0.8 1


3

2

1

0

1

2

3

go

L0

1

Nz 0.1

0.50.7

3rd Harmonic O mode

Graphics

ThirdXmode1 Plot Log 10, etaHarmX 3 . Nz 0.1, rho 0 ,






ECR_OptDepth.nb 8

129

Show ThirdXmode1, ThirdXmode2, ThirdXmode3, Graphics

Text "Nz 0.1", 0.5, 1.4 , Text "0.5", 0.35, 0.5 , Text "0.7", 0.15, 0.3 ,

Text "3rd Harmonic X mode ", 0.35, 2.5 , PlotRange 0, 0.7 , 3, 3 ,


0.1 0.2 0.3 0.4 0.5 0.6


2

1

0

1

2

3

go

L0

1

Nz 0.1

0.50.7

3rd Harmonic X mode

Graphics

ECR_OptDepth.nb 9

130

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.


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

143


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

9 Ray Tracing in Inhomogeneous Media

- On the basis of refraction and reflection- & mode-conversion, absorption∗ A magnetized plasma can usually support two or more modes at the samefrequency.∗ When the wave approaches a critical layer at which the refractive indexbecomes infinity,the wave may be reflected, transmitted, absorbed, and/or converted into acompanion mode.

- Geometric opticsConsidering a lossless dispersion relation

D(~r,~k, ω, t) = 0

in which D is slowly varying function of ~r and t

The set of equations of Hamiltonian form:

d~r

dτ=

∂D

∂~k−− 1©

d~k

dτ= −∂D

∂~r−− 2©

dt

dτ= −∂D

∂ω−− 3©

dω

dτ=

∂D

∂t−− 4©

The quantity τ is a measure of distance along the trajectory combining1© & 3©

d~r

dt= − ∂D/∂~k

∂D/∂ω=

∂ω

∂~k= ~vg

This is the group velocity ~vg

Combining 2© & 3©

d~k

dt=

∂D/∂~r

∂D/∂ω

Combining 4© & 3©dωdt = − ∂D/∂t

∂D/∂ω

Thus the wave evolves in the configuration and k-space according to

d~r

dt= − ∂D/∂~k

∂D/∂ω

d~k

dt=

∂D/∂~r

∂D/∂ω

147

dω

dt= − ∂D/∂t

∂D/∂ω

Taking the dispersion relation in the form ω = ω(~r,~k, t)

d~r

dt=

∂ω

∂~k

d~k

dt= −∂ω

∂~rdω

dt= −∂ω

∂t

∗ Excellent review article I. B. Bernstein, “Geometric Optics in Space-andTime-Varying Plasmas,”Phys. Fluids 18, 320(1975).

9.1 Electric and Magnetic fields of E-M waves in a Tokamakwith a cold Plasma

Define (Ex, iEy, Ez) = (Ex, IEy, Ez) ≡ (EX, IEY, EZ)

~∇× ~E = −∂ ~B

∂t

→ i~k × ~E = +iω ~B

→~k

ω× ~E = ~B ⇒ 1

c

~kc

ω× ~E = ~B

∗~∇× ~B = µ0ǫ0∂ ~E

∂t=

1

c2

∂ ~E

∂t

⇒ 1

c(Nxx + Nz z) × (Exx + iEyy + Ez z) = (Bz, By, Bz)

⇒ 1

c(−Nz)IEy = Bx

1

c[(−NxEz) + NzEx] = By

1

c[NxIEy] = Bz

148

∴ Bx = −Nz × (iEy)

(1

c

)

≡ (iBx)1

c(64)

By = (NzEx − NxEz)

(1

c

)

≡ (By)1

c(65)

Bz = NxiEy

(1

c

)

≡ (iBz)1

c(66)

~S =1

2~E × ~H =

1

2~E × 1

µ0

~B =1

2ǫ0c

2 ~E × ~B

=1

2ǫ0c

2 ((EyBz − ByEz)x + (EzBx − ExBz)y + (ExBy − EyBx)z)

Sx =1

2ǫ0c

2(EyBz − ByEz) =ǫ0c

2

2

(

iEy ·(

1

c

)

iBz − By

(1

c

)

Ez

)

=1

2ǫ0c(iEy · iBz − ByEz)

=1

2Z0(iEY · iBz − ByEz) ≡

1

2Z0SX

Sy = ǫ0c2(EzBx − ExBz)

= ǫ0c(Ez · iBx − Ex × iBz)

= ǫ0c(Ez × (−Nz) × (iEy) − Ex × (NxiEy)

= ǫ0c((−Nz)Ez × iEy − Nx × Ex × iEy)

= ǫ0c(−1)( ~N · ~E)iEy = 0

Sz =1

2ǫ0c

2(ExBy − EyBx)

=1

2ǫ0c(ExBy − (iEy)(iBx))

=1

2Z0(Ex × By − iEy × iBx ≡ 1

2Z0(SZ)

Where,SX = (iEy) × (iBz) − Ez · By

SZ = Ex × By − (iEy) · (iBx)

Z0 =

√µ0

ǫ0= 120π

From the cold plasma dispersion relation :

S − N2 cos2 θ −iD N2 cos θ sin θiD S − N2 0

N2 cos θ sin θ 0 P − N2 sin2 θ

Ex

Ey

Ez

= 0

⇒ D ≡

S − NxNz −iD NxNz

iD S − N2 0NxNz 0 P − N2

x

Ex

Ey

Ez

= 0

⇒ |D| = 0 for nonzero E fields.

149

⇒

(S − NxNz)Ex − iDEy + NxNzEz = 0iDEx + (S − N2)Ey = 0NxNzEx + (P − N2

x)Ez = 0

Where,

S =1

2(R + L), D =

1

2(R − L)

R ≡ 1 +∑

s

X−s = 1 −

∑

s

ω2ps

ω(ω + Ωs)

L ≡ 1 +∑

s

X+s = 1 −

∑

s

ω2ps

ω(ω − Ωs)

P ≡ 1 −∑

s

ω2ps

ω2

⇒ RL = S2 − D2

Ωs =qsBt

ms, ω2

ps =q2sne

msǫ0; qs =

Z|e| for ions−|e| for electrons

, ms =

AmH for ionsme for electrons

The perpendicular refractive index, Nx is solved with the determinant,|D| = 0.

⇒ AN4x + BN2

x + C = 0

⇒ A(N2x)2 + BN2

x + C = 0

Where,A = S

B = −(RL + PS − PN2z − SN2

z )

= −(S2 − D2 + PS − PN2z − SN2

z )

= D2 − S2 + SN2z + PN2

z − PS

= D2 − [S(S + P ) − (S + P )N2z ]

= −(S + P )(S − N2z ) + D2

C = P (RL − 2SN2z + N4

z )

= P (S2 − D2 − 2SN2z + N4

z )

= P ((S − N2z )2 − D2)

= P (S − N2z + D)(S − N2

z − D)

or

tan2 θ =−P (N2 − R)(N2 − L)

SN2 − RL)(N2 − P )

For propagation at θ = 0 and θ = π2 ,

1)θ = 0 : P = 0, N2 = R, N2 = L

2)θ =π

2: N2 =

RL

S=

S2 − D2

S, N2 = P

150

With the solutions, Nx, of dispersion relations, the electric fields canbe calculated. However, one more equation is necessary for exact solutionsof electric fields. Then, one component of the the electric fields is normal-ized to ±1, 0, 1(This value will be re-normalized with the input power ofelectromagnetic wave).

The wave equations are solved for the two cases, and there, they aresolved for the same conditions :

1. Nz ≃ 0 (near normal incidence, θ = 90) from Eqs. (7) and (8)

−(N2z − S +

D2

S)Ey = 0

and from Eq. (9)(P − N2

x)Ez = 0

(a) fast mode (or O-mode) : N2x = P (from Eq. (12))

From Eqs. (13) and (14), Ez 6= 0(Ez = −1, 0)Ey = 0From Eq. (8) Ex = 0

(b) slow mode (or X-mode) : N2x = S − D2

S (from Eq. (12))From Eqs. (13) and (14), Ez = 0iEy 6= 0 (iEy = 1, 0)

From Eq. (7) Ex =D(iEy)

S

2. Nz 6= 0

(a) |D| ≪ 1 or |P − N2x | ≪ 1 “Near Vacuum”, “Low Density”

From Eqs. (8) and (9),

Ez = −1, Ex = −(P − N2x)

NxNz, iEy =

D(Ex)

S − N2

(b) OtherwiseSet iEy = 1.From Eq. (8),

Ex =(S − N2)(iEy)

D

From Eq. (9),

Ez =−NxNzEx

P − N2x

• The magnetic fields are calculated with the Eqs. (1)-(3) : i.e.

iBx = −Nz(iEy) (67)

By = NzEx − NxEz (68)

iBz = Nx(iEy) (69)

Remember that above expressions are normalized with the speed oflight, c = 3 × 108m/s.

151

• The Poynting vector normalized with 2Z0 :From Eqs. (4) and (6),

Sx = iEyiBz − ByEz (70)

Sz = ExBy − iEyiBx (71)

or

Sx = iEy(NxiEy) − (NzEx − NxEz)Ez

= Nx(iEy)2 + NxE2

z − NzExEz

Sz = Ex(NxEx − NxEz) − iEy(−NziEy)

= NzE2x − NxExEz + Nz(iEy)

2

• Re-normalization with input RF power flux [W/m2]

Let the input RF power flux (Poynting vector) be S0[W/m2]

Then,

S′x =

1

2Z0Sx

S′z =

1

2Z0Sz

S′ =√

S′2x + S′2

z =1

2Z0

√

S2x + S2

z =1

2Z0S

The ratio power flux: S0/S′ = 2Z0S0/S = γ

where S =√

S2x + S2

z

Since S is proportional to E2, the normalized electric and magneticfields with the input RF power flux are given by

E′x =

√γ Ex

iE′y =

√γ iEy

E′z =

√γ Ez

iB′x =

1

c

√γ iBx

B′y =

1

c

√γ By

iB′z =

1

c

√γ iBz

and the re-normalized Poynting vector (S0x, S0z) is given by

S0x = S0 sin θ

S0z = S0 cos θ

Where

θ = tan−1

(Sx

|Sz|

)

152

9.2 Phase velocity and group velocity of EM waves in a toka-mak with a cold plasma

We consider that Nz is constant.

• Phase velocity:

The magnitude of the phase velocity over c is

| ~Vp|c

=1

N≡ vp,

where Nx =√

N2x + N2

z sin θ = N sin θ with constant Nz.

Then,

vpx = vp sin θ =1

Nsin θ

vpz = vp cos θ =1

Ncos θ

• Group velocity:

The group velocity is given by

~Vg =~S0

W ′

Where W ′ is the energy density of EM-wave and ~S is the Poyntingvector.

W ′ =1

4

[1

µ0

~B′∗ · ~B′ + ~E′∗ · ∂

∂ω(ω

↔ǫ ) · ~E′

]

S0 is already expressed above. The primed fields, ~E′ and ~B′ are re-normalized ones with the input Poynting flux S0.

Where↔ǫ is the dielectric tensor and is given by

↔ǫ = ǫ0

S −iD 0iD S 00 0 P

⊚The energy density calculation :

W ′ =1

4[1

µ0

~B′∗ · ~B′ + ~E′∗ · ∂

∂ω(ω

↔ǫ ) · ~E′]

a. The second term of the square bracket.

~E′∗ · ∂

∂ω(ω

↔ǫ ) · ~E′

153

ω↔ǫ = ωǫ0

S −iD 0iD S 00 0 P

= ǫ0

ωS −iωD 0iωD ωS 0

0 0 ωP

= ǫ0

S′ −iD′ 0iD′ S′ 00 0 P ′

where

S′ = ωS = ωR + L

2

D′ = ωD = ωR − L

2P ′ = ωP

and

R = 1 −∑

s

ω2ps

ω(ω + Ωs)

L = 1 −∑

s

ω2ps

ω(ω − Ωs)

P = 1 −∑

s

ω2ps

ω2

Then, S′ = ω − 1

2

(∑

s

ω2ps

ω + Ωs+

∑

s

ω2ps

ω − Ωs

)

D′ =1

2

(

−∑

s

ω2ps

ω + Ωs+

∑

s

ω2ps

ω − Ωs

)

P ′ = ω −∑

s

ω2ps

ω

~E∗ · ∂

∂ω(ω~ǫ) · ~E

= ǫ0(E′∗x E

′∗y E

′∗z )

∂S′

∂ω −i∂D′

∂ω 0

i∂P ′

∂ω∂S′

∂ω 0

0 0 ∂P ′

∂ω

E′x

E′y

E′z

= ǫ0

(

E′∗x

[∂S′

∂ωE′

x − ∂D′

∂ω(iE′

y)

]

+ E′∗y

[

i∂D′

∂ωE′

x +∂S′

∂ωE′

y

]

+ E′∗z

∂P ′

∂ωE′

z

)

= ǫ0

[∂S′

∂ω(E

′2x + E

′2y ) − E

′∗x

∂D′

∂ω(iE′

y) + E′∗x (−i)

∂D′

∂ωE′

y +∂P ′

∂ωE

′2z

]

= ǫ0

[∂S′

∂ω(E

′2x + (iE

′

y)2) − 2

∂D′

∂ωE′

xiE′y +

∂P ′

∂ωE

′2z

]

154

∗ ∂S′

∂ω= 1 +

1

2

(∑

s

ω2ps

(ω + Ωs)2+

∑

s

ω2ps

(ω − Ωs)2

)

∂D′

∂ω=

1

2

(∑

s

ω2ps

(ω + Ωs)2−

∑

s

ω2ps

(ω − Ωs)2

)

∂P ′

∂ω= 1 +

∑

s

ω2ps

ω2

b. The first term of the square bracket

~B′∗ · ~B′ = (iB′

x)2+ (B

′

y)2+ (iB

′

z)2

Thus, the energy density is

W′

=1

4[1

µ0(iB

′

x

2+ B

′

y

2+ iB

′

z

2) + ǫ0

∂S′

∂ω(E

′

x

2+ iE

′

y

2) − 2

∂D′

∂ωE

′

xiE′

y +∂D

′

∂ωE

′

z

2]

=1

4ǫ0[c

2(iB′

x

2+ B

′

y

2+ iB

′

z

2) +

∂S′

∂ω(E

′

x

2+ iE

′

y

2) − 2

∂D′

∂ωE

′

xiE′

y +∂D

′

∂ωE

′

z

2]

This energy density can be expressed with normalized fields,Ex, iEy, Ez, iBx, By, iBz

W′

=1

4ǫ0[c

2 γ

c2(iB

′

x

2+ B

′

y

2+ iB

′

z

2) + γ∂S

′

∂ω(E

′

x

2+ iE

′

y

2) − 2

∂D′

∂ωE

′

xiE′

y +∂D

′

∂ωE

′

z

2]

=1

4ǫ0γW =

1

4ǫ0

2Z0S0

SW

Where W = ∂S′

∂ω (E′

x2+ iE

′

y2) − 2∂D

′

∂ω E′

xiE′

y + ∂D′

∂ω E′

z2+ iB2

x + B2y + iB2

z

Thus, the group velocity | ~Vg|,

| ~Vg| =S0

W ′ =S0

14ǫ0

2Z0S0S W

=2S

ǫ0Z0W=

2S

Wc

(S =√

S2x + S2

z , c is the speed of light)

⇒ | ~Vg|c

=2S

W= vg

vgx = vg sinαvgz = vg cos α

α = tan−1 Sx

Sz

155

9.3 Raytracing of EC-wave in KSTAR tokamak

For the ray-trace of the EC-wave of 84 GHz, the launching an-gle (i.e., Nz) and the maximum plasma density are varied. Thetoroidal magnetic field at the major radius (R0 = 1.8m) is set to3.0 T.

There are two modes as the EC-wave is launched at the an-tenna: O-mode (Fast mode) and X-mode (Slow mode). For KSTAR,these two modes will be launched from outboard side (or lowtoroidal magnetic field side). In subsequent figures, the inboardside launches are also shown.

As the wave propagates into the plasma, it meets cut-off, tun-nelling, resonance, and mode-conversion. From the dispersion re-lation, a perpendicular refractive index is given by

Nx =−B ±

√B2 − 4AC

2A.

When the Nx becomes zero, the wave is cut-off or tunnelled fol-lowed by mode-conversion with the condition B2 − 4AC = 0. Inabove equation, + sign corresponds to O-mode and − sign corre-sponds to X-mode. If O-mode is launched at the outboard side, +sign is used for the ray propagation. However, when it meets themode-conversion, − sign is used for the ray propagation after themode-conversion point.

156

Nz=0.94, Ne(0) = 3 x 1019 m-3, BT = 3 T

O-mode Outboard launch

O-mode Inboard launch

X-mode Outboard launch

X-mode Inboard launch

Figure 18:

157

Nz=0.94, Ne(0) = 5 x 1019 m-3, BT = 3 T

O-mode Outboard launch X-mode Outboard launch

O-mode Inboard launch X-mode Inboard launch

Figure 19:

158

Nz=0.94, Ne(0) = 7 x 1019 m-3, BT = 3 T


O-mode Inboard launch X-mode Inboard launch

Figure 20:

159

Nz=0.94, Ne(0) = 9 x 1019 m-3, BT = 3 T


X-mode Inboard launchO-mode Inboard launch

Figure 21:

160

Nz=0.94, Ne(0) = 1 x 1020 m-3, BT = 3 T



Figure 22:

161

Nz=0.77, Ne(0) = 3 x 1019 m-3, BT = 3 T



Figure 23:

162

Nz=0.77, Ne(0) = 5 x 1019 m-3, BT = 3 T



Figure 24:

163

Nz=0.77, Ne(0) = 7 x 1019 m-3, BT = 3 T



Figure 25:

164

Nz=0.77, Ne(0) = 9 x 1019 m-3, BT = 3 T



Figure 26:

165

Nz=0.77, Ne(0) = 1 x 1020 m-3, BT = 3 T



Figure 27:

166

Nz=0.5, Ne(0) = 3 x 1019 m-3, BT = 3 T



Figure 28:

167

Nz=0.5, Ne(0) = 5 x 1019 m-3, BT = 3 T



Figure 29:

168

Nz=0.5, Ne(0) = 7 x 1019 m-3, BT = 3 T



Figure 30:

169

Nz=0.5, Ne(0) = 9 x 1019 m-3, BT = 3 T



Figure 31:

170

Nz=0.5, Ne(0) = 1 x 1020 m-3, BT = 3 T



Figure 32:

171

Nz=0.17, Ne(0) = 3 x 1019 m-3, BT = 3 T



Figure 33:

172

Nz=0.17, Ne(0) = 5 x 1019 m-3, BT = 3 T



Figure 34:

173

Nz=0.17, Ne(0) = 7 x 1019 m-3, BT = 3 T



Figure 35:

174

Nz=0.17, Ne(0) = 9 x 1019 m-3, BT = 3 T



Figure 36:

175

Nz=0.17, Ne(0) = 1 x 1020 m-3, BT = 3 T



Figure 37:

176

A Calculation of Sx

Sx =1

4µ0Re

[

~E × ~B∗ +1

2~E∗ · ∂

∂ ~N

↔ǫ · ~E

]

x

since ~∇× ~E = −∂ ~B

∂t→ ikxEy = (−iω)Bz

∴ Bz =kx

ωEy =

Nx

cEy

and,

ikzEx − ikxEz = iωBy,1

c(NzEx − NxEz) = By

Let ∂∂ ~N

↔ǫ · ~E = X

Sx =1

4µ0Re

[

(EyB∗z − EzB

∗y) +

1

2(E∗


zXzx)

]

But,

A:

X =∂

∂ ~N

↔ǫ · ~E

= x∂

∂Nx[x(ǫxxEx + ǫxyEy) + y(ǫyxEx + ǫyyEy + zǫzzEz]

+ z∂

∂Nz[x(ǫxxEx + ǫxyEy) + y(ǫyxEx + ǫyyEy + zǫzzEz]

Xxx = ∂∂Nx

(ǫxxEx + ǫxyEy)

Xyx = 0

Xzx = ∂∂Nz

(ǫxxEx + ǫxyEy)

before,

Ex =S − N2

DiEy, Ez = − NxNz

P − N2x

S − N2

DiEy

Then,

Xxx =∂

∂Nx

(

SS − N2

D− D

)

iEy = − S

D(2Nx)iEy

Xzx =∂

∂Nz

(

SS − N2

D− D

)

iEy = − S

D(2Nz)iEy

Therefore,

E∗xXxx =

S − N2

D(−iE∗

y)

(

− S

D

)

2NxiEy = − 2S

D2(S − N2)Nx|Ey|2

E∗yXyx = 0

E∗zXzx = − NxNz

P − N2x

S − N2

D(−iE∗

y)

(

− S

D

)

2NziEy =2NxN2

z

P − N2x

S(S − N2)

D2|Ey|2

177

=⇒ 1

2(E∗


zXzx)

=

[

−S(S − N2)

D2+

N2z

P − N2x

S

D2(S − N2)

]

Nx|Ey|2

= Nx|Ey|2[

−S(S − N2)

D2

](

1 − N2z

P − N2x

)

B:

EyB∗z − EzB

∗y =

1

c

[EyNxE∗

y − Ez(NzE∗x − NxE∗

z )]

=1

c

[

Nx|Ey|2 +NxN2

z

P − N2x

(S − N2

D

)2

|Ey|2 + NxN2

xN2z

(P − N2x)2

(S − N2)2

D2|Ey|2

]

=1

cNx|Ey|2

(

1 +(S − N2)2N2

z

D2(P − N2x)

+(S − N2)2N2

xN2z

D2(P − N2x)2

)

=1

cNx|Ey|2

D2(P − N2x)2 + (S − N2)2(P − N2

x)N2z + (S − N2)2N2

xN2z

D2(P − N2x)2

=1

cNx|Ey|2

D2(P − N2x)2 + (S − N2)2(PN2

z − N2xN2

z + N2xN2

z )

D2(P − N2x)2

=1

cNx|Ey|2

D2(P − N2x)2 + (S − N2)2(PN2

z )

D2(P − N2x)2

∴ Sx =1

4µ0cNx|Ey|2

[D2(P − N2

x)2 + (S − N2)2PN2z

D2(P − N2x)2

− S

D2(S − N2)

P − N2

P − N2x

]

≃ 1

4µ0cNx|Ey|2

D2(P − N2x)2 + (S − N2)2PN2

z

D2(P − N2x)2

178

B The reason of validity of cold plasmadielectric tensor in the calculationof harmonic damping rates(n ≥ 2)

Ans.)In the first harmonic damping ω = |Ω|. In this case, ǫxx, ǫxy, ǫyx is infinite.Thus, the finite electron temperature effect is included in the dielectric tensorin order to find the electric fields at resonance for the first harmonic damping.In other words, the finite cyclotron radius effect is included to the first orderin the temperature for those terms that are large near resonance. But, inthe higher harmonic damping, ω = n|Ω| (n ≥ 2). Then, the dielectric tensorS, D, P are not infinite at resonance. Thus, the cold plasma dielectric tensoris valid in the higher harmonic damping.

179

C Quasi-linear Theory

• Velocity-space diffusion rate

Dv =(∇v)2

2t∝ E2

where E is the amplitudes of the linear-theory modesSquares of E → “Quasi-linear”

• Motivation of the development of quasi-linear theory.

– Microinstabilities form an important part of wave theory.

– And the questions arise, how will the mode amplitude grow?

– What is the instability saturation mechanism?

– Bibliography

∗ W.E. Drummond and D.Pines (1961): “Nonlinear stabilityof Plasma Oscillations, General Atomic” GA-2386 (1961)

∗ A.A.Vedenov, E.P.Velikhov and R.Z. Sagdeev (1961): “Non-linear Oscillations of Rare field Plasma” Nuclear Fusion 1,82 (1961)

∗ Yu.A.Romanov and G.Filippov (1961): “The Interaction ofFast Electron Beams with Longitudinal Plasma Waves” sov.phys.-JETP 13, 87 (1961)

– In which it was found that a temporally growing micro-instabilityacts back on the zero-order velocity distribution function.

– Its effect on the distribution function is to produce velocity-spacediffusion.

– The diffusion tends to flatten f0(~v) in this region and drive theinstability growth rate to zero.

– This saturation process can be viewed as a continuous diffusionthrough which the zero-order distribution function evolves slowlyin time.

– Two assumptions in quasi-linear theory.

∗ The amplitudes of the perturbations in the plasma are notso large as to invalidate the use of zero-order orbits andof the spatially averaged distribution function, f0(~v, t) =<f0(~r,~v, t) >

∗ The effective wave spectrum should be sufficiently dense.Any appreciable coherence between modes will be destroyedby phase mixing.

• Electromagnetic Quasi-linear Theory

∂f

∂t+ ~v · ~∇f +

q

m~∇v ·

(

~E + ~v × ~B)

f = 0 (∗f = f(~r,~v, t))

Averaging over a number of space and time periods of the rapid fluc-tuations. In addition, in presence of ~B0, we also average over the

180

Figure 38: Velocity distribution for “bump-on-tail” instability. Real part ofunstable frequencies are such that v = ω0(k)/k lies in region where vdf0/dvis positive (opposite sense to Landau damping). Quasi-linear diffusion dueto these modes tends to flatten out the bump.

gyro-angle in velocity space:

∂f0(~v, t)

∂t=

⟨∂f

∂t

⟩

= −⟨

~v∂f

∂z

⟩

− q

m

⟨

~∇v ·(

~E + ~v × ~B)

f⟩

slow evolution of f0(~v, t) = 〈f(~r,~v, t)〉

Space averaging :⟨

∂f∂z

⟩

= 0

< E >= 0

Higher order contributions to⟨

~∇v ·(

~E + ~v × ~B)

f⟩

are neglected

∂f0

∂t≃ − q

m

⟨∫ 2π

0

dφ

2π~∇v ·

(

~E1 + ~v × ~B1

)

f1

⟩

= − q

m

∑

modes

1

V

∫ 2π

0

dφ

2π~∇v ·

(

~Ek + ~v × ~Bk

)

f−k

where V is the volume and we used Fourier formalism of

< A(t)B(t) >= limT→∞

1

T

∫ ∞

−∞dωA(−ω)B(ω)

< A(z)B(z) >= limV →∞

1

V

∫ ∞

−∞d3k(−k)B(k)

181

f−k = f(ω−k,−~k,~v) = f∗(ωk,~k,~v) (∵ f1(~r,~v, t) is real.)

since ~B1 =~kω × ~E1

~Ek + ~v × ~Bk = [1(1 −~k · ~vω

) +~k~v

ω] · ~Ek

Also,

~v = xv⊥ cos φ + yv⊥ sinφ + zv‖ = ρv⊥ + zv‖

~∇v = ρ∂

∂v⊥+ φ

1

v⊥

∂

∂vφ+ z

∂

∂v‖~k = xk⊥ cos θ + yk⊥ sin θ + zk‖

= ρk⊥ cos(φ − θ) − φk⊥ sin(φ − θ) + zk‖~E = xEx + yEy + zEz

= ρ(−Ex cos φ + Ey cos φ) + zEz

where ρ is the unit vector in the direction of ~v⊥, and φ = z × ρ.

182

~∇v · ( ~E + ~v × ~B)kf−k =1

v⊥

∂

∂v⊥v⊥[( ~E + ~v × ~B)kf−k]ρ +

1

v⊥

∂

∂vφv⊥[( ~E + ~v × ~B)kf−k]φ

+∂

∂v‖v⊥[( ~E + ~v × ~B)kf−k]‖

(1)[( ~E + ~v × ~B)kf−k]ρ = [(1(1 −~k · ~vω

) +~k~v

ω) · ~Ek]ρf−k

= [1 − 1

ω(k⊥v⊥ cos(φ − θ) + k‖v‖)](Ekx cos φ + Eky sinφ)f−k

+1

ω[v⊥(Ekx cos φ + Eky sinφ) + v‖Ez]kperp cos(φ − θ)f−k

∴1

v⊥

∂

∂v⊥v⊥[· · · ]ρ =

1

v⊥([· · · ]ρ) +

∂

∂v⊥[· · · ]ρ

=1

v⊥[· · · ]ρ + [−k⊥

ωcos(φ − θ)(Ekx cos φ + Eky sinφ)f−k]

+ [1 − 1

ω(k⊥v⊥ cos(φ − θ) + k‖v‖)](Ekx cos φ + Eky sinφ)

∂f−k

∂v⊥

+1

ω(Ekx cos φ + Eky sin φ)k⊥ cos(φ − θ)f−k

+1

ω[v⊥(Ekx cos φ + Eky sinφ) + v‖Ez]k⊥ cos(φ − θ)

∂f−k

∂v⊥

(2)[( ~E + ~v × ~B)kf−k]φ = [(1(1 −~k · ~vω

) +~k~v

ω) · ~Ek]φf−k

= [1 − 1

ω(k⊥v⊥ cos(φ − θ) + k‖v‖)](−Ekx sinφ + Eky cos φ)f−k

+1

ω[v⊥(Ekx cos φ + Eky sinφ) + v‖Ez](−k⊥ sin(φ − θ))f−k

∴1

v⊥

∂

∂vφ[· · · ]φ =

1

v⊥[1 − 1

ω(k⊥v⊥ cos(φ − θ) + k‖v‖)](−Ekx sinφ + Eky cos φ)

∂f−k

∂vφ

+1

v⊥

1

ω[v⊥(Ekx cos φ + Eky sin φ) + v‖Ez](−k⊥ sin(φ − θ))

∂f−k

∂vφ

(3)[( ~E + ~v × ~B)kf−k]‖ = [(1(1 −~k · ~vω

) +~k~v

ω) · ~Ek]‖f−k

= [1 − 1

ω(k⊥v⊥ cos(φ − θ) + k‖v‖)]Ekzf−k

+1

ω[v⊥(Ekx cos φ + Eky sinφ) + v‖Ekz]k‖f−k

∴∂

∂v‖[· · · ]‖ = k‖Ekzf−k + [1 − 1

ω(k⊥v⊥ cos(φ − θ) + k‖v‖)]Ekz

∂f−k

∂v‖

+1

ωEkzk‖f−k +

1

ω[v⊥(Ekx cos φ + Eky sinφ) + v‖Ekz]k‖

∂f−k

∂vφ

Thus,

~∇v · [( ~E + ~v × ~B)kf−k] = cos(φ − θ)[(Ek+ + Ek

−)Sf−k − EkzTf−k]

− i sin(φ − θ)(Ek+ + Ek

−)Sf−k + Ekz∂f−k

∂v‖+

1

v⊥

∂

∂vφ(· · · )

183

where

Sf−k = (1 −k‖v‖ω

)1

v⊥

∂

∂v⊥(v⊥f−k) +

k‖v⊥ω

∂f−k

∂v‖

Tf−k =k⊥v⊥

ω

∂f−k

∂v‖−

k⊥v‖ω

1

v⊥

∂

∂v⊥(v⊥f−k)

E±k =

1

2(Ekx ± iEky)e

∓iθ

andEkx ≡ Ex(~k), Eky ≡ Ey(~k), Ekz ≡ Ez(~k)

(ù Ekx cos φ + Eky sinφ = cos(φ − θ)(E+k + E−

k ) − i sin(φ − θ)(E+k − E−

k )

Ans.)

E+k + E−

k =1

2[Ekxe−iθ + iEkye

−iθ] +1

2[Ekxeiθ − iEkye

iθ]

=1

2[Ekx(cos θ − i sin θ) + iEky(cos θ − i sin θ)]

+1

2[Ekx(cos θ + i sin θ) − iEky(cos θ + i sin θ)]

= Ekx cos θ + Eky sin θ

E+k − E−

k = −iEkx sin θ + iEky cos θ

= −i(Ekx sin θ − Eky cos θ)

cos(φ − θ)(E+k + E−

k ) − i sin(φ − θ)(E+k − E−

k )

= (cos φ cos θ + sinφ sin θ)(Ekx cos θ + Eky sin θ)

−i(sinφ cos θ − cos φ sin θ)(−i)(Ekx sin θ − Eky cos θ)

= Ekx cos φ + Eky sinφ)

But,

fk = − q

m

∑

n

∑

m

e−i(n−m)(φ−θ)Jm(k⊥v⊥

Ω)Jn(

k⊥v⊥Ω

)

∫ ∞

0dτ exp [(ω − k‖v‖ − nΩ)τ ]

×cos(φ − θ + Ωτ)[(E+k + E−

k )U ′ − EkzV ]

−i sin(φ − θ + Ωτ)(E+k − E−

k )U ′ + Ekz∂f0

∂v‖

Remember f1 obtained for the calculation of the first-order Vlasov equationwith the simple replacement of φ by φ − θ and τ by −τwhere

U ′ =∂f0

∂v⊥+

k‖ω

(v⊥∂f0

∂v‖− v‖

∂f0

∂v⊥) =

1

ωU

V =k⊥ω

(v⊥∂f0

∂v‖− v‖

∂f0

∂v⊥)

W ′ = (1 − nΩ

ω)∂f0

∂v‖+

nΩ

ωv⊥vz

∂f0

∂v‖=

1

ωW

184

1

2π

∫ 2π

0dφ ~∇v · [( ~Ek + ~v × ~Bk)

∗fk]

=∞∑

n=−∞[(E+

k + E−k )S − EkzT ]

n

λJn(λ) + (E+

k − E−k )SJ ′

n(λ)

+EkzJn(λ)∂

∂v‖∗(− q

m

∫ ∞

0dτ ei(ω−k‖v‖−nΩ)τ )

×n

λJn(λ)[(E+

k + E−k )U ′ − EkzV ]

+J ′n(λ)(E+

k − E−k )U ′ + Jn(λ)Ekz

∂f0

∂v‖

ù φ integral∑

n

∑

m → ∑

n,∫

dτ → nλJn, J ′

n

But,

−n

λV +

∂f0

∂v‖= − nΩ

k⊥v⊥(k⊥v⊥

ω

∂f0

∂v‖−

k⊥v‖ω

∂f0

∂v⊥) +

∂f0

∂v‖

= (1 − nΩ

ω)∂f0

∂v‖+

nΩv‖ωv⊥

∂f0

∂v⊥= W ′

=v‖v⊥

U ′ +ω − k‖v‖ − nΩ

ω(∂f0

∂v‖−

v‖v⊥

∂f

∂v⊥)

=v‖v⊥

U ′

The Coefficient of E∗kzJn(λ)

−n

λT +

∂

∂v‖= − nΩ

k⊥v⊥(k⊥v⊥

ω

∂

∂v‖−

k⊥v‖ω

1

v⊥

∂

∂v⊥v⊥) +

∂

∂v‖

= (1 − nΩ

ω)

∂

∂v‖+

nΩv‖ωv⊥

1

v⊥

∂

∂v⊥v⊥

=v‖v⊥

S +ω − k‖v‖ − nΩ

ω(

∂

∂v‖−

v‖v⊥

1

v⊥

∂

∂v⊥v⊥)

=v‖v⊥

S

1

2π

∫ 2π

0dφ~∇v ·

[(

~Ek + ~ω × ~Bk

)∗fk

]

−→ −iq

m

∑

SA∗k

1

ω − k‖v‖ − nΩAkU

′

where Ak = v⊥

E+k

[n

λJn(λ) + J ′

n(λ)]

+ E−k

[n

λJn(λ) − J ′

n(λ)]

+ v‖EkzJn(λ)

= v⊥E+k Jn−1(λ) + v⊥E−

k Jn+1(λ) + v‖EkzJn(λ)

185

Thus, a remarkably compact expression for “quasi-linear evolution”

∂f0(v⊥, v‖, t)

∂t=

πq2

m2

∑

modes

1

ω2k

∞∑

−∞Lδ(ωk − k‖v‖ − nΩ)|A|2Lf0

L is the operator, such that

L = (ωk − k‖v‖)1

v⊥

∂

∂v⊥+ k‖

∂

∂v‖= nΩ

1

v⊥

∂

∂v⊥+ k‖

∂

∂v‖

We used Plemelj relation

1

ωkv= P

(1

ωkv

)

− iπδ(ω − kv)

Jn−1(x) =n

xJn(x) + J ′

n(x)

Jn+1(x) =n

xJn(x) − J ′

n(x)

Thus,

∂f0(~v, t)

∂t= π

( e

mω

)2∞∑

−∞

1

v⊥

∂

∂v⊥

[

nΩδ(ω − k‖v‖ − nΩ)|A|2(

nΩ

v⊥

∂f0

∂v⊥+ k‖

∂f0

∂v‖

)]

+∂

∂v‖

[

k‖δ(ω − k‖v‖ − nΩ)|A|2(

nΩ

v⊥

∂f0

∂v⊥+ k‖

∂f0

∂v‖

)]

Let E+ = Ex + iEy (left-hand polarization)

E− = Ex − iEy (right-hand polarization)

Ak =1

2

(

v⊥E+e−iθJn−1 + v⊥E−eiθJn+1 + 2v‖v⊥

EzJn

)

⇒ 1

2A

∴∂f0

∂t= π

( e

2mω

)2∞∑

−∞

1

v⊥

∂

∂v⊥

[

nΩδ(ω − k‖v‖ − nΩ)|A|2(

nΩ

v⊥

∂f0

∂v⊥+ k‖

∂f0

∂v‖

)]

+∂

∂v‖

[

k‖δ(ω − k‖v‖ − nΩ)|A|2(

nΩ

v⊥

∂f0

∂v⊥+ k‖

∂f0

∂v‖

)]

where

A = v⊥E+e−iθJN−1 + v⊥E−eiθJN+1 + 2v‖v⊥

EzJN

186

Documents

Plasma Sheath Lab - POSTECHpsl.postech.ac.kr › material › lecture › wavesinplasmas.pdf · 2004-09-24 · 1 Dispersion Relation in a Cold Uniform Plasmas As long as Te = Ti =