9 Reflection, Absorption, and Mode Conversion

9.1 Solution of Wave Equation Near a Turning Point

Assume a wave electric field of the form E(x, z, t) ∼ X(x)eikzz−iωt (i.e., ky = 0,kz = constant). The dispersion relation for a homogeneous plasma was givenby bk2 + c = 0. For the inhomogeneous case, the following differential waveequation must be solved

d2E

dx2+ k2(x)E = 0, where k2(x) = −c(x)

b(x).

When k2(x) is slowly varying, i.e., |k′′| ≪ |kk′| and |k′| ≪ |k2|, the WKBapproximation can be used. The solution is given by

E = (const.)1√k

exp(±i

∫ x

k dx

).

The WKB approximation is not valid when c → 0 or when b → 0. For c → 0,the wave equation is approximated by the linear turning point equation

d2E

dx2+ (x − x0 + iϵ)νE = 0

and for b → 0, by the singular turning point equation

d2E

dx2+

µ

x − x0 + iϵE = 0.

A small real constant ϵ is introduced to avoid the singularity at x = x0. It isseen later that ϵ describes damping or excitation of the wave. Solutions to theseturning point equations are given in terms of Bessel functions.

Fg. 1. A linear turning point (left) and a singular turning point (right).

9.2 Asymptotic Solutions

A useful approximation to the true solution can be obtained by joining thesolutions near the turning point to the solutions far away from the turning

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point (asymptotic solutions). The connection formula for the linear turningpoint is

1√k1

[(Ae−5iπ/12 + Be−iπ/12)eiξ1 + (Ae5iπ/12 + Beiπ/12)e−iξ1

]↔ 1√

|k2|

[(−A + B)e|ξ2| + (−Ae−5iπ/6 + Be−iπ/6)e−|ξ2|

]where ξ ≡

∫ xkdx, and the subscripts 1 and 2 refer to regions where the real

part of k2 is positive and negative, respectively. Since time dependence is e−iωt,eiξ1 describes waves travelling to the right, and e−iξ1 describes waves travellingto the left. The e|ξ2| term describes a growing solution away from the turningpoint whereas the e−|ξ2| term describes a decaying solution. When A = B,the incoming (incident) wave and the outgoing (reflected) wave have the sameamplitude, which describes total reflection. In this case the field amplitudedecays exponentially on the opposite side of the turning point.

The connection formula for the singular turning point is

1√k1

[(A − iB)ei[ξ1+(π/4)] + (A + iB)e−i[ξ1+(π/4)]

]↔ 1√

|k2|

[(A ± iB)e|ξ2| + 2Be−|ξ2|

]where upper or lower sign is chosen for ϵ positive or negative. Choosing thelower sign (i.e., ϵ < 0) requires A = iB to avoid a diverging solution, whichleaves only an incident wave, implying that total absorption takes place at theturning point. Choosing the upper sign (i.e., ϵ > 0) requires A = −iB, whichleaves only the outgoing wave, implying that a wave is emitted from the turningpoint.

9.3 Budden Tunneling Factors

Budden considered the case in which both linear and singular turning pointsexist close to each other, which ca be described by the following wave equation

d2E

dx2+

(β

x+

β2

η2

)E = 0.

Asymptotic solutions for x → +∞ are connected to asymptotic solutions forx → −∞. Consider a case in which the density increases to the right, so a wavemoving to the right passes through the R = 0 cutoff (linear turning point),through an evanescent region where k2 < 0, then past the upper-hybrid reso-nance (singular turning point). A wave incident from the left can be reflected,transmitted, or absorbed. The coefficients of wave reflection R and transmissionT are given by

|R| = 1 − exp(−πη)

|T | = exp(−πη

2

)

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|R|2 + |T |2 = 1 − exp(−πη) + exp(−2πη) < 1.

Fg. 2. A cutoff-resonance pair considered by Budden.

A wave incident from the right can either be transmitted or absorbed.

|R| = 0

|T | = exp(−πη

2

)|R|2 + |T |2 = exp(−πη) < 1.

For |x| → ∞, the wavenumber is given by

k2∞ =

β2

η2.

The distance ∆x between the cutoff (k2 = 0) and the resonance (k2 → ∞) is

|∆x| =η2

β.

Combining these givesη = |k∞∆x|

i.e., η is the distance between the cutoff and the resonance measured by thenumber of wavelengths at |x| → ∞. When η is small, appreciable tunnelingoccurs through the evanescent region. When η is large, complete reflectionoccurs for waves incident from the low density side, and complete absorptionoccurs for waves incident from the high density side.

9.4 Mode Conversion for Alfven Resonance

The cold plasma dispersion relation was given by

n2⊥ =

(R − n2∥)(L − n2

∥)

S − n2∥

.

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When a resonance is approached, the perpendicular wavenumber becomes large,and higher order derivatives can no longer be ignored. This introduces an addi-tional short wavelength mode, and conversion from the long wavelength mode tothe short wavelength mode becomes possible. Mode conversion can be modelledby the Standard Equation

d4E

dx4+ λ2(x − x0)

d2E

dx2+ γE = 0.

Fg. 3. The cold plasma dispersion relation showing the cutoff-resonance-cutofftriplet near the Alfven resonance.

Inclusion of finite electron mass and finite Larmor radius corrections changethe dispersion relation into a biquadratic equation. The dispersion relation canbe written

n4⊥

ϵxx

ϵzz− n2

⊥

[ϵxx − n2

∥ +ϵxx

ϵzz(ϵyy − n2

∥) −ϵxyϵyx

ϵzz

]+(ϵxx − n2

∥)(ϵyy − n2∥) − ϵxyϵyx = 0

where

ϵxx = S − n2⊥

β(i)⊥2

(ω2

Ω2i − ω2

− ω2

4Ω2i − ω2

)+ · · ·

and

β(i)⊥ ≡

2µ0niT(i)⊥

B20

.

Combining these, the dispersion relation can be rewritten as[S

P+

β(i)⊥2

(ω2

Ω2i − ω2

− ω2

4Ω2i − ω2

)]n4⊥ − (S − n2

∥)n2⊥ + (R − n2

∥)(L− n2∥) = 0.

39

The sign of the n4⊥ coefficient changes sign at

β(i)⊥ =

83

Zme

mi

(1 − ω2

4Ω2i

).

Below this value of β(i)⊥ there is a short-wavelength propagating mode on the

low-density side of the resonance (surface wave). This corresponds to the coolplasma case. Above this β

(i)⊥ there is a short-wavelength propagating mode on

the high-densty side of the resonance (kinetic Alfven mode). This correspondsto the warm plasma case.

Fg. 4. The cool plasma dispersion relation showing the surface wave propa-gating on the low-density side of the resonance.

Fg. 5. The warm plasma dispersion relation showing the kinetic Alfven wavepropagating on the high-density side of the resonance.

Finite electron temperature has a similar effect. In this case the transition

40

from cool plasma to warm plasma behavior occurs at

β(e)∥ =

Zme

mi

(1 − ω2

Ω2i

).

9.5 Hybrid Resonances

For perpendicular propagation (n∥ = 0), the warm plasma dispersion relation is

∑s

[β

(s)⊥2

(ω2

Ω2s − ω2

− ω2

4Ω2s − ω2

)]n4⊥ − Sn2

⊥ + RL = 0.

Close to an integral harmonic ω ∼ nΩs, higher order terms in n2⊥ must be

included. An approximate dispersion relation for Bernstein modes is

ω2

ω2ps

≃(

p3e−λIp(λ)λ

)Ωs

ω − pΩs.

Fg. 6. The warm plasma dispersion relation showing the electron Bernsteinwave.

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