9
9.1
Reflection, Absorption, and Mode Conversion
Solution of Wave Equation Near a Turning Point
Assume a wave electric field of the form E(x, z, t) ∼ X(x)eikz z−iωt (i.e., ky = 0,
kz = constant). The dispersion relation for a homogeneous plasma was given
by bk 2 + c = 0. For the inhomogeneous case, the following differential wave
equation must be solved
d2 E
+ k 2 (x)E = 0,
dx2
where k 2 (x) = −
c(x)
.
b(x)
When k 2 (x) is slowly varying, i.e., |k ′′ | ≪ |kk ′ | and |k ′ | ≪ |k 2 |, the WKB
approximation can be used. The solution is given by
( ∫ x
)
1
E = (const.) √ exp ±i
k dx .
k
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
d2 E
+ (x − x0 + iϵ)νE = 0
dx2
and for b → 0, by the singular turning point equation
d2 E
µ
+
E = 0.
dx2
x − x0 + iϵ
A small real constant ϵ is introduced to avoid the singularity at x = x0 . It is
seen later that ϵ describes damping or excitation of the wave. Solutions to these
turning 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 the
solutions near the turning point to the solutions far away from the turning
36
point (asymptotic solutions). The connection formula for the linear turning
point is
]
1 [
√
(Ae−5iπ/12 + Be−iπ/12 )eiξ1 + (Ae5iπ/12 + Beiπ/12 )e−iξ1
k1
]
1 [
↔√
(−A + B)e|ξ2 | + (−Ae−5iπ/6 + Be−iπ/6 )e−|ξ2 |
|k2 |
∫x
where ξ ≡
kdx, and the subscripts 1 and 2 refer to regions where the real
2
part of k is positive and negative, respectively. Since time dependence is e−iωt ,
eiξ1 describes waves travelling to the right, and e−iξ1 describes waves travelling
to the left. The e|ξ2 | term describes a growing solution away from the turning
point whereas the e−|ξ2 | term describes a decaying solution. When A = B,
the incoming (incident) wave and the outgoing (reflected) wave have the same
amplitude, which describes total reflection. In this case the field amplitude
decays exponentially on the opposite side of the turning point.
The connection formula for the singular turning point is
]
1 [
√
(A − iB)ei[ξ1 +(π/4)] + (A + iB)e−i[ξ1 +(π/4)]
k1
]
1 [
↔√
(A ± iB)e|ξ2 | + 2Be−|ξ2 |
|k2 |
where upper or lower sign is chosen for ϵ positive or negative. Choosing the
lower sign (i.e., ϵ < 0) requires A = iB to avoid a diverging solution, which
leaves only an incident wave, implying that total absorption takes place at the
turning point. Choosing the upper sign (i.e., ϵ > 0) requires A = −iB, which
leaves only the outgoing wave, implying that a wave is emitted from the turning
point.
9.3
Budden Tunneling Factors
Budden considered the case in which both linear and singular turning points
exist close to each other, which ca be described by the following wave equation
(
)
d2 E
β
β2
+
+ 2 E = 0.
dx2
x
η
Asymptotic solutions for x → +∞ are connected to asymptotic solutions for
x → −∞. Consider a case in which the density increases to the right, so a wave
moving to the right passes through the R = 0 cutoff (linear turning point),
through an evanescent region where k 2 < 0, then past the upper-hybrid resonance (singular turning point). A wave incident from the left can be reflected,
transmitted, or absorbed. The coefficients of wave reflection R and transmission
T are given by
|R| = 1 − exp(−πη)
(
)
−πη
|T | = exp
2
37
|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
(
)
−πη
|T | = exp
2
|R|2 + |T |2 = exp(−πη) < 1.
For |x| → ∞, the wavenumber is given by
2
=
k∞
β2
.
η2
The distance ∆x between the cutoff (k 2 = 0) and the resonance (k 2 → ∞) is
|∆x| =
η2
.
β
Combining these gives
η = |k∞ ∆x|
i.e., η is the distance between the cutoff and the resonance measured by the
number of wavelengths at |x| → ∞. When η is small, appreciable tunneling
occurs through the evanescent region. When η is large, complete reflection
occurs for waves incident from the low density side, and complete absorption
occurs for waves incident from the high density side.
9.4
Mode Conversion for Alfvén Resonance
The cold plasma dispersion relation was given by
n2⊥ =
(R − n2∥ )(L − n2∥ )
S − n2∥
38
.
When a resonance is approached, the perpendicular wavenumber becomes large,
and higher order derivatives can no longer be ignored. This introduces an additional short wavelength mode, and conversion from the long wavelength mode to
the short wavelength mode becomes possible. Mode conversion can be modelled
by the Standard Equation
d2 E
d4 E
2
+
λ
+ γE = 0.
(x
−
x
)
0
dx4
dx2
Fg. 3. The cold plasma dispersion relation showing the cutoff-resonance-cutoff
triplet near the Alfvén resonance.
Inclusion of finite electron mass and finite Larmor radius corrections change
the dispersion relation into a biquadratic equation. The dispersion relation can
be written
[
]
ϵxx
ϵxx
ϵxy ϵyx
n4⊥
− n2⊥ ϵxx − n2∥ +
(ϵyy − n2∥ ) −
ϵzz
ϵzz
ϵzz
+(ϵxx − n2∥ )(ϵyy − n2∥ ) − ϵxy ϵyx = 0
where
(i)
ϵxx = S − n2⊥
β⊥
2
(
ω2
ω2
−
2
2
2
Ωi − ω
4Ωi − ω 2
and
)
+ ···
(i)
(i)
β⊥ ≡
2µ0 ni T⊥
.
B02
Combining these, the dispersion relation can be rewritten as
[
)]
(i) (
β⊥
ω2
S
ω2
+
−
n4⊥ − (S − n2∥ )n2⊥ + (R − n2∥ )(L − n2∥ ) = 0.
P
2
Ω2i − ω 2
4Ω2i − ω 2
39
The sign of the n4⊥ coefficient changes sign at
(
)
ω2
8 Zme
(i)
β⊥ =
1−
.
3 mi
4Ω2i
(i)
Below this value of β⊥ there is a short-wavelength propagating mode on the
low-density side of the resonance (surface wave). This corresponds to the cool
(i)
plasma case. Above this β⊥ there is a short-wavelength propagating mode on
the high-densty side of the resonance (kinetic Alfvén mode). This corresponds
to the warm plasma case.
Fg. 4. The cool plasma dispersion relation showing the surface wave propagating on the low-density side of the resonance.
Fg. 5. The warm plasma dispersion relation showing the kinetic Alfvén wave
propagating 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
(
)
ω2
Zme
(e)
β∥ =
1− 2 .
mi
Ωi
9.5
Hybrid Resonances
For perpendicular propagation (n∥ = 0), the warm plasma dispersion relation is
[
)]
∑ β (s) ( ω 2
ω2
⊥
n4⊥ − Sn2⊥ + RL = 0.
−
2 − ω2
2 − ω2
2
Ω
4Ω
s
s
s
Close to an integral harmonic ω ∼ nΩs , higher order terms in n2⊥ must be
included. An approximate dispersion relation for Bernstein modes is
( 3 −λ
)
ω2
p e Ip (λ)
Ωs
≃
.
2
ωps
λ
ω − pΩs
Fg. 6. The warm plasma dispersion relation showing the electron Bernstein
wave.
41