14
Parametric Instability
14.1
Frequency Matching
The equation of motion for a simple harmonic oscillator x1 is
d2 x1
+ ω12 x1 = 0
dt2
where ω1 is the resonant frequency. If it is driven by a time-dependent force
which is proportional to the product of the amplitude E0 of the driver (pump)
and the amplitude x2 of a second oscillator, the equation of motion becomes
d2 x1
+ ω12 x1 = c1 x2 E0
dt2
where c1 is a constant indicating the strength of the coupling. A similar equation
holds for x2
d2 x2
+ ω22 x2 = c2 x1 E0 .
dt2
Writing x1 = x1 cos ωt, x2 = x2 cos ω ′ t, E0 = E 0 cos ω0 t,
(ω22 − ω ′2 )x2 cos ω ′ t = c2 E 0 x1 cos ω0 t cos ωt
1
= c2 E 0 x1 {cos [(ω0 + ω)t] + cos [(ω0 − ω)t]} .
2
The driving terms on the right can excite oscillators x2 with frequencies
ω ′ = ω0 ± ω.
The nonlinear driving terms can cause a frequency shift so ω ′ does not need
to be exactly ω2 , but only approximately equal to ω2 . Furthermore, ω ′ can be
complex because there can be damping or growth, so the oscillator x2 has finite
Q and can respond to a range of frequencies.
Let x1 = x1 cos ω ′′ t, x2 = x2 cos[(ω0 ± ω)t]. The equation of motion for x1
becomes
(ω12 − ω ′′2 )x1 cos ω ′′ t
1
= c1 E 0 x2 (cos {[ω0 + (ω0 ± ω)]t} + cos {[ω0 − (ω0 ± ω)]t})
2
1
= c1 E 0 x2 {cos [(2ω0 ± ω)t] + cos ωt} .
2
The driving terms can excite not only the original oscillation x1 (ω), but also new
frequencies ω ′′ = 2ω0 ± ω. Considering the case |ω0 | ≫ |ω1 | so that x1 (2ω0 ± ω)
can be neglected, coupled equations among x1 (ω), x2 (ω0 − ω), and x2 (ω0 + ω)
are derived
(ω12 − ω 2 )x1 (ω) − c1 E0 [x2 (ω0 − ω) + x2 (ω0 + ω)] = 0
[ 2
]
ω2 − (ω0 − ω)2 x2 (ω0 − ω) − c2 E0 (ω0 )x1 (ω) = 0
[ 2
]
ω2 − (ω0 + ω)2 x2 (ω0 + ω) − c2 E0 (ω0 )x1 (ω) = 0.
64
The dispersion relation is given by setting the determinant of the coefficients to
zero,
2
ω − ω12
c1 E0
c1 E0
2
2
c2 E0
= 0.
(ω0 − ω) − ω2
0
c E
2
2
0
(ω0 + ω) − ω2
2 0
For small frequency shifts and small damping or growth rates, ω and ω ′ can
be set approximately equal to the undisturbed frequencies ω1 and ω2 , so the
frequency matching condition can be written
ω0 ≃ ω2 ± ω1 .
When the oscillators are waves in a plasma, ωt should be replaced by ωt − ⃗k · ⃗r.
There is also a wavevector matching condition
⃗k0 ≃ ⃗k2 ± ⃗k1
describing spatial beating. These conditions can be interpreted as conservations
of energy and momentum.
The simultaneous satisfaction of frequency and wavevector matching conditions is possible only for certain combinations of waves. For one-dimensional
problems, the required relationships can be shown on an ω-k diagram. The figure shows the dispersion curves of ion acoustic waves (straight lines), electron
plasma waves (wide parabola), and electromagnetic waves (narrow parabola),
and the incident pump wave (ω0 ) and the two decay waves (ω1 and ω2 ). The
parallelogram construction ensures that the frequency and wavenumber matching conditions are satisfied simultaneously.
(A) Electron decay instability: A large amplitude electron plasma wave can
decay into a backward moving electron plasma wave and an ion acoustic wave.
The positions of the (ω0 , k0 ) and (ω2 , k2 ) on the electron plasma wave dispersion
curve must be adjusted so that the difference vector (ω1 , k1 ) lies on the ion
acoustic wave dispersion curve.
(B) Parametric decay instability: An incident electromagnetic wave of large
phase velocity (ω0 /k0 ≃ c) excites an electron plasma wave and an ion acoustic wave moving in opposite directions. Since k0 is small, k1 ≃ −k2 for this
instability.
(C) Parametric backscattering instability: A light wave excites an ion acoustic wave and another light wave moving in the opposite direction (stimulated
Brillouin backscattering). A light wave can also excite an electron plasma wave
and a backward moving light wave (stimulated Raman backscattering).
(D) Two-plasmon decay instability: An incident light wave decays into two
oppositely propagating electron plasma waves (plasmons). Frequency matching
can be satisfied only if ω0 ≃ 2ωp (i.e., ne = nc /4 where nc is the critical density
where ω0 = ωp ).
14.2
Instability Threshold
Even a small amount of damping (either collisional or collisionless) will prevent
parametric instability unless the pump wave is strong enough. To calculate the
65
Fig. 1. Parallelogram constructions showing the simultaneous matching of
frequency and wavenumber for (A) electron decay instability, (B) parametric
decay instability, (C) stimulated Brillouin backscattering instability, and (D)
two-plasmon decay instability.
threshold, damping rates Γ1 and Γ2 of the oscillators x1 and x2 are introduced,
d2 x1
dx1
+ ω12 x1 + 2Γ1
= c1 x2 E0
2
dt
dt
and similarly for x2 . The equations of motion become
(ω12 − ω 2 − 2iωΓ1 )x1 (ω) =
(ω22
− (ω − ω0 ) − 2i(ω − ω0 )Γ2 )x2 (ω − ω0 ) =
2
c1 x2 E0
c2 x1 E0 .
Take the case of two waves, i.e., when ω ≃ ω1 and ω0 − ω ≃ ω2 but ω0 + ω is
far enough from ω2 to be nonresonant, i.e., ϵ(ω0 + ω) ̸= 0. Expressing x1 , x2
and E0 in terms of their peak amplitudes,
[
] 1
2
(ω 2 − ω12 + 2iωΓ1 ) (ω0 − ω)2 − ω22 − 2i(ω0 − ω)Γ2 = c1 c2 E 0 .
4
At threshold, ℑ(ω) = 0. The lowest threshold will occur at exact frequency
matching, i.e., ω = ω1 , ω0 − ω = ω2 , giving
( 2)
= 16ω1 ω2 Γ1 Γ2
c1 c2 E 0
thresh
The threshold goes to zero with the damping of either wave goes to zero.
14.3
Physical Mechanism
Consider the case of an electromagnetic wave (ω0 , k0 ) driving an electron plasma
wave (ω2 , k2 ) and a low-frequency ion acoustic wave (ω1 , k1 ). Since ω1 is small,
66
ω0 must be close to ωp . The behavior is different for ω0 < ωp (oscillating
two-stream instability) and for ω0 > ωp (parametric decay instability).
Suppose there is a density perturbation in the plasma of the form n1 cos k1 x,
which can occur spontaneously as a component of the thermal noise. Let the
pump wave have an electric field ⃗x̂E0 cos ω0 t and assume there is no DC magnetic
⃗ 0 . The pump wave satisfies the dispersion relation ω 2 = ωp2 + c2 k 2 , so
field B
0
0
k0 ≃ 0 for ω0 ≃ ωp and E0 can be taken to be spatially uniform. If ω0 < ωp ,
the electrons will move in the direction opposite to E0 (ions do not move on
this time scale). The density perturbation causes a charge separation. The
electrostatic charge separation creates an electric field E1 which oscillates at
frequency ω0 . The ponderomotive force due to the total field is
⟩
⟨
ωp2
(E0 + E1 )2
⃗
ϵ0 .
FNL = − 2 ∇
ω0
2
Since E0 is spatially uniform and is much larger than E1 , only the cross term
is important
ωp2 ∂ ⟨2E0 E1 ⟩
F⃗NL ≃ − 2
ϵ0 .
ω0 ∂x
2
Since E1 changes sign with E0 , this force does not average to zero. The ponderomotive force F⃗NL is zero at the peaks and troughs of n1 , but is large where
∇n1 is large. This spatial distribution causes F⃗NL to push electrons from regions
of low density to regions of high density. The resulting DC electrid field drags
the ions and the density perturbation grows. The threshold FNL is the value
just sufficient to overcome the pressure gradient ∇ni1 (Ti + Te ) which tends to
flatten the density. The density perturbation does not propagate, so ℜω1 = 0.
This is called the oscillating two-stream instability (OTSI), because the sloshing
electrons have a time-averaged distribution function which is double-peaked.
⃗ 1 and FNL are reversed, and the ponderoIf ω0 > ωp , the directions of ⃗ve , E
motive force moves ions from dense regions to less dense regions. The density
perturbation would decay if it did not move, but could grow if it travelled at an
appropriate phase velocity, so that the inertial delay between the application of
FNL and the change of ion positions causes the density maxima to move into
the regions into which FNL is pushing the ions. This speed is the ion acoustic
speed cs , as described below. The phase of FNL is exactly the same as the phase
of the electrostatic restoring force in an ion acoustic wave, where the potential
is maximum at the density maximum. Consequently, FNL adds to the restoring
force. The electrons oscillate with large amplitude if the pump wave field is near
2
the natural frequency of the electron plasma wave, i.e., ω22 = ωp2 + 3k 2 vte
. The
pump wave cannot have exactly the frequency ω2 because the beat between ω0
and ω2 must be at the ion acoustic wave frequency ω1 = kcs . If this frequency
matching is satisfied, i.e., ω1 = ω0 − ω2 , both an ion acoustic wave and an
electron plasma wave are excited at the expense of the pump wave. This is the
mechanism of the parametric decay instability.
67
Fig. 2. Physical mechanism of the oscillating two-stream instability.
14.4
Oscillating Two-Stream Instability
For simplicity, let the temperature Ti and Te and the collision rates νi and νe
all vanish. The ion response is described by
∂vi1
∂t
∂ni1
∂vi1
+ n0
∂t
∂x
M n0
= en0 E = FNL
= 0.
Since the equilibrium is assumed to be spatially homogeneous, Fourier analysis
in space can be performed to yield
∂ 2 ni1
ik
+
FNL = 0.
2
∂t
M
The electron response is described by
)
(
∂
∂ve
+ ve ve = −e(E0 + E1 )
m
∂t
∂x
where E1 is related to the density ne1 by Poisson’s equation
ikϵ0 E1 = −ene1 .
The quantities E1 , ve and ne1 have both a high-frequency part, in which the
electrons move independently of the ions, and a low-frequency part, in which the
68
electrons move with the ions. To lowest order, the motion is a high-frequency
⃗0
one in response to the spatially uniform field E
∂ve0
e
e
= − E0 = − Ê0 cos ω0 t.
∂t
m
m
Linearizing about this oscillating equilibrium,
e
e
∂ve1
+ ikve0 ve1 = − E1 = − (E1h + E1l )
∂t
m
m
where the subscripts h and l denote the high- and low-frequency parts. The
first term consists mostly of the high-frequency velocity veh , given by
e
neh e2
∂veh
= − E1h =
.
∂t
m
ikϵ0 m
The low-frequency part is
ikve0 veh = −
e
Eil .
m
The right-hand side is the ponderomotive force to drive the ion acoustic waves,
resulting from the low-frequency beat between ve0 and veh . The left-hand side
is related to the electrostatic part of the ponderomotive force. The electron
continuity equation is
∂ne1
+ ikve0 ne1 + n0 ikve1 = 0
∂t
The high-frequency part is given by
∂neh
+ ikve0 ni1 + ikn0 veh = 0
∂t
where the middle term produces a high-frequency term only by beating of the
low-frequency density nel = ni1 with ve0 . Taking the time derivative, and
neglecting ∂ni1 /∂t gives
∂ 2 neh
ike
+ ωp2 neh =
ni1 E0 .
∂t2
m
Taking neh to vary as exp(−iωt),
(ωp2 − ω 2 )neh =
ike
ni1 E0 .
m
Combining with Poisson’s equation gives
E1h = −
e2 ni1 E0
e2 ni1 E0
≃−
.
2
2
ϵ0 m ωp − ω
ϵ0 m ωp2 − ω02
The ponderomotive force is
FNL ≃
ωp2 e2 ikni1 ⟨ 2 ⟩
E0 .
ω02 m ωp2 − ω02
69
Both E1h and FNL change sign with ωp2 − ω02 . The maximum response will occur
for ω02 ≃ ωp2 . The ion equation can be written
∂ 2 ni1
e2 k 2 Ê02 ni1
≃
.
∂t2
2M m ωp2 − ω02
Since the low-frequency perturbation does not propagate in this instability,
ni1 = ni1 exp γt, where γ is the growth rate, and
γ2 ≃
e2 k 2
Ê02
.
2M m ωp2 − ω02
The growth rate γ is real if ω02 < ωp2 . In the presence of finite damping, ωp2 − ω 2
will have an imaginary part proportional to 2Γ2 ωp , where Γ2 is the damping
rate of the electron oscillations. Then
Ê0
γ∝√ .
Γ2
Far above the threshold, the imaginary part of ω will be dominated by the
growth rate rather than Γ2 , so
2/3
γ ∝ Ê0 .
To solve the problem exactly, the following pair of equations are solved
∂ 2 ni1
∂t2
∂ 2 neh
+ ωp2 neh
∂t2
ike
neh E0
M
ike
ni1 E0 .
m
= −
=
The frequency ω1 vanishes because the ion acoustic speed is zero in the zerotemperature limit.
14.5
Example of Parametric Decay Instablity
A similar derivation for ω0 > ωp leads to the excitation of an electron plasma
wave and an ion acoustic wave. Frequency spectra of the waves measured in
a plasma are shown in the figure. Below threshold power, the high-frequency
spectrum shows only the pump wave, while the low-frequency spectrum shows
only a small amount of noise. When the pump wave ampltude is increased
above threshold, an ion acoustic wave appears in the low-frequency spectrum,
and an electron plasma wave appears in the high-frequency spectrum as the
lower sideband of the pump wave.
14.6
Parametric Dispersion Relation
⃗ = B0⃗ẑ and the pump wave electric field
Consider a magnetized plasma with B
⃗
⃗
⃗ 0 (⃗x, t) = (E0x x̂ + E0z ẑ) cos ω0 t. The 0-th order Vlasov equation is
E
)
∂f0
∂
q (⃗
⃗ · ∂ f0 = 0
+ ⃗v ·
f0 +
E0 cos ω0 t + ⃗v × B
∂t
∂⃗x
m
∂⃗v
70
Fig. 3. Frequency spectra showing the appearance of the electron plasma wave
and the ion acoustic wave excited by the pump wave above the threshold power.
where
d⃗x
= ⃗v ,
dt
vx
vy
vz
d⃗v
⃗ 0 cos ω0 t
⃗ + qE
= ⃗v × Ω
dt
m
q
E0x sin ω0 t
ω0 2
= ux + vDx
m
ω0 − Ω2
q E0x cos ω0 t
= uy + vDy
= uy + Ω 2
m
ω0 − Ω2
q E0z sin ω0 t
= uz +
= uz + vDz
m
ω0
= ux +
The solution for equilibrium distribution function is f0 (⃗v −⃗vD ). The generalized
driving term is
⃗k · ⃗xD = µ sin(ω0 t − β)
where
q
µ=
m
√(
E0z kz
E0x kx
+ 2
ω02
ω0 − Ω2
)2
+
(E0x ky )2 Ω2
.
(ω02 − Ω2 )ω02
The 1st order Vlasov equation is
)
∂f1
∂
q (⃗
⃗ · ∂ f1 = − q E
⃗ 1 · ∂ f0 .
+ ⃗v ·
f1 +
E0 cos ω0 t + ⃗v × B
∂t
∂⃗x
m
∂⃗v
m
∂⃗v
Fourier decompose
f1 =
∑
f1⃗k exp(i⃗k · ⃗x)
⃗
k
71
and write
∂f1k
=
∂t
(
f1⃗k = F (⃗x, ⃗u, t) exp(−i⃗k · ⃗xD )
)
∂F
∂F ∂u
∂xD
+
− ik
F exp(−i⃗k · ⃗xD )
∂t
∂u ∂t
∂t
The 1st order Vlasov equation can be rewritten
(
)
∂
q
∂
q ⃗
∂
∂F
⃗
+ ⃗u ·
F + (⃗u × B) ·
F =−
E1 ·
f0 exp(i⃗k · ⃗xD ).
∂t
∂⃗x
m
∂⃗u
m
∂⃗u
The solution is
F =
∫
q
m
t
dt′
−∞
∂φ′ (⃗x′ , t′ )
·
∂⃗x′
(
∂f0
∂⃗u′
)
exp[iµ sin(ω0 t′ − β)].
The exponential term can be expanded in terms of Bessel functions
exp[iµ sin(ω0 t′ − β)] =
∞
∑
Jn (µ) exp[in(ω0 t − β)].
−∞
Using
2u′
∂f0
= − 2 F0 (u′2 )
′
∂u
vt
and
d
∂
∂
= ′ + ⃗u′ ·
,
′
dt
∂t
∂⃗x
) [(
) ]
(
∫ t
q
d
∂
2
F (⃗x, ⃗u, t) = f0 (u2 )
−
φ exp[iµ sin(ω0 t − β)]
dt′ − 2
m −∞
vt
dt′
∂t′
which can be Fourier transformed to
∫
∑
⃗
dωF (k, u, ω)ei(k·⃗x−ωt) =
Nn In
n
where
Nn
=
In
=
2q f0 (u2 )
Jn (µ)e−inβ
m vt2
(
)∫
∫ t
′
′
d
∂
⃗ ′
dt′ einω0 t
−
dωφ(k, ω)ei(k·⃗x −ωt ) .
′
′
dt
∂t
−∞
−
After some algebra the following expression for F (k, u, ω) can be derived
F (k, u, ω)
=
−

2q
f0 (u2 )
mvt2
∑ Jl

1 − ω

l,p
∑
(
Jn (µ)e−inβ
n
) (
)

′
k⊥ u⊥
k⊥ u ⊥
Jp
ei(l−p)α 
Ω
Ω
 φ(k, ω + nω0 ).

ω − lΩ − k∥ uz
72
The charge density is given by
∫ ∞
∫ 2π
∫ ∞
ρ =
du∥
dα′
u⊥ du⊥ qF (k, u, ω)
−∞
0
0
[
]
2q 2 n0 ∑
ω ∑
−inβ
−b
1+
= −
Jn (µ)e
Il (b)e Z(ζl ) φ(k, ω + nω0 )
2
mv⊥
k z vt
n
l
where
ω − lΩ
.
kz vt
This can be rewritten using the susceptibility χ(ω) as
∑
Jn (µ)e−inβ χ(ω)φ(ω + nω0 )
ρ(ω) = −ϵ0 k 2
2 2
b = k⊥
rL ,
ζl =
n
where
]
[
∑
1
−b
χ(ω, k) = 2 2 1 + ζ0
Il (b)e Z(ζl ) .
k λD
l
Poisson’s equation can be expressed as
∫
1 ∑
⃗
φk =
q
d3 uF (u)e−ik·⃗vD
s
ϵ0 k 2 s
The potential can be written as
1 ∑∑
φ(ω) =
Jn (−µs )e−inβ ρs (ω + nω0 ).
ϵ0 k 2 s n
For µe < 1, Bessel functions can be expanded
J0 (µe ) = 1 −
µ2e
+ ···;
4
J±1 (µe ) = ±
µe
+ ···.
2
For ions, µi ≪ 1 and
J0 (µi ) = 1;
J1 (µi ) = 0.
Consider the case ω0 ≫ ωpi . In this case ion contributions can be ignored
for the sideband ω ± , and
ρi (ω) =
ρe (ω) =
φ(ω) =
−ϵ0 k 2 χi (ω)φ(ω)
[(
)
]
µ2
µ iβ
µ −iβ
2
−
+
−ϵ0 k χe (ω) 1 −
φ(ω) − e φ(ω ) + e φ(ω )
4
2
2
(
]
[
)
µ2
µ iβ
µ
1
−
−iβ
+
ρe (ω) 1 −
+ e ρe (ω ) − e ρe (ω ) + ρi (ω)
ϵ0 k 2
4
2
2
where µ = µe and ω ∓ = ω ∓ ω0 . Substituting φ(ω) into ρ(ω) gives
[(
)
]
χi
µ2
µ iβ − µ −iβ +
ρi = −
1−
ρe + e ρe − e ρe
1 + χi
4
2
2
(
)
2
χe
µ
ρe = −
1−
ρi
1 + χe
4
(µ
)
χ−
e
−iβ
ρ−
= −
e
ρ
i
e
2
1 + χ−
e
(
+
µ iβ )
χe
−
e ρi
ρ+
= −
e
2
1 + χ+
e
73
where χ = χ(ω) and χ± = χ(ω ± ), and similarly for ρ and ρ± . Substituting into
ρi yields
[(
]
)2
µ2 χ+
χi
µ2
χe
µ2 χ−
e
e
+
1=
1−
−
.
1 + χi
4
1 + χe
4 1 + χ−
4 1 + χ+
e
e
Assuming |ϵ− | ≪ 1 (the lower sideband is resonant) where ϵ = 1 + χe + χi , and
|χe | ≫ 1 (i.e., k 2 λ2D ≪ 1), the dispersion relation that describes parametric
instability can be written as
(
)
µ2
1
1
ϵ+
χi χe − + + = 0.
4
ϵ
ϵ
14.7
Resonant Decay
Consider the case in which both the low frequency mode (ω1 ) and the lower
sideband mode (ω2 = ω0 − ω1 ) are resonant, so that ϵRe (ω1 ) = 0 and ϵRe (ω2 ) =
0. The dielectric constant ϵ = ϵRe + iϵIm can be expanded as
ϵ(ω + iγ) =
=
ϵRe (ω) + iγ
i(γ + Γ)
where
Γ= (
∂ϵRe (ω)
+ iϵIm (ω)
∂ω
∂ϵRe
∂ω
ϵIm
)
∂ϵRe
∂ω
is the damping rate including both collisional damping and collisionless damping. The dielectric constants at the low frequency and at the lower sideband
are
ϵ(ω1 )
=
ϵ(ω1 − ω0 )
=
∂ϵRe
∂ω1
∂ϵRe
−i(γ + Γ2 )
.
∂ω2
i(γ + Γ1 )
Ignoring the upper sideband, which is off resonant, the dispersion relation can
be written as
(γ + Γ1 )(γ + Γ2 ) = −
µ2 χi (ω1 )χe (ω1 )
= A2 .
4 ∂ϵRe ∂ϵRe
∂ω1 ∂ω2
The threshold condition is given by setting γ = 0. For resonant decay into
electron plasma wave and ion acoustic wave in the absence of magnetic field,
µ=
and
eE0 k
mω02
( 2 2) (
)
/ω1 1/k 2 λ2De
e2 E02 k 2 ωpi
ϵ E2ω ω
( 2 3) ≃ 0 0 1 2
Γ1 Γ2 =
4
2
4m ω0 (2/ω2 ) 2ωpi /ω1
16n0 Te
74
where ω0 ≃ ωpe has been used. For resonant decay into lower hybrid wave and
a low frequency wave (ion acoustic wave or ion cyclotron wave) in the presence
of magnetic field, ωpi < ω0 ≪ ωce in which case the main driving term is the
⃗ ×B
⃗ drift
E
e E0x ky
V ky
µ≃
=
m ωce ω0
ω0
⃗ ×B
⃗ velocity. The dispersion relations are given by
where V = E0x /B is the E
)
)
(
(
2
2
2
k∥2
ωpe
ωpe
k⊥
−
+ 2 1−
ϵRe (ω ) =
1−
2
k2
k
ω− 2
ω − 2 − ωce
ϵRe (ω) =
1−
2
2 k2
2
ωpi
ωpi
k⊥
1
∥
−
+ 2 2 .
2
2
2
2
2
ω − ωci k
ω k
k λDe
For the lower hybrid wave, the dispersion relation is derived from ϵ(ω − ) = 0 as
(
)
2
k∥2 mi
ωpi
2
2
2
=
ω = ωLH 1 + 2
; ωLH
2
k me
ωpe
1+ 2
ωce
and 1 + (k∥2 /k 2 )(mi /me ) ∼ O(1), so k∥2 ≪ k 2 . For the ion acoustic wave ωs =
kcs ≫ ωci , and
2
ωpi
1
χi (ω) = − 2 2 = − 2 2 .
k cs
k λDe
Therefore,
1
µ2
4 (k 2 λ2De )2
µ2
ω ω
(
)
(2 s
).
=
(γ + Γ1 )(γ + Γ2 ) = (
)
2
2
16
ωpe
ωpe
2
2
2 2
1+ 2
k λDe 1 + 2
ωs k 2 λ2De ω2
ωce
ωce
Taking ky = k⊥ ≃ k for the lower hybrid wave, the threshold condition becomes
Γ1 Γ2
1 V2
=
ωs ωs
16 c2s
(
ω02
which can be rewritten
ω0
V
=4
cs
ωLH
√
2
ωpi
2
ωpe
1+ 2
ωce
)
Γ1 Γ2
.
ωs ω2
For typical damping rates of Γ1 Γ2 /(ωs ω2 ) ≃ 10−2 , V /cs ≃ O(1) is required for
instability if ω0 /ωLH ≃ 3. Essentially the same result can be obtained for decay
into lower hybrid wave and ion cyclotron wave.
Consider now that there is mismatch so ϵRe ̸= 0, and assume that damping
can be ignored (Γ1 = Γ2 = 0). Defining the frequency mismatch as
δj = (
ϵRe
);
∂ϵRe
∂ωj
75
j = 1, 2
the growth rate can be expressed as
γ=
−i(δ1 + δ2 ) ±
√
4A2 − (δ1 − δ2 )2
2
indicating that the frequency mismatch acts like damping to introduces an effective threshold.
14.8
Decay into Quasi-Modes
Quasi-modes do not exist naturally without the nonlinear drive. The low frequency mode (quasi-mode) does not satisfy the linear dispersion relation, so
ϵRe (ω) ̸= 0, but the lower sideband is assumed resonant ϵRe (ω − ) = 0 while the
upper sideband is nonresonant ϵRe (ω + ) ̸= 0. For the quasi-mode |χi (ω)| ≫ 1
and |χi (ω)| ≫ |χe (ω)|, so ϵ(ω) ≃ χi (ω). The parametric dispersion relation is
then
µ2
χe (ω)
1+
= 0.
∂ϵRe
4
ϵRe (ω2 ) − i(γ + Γ2 )
∂ω2
Consider the case in which the low frequency quasi-mode is strongly Landau
damped,
ϵ(ω1 ) = ϵRe (ω1 ) + iϵIm (ω1 ) ≃ ϵRe (ω1 ) + iχeIm (ω1 ).
The growth rate can be obtained by balancing the imaginary parts
γ = −Γ2 +
where
χeIm =
µ2 χeIm (ω1 )
(
)
∂ϵRe
4
∂ω2
1
ζℑZ(ζ).
k 2 λ2De
Resonant decay assumes that the low frequency mode is weakly damped.
In the example of decay into lower hybrid wave and ion acoustic wave, as the
pump wave frequency approaches ωLH , this assumption breaks down since
√
ωs
kcs
k me
=
≃
.
k∥ vte
k∥ vte
k ∥ mi
For the lower hybrid wave (k∥2 /k 2 )(mi /me ) ∼ O(1), so ω1 /(k∥ vte ) ≃ O(1). In
this case ωIm ≃ ωRe since ℑ[ζZ(ζ)] ≃ O(1) and the low frequency mode is
heavily electron Landau damped, indicating that the low frequency mode is a
quasi-mode, not a resonant mode. This is called the electron quasi-mode. The
most unstable situation occurs when χeIm maximizes at 0.76/(k 2 λ2De ),
γ + Γ2 =
V 2 k2
8ω02
ω2
V2
ω2
0.76
= 2 0.76ω2 LH
.
2 k 2 λ2
8cs
ω02
ωpe
De
1+ 2
ωce
76
The threshold is given by setting γ = 0, so
V2
10Γ2 ω02
≃
2 .
2
cs
ω2 ωLH
Note that the growth rate for quasi-mode decay increases like γ ∝ E02 compared
to γ ∝ E0 for resonant decay. Although the threshold is higher than that for
resonant decay, once the threshold is exceeded quasi-mode decay grows faster.
When the low frequency mode is strongly ion-cyclotron damped (ω1 = nωci ),
it is called the ion-cyclotron quasi-mode. In this case the frequency spectrum
typically exhibits many peaks at harmonics of the ion-cyclotron frequency.
When the low frequency mode is strongly ion Landau damped (ω1 = k∥ cs ≃
k∥ vti ), it is called the ion-acoustic quasi-mode. In this case the sideband frequency is usually not separated from the pump wave, and appears like frequency
broadening.
77