n e dx ¼ n 0 dx 0
ð3:9:5Þ
With (3.6.2), (3.9.5) reduces to:
∂x
∂x 0
¼ 1 þ V 0
k
ω p0
sin kx 0 À ω p0 t
À
Á
ð3:9:6Þ
The density profile is obtained by inserting (3.9.6) to (3.9.5):
n ¼
n 0
1 þ Δsin kx 0 À ω p0 t
À
Á
Δ ¼ V 0
k
ω p0
ð3:9:7Þ
(3.9.7) is valid as far as Δ < 1, namely, the density has negative part for Δ > 1. At the
time of Δ ¼ 1, the wavebreaking starts to happen to be described soon. It is
important to note that the condition Δ ¼ 1 means that the oscillation velocity is
equal to the phase velocity of the wave:
V 0 ¼
ω p0
k
ð3:9:8Þ
This means that the electrons at the maximum velocity point are trapped by the same
phase of the electric field of the wave and will be continuously feel the electric force
in the same direction.
These nonlinear wave profiles are given in Fig. 3.30. The parameter in Fig. 3.30 is
chosen so that the amplitude just before the wave- reaking happens. If the normalized amplitude Δ exceeds unity in (3.9.7), there is no solution physically acceptable
like the case the overtaking of the steepened shock front without viscosity. It is,
however, noted that the plasma is collisionless, and the wave-breaking is physically
acceptable. For Δ > 1, the density is not given by (3.9.5) where single value of the V
(x,t) is assumed along the axis of x. In such large amplitude, the solution of (3.9.2) is
known to provide the multi-value function of V(x,t) in x-space, and the particle
trapping is seen. This allows the acceleration of the trapped electrons to high energy
like a DC acceleration. This is one of the origins of the generation of hot electrons.
3.9.1 Wave-Breaking
It should be noted that (3.9.1) is an exact equation to the electron plasma oscillation
for the cold electron fluid. (3.6.5) is in the form of a forced oscillation of the electron
fluid in an inhomogeneous plasma density profile. The coordinate x appears only
through ω pe0, and x is the Lagrange coordinate x 0 defined in (3.9.2), and it is
independent of time t in (3.6.5).
Assume the initial condition
3.9 Large Amplitude Electron Plasma Waves
119
ð3:9:5Þ
With (3.6.2), (3.9.5) reduces to:
∂x
∂x 0
¼ 1 þ V 0
k
ω p0
sin kx 0 À ω p0 t
À
Á
ð3:9:6Þ
The density profile is obtained by inserting (3.9.6) to (3.9.5):
n ¼
n 0
1 þ Δsin kx 0 À ω p0 t
À
Á
Δ ¼ V 0
k
ω p0
ð3:9:7Þ
(3.9.7) is valid as far as Δ < 1, namely, the density has negative part for Δ > 1. At the
time of Δ ¼ 1, the wavebreaking starts to happen to be described soon. It is
important to note that the condition Δ ¼ 1 means that the oscillation velocity is
equal to the phase velocity of the wave:
V 0 ¼
ω p0
k
ð3:9:8Þ
This means that the electrons at the maximum velocity point are trapped by the same
phase of the electric field of the wave and will be continuously feel the electric force
in the same direction.
These nonlinear wave profiles are given in Fig. 3.30. The parameter in Fig. 3.30 is
chosen so that the amplitude just before the wave- reaking happens. If the normalized amplitude Δ exceeds unity in (3.9.7), there is no solution physically acceptable
like the case the overtaking of the steepened shock front without viscosity. It is,
however, noted that the plasma is collisionless, and the wave-breaking is physically
acceptable. For Δ > 1, the density is not given by (3.9.5) where single value of the V
(x,t) is assumed along the axis of x. In such large amplitude, the solution of (3.9.2) is
known to provide the multi-value function of V(x,t) in x-space, and the particle
trapping is seen. This allows the acceleration of the trapped electrons to high energy
like a DC acceleration. This is one of the origins of the generation of hot electrons.
3.9.1 Wave-Breaking
It should be noted that (3.9.1) is an exact equation to the electron plasma oscillation
for the cold electron fluid. (3.6.5) is in the form of a forced oscillation of the electron
fluid in an inhomogeneous plasma density profile. The coordinate x appears only
through ω pe0, and x is the Lagrange coordinate x 0 defined in (3.9.2), and it is
independent of time t in (3.6.5).
Assume the initial condition
3.9 Large Amplitude Electron Plasma Waves
119