mode-mode resonance a linear-nonlinear process. simple beam instability let us consider it is well...
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Mode-Mode Resonance
A linear-nonlinear process
Simple Beam Instability
• Let us consider
• It is well known that the equation supports reactive instability.
• What is the cause of instability?
2 2
220 0
1 0pe pebn
n kv
One may rewrite the equation
It indicates that Langmuir wave is coupled to a beam mode.
2 2
220 0
1 0pe pebn
n kv
22 2 2 20
0
bpe pe
nkv
n
Consequences depending on nature of coupling
• Propagation and evanescence
• Convective instability
• Absolute instability
Mode Evanescence and Instability
• Evanescence
• Instability
2 20pe kv
2 20pe kv
Graphical Description
Complex root
Beam mode
Stability and propagation
Stability and blocking
Convective Instability
Convective Instability
• The frequency is complex in certain range of k so that the system is unstable.
• The roots of the unstable roots are in the same half plane of k.
The instability is convective.
Absolute Instability
Absolute Instability
• The frequency is complex in certain range of k so that the system is unstable.
• The roots of the unstable roots are in opposite half planes of k.
Thus the instability is absolute.
Two Other Electron Beam Instabilities
• Beam mode coupled with right-hand polarized ion cyclotron wave
• Beam mode couple with left-hand polarized ion cyclotron wave
Ion cyclotron-beam instability
• The dispersion relation is
• Coupling of beam-cyclotron mode and the electromagnetic ion cyclotron mode leads to two different instabilities
2 2
2 2 2
0
pi be pe
i i
nk c kV kV
n
Two electron cyclotron-beam modes
• Left-hand polarized
• Right-hand polarized
0ekV
0ekV
Right-hand polarized beam mode
Absolute Instability
Left-hand polarized beam mode
Convective Instability
The two beam instabilities
• Have fundamentally different properties.
• The right-hand mode is absolutely unstable.
• The left-hand mode is convectively unstable
Modified Two Stream Instability
• The instability is related to shock wave study in the early 1970s.
• The instability theory is rather simple and the physics is fairly interesting.
• From the viewpoint of mode-coupling process it is obvious.
Dispersion Relation
• Consider electrostatic waves in a magnetized plasma
• Consider and obtain
2 2 2 2 2
2 2 2 20
cos sin1
( )pe pe pi
e k v
2cos /e pm m
2 2 2pe pe
Instability and Growth Rate
• Thus we obtain
2 2 2
2 2 20( )
pi pe e pi
UH p UH
m
k v m
0k UH kv
Mode Coupling and Modulation
• This is another important process in plasma physics.
• It is relevant to parametric excitation of waves.
An Oscillator with Modulation
• The equation that describes the motion is
• The modulation frequency is
201 2 cos 0X t X
0
Physical Parameters
• Natural frequency
• Pump or modulation frequency
• Modulation amplitude
• Oscillator with modulation
0
01 cos t
Fourier transform leads to• Two coupled oscillators if
where only terms close to the natural frequency are retained. Eventually we obtain the following dispersion equation
2 2 20
2 2 21
( ) ( )
( ) ( )
X X
X X
22 2 2 2 40
0 2
Two Cases of Interest
0
0
( ) 2
( )
a
b
22 2 2 2 40
2 40 0
0 2
2 4
2
4
Dispersion Equation• Eliminating X and Y we obtain the
dispersion equation
• Two cases of interest
1 2 0 2 0
21 2 0 2 0 2 0
( ) ( ) ( )
( ) ( )
D D D
Z D D
1 2 0( ) ( ) 0, ( ) 0a D D
1 0 2 0( ) ( ) 0, ( ) 0b D D
Further Discussion
Will be given later when we consider parametric instabilities. The details are similar to those discussed earlier.
Summary and Conclusions
• Mode coupling in general plays important roles.
• It can lead to reactive instabilities such as various types of beam instabilities.
• The coupled oscillator problem is an introduction of the theory of parametric instability.
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