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.