1. Mode-Mode Resonance A linear-nonlinear process

2. Simple Beam Instability Let us consider It is well known that the equation supports reactive instability. What is the cause of instability?

3. One may rewrite the equation It indicates that Langmuir wave is coupled to a beam mode.

4. Consequences depending on nature of coupling Propagation and evanescence Convective instability Absolute instability

5. Mode Evanescence and Instability Evanescence Instability

6. Graphical Description Complex root Beam mode

7. Stability and propagation

8. Stability and blocking

9. Convective Instability

10. 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.

11. Absolute Instability

12. 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.

13. 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

14. Ion cyclotron-beam instability The dispersion relation is Coupling of beam-cyclotron mode and the electromagnetic ion cyclotron mode leads to two different instabilities

15. Two electron cyclotron-beam modes Left-hand polarized Right-hand polarized

16. Right-hand polarized beam mode

17. Absolute Instability

18. Left-hand polarized beam mode

19. Convective Instability

20. The two beam instabilities Have fundamentally different properties. The right-hand mode is absolutely unstable. The left-hand mode is convectively unstable

21. 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.

22. Dispersion Relation Consider electrostatic waves in a magnetized plasma Consider and obtain

23. Instability and Growth Rate Thus we obtain

25. Mode Coupling and Modulation This is another important process in plasma physics. It is relevant to parametric excitation of waves.

26. An Oscillator with Modulation The equation that describes the motion is The modulation frequency is

27. Physical Parameters Natural frequency Pump or modulation frequency Modulation amplitude Oscillator with modulation

28. 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

29. Two Cases of Interest

31. Dispersion Equation Eliminating X and Y we obtain the dispersion equation Two cases of interest

32. Further Discussion Will be given later when we consider parametric instabilities. The details are similar to those discussed earlier.

33. 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.