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## Mode-Mode Resonance

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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?**One may rewrite the equation**It indicates that Langmuir wave is coupled to a beam mode.**Consequences depending on nature of coupling**• Propagation and evanescence • Convective instability • Absolute instability**Mode Evanescence andInstability**• Evanescence • Instability**Graphical Description**Beam mode Complex root**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**• 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**Two electron cyclotron-beam modes**Left-hand polarized Right-hand polarized**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**Instability and Growth Rate**• Thus we obtain**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**Physical Parameters**• Natural frequency • Pump or modulation frequency • Modulation amplitude • Oscillator with modulation**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**Dispersion Equation**• Eliminating X and Y we obtain the dispersion equation • Two cases of interest**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.