1. Physics 777 Plasma Physics and Magnetohydrodynamics (MHD)
Instructor: Gregory Fleishman
Lecture 9. Large-Amplitude Waves and Discontinuities
28 October 2008

2. Plan of the Lecture Simple Waves Nonlinear Waves, Solitons
Discontinuities in the Plasma
Shock Waves
Strong Turbulence

3. Section 1. Simple Waves

4. Set of MHD equations reads

5. & If If Section 1.1. Entropy simple waves - arbitrary
For other simple waves
where

6. Section 1.2. Alfven simple waves
&

7. If for some then for for all Section 1.3. Fast and slow simple waves
Consider simple case

9. Section 2. Nonlinear Waves, Solitons
Korteweg – de Vries – Burgers Equation, which includes nonlinearity, dispersion, and dissipation as additive terms:
where
Burgers Equation
Korteweg – de Vries (KdV) Equation

10. Section 2.1. Solitons
Consider KdV Equation
Seek solution

11. Necessary conditions:
Case I
For or
result in

12. Rarefaction!
Case II
Jakobi elliptic function;
- Cnoidal wave
Chain of solitons

13. Section 3. Discontinuities and Shocks in the Plasma

23. Upstream
Downstream

24. Section 4. Evolutionarity of the Discontinuities

26. Section 5. Strong Turbulence

27. Upstream
Downstream
&
&
&

28. Section 6. Homework
Complete Progress Reports on the research project. Include Introduction, the work done with plots, and plan for the remaining work. Be ready for brief presentation of the Progress Report.