Physics 777 Plasma Physics and Magnetohydrodynamics (MHD)

1. Physics 777Plasma Physics and Magnetohydrodynamics (MHD) Instructor: Gregory Fleishman Lecture 3. Plasma Dispersion 23 September 2008

2. Plan of the Lecture • Macroscopic Maxwell Equations. Linear response • Eigen-modes of anisotropic and Gyrotropic media; General Case • Cold Plasma Approximation • Multi-Component Plasma • Maxwellian plasma • Collisional and Collisionless Plasma • MHD approximation

3. Section 1. Macroscopic Maxwell Equations. Linear response Introduce polarization vector; continuity Eqn. is fulfilled: Form displacement vector: D=E+4pP; the most general (non-local) linear relation for statistically uniform medium reads:

4. Section 1. Macroscopic Maxwell Equations. Linear response Make the Fourier transform: where Excluding the magnetic field from Maxwell equations we obtain For the E Fourier component:

5. A convenient, more compact form: where is the Maxwellian tensor, j is an external electric current (including nonlinear plasma current in a general case). Note: for any medium without dissipation the dielectric tensor is Hermitian (and vice versa). If the dielectric tensor is Hermitian, the Maxwellian tensor is Hermitian too.

6. Section 2. Eigen-modes of anisotropic and Gyrotropic media; General Case Normalized polarization vector - Principal values - Refractive index

7. The dispersion relations are found from equating the determinant to zero: where Thus, if spatial dispersion is insignificant, the dispersion relations can be calculated as a solution of quadratic equation for any medium Note, that Hermitian (e.g., Maxwellian) tensor can be transformed to a diagonal form with real eigen-values on the basis of orthogonal complex eigen-vectors, i.e.:

8. In components:

9. Section 3.Cold plasma approximation. Subsection 3.1. Isotropic plasma Subsection 3.2. Magnetized plasma

10. As we had for the isotropic plasma, we can write: Section 4. Multi-component plasma Take into account both electron and ion contributions: If there are many specious of ions in the plasma, then each of them will contribute into the dielectric tensor components. The contributions of the heavier ions are especially important around corresponding gyroresonance. Another important example of the multi-component plasma is a plasma composed of normal (cold) component and high-energy (warm/hot) component, which will be studied later.

11. Section 5. Maxwellian plasma Fourier ?

12. Subsection 5.1. Isotropic Maxwellian plasma Taking into account both electron and ion contribution we obtain: Calculate this approximately for the cases of small and large arguments of F

13. Subsection 5.2. Magnetized Maxwellian plasma where

14. Section 6. Collisional and collisionless plasma =(f0-f)/t

15. Section 7. MHD approximation

16. Fourier transform yields:

17. Section 8. Homework • 1. Check and correct the homework from Lecture 1. • 2. Calculate the plasma permeability approximately for the cases of small and large arguments of F (see Sec. 5)