1. Fluid Theory: Magnetohydrodynamics (MHD)

4. Theoretical Wave Analyses
Oscillations: frequency, amplitude
Waves: amplitude, frequency, wavelength, (propagation speed)
at a fixed point: oscillation is time
a snapshot: oscillation in space
a movie shot: propagating oscillations
phase velocity: following a fixed phase
Eigen modes: coherent oscillation propagation
intrinsic to the medium (independent of source for a frequency)
dispersion relation: frequency dependence on wavelength, angle
perturbation relations: among different quantities
In finite space: Eigen modes can also refer to resonating frequencies

5. MHD Waves

6. MHD Waves, cont.

9. MHD Waves, continue Dispersion relations Phase velocity
Perturbation relations
Walén relation (for intermediate mode)
: for k antiparallel to B and parallel to B

10. Alfvén mode (Intermediate Mode)
In the Alfven mode, perturbations of B are perpendicular to both B and k. There is no change in density r or in the magnitude of B.
The changes involve only bends of B.
Alfvén mode waves work to reduce bending of the magnetic field. They carry field-aligned currents.

11. Fast and Slow Modes
The fast (+ sign) and slow (- sign) modes that make up the other two solutions are compressional (i.e. they do change density and field magnitude).
Fast waves are produced when the total pressure of the plasma (the sum of the particle pressure and field pressure) changes locally. The plasma and magnetic pressures are in phase. This wave propagates almost isotropically.

12. Fast and Slow Modes – cont.
For the slow mode the total pressure is approximately constant across the background field. Slow mode waves carry energy primarily along the background field. Field-aligned gradients in the total pressure drive slow mode waves.

13. Homework
3.1, 3.7, 4.10, 5.2,
3.10*, 3.12*, 4.8*
Hint, 4.10: the wave energy is the part of the energy that is associated with the wave or perturbations.
Hint, 4.10: equations (4.34) and (4.35) are incorrect.

14. 1-D discontinuities Shock Front v1 v2 Upstream (low entropy)
Downstream
(high entropy) v1 v2

15. Rankine-Hugoniot Relations

16. Types of Discontinuities in Ideal MHD
The R-H jump conditions are a set of 6 equations. If we want to find the downstream quantities given the upstream quantities then there are 6 unknowns ( ,vn,,vT,,p,Bn,BT).
The solutions to these equations are not necessarily shocks. These conservations laws and a multitude of other discontinuities can also be described by these equations.
Types of Discontinuities in Ideal MHD
Contact Discontinuity ,
Density jumps arbitrary, all others continuous. No plasma flow. Both sides flow together at vT.
Tangential Discontinuity
Complete separation. Plasma pressure and field change arbitrarily, but pressure balance
Rotational Discontinuity
Large amplitude intermediate wave, field and flow change direction but not magnitude.

17. Types of Shocks in Ideal MHD
Shock Waves
Flow crosses surface of discontinuity accompanied by compression.
Parallel Shock
B unchanged by shock.
Perpendicular Shock
P and B increase at shock
Oblique Shocks
Fast Shock
P, and B increase, B bends away from normal
Slow Shock
P increases, B decreases, B bends toward normal.
Intermediate Shock
B rotates 1800 in shock plane, density jump in anisotropic case

18. Configuration of magnetic field lines for fast and slow shocks
Configuration of magnetic field lines for fast and slow shocks. The lines are closer together for a fast shock, indicating that the field strength increases. [From Burgess, 1995].

19. For compressive fast-mode and slow-mode oblique shocks the upstream and downstream magnetic field directions and the shock normal all lie in the same plane: coplanarity theorem.
The transverse component of the momentum equation can be written as and Faraday’s Law gives
Therefore both and are parallel to and thus are parallel to each other.
Thus Expanding
If and must be parallel.
The plane containing one of these vectors and the normal contains both the upstream and downstream fields.
Since this means both and are perpendicular to the normal and

20. Formation of Sonic Shock