1. Boundaries, shocks, and discontinuities

2. How discontinuities form Often due to “wave steepening” Example in ordinary fluid: –V s 2 = dP/d  m –P/   m =constant (adiabatic equation of state) –Higher pressure leads to higher velocity –High pressure region “catches up” with low pressure region The following presentation draws from Basic Space Plasma Physics by Baumjohann and Treumann and http://www.solar-system- school.de/lectures/space_plasma_physics_2007/Lecture_8.ppt

3. Shock wave speed Usually between sound speed in two regions Thickness length scale –Mean free path in gas (but in collisionless plasma this is large) –Other length scale in plasma (ion gyroradius, for example).

4. Classification I.Contact Discontinuities Zero mass flux along normal direction (a) Tangential – B n zero, change in density across boundary (b) Contact – B n nonzero, no change in density across boundary II.Rotational Discontinuity Non-zero mass flux along normal direction Zero change in mass density across boundary III.Shock Non-zero mass flux along normal direction Non-zero change in density across boundary

5. Ia. Tangential Discontinuity B n = 0 Jump condition: [p+B 2 /2  0 ] = 0

6. Ib. Contact Discontinuity Jump conditions: [p]=0 [v t ]=0 [B n ]=0 [B t ]=0 B n not zero

7. 1b. Contact discontinuity Change in plasma density across boundary balanced by change in plasma temperature Temperature difference dissipates by electron heat flux along B. Bn not zero Jump conditions: –[p]=0 –[vt]=0 –[Bn]=0 –[Bt]=0

8. II Rotational Discontinuity Change in tangential flow velocity = change in tangential Alfvén velocity Occur frequently in the fast solar wind.

9. Finite normal mass flow Continuous  n Flux across boundary given by Flux continuity and and [  n ] => no jump in density. B n and  n are constant => tangential components must rotate together! Constant normal  n => constant  An the Walen relation II Rotational Discontinuity

10. III Shocks

11. Fast shock Magnetic field increases and is tilted toward the surface and bends away from the normal Fast shocks may evolve from fast mode waves. 21

12. Slow shock Magnetic field decreases and is tilted away from the surface and bends toward the normal. Slow shocks may evolve from slow mode waves.

13. Analysis How to arrive at three classes of discontinuities

14. Start with ideal MHD

15. and Assume ideal Ohm’s law: E = -v x B Equation of state: P/   m =constant Use special form of energy equation (w is enthalpy):

16. Draw thin box across boundary

17. Use Vector Calculus

18. Note that An integral over a conservation law is zero so gradient operations can be replaced by

19. Transform reference frame Transform to a frame moving with the discontinuity at local speed, U. Because of Galilean invariance, time derivative becomes:

20. Arrive at Rankine-Hugoniot conditions An additional equation expresses conservation of total energy across the D, whereby w denotes the specific internal energy in the plasma, w=c v T. R-H contain information about any discontinuity in MHD

21. Arrive at Rankine-Hugoniot conditions The normal component of the magnetic field is continuous: The mass flux across D is a constant: Using these two relations and splitting B and v into their normal (index n) and tangential (index t) components gives three remaining jump conditions: stress balance tangential electric field pressure balance

22. Next step: quasi-linearize by introducing and using the average of X across a discontinuity noting that introducing Specific volume V = (nm) -1 introducing normal mass flux, F = nm  n.

23. doing much algebra,... arrive at determinant for the modified system of R-H conditions (a seventh-order equation in F) Tangential and contact Rotational Shocks Next step: Algebra

24. Insert solutions for F = nmv n back into quasi-linearized R-H equations to arrive at three types of jump conditions. For example, for the Contact and Rotational Discontinuity: Finally