Boundaries , shocks , and discontinuities

1. Boundaries, shocks, anddiscontinuities

2. How discontinuities form • Often due to “wave steepening” • Example in ordinary fluid: • Vs2 = dP/drm • P/rgm=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 • Contact Discontinuities • Zero mass flux along normal direction • (a) Tangential – Bn zero, change in density across boundary • (b) Contact – Bn nonzero, no change in density across boundary • Rotational Discontinuity • Non-zero mass flux along normal direction • Zero change in mass density across boundary • Shock • Non-zero mass flux along normal direction • Non-zero change in density across boundary

5. Ia. Tangential Discontinuity Bn = 0 Jump condition: [p+B2/2m0] = 0

6. Ib. Contact Discontinuity Bn not zero • Jump conditions: • [p]=0 • [vt]=0 • [Bn]=0 • [Bt]=0

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. II Rotational Discontinuity • Finite normal massflow • Continuousn • Fluxacrossboundarygivenby • Fluxcontinuityandand [n] => no jump in density. • Bnandnareconstant => tangential components must rotatetogether! Constant normal n => constantAntheWalen relation

10. III Shocks

11. Fast shock 1 2 • Magnetic field increases and is tilted toward the surface and bends away from the normal • Fast shocks may evolve from fast mode waves.

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 = -vxB • Equation of state: P/rgm=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 framemovingwiththediscontinuityatlocalspeed, U. • BecauseofGalileaninvariance, time derivative becomes:

20. ArriveatRankine-Hugoniotconditions R-H contain information about any discontinuity in MHD An additional equationexpressesconservationof total energyacrossthe D, wherebywdenotesthespecificinternalenergy in theplasma, w=cvT.

21. ArriveatRankine-Hugoniotconditions 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 byintroducingandusingtheaverageofXacross a discontinuity notingthat introducingSpecificvolumeV = (nm)-1 introducing normal massflux, F = nmn.

23. Next step: Algebra doingmuchalgebra, ... arriveatdeterminantforthemodifiedsystemof R-H conditions (a seventh-order equation in F) Tangential andcontact Rotational Shocks

24. Finally Insertsolutionsfor F = nmvn back into quasi-linearized R-H equationstoarriveatthreetypesof jump conditions. Forexample, fortheContactandRotationalDiscontinuity: