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Boundaries , shocks , and discontinuities

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- 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

- 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).

- 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

Bn = 0

Jump condition: [p+B2/2m0] = 0

Bn not zero

- Jump conditions:
- [p]=0
- [vt]=0
- [Bn]=0
- [Bt]=0

- 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

Change in tangential flow velocity = change in tangential Alfvén velocity Occur frequently in the fast solar wind.

II Rotational Discontinuity

- Finite normal massflow
- Continuousn
- Fluxacrossboundarygivenby
- Fluxcontinuityandand [n] => no jump in density.
- Bnandnareconstant => tangential components must rotatetogether!

Constant normal n => constantAntheWalen relation

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.

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.

- How to arrive at three classes of discontinuities

- Assume ideal Ohm’s law: E = -vxB
- Equation of state: P/rgm=constant
- Use special form of energy equation (w is enthalpy):

An integral over a conservation law is zero so gradient operations can be replaced by

- Transform to a framemovingwiththediscontinuityatlocalspeed, U.
- BecauseofGalileaninvariance, time derivative becomes:

ArriveatRankine-Hugoniotconditions

R-H contain information about any discontinuity in MHD

An additional equationexpressesconservationof total energyacrossthe D, wherebywdenotesthespecificinternalenergy in theplasma, w=cvT.

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

Next step: quasi-linearize

byintroducingandusingtheaverageofXacross a discontinuity

notingthat

introducingSpecificvolumeV = (nm)-1

introducing normal massflux, F = nmn.

doingmuchalgebra, ... arriveatdeterminantforthemodifiedsystemof R-H conditions (a seventh-order equation in F)

Tangential andcontact

Rotational

Shocks

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