Boundaries , shocks , and discontinuities

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# Boundaries , shocks , and discontinuities - PowerPoint PPT Presentation

Boundaries , shocks , and discontinuities. How discontinuities form. Often due to “ wave steepening ” Example in ordinary fluid: V s 2 = dP/d r m P/ r g m =constant (adiabatic equation of state) Higher pressure leads to higher velocity

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Presentation Transcript
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

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).
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
Ia. Tangential Discontinuity

Bn = 0

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

Ib. Contact Discontinuity

Bn not zero

• Jump conditions:
• [p]=0
• [vt]=0
• [Bn]=0
• [Bt]=0
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
II Rotational Discontinuity

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

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

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.
Analysis
• How to arrive at three classes of discontinuities
and
• Assume ideal Ohm’s law: E = -vxB
• Equation of state: P/rgm=constant
• Use special form of energy equation (w is enthalpy):
Note that

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

Transform reference frame
• 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 = nmn.

Next step: Algebra

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

Tangential andcontact

Rotational

Shocks

Finally

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