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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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how discontinuities form
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
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
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
Ia. Tangential Discontinuity

Bn = 0

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

ib contact discontinuity
Ib. Contact Discontinuity

Bn not zero

  • Jump conditions:
    • [p]=0
    • [vt]=0
    • [Bn]=0
    • [Bt]=0
1b contact discontinuity
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
II Rotational Discontinuity

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

slide9

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

slide11

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.
slide12

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
Analysis
  • How to arrive at three classes of discontinuities
slide15
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
Note that

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

transform reference frame
Transform reference frame
  • Transform to a framemovingwiththediscontinuityatlocalspeed, U.
  • BecauseofGalileaninvariance, time derivative becomes:
slide20

ArriveatRankine-Hugoniotconditions

R-H contain information about any discontinuity in MHD

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

slide21

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

slide22

Next step: quasi-linearize

byintroducingandusingtheaverageofXacross a discontinuity

notingthat

introducingSpecificvolumeV = (nm)-1

introducing normal massflux, F = nmn.

next step algebra
Next step: Algebra

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

Tangential andcontact

Rotational

Shocks

finally
Finally

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

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