Boundaries shocks and discontinuities
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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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Boundaries , shocks , and discontinuities

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Boundaries shocks and discontinuities

Boundaries, shocks, anddiscontinuities


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.


Boundaries shocks and discontinuities

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


Iii shocks

III Shocks


Boundaries shocks and discontinuities

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.


Boundaries shocks and discontinuities

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


Start with ideal mhd

Start with ideal MHD


Boundaries shocks and discontinuities

and

  • Assume ideal Ohm’s law: E = -vxB

  • Equation of state: P/rgm=constant

  • Use special form of energy equation (w is enthalpy):


Draw thin box across boundary

Draw thin box across boundary


Use vector calculus

Use Vector Calculus


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:


Boundaries shocks and discontinuities

ArriveatRankine-Hugoniotconditions

R-H contain information about any discontinuity in MHD

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


Boundaries shocks and discontinuities

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


Boundaries shocks and discontinuities

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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