Boundaries, shocks, and discontinuities

How discontinuities form

• Often due to “wave steepening”

• Example in ordinary fluid:– Vs

2 = dP/dm

– P/m=constant (adiabatic equation of state)

– Higher pressure leads to higher velocity– High pressure region “catches up” with low

pressure regionThe 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).

ClassificationI. 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

II. Rotational Discontinuity

• Non-zero mass flux along normal direction

• Zero change in mass density across boundary

III. Shock

• Non-zero mass flux along normal direction

• Non-zero change in density across boundary

Ia. Tangential Discontinuity

Bn = 0 Jump condition: [p+B2/20] = 0

Ib. Contact Discontinuity

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

Bn not zero

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.

• Finite normal mass flow

• Continuous n

• Flux across boundary given by

• Flux continuity and and [n] => no jump in density.

• Bn and n are constant => tangential components must rotate together!

Constant normal n => constant An the Walen relation

II Rotational Discontinuity

III Shocks

Fast shock

• Magnetic field increases and is tilted toward the surface and bends away from the normal

• Fast shocks may evolve from fast mode waves.

21

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

Start with ideal MHD

and

• Assume ideal Ohm’s law: E = -v x B

• Equation of state: P/m=constant

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

Draw thin box across boundary

Use Vector Calculus

Note that

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

Transform reference frame

• Transform to a frame moving with the discontinuity at local speed, U.

• Because of Galilean invariance, time derivative becomes:

Arrive at Rankine-Hugoniot conditions

An additional equation expresses conservation of total energy across the D, whereby w denotes the specific internal energy in the plasma, w=cvT.

R-H contain information about any discontinuity in MHD

Arrive at Rankine-Hugoniot conditions

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

by introducing and using the average of X across a discontinuity

noting that

introducing Specific volume V = (nm)-1

introducing normal mass flux, F = nmn.

doing much algebra, ... arrive at determinant for the modified system of R-H conditions (a seventh-order equation in F)

Tangential and contact

Rotational Shocks

Next step: Algebra

Insert solutions for F = nmvn back into quasi-linearized R-H equations to arrive at three types of jump conditions. For example, for the Contact and Rotational Discontinuity:

Finally