Boundaries, shocks, and discontinuities
Dec 31, 2015
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.
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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