Boundaries shocks and discontinuities How discontinuities form Often
Boundaries, shocks, and discontinuities
How discontinuities form • Often due to “wave steepening” • Example in ordinary fluid: – Vs 2 = d. P/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-systemschool. 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 I. 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+B 2/2 m 0] = 0
Ib. Contact Discontinuity Bn not zero Jump conditions: [p]=0 [vt]=0 [Bn]=0 [Bt]=0
1 b. 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 mass flow • Continuous n • Flux across boundary given by • Flux continuity and [ n] => no jump in density. • Bn and n are constant => tangential components must rotate together! Constant normal n => constant An the Walen relation
III Shocks
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
Start with ideal MHD
and • Assume ideal Ohm’s law: E = -v x B • Equation of state: P/rgm=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 R-H contain information about any discontinuity in MHD An additional equation expresses conservation of total energy across the D, whereby w denotes the specific internal energy in the plasma, w=cv. T.
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 = nm n.
Next step: Algebra 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
Finally 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:
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