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
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
Bn = 0
Jump condition: [p+B2/2m0] = 0
Bn not zero
Change in tangential flow velocity = change in tangential Alfvén velocity Occur frequently in the fast solar wind.
Constant normal n => constantAntheWalen relation
An integral over a conservation law is zero so gradient operations can be replaced by
R-H contain information about any discontinuity in MHD
An additional equationexpressesconservationof total energyacrossthe D, wherebywdenotesthespecificinternalenergy in theplasma, w=cvT.
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:
tangential electric field
Next step: quasi-linearize
byintroducingandusingtheaverageofXacross a discontinuity
introducingSpecificvolumeV = (nm)-1
introducing normal massflux, F = nmn.
doingmuchalgebra, ... arriveatdeterminantforthemodifiedsystemof R-H conditions (a seventh-order equation in F)
Insertsolutionsfor F = nmvn back into quasi-linearized R-H equationstoarriveatthreetypesof jump conditions. Forexample, fortheContactandRotationalDiscontinuity: