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Perpendicular Shocks
The second special case is the socalled perpendicular shock in which both the
upstream and downstream plasma flows are perpendicular to the magnetic field, as well as the shock
front. In other
words,
Substitution into the general jump conditions (933)(938) yields
where

(960) 
and is a real positive root of the quadratic

(961) 
Here,
.
Now, if and are the two roots of Eq. (961) then

(962) 
Assuming that , we conclude that one of the roots is negative,
and, hence, that Eq. (961) only possesses one physical
solution: i.e., there is only one type of MHD shock which is
consistent with Eqs. (954) and (955). Now, it is easily
demonstrated that and
. Hence, the
physical root lies between and
.
Using similar analysis to that employed in the previous subsection, it
is easily demonstrated that the second law of thermodynamics requires a
perpendicular shock to be compressive: i.e., . It follows that a physical solution
is only obtained when , which reduces to

(963) 
This condition can also be written

(964) 
where
is the upstream
Alfvén velocity. Now,
can be recognized as the velocity of a fast wave propagating
perpendicular to the magnetic fieldsee Sect. 5.4. Thus, the
condition for the existence of a perpendicular shock is that the relative
upstream plasma velocity must be greater than the upstream fast wave velocity. Incidentally, it is easily
demonstrated that if this is the case then the downstream plasma velocity is less than the downstream
fast wave velocity. We can also deduce that, in a stationary plasma, a
perpendicular shock propagates across the magnetic field with
a velocity which exceeds the fast wave velocity.
In the strong shock limit, , Eqs. (960) and (961) become identical to Eqs. (945) and (946).
Hence, a strong perpendicular shock is very similar to a strong hydrodynamic shock (except that the former shock
propagates perpendicular, whereas the latter
shock propagates parallel, to the magnetic field). In particular, just like a hydrodynamic shock, a
perpendicular shock cannot
compress the density by more than a factor
. However, according to
Eq. (956), a perpendicular shock compresses the magnetic field by the same
factor that it compresses the plasma density. It follows that there is
also an upper limit to the factor by which a perpendicular shock can compress the magnetic field.
Next: Oblique Shocks
Up: Magnetohydrodynamic Fluids
Previous: Parallel Shocks
Richard Fitzpatrick
20110331