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AS551 General Plasma Physics 1 \\
December 2, 1996
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\bf In-class Exam II.
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\noindent 1. {\bf Fluid derivations (10 minutes).}

In the derivation of fluid equations from the Vlasov Eq., closure
approximations have to be made.  Briefly describe why this is so.  Briefly
describe 2 different closure approximations.

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\noindent 2. {\bf MHD overview (10 minutes).}

Write down a complete set of ideal MHD equations.

Briefly describe 2 important assumptions (other than fluid closure
approximations) in the derivation of the ideal MHD equations.  Briefly
mention what some of the consequences of these assumptions are.  [For
example, what phenomena are lost or ordered out of the equations by these
assumptions.]

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\noindent 3. {\bf Choice of a wave or equilibrium derivation (25 minutes).}

Do {\it either} problem 3A or problem 3B:

{\bf (3A) Shear-Alfv\'en wave.}  Starting from the ideal MHD equations,
derive the dispersion relation for the shear-Alfv\'en wave, propagating in
a uniform plasma with a uniform equilibrium magnetic field $\vec B_0 = B_0
\hat{z}$.

{\bf (3B)  Bennett pinch equilibrium.}

Consider a standard $z$-pinch equilibrium configuration (current in the
$z$-direction, magnetic field only in the $\theta$ direction), with
cylindrical symmetry ($\partial/\partial \theta = \partial/\partial z =0$).
Calculate the volume integrated plasma energy (per length in z):
$$
W = \int_0^a dr 2 \pi r {3 \over 2} p
$$
Assuming that ideal MHD holds, and that the plasma is confined to $r<a$ (so
$p=0$ at $r=a$), show that $W$ can be expressed in terms of the total
current carried by the plasma.

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\noindent {\bf 4. Quickie (5 minutes).}
Briefly describe what is meant by a flux surface.  Show that in ideal MHD
equilibria, surfaces of constant pressure are flux surfaces.

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