AS551 General Plasma Physics 1 December 2, 1996

In-class Exam II.

1. 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.

2. 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.]

3. Choice of a wave or equilibrium derivation (25 minutes).

Do either problem 3A or problem 3B:

(3A) Shear-Alfvén wave. Starting from the ideal MHD equations, derive the dispersion relation for the shear-Alfvén wave, propagating in a uniform plasma with a uniform equilibrium magnetic field B⃗_{0} = B_{0}ẑ.

(3B) Bennett pinch equilibrium.

Consider a standard z-pinch equilibrium configuration (current in the z-direction, magnetic field only in the θ direction), with cylindrical symmetry (∂/∂θ = ∂/∂z = 0). Calculate the volume integrated plasma energy (per length in z):

W = ∫_{0}^{a} dr2πr\frac{3}{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.

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.