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\begin{center}
AS551 General Plasma Physics 1 \\
January 18, 2001 \\
1:30 - 4:30 p.m. \\
\bf Final Exam \\
(closed book, except for NRL formulary given to you) \\
\end{center}

{\bf \noindent 180 points total: 1 point $\approx$ 1 minute.  Do not
spend too much time on the short (5-15 point) problems that need only brief
answers.  Note that you have a choice on the final question, do ONLY
problem 10A or 10B.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vskip 10pt
\noindent {\bf 1. [20 points]}  Write down a complete set of ideal MHD
equations.  Briefly describe each equation (a sentence or phrase or two for
each Eq. is sufficient).

Briefly describe 2 of the important assumptions made in the derivation of
these equations, and their impact on the properties of the MHD equations.

% Briefly: are $\omega_{pe}$ plasma oscillations included in ideal MHD? Why
% or why not?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \noindent {\bf 2. [15 points]}  Show how to derive the Debye length.

\noindent {\bf 2. [10 points]} Write down a phase-space conservation law
for the distribution function of particles responding to an arbitrary force
$\vec A(\vec x, \vec v, t)$.  What condition must the force satisfy in
order for the usual phase-space volume conservation property of the Vlasov
equation to hold?

\noindent {\bf 3. [5 points]} Express the ratio of the ion thermal
velocity to the Alfv\'en velocity in terms of another common plasma
parameter (assume equal ion and electron temperatures).

\noindent {\bf 4. [10 points]} Provide an order-of-magnitude estimate 
of the ratio of the distance of closest approach for a 90 degree scattering
event to the average distance between particles.  Express this ratio in
terms of a common plasma parameter.

\noindent {\bf 5. [10 points]} Provide an order-of-magnitude estimate of
the ratio of the ion mean-free-path to the electron mean-free-path
(assuming equal electron and ion temperatures and ion charge $Z=1$ for
simplicity).

\noindent {\bf 6. [10 points]} In MHD equilibrium, $\grad p = \vec j \times
\vec B / c$ implies the current is proportional to a pressure gradient.
But the single particle guiding center drifts depend only on the magnetic
fields and not on the pressure gradient of other particles.  Briefly
explain this apparent paradox and illustrate it with a sketch.

\noindent {\bf 7. [10 points]}  Consider an initially uniform magnetic
field pointing in the $x$ direction in an ideally conducting plasma.  Due
to some external force, the plasma develops a sheared flow with $\vec u =
\hat y u_0 \times (1-|x|/L)$ for $|x| < L$, and $\vec u =0$ otherwise.
(Assume this velocity remains fixed and neglect
the back reaction force of the magnetic field.) Sketch the magnetic field
at a later time.  Has the magnetic pressure in the $|x| < L$ region gone up
or down or stayed the same?

\noindent {\bf 8. [10 points]} Consider the magnetic field produced by a
small ring of current enclosing the $z$ axis.  The magnetic field far from
the ring has a dipole structure, so on the midplane $|B| \propto 1/R^3$,
where $R$ is the distance from the ring. Sketch the particle orbits in the
$(R,z)$ and $(x,y)$ planes in the guiding center approximation, $\rho \ll
R$, for a particle with $v_\Par \ll v_\Perp$.

The current in the ring is now increased very slowly (so all adiabatic
invariants are conserved).  Does this particle move in or out?

\noindent {\bf 9. [45 points] Shear Alfv\'en Waves}.  Start with the
equations of ideal MHD, but add a viscous drag term to the momentum
equation, of the form:
%
\be
\rho {d \vec u \over d t} = \dots + \chi \rho \grad^2 \vec u
\ee 
%
where $\dots$ are the usual terms in the MHD momentum equation.  (All of the
other MHD equations remain unchanged.)  Show how to derive the $\omega$
vs. $k$ dispersion relation for incompressible shear-Alfven waves. In the
limit of small but non-zero viscosity coefficient $\chi$, calculate what
the damping rate is for the wave.  With a sketch and a sentence or two,
briefly describe the role of the magnetic field in the dynamics of this
wave.

\ignore{
\noindent {\bf 5. \ [30 points]} Starting from fluid equations for ions,
derive the dispersion relation for electrostatic ion acoustic waves.  For
simplicity, you can assume quasineutrality and use a linearized adiabatic
equation of state for the ions, $\tilde{p} = \Gamma T_{i0} \tilde{n}_i$
(also, you can use 1-D fluid equations for the ions if you wish).  Use a
linearized Boltzmann response for the electrons, $\tilde{n}_e = n_{e0} (e
\Phi / T_{e0})$.  Under what conditions is such a Boltzmann electron
response appropriate?  Does this wave still propagate in the limit of
$T_i/T_e \rightarrow 0$?
} % end ignore

{\bf 10. \ [50 points] Instabilities.  Do {\large \bf EITHER} 10A or 10B,
not both.}

{\bf 10A. Two-Stream Instability}

Consider a uniform plasma of cold electrons and a beam of ions, where the
equilibrium distribution functions for electrons and ions vs.~velocity in
the $z$ direction are $f_e(v_z) = n_e \delta(v_z)$ and $f_i(v_z) = n_e
\delta(v_z - u_0)$.  Starting with the Vlasov equation, linearize it for
electrostatic perturbations $\propto \exp(i k z - i \omega t)$ and show how
to derive the dispersion relation
%
\be
1 - {\omega_{pe}^2 \over \omega^2} - {\omega_{pi}^2 \over (\omega - k
u_0)^2} = 0
\ee
%
In the limit of small but finite $|\omega_{pi}/\omega_{pe}|$ and
$|k u_0/ \omega_{pe}|$, calculate the growth rate of the instability.

{\bf 10B. Rayleigh-Taylor Instability}

Consider a plasma suspended against gravity by a magnetic field.  Denote
$g$ as the acceleration due to gravity (in the $-\hat{y}$ direction).
There is a stationary equilibrium where quantities vary only in the
$\hat{y}$ direction and the equilibrium magnetic field is in the $\hat{z}$
direction.  Consider small amplitude perturbations with the perturbed flow
in the $(x,y)$ plane so the magnetic field remains in the $\hat{z}$
direction.  Write down the linearized momentum equation.  Taking the curl
of this equation to eliminate the plasma and magnetic pressure, and looking
at its $\hat{z}$ component, show how to derive the equation,
%
\be
\rho_0 {\partial \over \partial x} {\partial u_y \over \partial t} -
{\partial \over \partial y} \left[ \rho_0 {\partial u_x \over \partial t}
\right] = - {\partial \tilde{\rho} \over \partial x} g
\ee
%
Assuming incompressibility and Fourier transforming in time and $x$, show
how to derive a second order differential equation for $u_y$ which
determines the eigenfrequency.  For an exponential density profile,
$\partial \rho_0 / \partial y = \rho_0/s$, calculate an instability growth
rate in the limit of very large wave number $k_x$.

\newpage

{\large \bf Answers:}

Overall this exam might be a bit on the easy side, but I think it did test
a broad range of plasma conceptual ideas and problem-solving skills.  Here
are answers to some of the conceptual issues that some people had.

1. Standard.

2.  Starting from a general conservation law for a distribution function
$f(\vec x_6, t)$ in a 6 dimensional phase space:
%
\be
{\partial f \over \partial t} + \grad_6 \cdot ( \dot{\vec x}_6 f) = 0
\ee
%
\be
{\partial f \over \partial t} + \vec x + \cdot \grad f + \vec A \cdot \grad_v
f + f \grad_v \cdot A = 0
\ee
and we get the usual form that conserves phase-space volume ($Df/Dt = 0$)
only if $\grad_v \cdot A = 0$.  This will be true for a Hamiltonian system.

3. $v_{ti} / v_A \approx (\beta / 2)^{1/2}$, where $\beta$ (ratio of
plasma to magnetic pressure) is {\it a} common plasma parameter.

4. Closest approach distance $b$ determined by $T \sim e^2/b$.  Average
interparticle spacing $n^{-1/3}$.  Their ratio scales as $1/\Lambda^{2/3}$,
where $\Lambda \sim n \lambda_d^3$ is {\it the} plasma parameter and is
usually a very big number.

5. $\nu_e/\nu_i \sim \nu_{ei}/\nu_{ii} \sim \sqrt{m_i/m_e}$, so the ion and
electron mean free path's turn out to be comparable.

6. Spitzer explained this apparent paradox, emphasizing that ``fluid flows
$\neq$ guiding center drifts''.  Picture is on p. 99 of Goldston and
Rutherford.

7. Standard frozen-in field line result.  $B_x$ is constant (to conserve
flux through an $x=$constant plane) while $B_y$ goes up, so the magnetic
pressure does go up.

8. Wanted you to sketch grad B and curvature drift around the ring (and some
bouncing motion along field line of the trapped particles).

The question of the direction of the particle motion is a fascinating
subtle problem that requires careful thinking.  Many people have the wrong
intuition about it, but about 25\% of you got the direction of the particle
motion right. [Because its somewhat non-intuitive, I didn't take off much
for those of you who got the sign wrong but invoked some of the right
physics.]  

It turns out that the particle moves outward.  In fact, it moves outwards
to regions of lower $|\vec B|$ faster than the magnetic field at a fixed
position is increasing, and the net result is that the energy of the
particle actually drops.  A harder variant of this problem is described
at \underline{http://w3.pppl.gov/$\sim$hammett/gpp1/counter-intuititive}.

9.  This is a straightforward calculational problem.  One should be careful
to note that the shear Alfven-wave is not restricted to have $k_\Perp=0$.
The dispersion relation $\omega^2 = k_\Par^2 v_A^2$ is valid for arbitrary
$\vec k$, it just turns out to depend only on the parallel component of
$\vec k$.

10 A.  This is a variant of the two-stream instability as done in class and
described in Goldston and Rutherford.  The derivation of the dispersion
relation is straightforward.  Writing it in terms of normalized quantities,
$\Omega = \omega/\omega_{pe}$, $\epsilon=\omega_{pi}^2/\omega_{pe}^2 =
m_e/m_i$, and $\alpha=k u_0 / \omega_{pe}$, it can be written as
%
\be
(\Omega^2 - 1) (\Omega - \alpha)^2 = \epsilon^2 \Omega^2
\ee
%
Taking the $\epsilon=0$ limit, to lowest order the roots are $\Omega = \pm
1$ and $\Omega=\alpha$ (a double root).  Substituting $\Omega = \Omega_0 +
\Delta \Omega$, where $\Omega_0=\alpha$ (one can show that the
$\Omega_0=\pm 1$ roots are stable), one finds that the disperion relation to
next order is
%
\be
(\alpha^2 + 2 \alpha \delta \Omega -1)( \delta \Omega)^2 = \epsilon^2 \alpha^2
\ee
%
As described in the lecture notes and Goldston and Rutherford, this will
give a growth rate as a function of $k$ (or $\alpha$) with the most
unstable mode $k$ such that $\alpha =1$, in which case this becomes a cubic
equation for $\delta \Omega$ to lowest order.  I was trying to simplify
this problem for you some by saying you should look at the small $\alpha$
limit, in which case this simplifies to $\delta \Omega = \pm i | \epsilon
\alpha|$, or in the original notation, the growth rate is $|\omega_{pi} k
u_0 / \omega_{pe} |$.

Knowing how to find approximate solutions to complicated equations (like a
quartic or complicated integrals) is a very powerful skill that you can
learn a lot more about from Prof. White's asymptotic math class (descended
from a course that Kruskal taught).

10 B. This problem is done in Goldston and Rutherford and in the lecture
notes.

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