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AS551 General Plasma Physics 1 \\
November 17, 1997 \\
\vspace{0.1in}
{\bf Problem Set \#7 (due Nov. 24)}
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For a plasma in a stationary equilibrium ($d/dt = 0$), with an isotropic
pressure, the MHD equilibrium relation is:
$$
\grad p = { \vec j \times \vec B \over c}
$$
In the following problems, you will explore some of the
properties of equilibria implied by this equation.

{\bf 1. General cylindrical pinch equilibria.}  Consider a plasma with
cylindrical symmetry ($\partial/\partial z=0$, $\partial/\partial \theta
=0$), with specified radial profiles for the pressure $p(r)$, azimuthal
field $B_\theta(r)$, and axial field $B_z(r)$.  From the MHD equation for
equilibria, derive the equation:
$$
p(r) + {B^2(r) \over 8 \pi} = 
p(0) + {B^2(0) \over 8 \pi} - {1 \over 4 \pi} \int_0 ^r dr {B_\theta^2
\over r}
$$
Assume the plasma extends to $r=a$, and is surrounded by a vacuum (with a
magnetic field) outside that radius.  For the case of uniform current
distribution $j_z =$ constant for $r<a$, and $j_z=0$ for $r>a$, show that
the plasma is paramagnetic, i.e. $B_z(0) > B_z(a)$, if $p(0) <
B_\theta^2(a) / (4 \pi)$. Is this result dependent, or not, on the current
and pressure distributions in the plasma?

{\bf 2. Bennet pinch condition.} Consider a standard $z$-pinch equilibrium
configuration (current in the $z$-direction, magnetic field only in the
$\theta$ direction).  The plasma is uniform in $z$ and $\theta$, and is
confined to $r<a$ by the magnetic field.  Calculate the volume integrated
plasma energy (per length in z):
$$
W = \int_0^a dr 2 \pi r {3 \over 2} p
$$
Integrate this by parts (assuming $p=0$ at $r=a$) to express this in terms
of $\partial p / \partial r$, and use the MHD force balance equation to
show that $W$ can be expressed in terms of the total current carried by the
plasma.  (This is known as the Bennett pinch condition.)

{\bf 3. Complete set of MHD waves.}  (This problem will require the most
algebra.) In the notes entitled ``MHD Waves and Instabilities'', dated
11/22/96, I derived the dispersion relation for compressional Alfv\'en
waves.  Half way down the 4th mini-page of the notes is the general wave
equation in MHD without making any compressional assumptions, which with a
minor modification can be written as:
$$
- \rho_0 \omega^2 \vec \xi = -\Gamma_0 p_0 \vec k \vec k \cdot \vec \xi
- {B_0^2 \over 4 \pi } \hat{z} \times [ \vec k \times ( \vec k \times
  (\hat{z} \times \vec \xi )) ]
$$
This can be written in the form
$$
[-\rho_o \omega^2 \vec{\vec{1}} + \vec{\vec{M}} ] \cdot \vec \xi 
= \vec{\vec{A}} \cdot \vec \xi = 0.
$$
Setting the determinant of the matrix $\vec{\vec{A}}$ to zero will
yield a cubic Eq. in $\omega^2$.  Find the 3 roots for $\omega^2$ and
identify each branch: the ``shear Alfv\'en wave'' (which is
incompressible), the ``compressional Alfv\'en wave'', and the ``sound
wave'' (whose propagation speed depends only on $c_s$ and not on $v_A$).
(To identify the separate branches, sometimes it is useful to consider the
limit of low $\beta= p / (B^2 / 8 \pi)$.)  Also find the corresponding
eigenvector $\vec \xi$ for each of the 3 eigenvalues for $\omega^2$, and
comment on important properties of the ``polarization'' $\vec \xi$ for each
mode.

{\large \bf Order-of-Magnitude Quickies} (Only rough order-of-magnitude
estimates are needed in this section, don't try to solve any differential
equations precisely in these problems.)

\ignore{
{\bf 4. Quasineutrality and $E^2/(8 \pi)$ energy density.}
Consider an electron-proton plasma which is not necessarily neutral.  As an
order of magnitude estimate, simplify the Poisson Eq. to:
$$
{E \over L} = 4 \pi e (n_i -n_e)
$$
where $L$ is a ``characteristic'' (or ``typical'') length.  Show that the
ratio of the electric field energy density $E^2/(8 \pi)$ to the thermal
energy density $3 n_0 T$ (where $n_0=(n_i+n_e)/2$) scales as:
$$
{E^2/(8 \pi) \over 3 n_0 T } \propto \left({n_i-n_e \over n_0} \right)^2
{L^2 \over \lambda_D^2}
$$
where $\lambda_D$ is the Debye length.  What is $L/\lambda_D$ for the
``typical'' laboratory and solar wind plasmas in problem 10 below?
[Implication:  If a plasma is initially neutral, $n_i=n_e$ and $E=0$, then
even if it later converted all of its thermal energy to electrical field
energy via some unspecified process, it would still be very close to neutral,
$(n_i-n_e) \ll n_0$, at least on scales large compared to $\lambda_D$.
}

{\bf 5.  Quasineutrality and $E \times B$ drifts.}  Assume the electric
field is large enough that the $E \times B$ velocity is comparable to the
ion thermal speed $v_{ti} = \sqrt{T_i/m_i}$. Again simplify Poisson's
Eq. using $\grad \cdot \vec E \sim E/L$.  Find the order of magnitude of the
relative charge imbalance $\Delta n/n_0 = (n_E-n_i)/n_0$ needed to produce
this electric field.  Express your result in terms of the fundamental
parameters of the Debye length $\lambda_D$ and the Alfv\'en speed $v_A$.

{\bf 6.} Express the ratio of the Alfv\'en speed to the ion thermal
velocity in terms of $\beta$, the ratio of the plasma pressure to magnetic
pressure.

{\bf 7.  Why magnetic fields don't decay on a $\nu_{ei}$ time scale.}
Assume the initial electron distribution function $f(v)$ is such that the
average electron velocity in any direction is zero.  A given current
density $j \sim n_e e u_e$ is then induced by accelerating all electrons
equally from their initial velocity $v$ to $v+u_e$ in a particular
direction.  What is the increase in their kinetic energy density $\Delta
W_e$?  Calculate the dimensionless ratio $\Delta W_e/ (B^2/(8 \pi))$, using
the rough order-of-magnitude estimate $\grad \times \vec B \sim B/L \sim
4\pi j/c$, where $B$ and $L$ are characteristic magnetic and length scales.
Express your final answer in terms of the collisionless skin depth $c /
\omega_{pe}$.  [Implication: You should find that this ratio is tiny for
typical parameters.  Thus, even though collisional drag would seem to want
to wipe out the electron current in a quick $\nu_{ei}$ time scale, only a
tiny fraction of the magnetic energy is needed during each collision time
to maintain the current.]  

Express the magnetic diffusion coefficient in terms of $\nu_{ei}$ and
$c/\omega_{pe}$.  How is this related to your calculation of $\Delta
W_e/(B^2/(8 \pi)$.

{\bf 8.}  Calculate the ratio of the collisionless skin depth to the ion
gyroradius and express the result in terms of $\beta$.

{\bf 9.}  Express the ratio of the collisionless skin depth to the Debye
length in terms of the electron temperature and the electron rest mass.

{\bf 10. Table of ``typical'' plasma parameters.}  Consider three types of
plasmas: 
(1) a ``typical'' laboratory fusion plasma with $n \sim 10^{14} /$cm$^3$,
$T \sim 10^4$ eV, $B \sim 5$ Tesla, and $L \sim 100$ cm, 
(2) a ``typical'' plasma used for semiconductor etching or materials
processing, with $n \sim 10^{12} /$cm$^3$,
$T \sim 1$ eV, $B \sim 0.1$ Tesla, and $L \sim 10$ cm, 
and (3) the solar wind plasma with $n \sim 5 /$cm$^3$, $T \sim 100$ eV, $B
\sim 5$ nTesla, and $L \sim 10^5$ km (based on the radius of the earth's
magnetopause, which is roughly $10 R_E$, where the earth radius $R_E =
6375$ km.
For these three plasmas calculate the following quantities, listing the
results in a table for comparison:

the plasma to magnetic pressure ratio $\beta$, the Debye length
$\lambda_D$, the collisionless skin depth $c/\omega_{pe}$, the ion
gyroradius $\rho_i$, the mean-free-path between collisions $\lambda_{mfp}$,
(and the ratio between these four lengths and $L$), the Alfv\'en-to-light
speed ratio $v_{A}/c$, and the number of particles in a Debye sphere $n
\lambda_D^3$.

{\bf End of order-of-magnitude section.}

{\bf 11. A puzzler requiring very careful thinking.}

In this problem we will assume that everything is uniform in the $y$ and
$z$ directions, and vary only in the $x$ direction.  The magnetic field
points only in the $z$ direction.  Assume the equations of ideal MHD hold,
with a scalar pressure, and the plasma is confined to the region from
$x=-a$ to $x=+a$ by the magnetic field.  The plasma pressure is uniform for
$|x| < a$. There are 2 ideal conducting walls at $|x| = a + L$, and there
is a vacuum (with a magnetic field) in the region between the plasma and
the walls.

Now suppose that the plasma is heated for a finite time (perhaps by
absorbing photons from a nearby supernova explosion).  As the pressure
increases, the plasma will expand until a new force balance is reached with
the external magnetic field.  (Ignoring the launching of Alfv\'en or sound
waves which might happen if the heating is too rapid.)  If so, what is this
new equilibrium state, and how does it scale with the amount of heating?
What happens in the limits where the initial $\beta$ is very small, or
where the initial $\beta$ (internal plasma pressure divided by external
magnetic field pressure) is 1?

Now redo all of this using a CGL tensor pressure instead of a scalar
pressure, assuming that the parallel temperature is fixed and that the
heating occurs only in $T_\pe$ (perhaps because the heating process uses
waves which resonate with the cyclotron frequency of the particles).
Assume that during any expansion process that $p_\pe/(nB)$ is constant
(ignoring heating).  [Stated another way including heating and dynamics
simultaneously, assume that $3 d(p_\pe/(nB))/dt = P/(nB)$ where $P$ is the
heating power density.] What is a possible physical argument for assuming
this?  Show that this assumption is equivalent to taking $\Gamma=2$ in the
``equation of state'', consistent with $\Gamma=(2+m)/m$, where $m$ is the
``number of degrees of freedom'' which can participate in the relevant
component of the pressure.

How do your answers to the above questions change in the limit as the walls
are moved to infinity ($L \rightarrow \infinity$).

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