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\begin{center}
{\Large{\bf Astrophysical Fluid Dynamics -- Final Exam}}

%{\large{\bf Due in class Wednesday May 2}}
\end{center}

\bigskip
\noindent
%Readings: However much of the hand-outs by Balbus \& Hawley that you like; Shu 93--98, 101--110, 120--123, 125--130

\bigskip

\begin{question}
{\large{\bf Problem 1.}} Adiabatic Wind

A spherically symmetric wind blows steadily
from a star. The wind behaves adiabatically
with the usual adiabatic index $\gamma$.
At infinite distance from the star, is the wind velocity
finite or infinite? Justify your answer.

\end{question}

\begin{question}
{\large{\bf Problem 2.}} Bonnor-Ebert Mass

A cloud of gas having sound speed $c_s$
is bounded by an external medium having pressure
$P_{\rm ext}$. Derive an order-of-magnitude
expression for the critical cloud mass above which
the cloud collapses.

\end{question}

\begin{question}
{\large{\bf Problem 3.}} The Br\"{u}nt in Disks

Consider a vertically thin, circumstellar
disk in vertical hydrostatic equilibrium. 
Neglect disk self-gravity.

Estimate, to order-of-magnitude, the frequency
of buoyant vertical oscillations in the disk,
$\omega_{\rm B-V}$.
Define whatever variables you need, but express
your answer in terms of the
local orbital angular frequency, $\Omega$.

\end{question}

\begin{question}
{\large{\bf Problem 4.}} Sub-Keplerian Magnetized Disks

Consider a circumstellar disk threaded with a poloidal magnetic field, i.e.,
a field whose only components are in the radial and vertical
directions. Work in the ideal MHD limit and neglect disk self-gravity.

The field geometry is such that magnetic tension
forces the disk to rotate at a substantially
sub-Keplerian rate. Its rotation velocity is, say, 0.5 that of the
Kepler velocity because of magnetic tension.

Sketch the disk and its field. Estimate the plasma $\beta$
(ratio of thermal pressure to magnetic pressure)
required to enforce substantially sub-Keplerian rotation.
Define whatever variables you need.

\end{question}

\begin{question}

{\large{\bf Problem 5.}} Instabilities

State as concisely but as completely as possible the requirements for
the following instabilities:

\begin{enumerate}
\item Toomre instability in disks
\item Magneto-rotational instability
\item Thermal convection
\item Richardson shear turbulence
\end{enumerate}

Credit awarded for quantitative statements.

\end{question}

\begin{question}
{\large{\bf Problem 6.}} Tabletop Dynamos

The magnetic diffusivity of liquid mercury is $7 \times 10^3 \cm^2 \s^{-1}$.

Explain quantitatively why it is so difficult to study magnetic dynamos
using table-top experiments.

\end{question}

\end{document}
