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\begin{document}
{\noindent\bf MAE224A: Environmental Fluid Mechanics}
\hfill Fall Quarter 2010
\bigskip

\hfill \textrm{http://maecourses.ucsd.edu/~sllewell/MAE224A\_2010/}
\bigskip

\begin{center}
\bf\LARGE Solution IV
\end{center}

\paragraph{1} See Cahn (1945).


\paragraph{2} In class, we saw that an internal wave will reflect
forward off a slope with angle less than the critical angle $\theta$,
but back off a slope with angles greater than the critical
angle. Hence an internal wave (packet) will just be reflected off a
wedge with a large opening angle. If the angle is small, it will
reflect into the wedge. The wave reflects specularly off the horizontal
boundary, so it continues into the wedge. Hence energy appears to
propagate into the wedge. The analysis in Wunsch (1968) finds the
appropriate mathematical solution, which has a singularity at the tip
of the wedge. In practise, viscous effects will become important as the
amplitude of the wave grows. See also SGLS (2004).


\paragraph{3} This can be found in a number of courses, e.g.~Kundu \&
Cohen or Pedlosky. I will save paper and disk space by not repeating
the analysis.


\paragraph{4} \emph{The most popular sources were Chapman \&
  Malanotte-Rizzoli and Kundu \& Cohen. Neither treatment was very
  clear about the range of $\omega$.} We can follow the treatment in
class with the following differences. The horizontal momentum equations
become
\[
u_t - fv = -\rho_0^{-1} p_x, \qquad v_t + fu = -\rho_0^{-1} p_y.
\]
Solving for $u$ and $v$ gives
\[
(\partial^2_{tt} + f^2) u = -\rho_0^{-1} p_{xt} - f \rho_0^{-1} p_y, \qquad
(\partial^2_{tt} + f^2) v = - \rho_0^{-1} p_{yt} + f \rho_0^{-1} p_x.
\]
Hence
\[
(\partial^2_{tt} + f^2) w_z = \rho_0^{-1} \nabla^2_H p_t.
\]
Take $\partial_z$ and eliminate $p$ using the same equation as in
class:
\[
(\partial^2_{tt} + N^2) w = -\rho_0^{-1} p_{tz}.
\]
Making the Boussinesq approximation gives
\[
\partial^2_{tt} \nabla^2 w + f^2 w_{zz} + N^2 \nabla^2_h w = 0.
\]
Substituting in the plane wave solution $\ee^{\ci(kx + ly + mz -
  \omega t)}$ gives the dispersion relation
\[
\omega^2 = \frac{N^2(k^2+l^2) + f^2m^2}{k^2 + l^2 + m^2} = N^2
\cos^2{\theta} + f^2 \sin^2{\theta}.
\]
In geophysical situations, $|f| < N$, so inertia-gravity waves occupy
the frequency band $f^2 \leq \omega^2 \leq N^2$.

Another approach that can be useful is as follows. Seek a plane wave
solution straight away. The governing equations then give five
homogeneous linear equations:
\[
\left( \begin{array}{rrrrr}
-\ci\omega & -f & 0 & \ci k\rho_0^{-1} & 0 \\
f & -\ci\omega &  0 & \ci l\rho_0^{-1} & 0 \\
0 & 0 & -\ci\omega &  \ci k\rho_0^{-1} & g \\
\ci k & \ci l & \ci m & 0 & 0 \\
0 & 0 & \dd\rho_0/\dd z & -\ci\omega
\end{array} \right)
\left( \begin{array}{r}
u \\
v \\
w \\
p \\
\rho
\end{array} \right) = \mathbf{0}.
\]
The determinant of the equation must vanish. Taking $f$ and $N$ to be
constant gives the dispersion relation.

\end{document}
