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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}

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\begin{document}

\title{Models of Weather and Climate}
\date{Summer Semester 2025}
\author{Models of geophysical fluid flow -- review questions and problems}
\maketitle

\begin{enumerate}

\item What is the meaning of the ``material derivative'' $D_t$?  Can you
show that $D_t = \partial_t + u \cdot \nabla$?  Which physical
quantities $\theta$, in typical geophysical models, satisfy
$D_t \theta = 0$ (``Material conservation of $\theta$'')?

\item How can you write Newton's second law for a fluid?  What
additional forces (per unit area or per unit volume) are particularly
relevant for large-scale geophysical flow?

\item Conservation laws for geophysical fluid flow:
\begin{enumerate}
\item Material conservation of potential vorticity.  For the shallow
water equation in non-dimensional variables, the potential vorticity is
\begin{equation}
  q = \frac{1 + \Ro \, \nabla^\bot \cdot u}{H_0 + h} \,.
\end{equation}
Can you interpret/explain the physical significance?
\item Conservation of volume (or ``incompressibility'').  How do you
derive the condition $\nabla \cdot u = 0$?  (Being able to give a
detailed derivation would be great, but you should at least know a
sketch of the proof.)
\item Conservation of mass.  How do you derive the ``continuity
equation''
\begin{equation}
  \partial_t \rho + \nabla \cdot (\rho u) = 0 \,?
\end{equation}
\item Conservation of total energy for fluid flow without viscosity
and forcing.  E.g., for the shallow water equations written in the
form
\begin{subequations}
  \label{rsw}
\begin{gather}
  \Ro \, (\partial_t u + u \cdot \nabla u) + u^\bot
  + \frac{\Bu}{\Ro} \, \nabla h = 0 \,, \\
  \partial_t h + \nabla \cdot (h u) + H_0 \, \nabla \cdot u = 0 \,,
\end{gather}
\end{subequations}
can you show that the total energy
\begin{equation}
  E = \frac12 \int_\Omega \Ro \, (H_0 + h) \, \lvert u \rvert^2
      + \frac{\Bu}{\Ro} \, h^2 \, \d x
\end{equation}
is conserved?
\item Local conservation laws for momentum (i.e., Newton's second law)
and energy: we did not discuss local conservation laws systematically,
but an idea of the conceptual distinction between ``global
conservation law'', ``material conservation law'', ``local
conservation law'' is helpful.

\item If you add a viscous force, i.e.\ $\nu \, \Delta u$, to the
right hand side of the momentum equation, what does this do to the
evolution of total energy?

\end{enumerate}

\item Geostrophic balance:
\begin{enumerate}
\item What is it and why is it important?
\item Can
you show that in planar geometry, e.g.\ in the context of the shallow
water system \eqref{rsw} above, a solution in geostrophic balance is
stationary?
\item Does geostrophic balance hold near the equator?
\end{enumerate}

\item Linear waves: Inertia-gravity waves vs.\ Rossby waves.
Inertia-gravity waves are dispersive.  What does this mean?  What is
the physical significance of the shallow water dispersion relation
(in dimensional variables)
\begin{gather*}
  \omega(k) = \sqrt{f^2 + g \, H \, \lvert k \rvert^2}
\end{gather*}

\item Baroclinic instability: What is it and why is it important?  In
the Eady model of baroclinic instability, what is the assumed
steady-state background flow?  What are the assumed background density
surfaces?  How are the unstable modes calculated?  What is the role of
baroclinic instability for the global energy balance of atmosphere and
ocean?

\item Non-dimensionalization and scaling: when written in the
nondimensionalized form \eqref{rsw}, which was derived by
non-dimensionalizing all variables, the characteristic scale height of
$h$ is not yet specifified.  Can you explain why there is still one
more scaling parameter to choose?  What are possible choices given
that a dominant balance of pressure gradient and Coriolis term is
consistent with observations (at large scales at the mid-latitudes)?

\item Hydrostatic balance: what is it?  Why is it often imposed, even
in operational models?  What is the underlying scaling limit that
leads to hydrostaticity?

\end{enumerate}
\end{document}
