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\textbf{Exercise 1}

\bigskip
Show that in a vortex in gradient wind balance and thermal wind balance, the pressure field satisfies the partial differential equation
\[
g\frac{{\partial p}}{{\partial r}} + C\frac{{\partial p}}{{\partial z}} = 0,
\]
where \[C = \frac{{v^2 }}{r} + fv.\]
Show also that an arbitrary function of pressure satisfies the same equation.

Given a tangential wind field $v(r,z)$ and the vertical pressure profile at large radius, $p_o(z)$,, explain how you could solve this equation to find the surface pressure distribution, $p(r,0)$.

\bigskip
\textbf{Exercise 2}
\bigskip

The thermal wind equation for a vortex with a tangential wind field $v(r,z)$ is:
\[
g\frac{{\partial (\ln \rho)}}{{\partial r}} + C\frac{{\partial (\ln \rho)}}{{\partial z}} = -\frac{{\partial C}}{{\partial z}}.
\]
where $\rho$ is the density and 
\[C = \frac{{v^2 }}{r} + fv.\]
Show that this equation may be rewritten as
\[
g\frac{{\partial (\ln \chi)}}{{\partial r}} + C\frac{{\partial (\ln \chi)}}{{\partial z}} = -\frac{{\partial C}}{{\partial z}}.
\]
where $\chi$ is the inverse of the potential temperature, $\theta$.

\bigskip
\textbf{Exercise 3}
\bigskip

Show that the tangential momentum equation for an inviscid, axisymmetric flow in cylindrical coordinates $(r,\lambda,z)$ may be written in the two forms:

\[
\frac{{\partial v}}{{\partial t}} + u\zeta_a + w\frac{{\partial v}}{{\partial z}}  = 0,
\]
or
\[
\frac{{\partial M}}{{\partial t}} + u\frac{{\partial M}}{{\partial r}} + w\frac{{\partial M}}{{\partial z}}  = 0,
\]
where $\zeta_a$ is the \textit{absolute vorticity} and $M = rv + {1 \over 2}fr^2$ is the \textit{absolute angular momentum} per unit mass. 
Explain why vortex spin up requires that air parcels move inwards.


\bigskip
\textbf{Exercise 4}
\bigskip

Show that the vertical momentum equation can be written in terms of \textit{the perturbation pressure} $p' = p_T + p_{ref}(z)$ and \textit{buoyancy} $b = -g(\rho - \rho_{ref}(z))/\rho$ as:
\[
\frac{{\partial w}}{{\partial t}} + u\frac{{\partial w}}{{\partial r}} + w\frac{{\partial w}}{{\partial z}} = -{1 \over \rho}\frac{{\partial p'}}{{\partial z}} + b,
\]
where $p_T$ is the total pressure, $p_{ref}(z)$ and $\rho_{ref}(z)$ are a \textit{reference pressure} and \textit{reference density}, respectively, that satisfy hydrostatic balance, and $g$ is the acceleration due to gravity.

Show that the same equation holds if $p_{ref}$ and $\rho_{ref}$ are functions of both radius $r$ and height $z$, such as the balanced pressure and density fields in a baroclinic vortex. How would you decide whether a cloud within a balanced warm-cored vortex had any buoyancy?

\bigskip
\textbf{Exercise 5}
\bigskip

The Boussinesq forms of the thermal wind equation, tangential momentum equation and thermodynamic equation are:
\begin{equation}
\frac{{\partial b}}{{\partial r}} = \frac{{\partial C}}{{\partial z}},
\label{Eq2.1}
\end{equation}
\begin{equation}
\frac{{\partial v}}{{\partial t}} + u\frac{{\partial v}}{{\partial r}} + w\frac{{\partial v}}{{\partial z}} + {{uv} \over r} + fu  = \dot V,
\label{Eq2.2}
\end{equation}
and
\begin{equation}
\frac{{\partial b}}{{\partial t}} + u\frac{{\partial b}}{{\partial r}} + wN^2 = \dot B,
\label{Eq2.3}
\end{equation}
respectively, where
\[\xi  = \frac{{2v}}{r} + f,\quad \quad S = \frac{{\partial v}}{{\partial z}},\quad \quad C = \frac{{v^2 }}{r} + fv .\]

Show that the Boussinesq form of the Sawyer-Eliassen equation for the streamfunction of the secondary circulation is
\begin{equation}\frac{{\partial}}{{\partial r }}\left[N^2\frac{1}{r}\frac{{\partial \psi }}{{\partial r}} - {S\xi \over r}{\partial \psi \over {\partial z}}\right] +
\frac{{\partial}}{{\partial z }}\left[{I^2 \over r} {\partial \psi \over \partial z} - {S\xi \over r}{\partial \psi \over {\partial r}}\right] = - \frac{{\partial \dot B}}{{\partial r}} - {\partial \over {\partial z}} (\xi\dot V), \label{Eq2.4}
\end{equation}
where $I^2 = \xi \zeta_a$ and $\zeta_a$ is the absolute vorticity.

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