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\textbf{Tropical cyclones. Problem Sheet 2, SS2011}

\bigskip
\textbf{Exercise 1}

\bigskip
Show that for a two-dimensional incompressible flow in the (x,y)-plane, the streamfunction $\psi$ and vorticity $\zeta$ are related by the equation
\[
\frac{{\partial^2 \psi}}{{\partial x^2}} + \frac{{\partial^2 \psi}}{{\partial y^2}} = \zeta.
\]
\begin{itemize}

\item Sketch the streamfunction field associated with two line vortices of equal strength separated by a distance $a$ in a large body of fluid. 

\item Show that the vortices will tend to rotate around each other.

\end{itemize}

[Hint: you may think of the vortices being represented by delta functions $\zeta_o \delta(x - {1 \over 2}a) \delta(y)$ and $\zeta_o \delta(x + {1 \over 2}a) \delta(y)$, where $\zeta_o$ is the strength of the vortices, and you may assume that $\psi \rightarrow 0$ as the distance from the vortices increases. Think in terms of the membrane analogy discussed in class.]

\begin{itemize}

\item Sketch the streamfunction field that would arise if the vortices had the same strength but opposite signs.

\item Show that, in the latter case, the vortices will tend to move together at right angles to the line joining them.

\end{itemize}

\bigskip
\textbf{Exercise 2}

\bigskip
Show that with the coordinate transformation
\[
X = x + v/f,\qquad Z = z,
\]
\[
{\partial^2 \psi \over {\partial z^2}} = {\partial^2 \psi \over {\partial Z^2}} + {2S \over f}{\partial^2 \psi \over {\partial X \partial Z}} + {S^2 \over f^2}{\partial^2 \psi \over {\partial X^2}} + {1 \over f}{\partial S \over {\partial z}}{\partial \psi \over {\partial X}}
\]

\bigskip
\textbf{Exercise 3}

\bigskip
In the slab-symmetric model for the Hadley circulation discussed in class, the Sawyer-Eliassen equation is:
\[
\left(N^2 + {\partial b \over {\partial z}}\right){\partial^2 \psi \over {\partial x^2}} - 2fS{\partial^2 \psi \over {\partial x \partial z}} + f\zeta_a{\partial^2 \psi \over {\partial z^2}} = {\partial \dot V \over {\partial z}} - {\partial \dot B \over {\partial x}}.
\]
\begin{itemize}

\item Write down the form of the equation for a barotropic vortex with zonal wind $v(x)$ and no diabatic heating when $\partial b / \partial z = 0$.

\item Sketch the circulation induced by a frictional force $\dot V = -Fz$ for $z < h$ and $\dot V = 0$ for $z > h$, where $h$ is typical of the depth of the atmospheric boundary layer (about 1 km) when $v = v_ox\exp(-x^2/x_o^2)$. Here $F$, $h$ $v_o$ and $x_o$ are positive constants.

\end{itemize}

\bigskip
\textbf{Exercise 4}

\bigskip

Show that the inertial stability parameter $I$ given by
\[
I^2 = {1 \over r^3}{\partial M^2 \over {\partial r}}
\]
is equal to $\xi\zeta_a$, where $\xi$ is twice the absolute angular velocity and $\zeta_a$ is the absolute vorticity.

Show that the potential vorticity of the axi-symmetric flow with tangential wind speed $v(r,z)$ and potential temperature $\theta(r,z)$ is positive if the slope of the M-surfaces is larger than that of the isentropic surfaces, where $M$ is the absolute angular momentum.


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