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\begin{abstract} 

     The subject of this study is flow around the Andes and the associated South American 
     Low-Level Jet (SALLJ).  
 
This file is on isabel in /papers/schubert/beta-plane/andes-f6.tex

\end{abstract}

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\section{Introduction}

Potential Vorticity in $z$ coordinates
\begin{equation}
P = \frac{1}{\rho}\left\{-\frac{\partial v}{\partial z}\left(\frac{\partial \theta}{\partial x}\right)_z+
       \frac{\partial u}{\partial z}\left(\frac{\partial\theta}{\partial y}\right)_z+
       \left[f+\left(\frac{\partial v}{\partial x}\right)_z-\left(\frac{\partial u}{\partial y}\right)_z\right]
       \frac{\partial\theta}{\partial z}
       \right\}
\end{equation}
Potential Vorticity in $p$ coordinates
\begin{equation}
P = g\left\{\frac{\partial v}{\partial p}\left(\frac{\partial \theta}{\partial x}\right)_p-
       \frac{\partial u}{\partial p}\left(\frac{\partial\theta}{\partial y}\right)_p+
       \left[f+\left(\frac{\partial v}{\partial x}\right)_p-\left(\frac{\partial u}{\partial y}\right)_p\right]
       \frac{\partial\theta}{\partial p}
       \right\}
\end{equation}
Potential Vorticity in $\theta$ coordinates
\begin{equation}
p = -g\left[f+\left(\frac{\partial v}{\partial x}\right)_{\theta}-\left(\frac{\partial u}{\partial y}\right)_{\theta}\right]\frac{\partial\theta}{\partial p}
\end{equation}

\end{document}