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\begin{document}

\raggedright
\footnotesize
\begin{multicols*}{3}


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\begin{center}
     \Large{Dynamics of Large-scale Atmospheric Flow} \\
    \small{\href{http://iacweb.ethz.ch/courses/id/701-1221-00L}{701-1221-00L}, HS 22} \\
    \small{Jannis Portmann} \\
    {\ccbysa}
\rule{\linewidth}{0.25pt}
\end{center}

\section{Equations}
\subsection{Fundamental equations}
\subsubsection{Navier-Stokes}
\begin{equation}
    \frac{D\vb{u}}{Dt} = \underbrace{-\frac{1}{\rho}\nabla p}_\mathrm{Pressure} - \underbrace{(2\Omega \times \vb{u})}_\mathrm{Coriolis} - \underbrace{g'K}_\mathrm{Gravity} + \underbrace{F^{**}}_\mathrm{Viscous}
\end{equation}

\subsubsection{Conservation of mass}
\begin{equation}
    \frac{D \rho}{Dt} + \rho(\nabla \vb{u}) = 0 
\end{equation}

\subsubsection{First law of thermodynamics}
\begin{equation}
    \frac{D\theta}{Dt} = \bigg(\frac{\theta}{c_p T} \bigg) \mathcal{H}
\end{equation}
if $\mathcal{H} = 0$, the process is \textit{adiabatic}

\subsubsection{Equation of state}
\begin{equation}
    p = \rho RT
\end{equation}

\subsection{Circulation}
\begin{equation}
    C = \oint_c \vec{v} \, dc = \oint (u \, dx + v \, dy + w \, dz) = \oint_0^{2\pi} \vec{v} \, r \, d\phi
\end{equation}

\subsection{Quasi geostrophic system of equations}
\begin{equation}
    \zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}
\end{equation}

\subsubsection*{Vorticity equation}
\begin{equation}
    \frac{D_h}{Dt} \zeta + \beta v = -f_0(\nabla_h \vec{v})
\end{equation}

\subsection{Geostrophic streamfunction}
\begin{equation}
    \nabla_h \psi = \zeta_G
\end{equation}

\subsection{Wave theory}
\subsubsection*{Pertubation tendency}
\begin{equation}
    \psi = \underbrace{\bar{\psi}(y,z)}_\mathrm{Mean \, meridional \, flow} + \underbrace{\psi'(x,y,z,t)}_\mathrm{Pertubation}
\end{equation}
assuming $|\psi'| \ll |\bar{\psi}|$


\begin{equation}
    \bigg(\frac{\partial}{\partial t} + \mathcal{U} \bigg) q' + v'(\frac{\partial}{\partial y} \bar{q} + \beta) = 0
\end{equation}
with $\mathcal{U} = - \frac{\partial \bar{\psi}}{\partial y}$

\section{Concepts}
\subsection{Thermal wind}
\textit{Thermal wind} describes the vertical change of geostrophic (i.e. horizontal) wind
\begin{equation}
    \frac{\partial}{\partial z} \vb{v_G} = \bigg(\frac{1}{f}\frac{g}{\theta_0}\bigg)(\vb{k} \times \nabla_h \theta^*)
\end{equation}
\subsection{$Q$-Vector}
The $Q$-Vector indicates if there is cyclogenesis ($\mathcal{F} < 0$, $\mathcal{F} \sim \nabla_h Q$)
\vspace{2mm} \\
How to determine the $Q$-Vector on weather charts:
\begin{enumerate}
    \item Locate regions with:
    \begin{itemize}
        \item Large temperature gradient
        \item Strong wind change
    \end{itemize}
    \item Determine wind-change vector along $\eta$ (warm to the right, see Figure~\ref{Q-eta})
    \item Rotate that vector by $-90^\circ$
\end{enumerate}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.1\textwidth]{eta.png}
    \caption{Direction of $\eta$}
    \label{Q-eta}
\end{figure}

\begin{equation}
    \vb{Q} = -\frac{g}{\theta_0}|\nabla_h\theta^{*}|(\vb{k} \wedge \frac{\partial}{\partial \xi}\vb{v_G})
\end{equation}

\subsection{PV streamer}
\begin{itemize}
    \item is an upper level positive PV anomaly
    \item induces cyclonal flow
\end{itemize}


\scriptsize

\section*{Copyleft}

\doclicenseImage \\
This document is released under (CC BY-SA 3.0) \\
\faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp/lsd-zf} \\
\faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/lsd-zf} \\
Jannis Portmann, \the\year

\section*{References}
\begin{itemize}
    \item Script, Heini Wernli and Lukas Papritz, 2022
\end{itemize}

\section*{Image sources}
\begin{itemize}
    \item Figure~\ref{Q-eta} (Wernli and Papritz 2022)
\end{itemize}

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