\documentclass[landscape]{article}
\usepackage[utf8]{inputenc}
\usepackage{multicol}
\usepackage{calc}
\usepackage{bookmark}
\usepackage{ifthen}
\usepackage[a4paper, landscape]{geometry}
\usepackage{hyperref}
\usepackage{ccicons}
\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{listings}
\usepackage{graphicx}
\usepackage{fontawesome5}
\usepackage{xcolor}
\usepackage{float}
\usepackage{apacite}
\usepackage{physics}
\usepackage[style=authoryear]{biblatex}
\usepackage[
    type={CC},
    modifier={by-sa},
    version={3.0}
]{doclicense}

\addbibresource{main.bib} 
\graphicspath{{./img/}} 

\definecolor{codegreen}{rgb}{0,0.6,0}
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
\definecolor{codepurple}{rgb}{0.58,0,0.82}
\definecolor{backcolour}{rgb}{0.95,0.95,0.92}

\lstdefinestyle{mystyle}{
    backgroundcolor=\color{backcolour},
    commentstyle=\color{codegreen},
    keywordstyle=\color{magenta},
    numberstyle=\tiny\color{codegray},
    stringstyle=\color{codepurple},
    basicstyle=\ttfamily\footnotesize,
    breakatwhitespace=false,
    breaklines=true,
    captionpos=b,
    keepspaces=true,
    numbers=left,
    numbersep=5pt,
    showspaces=false,
    showstringspaces=false,
    showtabs=false,
    tabsize=2
}

\lstset{style=mystyle}

% To make this come out properly in landscape mode, do one of the following
% 1.
%  pdflatex latexsheet.tex
%
% 2.
%  latex latexsheet.tex
%  dvips -P pdf  -t landscape latexsheet.dvi
%  ps2pdf latexsheet.ps


% If you're reading this, be prepared for confusion.  Making this was
% a learning experience for me, and it shows.  Much of the placement
% was hacked in; if you make it better, let me know...


% 2008-04
% Changed page margin code to use the geometry package. Also added code for
% conditional page margins, depending on paper size. Thanks to Uwe Ziegenhagen
% for the suggestions.

% 2006-08
% Made changes based on suggestions from Gene Cooperman. <gene at ccs.neu.edu>


% To Do:
% \listoffigures \listoftables
% \setcounter{secnumdepth}{0}


% This sets page margins to .5 inch if using letter paper, and to 1cm
% if using A4 paper. (This probably isn't strictly necessary.)
% If using another size paper, use default 1cm margins.
\ifthenelse{\lengthtest { \paperwidth = 11in}}
	{ \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} }
	{\ifthenelse{ \lengthtest{ \paperwidth = 297mm}}
		{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
		{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
	}

% Turn off header and footer
\pagestyle{empty}


% Redefine section commands to use less space
\makeatletter
\renewcommand{\section}{\@startsection{section}{1}{0mm}%
                                {-1ex plus -.5ex minus -.2ex}%
                                {0.5ex plus .2ex}%x
                                {\normalfont\large\bfseries}}
\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}%
                                {-1explus -.5ex minus -.2ex}%
                                {0.5ex plus .2ex}%
                                {\normalfont\normalsize\bfseries}}
\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}%
                                {-1ex plus -.5ex minus -.2ex}%
                                {1ex plus .2ex}%
                                {\normalfont\small\bfseries}}


\makeatother

% Define BibTeX command
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
    T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}

% Don't print section numbers
% \setcounter{secnumdepth}{0}


\setlength{\parindent}{0pt}
\setlength{\parskip}{0pt plus 0.5ex}

% -----------------------------------------------------------------------

\begin{document}

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


% multicol parameters
% These lengths are set only within the two main columns
%\setlength{\columnseprule}{0.25pt}
\setlength{\premulticols}{1pt}
\setlength{\postmulticols}{1pt}
\setlength{\multicolsep}{1pt}
\setlength{\columnsep}{2pt}

\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, HS22

\printbibliography

\section*{Image sources}
\begin{itemize}
    \item Figure~\ref{Q-eta} (Skript, \cite{wernliLectureScript})
\end{itemize}

\end{multicols*}

\end{document}
