Add some things

main.bib

@@ -0,0 +1,7 @@
+@misc{wernliLectureScript,
+  title   = {Lecture {{Script}}},
+  author  = {Wernli, Heini and Papritz, Lukas},
+  url     = {https://ethz.ch/content/dam/ethz/special-interest/usys/iac/iac-dam/documents/edu/courses/dynamics_of_large_scale_flow/LSD_2022.pdf},
+  urldate = {2022-09-10},
+  file    = {/Users/thisfro/Zotero/storage/ERN2IM82/Lecture Script.pdf}
+}

main.tex

@@ -1,4 +1,5 @@
-\documentclass[8pt,landscape]{article}
+\documentclass[landscape]{article}
+\usepackage[utf8]{inputenc}
 \usepackage{multicol}
 \usepackage{calc}
 \usepackage{bookmark}
@@ -13,12 +14,15 @@
 \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}
@@ -148,12 +152,12 @@
 \subsection{Fundamental equations}
 \subsubsection{Navier-Stokes}
 \begin{equation}
-    \frac{Du}{Dt} = \underbrace{-\frac{1}{\rho}\nabla p}_\mathrm{Pressure} - \underbrace{(2\Omega \times u)}_\mathrm{Coriolis} - \underbrace{g'K}_\mathrm{Gravity} + \underbrace{F^{**}}_\mathrm{Viscous}
+    \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 u) = 0 
+    \frac{D \rho}{Dt} + \rho(\nabla \vb{u}) = 0 
 \end{equation}
 
 \subsubsection{First law of thermodynamics}
@@ -182,13 +186,30 @@ if $\mathcal{H} = 0$, the process is \textit{adiabatic}
     \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} v_G = \bigg(\frac{1}{f}\frac{g}{\theta_0}\bigg)(k \times \nabla_h \theta^*)
+    \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} \\
@@ -199,10 +220,28 @@ How to determine the $Q$-Vector on weather charts:
         \item Large temperature gradient
         \item Strong wind change
     \end{itemize}
-    \item Determine wind-change vector
+    \item Determine wind-change vector along $\eta$ (warm to the right, see Figure~\ref{Q-eta})
-    \item Rotate that vector by $+90^\circ$
+    \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}
@@ -213,14 +252,11 @@ This document is released under (CC BY-SA 3.0) \\
 \faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/lsd-zf} \\
 Jannis Portmann, HS22
 
-\section*{References}
+\printbibliography
-\begin{enumerate}
-    \item Script: LSD\_22.pdf
-\end{enumerate}
 
 \section*{Image sources}
 \begin{itemize}
-    \item Bild
+    \item Figure~\ref{Q-eta} (Skript, \cite{wernliLectureScript})
 \end{itemize}
 
 \end{multicols*}