From c1572375face134db26ec2179fa834e8c4595c36 Mon Sep 17 00:00:00 2001
From: Jannis Portmann <jannisp@student.ethz.ch>
Date: Fri, 3 Feb 2023 18:06:51 +0100
Subject: [PATCH] Some additions

---
 main.tex | 47 ++++++++++++++++++++++++++++++++++++++++++++---
 1 file changed, 44 insertions(+), 3 deletions(-)

diff --git a/main.tex b/main.tex
index f2ebbef..7922f38 100644
--- a/main.tex
+++ b/main.tex
@@ -145,14 +145,41 @@
 \end{center}
 
 \section{Equations}
-\subsection{Navier-Stokes}
+\subsection{Fundamental equations}
+\subsubsection{Navier-Stokes}
 \begin{equation}
-    \frac{D}{Dt}
+    \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}
+\end{equation}
+
+\subsubsection{Conservation of mass}
+\begin{equation}
+    \frac{D \rho}{Dt} + \rho(\nabla 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 u \, dc = \oint (u \, dx + v \, dy + w \, dz)
+    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}
 
 \section{Concepts}
@@ -162,6 +189,20 @@
     \frac{\partial}{\partial z} v_G = \bigg(\frac{1}{f}\frac{g}{\theta_0}\bigg)(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
+    \item Rotate that vector by $+90^\circ$
+\end{enumerate}
+
 \scriptsize
 
 \section*{Copyleft}