lsd-zf/main.tex

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\documentclass[landscape]{article}
\usepackage[utf8]{inputenc}
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\usepackage{multicol}
\usepackage{calc}
\usepackage{bookmark}
\usepackage{ifthen}
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\usepackage{hyperref}
\usepackage{ccicons}
\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{listings}
\usepackage{graphicx}
\usepackage{fontawesome5}
\usepackage{xcolor}
\usepackage{float}
\usepackage{apacite}
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\usepackage{physics}
\usepackage[style=authoryear]{biblatex}
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\usepackage[
type={CC},
modifier={by-sa},
version={3.0}
]{doclicense}
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\addbibresource{main.bib}
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% latex latexsheet.tex
% dvips -P pdf -t landscape latexsheet.dvi
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% If you're reading this, be prepared for confusion. Making this was
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% \setcounter{secnumdepth}{0}
\setlength{\parindent}{0pt}
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% -----------------------------------------------------------------------
\begin{document}
\raggedright
\footnotesize
\begin{multicols*}{3}
% multicol parameters
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\begin{center}
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\Large{Dynamics of Large-scale Atmospheric Flow} \\
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\small{\href{http://iacweb.ethz.ch/courses/id/701-1221-00L}{701-1221-00L}, HS 22} \\
\small{Jannis Portmann} \\
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{\ccbysa}
\rule{\linewidth}{0.25pt}
\end{center}
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\section{Equations}
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\subsection{Fundamental equations}
\subsubsection{Navier-Stokes}
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\begin{equation}
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\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}
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\end{equation}
\subsubsection{Conservation of mass}
\begin{equation}
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\frac{D \rho}{Dt} + \rho(\nabla \vb{u}) = 0
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\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
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\end{equation}
\subsection{Circulation}
\begin{equation}
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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})
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\end{equation}
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\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 $\mathcal{U} = - \frac{\partial \bar{\psi}}{\partial y}$
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\section{Concepts}
\subsection{Thermal wind}
\textit{Thermal wind} describes the vertical change of geostrophic (i.e. horizontal) wind
\begin{equation}
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\frac{\partial}{\partial z} \vb{v_G} = \bigg(\frac{1}{f}\frac{g}{\theta_0}\bigg)(\vb{k} \times \nabla_h \theta^*)
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\end{equation}
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\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}
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\item Determine wind-change vector along $\eta$ (warm to the right, see Figure~\ref{Q-eta})
\item Rotate that vector by $-90^\circ$
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\end{enumerate}
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\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}
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\scriptsize
\section*{Copyleft}
\doclicenseImage \\
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This document is released under (CC BY-SA 3.0) \\
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\faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp/lsd-zf} \\
\faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/lsd-zf} \\
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Jannis Portmann, HS22
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\printbibliography
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\section*{Image sources}
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\begin{itemize}
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\item Figure~\ref{Q-eta} (Skript, \cite{wernliLectureScript})
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\end{itemize}
\end{multicols*}
\end{document}