265 lines
7.4 KiB
TeX
265 lines
7.4 KiB
TeX
\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{physics}
|
|
\usepackage[
|
|
type={CC},
|
|
modifier={by-sa},
|
|
version={3.0}
|
|
]{doclicense}
|
|
|
|
\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, \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}
|
|
|
|
\end{multicols*}
|
|
|
|
\end{document}
|