Add some things

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Jannis Portmann 2023-02-06 23:36:59 +01:00
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commit 20ccc2282a
3 changed files with 55 additions and 12 deletions

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@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}
}

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@ -1,4 +1,5 @@
\documentclass[8pt,landscape]{article} \documentclass[landscape]{article}
\usepackage[utf8]{inputenc}
\usepackage{multicol} \usepackage{multicol}
\usepackage{calc} \usepackage{calc}
\usepackage{bookmark} \usepackage{bookmark}
@ -13,12 +14,15 @@
\usepackage{xcolor} \usepackage{xcolor}
\usepackage{float} \usepackage{float}
\usepackage{apacite} \usepackage{apacite}
\usepackage{physics}
\usepackage[style=authoryear]{biblatex}
\usepackage[ \usepackage[
type={CC}, type={CC},
modifier={by-sa}, modifier={by-sa},
version={3.0} version={3.0}
]{doclicense} ]{doclicense}
\addbibresource{main.bib}
\graphicspath{{./img/}} \graphicspath{{./img/}}
\definecolor{codegreen}{rgb}{0,0.6,0} \definecolor{codegreen}{rgb}{0,0.6,0}
@ -148,12 +152,12 @@
\subsection{Fundamental equations} \subsection{Fundamental equations}
\subsubsection{Navier-Stokes} \subsubsection{Navier-Stokes}
\begin{equation} \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} \end{equation}
\subsubsection{Conservation of mass} \subsubsection{Conservation of mass}
\begin{equation} \begin{equation}
\frac{D \rho}{Dt} + \rho(\nabla u) = 0 \frac{D \rho}{Dt} + \rho(\nabla \vb{u}) = 0
\end{equation} \end{equation}
\subsubsection{First law of thermodynamics} \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}) \frac{D_h}{Dt} \zeta + \beta v = -f_0(\nabla_h \vec{v})
\end{equation} \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} \section{Concepts}
\subsection{Thermal wind} \subsection{Thermal wind}
\textit{Thermal wind} describes the vertical change of geostrophic (i.e. horizontal) wind \textit{Thermal wind} describes the vertical change of geostrophic (i.e. horizontal) wind
\begin{equation} \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} \end{equation}
\subsection{$Q$-Vector} \subsection{$Q$-Vector}
The $Q$-Vector indicates if there is cyclogenesis ($\mathcal{F} < 0$, $\mathcal{F} \sim \nabla_h Q$) The $Q$-Vector indicates if there is cyclogenesis ($\mathcal{F} < 0$, $\mathcal{F} \sim \nabla_h Q$)
\vspace{2mm} \\ \vspace{2mm} \\
@ -199,10 +220,28 @@ How to determine the $Q$-Vector on weather charts:
\item Large temperature gradient \item Large temperature gradient
\item Strong wind change \item Strong wind change
\end{itemize} \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} \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 \scriptsize
\section*{Copyleft} \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} \\ \faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/lsd-zf} \\
Jannis Portmann, HS22 Jannis Portmann, HS22
\section*{References} \printbibliography
\begin{enumerate}
\item Script: LSD\_22.pdf
\end{enumerate}
\section*{Image sources} \section*{Image sources}
\begin{itemize} \begin{itemize}
\item Bild \item Figure~\ref{Q-eta} (Skript, \cite{wernliLectureScript})
\end{itemize} \end{itemize}
\end{multicols*} \end{multicols*}