%% BEGIN semsamp2.tex
% This is a sample document for seminar.sty, v0.93 (and maybe later).
%
% This file contains both landscape and portrait mode slides.
% Choose one of the following to print them out:
%  - If using PSTricks, try the semcolor style option.
%  - If using Rokicki's dvips, try the semrot style option.
%  - To print the landscape slides, put \landscapeonly in the preamble.
%    To print the portrait slides, include the portrait style option and
%    put \portraitonly in the preamble.
%
%
\documentstyle[%
  fontenc,inputenc,
  pstricks,
  %slidesonly,%  Try notes or notesonly instead.
  %notes,%      Use instead of slidesonly to typeset the notes.
  %notesonly,%  Use instead of slidesonly to typeset notes and slides.
  semcolor,%   Try me if using PSTricks.
  %semrot,%     Try me if using Rokicki's dvips.
  %semhelv,%    Try me if using a PostScript printer.
  %article,%    Try me.
  %portrait,%   Try me.
  %sem-a4,%     Try me if using A4 paper.
  semlayer%     This must be included, but you need the semcolor option to
  ]{seminar}                                  % actually see the overlays.

\slidesmag{5}
\articlemag{1}

%\twoup                     % Try me for twoup printing.

%\portraitonly              % To print only portrait slides
%\landscapeonly             % To print only landscape slides

%\notslides{\ref{questions}-7,1}   %Try me: The slides are omitted.
%\onlyslides{\ref{questions}-7,1}  %Try me: Only these slides are included.
%\onlynotestoo                     %Try me: For selecting notes as well.

%\colorlayers{red,blue}      % Try deleting this if using the semcolor option,
                            % to get \blue and \red to use PostScript color.

\overlaysfalse             % Suppress overlays with semcolor option.
\layersfalse               % Suppress color layers with semcolor option.

\rotateheaderstrue          % Try this out if using rotation macros.


\title{Why the Rhine's length is crucially important to every living
organism on Earth}
\author{Ryan Woodard}
\date{12 December 2001}

\newcommand{\sref}[1]{SLIDE \ref{#1}}
\newcommand{\heading}[1]{\begin{center}\large\bf #1\end{center}}

\newpagestyle{MH}%
  {University of Alaska Fairbanks, 12 December 2001\hfil\thepage}{}
\pagestyle{MH}

\begin{document}

\maketitle          % This won't show up when \onlynotestoo is in effect.

%%% begin 
\begin{slide*}
  \ifslidesonly              % Title slide only for slidesonly selection.
    \maketitle
    \addtocounter{slide}{-1}
    \slidepagestyle{empty}
  \fi
\end{slide*}

%%% begin 
\begin{slide*} \label{Introduction}
\heading{Introduction}

\begin{itemize}
  \item Eddies get larger and move around.
  \item At a certain scale (the Rhine's length), the changing Coriolis parameter ends eddy growth.
  \item Energy goes into Rossby wave production.
  \item Rossby waves decay to smaller frequencies; this produces zonal flow.
\end{itemize}
\end{slide*}

%%% begin 
\begin{slide*}\label{too_much}%
\begin{center}
  \Large\bf
  Length scale at which Earth's rotation is important
\smallskip

{\small \centering \begin{tabular}{p{.9in}cp{.9in}}
\centering time for one rotation of Earth&
\( \approx  \)&
\centering time for fluid parcel to move \( L \) at \( U \)\\
\end{tabular}\par}

\large
\( \frac{L}{U}\approx \frac{2\pi }{\Omega } \)
\smallskip

For \( U\approx 1 \) m/s , \( L\approx 100 \) km
\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{overload}
\ptsize{12}
In 2D turbulence, the inverse energy cascade produces larger eddies:

\begin{center}
\setlength{\unitlength}{1pt}
\begin{picture}(200,50)(1,-10)
\put(1,25){\circle{10}}
\put(20,25){\vector(1,0){30}}
\put(70,25){\circle{20}}
\put(90,25){\vector(1,0){30}}
\put(150,25){\circle{40}}
\put(180,22){$U$}
\end{picture}

Relative vorticity of such an eddy is\\
\( \zeta =\nabla x\vec{u} \)

For eddies with a velocity \( U \) at the perimeter,\\
\( \zeta =\frac{2U}{r} \)

\end{center}
\end{slide*}

%%% begin 
\begin{slide*} \label{informed}
\begin{center}
\Large {\bf Potential vorticity must be conserved}

\( q=\frac{\zeta +f}{h} \)

For constant \( h \),\\
\( \zeta +f\approx  \) constant

\medskip
For an eddy\\
\( q=\frac{2U}{r}+f\approx  \) constant

\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\Large
{\bf On the \( \beta  \) plane, small eddies}

$q=$ {\blue $\frac{2U}{r}$} $+f_0+$ {\blue $\beta _{0}r$}

\( r\approx \delta y\ll L \), {\blue $\frac{2U}{r}$}$\gg${\blue $\beta _{0}r$}

\setlength{\unitlength}{1pt}
\begin{picture}(200,75)(0,5)
\put(1,1){\line(0,1){75}}
\put(1,76){\line(1,0){150}}
\put(151,76){\line(0,-1){75}}
\put(151,1){\line(-1,0){150}}
\put(153,76){\normalsize $y=L$}
\put(153,1){\normalsize $y=0$}
\put(20,20){\circle{10}}
\put(40,25){\circle{10}}
\put(47,20){\line(1,0){5}}
\put(47,25){\line(1,0){5}}
\put(55,20){\normalsize $\delta y \approx r$, $q\approx$ constant}
\end{picture}
\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\Large
{\bf On the \( \beta  \) plane, large eddies}

$q=$ {\blue $\frac{2U}{r}$} $+f_0+$ {\blue $\beta _{0}r$}

\( r\approx \delta y\approx L \), {\blue $\frac{2U}{r}$}$\approx${\blue $\beta _{0}r$}

\setlength{\unitlength}{1pt}
\begin{picture}(200,75)(0,5)
\put(1,1){\line(0,1){75}}
\put(1,76){\line(1,0){150}}
\put(151,76){\line(0,-1){75}}
\put(151,1){\line(-1,0){150}}
\put(153,76){\normalsize $y=L$}
\put(153,1){\normalsize $y=0$}
\put(30,25){\circle{40}}
\put(70,45){\circle{40}}
\put(93,25){\line(1,0){10}}
\put(93,45){\line(1,0){10}}
\put(106,37){\normalsize $\delta y \approx L$}
\put(106,27){\normalsize $q\approx$ not constant}
\end{picture}
\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\Large
{\bf On the \( \beta  \) plane, large eddies}

\( q=\frac{2U}{r}+f_0+\beta _{0}r \)\\
\( \frac{2U}{r}\approx \beta _{0}r \) \( \Longrightarrow  \) \( r=\sqrt{\frac{2U}{\beta _{0}}}\equiv L_{R} \)\\

\medskip
{\blue $L_{R}$ is the Rhine's length}

\medskip
in ocean, \( L_{R}\approx 100 \) km

in atmos., \( L_{R}\approx 1000 \) km

\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\Large
{\bf At $L_{R}$, eddies feel rotation effects}

\( \frac{L_R}{U}\approx \tau \approx \frac{2\pi }{\Omega } \)

So flow takes so long to go around eddy, it begins to notice the Coriolis.
\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\Large
{\bf What are the effects of rotation?}

Remember our first, simplest solution on the $\beta$ plane:

{\bf Rossby waves}

Eddys at size limit ($L_R$) still contain energy.

That energy goes to drive
Rossby waves.

\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\Large
{\bf Rossby wave solutions}

\( \eta \sim e^{i(lx+my-\omega t)} \)

\( u=u_{0}+u_{c}e^{i(lx+my-\omega t)} \)

\( v=v_{0}+v_{c}e^{i(lx+my-\omega t)} \)

Dispersion relation:\\
\( \omega \sim \frac{-\beta _{0}l}{l^{2}+m^{2}} \)

\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\Large
{\bf Rossby waves are unstable}

They decay to Rossby waves with smaller frequencies.

\( \omega _{1}\rightarrow \omega _{2}, \omega _{3}  \) 

\( \omega _{2} < \omega _{1} \) and 
\( \omega _{3} < \omega _{1} \)

\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\Large
{\bf This decay brings about zonal flow}

\( \omega \rightarrow 0 \), \( l\rightarrow 0 \) \\(remember: 
\( \omega \sim \frac{-\beta _{0}l}{l^{2}+m^{2}} \))

\( u\rightarrow u_{0}+e^{imy} \) \( \Longrightarrow  \) \( u \)
is independent of \( x \), so is a constant zonal flow
\end{center}
\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\heading{Conclusions}

\begin{itemize}
  \item Eddies get larger and move around.
  \item At Rhine's length, the changing Coriolis parameter ends eddy growth.
  \item Energy goes into Rossby wave production.
  \item Rossby waves decay to smaller frequencies which produces zonal
  flow.
  \item {\blue Could produce the zonal flows seen on Jupiter (see web project).}
  \item {\blue Is why we don't have system size eddies on Earth.}
  \item {\blue Allows trade winds and westerlies that were 
  important for\ldots{}}
\end{itemize}

\end{center}
\end{slide*}

%%% begin 
%\begin{slide*}\label{akbar}
%\begin{center}
%\heading{Jupiter}
%{http://space.about.com/library/solarsys/images/bljupiterimages.htm?iam=dpile&terms=jupiter+hubble}
%\end{center}
%\end{slide*}
%\end{center}
%\end{slide*}

%%% begin 
\begin{slide*}\label{akbar}
\begin{center}
\heading{\ldots{}conclusions}

{\blue Columbus to `discover' this place, thereby setting into motion
the genocide of native people, the slave trade, the destruction of
healthy ecosystems, multitudes of extinctions and the inevitable
control of the world by right-wing multinational conglomerate
corporate puppet politicians and power-hungry CEOs.}

\bigskip
{\red That is why the Rhine's length is crucially important 
to every living organism on Earth.}
\end{center}
\end{slide*}

\end{document}
%% END semsamp2.tex
