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--- manual/s_overview/text/manual.tex 2001/09/27 17:45:03 1.1
+++ manual/s_overview/text/manual.tex 2001/10/09 10:48:03 1.2
@@ -1,27 +1,27 @@
-%%%% % $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
-%%%% % $Name: $
-%%%% %\usepackage{oldgerm}
-%%%% % I commented the following because it introduced excessive white space
-%%%% %\usepackage{palatcm} % better PDF
-%%%% % page headers and footers
-%%%% %\pagestyle{fancy}
-%%%% % referencing
-%%%% %% \newcommand{\refequ}[1]{equation (\ref{equ:#1})}
-%%%% %% \newcommand{\refequbig}[1]{Equation (\ref{equ:#1})}
-%%%% %% \newcommand{\reftab}[1]{Tab.~\ref{tab:#1}}
-%%%% %% \newcommand{\reftabno}[1]{\ref{tab:#1}}
-%%%% %% \newcommand{\reffig}[1]{Fig.~\ref{fig:#1}}
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-%%%% % stuff for psfrag
-%%%% %% \newcommand{\textinfigure}[1]{{\footnotesize\textbf{\textsf{#1}}}}
-%%%% %% \newcommand{\mathinfigure}[1]{\small\ensuremath{{#1}}}
-%%%% % This allows numbering of subsubsections
-%%%% % This changes the the chapter title
-%%%% %\renewcommand{\chaptername}{Section}
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
+% $Name: $
+%\usepackage{oldgerm}
+% I commented the following because it introduced excessive white space
+%\usepackage{palatcm} % better PDF
+% page headers and footers
+%\pagestyle{fancy}
+% referencing
+%% \newcommand{\refequ}[1]{equation (\ref{equ:#1})}
+%% \newcommand{\refequbig}[1]{Equation (\ref{equ:#1})}
+%% \newcommand{\reftab}[1]{Tab.~\ref{tab:#1}}
+%% \newcommand{\reftabno}[1]{\ref{tab:#1}}
+%% \newcommand{\reffig}[1]{Fig.~\ref{fig:#1}}
+%% \newcommand{\reffigno}[1]{\ref{fig:#1}}
+% stuff for psfrag
+%% \newcommand{\textinfigure}[1]{{\footnotesize\textbf{\textsf{#1}}}}
+%% \newcommand{\mathinfigure}[1]{\small\ensuremath{{#1}}}
+% This allows numbering of subsubsections
+% This changes the the chapter title
+%\renewcommand{\chaptername}{Section}
%%%% \documentclass[12pt]{book}
-%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% \usepackage{amsmath}
%%%% \usepackage{html}
%%%% \usepackage{epsfig}
@@ -30,12 +30,12 @@
%%%% \usepackage{multirow}
%%%% \usepackage{fancyhdr}
%%%% \usepackage{psfrag}
-%%%%
+
%%%% %TCIDATA{OutputFilter=Latex.dll}
-%%%% %TCIDATA{LastRevised=Thursday, September 27, 2001 10:59:02}
+%%%% %TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22}
%%%% %TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%%%% %TCIDATA{Language=American English}
-%%%%
+
%%%% \fancyhead{}
%%%% \fancyhead[LO]{\slshape \rightmark}
%%%% \fancyhead[RE]{\slshape \leftmark}
@@ -45,20 +45,19 @@
%%%% \renewcommand{\headrulewidth}{0.4pt}
%%%% \renewcommand{\footrulewidth}{0.4pt}
%%%% \setcounter{secnumdepth}{3}
-%%%%
%%%% \input{tcilatex}
%%%%
%%%% \begin{document}
%%%%
%%%% \tableofcontents
+%%%%
+%%%% \pagebreak
-\pagebreak
-
-\part{MITgcm basics}
+%%%% \part{MIT GCM basics}
% Section: Overview
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
% $Name: $
\section{Introduction}
@@ -82,39 +81,11 @@
\marginpar{
Fig.1 One model}\ref{fig:onemodel}
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps}
- }
- }
-}
-\end{center}
-\label{fig:onemodel}
-\end{figure}
-
\item it has a non-hydrostatic capability and so can be used to study both
small-scale and large scale processes - see fig.2%
\marginpar{
Fig.2 All scales}\ref{fig:all-scales}
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/scales.eps}
- }
- }
-}
-\end{center}
-\label{fig:scales}
-\end{figure}
-
-
\item finite volume techniques are employed yielding an intuitive
discretization and support for the treatment of irregular geometries using
orthogonal curvilinear grids and shaved cells - see fig.3%
@@ -136,7 +107,7 @@
give a feel for the wide range of problems that can be addressed using it.
\pagebreak
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
% $Name: $
\section{Illustrations of the model in action}
@@ -147,115 +118,149 @@
kinds of problems the model has been used to study, we briefly describe some
of them here. A more detailed description of the underlying formulation,
numerical algorithm and implementation that lie behind these calculations is
-given later. Indeed it is easy to reproduce the results shown here: simply
-download the model (the minimum you need is a PC running linux, together
-with a FORTRAN\ 77 compiler) and follow the examples.
+given later. Indeed many of the illustrative examples shown below can be
+easily reproduced: simply download the model (the minimum you need is a PC
+running linux, together with a FORTRAN\ 77 compiler) and follow the examples
+described in detail in the documentation.
\subsection{Global atmosphere: `Held-Suarez' benchmark}
-Fig.E1a.\ref{fig:Held-Suarez} is an instaneous plot of the 500$mb$ height
-field obtained using a 5-level version of the atmospheric pressure isomorph
-run at 2.8$^{\circ }$ resolution. We see fully developed baroclinic eddies
-along the northern hemisphere storm track. There are no mountains or
-land-sea contrast in this calculation, but you can easily put them in. The
-model is driven by relaxation to a radiative-convective equilibrium profile,
-following the description set out in Held and Suarez; 1994 designed to test
-atmospheric hydrodynamical cores - there are no mountains or land-sea
-contrast. As decribed in Adcroft (2001), a `cubed sphere' is used to
-descretize the globe permitting a uniform gridding and obviated the need to
-fourier filter.
+A novel feature of MITgcm is its ability to simulate both atmospheric and
+oceanographic flows at both small and large scales.
-Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
-wind and meridional overturning streamfunction from the 5-level model.
-
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps}
- }
-}
-\end{center}
-\label{fig:hscs}
-\end{figure}
+Fig.E1a.\ref{fig:Held-Suarez} shows an instantaneous plot of the 500$mb$
+temperature field obtained using the atmospheric isomorph of MITgcm run at
+2.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
+(blue) and warm air along an equatorial band (red). Fully developed
+baroclinic eddies spawned in the northern hemisphere storm track are
+evident. There are no mountains or land-sea contrast in this calculation,
+but you can easily put them in. The model is driven by relaxation to a
+radiative-convective equilibrium profile, following the description set out
+in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
+there are no mountains or land-sea contrast.
+
+As described in Adcroft (2001), a `cubed sphere' is used to discretize the
+globe permitting a uniform gridding and obviated the need to fourier filter.
+The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
+grid, of which the cubed sphere is just one of many choices.
+Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
+wind and meridional overturning streamfunction from a 20-level version of
+the model. It compares favorable with more conventional spatial
+discretization approaches.
A regular spherical lat-lon grid can also be used.
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps}
- }
-}
-\end{center}
-\label{fig:hslatlon}
-\end{figure}
-
\subsection{Ocean gyres}
+Baroclinic instability is a ubiquitous process in the ocean, as well as the
+atmosphere. Ocean eddies play an important role in modifying the
+hydrographic structure and current systems of the oceans. Coarse resolution
+models of the oceans cannot resolve the eddy field and yield rather broad,
+diffusive patterns of ocean currents. But if the resolution of our models is
+increased until the baroclinic instability process is resolved, numerical
+solutions of a different and much more realistic kind, can be obtained.
+
+Fig. ?.? shows the surface temperature and velocity field obtained from
+MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
+grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
+(to avoid the converging of meridian in northern latitudes). 21 vertical
+levels are used in the vertical with a `lopped cell' representation of
+topography. The development and propagation of anomalously warm and cold
+eddies can be clearly been seen in the Gulf Stream region. The transport of
+warm water northward by the mean flow of the Gulf Stream is also clearly
+visible.
+
\subsection{Global ocean circulation}
Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
-global ocean model run with 15 vertical levels. The model is driven using
-monthly-mean winds with mixed boundary conditions on temperature and
-salinity at the surface. Fig.E2b shows the overturning (thermohaline)
-circulation. Lopped cells are used to represent topography on a regular $%
-lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$.
-
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
-% \rotatebox{90}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps}
-% }
-}
-\end{center}
-\label{fig:horizcirc}
-\end{figure}
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps}
- }
- }
-}
-\end{center}
-\label{fig:moc}
-\end{figure}
-
-
-\subsection{Flow over topography}
-
-\subsection{Ocean convection}
-
-Fig.E3 shows convection over a slope using the non-hydrostatic ocean
-isomorph and lopped cells to respresent topography. .....The grid resolution
-is
+global ocean model run with 15 vertical levels. Lopped cells are used to
+represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
+}N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
+mixed boundary conditions on temperature and salinity at the surface. The
+transfer properties of ocean eddies, convection and mixing is parameterized
+in this model.
+
+Fig.E2b shows the meridional overturning circulation of the global ocean in
+Sverdrups.
+
+\subsection{Convection and mixing over topography}
+
+Dense plumes generated by localized cooling on the continental shelf of the
+ocean may be influenced by rotation when the deformation radius is smaller
+than the width of the cooling region. Rather than gravity plumes, the
+mechanism for moving dense fluid down the shelf is then through geostrophic
+eddies. The simulation shown in the figure (blue is cold dense fluid, red is
+warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
+trigger convection by surface cooling. The cold, dense water falls down the
+slope but is deflected along the slope by rotation. It is found that
+entrainment in the vertical plane is reduced when rotational control is
+strong, and replaced by lateral entrainment due to the baroclinic
+instability of the along-slope current.
\subsection{Boundary forced internal waves}
-\subsection{Carbon outgassing sensitivity}
-
-Fig.E4 shows....
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps}
-}
-\end{center}
-\label{fig:co2mrt}
-\end{figure}
+The unique ability of MITgcm to treat non-hydrostatic dynamics in the
+presence of complex geometry makes it an ideal tool to study internal wave
+dynamics and mixing in oceanic canyons and ridges driven by large amplitude
+barotropic tidal currents imposed through open boundary conditions.
+
+Fig. ?.? shows the influence of cross-slope topographic variations on
+internal wave breaking - the cross-slope velocity is in color, the density
+contoured. The internal waves are excited by application of open boundary
+conditions on the left.\ They propagate to the sloping boundary (represented
+using MITgcm's finite volume spatial discretization) where they break under
+nonhydrostatic dynamics.
+
+\subsection{Parameter sensitivity using the adjoint of MITgcm}
+
+Forward and tangent linear counterparts of MITgcm are supported using an
+`automatic adjoint compiler'. These can be used in parameter sensitivity and
+data assimilation studies.
+
+As one example of application of the MITgcm adjoint, Fig.E4 maps the
+gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $%
+\mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
+sensitive to heat fluxes over the Labrador Sea, one of the important sources
+of deep water for the thermohaline circulations. This calculation also
+yields sensitivities to all other model parameters.
+
+\subsection{Global state estimation of the ocean}
+
+An important application of MITgcm is in state estimation of the global
+ocean circulation. An appropriately defined `cost function', which measures
+the departure of the model from observations (both remotely sensed and
+insitu) over an interval of time, is minimized by adjusting `control
+parameters' such as air-sea fluxes, the wind field, the initial conditions
+etc. Figure ?.? shows an estimate of the time-mean surface elevation of the
+ocean obtained by bringing the model in to consistency with altimetric and
+in-situ observations over the period 1992-1997.
+
+\subsection{Ocean biogeochemical cycles}
+
+MITgcm is being used to study global biogeochemical cycles in the ocean. For
+example one can study the effects of interannual changes in meteorological
+forcing and upper ocean circulation on the fluxes of carbon dioxide and
+oxygen between the ocean and atmosphere. The figure shows the annual air-sea
+flux of oxygen and its relation to density outcrops in the southern oceans
+from a single year of a global, interannually varying simulation.
+
+Chris - get figure here: http://puddle.mit.edu/\symbol{126}%
+mick/biogeochem.html
+
+\subsection{Simulations of laboratory experiments}
+
+Figure ?.? shows MITgcm being used to simulate a laboratory experiment
+enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
+initially homogeneous tank of water ($1m$ in diameter) is driven from its
+free surface by a rotating heated disk. The combined action of mechanical
+and thermal forcing creates a lens of fluid which becomes baroclinically
+unstable. The stratification and depth of penetration of the lens is
+arrested by its instability in a process analogous to that whic sets the
+stratification of the ACC.
-
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
% $Name: $
\section{Continuous equations in `r' coordinates}
@@ -280,19 +285,6 @@
\marginpar{
Fig.5 The vertical coordinate of model}:
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/vertcoord.eps}
- }
- }
-}
-\end{center}
-\label{fig:vertcoord}
-\end{figure}
-
\begin{equation*}
\frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}%
\right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}%
@@ -311,63 +303,63 @@
\end{equation}
\begin{equation*}
-b=b(\theta ,S,r)\text{ equation of state}
+b=b(\theta ,S,r)\text{ equation of state}
\end{equation*}
\begin{equation*}
-\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
+\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
\end{equation*}
\begin{equation*}
-\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
+\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
\end{equation*}
Here:
\begin{equation*}
-r\text{ is the vertical coordinate}
+r\text{ is the vertical coordinate}
\end{equation*}
\begin{equation*}
\frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{
-is the total derivative}
+is the total derivative}
\end{equation*}
\begin{equation*}
\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%
-\text{ is the `grad' operator}
+\text{ is the `grad' operator}
\end{equation*}
with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%
\frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
is a unit vector in the vertical
\begin{equation*}
-t\text{ is time}
+t\text{ is time}
\end{equation*}
\begin{equation*}
\vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the
-velocity}
+velocity}
\end{equation*}
\begin{equation*}
-\phi \text{ is the `pressure'/`geopotential'}
+\phi \text{ is the `pressure'/`geopotential'}
\end{equation*}
\begin{equation*}
-\vec{\Omega}\text{ is the Earth's rotation}
+\vec{\Omega}\text{ is the Earth's rotation}
\end{equation*}
\begin{equation*}
-b\text{ is the `buoyancy'}
+b\text{ is the `buoyancy'}
\end{equation*}
\begin{equation*}
-\theta \text{ is potential temperature}
+\theta \text{ is potential temperature}
\end{equation*}
\begin{equation*}
-S\text{ is specific humidity in the atmosphere; salinity in the ocean}
+S\text{ is specific humidity in the atmosphere; salinity in the ocean}
\end{equation*}
\begin{equation*}
@@ -376,8 +368,7 @@
\end{equation*}
\begin{equation*}
-\mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }%
-\theta
+\mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
\end{equation*}
\begin{equation*}
@@ -406,7 +397,7 @@
Here
\begin{equation*}
-R_{moving}=R_{o}+\eta
+R_{moving}=R_{o}+\eta
\end{equation*}
where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on
whether we are in the atmosphere or ocean) of the `moving surface' in the
@@ -474,12 +465,12 @@
At the top of the atmosphere (which is `fixed' in our $r$ coordinate):
\begin{equation*}
-R_{fixed}=p_{top}=0
+R_{fixed}=p_{top}=0
\end{equation*}
In a resting atmosphere the elevation of the mountains at the bottom is
given by
\begin{equation*}
-R_{moving}=R_{o}(x,y)=p_{o}(x,y)
+R_{moving}=R_{o}(x,y)=p_{o}(x,y)
\end{equation*}
i.e. the (hydrostatic) pressure at the top of the mountains in a resting
atmosphere.
@@ -589,7 +580,7 @@
\textit{Coriolis} \\
\textit{\ Forcing/Dissipation}%
\end{tabular}%
-\ \right. \qquad \label{eq:gu-speherical}
+\ \right. \qquad \label{eq:gu-speherical}
\end{equation}
\begin{equation}
@@ -608,9 +599,9 @@
\textit{Coriolis} \\
\textit{\ Forcing/Dissipation}%
\end{tabular}%
-\ \right. \qquad \label{eq:gv-spherical}
+\ \right. \qquad \label{eq:gv-spherical}
\end{equation}%
-\qquad \qquad \qquad \qquad \qquad
+\qquad \qquad \qquad \qquad \qquad
\begin{equation}
\left.
@@ -627,9 +618,9 @@
\textit{Coriolis} \\
\textit{\ Forcing/Dissipation}%
\end{tabular}%
-\ \right. \label{eq:gw-spherical}
+\ \right. \label{eq:gw-spherical}
\end{equation}%
-\qquad \qquad \qquad \qquad \qquad
+\qquad \qquad \qquad \qquad \qquad
In the above `${r}$' is the distance from the center of the earth and `$lat$%
' is latitude.
@@ -639,20 +630,6 @@
\marginpar{
Fig.6 Spherical polar coordinate system.}
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps}
- }
- }
-}
-\end{center}
-\label{fig:spcoord}
-\end{figure}
-
-
\subsubsection{Shallow atmosphere approximation}
Most models are based on the `hydrostatic primitive equations' (HPE's) in
@@ -661,8 +638,8 @@
Coriolis force is treated approximately and the shallow atmosphere
approximation is made.\ The MITgcm need not make the `traditional
approximation'. To be able to support consistent non-hydrostatic forms the
-shallow atmosphere approximation can be relaxed - when dividing through by $r
-$ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
+shallow atmosphere approximation can be relaxed - when dividing through by $%
+r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
the radius of the earth.
\subsubsection{Hydrostatic and quasi-hydrostatic forms}
@@ -688,7 +665,7 @@
vertical momentum equation (\ref{eq:mom-w}) becomes:
\begin{equation*}
-\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat
+\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat
\end{equation*}
making a small correction to the hydrostatic pressure.
@@ -769,20 +746,6 @@
stepping forward the horizontal momentum equations; $\dot{r}$ is found by
stepping forward the vertical momentum equation.
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
- \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps}
- }
- }
-}
-\end{center}
-\label{fig:solnstart}
-\end{figure}
-
-
There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
course, some complication that goes with the inclusion of $\cos \phi \ $%
Coriolis terms and the relaxation of the shallow atmosphere approximation.
@@ -809,7 +772,7 @@
\begin{equation*}
\int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}%
-\right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
+\right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
\end{equation*}
and so
@@ -831,7 +794,7 @@
\begin{equation*}
\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}%
-}_{h}+\partial _{r}\dot{r}\right) dr=0
+}_{h}+\partial _{r}\dot{r}\right) dr=0
\end{equation*}
Thus:
@@ -839,7 +802,7 @@
\begin{equation*}
\frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
+\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%
-_{h}dr=0
+_{h}dr=0
\end{equation*}
where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $%
r $. The above can be rearranged to yield, using Leibnitz's theorem:
@@ -855,7 +818,7 @@
(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
be written
\begin{equation}
-\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
+\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
\label{eq:phi-surf}
\end{equation}%
where $b_{s}$ is the buoyancy at the surface.
@@ -914,18 +877,18 @@
presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
(\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
exploit classical 3D potential theory and, by introducing an appropriately
-chosen $\delta $-function sheet of `source-charge', replace the inhomogenous
-boundary condition on pressure by a homogeneous one. The source term $rhs$
-in (\ref{eq:3d-invert}) is the divergence of the vector $\vec{\mathbf{F}}.$
-By simultaneously setting $%
+chosen $\delta $-function sheet of `source-charge', replace the
+inhomogeneous boundary condition on pressure by a homogeneous one. The
+source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $%
+\vec{\mathbf{F}}.$ By simultaneously setting $%
\begin{array}{l}
\widehat{n}.\vec{\mathbf{F}}%
\end{array}%
=0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
-self-consistent but simpler homogenised Elliptic problem is obtained:
+self-consistent but simpler homogenized Elliptic problem is obtained:
\begin{equation*}
-\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
+\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
\end{equation*}%
where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref%
@@ -1000,7 +963,7 @@
\begin{equation}
\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}%
+\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %
-\left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
+\left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
\label{eq:vi-identity}
\end{equation}%
This permits alternative numerical treatments of the non-linear terms based
@@ -1013,10 +976,10 @@
\subsection{Adjoint}
-Tangent linear and adoint counterparts of the forward model and described in
-Chapter 5.
+Tangent linear and adjoint counterparts of the forward model and described
+in Chapter 5.
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
% $Name: $
\section{Appendix ATMOSPHERE}
@@ -1029,12 +992,12 @@
The hydrostatic primitive equations (HPEs) in p-coordinates are:
\begin{eqnarray}
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
-_{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
+_{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
\label{eq:atmos-mom} \\
-\frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\
+\frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\
\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%
\partial p} &=&0 \label{eq:atmos-cont} \\
-p\alpha &=&RT \label{eq:atmos-eos} \\
+p\alpha &=&RT \label{eq:atmos-eos} \\
c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} \label{eq:atmos-heat}
\end{eqnarray}%
where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
@@ -1081,7 +1044,7 @@
The heat equation is obtained by noting that
\begin{equation*}
c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta
-\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
+\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
\end{equation*}
and on substituting into (\ref{eq-p-heat-interim}) gives:
\begin{equation}
@@ -1143,7 +1106,7 @@
\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \label{eq:atmos-prime}
\end{eqnarray}
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
% $Name: $
\section{Appendix OCEAN}
@@ -1160,7 +1123,7 @@
&=&\epsilon _{nh}\mathcal{F}_{w} \\
\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%
_{h}+\frac{\partial w}{\partial z} &=&0 \\
-\rho &=&\rho (\theta ,S,p) \\
+\rho &=&\rho (\theta ,S,p) \\
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:non-boussinesq}
\end{eqnarray}%
@@ -1242,14 +1205,14 @@
pressure in the EOS by splitting the pressure into a reference function of
height and a perturbation:
\begin{equation*}
-\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
+\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
\end{equation*}
Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
differentiating the EOS, the continuity equation then becomes:
\begin{equation*}
\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%
Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%
-\frac{\partial w}{\partial z}=0
+\frac{\partial w}{\partial z}=0
\end{equation*}
If the time- and space-scales of the motions of interest are longer than
those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%
@@ -1359,7 +1322,7 @@
_{nh}=0$ form of these equations that are used throughout the ocean modeling
community and referred to as the primitive equations (HPE).
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
% $Name: $
\section{Appendix:OPERATORS}
@@ -1372,16 +1335,16 @@
and vertical direction respectively, are given by (see Fig.2) :
\begin{equation*}
-u=r\cos \phi \frac{D\lambda }{Dt}
+u=r\cos \phi \frac{D\lambda }{Dt}
\end{equation*}
\begin{equation*}
-v=r\frac{D\phi }{Dt}\qquad
+v=r\frac{D\phi }{Dt}\qquad
\end{equation*}
$\qquad \qquad \qquad \qquad $
\begin{equation*}
-\dot{r}=\frac{Dr}{Dt}
+\dot{r}=\frac{Dr}{Dt}
\end{equation*}
Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial
@@ -1394,13 +1357,13 @@
\begin{equation*}
\nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%
,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%
-\right)
+\right)
\end{equation*}
\begin{equation*}
\nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial
\lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}
-+\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
++\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
\end{equation*}
%%%% \end{document}
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