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--- manual/s_overview/text/manual.tex 2001/10/25 12:06:56 1.7
+++ manual/s_overview/text/manual.tex 2001/10/25 15:24:01 1.8
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.8 2001/10/25 15:24:01 cnh Exp $
% $Name: $
%tci%\documentclass[12pt]{book}
@@ -34,7 +34,7 @@
% Section: Overview
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.8 2001/10/25 15:24:01 cnh Exp $
% $Name: $
\section{Introduction}
@@ -89,7 +89,7 @@
We begin by briefly showing some of the results of the model in action to
give a feel for the wide range of problems that can be addressed using it.
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.8 2001/10/25 15:24:01 cnh Exp $
% $Name: $
\section{Illustrations of the model in action}
@@ -289,7 +289,7 @@
\input{part1/lab_figure}
%%CNHend
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.8 2001/10/25 15:24:01 cnh Exp $
% $Name: $
\section{Continuous equations in `r' coordinates}
@@ -326,31 +326,33 @@
\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}}}
-\text{ horizontal mtm}
+\text{ horizontal mtm} \label{eq:horizontal_mtm}
\end{equation*}
-\begin{equation*}
+\begin{equation}
\frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
-vertical mtm}
-\end{equation*}
+vertical mtm} \label{eq:vertical_mtm}
+\end{equation}
\begin{equation}
\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
-\partial r}=0\text{ continuity} \label{eq:continuous}
+\partial r}=0\text{ continuity} \label{eq:continuity}
\end{equation}
-\begin{equation*}
-b=b(\theta ,S,r)\text{ equation of state}
-\end{equation*}
+\begin{equation}
+b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
+\end{equation}
-\begin{equation*}
+\begin{equation}
\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
-\end{equation*}
+\label{eq:potential_temperature}
+\end{equation}
-\begin{equation*}
+\begin{equation}
\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
-\end{equation*}
+\label{eq:humidtity_salt}
+\end{equation}
Here:
@@ -523,8 +525,8 @@
atmosphere)} \label{eq:moving-bc-atmos}
\end{eqnarray}
-Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent
-set of atmospheric equations which, for convenience, are written out in $p$
+Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_slainty})
+yields a consistent set of atmospheric equations which, for convenience, are written out in $p$
coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).
\subsection{Ocean}
@@ -560,7 +562,8 @@
\end{eqnarray}
where $\eta $ is the elevation of the free surface.
-Then eq(\ref{eq:continuous}) yields a consistent set of oceanic equations
+Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_slainty}) yield a consistent set
+of oceanic equations
which, for convenience, are written out in $z$ coordinates in Appendix Ocean
- see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
@@ -573,7 +576,7 @@
\phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
\label{eq:phi-split}
\end{equation}
-and write eq(\ref{incompressible}a,b) in the form:
+and write eq(\ref{eq:incompressible}) in the form:
\begin{equation}
\frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
@@ -666,8 +669,6 @@
Grad and div operators in spherical coordinates are defined in appendix
OPERATORS.
-\marginpar{
-Fig.6 Spherical polar coordinate system.}
%%CNHbegin
\input{part1/sphere_coord_figure.tex}
@@ -730,7 +731,7 @@
three dimensional elliptic equation must be solved subject to Neumann
boundary conditions (see below). It is important to note that use of the
full \textbf{NH} does not admit any new `fast' waves in to the system - the
-incompressible condition eq(\ref{eq:continuous})c has already filtered out
+incompressible condition eq(\ref{eq:continuity}) has already filtered out
acoustic modes. It does, however, ensure that the gravity waves are treated
accurately with an exact dispersion relation. The \textbf{NH} set has a
complete angular momentum principle and consistent energetics - see White
@@ -779,9 +780,8 @@
\subsection{Solution strategy}
The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
-NH} models is summarized in Fig.7.
-\marginpar{
-Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
+NH} models is summarized in Figure \ref{fig:solution-strategy}.
+Under all dynamics, a 2-d elliptic equation is
first solved to find the surface pressure and the hydrostatic pressure at
any level computed from the weight of fluid above. Under \textbf{HPE} and
\textbf{QH} dynamics, the horizontal momentum equations are then stepped
@@ -838,8 +838,8 @@
\subsubsection{Surface pressure}
-The surface pressure equation can be obtained by integrating continuity, (
-\ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
+The surface pressure equation can be obtained by integrating continuity,
+(\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
\begin{equation*}
\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
@@ -864,7 +864,7 @@
where we have incorporated a source term.
Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
-(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
+(atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
be written
\begin{equation}
\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
@@ -872,16 +872,16 @@
\end{equation}
where $b_{s}$ is the buoyancy at the surface.
-In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref
+In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref
{eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
surface' and `rigid lid' approaches are available.
\subsubsection{Non-hydrostatic pressure}
-Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
-\partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
-(\ref{incompressible}), we deduce that:
+Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
+$\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
+(\ref{eq:continuity}), we deduce that:
\begin{equation}
\nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
@@ -911,7 +911,7 @@
depending on the form chosen for the dissipative terms in the momentum
equations - see below.
-Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that:
+Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
\begin{equation}
\widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
@@ -951,7 +951,7 @@
converges rapidly because $\phi _{nh}\ $is then only a small correction to
the hydrostatic pressure field (see the discussion in Marshall et al, a,b).
-The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{homneuman})
+The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh})
does not vanish at $r=R_{moving}$, and so refines the pressure there.
\subsection{Forcing/dissipation}
@@ -959,7 +959,7 @@
\subsubsection{Forcing}
The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by
-`physics packages' described in detail in chapter ??.
+`physics packages' and forcing packages. These are described later on.
\subsubsection{Dissipation}
@@ -1007,7 +1007,7 @@
\subsection{Vector invariant form}
For some purposes it is advantageous to write momentum advection in eq(\ref
-{hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
+{eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the (so-called) `vector invariant' form:
\begin{equation}
\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
@@ -1025,10 +1025,10 @@
\subsection{Adjoint}
-Tangent linear and adjoint counterparts of the forward model and described
+Tangent linear and adjoint counterparts of the forward model are described
in Chapter 5.
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.8 2001/10/25 15:24:01 cnh Exp $
% $Name: $
\section{Appendix ATMOSPHERE}
@@ -1147,15 +1147,15 @@
The final form of the HPE's in p coordinates is then:
\begin{eqnarray}
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
-_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\
+_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-prime} \\
\frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
\partial p} &=&0 \\
\frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
-\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \label{eq:atmos-prime}
+\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }
\end{eqnarray}
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.8 2001/10/25 15:24:01 cnh Exp $
% $Name: $
\section{Appendix OCEAN}
@@ -1171,10 +1171,11 @@
\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
&=&\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) \\
-\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
-\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:non-boussinesq}
+_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
+\rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
+\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
+\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zns-salt}
+\label{eq:non-boussinesq}
\end{eqnarray}
These equations permit acoustics modes, inertia-gravity waves,
non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
@@ -1193,8 +1194,7 @@
\end{equation}
Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
-reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
-{eq-zns-cont} gives:
+reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref{eq-zns-cont} gives:
\begin{equation}
\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure}
@@ -1371,7 +1371,7 @@
_{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 1.7 2001/10/25 12:06:56 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.8 2001/10/25 15:24:01 cnh Exp $
% $Name: $
\section{Appendix:OPERATORS}
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