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--- manual/s_overview/continuous_eqns.tex 2001/08/08 16:16:18 1.1
+++ manual/s_overview/continuous_eqns.tex 2001/09/11 14:03:33 1.2
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/Attic/continuous_eqns.tex,v 1.1 2001/08/08 16:16:18 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/Attic/continuous_eqns.tex,v 1.2 2001/09/11 14:03:33 cnh Exp $
% $Name: $
\section{Continuous equations in `r' coordinates}
@@ -202,9 +202,11 @@
The boundary conditions at top and bottom are given by:
\begin{eqnarray*}
-&&\omega =0~\text{at }r=R_{fixed}\text{ (top of the atmosphere)} \\
+&&\omega =0~\text{at }r=R_{fixed} \label{eq:fixed-bc-atmos}
+\text{ (top of the atmosphere)} \\
\omega &=&\frac{Dp_{s}}{Dt}\text{; at }r=R_{moving}\text{ (bottom of the
atmosphere)}
+\label{eq:moving-bc-atmos}
\end{eqnarray*}
Then the (hydrostatic form of) eq(\ref{incompressible}) yields a consistent
@@ -216,11 +218,15 @@
In the ocean we interpret:
\begin{eqnarray}
-r &=&z\text{ is the height} \\
-\dot{r} &=&\frac{Dz}{Dt}=w\text{ is the vertical velocity} \\
-\phi &=&\frac{p}{\rho _{c}}\text{ is the pressure} \\
+r &=&z\text{ is the height}
+\label{eq:ocean-z}\\
+\dot{r} &=&\frac{Dz}{Dt}=w\text{ is the vertical velocity}
+\label{eq:ocean-w}\\
+\phi &=&\frac{p}{\rho _{c}}\text{ is the pressure}
+\label{eq:ocean-p}\\
b(\theta ,S,r) &=&\frac{g}{\rho _{c}}\left( \rho (\theta ,S,r)-\rho
_{c}\right) \text{ is the buoyancy}
+\label{eq:ocean-b}
\end{eqnarray}
where $\rho _{c}$ is a fixed reference density of water and $g$ is the
acceleration due to gravity.\noindent
@@ -237,8 +243,10 @@
Boundary conditions are:
\begin{eqnarray*}
-w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)} \\
-w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)}
+w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)}
+\label{eq:fixed-bc-ocean}\\
+w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)
+\label{eq:moving-bc-ocean}}
\end{eqnarray*}
where $\eta $ is the elevation of the free surface.
@@ -252,23 +260,23 @@
\begin{equation}
\phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
-\label{pressuresplit}
+\label{eq:phi-split}
\end{equation}
and write eq(\ref{incompressible}a) in the form:
\begin{equation}
\frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
_{s}+\mathbf{\nabla }_{h}\phi _{hyd}+\epsilon _{nh}\mathbf{\nabla }_{h}\phi
-_{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}} \label{hor-mtm}
+_{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}} \label{eq:mom-h}
\end{equation}
\begin{equation}
-\frac{\partial \phi _{hyd}}{\partial r}=-b \label{hydro}
+\frac{\partial \phi _{hyd}}{\partial r}=-b \label{eq:hydrostatic}
\end{equation}
\begin{equation}
\epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{%
-\partial r}=G_{\dot{r}} \label{vertmtm}
+\partial r}=G_{\dot{r}} \label{eq:mom-w}
\end{equation}
Here $\epsilon _{nh}$ is a non-hydrostatic parameter.
@@ -298,7 +306,7 @@
\textit{Coriolis} \\
\textit{\ Forcing/Dissipation}
\end{tabular}
-\right. \qquad \label{Gu}
+\right. \qquad \label{eq:gu-speherical}
\end{equation}
\begin{equation}
@@ -317,7 +325,7 @@
\textit{Coriolis} \\
\textit{\ Forcing/Dissipation}
\end{tabular}
-\right. \qquad \label{Gv}
+\right. \qquad \label{eq:gv-spherical}
\end{equation}
\qquad \qquad \qquad \qquad \qquad
@@ -337,7 +345,7 @@
\textit{Coriolis} \\
\textit{\ Forcing/Dissipation}
\end{tabular}
-\right. \label{Gw}
+\right. \label{eq:gw-spherical}
\end{equation}
\qquad \qquad \qquad \qquad \qquad
@@ -409,7 +417,7 @@
here) by:
\begin{equation}
-\dot{r}=\frac{Dp}{Dt}=\frac{Dp_{hyd}}{Dt} \label{quasinonhydro}
+\dot{r}=\frac{Dp}{Dt}=\frac{Dp_{hyd}}{Dt} \label{eq:quasi-nh-w}
\end{equation}
where $p_{hy}$ is the hydrostatic pressure.
@@ -469,13 +477,8 @@
dividing the total (pressure/geo) potential in to three parts, a surface
part, $\phi _{s}(x,y)$, a hydrostatic part $\phi _{hyd}(x,y,r)$ and a
non-hydrostatic part $\phi _{nh}(x,y,r)$, as in (\ref{pressuresplit}), and
-writing the momentum equation in the form
-\begin{equation}
-\frac{\partial }{\partial t}\vec{\mathbf{v}_{h}}+\mathbf{\nabla }_{h}\phi
-_{s}+\mathbf{\nabla }_{h}\phi _{hyd}+\epsilon _{nh}\mathbf{\nabla }\phi
-_{nh}=\vec{\mathbf{G}}_{\vec{v}} \label{mtm-split}
-\end{equation}
-as in (\ref{hor-mtm}).
+writing the momentum equation
+as in (\ref{eq:mom-h}).
\subsubsection{Hydrostatic pressure}
@@ -488,9 +491,10 @@
\]
and so
-\[
+\begin{equation}
\phi _{hyd}(x,y,r)=\int_{r}^{R_{o}}bdr
-\]
+\label{eq:hydro-phi}
+\end{equation}
\subsubsection{Surface pressure}
@@ -515,14 +519,15 @@
\begin{equation}
\frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot
\int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=0
-\label{integralcontinuity}
+\label{eq:free-surface}
\end{equation}
Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
be written
\begin{equation}
-\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}b\eta \label{link}
+\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}b\eta
+\label{eq:phi-surf}
\end{equation}
where $b$ is the buoyancy.
@@ -540,7 +545,7 @@
\begin{equation}
\nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{%
\nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .%
-\vec{\mathbf{F}} \label{3dinvert}
+\vec{\mathbf{F}} \label{eq:3d-invert}
\end{equation}
For a given rhs this 3-d elliptic equation must be inverted for $\phi _{nh}$
@@ -569,7 +574,7 @@
\begin{equation}
\widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
-\label{inhomneumann}
+\label{eq:inhom-neumann-nh}
\end{equation}
where
@@ -598,7 +603,7 @@
{inhomneumann}) the modified boundary condition becomes:
\begin{equation}
-\widehat{n}.\nabla \phi _{nh}=0 \label{homneuman}
+\widehat{n}.\nabla \phi _{nh}=0 \label{eq:hom-neumann-nh}
\end{equation}
If the flow is `close' to hydrostatic balance then the 3-d inversion
@@ -622,10 +627,11 @@
Many forms of momentum dissipation are available in the model. Laplacian and
biharmonic frictions are commonly used:
-\[
+\begin{equation}
D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}%
+A_{4}\nabla _{h}^{4}v
-\]
+\label{eq:dissipation}
+\end{equation}
where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity
coefficients and $A_{4}\ $is the horizontal coefficient for biharmonic
friction. These coefficients are the same for all velocity components.
@@ -635,17 +641,18 @@
The mixing terms for the temperature and salinity equations have a similar
form to that of momentum except that the diffusion tensor can be
non-diagonal and have varying coefficients. $\qquad $%
-\[
+\begin{equation}
D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
_{h}^{4}(T,S)
-\]
+\label{eq:diffusion}
+\end{equation}
where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%
horizontal coefficient for biharmonic diffusion. In the simplest case where
the subgrid-scale fluxes of heat and salt are parameterized with constant
horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,
reduces to a diagonal matrix with constant coefficients:
-\[
+\begin{equation}
\qquad \qquad \qquad \qquad K=\left(
\begin{array}{ccc}
K_{h} & 0 & 0 \\
@@ -653,7 +660,8 @@
0 & 0 & K_{v}
\end{array}
\right) \qquad \qquad \qquad
-\]
+\label{eq:diagonal-diffusion-tensor}
+\end{equation}
where $K_{h}\ $and $K_{v}\ $are the horizontal and vertical diffusion
coefficients. These coefficients are the same for all tracers (temperature,
salinity ... ).
@@ -667,7 +675,7 @@
\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]
-\label{vecinvariant}
+\label{eq:vi-identity}
\end{equation}
This permits alternative numerical treatments of the non-linear terms based
on their representation as a vorticity flux. Because gradients of coordinate
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