manual/s_overview/appendix_ocean.tex

% $Header: $
% $Name: $

\section{Appendix OCEAN}

\subsection{Equations of motion for the ocean}

We review here the method by which the standard (Boussinesq, incompressible)
HPE's for the ocean written in z-coordinates are obtained. The
non-Boussinesq equations for oceanic motion are: 
\begin{eqnarray}
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
_{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
\label{eq-zns-hmom} \\
\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
&=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-zns-hydro} \\
\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%
_{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}
\end{eqnarray}
These equations permit acoustics modes, inertia-gravity waves,
non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
mode. As written, they cannot be integrated forward consistently - if we
step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
{eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
therefore necessary to manipulate the system as follows. Differentiating the
EOS (equation of state) gives:

\begin{equation}
\frac{D\rho }{Dt}=\left. \frac{\partial \rho }{\partial \theta }\right|
_{S,p}\frac{D\theta }{Dt}+\left. \frac{\partial \rho }{\partial S}\right|
_{\theta ,p}\frac{DS}{Dt}+\left. \frac{\partial \rho }{\partial p}\right|
_{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}
\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: 
\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}
\end{equation}
where we have used an approximation sign to indicate that we have assumed
adiabatic motion, dropping the $\frac{D\theta }{Dt}$ and $\frac{DS}{Dt}$.
Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that
can be explicitly integrated forward: 
\begin{eqnarray}
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
_{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
\label{eq-cns-hmom} \\
\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
&=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\
\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%
v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\
\rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\
\frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-cns-salt}
\end{eqnarray}

\subsubsection{Compressible z-coordinate equations}

Here we linearize the acoustic modes by replacing $\rho $ with $\rho _{o}(z)$
wherever it appears in a product (ie. non-linear term) - this is the
`Boussinesq assumption'. The only term that then retains the full variation
in $\rho $ is the gravitational acceleration: 
\begin{eqnarray}
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
_{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
\label{eq-zcb-hmom} \\
\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%
\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
\label{eq-zcb-hydro} \\
\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{%
\mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-zcb-cont} \\
\rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\
\frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}
\end{eqnarray}
These equations still retain acoustic modes. But, because the
``compressible'' terms are linearized, the pressure equation \ref
{eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
term appears as a Helmholtz term in the non-hydrostatic pressure equation).
These are the \emph{truly} compressible Boussinesq equations. Note that the
EOS must have the same pressure dependency as the linearized pressure term,
ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{%
c_{s}^{2}}$, for consistency.

\subsubsection{`Anelastic' z-coordinate equations}

The anelastic approximation filters the acoustic mode by removing the
time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}%
). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}%
\frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
continuity and EOS. A better solution is to change the dependency on
pressure in the EOS by splitting the pressure into a reference function of
height and a perturbation: 
\[
\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime }) 
\]
Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
differentiating the EOS, the continuity equation then becomes: 
\[
\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%
Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%
\frac{\partial w}{\partial z}=0 
\]
If the time- and space-scales of the motions of interest are longer than
those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%
\mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and 
$\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{%
Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
_{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the
anelastic continuity equation: 
\begin{equation}
\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-%
\frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}
\end{equation}
A slightly different route leads to the quasi-Boussinesq continuity equation
where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+%
\mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%
_{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding: 
\begin{equation}
\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%
\partial \left( \rho _{o}w\right) }{\partial z}=0  \label{eq-za-cont2}
\end{equation}
Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same
equation if: 
\begin{equation}
\frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}
\end{equation}
Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{%
g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
then: 
\begin{eqnarray}
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
_{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
\label{eq-zab-hmom} \\
\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%
\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
\label{eq-zab-hydro} \\
\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%
\partial \left( \rho _{o}w\right) }{\partial z} &=&0  \label{eq-zab-cont} \\
\rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\
\frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zab-salt}
\end{eqnarray}

\subsubsection{Incompressible z-coordinate equations}

Here, the objective is to drop the depth dependence of $\rho _{o}$ and so,
technically, to also remove the dependence of $\rho $ on $p_{o}$. This would
yield the ``truly'' incompressible Boussinesq equations: 
\begin{eqnarray}
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
\label{eq-ztb-hmom} \\
\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}%
\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
\label{eq-ztb-hydro} \\
\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
&=&0  \label{eq-ztb-cont} \\
\rho &=&\rho (\theta ,S)  \label{eq-ztb-eos} \\
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-ztb-heat} \\
\frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-ztb-salt}
\end{eqnarray}
where $\rho _{c}$ is a constant reference density of water.

\subsubsection{Compressible non-divergent equations}

The above ``incompressible'' equations are incompressible in both the flow
and the density. In many oceanic applications, however, it is important to
retain compressibility effects in the density. To do this we must split the
density thus: 
\[
\rho =\rho _{o}+\rho ^{\prime } 
\]
We then assert that variations with depth of $\rho _{o}$ are unimportant
while the compressible effects in $\rho ^{\prime }$ are: 
\[
\rho _{o}=\rho _{c} 
\]
\[
\rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o} 
\]
This then yields what we can call the semi-compressible Boussinesq
equations: 
\begin{eqnarray}
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%
\mathcal{F}}}  \label{eq-zpe-hmom} \\
\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
_{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
\label{eq-zpe-hydro} \\
\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
&=&0  \label{eq-zpe-cont} \\
\rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c}  \label{eq-zpe-eos} \\
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zpe-heat} \\
\frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zpe-salt}
\end{eqnarray}
Note that the hydrostatic pressure of the resting fluid, including that
associated with $\rho _{c}$, is subtracted out since it has no effect on the
dynamics.

Though necessary, the assumptions that go into these equations are messy
since we essentially assume a different EOS for the reference density and
the perturbation density. Nevertheless, it is the hydrostatic ($\epsilon
_{nh}=0$ form of these equations that are used throughout the ocean modeling
community and referred to as the primitive equations (HPE).
1	adcroft	1.1	% $Header: $
2			% $Name: $
3
4			\section{Appendix OCEAN}
5
6			\subsection{Equations of motion for the ocean}
7
8			We review here the method by which the standard (Boussinesq, incompressible)
9			HPE's for the ocean written in z-coordinates are obtained. The
10			non-Boussinesq equations for oceanic motion are:
11			\begin{eqnarray}
12			\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
13			_{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
14			\label{eq-zns-hmom} \\
15			\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
16			&=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-zns-hydro} \\
17			\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%
18			_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont} \\
19			\rho &=&\rho (\theta ,S,p) \label{eq-zns-eos} \\
20			\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat} \\
21			\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zns-salt}
22			\end{eqnarray}
23			These equations permit acoustics modes, inertia-gravity waves,
24			non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
25			mode. As written, they cannot be integrated forward consistently - if we
26			step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
27			consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
28			{eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
29			therefore necessary to manipulate the system as follows. Differentiating the
30			EOS (equation of state) gives:
31
32			\begin{equation}
33			\frac{D\rho }{Dt}=\left. \frac{\partial \rho }{\partial \theta }\right\|
34			_{S,p}\frac{D\theta }{Dt}+\left. \frac{\partial \rho }{\partial S}\right\|
35			_{\theta ,p}\frac{DS}{Dt}+\left. \frac{\partial \rho }{\partial p}\right\|
36			_{\theta ,S}\frac{Dp}{Dt} \label{EOSexpansion}
37			\end{equation}
38
39			Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
40			reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
41			{eq-zns-cont} gives:
42			\begin{equation}
43			\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%
44			v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure}
45			\end{equation}
46			where we have used an approximation sign to indicate that we have assumed
47			adiabatic motion, dropping the $\frac{D\theta }{Dt}$ and $\frac{DS}{Dt}$.
48			Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that
49			can be explicitly integrated forward:
50			\begin{eqnarray}
51			\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
52			_{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
53			\label{eq-cns-hmom} \\
54			\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
55			&=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-cns-hydro} \\
56			\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%
57			v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-cns-cont} \\
58			\rho &=&\rho (\theta ,S,p) \label{eq-cns-eos} \\
59			\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-cns-heat} \\
60			\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-cns-salt}
61			\end{eqnarray}
62
63			\subsubsection{Compressible z-coordinate equations}
64
65			Here we linearize the acoustic modes by replacing $\rho $ with $\rho _{o}(z)$
66			wherever it appears in a product (ie. non-linear term) - this is the
67			`Boussinesq assumption'. The only term that then retains the full variation
68			in $\rho $ is the gravitational acceleration:
69			\begin{eqnarray}
70			\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
71			_{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
72			\label{eq-zcb-hmom} \\
73			\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%
74			\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
75			\label{eq-zcb-hydro} \\
76			\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{%
77			\mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zcb-cont} \\
78			\rho &=&\rho (\theta ,S,p) \label{eq-zcb-eos} \\
79			\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zcb-heat} \\
80			\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zcb-salt}
81			\end{eqnarray}
82			These equations still retain acoustic modes. But, because the
83			``compressible'' terms are linearized, the pressure equation \ref
84			{eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
85			term appears as a Helmholtz term in the non-hydrostatic pressure equation).
86			These are the \emph{truly} compressible Boussinesq equations. Note that the
87			EOS must have the same pressure dependency as the linearized pressure term,
88			ie. $\left. \frac{\partial \rho }{\partial p}\right\| _{\theta ,S}=\frac{1}{%
89			c_{s}^{2}}$, for consistency.
90
91			\subsubsection{`Anelastic' z-coordinate equations}
92
93			The anelastic approximation filters the acoustic mode by removing the
94			time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}%
95			). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}%
96			\frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
97			continuity and EOS. A better solution is to change the dependency on
98			pressure in the EOS by splitting the pressure into a reference function of
99			height and a perturbation:
100			\[
101			\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
102			\]
103			Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
104			differentiating the EOS, the continuity equation then becomes:
105			\[
106			\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%
107			Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%
108			\frac{\partial w}{\partial z}=0
109			\]
110			If the time- and space-scales of the motions of interest are longer than
111			those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%
112			\mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and
113			$\left. \frac{\partial \rho }{\partial p}\right\| _{\theta ,S}\frac{%
114			Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right\| _{\theta
115			,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
116			_{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
117			and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the
118			anelastic continuity equation:
119			\begin{equation}
120			\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-%
121			\frac{g}{c_{s}^{2}}w=0 \label{eq-za-cont1}
122			\end{equation}
123			A slightly different route leads to the quasi-Boussinesq continuity equation
124			where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+%
125			\mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%
126			_{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:
127			\begin{equation}
128			\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%
129			\partial \left( \rho _{o}w\right) }{\partial z}=0 \label{eq-za-cont2}
130			\end{equation}
131			Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same
132			equation if:
133			\begin{equation}
134			\frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}
135			\end{equation}
136			Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
137			and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{%
138			g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
139			full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
140			then:
141			\begin{eqnarray}
142			\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
143			_{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
144			\label{eq-zab-hmom} \\
145			\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%
146			\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
147			\label{eq-zab-hydro} \\
148			\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%
149			\partial \left( \rho _{o}w\right) }{\partial z} &=&0 \label{eq-zab-cont} \\
150			\rho &=&\rho (\theta ,S,p_{o}(z)) \label{eq-zab-eos} \\
151			\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zab-heat} \\
152			\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zab-salt}
153			\end{eqnarray}
154
155			\subsubsection{Incompressible z-coordinate equations}
156
157			Here, the objective is to drop the depth dependence of $\rho _{o}$ and so,
158			technically, to also remove the dependence of $\rho $ on $p_{o}$. This would
159			yield the ``truly'' incompressible Boussinesq equations:
160			\begin{eqnarray}
161			\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
162			_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
163			\label{eq-ztb-hmom} \\
164			\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}%
165			\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
166			\label{eq-ztb-hydro} \\
167			\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
168			&=&0 \label{eq-ztb-cont} \\
169			\rho &=&\rho (\theta ,S) \label{eq-ztb-eos} \\
170			\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-ztb-heat} \\
171			\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-ztb-salt}
172			\end{eqnarray}
173			where $\rho _{c}$ is a constant reference density of water.
174
175			\subsubsection{Compressible non-divergent equations}
176
177			The above ``incompressible'' equations are incompressible in both the flow
178			and the density. In many oceanic applications, however, it is important to
179			retain compressibility effects in the density. To do this we must split the
180			density thus:
181			\[
182			\rho =\rho _{o}+\rho ^{\prime }
183			\]
184			We then assert that variations with depth of $\rho _{o}$ are unimportant
185			while the compressible effects in $\rho ^{\prime }$ are:
186			\[
187			\rho _{o}=\rho _{c}
188			\]
189			\[
190			\rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}
191			\]
192			This then yields what we can call the semi-compressible Boussinesq
193			equations:
194			\begin{eqnarray}
195			\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
196			_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%
197			\mathcal{F}}} \label{eq-zpe-hmom} \\
198			\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
199			_{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
200			\label{eq-zpe-hydro} \\
201			\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
202			&=&0 \label{eq-zpe-cont} \\
203			\rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c} \label{eq-zpe-eos} \\
204			\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zpe-heat} \\
205			\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zpe-salt}
206			\end{eqnarray}
207			Note that the hydrostatic pressure of the resting fluid, including that
208			associated with $\rho _{c}$, is subtracted out since it has no effect on the
209			dynamics.
210
211			Though necessary, the assumptions that go into these equations are messy
212			since we essentially assume a different EOS for the reference density and
213			the perturbation density. Nevertheless, it is the hydrostatic ($\epsilon
214			_{nh}=0$ form of these equations that are used throughout the ocean modeling
215			community and referred to as the primitive equations (HPE).