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revision 1.1.1.1 by adcroft, Wed Aug 8 16:16:19 2001 UTC revision 1.4 by cnh, Wed Oct 10 16:48:38 2001 UTC
# Line 9  We review here the method by which the s Line 9  We review here the method by which the s
9  HPE's for the ocean written in z-coordinates are obtained. The  HPE's for the ocean written in z-coordinates are obtained. The
10  non-Boussinesq equations for oceanic motion are:  non-Boussinesq equations for oceanic motion are:
11  \begin{eqnarray}  \begin{eqnarray}
12  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \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}}}  _{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}  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
16  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-zns-hydro} \\  &=&\epsilon _{nh}\mathcal{F}_{w}  \\
17  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%  \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} \\  _{h}+\frac{\partial w}{\partial z} &=&0  \\
19  \rho &=&\rho (\theta ,S,p)  \label{eq-zns-eos} \\  \rho &=&\rho (\theta ,S,p)  \\
20  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zns-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \\
21  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zns-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}
22    \label{eq:non-boussinesq}
23  \end{eqnarray}  \end{eqnarray}
24  These equations permit acoustics modes, inertia-gravity waves,  These equations permit acoustics modes, inertia-gravity waves,
25  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
# Line 40  Note that $\frac{\partial \rho }{\partia Line 41  Note that $\frac{\partial \rho }{\partia
41  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
42  {eq-zns-cont} gives:  {eq-zns-cont} gives:
43  \begin{equation}  \begin{equation}
44  \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%  \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
45  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
46  \end{equation}  \end{equation}
47  where we have used an approximation sign to indicate that we have assumed  where we have used an approximation sign to indicate that we have assumed
# Line 48  adiabatic motion, dropping the $\frac{D\ Line 49  adiabatic motion, dropping the $\frac{D\
49  Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that  Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that
50  can be explicitly integrated forward:  can be explicitly integrated forward:
51  \begin{eqnarray}  \begin{eqnarray}
52  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
53  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
54  \label{eq-cns-hmom} \\  \label{eq-cns-hmom} \\
55  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
56  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\
57  \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%  \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
58  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\
59  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\
60  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\
# Line 67  wherever it appears in a product (ie. no Line 68  wherever it appears in a product (ie. no
68  `Boussinesq assumption'. The only term that then retains the full variation  `Boussinesq assumption'. The only term that then retains the full variation
69  in $\rho $ is the gravitational acceleration:  in $\rho $ is the gravitational acceleration:
70  \begin{eqnarray}  \begin{eqnarray}
71  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
72  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
73  \label{eq-zcb-hmom} \\  \label{eq-zcb-hmom} \\
74  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
75  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
76  \label{eq-zcb-hydro} \\  \label{eq-zcb-hydro} \\
77  \frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{%  \frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{
78  \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-zcb-cont} \\  \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-zcb-cont} \\
79  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\
80  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\
# Line 85  These equations still retain acoustic mo Line 86  These equations still retain acoustic mo
86  term appears as a Helmholtz term in the non-hydrostatic pressure equation).  term appears as a Helmholtz term in the non-hydrostatic pressure equation).
87  These are the \emph{truly} compressible Boussinesq equations. Note that the  These are the \emph{truly} compressible Boussinesq equations. Note that the
88  EOS must have the same pressure dependency as the linearized pressure term,  EOS must have the same pressure dependency as the linearized pressure term,
89  ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{%  ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{
90  c_{s}^{2}}$, for consistency.  c_{s}^{2}}$, for consistency.
91    
92  \subsubsection{`Anelastic' z-coordinate equations}  \subsubsection{`Anelastic' z-coordinate equations}
93    
94  The anelastic approximation filters the acoustic mode by removing the  The anelastic approximation filters the acoustic mode by removing the
95  time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}%  time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}
96  ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}%  ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}
97  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
98  continuity and EOS. A better solution is to change the dependency on  continuity and EOS. A better solution is to change the dependency on
99  pressure in the EOS by splitting the pressure into a reference function of  pressure in the EOS by splitting the pressure into a reference function of
# Line 103  height and a perturbation: Line 104  height and a perturbation:
104  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
105  differentiating the EOS, the continuity equation then becomes:  differentiating the EOS, the continuity equation then becomes:
106  \[  \[
107  \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%  \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{
108  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+
109  \frac{\partial w}{\partial z}=0  \frac{\partial w}{\partial z}=0
110  \]  \]
111  If the time- and space-scales of the motions of interest are longer than  If the time- and space-scales of the motions of interest are longer than
112  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},
113  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and
114  $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{%  $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{
115  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
116  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
117  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
118  and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the  and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the
119  anelastic continuity equation:  anelastic continuity equation:
120  \begin{equation}  \begin{equation}
121  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-%  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-
122  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}
123  \end{equation}  \end{equation}
124  A slightly different route leads to the quasi-Boussinesq continuity equation  A slightly different route leads to the quasi-Boussinesq continuity equation
125  where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+%  where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
126  \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%  \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
127  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:
128  \begin{equation}  \begin{equation}
129  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
130  \partial \left( \rho _{o}w\right) }{\partial z}=0  \label{eq-za-cont2}  \partial \left( \rho _{o}w\right) }{\partial z}=0  \label{eq-za-cont2}
131  \end{equation}  \end{equation}
132  Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same  Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same
# Line 134  equation if: Line 135  equation if:
135  \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}  \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}
136  \end{equation}  \end{equation}
137  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
138  and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{%  and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{
139  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
140  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
141  then:  then:
142  \begin{eqnarray}  \begin{eqnarray}
143  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
144  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
145  \label{eq-zab-hmom} \\  \label{eq-zab-hmom} \\
146  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
147  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
148  \label{eq-zab-hydro} \\  \label{eq-zab-hydro} \\
149  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
150  \partial \left( \rho _{o}w\right) }{\partial z} &=&0  \label{eq-zab-cont} \\  \partial \left( \rho _{o}w\right) }{\partial z} &=&0  \label{eq-zab-cont} \\
151  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\
152  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\
# Line 158  Here, the objective is to drop the depth Line 159  Here, the objective is to drop the depth
159  technically, to also remove the dependence of $\rho $ on $p_{o}$. This would  technically, to also remove the dependence of $\rho $ on $p_{o}$. This would
160  yield the ``truly'' incompressible Boussinesq equations:  yield the ``truly'' incompressible Boussinesq equations:
161  \begin{eqnarray}  \begin{eqnarray}
162  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
163  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
164  \label{eq-ztb-hmom} \\  \label{eq-ztb-hmom} \\
165  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}%  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}
166  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
167  \label{eq-ztb-hydro} \\  \label{eq-ztb-hydro} \\
168  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
# Line 192  while the compressible effects in $\rho Line 193  while the compressible effects in $\rho
193  This then yields what we can call the semi-compressible Boussinesq  This then yields what we can call the semi-compressible Boussinesq
194  equations:  equations:
195  \begin{eqnarray}  \begin{eqnarray}
196  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
197  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{
198  \mathcal{F}}}  \label{eq-zpe-hmom} \\  \mathcal{F}}}  \label{eq:ocean-mom} \\
199  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
200  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
201  \label{eq-zpe-hydro} \\  \label{eq:ocean-wmom} \\
202  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
203  &=&0  \label{eq-zpe-cont} \\  &=&0  \label{eq:ocean-cont} \\
204  \rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c}  \label{eq-zpe-eos} \\  \rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c}  \label{eq:ocean-eos} \\
205  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zpe-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\
206  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zpe-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}
207  \end{eqnarray}  \end{eqnarray}
208  Note that the hydrostatic pressure of the resting fluid, including that  Note that the hydrostatic pressure of the resting fluid, including that
209  associated with $\rho _{c}$, is subtracted out since it has no effect on the  associated with $\rho _{c}$, is subtracted out since it has no effect on the
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