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revision 1.3 by cnh, Wed Oct 10 16:48:45 2001 UTC revision 1.6 by cnh, Wed Oct 24 15:21:27 2001 UTC
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1  % $Header$  % $Header$
2  % $Name$  % $Name$
 %\usepackage{oldgerm}  
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3    
4  %%%% \part{MIT GCM basics}  %tci%\documentclass[12pt]{book}
5    %tci%\usepackage{amsmath}
6    %tci%\usepackage{html}
7    %tci%\usepackage{epsfig}
8    %tci%\usepackage{graphics,subfigure}
9    %tci%\usepackage{array}
10    %tci%\usepackage{multirow}
11    %tci%\usepackage{fancyhdr}
12    %tci%\usepackage{psfrag}
13    
14    %tci%%TCIDATA{OutputFilter=Latex.dll}
15    %tci%%TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22}
16    %tci%%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
17    %tci%%TCIDATA{Language=American English}
18    
19    %tci%\fancyhead{}
20    %tci%\fancyhead[LO]{\slshape \rightmark}
21    %tci%\fancyhead[RE]{\slshape \leftmark}
22    %tci%\fancyhead[RO,LE]{\thepage}
23    %tci%\fancyfoot[CO,CE]{\today}
24    %tci%\fancyfoot[RO,LE]{ }
25    %tci%\renewcommand{\headrulewidth}{0.4pt}
26    %tci%\renewcommand{\footrulewidth}{0.4pt}
27    %tci%\setcounter{secnumdepth}{3}
28    %tci%\input{tcilatex}
29    
30    %tci%\begin{document}
31    
32    %tci%\tableofcontents
33    
34    
35  % Section: Overview  % Section: Overview
36    
# Line 77  MITgcm has a number of novel aspects: Line 54  MITgcm has a number of novel aspects:
54  \begin{itemize}  \begin{itemize}
55  \item it can be used to study both atmospheric and oceanic phenomena; one  \item it can be used to study both atmospheric and oceanic phenomena; one
56  hydrodynamical kernel is used to drive forward both atmospheric and oceanic  hydrodynamical kernel is used to drive forward both atmospheric and oceanic
57  models - see fig%  models - see fig
58  \marginpar{  \marginpar{
59  Fig.1 One model}\ref{fig:onemodel}  Fig.1 One model}\ref{fig:onemodel}
60    
# Line 86  Fig.1 One model}\ref{fig:onemodel} Line 63  Fig.1 One model}\ref{fig:onemodel}
63  %% CNHend  %% CNHend
64    
65  \item it has a non-hydrostatic capability and so can be used to study both  \item it has a non-hydrostatic capability and so can be used to study both
66  small-scale and large scale processes - see fig %  small-scale and large scale processes - see fig
67  \marginpar{  \marginpar{
68  Fig.2 All scales}\ref{fig:all-scales}  Fig.2 All scales}\ref{fig:all-scales}
69    
# Line 96  Fig.2 All scales}\ref{fig:all-scales} Line 73  Fig.2 All scales}\ref{fig:all-scales}
73    
74  \item finite volume techniques are employed yielding an intuitive  \item finite volume techniques are employed yielding an intuitive
75  discretization and support for the treatment of irregular geometries using  discretization and support for the treatment of irregular geometries using
76  orthogonal curvilinear grids and shaved cells - see fig %  orthogonal curvilinear grids and shaved cells - see fig
77  \marginpar{  \marginpar{
78  Fig.3 Finite volumes}\ref{fig:finite-volumes}  Fig.3 Finite volumes}\ref{fig:finite-volumes}
79    
# Line 117  listed in an Appendix. Line 94  listed in an Appendix.
94    
95  We begin by briefly showing some of the results of the model in action to  We begin by briefly showing some of the results of the model in action to
96  give a feel for the wide range of problems that can be addressed using it.  give a feel for the wide range of problems that can be addressed using it.
 \pagebreak  
97    
98  % $Header$  % $Header$
99  % $Name$  % $Name$
# Line 257  data assimilation studies. Line 233  data assimilation studies.
233    
234  As one example of application of the MITgcm adjoint, Fig.E4 maps the  As one example of application of the MITgcm adjoint, Fig.E4 maps the
235  gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude  gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
236  of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $%  of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $
237  \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is  \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
238  sensitive to heat fluxes over the Labrador Sea, one of the important sources  sensitive to heat fluxes over the Labrador Sea, one of the important sources
239  of deep water for the thermohaline circulations. This calculation also  of deep water for the thermohaline circulations. This calculation also
# Line 317  stratification of the ACC. Line 293  stratification of the ACC.
293    
294  To render atmosphere and ocean models from one dynamical core we exploit  To render atmosphere and ocean models from one dynamical core we exploit
295  `isomorphisms' between equation sets that govern the evolution of the  `isomorphisms' between equation sets that govern the evolution of the
296  respective fluids - see fig.4%  respective fluids - see fig.4
297  \marginpar{  \marginpar{
298  Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down  Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down
299  and encoded. The model variables have different interpretations depending on  and encoded. The model variables have different interpretations depending on
# Line 335  velocity $\vec{\mathbf{v}}$, active trac Line 311  velocity $\vec{\mathbf{v}}$, active trac
311  depend on $\theta $, $S$, and $p$. The equations that govern the evolution  depend on $\theta $, $S$, and $p$. The equations that govern the evolution
312  of these fields, obtained by applying the laws of classical mechanics and  of these fields, obtained by applying the laws of classical mechanics and
313  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
314  a generic vertical coordinate, $r$, see fig.5%  a generic vertical coordinate, $r$, see fig.5
315  \marginpar{  \marginpar{
316  Fig.5 The vertical coordinate of model}:  Fig.5 The vertical coordinate of model}:
317    
# Line 344  Fig.5 The vertical coordinate of model}: Line 320  Fig.5 The vertical coordinate of model}:
320  %%CNHend  %%CNHend
321    
322  \begin{equation*}  \begin{equation*}
323  \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}
324  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}%  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
325  \text{ horizontal mtm}  \text{ horizontal mtm}
326  \end{equation*}  \end{equation*}
327    
328  \begin{equation*}  \begin{equation*}
329  \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{%  \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
330  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
331  vertical mtm}  vertical mtm}
332  \end{equation*}  \end{equation*}
333    
334  \begin{equation}  \begin{equation}
335  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{%  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
336  \partial r}=0\text{ continuity}  \label{eq:continuous}  \partial r}=0\text{ continuity}  \label{eq:continuous}
337  \end{equation}  \end{equation}
338    
# Line 384  is the total derivative} Line 360  is the total derivative}
360  \end{equation*}  \end{equation*}
361    
362  \begin{equation*}  \begin{equation*}
363  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
364  \text{ is the `grad' operator}  \text{ is the `grad' operator}
365  \end{equation*}  \end{equation*}
366  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}
367  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
368  is a unit vector in the vertical  is a unit vector in the vertical
369    
# Line 421  S\text{ is specific humidity in the atmo Line 397  S\text{ is specific humidity in the atmo
397  \end{equation*}  \end{equation*}
398    
399  \begin{equation*}  \begin{equation*}
400  \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{%  \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{
401  \mathbf{v}}  \mathbf{v}}
402  \end{equation*}  \end{equation*}
403    
# Line 466  of motion. Line 442  of motion.
442    
443  \begin{equation}  \begin{equation}
444  \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0  \label{eq:noflow}  \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0  \label{eq:noflow}
445  \end{equation}%  \end{equation}
446  where $\vec{\mathbf{n}}$ is the normal to a solid boundary.  where $\vec{\mathbf{n}}$ is the normal to a solid boundary.
447    
448  \subsection{Atmosphere}  \subsection{Atmosphere}
# Line 503  where Line 479  where
479    
480  \begin{equation*}  \begin{equation*}
481  T\text{ is absolute temperature}  T\text{ is absolute temperature}
482  \end{equation*}%  \end{equation*}
483  \begin{equation*}  \begin{equation*}
484  p\text{ is the pressure}  p\text{ is the pressure}
485  \end{equation*}%  \end{equation*}
486  \begin{eqnarray*}  \begin{eqnarray*}
487  &&z\text{ is the height of the pressure surface} \\  &&z\text{ is the height of the pressure surface} \\
488  &&g\text{ is the acceleration due to gravity}  &&g\text{ is the acceleration due to gravity}
# Line 516  In the above the ideal gas law, $p=\rho Line 492  In the above the ideal gas law, $p=\rho
492  the Exner function $\Pi (p)$ given by (see Appendix Atmosphere)  the Exner function $\Pi (p)$ given by (see Appendix Atmosphere)
493  \begin{equation}  \begin{equation}
494  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{eq:exner}  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{eq:exner}
495  \end{equation}%  \end{equation}
496  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas
497  constant and $c_{p}$ the specific heat of air at constant pressure.  constant and $c_{p}$ the specific heat of air at constant pressure.
498    
# Line 566  At the bottom of the ocean: $R_{fixed}(x Line 542  At the bottom of the ocean: $R_{fixed}(x
542    
543  The surface of the ocean is given by: $R_{moving}=\eta $  The surface of the ocean is given by: $R_{moving}=\eta $
544    
545  The position of the resting free surface of the ocean is given by $%  The position of the resting free surface of the ocean is given by $
546  R_{o}=Z_{o}=0$.  R_{o}=Z_{o}=0$.
547    
548  Boundary conditions are:  Boundary conditions are:
# Line 574  Boundary conditions are: Line 550  Boundary conditions are:
550  \begin{eqnarray}  \begin{eqnarray}
551  w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)}  \label{eq:fixed-bc-ocean}  w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)}  \label{eq:fixed-bc-ocean}
552  \\  \\
553  w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) %  w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)
554  \label{eq:moving-bc-ocean}}  \label{eq:moving-bc-ocean}}
555  \end{eqnarray}  \end{eqnarray}
556  where $\eta $ is the elevation of the free surface.  where $\eta $ is the elevation of the free surface.
# Line 591  Let us separate $\phi $ in to surface, h Line 567  Let us separate $\phi $ in to surface, h
567  \begin{equation}  \begin{equation}
568  \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)  \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
569  \label{eq:phi-split}  \label{eq:phi-split}
570  \end{equation}%  \end{equation}
571  and write eq(\ref{incompressible}a,b) in the form:  and write eq(\ref{incompressible}a,b) in the form:
572    
573  \begin{equation}  \begin{equation}
# Line 605  _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l Line 581  _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
581  \end{equation}  \end{equation}
582    
583  \begin{equation}  \begin{equation}
584  \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{%  \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{
585  \partial r}=G_{\dot{r}}  \label{eq:mom-w}  \partial r}=G_{\dot{r}}  \label{eq:mom-w}
586  \end{equation}  \end{equation}
587  Here $\epsilon _{nh}$ is a non-hydrostatic parameter.  Here $\epsilon _{nh}$ is a non-hydrostatic parameter.
588    
589  The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref%  The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref
590  {eq:mom-h}) and (\ref{eq:mom-w}) represent advective, metric and Coriolis  {eq:mom-h}) and (\ref{eq:mom-w}) represent advective, metric and Coriolis
591  terms in the momentum equations. In spherical coordinates they take the form%  terms in the momentum equations. In spherical coordinates they take the form
592  \footnote{%  \footnote{
593  In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms  In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms
594  in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref%  in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref
595  {eq:gw-spherical}) are omitted; the singly-underlined terms are included in  {eq:gw-spherical}) are omitted; the singly-underlined terms are included in
596  the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (%  the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (
597  \textbf{NH}) includes all terms.} - see Marshall et al 1997a for a full  \textbf{NH}) includes all terms.} - see Marshall et al 1997a for a full
598  discussion:  discussion:
599    
# Line 625  discussion: Line 601  discussion:
601  \left.  \left.
602  \begin{tabular}{l}  \begin{tabular}{l}
603  $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\  $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\
604  $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan lat}{{r}}\right\} $  $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} $
605  \\  \\
606  $-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\right\} $  $-\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\right\} $
607  \\  \\
608  $+\mathcal{F}_{u}$%  $+\mathcal{F}_{u}$
609  \end{tabular}%  \end{tabular}
610  \ \right\} \left\{  \ \right\} \left\{
611  \begin{tabular}{l}  \begin{tabular}{l}
612  \textit{advection} \\  \textit{advection} \\
613  \textit{metric} \\  \textit{metric} \\
614  \textit{Coriolis} \\  \textit{Coriolis} \\
615  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
616  \end{tabular}%  \end{tabular}
617  \ \right. \qquad  \label{eq:gu-speherical}  \ \right. \qquad  \label{eq:gu-speherical}
618  \end{equation}  \end{equation}
619    
# Line 645  $+\mathcal{F}_{u}$% Line 621  $+\mathcal{F}_{u}$%
621  \left.  \left.
622  \begin{tabular}{l}  \begin{tabular}{l}
623  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
624  $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\}  $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\}
625  $ \\  $ \\
626  $-\left\{ -2\Omega u\sin lat\right\} $ \\  $-\left\{ -2\Omega u\sin \varphi \right\} $ \\
627  $+\mathcal{F}_{v}$%  $+\mathcal{F}_{v}$
628  \end{tabular}%  \end{tabular}
629  \ \right\} \left\{  \ \right\} \left\{
630  \begin{tabular}{l}  \begin{tabular}{l}
631  \textit{advection} \\  \textit{advection} \\
632  \textit{metric} \\  \textit{metric} \\
633  \textit{Coriolis} \\  \textit{Coriolis} \\
634  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
635  \end{tabular}%  \end{tabular}
636  \ \right. \qquad  \label{eq:gv-spherical}  \ \right. \qquad  \label{eq:gv-spherical}
637  \end{equation}%  \end{equation}
638  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
639    
640  \begin{equation}  \begin{equation}
# Line 666  $+\mathcal{F}_{v}$% Line 642  $+\mathcal{F}_{v}$%
642  \begin{tabular}{l}  \begin{tabular}{l}
643  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
644  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
645  ${+}\underline{{2\Omega u\cos lat}}$ \\  ${+}\underline{{2\Omega u\cos \varphi}}$ \\
646  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$%  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
647  \end{tabular}%  \end{tabular}
648  \ \right\} \left\{  \ \right\} \left\{
649  \begin{tabular}{l}  \begin{tabular}{l}
650  \textit{advection} \\  \textit{advection} \\
651  \textit{metric} \\  \textit{metric} \\
652  \textit{Coriolis} \\  \textit{Coriolis} \\
653  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
654  \end{tabular}%  \end{tabular}
655  \ \right.  \label{eq:gw-spherical}  \ \right.  \label{eq:gw-spherical}
656  \end{equation}%  \end{equation}
657  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
658    
659  In the above `${r}$' is the distance from the center of the earth and `$lat$%  In the above `${r}$' is the distance from the center of the earth and `$\varphi$
660  ' is latitude.  ' is latitude.
661    
662  Grad and div operators in spherical coordinates are defined in appendix  Grad and div operators in spherical coordinates are defined in appendix
663  OPERATORS.%  OPERATORS.
664  \marginpar{  \marginpar{
665  Fig.6 Spherical polar coordinate system.}  Fig.6 Spherical polar coordinate system.}
666    
# Line 700  hydrostatic balance and the `traditional Line 676  hydrostatic balance and the `traditional
676  Coriolis force is treated approximately and the shallow atmosphere  Coriolis force is treated approximately and the shallow atmosphere
677  approximation is made.\ The MITgcm need not make the `traditional  approximation is made.\ The MITgcm need not make the `traditional
678  approximation'. To be able to support consistent non-hydrostatic forms the  approximation'. To be able to support consistent non-hydrostatic forms the
679  shallow atmosphere approximation can be relaxed - when dividing through by $%  shallow atmosphere approximation can be relaxed - when dividing through by $
680  r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,  r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
681  the radius of the earth.  the radius of the earth.
682    
# Line 713  terms in Eqs. (\ref{eq:gu-speherical} $\ Line 689  terms in Eqs. (\ref{eq:gu-speherical} $\
689  are neglected and `${r}$' is replaced by `$a$', the mean radius of the  are neglected and `${r}$' is replaced by `$a$', the mean radius of the
690  earth. Once the pressure is found at one level - e.g. by inverting a 2-d  earth. Once the pressure is found at one level - e.g. by inverting a 2-d
691  Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be  Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be
692  computed at all other levels by integration of the hydrostatic relation, eq(%  computed at all other levels by integration of the hydrostatic relation, eq(
693  \ref{eq:hydrostatic}).  \ref{eq:hydrostatic}).
694    
695  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
696  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
697  \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic  \varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
698  contribution to the pressure field: only the terms underlined twice in Eqs. (%  contribution to the pressure field: only the terms underlined twice in Eqs. (
699  \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero  \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero
700  and, simultaneously, the shallow atmosphere approximation is relaxed. In  and, simultaneously, the shallow atmosphere approximation is relaxed. In
701  \textbf{QH}\ \textit{all} the metric terms are retained and the full  \textbf{QH}\ \textit{all} the metric terms are retained and the full
# Line 727  variation of the radial position of a pa Line 703  variation of the radial position of a pa
703  vertical momentum equation (\ref{eq:mom-w}) becomes:  vertical momentum equation (\ref{eq:mom-w}) becomes:
704    
705  \begin{equation*}  \begin{equation*}
706  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi
707  \end{equation*}  \end{equation*}
708  making a small correction to the hydrostatic pressure.  making a small correction to the hydrostatic pressure.
709    
# Line 743  only a quasi-non-hydrostatic atmospheric Line 719  only a quasi-non-hydrostatic atmospheric
719    
720  \paragraph{Non-hydrostatic Ocean}  \paragraph{Non-hydrostatic Ocean}
721    
722  In the non-hydrostatic ocean model all terms in equations Eqs.(\ref%  In the non-hydrostatic ocean model all terms in equations Eqs.(\ref
723  {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A  {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A
724  three dimensional elliptic equation must be solved subject to Neumann  three dimensional elliptic equation must be solved subject to Neumann
725  boundary conditions (see below). It is important to note that use of the  boundary conditions (see below). It is important to note that use of the
# Line 756  and Bromley, 1995; Marshall et.al.\ 1997 Line 732  and Bromley, 1995; Marshall et.al.\ 1997
732    
733  \paragraph{Quasi-nonhydrostatic Atmosphere}  \paragraph{Quasi-nonhydrostatic Atmosphere}
734    
735  In the non-hydrostatic version of our atmospheric model we approximate $\dot{%  In the non-hydrostatic version of our atmospheric model we approximate $\dot{
736  r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical})  r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical})
737  (but only here) by:  (but only here) by:
738    
739  \begin{equation}  \begin{equation}
740  \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt}  \label{eq:quasi-nh-w}  \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt}  \label{eq:quasi-nh-w}
741  \end{equation}%  \end{equation}
742  where $p_{hy}$ is the hydrostatic pressure.  where $p_{hy}$ is the hydrostatic pressure.
743    
744  \subsubsection{Summary of equation sets supported by model}  \subsubsection{Summary of equation sets supported by model}
# Line 790  equations in $z-$coordinates are support Line 766  equations in $z-$coordinates are support
766    
767  \subparagraph{Non-hydrostatic}  \subparagraph{Non-hydrostatic}
768    
769  Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$%  Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
770  coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref%  coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref
771  {eq:ocean-salt}).  {eq:ocean-salt}).
772    
773  \subsection{Solution strategy}  \subsection{Solution strategy}
774    
775  The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{%  The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
776  NH} models is summarized in Fig.7.%  NH} models is summarized in Fig.7.
777  \marginpar{  \marginpar{
778  Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is  Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
779  first solved to find the surface pressure and the hydrostatic pressure at  first solved to find the surface pressure and the hydrostatic pressure at
# Line 813  stepping forward the vertical momentum e Line 789  stepping forward the vertical momentum e
789  %%CNHend  %%CNHend
790    
791  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
792  course, some complication that goes with the inclusion of $\cos \phi \ $%  course, some complication that goes with the inclusion of $\cos \varphi \ $
793  Coriolis terms and the relaxation of the shallow atmosphere approximation.  Coriolis terms and the relaxation of the shallow atmosphere approximation.
794  But this leads to negligible increase in computation. In \textbf{NH}, in  But this leads to negligible increase in computation. In \textbf{NH}, in
795  contrast, one additional elliptic equation - a three-dimensional one - must  contrast, one additional elliptic equation - a three-dimensional one - must
# Line 837  Hydrostatic pressure is obtained by inte Line 813  Hydrostatic pressure is obtained by inte
813  vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:  vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:
814    
815  \begin{equation*}  \begin{equation*}
816  \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}%  \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}
817  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
818  \end{equation*}  \end{equation*}
819  and so  and so
# Line 855  atmospheric pressure pushing down on the Line 831  atmospheric pressure pushing down on the
831    
832  \subsubsection{Surface pressure}  \subsubsection{Surface pressure}
833    
834  The surface pressure equation can be obtained by integrating continuity, (%  The surface pressure equation can be obtained by integrating continuity, (
835  \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$  \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
836    
837  \begin{equation*}  \begin{equation*}
838  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}%  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
839  }_{h}+\partial _{r}\dot{r}\right) dr=0  }_{h}+\partial _{r}\dot{r}\right) dr=0
840  \end{equation*}  \end{equation*}
841    
# Line 867  Thus: Line 843  Thus:
843    
844  \begin{equation*}  \begin{equation*}
845  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
846  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}
847  _{h}dr=0  _{h}dr=0
848  \end{equation*}  \end{equation*}
849  where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $%  where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $
850  r $. The above can be rearranged to yield, using Leibnitz's theorem:  r $. The above can be rearranged to yield, using Leibnitz's theorem:
851    
852  \begin{equation}  \begin{equation}
853  \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot  \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot
854  \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}  \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}
855  \label{eq:free-surface}  \label{eq:free-surface}
856  \end{equation}%  \end{equation}
857  where we have incorporated a source term.  where we have incorporated a source term.
858    
859  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
# Line 886  be written Line 862  be written
862  \begin{equation}  \begin{equation}
863  \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)
864  \label{eq:phi-surf}  \label{eq:phi-surf}
865  \end{equation}%  \end{equation}
866  where $b_{s}$ is the buoyancy at the surface.  where $b_{s}$ is the buoyancy at the surface.
867    
868  In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref%  In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref
869  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
870  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
871  surface' and `rigid lid' approaches are available.  surface' and `rigid lid' approaches are available.
872    
873  \subsubsection{Non-hydrostatic pressure}  \subsubsection{Non-hydrostatic pressure}
874    
875  Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{%  Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
876  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
877  (\ref{incompressible}), we deduce that:  (\ref{incompressible}), we deduce that:
878    
879  \begin{equation}  \begin{equation}
880  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{%  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
881  \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .%  \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .
882  \vec{\mathbf{F}}  \label{eq:3d-invert}  \vec{\mathbf{F}}  \label{eq:3d-invert}
883  \end{equation}  \end{equation}
884    
# Line 922  coasts (in the ocean) and the bottom: Line 898  coasts (in the ocean) and the bottom:
898  \end{equation}  \end{equation}
899  where $\widehat{n}$ is a vector of unit length normal to the boundary. The  where $\widehat{n}$ is a vector of unit length normal to the boundary. The
900  kinematic condition (\ref{nonormalflow}) is also applied to the vertical  kinematic condition (\ref{nonormalflow}) is also applied to the vertical
901  velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $%  velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $
902  \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the  \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the
903  tangential component of velocity, $v_{T}$, at all solid boundaries,  tangential component of velocity, $v_{T}$, at all solid boundaries,
904  depending on the form chosen for the dissipative terms in the momentum  depending on the form chosen for the dissipative terms in the momentum
# Line 939  where Line 915  where
915  \begin{equation*}  \begin{equation*}
916  \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi  \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi
917  _{s}+\mathbf{\nabla }\phi _{hyd}\right)  _{s}+\mathbf{\nabla }\phi _{hyd}\right)
918  \end{equation*}%  \end{equation*}
919  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
920  (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can  (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
921  exploit classical 3D potential theory and, by introducing an appropriately  exploit classical 3D potential theory and, by introducing an appropriately
922  chosen $\delta $-function sheet of `source-charge', replace the  chosen $\delta $-function sheet of `source-charge', replace the
923  inhomogeneous boundary condition on pressure by a homogeneous one. The  inhomogeneous boundary condition on pressure by a homogeneous one. The
924  source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $%  source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $
925  \vec{\mathbf{F}}.$ By simultaneously setting $%  \vec{\mathbf{F}}.$ By simultaneously setting $
926  \begin{array}{l}  \begin{array}{l}
927  \widehat{n}.\vec{\mathbf{F}}%  \widehat{n}.\vec{\mathbf{F}}
928  \end{array}%  \end{array}
929  =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following  =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
930  self-consistent but simpler homogenized Elliptic problem is obtained:  self-consistent but simpler homogenized Elliptic problem is obtained:
931    
932  \begin{equation*}  \begin{equation*}
933  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
934  \end{equation*}%  \end{equation*}
935  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
936  that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref%  that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref
937  {eq:inhom-neumann-nh}) the modified boundary condition becomes:  {eq:inhom-neumann-nh}) the modified boundary condition becomes:
938    
939  \begin{equation}  \begin{equation}
# Line 986  Many forms of momentum dissipation are a Line 962  Many forms of momentum dissipation are a
962  biharmonic frictions are commonly used:  biharmonic frictions are commonly used:
963    
964  \begin{equation}  \begin{equation}
965  D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}%  D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}
966  +A_{4}\nabla _{h}^{4}v  \label{eq:dissipation}  +A_{4}\nabla _{h}^{4}v  \label{eq:dissipation}
967  \end{equation}  \end{equation}
968  where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity  where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity
# Line 997  friction. These coefficients are the sam Line 973  friction. These coefficients are the sam
973    
974  The mixing terms for the temperature and salinity equations have a similar  The mixing terms for the temperature and salinity equations have a similar
975  form to that of momentum except that the diffusion tensor can be  form to that of momentum except that the diffusion tensor can be
976  non-diagonal and have varying coefficients. $\qquad $%  non-diagonal and have varying coefficients. $\qquad $
977  \begin{equation}  \begin{equation}
978  D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla  D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
979  _{h}^{4}(T,S)  \label{eq:diffusion}  _{h}^{4}(T,S)  \label{eq:diffusion}
980  \end{equation}  \end{equation}
981  where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%  where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $
982  horizontal coefficient for biharmonic diffusion. In the simplest case where  horizontal coefficient for biharmonic diffusion. In the simplest case where
983  the subgrid-scale fluxes of heat and salt are parameterized with constant  the subgrid-scale fluxes of heat and salt are parameterized with constant
984  horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,  horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,
# Line 1013  reduces to a diagonal matrix with consta Line 989  reduces to a diagonal matrix with consta
989  \begin{array}{ccc}  \begin{array}{ccc}
990  K_{h} & 0 & 0 \\  K_{h} & 0 & 0 \\
991  0 & K_{h} & 0 \\  0 & K_{h} & 0 \\
992  0 & 0 & K_{v}%  0 & 0 & K_{v}
993  \end{array}  \end{array}
994  \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}  \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}
995  \end{equation}  \end{equation}
# Line 1023  salinity ... ). Line 999  salinity ... ).
999    
1000  \subsection{Vector invariant form}  \subsection{Vector invariant form}
1001    
1002  For some purposes it is advantageous to write momentum advection in eq(\ref%  For some purposes it is advantageous to write momentum advection in eq(\ref
1003  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
1004    
1005  \begin{equation}  \begin{equation}
1006  \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}%  \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
1007  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla
1008  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
1009  \label{eq:vi-identity}  \label{eq:vi-identity}
1010  \end{equation}%  \end{equation}
1011  This permits alternative numerical treatments of the non-linear terms based  This permits alternative numerical treatments of the non-linear terms based
1012  on their representation as a vorticity flux. Because gradients of coordinate  on their representation as a vorticity flux. Because gradients of coordinate
1013  vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit  vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit
1014  representation of the metric terms in (\ref{eq:gu-speherical}), (\ref%  representation of the metric terms in (\ref{eq:gu-speherical}), (\ref
1015  {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information  {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information
1016  about the geometry is contained in the areas and lengths of the volumes used  about the geometry is contained in the areas and lengths of the volumes used
1017  to discretize the model.  to discretize the model.
# Line 1057  coordinates} Line 1033  coordinates}
1033    
1034  The hydrostatic primitive equations (HPEs) in p-coordinates are:  The hydrostatic primitive equations (HPEs) in p-coordinates are:
1035  \begin{eqnarray}  \begin{eqnarray}
1036  \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}}
1037  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
1038  \label{eq:atmos-mom} \\  \label{eq:atmos-mom} \\
1039  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
1040  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1041  \partial p} &=&0  \label{eq:atmos-cont} \\  \partial p} &=&0  \label{eq:atmos-cont} \\
1042  p\alpha &=&RT  \label{eq:atmos-eos} \\  p\alpha &=&RT  \label{eq:atmos-eos} \\
1043  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}
1044  \end{eqnarray}%  \end{eqnarray}
1045  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
1046  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
1047  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
1048  derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is  derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter, $\phi =gz$ is
1049  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp%  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
1050  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref%  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
1051  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $%  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
1052  e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $%  e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $
1053  p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.  p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.
1054    
1055  It is convenient to cast the heat equation in terms of potential temperature  It is convenient to cast the heat equation in terms of potential temperature
# Line 1081  $\theta $ so that it looks more like a g Line 1057  $\theta $ so that it looks more like a g
1057  Differentiating (\ref{eq:atmos-eos}) we get:  Differentiating (\ref{eq:atmos-eos}) we get:
1058  \begin{equation*}  \begin{equation*}
1059  p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}  p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
1060  \end{equation*}%  \end{equation*}
1061  which, when added to the heat equation (\ref{eq:atmos-heat}) and using $%  which, when added to the heat equation (\ref{eq:atmos-heat}) and using $
1062  c_{p}=c_{v}+R$, gives:  c_{p}=c_{v}+R$, gives:
1063  \begin{equation}  \begin{equation}
1064  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
1065  \label{eq-p-heat-interim}  \label{eq-p-heat-interim}
1066  \end{equation}%  \end{equation}
1067  Potential temperature is defined:  Potential temperature is defined:
1068  \begin{equation}  \begin{equation}
1069  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}
1070  \end{equation}%  \end{equation}
1071  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience
1072  we will make use of the Exner function $\Pi (p)$ which defined by:  we will make use of the Exner function $\Pi (p)$ which defined by:
1073  \begin{equation}  \begin{equation}
1074  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{Exner}  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{Exner}
1075  \end{equation}%  \end{equation}
1076  The following relations will be useful and are easily expressed in terms of  The following relations will be useful and are easily expressed in terms of
1077  the Exner function:  the Exner function:
1078  \begin{equation*}  \begin{equation*}
1079  c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi  c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi
1080  }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{%  }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
1081  \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}%  \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}
1082  \frac{Dp}{Dt}  \frac{Dp}{Dt}
1083  \end{equation*}%  \end{equation*}
1084  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.
1085    
1086  The heat equation is obtained by noting that  The heat equation is obtained by noting that
# Line 1119  and on substituting into (\ref{eq-p-heat Line 1095  and on substituting into (\ref{eq-p-heat
1095  \end{equation}  \end{equation}
1096  which is in conservative form.  which is in conservative form.
1097    
1098  For convenience in the model we prefer to step forward (\ref%  For convenience in the model we prefer to step forward (\ref
1099  {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).  {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).
1100    
1101  \subsubsection{Boundary conditions}  \subsubsection{Boundary conditions}
# Line 1163  _{o}(p_{o})=g~Z_{topo}$, defined: Line 1139  _{o}(p_{o})=g~Z_{topo}$, defined:
1139    
1140  The final form of the HPE's in p coordinates is then:  The final form of the HPE's in p coordinates is then:
1141  \begin{eqnarray}  \begin{eqnarray}
1142  \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}}
1143  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\
1144  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
1145  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1146  \partial p} &=&0 \\  \partial p} &=&0 \\
1147  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
1148  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}
# Line 1183  We review here the method by which the s Line 1159  We review here the method by which the s
1159  HPE's for the ocean written in z-coordinates are obtained. The  HPE's for the ocean written in z-coordinates are obtained. The
1160  non-Boussinesq equations for oceanic motion are:  non-Boussinesq equations for oceanic motion are:
1161  \begin{eqnarray}  \begin{eqnarray}
1162  \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}}
1163  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\
1164  \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}
1165  &=&\epsilon _{nh}\mathcal{F}_{w} \\  &=&\epsilon _{nh}\mathcal{F}_{w} \\
1166  \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}}
1167  _{h}+\frac{\partial w}{\partial z} &=&0 \\  _{h}+\frac{\partial w}{\partial z} &=&0 \\
1168  \rho &=&\rho (\theta ,S,p) \\  \rho &=&\rho (\theta ,S,p) \\
1169  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
1170  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
1171  \end{eqnarray}%  \end{eqnarray}
1172  These equations permit acoustics modes, inertia-gravity waves,  These equations permit acoustics modes, inertia-gravity waves,
1173  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
1174  mode. As written, they cannot be integrated forward consistently - if we  mode. As written, they cannot be integrated forward consistently - if we
1175  step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be  step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
1176  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref%  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
1177  {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is  {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
1178  therefore necessary to manipulate the system as follows. Differentiating the  therefore necessary to manipulate the system as follows. Differentiating the
1179  EOS (equation of state) gives:  EOS (equation of state) gives:
# Line 1210  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp Line 1186  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
1186  \end{equation}  \end{equation}
1187    
1188  Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the  Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
1189  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref%  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
1190  {eq-zns-cont} gives:  {eq-zns-cont} gives:
1191  \begin{equation}  \begin{equation}
1192  \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{
1193  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
1194  \end{equation}  \end{equation}
1195  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 1221  adiabatic motion, dropping the $\frac{D\ Line 1197  adiabatic motion, dropping the $\frac{D\
1197  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
1198  can be explicitly integrated forward:  can be explicitly integrated forward:
1199  \begin{eqnarray}  \begin{eqnarray}
1200  \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}}
1201  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1202  \label{eq-cns-hmom} \\  \label{eq-cns-hmom} \\
1203  \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}
1204  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\
1205  \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{
1206  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\
1207  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\
1208  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\
# Line 1240  wherever it appears in a product (ie. no Line 1216  wherever it appears in a product (ie. no
1216  `Boussinesq assumption'. The only term that then retains the full variation  `Boussinesq assumption'. The only term that then retains the full variation
1217  in $\rho $ is the gravitational acceleration:  in $\rho $ is the gravitational acceleration:
1218  \begin{eqnarray}  \begin{eqnarray}
1219  \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}}
1220  _{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}}}
1221  \label{eq-zcb-hmom} \\  \label{eq-zcb-hmom} \\
1222  \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}}
1223  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1224  \label{eq-zcb-hydro} \\  \label{eq-zcb-hydro} \\
1225  \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{
1226  \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} \\
1227  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\
1228  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\
1229  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}
1230  \end{eqnarray}  \end{eqnarray}
1231  These equations still retain acoustic modes. But, because the  These equations still retain acoustic modes. But, because the
1232  ``compressible'' terms are linearized, the pressure equation \ref%  ``compressible'' terms are linearized, the pressure equation \ref
1233  {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent  {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
1234  term appears as a Helmholtz term in the non-hydrostatic pressure equation).  term appears as a Helmholtz term in the non-hydrostatic pressure equation).
1235  These are the \emph{truly} compressible Boussinesq equations. Note that the  These are the \emph{truly} compressible Boussinesq equations. Note that the
1236  EOS must have the same pressure dependency as the linearized pressure term,  EOS must have the same pressure dependency as the linearized pressure term,
1237  ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{%  ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{
1238  c_{s}^{2}}$, for consistency.  c_{s}^{2}}$, for consistency.
1239    
1240  \subsubsection{`Anelastic' z-coordinate equations}  \subsubsection{`Anelastic' z-coordinate equations}
1241    
1242  The anelastic approximation filters the acoustic mode by removing the  The anelastic approximation filters the acoustic mode by removing the
1243  time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}%  time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}
1244  ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}%  ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}
1245  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
1246  continuity and EOS. A better solution is to change the dependency on  continuity and EOS. A better solution is to change the dependency on
1247  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 1276  height and a perturbation: Line 1252  height and a perturbation:
1252  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
1253  differentiating the EOS, the continuity equation then becomes:  differentiating the EOS, the continuity equation then becomes:
1254  \begin{equation*}  \begin{equation*}
1255  \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{
1256  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+
1257  \frac{\partial w}{\partial z}=0  \frac{\partial w}{\partial z}=0
1258  \end{equation*}  \end{equation*}
1259  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
1260  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},
1261  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and
1262  $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{%  $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{
1263  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
1264  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
1265  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
1266  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
1267  anelastic continuity equation:  anelastic continuity equation:
1268  \begin{equation}  \begin{equation}
1269  \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}-
1270  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}
1271  \end{equation}  \end{equation}
1272  A slightly different route leads to the quasi-Boussinesq continuity equation  A slightly different route leads to the quasi-Boussinesq continuity equation
1273  where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+%  where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
1274  \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%  \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
1275  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:
1276  \begin{equation}  \begin{equation}
1277  \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{
1278  \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}
1279  \end{equation}  \end{equation}
1280  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 1307  equation if: Line 1283  equation if:
1283  \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}}
1284  \end{equation}  \end{equation}
1285  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
1286  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 $\frac{
1287  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
1288  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
1289  then:  then:
1290  \begin{eqnarray}  \begin{eqnarray}
1291  \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}}
1292  _{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}}}
1293  \label{eq-zab-hmom} \\  \label{eq-zab-hmom} \\
1294  \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}}
1295  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1296  \label{eq-zab-hydro} \\  \label{eq-zab-hydro} \\
1297  \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{
1298  \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} \\
1299  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\
1300  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\
# Line 1331  Here, the objective is to drop the depth Line 1307  Here, the objective is to drop the depth
1307  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
1308  yield the ``truly'' incompressible Boussinesq equations:  yield the ``truly'' incompressible Boussinesq equations:
1309  \begin{eqnarray}  \begin{eqnarray}
1310  \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}}
1311  _{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}}}
1312  \label{eq-ztb-hmom} \\  \label{eq-ztb-hmom} \\
1313  \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}}
1314  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1315  \label{eq-ztb-hydro} \\  \label{eq-ztb-hydro} \\
1316  \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 1353  retain compressibility effects in the de Line 1329  retain compressibility effects in the de
1329  density thus:  density thus:
1330  \begin{equation*}  \begin{equation*}
1331  \rho =\rho _{o}+\rho ^{\prime }  \rho =\rho _{o}+\rho ^{\prime }
1332  \end{equation*}%  \end{equation*}
1333  We then assert that variations with depth of $\rho _{o}$ are unimportant  We then assert that variations with depth of $\rho _{o}$ are unimportant
1334  while the compressible effects in $\rho ^{\prime }$ are:  while the compressible effects in $\rho ^{\prime }$ are:
1335  \begin{equation*}  \begin{equation*}
1336  \rho _{o}=\rho _{c}  \rho _{o}=\rho _{c}
1337  \end{equation*}%  \end{equation*}
1338  \begin{equation*}  \begin{equation*}
1339  \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}  \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}
1340  \end{equation*}%  \end{equation*}
1341  This then yields what we can call the semi-compressible Boussinesq  This then yields what we can call the semi-compressible Boussinesq
1342  equations:  equations:
1343  \begin{eqnarray}  \begin{eqnarray}
1344  \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}}
1345  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{
1346  \mathcal{F}}}  \label{eq:ocean-mom} \\  \mathcal{F}}}  \label{eq:ocean-mom} \\
1347  \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
1348  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
# Line 1377  _{c}}\frac{\partial p^{\prime }}{\partia Line 1353  _{c}}\frac{\partial p^{\prime }}{\partia
1353  \\  \\
1354  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\
1355  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}
1356  \end{eqnarray}%  \end{eqnarray}
1357  Note that the hydrostatic pressure of the resting fluid, including that  Note that the hydrostatic pressure of the resting fluid, including that
1358  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
1359  dynamics.  dynamics.
# Line 1401  In spherical coordinates, the velocity c Line 1377  In spherical coordinates, the velocity c
1377  and vertical direction respectively, are given by (see Fig.2) :  and vertical direction respectively, are given by (see Fig.2) :
1378    
1379  \begin{equation*}  \begin{equation*}
1380  u=r\cos \phi \frac{D\lambda }{Dt}  u=r\cos \varphi \frac{D\lambda }{Dt}
1381  \end{equation*}  \end{equation*}
1382    
1383  \begin{equation*}  \begin{equation*}
1384  v=r\frac{D\phi }{Dt}\qquad  v=r\frac{D\varphi }{Dt}\qquad
1385  \end{equation*}  \end{equation*}
1386  $\qquad \qquad \qquad \qquad $  $\qquad \qquad \qquad \qquad $
1387    
# Line 1413  $\qquad \qquad \qquad \qquad $ Line 1389  $\qquad \qquad \qquad \qquad $
1389  \dot{r}=\frac{Dr}{Dt}  \dot{r}=\frac{Dr}{Dt}
1390  \end{equation*}  \end{equation*}
1391    
1392  Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial  Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
1393  distance of the particle from the center of the earth, $\Omega $ is the  distance of the particle from the center of the earth, $\Omega $ is the
1394  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
1395    
# Line 1421  The `grad' ($\nabla $) and `div' ($\nabl Line 1397  The `grad' ($\nabla $) and `div' ($\nabl
1397  spherical coordinates:  spherical coordinates:
1398    
1399  \begin{equation*}  \begin{equation*}
1400  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%  \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda }
1401  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%  ,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r}
1402  \right)  \right)
1403  \end{equation*}  \end{equation*}
1404    
1405  \begin{equation*}  \begin{equation*}
1406  \nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial  \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
1407  \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}  \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
1408  +\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}
1409  \end{equation*}  \end{equation*}
1410    
1411  %%%% \end{document}  %tci%\end{document}
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