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revision 1.3 by cnh, Wed Oct 10 16:48:45 2001 UTC revision 1.4 by adcroft, Thu Oct 11 19:36:56 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}
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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}
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27    %tci%\setcounter{secnumdepth}{3}
28    %tci%\input{tcilatex}
29    
30    %tci%\begin{document}
31    
32    %tci%\tableofcontents
33    
34    
35    \part{MIT GCM basics}
36    
37  % Section: Overview  % Section: Overview
38    
# Line 77  MITgcm has a number of novel aspects: Line 56  MITgcm has a number of novel aspects:
56  \begin{itemize}  \begin{itemize}
57  \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
58  hydrodynamical kernel is used to drive forward both atmospheric and oceanic  hydrodynamical kernel is used to drive forward both atmospheric and oceanic
59  models - see fig%  models - see fig
60  \marginpar{  \marginpar{
61  Fig.1 One model}\ref{fig:onemodel}  Fig.1 One model}\ref{fig:onemodel}
62    
# Line 86  Fig.1 One model}\ref{fig:onemodel} Line 65  Fig.1 One model}\ref{fig:onemodel}
65  %% CNHend  %% CNHend
66    
67  \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
68  small-scale and large scale processes - see fig %  small-scale and large scale processes - see fig
69  \marginpar{  \marginpar{
70  Fig.2 All scales}\ref{fig:all-scales}  Fig.2 All scales}\ref{fig:all-scales}
71    
# Line 96  Fig.2 All scales}\ref{fig:all-scales} Line 75  Fig.2 All scales}\ref{fig:all-scales}
75    
76  \item finite volume techniques are employed yielding an intuitive  \item finite volume techniques are employed yielding an intuitive
77  discretization and support for the treatment of irregular geometries using  discretization and support for the treatment of irregular geometries using
78  orthogonal curvilinear grids and shaved cells - see fig %  orthogonal curvilinear grids and shaved cells - see fig
79  \marginpar{  \marginpar{
80  Fig.3 Finite volumes}\ref{fig:finite-volumes}  Fig.3 Finite volumes}\ref{fig:finite-volumes}
81    
# Line 257  data assimilation studies. Line 236  data assimilation studies.
236    
237  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
238  gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude  gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
239  of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $%  of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $
240  \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
241  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
242  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 296  stratification of the ACC.
296    
297  To render atmosphere and ocean models from one dynamical core we exploit  To render atmosphere and ocean models from one dynamical core we exploit
298  `isomorphisms' between equation sets that govern the evolution of the  `isomorphisms' between equation sets that govern the evolution of the
299  respective fluids - see fig.4%  respective fluids - see fig.4
300  \marginpar{  \marginpar{
301  Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down  Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down
302  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 314  velocity $\vec{\mathbf{v}}$, active trac
314  depend on $\theta $, $S$, and $p$. The equations that govern the evolution  depend on $\theta $, $S$, and $p$. The equations that govern the evolution
315  of these fields, obtained by applying the laws of classical mechanics and  of these fields, obtained by applying the laws of classical mechanics and
316  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
317  a generic vertical coordinate, $r$, see fig.5%  a generic vertical coordinate, $r$, see fig.5
318  \marginpar{  \marginpar{
319  Fig.5 The vertical coordinate of model}:  Fig.5 The vertical coordinate of model}:
320    
# Line 344  Fig.5 The vertical coordinate of model}: Line 323  Fig.5 The vertical coordinate of model}:
323  %%CNHend  %%CNHend
324    
325  \begin{equation*}  \begin{equation*}
326  \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}}
327  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}%  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
328  \text{ horizontal mtm}  \text{ horizontal mtm}
329  \end{equation*}  \end{equation*}
330    
331  \begin{equation*}  \begin{equation*}
332  \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{
333  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{
334  vertical mtm}  vertical mtm}
335  \end{equation*}  \end{equation*}
336    
337  \begin{equation}  \begin{equation}
338  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{%  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
339  \partial r}=0\text{ continuity}  \label{eq:continuous}  \partial r}=0\text{ continuity}  \label{eq:continuous}
340  \end{equation}  \end{equation}
341    
# Line 384  is the total derivative} Line 363  is the total derivative}
363  \end{equation*}  \end{equation*}
364    
365  \begin{equation*}  \begin{equation*}
366  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
367  \text{ is the `grad' operator}  \text{ is the `grad' operator}
368  \end{equation*}  \end{equation*}
369  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}
370  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
371  is a unit vector in the vertical  is a unit vector in the vertical
372    
# Line 421  S\text{ is specific humidity in the atmo Line 400  S\text{ is specific humidity in the atmo
400  \end{equation*}  \end{equation*}
401    
402  \begin{equation*}  \begin{equation*}
403  \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{%  \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{
404  \mathbf{v}}  \mathbf{v}}
405  \end{equation*}  \end{equation*}
406    
# Line 466  of motion. Line 445  of motion.
445    
446  \begin{equation}  \begin{equation}
447  \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0  \label{eq:noflow}  \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0  \label{eq:noflow}
448  \end{equation}%  \end{equation}
449  where $\vec{\mathbf{n}}$ is the normal to a solid boundary.  where $\vec{\mathbf{n}}$ is the normal to a solid boundary.
450    
451  \subsection{Atmosphere}  \subsection{Atmosphere}
# Line 503  where Line 482  where
482    
483  \begin{equation*}  \begin{equation*}
484  T\text{ is absolute temperature}  T\text{ is absolute temperature}
485  \end{equation*}%  \end{equation*}
486  \begin{equation*}  \begin{equation*}
487  p\text{ is the pressure}  p\text{ is the pressure}
488  \end{equation*}%  \end{equation*}
489  \begin{eqnarray*}  \begin{eqnarray*}
490  &&z\text{ is the height of the pressure surface} \\  &&z\text{ is the height of the pressure surface} \\
491  &&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 495  In the above the ideal gas law, $p=\rho
495  the Exner function $\Pi (p)$ given by (see Appendix Atmosphere)  the Exner function $\Pi (p)$ given by (see Appendix Atmosphere)
496  \begin{equation}  \begin{equation}
497  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{eq:exner}  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{eq:exner}
498  \end{equation}%  \end{equation}
499  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
500  constant and $c_{p}$ the specific heat of air at constant pressure.  constant and $c_{p}$ the specific heat of air at constant pressure.
501    
# Line 566  At the bottom of the ocean: $R_{fixed}(x Line 545  At the bottom of the ocean: $R_{fixed}(x
545    
546  The surface of the ocean is given by: $R_{moving}=\eta $  The surface of the ocean is given by: $R_{moving}=\eta $
547    
548  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 $
549  R_{o}=Z_{o}=0$.  R_{o}=Z_{o}=0$.
550    
551  Boundary conditions are:  Boundary conditions are:
# Line 574  Boundary conditions are: Line 553  Boundary conditions are:
553  \begin{eqnarray}  \begin{eqnarray}
554  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}
555  \\  \\
556  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)
557  \label{eq:moving-bc-ocean}}  \label{eq:moving-bc-ocean}}
558  \end{eqnarray}  \end{eqnarray}
559  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 570  Let us separate $\phi $ in to surface, h
570  \begin{equation}  \begin{equation}
571  \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)
572  \label{eq:phi-split}  \label{eq:phi-split}
573  \end{equation}%  \end{equation}
574  and write eq(\ref{incompressible}a,b) in the form:  and write eq(\ref{incompressible}a,b) in the form:
575    
576  \begin{equation}  \begin{equation}
# Line 605  _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l Line 584  _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
584  \end{equation}  \end{equation}
585    
586  \begin{equation}  \begin{equation}
587  \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{%  \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{
588  \partial r}=G_{\dot{r}}  \label{eq:mom-w}  \partial r}=G_{\dot{r}}  \label{eq:mom-w}
589  \end{equation}  \end{equation}
590  Here $\epsilon _{nh}$ is a non-hydrostatic parameter.  Here $\epsilon _{nh}$ is a non-hydrostatic parameter.
591    
592  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
593  {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
594  terms in the momentum equations. In spherical coordinates they take the form%  terms in the momentum equations. In spherical coordinates they take the form
595  \footnote{%  \footnote{
596  In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms  In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms
597  in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref%  in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref
598  {eq:gw-spherical}) are omitted; the singly-underlined terms are included in  {eq:gw-spherical}) are omitted; the singly-underlined terms are included in
599  the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (%  the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (
600  \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
601  discussion:  discussion:
602    
# Line 629  $-\left\{ \underline{\frac{u\dot{r}}{{r} Line 608  $-\left\{ \underline{\frac{u\dot{r}}{{r}
608  \\  \\
609  $-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\right\} $  $-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\right\} $
610  \\  \\
611  $+\mathcal{F}_{u}$%  $+\mathcal{F}_{u}$
612  \end{tabular}%  \end{tabular}
613  \ \right\} \left\{  \ \right\} \left\{
614  \begin{tabular}{l}  \begin{tabular}{l}
615  \textit{advection} \\  \textit{advection} \\
616  \textit{metric} \\  \textit{metric} \\
617  \textit{Coriolis} \\  \textit{Coriolis} \\
618  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
619  \end{tabular}%  \end{tabular}
620  \ \right. \qquad  \label{eq:gu-speherical}  \ \right. \qquad  \label{eq:gu-speherical}
621  \end{equation}  \end{equation}
622    
# Line 648  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\ Line 627  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
627  $-\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 lat}{{r}}\right\}
628  $ \\  $ \\
629  $-\left\{ -2\Omega u\sin lat\right\} $ \\  $-\left\{ -2\Omega u\sin lat\right\} $ \\
630  $+\mathcal{F}_{v}$%  $+\mathcal{F}_{v}$
631  \end{tabular}%  \end{tabular}
632  \ \right\} \left\{  \ \right\} \left\{
633  \begin{tabular}{l}  \begin{tabular}{l}
634  \textit{advection} \\  \textit{advection} \\
635  \textit{metric} \\  \textit{metric} \\
636  \textit{Coriolis} \\  \textit{Coriolis} \\
637  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
638  \end{tabular}%  \end{tabular}
639  \ \right. \qquad  \label{eq:gv-spherical}  \ \right. \qquad  \label{eq:gv-spherical}
640  \end{equation}%  \end{equation}
641  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
642    
643  \begin{equation}  \begin{equation}
# Line 667  $+\mathcal{F}_{v}$% Line 646  $+\mathcal{F}_{v}$%
646  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
647  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
648  ${+}\underline{{2\Omega u\cos lat}}$ \\  ${+}\underline{{2\Omega u\cos lat}}$ \\
649  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$%  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
650  \end{tabular}%  \end{tabular}
651  \ \right\} \left\{  \ \right\} \left\{
652  \begin{tabular}{l}  \begin{tabular}{l}
653  \textit{advection} \\  \textit{advection} \\
654  \textit{metric} \\  \textit{metric} \\
655  \textit{Coriolis} \\  \textit{Coriolis} \\
656  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
657  \end{tabular}%  \end{tabular}
658  \ \right.  \label{eq:gw-spherical}  \ \right.  \label{eq:gw-spherical}
659  \end{equation}%  \end{equation}
660  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
661    
662  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 `$lat$
663  ' is latitude.  ' is latitude.
664    
665  Grad and div operators in spherical coordinates are defined in appendix  Grad and div operators in spherical coordinates are defined in appendix
666  OPERATORS.%  OPERATORS.
667  \marginpar{  \marginpar{
668  Fig.6 Spherical polar coordinate system.}  Fig.6 Spherical polar coordinate system.}
669    
# Line 700  hydrostatic balance and the `traditional Line 679  hydrostatic balance and the `traditional
679  Coriolis force is treated approximately and the shallow atmosphere  Coriolis force is treated approximately and the shallow atmosphere
680  approximation is made.\ The MITgcm need not make the `traditional  approximation is made.\ The MITgcm need not make the `traditional
681  approximation'. To be able to support consistent non-hydrostatic forms the  approximation'. To be able to support consistent non-hydrostatic forms the
682  shallow atmosphere approximation can be relaxed - when dividing through by $%  shallow atmosphere approximation can be relaxed - when dividing through by $
683  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 $a$,
684  the radius of the earth.  the radius of the earth.
685    
# Line 713  terms in Eqs. (\ref{eq:gu-speherical} $\ Line 692  terms in Eqs. (\ref{eq:gu-speherical} $\
692  are neglected and `${r}$' is replaced by `$a$', the mean radius of the  are neglected and `${r}$' is replaced by `$a$', the mean radius of the
693  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
694  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
695  computed at all other levels by integration of the hydrostatic relation, eq(%  computed at all other levels by integration of the hydrostatic relation, eq(
696  \ref{eq:hydrostatic}).  \ref{eq:hydrostatic}).
697    
698  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
699  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
700  \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic  \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
701  contribution to the pressure field: only the terms underlined twice in Eqs. (%  contribution to the pressure field: only the terms underlined twice in Eqs. (
702  \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
703  and, simultaneously, the shallow atmosphere approximation is relaxed. In  and, simultaneously, the shallow atmosphere approximation is relaxed. In
704  \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 743  only a quasi-non-hydrostatic atmospheric Line 722  only a quasi-non-hydrostatic atmospheric
722    
723  \paragraph{Non-hydrostatic Ocean}  \paragraph{Non-hydrostatic Ocean}
724    
725  In the non-hydrostatic ocean model all terms in equations Eqs.(\ref%  In the non-hydrostatic ocean model all terms in equations Eqs.(\ref
726  {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A  {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A
727  three dimensional elliptic equation must be solved subject to Neumann  three dimensional elliptic equation must be solved subject to Neumann
728  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 735  and Bromley, 1995; Marshall et.al.\ 1997
735    
736  \paragraph{Quasi-nonhydrostatic Atmosphere}  \paragraph{Quasi-nonhydrostatic Atmosphere}
737    
738  In the non-hydrostatic version of our atmospheric model we approximate $\dot{%  In the non-hydrostatic version of our atmospheric model we approximate $\dot{
739  r}$ 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})
740  (but only here) by:  (but only here) by:
741    
742  \begin{equation}  \begin{equation}
743  \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}
744  \end{equation}%  \end{equation}
745  where $p_{hy}$ is the hydrostatic pressure.  where $p_{hy}$ is the hydrostatic pressure.
746    
747  \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 769  equations in $z-$coordinates are support
769    
770  \subparagraph{Non-hydrostatic}  \subparagraph{Non-hydrostatic}
771    
772  Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$%  Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
773  coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref%  coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref
774  {eq:ocean-salt}).  {eq:ocean-salt}).
775    
776  \subsection{Solution strategy}  \subsection{Solution strategy}
777    
778  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{
779  NH} models is summarized in Fig.7.%  NH} models is summarized in Fig.7.
780  \marginpar{  \marginpar{
781  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
782  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 792  stepping forward the vertical momentum e
792  %%CNHend  %%CNHend
793    
794  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
795  course, some complication that goes with the inclusion of $\cos \phi \ $%  course, some complication that goes with the inclusion of $\cos \phi \ $
796  Coriolis terms and the relaxation of the shallow atmosphere approximation.  Coriolis terms and the relaxation of the shallow atmosphere approximation.
797  But this leads to negligible increase in computation. In \textbf{NH}, in  But this leads to negligible increase in computation. In \textbf{NH}, in
798  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 816  Hydrostatic pressure is obtained by inte
816  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:
817    
818  \begin{equation*}  \begin{equation*}
819  \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}
820  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
821  \end{equation*}  \end{equation*}
822  and so  and so
# Line 855  atmospheric pressure pushing down on the Line 834  atmospheric pressure pushing down on the
834    
835  \subsubsection{Surface pressure}  \subsubsection{Surface pressure}
836    
837  The surface pressure equation can be obtained by integrating continuity, (%  The surface pressure equation can be obtained by integrating continuity, (
838  \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$  \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
839    
840  \begin{equation*}  \begin{equation*}
841  \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}
842  }_{h}+\partial _{r}\dot{r}\right) dr=0  }_{h}+\partial _{r}\dot{r}\right) dr=0
843  \end{equation*}  \end{equation*}
844    
# Line 867  Thus: Line 846  Thus:
846    
847  \begin{equation*}  \begin{equation*}
848  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
849  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}
850  _{h}dr=0  _{h}dr=0
851  \end{equation*}  \end{equation*}
852  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 $
853  r $. The above can be rearranged to yield, using Leibnitz's theorem:  r $. The above can be rearranged to yield, using Leibnitz's theorem:
854    
855  \begin{equation}  \begin{equation}
856  \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot  \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot
857  \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}  \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}
858  \label{eq:free-surface}  \label{eq:free-surface}
859  \end{equation}%  \end{equation}
860  where we have incorporated a source term.  where we have incorporated a source term.
861    
862  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 865  be written
865  \begin{equation}  \begin{equation}
866  \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)
867  \label{eq:phi-surf}  \label{eq:phi-surf}
868  \end{equation}%  \end{equation}
869  where $b_{s}$ is the buoyancy at the surface.  where $b_{s}$ is the buoyancy at the surface.
870    
871  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
872  {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
873  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
874  surface' and `rigid lid' approaches are available.  surface' and `rigid lid' approaches are available.
875    
876  \subsubsection{Non-hydrostatic pressure}  \subsubsection{Non-hydrostatic pressure}
877    
878  Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{%  Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
879  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
880  (\ref{incompressible}), we deduce that:  (\ref{incompressible}), we deduce that:
881    
882  \begin{equation}  \begin{equation}
883  \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{
884  \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .%  \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .
885  \vec{\mathbf{F}}  \label{eq:3d-invert}  \vec{\mathbf{F}}  \label{eq:3d-invert}
886  \end{equation}  \end{equation}
887    
# Line 922  coasts (in the ocean) and the bottom: Line 901  coasts (in the ocean) and the bottom:
901  \end{equation}  \end{equation}
902  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
903  kinematic condition (\ref{nonormalflow}) is also applied to the vertical  kinematic condition (\ref{nonormalflow}) is also applied to the vertical
904  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 $
905  \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
906  tangential component of velocity, $v_{T}$, at all solid boundaries,  tangential component of velocity, $v_{T}$, at all solid boundaries,
907  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 918  where
918  \begin{equation*}  \begin{equation*}
919  \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
920  _{s}+\mathbf{\nabla }\phi _{hyd}\right)  _{s}+\mathbf{\nabla }\phi _{hyd}\right)
921  \end{equation*}%  \end{equation*}
922  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
923  (\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
924  exploit classical 3D potential theory and, by introducing an appropriately  exploit classical 3D potential theory and, by introducing an appropriately
925  chosen $\delta $-function sheet of `source-charge', replace the  chosen $\delta $-function sheet of `source-charge', replace the
926  inhomogeneous boundary condition on pressure by a homogeneous one. The  inhomogeneous boundary condition on pressure by a homogeneous one. The
927  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 $
928  \vec{\mathbf{F}}.$ By simultaneously setting $%  \vec{\mathbf{F}}.$ By simultaneously setting $
929  \begin{array}{l}  \begin{array}{l}
930  \widehat{n}.\vec{\mathbf{F}}%  \widehat{n}.\vec{\mathbf{F}}
931  \end{array}%  \end{array}
932  =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
933  self-consistent but simpler homogenized Elliptic problem is obtained:  self-consistent but simpler homogenized Elliptic problem is obtained:
934    
935  \begin{equation*}  \begin{equation*}
936  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
937  \end{equation*}%  \end{equation*}
938  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
939  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
940  {eq:inhom-neumann-nh}) the modified boundary condition becomes:  {eq:inhom-neumann-nh}) the modified boundary condition becomes:
941    
942  \begin{equation}  \begin{equation}
# Line 986  Many forms of momentum dissipation are a Line 965  Many forms of momentum dissipation are a
965  biharmonic frictions are commonly used:  biharmonic frictions are commonly used:
966    
967  \begin{equation}  \begin{equation}
968  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}}
969  +A_{4}\nabla _{h}^{4}v  \label{eq:dissipation}  +A_{4}\nabla _{h}^{4}v  \label{eq:dissipation}
970  \end{equation}  \end{equation}
971  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 976  friction. These coefficients are the sam
976    
977  The mixing terms for the temperature and salinity equations have a similar  The mixing terms for the temperature and salinity equations have a similar
978  form to that of momentum except that the diffusion tensor can be  form to that of momentum except that the diffusion tensor can be
979  non-diagonal and have varying coefficients. $\qquad $%  non-diagonal and have varying coefficients. $\qquad $
980  \begin{equation}  \begin{equation}
981  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
982  _{h}^{4}(T,S)  \label{eq:diffusion}  _{h}^{4}(T,S)  \label{eq:diffusion}
983  \end{equation}  \end{equation}
984  where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%  where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $
985  horizontal coefficient for biharmonic diffusion. In the simplest case where  horizontal coefficient for biharmonic diffusion. In the simplest case where
986  the subgrid-scale fluxes of heat and salt are parameterized with constant  the subgrid-scale fluxes of heat and salt are parameterized with constant
987  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 992  reduces to a diagonal matrix with consta
992  \begin{array}{ccc}  \begin{array}{ccc}
993  K_{h} & 0 & 0 \\  K_{h} & 0 & 0 \\
994  0 & K_{h} & 0 \\  0 & K_{h} & 0 \\
995  0 & 0 & K_{v}%  0 & 0 & K_{v}
996  \end{array}  \end{array}
997  \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}  \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}
998  \end{equation}  \end{equation}
# Line 1023  salinity ... ). Line 1002  salinity ... ).
1002    
1003  \subsection{Vector invariant form}  \subsection{Vector invariant form}
1004    
1005  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
1006  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
1007    
1008  \begin{equation}  \begin{equation}
1009  \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}
1010  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla
1011  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
1012  \label{eq:vi-identity}  \label{eq:vi-identity}
1013  \end{equation}%  \end{equation}
1014  This permits alternative numerical treatments of the non-linear terms based  This permits alternative numerical treatments of the non-linear terms based
1015  on their representation as a vorticity flux. Because gradients of coordinate  on their representation as a vorticity flux. Because gradients of coordinate
1016  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
1017  representation of the metric terms in (\ref{eq:gu-speherical}), (\ref%  representation of the metric terms in (\ref{eq:gu-speherical}), (\ref
1018  {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information  {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information
1019  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
1020  to discretize the model.  to discretize the model.
# Line 1057  coordinates} Line 1036  coordinates}
1036    
1037  The hydrostatic primitive equations (HPEs) in p-coordinates are:  The hydrostatic primitive equations (HPEs) in p-coordinates are:
1038  \begin{eqnarray}  \begin{eqnarray}
1039  \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}}
1040  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
1041  \label{eq:atmos-mom} \\  \label{eq:atmos-mom} \\
1042  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
1043  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1044  \partial p} &=&0  \label{eq:atmos-cont} \\  \partial p} &=&0  \label{eq:atmos-cont} \\
1045  p\alpha &=&RT  \label{eq:atmos-eos} \\  p\alpha &=&RT  \label{eq:atmos-eos} \\
1046  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}
1047  \end{eqnarray}%  \end{eqnarray}
1048  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
1049  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
1050  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
1051  derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is  derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is
1052  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp%  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
1053  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref%  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
1054  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $%  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
1055  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 $
1056  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.
1057    
1058  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 1060  $\theta $ so that it looks more like a g
1060  Differentiating (\ref{eq:atmos-eos}) we get:  Differentiating (\ref{eq:atmos-eos}) we get:
1061  \begin{equation*}  \begin{equation*}
1062  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}
1063  \end{equation*}%  \end{equation*}
1064  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 $
1065  c_{p}=c_{v}+R$, gives:  c_{p}=c_{v}+R$, gives:
1066  \begin{equation}  \begin{equation}
1067  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
1068  \label{eq-p-heat-interim}  \label{eq-p-heat-interim}
1069  \end{equation}%  \end{equation}
1070  Potential temperature is defined:  Potential temperature is defined:
1071  \begin{equation}  \begin{equation}
1072  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}
1073  \end{equation}%  \end{equation}
1074  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
1075  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:
1076  \begin{equation}  \begin{equation}
1077  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{Exner}  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{Exner}
1078  \end{equation}%  \end{equation}
1079  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
1080  the Exner function:  the Exner function:
1081  \begin{equation*}  \begin{equation*}
1082  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
1083  }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{%  }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
1084  \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}%  \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}
1085  \frac{Dp}{Dt}  \frac{Dp}{Dt}
1086  \end{equation*}%  \end{equation*}
1087  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.
1088    
1089  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 1098  and on substituting into (\ref{eq-p-heat
1098  \end{equation}  \end{equation}
1099  which is in conservative form.  which is in conservative form.
1100    
1101  For convenience in the model we prefer to step forward (\ref%  For convenience in the model we prefer to step forward (\ref
1102  {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).  {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).
1103    
1104  \subsubsection{Boundary conditions}  \subsubsection{Boundary conditions}
# Line 1163  _{o}(p_{o})=g~Z_{topo}$, defined: Line 1142  _{o}(p_{o})=g~Z_{topo}$, defined:
1142    
1143  The final form of the HPE's in p coordinates is then:  The final form of the HPE's in p coordinates is then:
1144  \begin{eqnarray}  \begin{eqnarray}
1145  \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}}
1146  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\
1147  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
1148  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1149  \partial p} &=&0 \\  \partial p} &=&0 \\
1150  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
1151  \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 1162  We review here the method by which the s
1162  HPE's for the ocean written in z-coordinates are obtained. The  HPE's for the ocean written in z-coordinates are obtained. The
1163  non-Boussinesq equations for oceanic motion are:  non-Boussinesq equations for oceanic motion are:
1164  \begin{eqnarray}  \begin{eqnarray}
1165  \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}}
1166  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\
1167  \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}
1168  &=&\epsilon _{nh}\mathcal{F}_{w} \\  &=&\epsilon _{nh}\mathcal{F}_{w} \\
1169  \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}}
1170  _{h}+\frac{\partial w}{\partial z} &=&0 \\  _{h}+\frac{\partial w}{\partial z} &=&0 \\
1171  \rho &=&\rho (\theta ,S,p) \\  \rho &=&\rho (\theta ,S,p) \\
1172  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
1173  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
1174  \end{eqnarray}%  \end{eqnarray}
1175  These equations permit acoustics modes, inertia-gravity waves,  These equations permit acoustics modes, inertia-gravity waves,
1176  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
1177  mode. As written, they cannot be integrated forward consistently - if we  mode. As written, they cannot be integrated forward consistently - if we
1178  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
1179  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref%  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
1180  {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
1181  therefore necessary to manipulate the system as follows. Differentiating the  therefore necessary to manipulate the system as follows. Differentiating the
1182  EOS (equation of state) gives:  EOS (equation of state) gives:
# Line 1210  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp Line 1189  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
1189  \end{equation}  \end{equation}
1190    
1191  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
1192  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref%  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
1193  {eq-zns-cont} gives:  {eq-zns-cont} gives:
1194  \begin{equation}  \begin{equation}
1195  \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{
1196  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
1197  \end{equation}  \end{equation}
1198  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 1200  adiabatic motion, dropping the $\frac{D\
1200  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
1201  can be explicitly integrated forward:  can be explicitly integrated forward:
1202  \begin{eqnarray}  \begin{eqnarray}
1203  \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}}
1204  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1205  \label{eq-cns-hmom} \\  \label{eq-cns-hmom} \\
1206  \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}
1207  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\
1208  \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{
1209  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\
1210  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\
1211  \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 1219  wherever it appears in a product (ie. no
1219  `Boussinesq assumption'. The only term that then retains the full variation  `Boussinesq assumption'. The only term that then retains the full variation
1220  in $\rho $ is the gravitational acceleration:  in $\rho $ is the gravitational acceleration:
1221  \begin{eqnarray}  \begin{eqnarray}
1222  \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}}
1223  _{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}}}
1224  \label{eq-zcb-hmom} \\  \label{eq-zcb-hmom} \\
1225  \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}}
1226  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1227  \label{eq-zcb-hydro} \\  \label{eq-zcb-hydro} \\
1228  \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{
1229  \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} \\
1230  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\
1231  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\
1232  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}
1233  \end{eqnarray}  \end{eqnarray}
1234  These equations still retain acoustic modes. But, because the  These equations still retain acoustic modes. But, because the
1235  ``compressible'' terms are linearized, the pressure equation \ref%  ``compressible'' terms are linearized, the pressure equation \ref
1236  {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent  {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
1237  term appears as a Helmholtz term in the non-hydrostatic pressure equation).  term appears as a Helmholtz term in the non-hydrostatic pressure equation).
1238  These are the \emph{truly} compressible Boussinesq equations. Note that the  These are the \emph{truly} compressible Boussinesq equations. Note that the
1239  EOS must have the same pressure dependency as the linearized pressure term,  EOS must have the same pressure dependency as the linearized pressure term,
1240  ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{%  ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{
1241  c_{s}^{2}}$, for consistency.  c_{s}^{2}}$, for consistency.
1242    
1243  \subsubsection{`Anelastic' z-coordinate equations}  \subsubsection{`Anelastic' z-coordinate equations}
1244    
1245  The anelastic approximation filters the acoustic mode by removing the  The anelastic approximation filters the acoustic mode by removing the
1246  time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}%  time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}
1247  ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}%  ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}
1248  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
1249  continuity and EOS. A better solution is to change the dependency on  continuity and EOS. A better solution is to change the dependency on
1250  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 1255  height and a perturbation:
1255  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
1256  differentiating the EOS, the continuity equation then becomes:  differentiating the EOS, the continuity equation then becomes:
1257  \begin{equation*}  \begin{equation*}
1258  \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{
1259  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+
1260  \frac{\partial w}{\partial z}=0  \frac{\partial w}{\partial z}=0
1261  \end{equation*}  \end{equation*}
1262  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
1263  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},
1264  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and
1265  $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{%  $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{
1266  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
1267  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
1268  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
1269  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
1270  anelastic continuity equation:  anelastic continuity equation:
1271  \begin{equation}  \begin{equation}
1272  \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}-
1273  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}
1274  \end{equation}  \end{equation}
1275  A slightly different route leads to the quasi-Boussinesq continuity equation  A slightly different route leads to the quasi-Boussinesq continuity equation
1276  where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+%  where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
1277  \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%  \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
1278  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:
1279  \begin{equation}  \begin{equation}
1280  \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{
1281  \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}
1282  \end{equation}  \end{equation}
1283  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 1286  equation if:
1286  \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}}
1287  \end{equation}  \end{equation}
1288  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
1289  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 $\frac{
1290  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
1291  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
1292  then:  then:
1293  \begin{eqnarray}  \begin{eqnarray}
1294  \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}}
1295  _{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}}}
1296  \label{eq-zab-hmom} \\  \label{eq-zab-hmom} \\
1297  \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}}
1298  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1299  \label{eq-zab-hydro} \\  \label{eq-zab-hydro} \\
1300  \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{
1301  \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} \\
1302  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\
1303  \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 1310  Here, the objective is to drop the depth
1310  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
1311  yield the ``truly'' incompressible Boussinesq equations:  yield the ``truly'' incompressible Boussinesq equations:
1312  \begin{eqnarray}  \begin{eqnarray}
1313  \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}}
1314  _{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}}}
1315  \label{eq-ztb-hmom} \\  \label{eq-ztb-hmom} \\
1316  \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}}
1317  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1318  \label{eq-ztb-hydro} \\  \label{eq-ztb-hydro} \\
1319  \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 1332  retain compressibility effects in the de
1332  density thus:  density thus:
1333  \begin{equation*}  \begin{equation*}
1334  \rho =\rho _{o}+\rho ^{\prime }  \rho =\rho _{o}+\rho ^{\prime }
1335  \end{equation*}%  \end{equation*}
1336  We then assert that variations with depth of $\rho _{o}$ are unimportant  We then assert that variations with depth of $\rho _{o}$ are unimportant
1337  while the compressible effects in $\rho ^{\prime }$ are:  while the compressible effects in $\rho ^{\prime }$ are:
1338  \begin{equation*}  \begin{equation*}
1339  \rho _{o}=\rho _{c}  \rho _{o}=\rho _{c}
1340  \end{equation*}%  \end{equation*}
1341  \begin{equation*}  \begin{equation*}
1342  \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}  \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}
1343  \end{equation*}%  \end{equation*}
1344  This then yields what we can call the semi-compressible Boussinesq  This then yields what we can call the semi-compressible Boussinesq
1345  equations:  equations:
1346  \begin{eqnarray}  \begin{eqnarray}
1347  \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}}
1348  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{
1349  \mathcal{F}}}  \label{eq:ocean-mom} \\  \mathcal{F}}}  \label{eq:ocean-mom} \\
1350  \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
1351  _{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 1356  _{c}}\frac{\partial p^{\prime }}{\partia
1356  \\  \\
1357  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\
1358  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}
1359  \end{eqnarray}%  \end{eqnarray}
1360  Note that the hydrostatic pressure of the resting fluid, including that  Note that the hydrostatic pressure of the resting fluid, including that
1361  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
1362  dynamics.  dynamics.
# Line 1421  The `grad' ($\nabla $) and `div' ($\nabl Line 1400  The `grad' ($\nabla $) and `div' ($\nabl
1400  spherical coordinates:  spherical coordinates:
1401    
1402  \begin{equation*}  \begin{equation*}
1403  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }
1404  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}
1405  \right)  \right)
1406  \end{equation*}  \end{equation*}
1407    
# Line 1432  spherical coordinates: Line 1411  spherical coordinates:
1411  +\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}
1412  \end{equation*}  \end{equation*}
1413    
1414  %%%% \end{document}  %tci%\end{document}
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