--- manual/s_overview/text/manual.tex 2001/10/09 10:48:03 1.2 +++ manual/s_overview/text/manual.tex 2006/06/28 15:22:13 1.26 @@ -1,68 +1,43 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.26 2006/06/28 15:22:13 edhill Exp $ % $Name: $ -%\usepackage{oldgerm} -% I commented the following because it introduced excessive white space -%\usepackage{palatcm} % better PDF -% page headers and footers -%\pagestyle{fancy} -% referencing -%% \newcommand{\refequ}[1]{equation (\ref{equ:#1})} -%% \newcommand{\refequbig}[1]{Equation (\ref{equ:#1})} -%% \newcommand{\reftab}[1]{Tab.~\ref{tab:#1}} -%% \newcommand{\reftabno}[1]{\ref{tab:#1}} -%% \newcommand{\reffig}[1]{Fig.~\ref{fig:#1}} -%% \newcommand{\reffigno}[1]{\ref{fig:#1}} -% stuff for psfrag -%% \newcommand{\textinfigure}[1]{{\footnotesize\textbf{\textsf{#1}}}} -%% \newcommand{\mathinfigure}[1]{\small\ensuremath{{#1}}} -% This allows numbering of subsubsections -% This changes the the chapter title -%\renewcommand{\chaptername}{Section} - - -%%%% \documentclass[12pt]{book} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%% \usepackage{amsmath} -%%%% \usepackage{html} -%%%% \usepackage{epsfig} -%%%% \usepackage{graphics,subfigure} -%%%% \usepackage{array} -%%%% \usepackage{multirow} -%%%% \usepackage{fancyhdr} -%%%% \usepackage{psfrag} - -%%%% %TCIDATA{OutputFilter=Latex.dll} -%%%% %TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22} -%%%% %TCIDATA{} -%%%% %TCIDATA{Language=American English} - -%%%% \fancyhead{} -%%%% \fancyhead[LO]{\slshape \rightmark} -%%%% \fancyhead[RE]{\slshape \leftmark} -%%%% \fancyhead[RO,LE]{\thepage} -%%%% \fancyfoot[CO,CE]{\today} -%%%% \fancyfoot[RO,LE]{ } -%%%% \renewcommand{\headrulewidth}{0.4pt} -%%%% \renewcommand{\footrulewidth}{0.4pt} -%%%% \setcounter{secnumdepth}{3} -%%%% \input{tcilatex} -%%%% -%%%% \begin{document} -%%%% -%%%% \tableofcontents -%%%% -%%%% \pagebreak -%%%% \part{MIT GCM basics} +%tci%\documentclass[12pt]{book} +%tci%\usepackage{amsmath} +%tci%\usepackage{html} +%tci%\usepackage{epsfig} +%tci%\usepackage{graphics,subfigure} +%tci%\usepackage{array} +%tci%\usepackage{multirow} +%tci%\usepackage{fancyhdr} +%tci%\usepackage{psfrag} + +%tci%%TCIDATA{OutputFilter=Latex.dll} +%tci%%TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22} +%tci%%TCIDATA{} +%tci%%TCIDATA{Language=American English} + +%tci%\fancyhead{} +%tci%\fancyhead[LO]{\slshape \rightmark} +%tci%\fancyhead[RE]{\slshape \leftmark} +%tci%\fancyhead[RO,LE]{\thepage} +%tci%\fancyfoot[CO,CE]{\today} +%tci%\fancyfoot[RO,LE]{ } +%tci%\renewcommand{\headrulewidth}{0.4pt} +%tci%\renewcommand{\footrulewidth}{0.4pt} +%tci%\setcounter{secnumdepth}{3} +%tci%\input{tcilatex} + +%tci%\begin{document} + +%tci%\tableofcontents + % Section: Overview -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.26 2006/06/28 15:22:13 edhill Exp $ % $Name: $ -\section{Introduction} - -This documentation provides the reader with the information necessary to +This document provides the reader with the information necessary to carry out numerical experiments using MITgcm. It gives a comprehensive description of the continuous equations on which the model is based, the numerical algorithms the model employs and a description of the associated @@ -72,25 +47,37 @@ both process and general circulation studies of the atmosphere and ocean are also presented. +\section{Introduction} +\begin{rawhtml} + +\end{rawhtml} + + MITgcm has a number of novel aspects: \begin{itemize} \item it can be used to study both atmospheric and oceanic phenomena; one hydrodynamical kernel is used to drive forward both atmospheric and oceanic -models - see fig.1% -\marginpar{ -Fig.1 One model}\ref{fig:onemodel} +models - see fig \ref{fig:onemodel} + +%% CNHbegin +\input{part1/one_model_figure} +%% CNHend \item it has a non-hydrostatic capability and so can be used to study both -small-scale and large scale processes - see fig.2% -\marginpar{ -Fig.2 All scales}\ref{fig:all-scales} +small-scale and large scale processes - see fig \ref{fig:all-scales} + +%% CNHbegin +\input{part1/all_scales_figure} +%% CNHend \item finite volume techniques are employed yielding an intuitive discretization and support for the treatment of irregular geometries using -orthogonal curvilinear grids and shaved cells - see fig.3% -\marginpar{ -Fig.3 Finite volumes}\ref{fig:Finite volumes} +orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes} + +%% CNHbegin +\input{part1/fvol_figure} +%% CNHend \item tangent linear and adjoint counterparts are automatically maintained along with the forward model, permitting sensitivity and optimization @@ -101,36 +88,84 @@ \end{itemize} Key publications reporting on and charting the development of the model are -listed in an Appendix. +\cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04}: + +\begin{verbatim} +Hill, C. and J. Marshall, (1995) +Application of a Parallel Navier-Stokes Model to Ocean Circulation in +Parallel Computational Fluid Dynamics +In Proceedings of Parallel Computational Fluid Dynamics: Implementations +and Results Using Parallel Computers, 545-552. +Elsevier Science B.V.: New York + +Marshall, J., C. Hill, L. Perelman, and A. Adcroft, (1997) +Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling +J. Geophysical Res., 102(C3), 5733-5752. + +Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, (1997) +A finite-volume, incompressible Navier Stokes model for studies of the ocean +on parallel computers, +J. Geophysical Res., 102(C3), 5753-5766. + +Adcroft, A.J., Hill, C.N. and J. Marshall, (1997) +Representation of topography by shaved cells in a height coordinate ocean +model +Mon Wea Rev, vol 125, 2293-2315 + +Marshall, J., Jones, H. and C. Hill, (1998) +Efficient ocean modeling using non-hydrostatic algorithms +Journal of Marine Systems, 18, 115-134 + +Adcroft, A., Hill C. and J. Marshall: (1999) +A new treatment of the Coriolis terms in C-grid models at both high and low +resolutions, +Mon. Wea. Rev. Vol 127, pages 1928-1936 + +Hill, C, Adcroft,A., Jamous,D., and J. Marshall, (1999) +A Strategy for Terascale Climate Modeling. +In Proceedings of the Eighth ECMWF Workshop on the Use of Parallel Processors +in Meteorology, pages 406-425 +World Scientific Publishing Co: UK + +Marotzke, J, Giering,R., Zhang, K.Q., Stammer,D., Hill,C., and T.Lee, (1999) +Construction of the adjoint MIT ocean general circulation model and +application to Atlantic heat transport variability +J. Geophysical Res., 104(C12), 29,529-29,547. + +\end{verbatim} We begin by briefly showing some of the results of the model in action to give a feel for the wide range of problems that can be addressed using it. -\pagebreak -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.26 2006/06/28 15:22:13 edhill Exp $ % $Name: $ \section{Illustrations of the model in action} -The MITgcm has been designed and used to model a wide range of phenomena, +MITgcm has been designed and used to model a wide range of phenomena, from convection on the scale of meters in the ocean to the global pattern of -atmospheric winds - see fig.2\ref{fig:all-scales}. To give a flavor of the +atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the kinds of problems the model has been used to study, we briefly describe some of them here. A more detailed description of the underlying formulation, numerical algorithm and implementation that lie behind these calculations is given later. Indeed many of the illustrative examples shown below can be easily reproduced: simply download the model (the minimum you need is a PC -running linux, together with a FORTRAN\ 77 compiler) and follow the examples +running Linux, together with a FORTRAN\ 77 compiler) and follow the examples described in detail in the documentation. \subsection{Global atmosphere: `Held-Suarez' benchmark} +\begin{rawhtml} + +\end{rawhtml} + + -A novel feature of MITgcm is its ability to simulate both atmospheric and -oceanographic flows at both small and large scales. +A novel feature of MITgcm is its ability to simulate, using one basic algorithm, +both atmospheric and oceanographic flows at both small and large scales. -Fig.E1a.\ref{fig:Held-Suarez} shows an instantaneous plot of the 500$mb$ +Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$ temperature field obtained using the atmospheric isomorph of MITgcm run at -2.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole +$2.8^{\circ }$ resolution on the cubed sphere. We see cold air over the pole (blue) and warm air along an equatorial band (red). Fully developed baroclinic eddies spawned in the northern hemisphere storm track are evident. There are no mountains or land-sea contrast in this calculation, @@ -139,19 +174,33 @@ in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores - there are no mountains or land-sea contrast. +%% CNHbegin +\input{part1/cubic_eddies_figure} +%% CNHend + As described in Adcroft (2001), a `cubed sphere' is used to discretize the -globe permitting a uniform gridding and obviated the need to fourier filter. +globe permitting a uniform griding and obviated the need to Fourier filter. The `vector-invariant' form of MITgcm supports any orthogonal curvilinear grid, of which the cubed sphere is just one of many choices. -Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal -wind and meridional overturning streamfunction from a 20-level version of +Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal +wind from a 20-level configuration of the model. It compares favorable with more conventional spatial -discretization approaches. - -A regular spherical lat-lon grid can also be used. +discretization approaches. The two plots show the field calculated using the +cube-sphere grid and the flow calculated using a regular, spherical polar +latitude-longitude grid. Both grids are supported within the model. + +%% CNHbegin +\input{part1/hs_zave_u_figure} +%% CNHend \subsection{Ocean gyres} +\begin{rawhtml} + +\end{rawhtml} +\begin{rawhtml} + +\end{rawhtml} Baroclinic instability is a ubiquitous process in the ocean, as well as the atmosphere. Ocean eddies play an important role in modifying the @@ -161,36 +210,55 @@ increased until the baroclinic instability process is resolved, numerical solutions of a different and much more realistic kind, can be obtained. -Fig. ?.? shows the surface temperature and velocity field obtained from -MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$ -grid in which the pole has been rotated by 90$^{\circ }$ on to the equator -(to avoid the converging of meridian in northern latitudes). 21 vertical -levels are used in the vertical with a `lopped cell' representation of -topography. The development and propagation of anomalously warm and cold -eddies can be clearly been seen in the Gulf Stream region. The transport of -warm water northward by the mean flow of the Gulf Stream is also clearly -visible. +Figure \ref{fig:ocean-gyres} shows the surface temperature and +velocity field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ +horizontal resolution on a \textit{lat-lon} grid in which the pole has +been rotated by $90^{\circ }$ on to the equator (to avoid the +converging of meridian in northern latitudes). 21 vertical levels are +used in the vertical with a `lopped cell' representation of +topography. The development and propagation of anomalously warm and +cold eddies can be clearly seen in the Gulf Stream region. The +transport of warm water northward by the mean flow of the Gulf Stream +is also clearly visible. + +%% CNHbegin +\input{part1/atl6_figure} +%% CNHend -\subsection{Global ocean circulation} -Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$ -global ocean model run with 15 vertical levels. Lopped cells are used to -represent topography on a regular $lat-lon$ grid extending from 70$^{\circ -}N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with -mixed boundary conditions on temperature and salinity at the surface. The -transfer properties of ocean eddies, convection and mixing is parameterized -in this model. - -Fig.E2b shows the meridional overturning circulation of the global ocean in -Sverdrups. +\subsection{Global ocean circulation} +\begin{rawhtml} + +\end{rawhtml} + +Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean +currents at the surface of a $4^{\circ }$ global ocean model run with +15 vertical levels. Lopped cells are used to represent topography on a +regular \textit{lat-lon} grid extending from $70^{\circ }N$ to +$70^{\circ }S$. The model is driven using monthly-mean winds with +mixed boundary conditions on temperature and salinity at the surface. +The transfer properties of ocean eddies, convection and mixing is +parameterized in this model. + +Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning +circulation of the global ocean in Sverdrups. + +%%CNHbegin +\input{part1/global_circ_figure} +%%CNHend \subsection{Convection and mixing over topography} +\begin{rawhtml} + +\end{rawhtml} + Dense plumes generated by localized cooling on the continental shelf of the ocean may be influenced by rotation when the deformation radius is smaller than the width of the cooling region. Rather than gravity plumes, the mechanism for moving dense fluid down the shelf is then through geostrophic -eddies. The simulation shown in the figure (blue is cold dense fluid, red is +eddies. The simulation shown in the figure \ref{fig:convect-and-topo} +(blue is cold dense fluid, red is warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to trigger convection by surface cooling. The cold, dense water falls down the slope but is deflected along the slope by rotation. It is found that @@ -198,82 +266,136 @@ strong, and replaced by lateral entrainment due to the baroclinic instability of the along-slope current. +%%CNHbegin +\input{part1/convect_and_topo} +%%CNHend + \subsection{Boundary forced internal waves} +\begin{rawhtml} + +\end{rawhtml} The unique ability of MITgcm to treat non-hydrostatic dynamics in the presence of complex geometry makes it an ideal tool to study internal wave dynamics and mixing in oceanic canyons and ridges driven by large amplitude barotropic tidal currents imposed through open boundary conditions. -Fig. ?.? shows the influence of cross-slope topographic variations on +Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope +topographic variations on internal wave breaking - the cross-slope velocity is in color, the density contoured. The internal waves are excited by application of open boundary -conditions on the left.\ They propagate to the sloping boundary (represented +conditions on the left. They propagate to the sloping boundary (represented using MITgcm's finite volume spatial discretization) where they break under nonhydrostatic dynamics. +%%CNHbegin +\input{part1/boundary_forced_waves} +%%CNHend + \subsection{Parameter sensitivity using the adjoint of MITgcm} +\begin{rawhtml} + +\end{rawhtml} Forward and tangent linear counterparts of MITgcm are supported using an `automatic adjoint compiler'. These can be used in parameter sensitivity and data assimilation studies. -As one example of application of the MITgcm adjoint, Fig.E4 maps the -gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude -of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $% -\mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is -sensitive to heat fluxes over the Labrador Sea, one of the important sources -of deep water for the thermohaline circulations. This calculation also +As one example of application of the MITgcm adjoint, Figure +\ref{fig:hf-sensitivity} maps the gradient $\frac{\partial J}{\partial + \mathcal{H}}$where $J$ is the magnitude of the overturning +stream-function shown in figure \ref{fig:large-scale-circ} at +$60^{\circ }N$ and $ \mathcal{H}(\lambda,\varphi)$ is the mean, local +air-sea heat flux over a 100 year period. We see that $J$ is sensitive +to heat fluxes over the Labrador Sea, one of the important sources of +deep water for the thermohaline circulations. This calculation also yields sensitivities to all other model parameters. +%%CNHbegin +\input{part1/adj_hf_ocean_figure} +%%CNHend + \subsection{Global state estimation of the ocean} +\begin{rawhtml} + +\end{rawhtml} + An important application of MITgcm is in state estimation of the global ocean circulation. An appropriately defined `cost function', which measures the departure of the model from observations (both remotely sensed and -insitu) over an interval of time, is minimized by adjusting `control +in-situ) over an interval of time, is minimized by adjusting `control parameters' such as air-sea fluxes, the wind field, the initial conditions -etc. Figure ?.? shows an estimate of the time-mean surface elevation of the -ocean obtained by bringing the model in to consistency with altimetric and -in-situ observations over the period 1992-1997. +etc. Figure \ref{fig:assimilated-globes} shows the large scale planetary +circulation and a Hopf-Muller plot of Equatorial sea-surface height. +Both are obtained from assimilation bringing the model in to +consistency with altimetric and in-situ observations over the period +1992-1997. + +%% CNHbegin +\input{part1/assim_figure} +%% CNHend \subsection{Ocean biogeochemical cycles} - -MITgcm is being used to study global biogeochemical cycles in the ocean. For -example one can study the effects of interannual changes in meteorological -forcing and upper ocean circulation on the fluxes of carbon dioxide and -oxygen between the ocean and atmosphere. The figure shows the annual air-sea -flux of oxygen and its relation to density outcrops in the southern oceans -from a single year of a global, interannually varying simulation. - -Chris - get figure here: http://puddle.mit.edu/\symbol{126}% -mick/biogeochem.html +\begin{rawhtml} + +\end{rawhtml} + +MITgcm is being used to study global biogeochemical cycles in the +ocean. For example one can study the effects of interannual changes in +meteorological forcing and upper ocean circulation on the fluxes of +carbon dioxide and oxygen between the ocean and atmosphere. Figure +\ref{fig:biogeo} shows the annual air-sea flux of oxygen and its +relation to density outcrops in the southern oceans from a single year +of a global, interannually varying simulation. The simulation is run +at $1^{\circ}\times1^{\circ}$ resolution telescoping to +$\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not +shown). + +%%CNHbegin +\input{part1/biogeo_figure} +%%CNHend \subsection{Simulations of laboratory experiments} +\begin{rawhtml} + +\end{rawhtml} -Figure ?.? shows MITgcm being used to simulate a laboratory experiment -enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An +Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a +laboratory experiment inquiring into the dynamics of the Antarctic Circumpolar Current (ACC). An initially homogeneous tank of water ($1m$ in diameter) is driven from its free surface by a rotating heated disk. The combined action of mechanical and thermal forcing creates a lens of fluid which becomes baroclinically unstable. The stratification and depth of penetration of the lens is -arrested by its instability in a process analogous to that whic sets the +arrested by its instability in a process analogous to that which sets the stratification of the ACC. -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $ +%%CNHbegin +\input{part1/lab_figure} +%%CNHend + +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.26 2006/06/28 15:22:13 edhill Exp $ % $Name: $ \section{Continuous equations in `r' coordinates} +\begin{rawhtml} + +\end{rawhtml} To render atmosphere and ocean models from one dynamical core we exploit `isomorphisms' between equation sets that govern the evolution of the -respective fluids - see fig.4% -\marginpar{ -Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down +respective fluids - see figure \ref{fig:isomorphic-equations}. +One system of hydrodynamical equations is written down and encoded. The model variables have different interpretations depending on whether the atmosphere or ocean is being studied. Thus, for example, the vertical coordinate `$r$' is interpreted as pressure, $p$, if we are -modeling the atmosphere and height, $z$, if we are modeling the ocean. +modeling the atmosphere (right hand side of figure \ref{fig:isomorphic-equations}) +and height, $z$, if we are modeling the ocean (left hand side of figure +\ref{fig:isomorphic-equations}). + +%%CNHbegin +\input{part1/zandpcoord_figure.tex} +%%CNHend The state of the fluid at any time is characterized by the distribution of velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a @@ -281,38 +403,44 @@ depend on $\theta $, $S$, and $p$. The equations that govern the evolution of these fields, obtained by applying the laws of classical mechanics and thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of -a generic vertical coordinate, $r$, see fig.5% -\marginpar{ -Fig.5 The vertical coordinate of model}: +a generic vertical coordinate, $r$, so that the appropriate +kinematic boundary conditions can be applied isomorphically +see figure \ref{fig:zandp-vert-coord}. -\begin{equation*} -\frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}% -\right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}% -\text{ horizontal mtm} -\end{equation*} +%%CNHbegin +\input{part1/vertcoord_figure.tex} +%%CNHend -\begin{equation*} -\frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{% +\begin{equation} +\frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}} +\right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}} +\text{ horizontal mtm} \label{eq:horizontal_mtm} +\end{equation} + +\begin{equation} +\frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{ v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{ -vertical mtm} -\end{equation*} +vertical mtm} \label{eq:vertical_mtm} +\end{equation} \begin{equation} -\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{% -\partial r}=0\text{ continuity} \label{eq:continuous} +\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{ +\partial r}=0\text{ continuity} \label{eq:continuity} \end{equation} -\begin{equation*} -b=b(\theta ,S,r)\text{ equation of state} -\end{equation*} +\begin{equation} +b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state} +\end{equation} -\begin{equation*} +\begin{equation} \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature} -\end{equation*} +\label{eq:potential_temperature} +\end{equation} -\begin{equation*} +\begin{equation} \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity} -\end{equation*} +\label{eq:humidity_salt} +\end{equation} Here: @@ -326,10 +454,10 @@ \end{equation*} \begin{equation*} -\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}% +\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r} \text{ is the `grad' operator} \end{equation*} -with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}% +with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k} \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$ is a unit vector in the vertical @@ -363,7 +491,7 @@ \end{equation*} \begin{equation*} -\mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{% +\mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{ \mathbf{v}} \end{equation*} @@ -376,22 +504,23 @@ \end{equation*} The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by -extensive `physics' packages for atmosphere and ocean described in Chapter 6. +`physics' and forcing packages for atmosphere and ocean. These are described +in later chapters. \subsection{Kinematic Boundary conditions} \subsubsection{vertical} -at fixed and moving $r$ surfaces we set (see fig.5): +at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}): \begin{equation} -\dot{r}=0atr=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)} +\dot{r}=0 \text{\ at\ } r=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)} \label{eq:fixedbc} \end{equation} \begin{equation} -\dot{r}=\frac{Dr}{Dt}atr=R_{moving}\text{ \ -(oceansurface,bottomoftheatmosphere)} \label{eq:movingbc} +\dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \ +(ocean surface,bottom of the atmosphere)} \label{eq:movingbc} \end{equation} Here @@ -408,12 +537,12 @@ \begin{equation} \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0 \label{eq:noflow} -\end{equation}% +\end{equation} where $\vec{\mathbf{n}}$ is the normal to a solid boundary. \subsection{Atmosphere} -In the atmosphere, see fig.5, we interpret: +In the atmosphere, (see figure \ref{fig:zandp-vert-coord}), we interpret: \begin{equation} r=p\text{ is the pressure} \label{eq:atmos-r} @@ -445,10 +574,10 @@ \begin{equation*} T\text{ is absolute temperature} -\end{equation*}% +\end{equation*} \begin{equation*} p\text{ is the pressure} -\end{equation*}% +\end{equation*} \begin{eqnarray*} &&z\text{ is the height of the pressure surface} \\ &&g\text{ is the acceleration due to gravity} @@ -458,7 +587,7 @@ the Exner function $\Pi (p)$ given by (see Appendix Atmosphere) \begin{equation} \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa } \label{eq:exner} -\end{equation}% +\end{equation} where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas constant and $c_{p}$ the specific heat of air at constant pressure. @@ -484,9 +613,11 @@ atmosphere)} \label{eq:moving-bc-atmos} \end{eqnarray} -Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent -set of atmospheric equations which, for convenience, are written out in $p$ -coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}). +Then the (hydrostatic form of) equations +(\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yields a consistent +set of atmospheric equations which, for convenience, are written out +in $p$ coordinates in Appendix Atmosphere - see +eqs(\ref{eq:atmos-prime}). \subsection{Ocean} @@ -508,7 +639,7 @@ The surface of the ocean is given by: $R_{moving}=\eta $ -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 $ R_{o}=Z_{o}=0$. Boundary conditions are: @@ -516,25 +647,32 @@ \begin{eqnarray} w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)} \label{eq:fixed-bc-ocean} \\ -w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) % +w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) \label{eq:moving-bc-ocean}} \end{eqnarray} where $\eta $ is the elevation of the free surface. -Then eq(\ref{eq:continuous}) yields a consistent set of oceanic equations +Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yield a consistent set +of oceanic equations which, for convenience, are written out in $z$ coordinates in Appendix Ocean - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}). \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and Non-hydrostatic forms} +\begin{rawhtml} + +\end{rawhtml} + Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms: \begin{equation} \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r) \label{eq:phi-split} -\end{equation}% -and write eq(\ref{incompressible}a,b) in the form: +\end{equation} +%and write eq(\ref{eq:incompressible}) in the form: +% ^- this eq is missing (jmc) ; replaced with: +and write eq( \ref{eq:horizontal_mtm}) in the form: \begin{equation} \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi @@ -547,19 +685,19 @@ \end{equation} \begin{equation} -\epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{% +\epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{ \partial r}=G_{\dot{r}} \label{eq:mom-w} \end{equation} Here $\epsilon _{nh}$ is a non-hydrostatic parameter. -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 {eq:mom-h}) and (\ref{eq:mom-w}) represent advective, metric and Coriolis -terms in the momentum equations. In spherical coordinates they take the form% -\footnote{% +terms in the momentum equations. In spherical coordinates they take the form +\footnote{ In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms -in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref% +in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref {eq:gw-spherical}) are omitted; the singly-underlined terms are included in -the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (% +the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model ( \textbf{NH}) includes all terms.} - see Marshall et al 1997a for a full discussion: @@ -567,19 +705,19 @@ \left. \begin{tabular}{l} $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\ -$-\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\} $ \\ -$-\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\} $ \\ -$+\mathcal{F}_{u}$% -\end{tabular}% +$+\mathcal{F}_{u}$ +\end{tabular} \ \right\} \left\{ \begin{tabular}{l} \textit{advection} \\ \textit{metric} \\ \textit{Coriolis} \\ -\textit{\ Forcing/Dissipation}% -\end{tabular}% +\textit{\ Forcing/Dissipation} +\end{tabular} \ \right. \qquad \label{eq:gu-speherical} \end{equation} @@ -587,20 +725,20 @@ \left. \begin{tabular}{l} $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\ -$-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\} +$-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\} $ \\ -$-\left\{ -2\Omega u\sin lat\right\} $ \\ -$+\mathcal{F}_{v}$% -\end{tabular}% +$-\left\{ -2\Omega u\sin \varphi \right\} $ \\ +$+\mathcal{F}_{v}$ +\end{tabular} \ \right\} \left\{ \begin{tabular}{l} \textit{advection} \\ \textit{metric} \\ \textit{Coriolis} \\ -\textit{\ Forcing/Dissipation}% -\end{tabular}% +\textit{\ Forcing/Dissipation} +\end{tabular} \ \right. \qquad \label{eq:gv-spherical} -\end{equation}% +\end{equation} \qquad \qquad \qquad \qquad \qquad \begin{equation} @@ -608,41 +746,45 @@ \begin{tabular}{l} $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\ $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\ -${+}\underline{{2\Omega u\cos lat}}$ \\ -$\underline{\underline{\mathcal{F}_{\dot{r}}}}$% -\end{tabular}% +${+}\underline{{2\Omega u\cos \varphi}}$ \\ +$\underline{\underline{\mathcal{F}_{\dot{r}}}}$ +\end{tabular} \ \right\} \left\{ \begin{tabular}{l} \textit{advection} \\ \textit{metric} \\ \textit{Coriolis} \\ -\textit{\ Forcing/Dissipation}% -\end{tabular}% +\textit{\ Forcing/Dissipation} +\end{tabular} \ \right. \label{eq:gw-spherical} -\end{equation}% +\end{equation} \qquad \qquad \qquad \qquad \qquad -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 `$\varphi$ ' is latitude. Grad and div operators in spherical coordinates are defined in appendix -OPERATORS.% -\marginpar{ -Fig.6 Spherical polar coordinate system.} +OPERATORS. + +%%CNHbegin +\input{part1/sphere_coord_figure.tex} +%%CNHend \subsubsection{Shallow atmosphere approximation} -Most models are based on the `hydrostatic primitive equations' (HPE's) in -which the vertical momentum equation is reduced to a statement of -hydrostatic balance and the `traditional approximation' is made in which the -Coriolis force is treated approximately and the shallow atmosphere -approximation is made.\ The MITgcm need not make the `traditional -approximation'. To be able to support consistent non-hydrostatic forms the -shallow atmosphere approximation can be relaxed - when dividing through by $% -r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$, -the radius of the earth. +Most models are based on the `hydrostatic primitive equations' (HPE's) +in which the vertical momentum equation is reduced to a statement of +hydrostatic balance and the `traditional approximation' is made in +which the Coriolis force is treated approximately and the shallow +atmosphere approximation is made. MITgcm need not make the +`traditional approximation'. To be able to support consistent +non-hydrostatic forms the shallow atmosphere approximation can be +relaxed - when dividing through by $ r $ in, for example, +(\ref{eq:gu-speherical}), we do not replace $r$ by $a$, the radius of +the earth. \subsubsection{Hydrostatic and quasi-hydrostatic forms} +\label{sec:hydrostatic_and_quasi-hydrostatic_forms} These are discussed at length in Marshall et al (1997a). @@ -651,13 +793,13 @@ are neglected and `${r}$' is replaced by `$a$', the mean radius of the earth. Once the pressure is found at one level - e.g. by inverting a 2-d Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be -computed at all other levels by integration of the hydrostatic relation, eq(% +computed at all other levels by integration of the hydrostatic relation, eq( \ref{eq:hydrostatic}). In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos -\phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic -contribution to the pressure field: only the terms underlined twice in Eqs. (% +\varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic +contribution to the pressure field: only the terms underlined twice in Eqs. ( \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero and, simultaneously, the shallow atmosphere approximation is relaxed. In \textbf{QH}\ \textit{all} the metric terms are retained and the full @@ -665,7 +807,7 @@ vertical momentum equation (\ref{eq:mom-w}) becomes: \begin{equation*} -\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat +\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi \end{equation*} making a small correction to the hydrostatic pressure. @@ -676,17 +818,17 @@ \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms} -The MIT model presently supports a full non-hydrostatic ocean isomorph, but +MITgcm presently supports a full non-hydrostatic ocean isomorph, but only a quasi-non-hydrostatic atmospheric isomorph. \paragraph{Non-hydrostatic Ocean} -In the non-hydrostatic ocean model all terms in equations Eqs.(\ref% +In the non-hydrostatic ocean model all terms in equations Eqs.(\ref {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A three dimensional elliptic equation must be solved subject to Neumann boundary conditions (see below). It is important to note that use of the full \textbf{NH} does not admit any new `fast' waves in to the system - the -incompressible condition eq(\ref{eq:continuous})c has already filtered out +incompressible condition eq(\ref{eq:continuity}) has already filtered out acoustic modes. It does, however, ensure that the gravity waves are treated accurately with an exact dispersion relation. The \textbf{NH} set has a complete angular momentum principle and consistent energetics - see White @@ -694,13 +836,13 @@ \paragraph{Quasi-nonhydrostatic Atmosphere} -In the non-hydrostatic version of our atmospheric model we approximate $\dot{% +In the non-hydrostatic version of our atmospheric model we approximate $\dot{ r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical}) (but only here) by: \begin{equation} \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt} \label{eq:quasi-nh-w} -\end{equation}% +\end{equation} where $p_{hy}$ is the hydrostatic pressure. \subsubsection{Summary of equation sets supported by model} @@ -728,16 +870,15 @@ \subparagraph{Non-hydrostatic} -Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$% -coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref% +Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$ +coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref {eq:ocean-salt}). \subsection{Solution strategy} -The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{% -NH} models is summarized in Fig.7.% -\marginpar{ -Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is +The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{ +NH} models is summarized in Figure \ref{fig:solution-strategy}. +Under all dynamics, a 2-d elliptic equation is first solved to find the surface pressure and the hydrostatic pressure at any level computed from the weight of fluid above. Under \textbf{HPE} and \textbf{QH} dynamics, the horizontal momentum equations are then stepped @@ -746,8 +887,12 @@ stepping forward the horizontal momentum equations; $\dot{r}$ is found by stepping forward the vertical momentum equation. +%%CNHbegin +\input{part1/solution_strategy_figure.tex} +%%CNHend + There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of -course, some complication that goes with the inclusion of $\cos \phi \ $% +course, some complication that goes with the inclusion of $\cos \varphi \ $ Coriolis terms and the relaxation of the shallow atmosphere approximation. But this leads to negligible increase in computation. In \textbf{NH}, in contrast, one additional elliptic equation - a three-dimensional one - must @@ -757,6 +902,7 @@ hydrostatic limit, is as computationally economic as the \textbf{HPEs}. \subsection{Finding the pressure field} +\label{sec:finding_the_pressure_field} Unlike the prognostic variables $u$, $v$, $w$, $\theta $ and $S$, the pressure field must be obtained diagnostically. We proceed, as before, by @@ -771,7 +917,7 @@ vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield: \begin{equation*} -\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} \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr \end{equation*} and so @@ -789,11 +935,11 @@ \subsubsection{Surface pressure} -The surface pressure equation can be obtained by integrating continuity, (% -\ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$ +The surface pressure equation can be obtained by integrating continuity, +(\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$ \begin{equation*} -\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} }_{h}+\partial _{r}\dot{r}\right) dr=0 \end{equation*} @@ -801,42 +947,42 @@ \begin{equation*} \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta -+\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}% ++\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}} _{h}dr=0 \end{equation*} -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 $ r $. The above can be rearranged to yield, using Leibnitz's theorem: \begin{equation} \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source} \label{eq:free-surface} -\end{equation}% +\end{equation} where we have incorporated a source term. Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential -(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can +(atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can be written \begin{equation} \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right) \label{eq:phi-surf} -\end{equation}% +\end{equation} where $b_{s}$ is the buoyancy at the surface. -In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref% +In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free surface' and `rigid lid' approaches are available. \subsubsection{Non-hydrostatic pressure} -Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{% -\partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation -(\ref{incompressible}), we deduce that: +Taking the horizontal divergence of (\ref{eq:mom-h}) and adding +$\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation +(\ref{eq:continuity}), we deduce that: \begin{equation} -\nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{% -\nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .% +\nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{ +\nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla . \vec{\mathbf{F}} \label{eq:3d-invert} \end{equation} @@ -856,13 +1002,13 @@ \end{equation} where $\widehat{n}$ is a vector of unit length normal to the boundary. The kinematic condition (\ref{nonormalflow}) is also applied to the vertical -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 $ \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the tangential component of velocity, $v_{T}$, at all solid boundaries, depending on the form chosen for the dissipative terms in the momentum equations - see below. -Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that: +Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that: \begin{equation} \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}} @@ -873,25 +1019,25 @@ \begin{equation*} \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi _{s}+\mathbf{\nabla }\phi _{hyd}\right) -\end{equation*}% +\end{equation*} presenting inhomogeneous Neumann boundary conditions to the Elliptic problem (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can exploit classical 3D potential theory and, by introducing an appropriately chosen $\delta $-function sheet of `source-charge', replace the inhomogeneous boundary condition on pressure by a homogeneous one. The -source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $% -\vec{\mathbf{F}}.$ By simultaneously setting $% +source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $ +\vec{\mathbf{F}}.$ By simultaneously setting $ \begin{array}{l} -\widehat{n}.\vec{\mathbf{F}}% -\end{array}% +\widehat{n}.\vec{\mathbf{F}} +\end{array} =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following self-consistent but simpler homogenized Elliptic problem is obtained: \begin{equation*} \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad -\end{equation*}% +\end{equation*} where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such -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 {eq:inhom-neumann-nh}) the modified boundary condition becomes: \begin{equation} @@ -902,7 +1048,7 @@ converges rapidly because $\phi _{nh}\ $is then only a small correction to the hydrostatic pressure field (see the discussion in Marshall et al, a,b). -The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{homneuman}) +The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh}) does not vanish at $r=R_{moving}$, and so refines the pressure there. \subsection{Forcing/dissipation} @@ -910,7 +1056,7 @@ \subsubsection{Forcing} The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by -`physics packages' described in detail in chapter ??. +`physics packages' and forcing packages. These are described later on. \subsubsection{Dissipation} @@ -920,7 +1066,7 @@ biharmonic frictions are commonly used: \begin{equation} -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}} +A_{4}\nabla _{h}^{4}v \label{eq:dissipation} \end{equation} where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity @@ -931,12 +1077,12 @@ The mixing terms for the temperature and salinity equations have a similar form to that of momentum except that the diffusion tensor can be -non-diagonal and have varying coefficients. $\qquad $% +non-diagonal and have varying coefficients. \begin{equation} D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla _{h}^{4}(T,S) \label{eq:diffusion} \end{equation} -where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $% +where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $ horizontal coefficient for biharmonic diffusion. In the simplest case where the subgrid-scale fluxes of heat and salt are parameterized with constant horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$, @@ -947,7 +1093,7 @@ \begin{array}{ccc} K_{h} & 0 & 0 \\ 0 & K_{h} & 0 \\ -0 & 0 & K_{v}% +0 & 0 & K_{v} \end{array} \right) \qquad \qquad \qquad \label{eq:diagonal-diffusion-tensor} \end{equation} @@ -957,29 +1103,30 @@ \subsection{Vector invariant form} -For some purposes it is advantageous to write momentum advection in eq(\ref% -{hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form: +For some purposes it is advantageous to write momentum advection in +eq(\ref {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the +(so-called) `vector invariant' form: \begin{equation} -\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}% -+\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla % +\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t} ++\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right] \label{eq:vi-identity} -\end{equation}% +\end{equation} This permits alternative numerical treatments of the non-linear terms based on their representation as a vorticity flux. Because gradients of coordinate vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit -representation of the metric terms in (\ref{eq:gu-speherical}), (\ref% +representation of the metric terms in (\ref{eq:gu-speherical}), (\ref {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information about the geometry is contained in the areas and lengths of the volumes used to discretize the model. \subsection{Adjoint} -Tangent linear and adjoint counterparts of the forward model and described +Tangent linear and adjoint counterparts of the forward model are described in Chapter 5. -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.26 2006/06/28 15:22:13 edhill Exp $ % $Name: $ \section{Appendix ATMOSPHERE} @@ -991,23 +1138,23 @@ The hydrostatic primitive equations (HPEs) in p-coordinates are: \begin{eqnarray} -\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}} _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-mom} \\ \frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\ -\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{% +\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ \partial p} &=&0 \label{eq:atmos-cont} \\ p\alpha &=&RT \label{eq:atmos-eos} \\ c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} \label{eq:atmos-heat} -\end{eqnarray}% +\end{eqnarray} where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total -derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is -the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp% -}{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref% -{eq:atmos-heat}) is the first law of thermodynamics where internal energy $% -e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $% +derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter, $\phi =gz$ is +the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp +}{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref +{eq:atmos-heat}) is the first law of thermodynamics where internal energy $ +e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $ p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing. It is convenient to cast the heat equation in terms of potential temperature @@ -1015,30 +1162,30 @@ Differentiating (\ref{eq:atmos-eos}) we get: \begin{equation*} p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt} -\end{equation*}% -which, when added to the heat equation (\ref{eq:atmos-heat}) and using $% +\end{equation*} +which, when added to the heat equation (\ref{eq:atmos-heat}) and using $ c_{p}=c_{v}+R$, gives: \begin{equation} c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q} \label{eq-p-heat-interim} -\end{equation}% +\end{equation} Potential temperature is defined: \begin{equation} \theta =T(\frac{p_{c}}{p})^{\kappa } \label{eq:potential-temp} -\end{equation}% +\end{equation} where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience we will make use of the Exner function $\Pi (p)$ which defined by: \begin{equation} \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa } \label{Exner} -\end{equation}% +\end{equation} The following relations will be useful and are easily expressed in terms of the Exner function: \begin{equation*} c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi -}{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{% -\partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}% +}{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{ +\partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p} \frac{Dp}{Dt} -\end{equation*}% +\end{equation*} where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy. The heat equation is obtained by noting that @@ -1053,7 +1200,7 @@ \end{equation} which is in conservative form. -For convenience in the model we prefer to step forward (\ref% +For convenience in the model we prefer to step forward (\ref {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}). \subsubsection{Boundary conditions} @@ -1069,6 +1216,7 @@ surface ($\phi $ is imposed and $\omega \neq 0$). \subsubsection{Splitting the geo-potential} +\label{sec:hpe-p-geo-potential-split} For the purposes of initialization and reducing round-off errors, the model deals with perturbations from reference (or ``standard'') profiles. For @@ -1097,16 +1245,17 @@ The final form of the HPE's in p coordinates is then: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% -_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\ +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} +_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} +\label{eq:atmos-prime} \\ \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\ -\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{% +\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ \partial p} &=&0 \\ \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\ -\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \label{eq:atmos-prime} +\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \end{eqnarray} -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.26 2006/06/28 15:22:13 edhill Exp $ % $Name: $ \section{Appendix OCEAN} @@ -1117,21 +1266,22 @@ HPE's for the ocean written in z-coordinates are obtained. The non-Boussinesq equations for oceanic motion are: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\ \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} \\ -\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}% -_{h}+\frac{\partial w}{\partial z} &=&0 \\ -\rho &=&\rho (\theta ,S,p) \\ -\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\ -\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:non-boussinesq} -\end{eqnarray}% +\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}} +_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\ +\rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\ +\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\ +\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zns-salt} +\label{eq:non-boussinesq} +\end{eqnarray} These equations permit acoustics modes, inertia-gravity waves, -non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline +non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline mode. As written, they cannot be integrated forward consistently - if we step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be -consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref% +consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is therefore necessary to manipulate the system as follows. Differentiating the EOS (equation of state) gives: @@ -1143,11 +1293,11 @@ _{\theta ,S}\frac{Dp}{Dt} \label{EOSexpansion} \end{equation} -Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the -reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref% -{eq-zns-cont} gives: +Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is +the reciprocal of the sound speed ($c_{s}$) squared. Substituting into +\ref{eq-zns-cont} gives: \begin{equation} -\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{% +\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{ v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure} \end{equation} where we have used an approximation sign to indicate that we have assumed @@ -1155,12 +1305,12 @@ Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that can be explicitly integrated forward: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \label{eq-cns-hmom} \\ \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-cns-hydro} \\ -\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{% +\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{ v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-cns-cont} \\ \rho &=&\rho (\theta ,S,p) \label{eq-cns-eos} \\ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-cns-heat} \\ @@ -1174,32 +1324,32 @@ `Boussinesq assumption'. The only term that then retains the full variation in $\rho $ is the gravitational acceleration: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \label{eq-zcb-hmom} \\ -\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}% +\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}} \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-zcb-hydro} \\ -\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{% +\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{ \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zcb-cont} \\ \rho &=&\rho (\theta ,S,p) \label{eq-zcb-eos} \\ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zcb-heat} \\ \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zcb-salt} \end{eqnarray} These equations still retain acoustic modes. But, because the -``compressible'' terms are linearized, the pressure equation \ref% +``compressible'' terms are linearized, the pressure equation \ref {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent term appears as a Helmholtz term in the non-hydrostatic pressure equation). These are the \emph{truly} compressible Boussinesq equations. Note that the EOS must have the same pressure dependency as the linearized pressure term, -ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{% +ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{ c_{s}^{2}}$, for consistency. \subsubsection{`Anelastic' z-coordinate equations} The anelastic approximation filters the acoustic mode by removing the -time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}% -). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}% +time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont} +). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o} \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between continuity and EOS. A better solution is to change the dependency on pressure in the EOS by splitting the pressure into a reference function of @@ -1210,29 +1360,29 @@ Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from differentiating the EOS, the continuity equation then becomes: \begin{equation*} -\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{% -Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+% +\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{ +Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+ \frac{\partial w}{\partial z}=0 \end{equation*} If the time- and space-scales of the motions of interest are longer than -those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},% +those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt}, \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and -$\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{% +$\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{ Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the anelastic continuity equation: \begin{equation} -\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-% +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}- \frac{g}{c_{s}^{2}}w=0 \label{eq-za-cont1} \end{equation} A slightly different route leads to the quasi-Boussinesq continuity equation -where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+% -\mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }% +where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+ +\mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla } _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding: \begin{equation} -\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{% +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{ \partial \left( \rho _{o}w\right) }{\partial z}=0 \label{eq-za-cont2} \end{equation} Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same @@ -1241,18 +1391,18 @@ \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}} \end{equation} Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$ -and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{% +and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{ g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are then: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \label{eq-zab-hmom} \\ -\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}% +\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}} \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-zab-hydro} \\ -\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{% +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{ \partial \left( \rho _{o}w\right) }{\partial z} &=&0 \label{eq-zab-cont} \\ \rho &=&\rho (\theta ,S,p_{o}(z)) \label{eq-zab-eos} \\ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zab-heat} \\ @@ -1265,10 +1415,10 @@ technically, to also remove the dependence of $\rho $ on $p_{o}$. This would yield the ``truly'' incompressible Boussinesq equations: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \label{eq-ztb-hmom} \\ -\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}% +\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}} \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-ztb-hydro} \\ \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} @@ -1287,20 +1437,20 @@ density thus: \begin{equation*} \rho =\rho _{o}+\rho ^{\prime } -\end{equation*}% +\end{equation*} We then assert that variations with depth of $\rho _{o}$ are unimportant while the compressible effects in $\rho ^{\prime }$ are: \begin{equation*} \rho _{o}=\rho _{c} -\end{equation*}% +\end{equation*} \begin{equation*} \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o} -\end{equation*}% +\end{equation*} This then yields what we can call the semi-compressible Boussinesq equations: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% -_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{% +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} +_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{ \mathcal{F}}} \label{eq:ocean-mom} \\ \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} @@ -1311,7 +1461,7 @@ \\ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq:ocean-theta} \\ \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:ocean-salt} -\end{eqnarray}% +\end{eqnarray} Note that the hydrostatic pressure of the resting fluid, including that associated with $\rho _{c}$, is subtracted out since it has no effect on the dynamics. @@ -1322,7 +1472,7 @@ _{nh}=0$ form of these equations that are used throughout the ocean modeling community and referred to as the primitive equations (HPE). -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.26 2006/06/28 15:22:13 edhill Exp $ % $Name: $ \section{Appendix:OPERATORS} @@ -1335,35 +1485,34 @@ and vertical direction respectively, are given by (see Fig.2) : \begin{equation*} -u=r\cos \phi \frac{D\lambda }{Dt} +u=r\cos \varphi \frac{D\lambda }{Dt} \end{equation*} \begin{equation*} -v=r\frac{D\phi }{Dt}\qquad +v=r\frac{D\varphi }{Dt} \end{equation*} -$\qquad \qquad \qquad \qquad $ \begin{equation*} \dot{r}=\frac{Dr}{Dt} \end{equation*} -Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial +Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial distance of the particle from the center of the earth, $\Omega $ is the angular speed of rotation of the Earth and $D/Dt$ is the total derivative. -The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in +The `grad' ($\nabla $) and `div' ($\nabla\cdot$) operators are defined by, in spherical coordinates: \begin{equation*} -\nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }% -,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}% +\nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda } +,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r} \right) \end{equation*} \begin{equation*} -\nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial -\lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\} +\nabla\cdot v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial +\lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\} +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r} \end{equation*} -%%%% \end{document} +%tci%\end{document}