--- manual/s_overview/text/manual.tex 2001/09/27 17:45:03 1.1 +++ manual/s_overview/text/manual.tex 2001/10/24 15:21:27 1.6 @@ -1,64 +1,40 @@ -%%%% % $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh 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, September 27, 2001 10:59:02} -%%%% %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 +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.6 2001/10/24 15:21:27 cnh Exp $ +% $Name: $ + +%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} -\pagebreak +%tci%\tableofcontents -\part{MITgcm basics} % Section: Overview -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.6 2001/10/24 15:21:27 cnh Exp $ % $Name: $ \section{Introduction} @@ -78,48 +54,32 @@ \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% +models - see fig \marginpar{ Fig.1 One model}\ref{fig:onemodel} -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \rotatebox{90}{ - \rotatebox{180}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps} - } - } -} -\end{center} -\label{fig:onemodel} -\end{figure} +%% 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% +small-scale and large scale processes - see fig \marginpar{ Fig.2 All scales}\ref{fig:all-scales} - -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \rotatebox{90}{ - \rotatebox{180}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/scales.eps} - } - } -} -\end{center} -\label{fig:scales} -\end{figure} - +%% 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% +orthogonal curvilinear grids and shaved cells - see fig \marginpar{ -Fig.3 Finite volumes}\ref{fig:Finite volumes} +Fig.3 Finite volumes}\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 @@ -134,9 +94,8 @@ 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 1.1 2001/09/27 17:45:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.6 2001/10/24 15:21:27 cnh Exp $ % $Name: $ \section{Illustrations of the model in action} @@ -147,122 +106,194 @@ 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 it is easy to reproduce the results shown here: simply -download the model (the minimum you need is a PC running linux, together -with a FORTRAN\ 77 compiler) and follow the examples. +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 +described in detail in the documentation. \subsection{Global atmosphere: `Held-Suarez' benchmark} -Fig.E1a.\ref{fig:Held-Suarez} is an instaneous plot of the 500$mb$ height -field obtained using a 5-level version of the atmospheric pressure isomorph -run at 2.8$^{\circ }$ resolution. We see fully developed baroclinic eddies -along the northern hemisphere storm track. There are no mountains or -land-sea contrast in this calculation, but you can easily put them in. The -model is driven by relaxation to a radiative-convective equilibrium profile, -following the description set out in Held and Suarez; 1994 designed to test -atmospheric hydrodynamical cores - there are no mountains or land-sea -contrast. As decribed in Adcroft (2001), a `cubed sphere' is used to -descretize the globe permitting a uniform gridding and obviated the need to -fourier filter. +A novel feature of MITgcm is its ability to simulate both atmospheric and +oceanographic flows at both small and large scales. -Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal -wind and meridional overturning streamfunction from the 5-level model. - - -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \rotatebox{90}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps} - } -} -\end{center} -\label{fig:hscs} -\end{figure} +Fig.E1a.\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 +(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, +but you can easily put them in. The model is driven by relaxation to a +radiative-convective equilibrium profile, following the description set out +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. +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 +the model. It compares favorable with more conventional spatial +discretization approaches. A regular spherical lat-lon grid can also be used. -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \rotatebox{90}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps} - } -} -\end{center} -\label{fig:hslatlon} -\end{figure} +%% CNHbegin +\input{part1/hs_zave_u_figure} +%% CNHend \subsection{Ocean gyres} +Baroclinic instability is a ubiquitous process in the ocean, as well as the +atmosphere. Ocean eddies play an important role in modifying the +hydrographic structure and current systems of the oceans. Coarse resolution +models of the oceans cannot resolve the eddy field and yield rather broad, +diffusive patterns of ocean currents. But if the resolution of our models is +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. + +%% CNHbegin +\input{part1/ocean_gyres_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. The model is driven using -monthly-mean winds with mixed boundary conditions on temperature and -salinity at the surface. Fig.E2b shows the overturning (thermohaline) -circulation. Lopped cells are used to represent topography on a regular $% -lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$. - - -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ -% \rotatebox{90}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps} -% } -} -\end{center} -\label{fig:horizcirc} -\end{figure} - -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \rotatebox{90}{ - \rotatebox{180}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps} - } - } -} -\end{center} -\label{fig:moc} -\end{figure} - - -\subsection{Flow over topography} - -\subsection{Ocean convection} - -Fig.E3 shows convection over a slope using the non-hydrostatic ocean -isomorph and lopped cells to respresent topography. .....The grid resolution -is +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. + +%%CNHbegin +\input{part1/global_circ_figure} +%%CNHend + +\subsection{Convection and mixing over topography} + +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 +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 +entrainment in the vertical plane is reduced when rotational control is +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} -\subsection{Carbon outgassing sensitivity} - -Fig.E4 shows.... - -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps} -} -\end{center} -\label{fig:co2mrt} -\end{figure} +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 +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 +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} + +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 +yields sensitivities to all other model parameters. + +%%CNHbegin +\input{part1/adj_hf_ocean_figure} +%%CNHend + +\subsection{Global state estimation of the ocean} + +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 +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. + +%% CNHbegin +\input{part1/globes_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. + +%%CNHbegin +\input{part1/biogeo_figure} +%%CNHend + +\subsection{Simulations of laboratory experiments} + +Figure ?.? shows MITgcm being used to simulate a laboratory experiment +enquiring in to 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 +stratification of the ACC. + +%%CNHbegin +\input{part1/lab_figure} +%%CNHend - -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.6 2001/10/24 15:21:27 cnh Exp $ % $Name: $ \section{Continuous equations in `r' coordinates} 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% +respective fluids - see fig.4 \marginpar{ Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down and encoded. The model variables have different interpretations depending on @@ -270,114 +301,108 @@ vertical coordinate `$r$' is interpreted as pressure, $p$, if we are modeling the atmosphere and height, $z$, if we are modeling the ocean. +%%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 `geopotential' $\phi $ and density $\rho =\rho (\theta ,S,p)$ which may 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% +a generic vertical coordinate, $r$, see fig.5 \marginpar{ Fig.5 The vertical coordinate of model}: -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \rotatebox{90}{ - \rotatebox{180}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/vertcoord.eps} - } - } -} -\end{center} -\label{fig:vertcoord} -\end{figure} +%%CNHbegin +\input{part1/vertcoord_figure.tex} +%%CNHend \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}}}% +\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*} \begin{equation*} -\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{ v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{ vertical mtm} \end{equation*} \begin{equation} -\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{% +\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{ \partial r}=0\text{ continuity} \label{eq:continuous} \end{equation} \begin{equation*} -b=b(\theta ,S,r)\text{ equation of state} +b=b(\theta ,S,r)\text{ equation of state} \end{equation*} \begin{equation*} -\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature} +\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature} \end{equation*} \begin{equation*} -\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity} +\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity} \end{equation*} Here: \begin{equation*} -r\text{ is the vertical coordinate} +r\text{ is the vertical coordinate} \end{equation*} \begin{equation*} \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{ -is the total derivative} +is the total derivative} \end{equation*} \begin{equation*} -\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}% -\text{ is the `grad' operator} +\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 \begin{equation*} -t\text{ is time} +t\text{ is time} \end{equation*} \begin{equation*} \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the -velocity} +velocity} \end{equation*} \begin{equation*} -\phi \text{ is the `pressure'/`geopotential'} +\phi \text{ is the `pressure'/`geopotential'} \end{equation*} \begin{equation*} -\vec{\Omega}\text{ is the Earth's rotation} +\vec{\Omega}\text{ is the Earth's rotation} \end{equation*} \begin{equation*} -b\text{ is the `buoyancy'} +b\text{ is the `buoyancy'} \end{equation*} \begin{equation*} -\theta \text{ is potential temperature} +\theta \text{ is potential temperature} \end{equation*} \begin{equation*} -S\text{ is specific humidity in the atmosphere; salinity in the ocean} +S\text{ is specific humidity in the atmosphere; salinity in the ocean} \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*} \begin{equation*} -\mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }% -\theta +\mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta \end{equation*} \begin{equation*} @@ -406,7 +431,7 @@ Here \begin{equation*} -R_{moving}=R_{o}+\eta +R_{moving}=R_{o}+\eta \end{equation*} where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on whether we are in the atmosphere or ocean) of the `moving surface' in the @@ -417,7 +442,7 @@ \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} @@ -454,10 +479,10 @@ \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} @@ -467,19 +492,19 @@ 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. At the top of the atmosphere (which is `fixed' in our $r$ coordinate): \begin{equation*} -R_{fixed}=p_{top}=0 +R_{fixed}=p_{top}=0 \end{equation*} In a resting atmosphere the elevation of the mountains at the bottom is given by \begin{equation*} -R_{moving}=R_{o}(x,y)=p_{o}(x,y) +R_{moving}=R_{o}(x,y)=p_{o}(x,y) \end{equation*} i.e. the (hydrostatic) pressure at the top of the mountains in a resting atmosphere. @@ -517,7 +542,7 @@ 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: @@ -525,7 +550,7 @@ \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. @@ -542,7 +567,7 @@ \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}% +\end{equation} and write eq(\ref{incompressible}a,b) in the form: \begin{equation} @@ -556,19 +581,19 @@ \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: @@ -576,82 +601,72 @@ \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}% -\ \right. \qquad \label{eq:gu-speherical} +\textit{\ Forcing/Dissipation} +\end{tabular} +\ \right. \qquad \label{eq:gu-speherical} \end{equation} \begin{equation} \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}% -\ \right. \qquad \label{eq:gv-spherical} -\end{equation}% -\qquad \qquad \qquad \qquad \qquad +\textit{\ Forcing/Dissipation} +\end{tabular} +\ \right. \qquad \label{eq:gv-spherical} +\end{equation} +\qquad \qquad \qquad \qquad \qquad \begin{equation} \left. \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}% -\ \right. \label{eq:gw-spherical} -\end{equation}% -\qquad \qquad \qquad \qquad \qquad +\textit{\ Forcing/Dissipation} +\end{tabular} +\ \right. \label{eq:gw-spherical} +\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.% +OPERATORS. \marginpar{ Fig.6 Spherical polar coordinate system.} -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \rotatebox{90}{ - \rotatebox{180}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps} - } - } -} -\end{center} -\label{fig:spcoord} -\end{figure} - +%%CNHbegin +\input{part1/sphere_coord_figure.tex} +%%CNHend \subsubsection{Shallow atmosphere approximation} @@ -661,8 +676,8 @@ 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 $a$, +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} @@ -674,13 +689,13 @@ 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 @@ -688,7 +703,7 @@ 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. @@ -704,7 +719,7 @@ \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 @@ -717,13 +732,13 @@ \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} @@ -751,14 +766,14 @@ \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.% +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 first solved to find the surface pressure and the hydrostatic pressure at @@ -769,22 +784,12 @@ stepping forward the horizontal momentum equations; $\dot{r}$ is found by stepping forward the vertical momentum equation. -\begin{figure} -\begin{center} -\resizebox{!}{4in}{ - \rotatebox{90}{ - \rotatebox{180}{ - \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps} - } - } -} -\end{center} -\label{fig:solnstart} -\end{figure} - +%%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 @@ -808,8 +813,8 @@ 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}% -\right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr +\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 @@ -826,54 +831,54 @@ \subsubsection{Surface pressure} -The surface pressure equation can be obtained by integrating continuity, (% +The surface pressure equation can be obtained by integrating continuity, ( \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$ \begin{equation*} -\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}% -}_{h}+\partial _{r}\dot{r}\right) dr=0 +\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v} +}_{h}+\partial _{r}\dot{r}\right) dr=0 \end{equation*} Thus: \begin{equation*} \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta -+\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}% -_{h}dr=0 ++\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 be written \begin{equation} -\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) \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 ($\epsilon _{nh}=0$), Eqs(\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{% +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: \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} @@ -893,7 +898,7 @@ \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 @@ -910,25 +915,25 @@ \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 inhomogenous -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 $% +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 $ \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 homogenised Elliptic problem is obtained: +self-consistent but simpler homogenized Elliptic problem is obtained: \begin{equation*} -\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad -\end{equation*}% +\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad +\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} @@ -957,7 +962,7 @@ 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 @@ -968,12 +973,12 @@ 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. $\qquad $ \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, $\underline{\underline{K}}$, @@ -984,7 +989,7 @@ \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} @@ -994,29 +999,29 @@ \subsection{Vector invariant form} -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 {hor-mtm}) and (\ref{vertmtm}) 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 % -\left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right] +\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 adoint counterparts of the forward model and described in -Chapter 5. +Tangent linear and adjoint counterparts of the forward model and described +in Chapter 5. -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.6 2001/10/24 15:21:27 cnh Exp $ % $Name: $ \section{Appendix ATMOSPHERE} @@ -1028,23 +1033,23 @@ 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}}% -_{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}} +\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 }{% +\frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\ +\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} \\ +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 @@ -1052,36 +1057,36 @@ 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 \begin{equation*} c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta -\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt} +\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt} \end{equation*} and on substituting into (\ref{eq-p-heat-interim}) gives: \begin{equation} @@ -1090,7 +1095,7 @@ \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} @@ -1134,16 +1139,16 @@ 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}}% +\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{\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} \end{eqnarray} -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.6 2001/10/24 15:21:27 cnh Exp $ % $Name: $ \section{Appendix OCEAN} @@ -1154,21 +1159,21 @@ 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}}% +\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) \\ +\rho &=&\rho (\theta ,S,p) \\ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\ \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:non-boussinesq} -\end{eqnarray}% +\end{eqnarray} These equations permit acoustics modes, inertia-gravity waves, non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline mode. As written, they cannot be integrated forward consistently - if we step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be -consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref% +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: @@ -1181,10 +1186,10 @@ \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% +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 @@ -1192,12 +1197,12 @@ 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} \\ @@ -1211,65 +1216,65 @@ `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 height and a perturbation: \begin{equation*} -\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime }) +\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime }) \end{equation*} 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{\partial w}{\partial z}=0 +\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 @@ -1278,18 +1283,18 @@ \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} \\ @@ -1302,10 +1307,10 @@ 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} @@ -1324,20 +1329,20 @@ 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} @@ -1348,7 +1353,7 @@ \\ \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. @@ -1359,7 +1364,7 @@ _{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 1.1 2001/09/27 17:45:03 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.6 2001/10/24 15:21:27 cnh Exp $ % $Name: $ \section{Appendix:OPERATORS} @@ -1372,19 +1377,19 @@ 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}\qquad \end{equation*} $\qquad \qquad \qquad \qquad $ \begin{equation*} -\dot{r}=\frac{Dr}{Dt} +\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. @@ -1392,15 +1397,15 @@ 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}% -\right) +\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\} -+\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r} +\nabla .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}