--- manual/s_overview/text/manual.tex 2001/11/21 14:13:17 1.14 +++ manual/s_overview/text/manual.tex 2010/08/30 23:09:21 1.29 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.14 2001/11/21 14:13:17 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.29 2010/08/30 23:09:21 jmc Exp $ % $Name: $ %tci%\documentclass[12pt]{book} @@ -34,12 +34,10 @@ % Section: Overview -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.14 2001/11/21 14:13:17 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.29 2010/08/30 23:09:21 jmc 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 @@ -49,6 +47,12 @@ 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} @@ -57,14 +61,14 @@ models - see fig \ref{fig:onemodel} %% CNHbegin -\input{part1/one_model_figure} +\input{s_overview/text/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 \ref{fig:all-scales} %% CNHbegin -\input{part1/all_scales_figure} +\input{s_overview/text/all_scales_figure} %% CNHend \item finite volume techniques are employed yielding an intuitive @@ -72,7 +76,7 @@ orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes} %% CNHbegin -\input{part1/fvol_figure} +\input{s_overview/text/fvol_figure} %% CNHend \item tangent linear and adjoint counterparts are automatically maintained @@ -83,10 +87,12 @@ computational platforms. \end{itemize} -Key publications reporting on and charting the development of the model are: -\begin{verbatim} +Key publications reporting on and charting the development of the model are +\cite{hill:95,marshall:97a,marshall:97b,adcroft:97,mars-eta:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04} +(an overview on the model formulation can also be found in \cite{adcroft:04c}): +\begin{verbatim} Hill, C. and J. Marshall, (1995) Application of a Parallel Navier-Stokes Model to Ocean Circulation in Parallel Computational Fluid Dynamics @@ -95,7 +101,7 @@ Elsevier Science B.V.: New York Marshall, J., C. Hill, L. Perelman, and A. Adcroft, (1997) -Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling, +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) @@ -128,18 +134,17 @@ 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. -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.14 2001/11/21 14:13:17 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.29 2010/08/30 23:09:21 jmc 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 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 @@ -151,13 +156,18 @@ 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, using one basic algorithm, both atmospheric and oceanographic flows at both small and large scales. 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, @@ -167,7 +177,7 @@ there are no mountains or land-sea contrast. %% CNHbegin -\input{part1/cubic_eddies_figure} +\input{s_overview/text/cubic_eddies_figure} %% CNHend As described in Adcroft (2001), a `cubed sphere' is used to discretize the @@ -183,10 +193,16 @@ latitude-longitude grid. Both grids are supported within the model. %% CNHbegin -\input{part1/hs_zave_u_figure} +\input{s_overview/text/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 @@ -196,41 +212,48 @@ increased until the baroclinic instability process is resolved, numerical solutions of a different and much more realistic kind, can be obtained. -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 $lat-lon$ -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. +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} +\input{s_overview/text/atl6_figure} %% CNHend \subsection{Global ocean circulation} - -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 $lat-lon$ 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. +\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} +\input{s_overview/text/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 @@ -246,10 +269,13 @@ instability of the along-slope current. %%CNHbegin -\input{part1/convect_and_topo} +\input{s_overview/text/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 @@ -265,64 +291,80 @@ nonhydrostatic dynamics. %%CNHbegin -\input{part1/boundary_forced_waves} +\input{s_overview/text/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, 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 +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} +\input{s_overview/text/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 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 \ref{fig:assimilated-globes} shows an estimate of the time-mean -surface elevation of the ocean obtained by bringing the model in to +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. {\bf CHANGE THIS TEXT - FIG FROM PATRICK/CARL/DETLEF} +1992-1997. %% CNHbegin -\input{part1/assim_figure} +\input{s_overview/text/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. 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). +\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} +\input{s_overview/text/biogeo_figure} %%CNHend \subsection{Simulations of laboratory experiments} +\begin{rawhtml} + +\end{rawhtml} Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a -laboratory experiment inquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An +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 @@ -331,13 +373,16 @@ stratification of the ACC. %%CNHbegin -\input{part1/lab_figure} +\input{s_overview/text/lab_figure} %%CNHend -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.14 2001/11/21 14:13:17 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.29 2010/08/30 23:09:21 jmc 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 @@ -346,12 +391,12 @@ 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 (left hand side of figure \ref{fig:isomorphic-equations}) -and height, $z$, if we are modeling the ocean (right hand side of figure +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} +\input{s_overview/text/zandpcoord_figure.tex} %%CNHend The state of the fluid at any time is characterized by the distribution of @@ -365,14 +410,14 @@ see figure \ref{fig:zandp-vert-coord}. %%CNHbegin -\input{part1/vertcoord_figure.tex} +\input{s_overview/text/vertcoord_figure.tex} %%CNHend -\begin{equation*} +\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*} +\end{equation} \begin{equation} \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{ @@ -471,12 +516,12 @@ 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{ \ +\dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \ (ocean surface,bottom of the atmosphere)} \label{eq:movingbc} \end{equation} @@ -570,9 +615,11 @@ atmosphere)} \label{eq:moving-bc-atmos} \end{eqnarray} -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}). +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} @@ -614,6 +661,11 @@ \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and Non-hydrostatic forms} +\label{sec:all_hydrostatic_forms} +\begin{rawhtml} + +\end{rawhtml} + Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms: @@ -621,7 +673,9 @@ \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{eq:incompressible}) in the form: +%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 @@ -716,20 +770,21 @@ OPERATORS. %%CNHbegin -\input{part1/sphere_coord_figure.tex} +\input{s_overview/text/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} @@ -766,7 +821,7 @@ \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} @@ -836,7 +891,7 @@ stepping forward the vertical momentum equation. %%CNHbegin -\input{part1/solution_strategy_figure.tex} +\input{s_overview/text/solution_strategy_figure.tex} %%CNHend There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of @@ -1025,7 +1080,7 @@ 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} @@ -1051,8 +1106,9 @@ \subsection{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: +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} @@ -1073,7 +1129,7 @@ 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 1.14 2001/11/21 14:13:17 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.29 2010/08/30 23:09:21 jmc Exp $ % $Name: $ \section{Appendix ATMOSPHERE} @@ -1163,6 +1219,7 @@ 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 @@ -1192,7 +1249,8 @@ 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}}} \label{eq:atmos-prime} \\ +_{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 }{ \partial p} &=&0 \\ @@ -1200,7 +1258,7 @@ \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \end{eqnarray} -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.14 2001/11/21 14:13:17 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.29 2010/08/30 23:09:21 jmc Exp $ % $Name: $ \section{Appendix OCEAN} @@ -1238,8 +1296,9 @@ _{\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{ v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure} @@ -1416,7 +1475,7 @@ _{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 1.14 2001/11/21 14:13:17 cnh Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.29 2010/08/30 23:09:21 jmc Exp $ % $Name: $ \section{Appendix:OPERATORS} @@ -1433,9 +1492,8 @@ \end{equation*} \begin{equation*} -v=r\frac{D\varphi }{Dt}\qquad +v=r\frac{D\varphi }{Dt} \end{equation*} -$\qquad \qquad \qquad \qquad $ \begin{equation*} \dot{r}=\frac{Dr}{Dt} @@ -1445,7 +1503,7 @@ 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*} @@ -1455,7 +1513,7 @@ \end{equation*} \begin{equation*} -\nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial +\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*}