/[MITgcm]/manual/s_overview/text/manual.tex

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-revision 1.16 by cnh,
Thu Feb 28 19:32:19 2002 UTC
+revision 1.29 by jmc,
Mon Aug 30 23:09:21 2010 UTC
 Line 48 
 both process and general circulation stu
  also presented.
  \section{Introduction}
+ \begin{rawhtml}
+ <!-- CMIREDIR:innovations: -->
+ \end{rawhtml}
  MITgcm has a number of novel aspects:
-Line 57 
 hydrodynamical kernel is used to drive f
+Line 61 
 hydrodynamical kernel is used to drive f
  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
-Line 72 
 discretization and support for the treat
+Line 76 
 discretization and support for the treat
  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
-Line 83 
 studies.
+Line 87 
 studies.
  computational platforms.
  \end{itemize}
  Key publications reporting on and charting the development of the model are
- \cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99}:
+ \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)
-Line 138 
 give a feel for the wide range of proble
+Line 144 
 give a feel for the wide range of proble
  \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
-Line 150 
 running Linux, together with a FORTRAN\
+Line 156 
 running Linux, together with a FORTRAN\
  described in detail in the documentation.
  \subsection{Global atmosphere: `Held-Suarez' benchmark}
+ \begin{rawhtml}
+ <!-- CMIREDIR:atmospheric_example: -->
+ \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
-.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,
-Line 166 
 in Held and Suarez; 1994 designed to tes
+Line 177 
 in Held and Suarez; 1994 designed to tes
  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
-Line 182 
 cube-sphere grid and the flow calculated
+Line 193 
 cube-sphere grid and the flow calculated
  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}
+ <!-- CMIREDIR:oceanic_example: -->
+ \end{rawhtml}
+ \begin{rawhtml}
+ <!-- CMIREDIR:ocean_gyres: -->
+ \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
-Line 195 
 diffusive patterns of ocean currents. Bu
+Line 212 
 diffusive patterns of ocean currents. Bu
  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
+ Figure \ref{fig:ocean-gyres} shows the surface temperature and
- field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal
+ velocity field obtained from MITgcm run at $\frac{1}{6}^{\circ }$
- resolution on a $lat-lon$
+ horizontal resolution on a \textit{lat-lon} grid in which the pole has
- grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
+ been rotated by $90^{\circ }$ on to the equator (to avoid the
- (to avoid the converging of meridian in northern latitudes). 21 vertical
+ converging of meridian in northern latitudes). 21 vertical levels are
- levels are used in the vertical with a `lopped cell' representation of
+ used in the vertical with a `lopped cell' representation of
- topography. The development and propagation of anomalously warm and cold
+ topography. The development and propagation of anomalously warm and
- eddies can be clearly seen in the Gulf Stream region. The transport of
+ cold eddies can be clearly seen in the Gulf Stream region. The
- warm water northward by the mean flow of the Gulf Stream is also clearly
+ transport of warm water northward by the mean flow of the Gulf Stream
- visible.
+ is also clearly visible.
  %% CNHbegin
- \input{part1/atl6_figure}
+ \input{s_overview/text/atl6_figure}
  %% CNHend
  \subsection{Global ocean circulation}
+ \begin{rawhtml}
- Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
+ <!-- CMIREDIR:global_ocean_circulation: -->
- the surface of a 4$^{\circ }$
+ \end{rawhtml}
- 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
+ Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean
- }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
+ currents at the surface of a $4^{\circ }$ global ocean model run with
- mixed boundary conditions on temperature and salinity at the surface. The
+vertical levels. Lopped cells are used to represent topography on a
- transfer properties of ocean eddies, convection and mixing is parameterized
+ regular \textit{lat-lon} grid extending from $70^{\circ }N$ to
- in this model.
+ $70^{\circ }S$. 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}
+ <!-- CMIREDIR:mixing_over_topography: -->
+ \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
-Line 245 
 strong, and replaced by lateral entrainm
+Line 269 
 strong, and replaced by lateral entrainm
  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}
+ <!-- CMIREDIR:boundary_forced_internal_waves: -->
+ \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
-Line 264 
 using MITgcm's finite volume spatial dis
+Line 291 
 using MITgcm's finite volume spatial dis
  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}
+ <!-- CMIREDIR:parameter_sensitivity: -->
+ \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}
+ As one example of application of the MITgcm adjoint, Figure
- maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+ \ref{fig:hf-sensitivity} maps the gradient $\frac{\partial J}{\partial
- of the overturning stream-function shown in figure \ref{fig:large-scale-circ}
+   \mathcal{H}}$where $J$ is the magnitude of the overturning
- at 60$^{\circ }$N and $
+ stream-function shown in figure \ref{fig:large-scale-circ} at
- \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
+ $60^{\circ }N$ and $ \mathcal{H}(\lambda,\varphi)$ is the mean, local
- a 100 year period. We see that $J$ is
+ air-sea heat flux over a 100 year period. We see that $J$ is sensitive
- sensitive to heat fluxes over the Labrador Sea, one of the important sources
+ to heat fluxes over the Labrador Sea, one of the important sources of
- of deep water for the thermohaline circulations. This calculation also
+ 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}
+ <!-- CMIREDIR:global_state_estimation: -->
+ \end{rawhtml}
  An important application of MITgcm is in state estimation of the global
  ocean circulation. An appropriately defined `cost function', which measures
-Line 301 
 consistency with altimetric and in-situ
+Line 335 
 consistency with altimetric and in-situ
 -1997.
  %% CNHbegin
- \input{part1/assim_figure}
+ \input{s_overview/text/assim_figure}
  %% CNHend
  \subsection{Ocean biogeochemical cycles}
+ \begin{rawhtml}
- MITgcm is being used to study global biogeochemical cycles in the ocean. For
+ <!-- CMIREDIR:ocean_biogeo_cycles: -->
- example one can study the effects of interannual changes in meteorological
+ \end{rawhtml}
- forcing and upper ocean circulation on the fluxes of carbon dioxide and
- oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows
+ MITgcm is being used to study global biogeochemical cycles in the
- the annual air-sea flux of oxygen and its relation to density outcrops in
+ ocean. For example one can study the effects of interannual changes in
- the southern oceans from a single year of a global, interannually varying
+ meteorological forcing and upper ocean circulation on the fluxes of
- simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution
+ carbon dioxide and oxygen between the ocean and atmosphere. Figure
- telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).
+ \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}
+ <!-- CMIREDIR:classroom_exp: -->
+ \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
-Line 331 
 arrested by its instability in a process
+Line 373 
 arrested by its instability in a process
  stratification of the ACC.
  %%CNHbegin
- \input{part1/lab_figure}
+ \input{s_overview/text/lab_figure}
  %%CNHend
  % $Header$
  % $Name$
  \section{Continuous equations in `r' coordinates}
+ \begin{rawhtml}
+ <!-- CMIREDIR:z-p_isomorphism: -->
+ \end{rawhtml}
  To render atmosphere and ocean models from one dynamical core we exploit
  `isomorphisms' between equation sets that govern the evolution of the
-Line 346 
 One system of hydrodynamical equations i
+Line 391 
 One system of hydrodynamical equations i
  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})
+ modeling the atmosphere (right hand side of figure \ref{fig:isomorphic-equations})
- and height, $z$, if we are modeling the ocean (right hand side of figure
+ 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
-Line 365 
 kinematic boundary conditions can be app
+Line 410 
 kinematic boundary conditions can be app
  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{
-Line 471 
 in later chapters.
+Line 516 
 in later chapters.
  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}
-Line 570 
 The boundary conditions at top and botto
+Line 615 
 The boundary conditions at top and botto
  atmosphere)}  \label{eq:moving-bc-atmos}
  \end{eqnarray}
- Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt})
+ Then the (hydrostatic form of) equations
- yields a consistent set of atmospheric equations which, for convenience, are written out in $p$
+ (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yields a consistent
- coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).
+ set of atmospheric equations which, for convenience, are written out
+ in $p$ coordinates in Appendix Atmosphere - see
+ eqs(\ref{eq:atmos-prime}).
  \subsection{Ocean}
-Line 614 
 which, for convenience, are written out
+Line 661 
 which, for convenience, are written out
  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
  Non-hydrostatic forms}
+ \label{sec:all_hydrostatic_forms}
+ \begin{rawhtml}
+ <!-- CMIREDIR:non_hydrostatic: -->
+ \end{rawhtml}
  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
-Line 621 
 Let us separate $\phi $ in to surface, h
+Line 673 
 Let us separate $\phi $ in to surface, h
  \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
-Line 716 
 Grad and div operators in spherical coor
+Line 770 
 Grad and div operators in spherical coor
  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
+ Most models are based on the `hydrostatic primitive equations' (HPE's)
- which the vertical momentum equation is reduced to a statement of
+ in which the vertical momentum equation is reduced to a statement of
- hydrostatic balance and the `traditional approximation' is made in which the
+ hydrostatic balance and the `traditional approximation' is made in
- Coriolis force is treated approximately and the shallow atmosphere
+ which the Coriolis force is treated approximately and the shallow
- approximation is made.\ The MITgcm need not make the `traditional
+ atmosphere approximation is made.  MITgcm need not make the
- approximation'. To be able to support consistent non-hydrostatic forms the
+ `traditional approximation'. To be able to support consistent
- shallow atmosphere approximation can be relaxed - when dividing through by $
+ non-hydrostatic forms the shallow atmosphere approximation can be
- r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
+ relaxed - when dividing through by $ r $ in, for example,
- the radius of the earth.
+ (\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}
-Line 766 
 et.al., 1997a. As in \textbf{HPE }only a
+Line 821 
 et.al., 1997a. As in \textbf{HPE }only a
  \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}
-Line 836 
 stepping forward the horizontal momentum
+Line 891 
 stepping forward the horizontal momentum
  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
-Line 1025 
 friction. These coefficients are the sam
+Line 1080 
 friction. These coefficients are the sam
  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}
-Line 1051 
 salinity ... ).
+Line 1106 
 salinity ... ).
  \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:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the (so-called) `vector invariant' form:
+ 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}
-Line 1163 
 In $p$-coordinates, the upper boundary a
+Line 1219 
 In $p$-coordinates, the upper boundary a
  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
-Line 1192 
 _{o}(p_{o})=g~Z_{topo}$, defined:
+Line 1249 
 _{o}(p_{o})=g~Z_{topo}$, defined:
  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 \\
-Line 1238 
 _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
+Line 1296 
 _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}
  \end{equation}
- Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
+ Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is
- reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref{eq-zns-cont} gives:
+ 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}
-Line 1433 
 u=r\cos \varphi \frac{D\lambda }{Dt}
+Line 1492 
 u=r\cos \varphi \frac{D\lambda }{Dt}
  \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}
-Line 1445 
 Here $\varphi $ is the latitude, $\lambd
+Line 1503 
 Here $\varphi $ is the latitude, $\lambd
  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*}
-Line 1455 
 spherical coordinates:
+Line 1513 
 spherical coordinates:
  \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*}

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