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-revision 1.12 by cnh,
Mon Nov 19 14:33:44 2001 UTC
+revision 1.26 by edhill,
Wed Jun 28 15:22:13 2006 UTC
 Line 37
  % $Header$
  % $Name$
- \section{Introduction}
+ This document provides the reader with the information necessary to
- This documentation 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
-Line 49 
 are available. A number of examples illu
+Line 47 
 are available. A number of examples illu
  both process and general circulation studies of the atmosphere and ocean are
  also presented.
+ \section{Introduction}
+ \begin{rawhtml}
+ <!-- CMIREDIR:innovations: -->
+ \end{rawhtml}
  MITgcm has a number of novel aspects:
  \begin{itemize}
-Line 83 
 studies.
+Line 87 
 studies.
  computational platforms.
  \end{itemize}
- Key publications reporting on and charting the development of the model are:
+ 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,adcroft:04a,adcroft:04b,marshall:04}:
  \begin{verbatim}
  Hill, C. and J. Marshall, (1995)
  Application of a Parallel Navier-Stokes Model to Ocean Circulation in
  Parallel Computational Fluid Dynamics
-Line 95 
 and Results Using Parallel Computers, 54
+Line 99 
 and Results Using Parallel Computers, 54
  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)
-Line 119 
 Mon. Wea. Rev. Vol 127, pages 1928-1936
+Line 123 
 Mon. Wea. Rev. Vol 127, pages 1928-1936
  Hill, C, Adcroft,A., Jamous,D., and J. Marshall, (1999)
  A Strategy for Terascale Climate Modeling.
- In Proceedings of the Eight ECMWF Workshop on the Use of Parallel Processors
+ In Proceedings of the Eighth ECMWF Workshop on the Use of Parallel Processors
- in Meteorology
+ in Meteorology, pages 406-425
+ World Scientific Publishing Co: UK
  Marotzke, J, Giering,R., Zhang, K.Q., Stammer,D., Hill,C., and T.Lee, (1999)
  Construction of the adjoint MIT ocean general circulation model and
  application to Atlantic heat transport variability
  J. Geophysical Res., 104(C12), 29,529-29,547.
  \end{verbatim}
  We begin by briefly showing some of the results of the model in action to
-Line 138 
 give a feel for the wide range of proble
+Line 142 
 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 154 
 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 186 
 latitude-longitude grid. Both grids are
+Line 195 
 latitude-longitude grid. Both grids are
  %% 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 210 
 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}
-Line 212 
 visible.
+Line 227 
 visible.
  \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.
-Line 230 
 circulation of the global ocean in Sverd
+Line 248 
 circulation of the global ocean in Sverd
  %%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 249 
 instability of the along-slope current.
+Line 271 
 instability of the along-slope current.
  %%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 268 
 nonhydrostatic dynamics.
+Line 293 
 nonhydrostatic dynamics.
  %%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
-Line 288 
 yields sensitivities to all other model
+Line 316 
 yields sensitivities to all other model
  %%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
  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
+ etc. Figure \ref{fig:assimilated-globes} shows the large scale planetary
- surface elevation of the ocean obtained by bringing the model in to
+ 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
--1997. {\bf CHANGE THIS TEXT - FIG FROM PATRICK/CARL/DETLEF}
+-1997.
  %% CNHbegin
- \input{part1/globes_figure}
+ \input{part1/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}
  %%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 337 
 stratification of the ACC.
+Line 378 
 stratification of the ACC.
  % $Name$
  \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 345 
 One system of hydrodynamical equations i
+Line 389 
 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
-Line 367 
 see figure \ref{fig:zandp-vert-coord}.
+Line 411 
 see figure \ref{fig:zandp-vert-coord}.
  \input{part1/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 470 
 in later chapters.
+Line 514 
 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 569 
 The boundary conditions at top and botto
+Line 613 
 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 613 
 which, for convenience, are written out
+Line 659 
 which, for convenience, are written out
  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
  Non-hydrostatic forms}
+ \begin{rawhtml}
+ <!-- CMIREDIR:non_hydrostatic: -->
+ \end{rawhtml}
  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
-Line 620 
 Let us separate $\phi $ in to surface, h
+Line 670 
 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 720 
 OPERATORS.
+Line 772 
 OPERATORS.
  \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 765 
 et.al., 1997a. As in \textbf{HPE }only a
+Line 818 
 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 1024 
 friction. These coefficients are the sam
+Line 1077 
 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 1050 
 salinity ... ).
+Line 1103 
 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 1162 
 In $p$-coordinates, the upper boundary a
+Line 1216 
 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 1191 
 _{o}(p_{o})=g~Z_{topo}$, defined:
+Line 1246 
 _{o}(p_{o})=g~Z_{topo}$, 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 1237 
 _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
+Line 1293 
 _{\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 1432 
 u=r\cos \varphi \frac{D\lambda }{Dt}
+Line 1489 
 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 1444 
 Here $\varphi $ is the latitude, $\lambd
+Line 1500 
 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 1454 
 spherical coordinates:
+Line 1510 
 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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