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

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-revision 1.7 by cnh,
Thu Oct 25 12:06:56 2001 UTC
+revision 1.11 by cnh,
Fri Nov 16 21:18:36 2001 UTC
 Line 102 
 of them here. A more detailed descriptio
  numerical algorithm and implementation that lie behind these calculations is
  given later. Indeed many of the illustrative examples shown below can be
  easily reproduced: simply download the model (the minimum you need is a PC
- running linux, together with a FORTRAN\ 77 compiler) and follow the examples
+ running Linux, together with a FORTRAN\ 77 compiler) and follow the examples
  described in detail in the documentation.
  \subsection{Global atmosphere: `Held-Suarez' benchmark}
 Line 126 
 there are no mountains or land-sea contr
  %% CNHend
  As described in Adcroft (2001), a `cubed sphere' is used to discretize the
- globe permitting a uniform gridding and obviated the need to Fourier filter.
+ globe permitting a uniform griding and obviated the need to Fourier filter.
  The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
  grid, of which the cubed sphere is just one of many choices.
 Line 163 
 warm water northward by the mean flow of
  visible.
  %% CNHbegin
- \input{part1/ocean_gyres_figure}
+ \input{part1/atl6_figure}
  %% CNHend
 Line 191 
 Dense plumes generated by localized cool
  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 \ref{fig::convect-and-topo}
+ eddies. The simulation shown in the figure \ref{fig:convect-and-topo}
  (blue is cold dense fluid, red is
  warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
  trigger convection by surface cooling. The cold, dense water falls down the
 Line 231 
 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 streamfunction shown in figure \ref{fig:large-scale-circ}
+ 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
 Line 248 
 yields sensitivities to all other model
  An important application of MITgcm is in state estimation of the global
  ocean circulation. An appropriately defined `cost function', which measures
  the departure of the model from observations (both remotely sensed and
- insitu) over an interval of time, is minimized by adjusting `control
+ in-situ) over an interval of time, is minimized by adjusting `control
  parameters' such as air-sea fluxes, the wind field, the initial conditions
  etc. Figure \ref{fig:assimilated-globes} shows an estimate of the time-mean
  surface elevation of the ocean obtained by bringing the model in to
 Line 277 
 telescoping to $\frac{1}{3}^{\circ}\time
  \subsection{Simulations of laboratory experiments}
  Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
- laboratory experiment enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
+ laboratory experiment inquiring 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
 Line 326 
 see figure \ref{fig:zandp-vert-coord}.
  \begin{equation*}
  \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}
  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
- \text{ horizontal mtm}
+ \text{ horizontal mtm} \label{eq:horizontal_mtm}
  \end{equation*}
- \begin{equation*}
+ \begin{equation}
  \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
- vertical mtm}
+ vertical mtm} \label{eq:vertical_mtm}
- \end{equation*}
+ \end{equation}
  \begin{equation}
  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
- \partial r}=0\text{ continuity}  \label{eq:continuous}
+ \partial r}=0\text{ continuity}  \label{eq:continuity}
  \end{equation}
- \begin{equation*}
+ \begin{equation}
- b=b(\theta ,S,r)\text{ equation of state}
+ b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
- \end{equation*}
+ \end{equation}
- \begin{equation*}
+ \begin{equation}
  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
- \end{equation*}
+ \label{eq:potential_temperature}
+ \end{equation}
- \begin{equation*}
+ \begin{equation}
  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
- \end{equation*}
+ \label{eq:humidity_salt}
+ \end{equation}
  Here:
-Line 430 
 at fixed and moving $r$ surfaces we set
+Line 432 
 at fixed and moving $r$ surfaces we set
  \begin{equation}
  \dot{r}=\frac{Dr}{Dt}atr=R_{moving}\text{ \
- (oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}
+ (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
  \end{equation}
  Here
-Line 523 
 The boundary conditions at top and botto
+Line 525 
 The boundary conditions at top and botto
  atmosphere)}  \label{eq:moving-bc-atmos}
  \end{eqnarray}
- Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent
+ Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt})
- set of atmospheric equations which, for convenience, are written out in $p$
+ 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}
-Line 560 
 w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo
+Line 562 
 w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo
  \end{eqnarray}
  where $\eta $ is the elevation of the free surface.
- Then eq(\ref{eq:continuous}) yields a consistent set of oceanic equations
+ Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yield a consistent set
+ of oceanic equations
  which, for convenience, are written out in $z$ coordinates in Appendix Ocean
  - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
-Line 573 
 Let us separate $\phi $ in to surface, h
+Line 576 
 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{incompressible}a,b) in the form:
+ and write eq(\ref{eq:incompressible}) in the form:
  \begin{equation}
  \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
-Line 666 
 In the above `${r}$' is the distance fro
+Line 669 
 In the above `${r}$' is the distance fro
  Grad and div operators in spherical coordinates are defined in appendix
  OPERATORS.
- \marginpar{
- Fig.6 Spherical polar coordinate system.}
  %%CNHbegin
  \input{part1/sphere_coord_figure.tex}
-Line 730 
 In the non-hydrostatic ocean model all t
+Line 731 
 In the non-hydrostatic ocean model all t
  three dimensional elliptic equation must be solved subject to Neumann
  boundary conditions (see below). It is important to note that use of the
  full \textbf{NH} does not admit any new `fast' waves in to the system - the
- incompressible condition eq(\ref{eq:continuous})c has already filtered out
+ incompressible condition eq(\ref{eq:continuity}) has already filtered out
  acoustic modes. It does, however, ensure that the gravity waves are treated
  accurately with an exact dispersion relation. The \textbf{NH} set has a
  complete angular momentum principle and consistent energetics - see White
-Line 779 
 coordinates are supported - see eqs(\ref
+Line 780 
 coordinates are supported - see eqs(\ref
  \subsection{Solution strategy}
  The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
- NH} models is summarized in Fig.7.
+ NH} models is summarized in Figure \ref{fig:solution-strategy}.
- \marginpar{
+ Under all dynamics, a 2-d elliptic equation is
- 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
  any level computed from the weight of fluid above. Under \textbf{HPE} and
  \textbf{QH} dynamics, the horizontal momentum equations are then stepped
 Line 838 
 atmospheric pressure pushing down on the
  \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 $r=R_{moving}$
+ (\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
  \begin{equation*}
  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
 Line 864 
 r $. The above can be rearranged to yiel
  where we have incorporated a source term.
  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
- (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
+ (atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
  be written
  \begin{equation}
  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
 Line 872 
 be written
  \end{equation}
  where $b_{s}$ is the buoyancy at the surface.
- In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref
+ In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref
  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
  surface' and `rigid lid' approaches are available.
  \subsubsection{Non-hydrostatic pressure}
- Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
+ Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
- \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
+ $\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
- (\ref{incompressible}), we deduce that:
+ (\ref{eq:continuity}), we deduce that:
  \begin{equation}
  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
 Line 911 
 tangential component of velocity, $v_{T}
  depending on the form chosen for the dissipative terms in the momentum
  equations - see below.
- Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that:
+ Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
  \begin{equation}
  \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
 Line 951 
 If the flow is `close' to hydrostatic ba
  converges rapidly because $\phi _{nh}\ $is then only a small correction to
  the hydrostatic pressure field (see the discussion in Marshall et al, a,b).
- The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{homneuman})
+ The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh})
  does not vanish at $r=R_{moving}$, and so refines the pressure there.
  \subsection{Forcing/dissipation}
 Line 959 
 does not vanish at $r=R_{moving}$, and s
  \subsubsection{Forcing}
  The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by
- `physics packages' described in detail in chapter ??.
+ `physics packages' and forcing packages. These are described later on.
  \subsubsection{Dissipation}
 Line 1007 
 salinity ... ).
  \subsection{Vector invariant form}
  For some purposes it is advantageous to write momentum advection in eq(\ref
- {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
+ {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 1025 
 to discretize the model.
  \subsection{Adjoint}
- Tangent linear and adjoint counterparts of the forward model and described
+ Tangent linear and adjoint counterparts of the forward model are described
  in Chapter 5.
  % $Header$
 Line 1147 
 _{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}}} \\
+ _{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 \\
  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
- \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}
+ \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }
  \end{eqnarray}
  % $Header$
 Line 1171 
 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}
  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
  &=&\epsilon _{nh}\mathcal{F}_{w} \\
  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
- _{h}+\frac{\partial w}{\partial z} &=&0 \\
+ _{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
- \rho &=&\rho (\theta ,S,p) \\
+ \rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
- \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
+ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
- \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
+ \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zns-salt}
+ \label{eq:non-boussinesq}
  \end{eqnarray}
  These equations permit acoustics modes, inertia-gravity waves,
- non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
+ non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
  mode. As written, they cannot be integrated forward consistently - if we
  step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
-Line 1193 
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
+Line 1194 
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
  \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:
- {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}

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