/[MITgcm]/manual/s_examples/baroclinic_gyre/fourlayer.tex

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-revision 1.5 by cnh,
Wed Oct 24 23:14:44 2001 UTC
+revision 1.12 by cnh,
Thu Feb 28 19:32:19 2002 UTC
 Line 1
  % $Header$
  % $Name$
- \section{Example: Four layer Baroclinic Ocean Gyre In Spherical Coordinates}
+ \section{Four Layer Baroclinic Ocean Gyre In Spherical Coordinates}
- \label{sec:eg-fourlayer}
+ \label{sect:eg-fourlayer}
  \bodytext{bgcolor="#FFFFFFFF"}
 Line 19
  This document describes an example experiment using MITgcm
  to simulate a baroclinic ocean gyre in spherical
  polar coordinates. The barotropic
- example experiment in section \ref{sec:eg-baro}
+ example experiment in section \ref{sect:eg-baro}
- ilustrated how to configure the code for a single layer
+ illustrated how to configure the code for a single layer
- simulation in a cartesian grid. In this example a similar physical problem
+ simulation in a Cartesian grid. In this example a similar physical problem
  is simulated, but the code is now configured
  for four layers and in a spherical polar coordinate system.
 Line 29 
 for four layers and in a spherical polar
  This example experiment demonstrates using the MITgcm to simulate
  a baroclinic, wind-forced, ocean gyre circulation. The experiment
- is a numerical rendition of the gyre circulation problem simliar
+ is a numerical rendition of the gyre circulation problem similar
  to the problems described analytically by Stommel in 1966
  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.
  \\
 Line 43 
 domain is a sector on a sphere and the c
  according to latitude, $\varphi$
  \begin{equation}
- \label{EQ:fcori}
+ \label{EQ:eg-fourlayer-fcori}
  f(\varphi) = 2 \Omega \sin( \varphi )
  \end{equation}
 Line 61 
 f(\varphi) = 2 \Omega \sin( \varphi )
  $\tau_0$ is set to $0.1N m^{-2}$.
  \\
- Figure \ref{FIG:simulation_config}
+ Figure \ref{FIG:eg-fourlayer-simulation_config}
- summarises the configuration simulated.
+ summarizes the configuration simulated.
- In contrast to the example in section \ref{sec:eg-baro}, the
+ In contrast to the example in section \ref{sect:eg-baro}, the
  current experiment simulates a spherical polar domain. As indicated
  by the axes in the lower left of the figure the model code works internally
- in a locally orthoganal coordinate $(x,y,z)$. For this experiment description
+ in a locally orthogonal coordinate $(x,y,z)$. For this experiment description
- of this document the local orthogonal model coordinate $(x,y,z)$ is synonomous
+ the local orthogonal model coordinate $(x,y,z)$ is synonymous
- with the spherical polar coordinate shown in figure
+ with the coordinates $(\lambda,\varphi,r)$ shown in figure
  \ref{fig:spherical-polar-coord}
  \\
 Line 82 
 $\theta_{1750}=6^{\circ}$~C. The equatio
  linear
  \begin{equation}
- \label{EQ:linear1_eos}
+ \label{EQ:eg-fourlayer-linear1_eos}
  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )
  \end{equation}
  \noindent which is implemented in the model as a density anomaly equation
  \begin{equation}
- \label{EQ:linear1_eos_pert}
+ \label{EQ:eg-fourlayer-linear1_eos_pert}
  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}
  \end{equation}
 Line 114 
 An initial stratification is
  imposed by setting the potential temperature, $\theta$, in each layer.
  The vertical spacing, $\Delta z$, is constant and equal to $500$m.
  }
- \label{FIG:simulation_config}
+ \label{FIG:eg-fourlayer-simulation_config}
  \end{figure}
  \subsection{Equations solved}
+ For this problem
- The implicit free surface {\bf HPE} form of the
+ the implicit free surface, {\bf HPE} (see section \ref{sect:hydrostatic_and_quasi-hydrostatic_forms}) form of the
- equations described in Marshall et. al \cite{Marshall97a} is
+ equations described in Marshall et. al \cite{marshall:97a} are
  employed. The flow is three-dimensional with just temperature, $\theta$, as
  an active tracer.  The equation of state is linear.
- A horizontal laplacian operator $\nabla_{h}^2$ provides viscous
+ A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous
  dissipation and provides a diffusive sub-grid scale closure for the
  temperature equation. A wind-stress momentum forcing is added to the momentum
  equation for the zonal flow, $u$. Other terms in the model
- are explicitly switched off for this experiement configuration (see section
+ are explicitly switched off for this experiment configuration (see section
  \ref{SEC:eg_fourl_code_config} ). This yields an active set of equations
  solved in this configuration, written in spherical polar coordinates as
  follows
  \begin{eqnarray}
- \label{EQ:model_equations}
+ \label{EQ:eg-fourlayer-model_equations}
  \frac{Du}{Dt} - fv +
    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -
    A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}
 Line 185 
 due to variations in sea-surface height,
  part due to variations in density, $\rho^{\prime}$, integrated
  through the water column.
- The suffices ${s},{i}$ indicate surface and interior of the domain.
+ The suffices ${s},{i}$ indicate surface layer and the interior of the domain.
  The windstress forcing, ${\cal F}_{\lambda}$, is applied in the surface layer
  by a source term in the zonal momentum equation. In the ocean interior
  this term is zero.
 Line 210 
 that there are sixty grid cells in the z
  Vertically the
  model is configured with four layers with constant depth,
  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate
- variables $x$ and $y$ are initialised from the values of
+ variables $x$ and $y$ are initialized from the values of
  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in
  radians according to
 Line 221 
 y=r\varphi,~\Delta y &= &r\Delta \varphi
  The procedure for generating a set of internal grid variables from a
  spherical polar grid specification is discussed in section
- \ref{sec:spatial_discrete_horizontal_grid}.
+ \ref{sect:spatial_discrete_horizontal_grid}.
  \noindent\fbox{ \begin{minipage}{5.5in}
  {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em
 Line 242 
 $\Delta x_v$, $\Delta y_u$: {\bf DXv}, {
- As described in \ref{sec:tracer_equations}, the time evolution of potential
+ As described in \ref{sect:tracer_equations}, the time evolution of potential
  temperature,
  $\theta$, (equation \ref{eq:eg_fourl_theta})
  is evaluated prognostically. The centered second-order scheme with
  Adams-Bashforth time stepping described in section
- \ref{sec:tracer_equations_abII} is used to step forward the temperature
+ \ref{sect:tracer_equations_abII} is used to step forward the temperature
- equation. The pressure forces that drive the fluid motions, (
+ equation. Prognostic terms in
+ the momentum equations are solved using flux form as
+ described in section \ref{sect:flux-form_momentum_eqautions}.
+ The pressure forces that drive the fluid motions, (
  $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface
  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the
- pressure is evaluated explicitly by integrating density. The sea-surface
+ pressure is diagnosed explicitly by integrating density. The sea-surface
- height, $\eta$, is solved for implicitly as described in section
+ height, $\eta$, is diagnosed using an implicit scheme. The pressure
- \ref{sect:pressure-method-linear-backward}.
+ field solution method is described in sections
+ \ref{sect:pressure-method-linear-backward} and
+ \ref{sect:finding_the_pressure_field}.
  \subsubsection{Numerical Stability Criteria}
- The laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.
+ The Laplacian viscosity coefficient, $A_{h}$, is set to $400 m s^{-1}$.
- This value is chosen to yield a Munk layer width \cite{Adcroft_thesis},
+ This value is chosen to yield a Munk layer width,
  \begin{eqnarray}
- \label{EQ:munk_layer}
+ \label{EQ:eg-fourlayer-munk_layer}
  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
  \end{eqnarray}
  \noindent  of $\approx 100$km. This is greater than the model
- resolution in mid-latitudes $\Delta x$, ensuring that the frictional
+ resolution in mid-latitudes
+ $\Delta x=r \cos(\varphi) \Delta \lambda \approx 80~{\rm km}$ at
+ $\varphi=45^{\circ}$, ensuring that the frictional
  boundary layer is well resolved.
  \\
  \noindent The model is stepped forward with a
  time step $\delta t=1200$secs. With this time step the stability
- parameter to the horizontal laplacian friction \cite{Adcroft_thesis}
+ parameter to the horizontal Laplacian friction
  \begin{eqnarray}
- \label{EQ:laplacian_stability}
+ \label{EQ:eg-fourlayer-laplacian_stability}
  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
  \end{eqnarray}
  \noindent evaluates to 0.012, which is well below the 0.3 upper limit
- for stability.
+ for stability for this term under ABII time-stepping.
  \\
  \noindent The vertical dissipation coefficient, $A_{z}$, is set to
  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
  \begin{eqnarray}
- \label{EQ:laplacian_stability_z}
+ \label{EQ:eg-fourlayer-laplacian_stability_z}
  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}
  \end{eqnarray}
-Line 298 
 and vertical ($K_{z}$) diffusion coeffic
+Line 305 
 and vertical ($K_{z}$) diffusion coeffic
  \\
  \noindent The numerical stability for inertial oscillations
- \cite{Adcroft_thesis}
  \begin{eqnarray}
- \label{EQ:inertial_stability}
+ \label{EQ:eg-fourlayer-inertial_stability}
  S_{i} = f^{2} {\delta t}^2
  \end{eqnarray}
-Line 309 
 S_{i} = f^{2} {\delta t}^2
+Line 315 
 S_{i} = f^{2} {\delta t}^2
  limit for stability.
  \\
- \noindent The advective CFL \cite{Adcroft_thesis} for a extreme maximum
+ \noindent The advective CFL for a extreme maximum
  horizontal flow
  speed of $ | \vec{u} | = 2 ms^{-1}$
  \begin{eqnarray}
- \label{EQ:cfl_stability}
+ \label{EQ:eg-fourlayer-cfl_stability}
- S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
+ C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
  \end{eqnarray}
  \noindent evaluates to $5 \times 10^{-2}$. This is well below the stability
  limit of 0.5.
  \\
  \noindent The stability parameter for internal gravity waves
- \cite{Adcroft_thesis}
+ propagating at $2~{\rm m}~{\rm s}^{-1}$
  \begin{eqnarray}
- \label{EQ:igw_stability}
+ \label{EQ:eg-fourlayer-igw_stability}
  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
  \end{eqnarray}
- \noindent evaluates to $5 \times 10^{-2}$. This is well below the linear
+ \noindent evaluates to $\approx 5 \times 10^{-2}$. This is well below the linear
  stability limit of 0.25.
  \subsection{Code Configuration}
  \label{SEC:eg_fourl_code_config}
  The model configuration for this experiment resides under the
- directory {\it verification/exp1/}.  The experiment files
+ directory {\it verification/exp2/}.  The experiment files
  \begin{itemize}
  \item {\it input/data}
  \item {\it input/data.pkg}
-Line 349 
 directory {\it verification/exp1/}.  The
+Line 355 
 directory {\it verification/exp1/}.  The
  \item {\it code/SIZE.h}.
  \end{itemize}
  contain the code customisations and parameter settings for this
- experiements. Below we describe the customisations
+ experiments. Below we describe the customisations
  to these files associated with this experiment.
  \subsubsection{File {\it input/data}}
-Line 366 
 this line sets
+Line 372 
 this line sets
  the initial and reference values of potential temperature at each model
  level in units of $^{\circ}$C.
  The entries are ordered from surface to depth. For each
- depth level the inital and reference profiles will be uniform in
+ depth level the initial and reference profiles will be uniform in
  $x$ and $y$. The values specified here are read into the
  variable
  {\bf
-Line 412 
 goto code
+Line 418 
 goto code
  \item Line 6,
  \begin{verbatim} viscAz=1.E-2, \end{verbatim}
- this line sets the vertical laplacian dissipation coefficient to
+ this line sets the vertical Laplacian dissipation coefficient to
  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions
  for this operator are specified later.
  The variable
-Line 432 
 and is copied into model general vertica
+Line 438 
 and is copied into model general vertica
  \begin{rawhtml} <A href=../../../code_reference/vdb/names/PF.htm> \end{rawhtml}
  viscAr
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ }. At each time step, the viscous term contribution to the momentum equations
+ is calculated in routine
+ {\it S/R CALC\_DIFFUSIVITY}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 463 
 is read in the routine
+Line 471 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } and applied in routines {\it CALC\_MOM\_RHS} and {\it CALC\_GW}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 506 
 is read in the routine
+Line 514 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } and the boundary condition is evaluated in routine
+ {\it S/R CALC\_MOM\_RHS}.
  \fbox{
-Line 538 
 is read in the routine
+Line 547 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } and is applied in the routine {\it S/R CALC\_MOM\_RHS}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 570 
 is read in the routine
+Line 579 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } and used in routine {\it S/R CALC\_GT}.
  \fbox{ \begin{minipage}{5.0in}
  {\it S/R CALC\_GT}({\it calc\_gt.F})
-Line 606 
 It is copied into model general vertical
+Line 615 
 It is copied into model general vertical
  \begin{rawhtml} <A href=../../../code_reference/vdb/names/PD.htm> \end{rawhtml}
  diffKrT
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } which is used in routine {\it S/R CALC\_DIFFUSIVITY}.
  \fbox{ \begin{minipage}{5.0in}
  {\it S/R CALC\_DIFFUSIVITY}({\it calc\_diffusivity.F})
-Line 637 
 is read in the routine
+Line 646 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ }. The routine {\it S/R FIND\_RHO} makes use of {\bf tAlpha}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 666 
 is read in the routine
+Line 675 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ }. The values of {\bf eosType} sets which formula in routine
+ {\it FIND\_RHO} is used to calculate density.
  \fbox{
  \begin{minipage}{5.0in}
-Line 687 
 usingSphericalPolarGrid=.TRUE.,
+Line 697 
 usingSphericalPolarGrid=.TRUE.,
  \end{verbatim}
  This line requests that the simulation be performed in a
  spherical polar coordinate system. It affects the interpretation of
- grid inoput parameters, for exampl {\bf delX} and {\bf delY} and
+ grid input parameters, for example {\bf delX} and {\bf delY} and
- causes the grid generation routines to initialise an internal grid based
+ causes the grid generation routines to initialize an internal grid based
  on spherical polar geometry.
  The variable
  {\bf
-Line 701 
 is read in the routine
+Line 711 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ }. When set to {\bf .TRUE.} the settings of {\bf delX} and {\bf delY} are
+ taken to be in degrees. These values are used in the
+ routine {\it INI\_SPEHRICAL\_POLAR\_GRID}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 721 
 phiMin=0.,
+Line 733 
 phiMin=0.,
  This line sets the southern boundary of the modeled
  domain to $0^{\circ}$ latitude. This value affects both the
  generation of the locally orthogonal grid that the model
- uses internally and affects the initialisation of the coriolis force.
+ uses internally and affects the initialization of the coriolis force.
  Note - it is not required to set
  a longitude boundary, since the absolute longitude does
  not alter the kernel equation discretisation.
-Line 736 
 is read in the routine
+Line 748 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } and is used in routine {\it INI\_SPEHRICAL\_POLAR\_GRID}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 766 
 is read in the routine
+Line 778 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } and is used in routine {\it INI\_SPEHRICAL\_POLAR\_GRID}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 796 
 is read in the routine
+Line 808 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } and is used in routine {\it INI\_SPEHRICAL\_POLAR\_GRID}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 834 
 model coordinate variable
+Line 846 
 model coordinate variable
  \begin{rawhtml} <A href=../../../code_reference/vdb/names/10Y.htm> \end{rawhtml}
  delR
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ } which is used in routine {\it INI\_VERTICAL\_GRID}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 873 
 is read in the routine
+Line 885 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ }. The bathymetry file is read in the routine {\it INI\_DEPTHS}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 892 
 goto code
+Line 904 
 goto code
  zonalWindFile='windx.sin_y'
  \end{verbatim}
  This line specifies the name of the file from which the x-direction
- surface wind stress is read. This file is also a two-dimensional
+ (zonal) surface wind stress is read. This file is also a two-dimensional
  ($x,y$) map and is enumerated and formatted in the same manner as the
  bathymetry file. The matlab program {\it input/gendata.m} includes example
  code to generate a valid
-Line 909 
 is read in the routine
+Line 921 
 is read in the routine
  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
  INI\_PARMS
  \begin{rawhtml} </A>\end{rawhtml}
- }.
+ }.  The wind-stress file is read in the routine
+ {\it EXTERNAL\_FIELDS\_LOAD}.
  \fbox{
  \begin{minipage}{5.0in}
-Line 924 
 goto code
+Line 937 
 goto code
  \end{itemize}
- \noindent other lines in the file {\it input/data} are standard values
+ \noindent other lines in the file {\it input/data} are standard values.
- that are described in the MITgcm Getting Started and MITgcm Parameters
- notes.
  \begin{rawhtml}<PRE>\end{rawhtml}
  \begin{small}
-Line 947 
 customisations for this experiment.
+Line 958 
 customisations for this experiment.
  \subsubsection{File {\it input/windx.sin\_y}}
  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
- map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.
+ map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$ (the
- Although $\tau_{x}$ is only a function of $y$n in this experiment
+ default for MITgcm).
+ Although $\tau_{x}$ is only a function of latitude, $y$,
+ in this experiment
  this file must still define a complete two-dimensional map in order
  to be compatible with the standard code for loading forcing fields
- in MITgcm. The included matlab program {\it input/gendata.m} gives a complete
+ in MITgcm (routine {\it EXTERNAL\_FIELDS\_LOAD}.
+ The included matlab program {\it input/gendata.m} gives a complete
  code for creating the {\it input/windx.sin\_y} file.
  \subsubsection{File {\it input/topog.box}}
-Line 959 
 code for creating the {\it input/windx.s
+Line 973 
 code for creating the {\it input/windx.s
  The {\it input/topog.box} file specifies a two-dimensional ($x,y$)
  map of depth values. For this experiment values are either
- $0m$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep
+ $0~{\rm m}$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep
  ocean. The file contains a raw binary stream of data that is enumerated
  in the same way as standard MITgcm two-dimensional, horizontal arrays.
  The included matlab program {\it input/gendata.m} gives a complete
-Line 1020 
 dxF, dyF, dxG, dyG, dxC, dyC}.
+Line 1034 
 dxF, dyF, dxG, dyG, dxC, dyC}.
  \subsubsection{Code Download}
   In order to run the examples you must first download the code distribution.
- Instructions for downloading the code can be found in the Getting Started
+ Instructions for downloading the code can be found in section
- Guide \cite{MITgcm_Getting_Started}.
+ \ref{sect:obtainingCode}.
  \subsubsection{Experiment Location}
   This example experiments is located under the release sub-directory
  \vspace{5mm}
- {\it verification/exp1/ }
+ {\it verification/exp2/ }
  \subsubsection{Running the Experiment}
-Line 1047 
 Guide \cite{MITgcm_Getting_Started}.
+Line 1061 
 Guide \cite{MITgcm_Getting_Started}.
  % pwd
  \end{verbatim}
-  You shold see a response on the screen ending in
+  You should see a response on the screen ending in
- {\it verification/exp1/input }
+ {\it verification/exp2/input }
  \item Run the genmake script to create the experiment {\it Makefile}

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