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--- manual/s_examples/baroclinic_gyre/fourlayer.tex 2001/09/27 00:58:17 1.2
+++ manual/s_examples/baroclinic_gyre/fourlayer.tex 2001/10/24 15:21:27 1.3
@@ -1,7 +1,8 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_examples/baroclinic_gyre/fourlayer.tex,v 1.2 2001/09/27 00:58:17 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_examples/baroclinic_gyre/fourlayer.tex,v 1.3 2001/10/24 15:21:27 cnh Exp $
% $Name: $
\section{Example: Four layer Baroclinic Ocean Gyre In Spherical Coordinates}
+\label{sec:eg-fourlayer}
\bodytext{bgcolor="#FFFFFFFF"}
@@ -15,10 +16,11 @@
%{\large May 2001}
%\end{center}
-\subsection{Introduction}
-
-This document describes the second example MITgcm experiment. The first
-example experiment ilustrated how to configure the code for a single layer
+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}
+ilustrated how to configure the code for a single layer
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.
@@ -38,11 +40,11 @@
by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally
in the north-south direction. Topologically the simulated
domain is a sector on a sphere and the coriolis parameter, $f$, is defined
-according to latitude, $\phi$
+according to latitude, $\varphi$
\begin{equation}
\label{EQ:fcori}
-f(\phi) = 2 \Omega \sin( \phi )
+f(\varphi) = 2 \Omega \sin( \varphi )
\end{equation}
\noindent with the rotation rate, $\Omega$ set to $\frac{2 \pi}{86400s}$.
@@ -52,20 +54,22 @@
\begin{equation}
\label{EQ:taux}
-\tau_x(\phi) = \tau_{0}\sin(\pi \frac{\phi}{L_{\phi}})
+\tau_x(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}})
\end{equation}
-\noindent where $L_{\phi}$ is the lateral domain extent ($60^{\circ}$) and
+\noindent where $L_{\varphi}$ is the lateral domain extent ($60^{\circ}$) and
$\tau_0$ is set to $0.1N m^{-2}$.
\\
Figure \ref{FIG:simulation_config}
summarises the configuration simulated.
-In contrast to example (1) \cite{baro_gyre_case_study}, the current
-experiment simulates a spherical polar domain. However, as indicated
+In contrast to the example in section \ref{sec:eg-baro}, the
+current experiment simulates a spherical polar domain. However, as indicated
by the axes in the lower left of the figure the model code works internally
-in a locally orthoganal coordinate $(x,y,z)$. In the remainder of this
-document the local coordinate $(x,y,z)$ will be adopted.
+in a locally orthoganal coordinate $(x,y,z)$. For this experiment description
+of this document the local orthogonal model coordinate $(x,y,z)$ is synonomous
+with the spherical polar coordinate shown in figure
+\ref{fig:spherical-polar-coord}
\\
The experiment has four levels in the vertical, each of equal thickness,
@@ -113,14 +117,8 @@
\label{FIG:simulation_config}
\end{figure}
-\subsection{Discrete Numerical Configuration}
+\subsection{Equations solved}
- The model is configured in hydrostatic form. The domain is discretised with
-a uniform grid spacing in latitude and longitude
- $\Delta x=\Delta y=1^{\circ}$, so
-that there are sixty grid cells in the $x$ and $y$ directions. Vertically the
-model is configured with a four layers with constant depth,
-$\Delta z$, of $500$~m.
The implicit free surface form of the
pressure equation described in Marshall et. al \cite{Marshall97a} is
employed.
@@ -128,24 +126,26 @@
dissipation. The wind-stress momentum input 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
-\ref{SEC:code_config} ), yielding an active set of equations solved in this
-configuration as follows
+\ref{SEC:code_config} ). This yields an active set of equations in
+solved in this configuration, written in spherical polar coordinates as
+follows
\begin{eqnarray}
\label{EQ:model_equations}
\frac{Du}{Dt} - fv +
- \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -
+ \frac{1}{\rho}\frac{\partial p^{'}}{\partial \lambda} -
A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}
& = &
\cal{F}
\\
\frac{Dv}{Dt} + fu +
- \frac{1}{\rho}\frac{\partial p^{'}}{\partial y} -
+ \frac{1}{\rho}\frac{\partial p^{'}}{\partial \varphi} -
A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}
& = &
0
\\
-\frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}
+\frac{\partial \eta}{\partial t} + \frac{\partial H \hat{u}}{\partial \lambda} +
+\frac{\partial H \hat{v}}{\partial \varphi}
&=&
0
\\
@@ -154,21 +154,71 @@
& = &
0
\\
-g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz & = & p^{'}
+p^{'} & = &
+g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz
+\\
+\rho^{'} & = &- \alpha_{\theta}\rho_{0}\theta^{'}
\\
{\cal F} |_{s} & = & \frac{\tau_{x}}{\rho_{0}\Delta z_{s}}
\\
{\cal F} |_{i} & = & 0
\end{eqnarray}
-\noindent where $u$ and $v$ are the $x$ and $y$ components of the
-flow vector $\vec{u}$. The suffices ${s},{i}$ indicate surface and
-interior model levels respectively. As described in
-MITgcm Numerical Solution Procedure \cite{MITgcm_Numerical_Scheme}, the time
-evolution of potential temperature, $\theta$, equation is solved prognostically.
-The total pressure, $p$, is diagnosed by summing pressure due to surface
+\noindent where $u$ and $v$ are the components of the horizontal
+flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).
+The suffices ${s},{i}$ indicate surface and interior of the domain.
+The forcing $\cal F$ is only applied at the surface.
+The pressure field $p^{'}$ is separated into a barotropic part
+due to variations in sea-surface height, $\eta$, and a hydrostatic
+part due to variations in density, $\rho^{'}$, over the water column.
+
+\subsection{Discrete Numerical Configuration}
+
+ The model is configured in hydrostatic form. The domain is discretised with
+a uniform grid spacing in latitude and longitude
+ $\Delta \lambda=\Delta \varphi=1^{\circ}$, so
+that there are sixty grid cells in the zonal and meridional directions.
+Vertically the
+model is configured with a 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
+$\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in
+radians according to
+
+\begin{eqnarray}
+x=r\cos(\varphi)\lambda,~\Delta x & = &r\cos(\varphi)\Delta \lambda \\
+y=r\varphi,~\Delta y &= &r\Delta \varphi
+\end{eqnarray}
+
+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}.
+
+\noindent\fbox{ \begin{minipage}{5.5in}
+{\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em
+model/src/ini\_spherical\_polar\_grid.F})
+
+$A_c$, $A_\zeta$, $A_w$, $A_s$: {\bf rAc}, {\bf rAz}, {\bf rAw}, {\bf rAs}
+({\em GRID.h})
+
+$\Delta x_g$, $\Delta y_g$: {\bf DXg}, {\bf DYg} ({\em GRID.h})
+
+$\Delta x_c$, $\Delta y_c$: {\bf DXc}, {\bf DYc} ({\em GRID.h})
+
+$\Delta x_f$, $\Delta y_f$: {\bf DXf}, {\bf DYf} ({\em GRID.h})
+
+$\Delta x_v$, $\Delta y_u$: {\bf DXv}, {\bf DYu} ({\em GRID.h})
+
+\end{minipage} }\\
+
+
+
+As described in \ref{sec:tracer_equations}, the time evolution of potential
+temperature,
+$\theta$, equation is solved prognostically.
+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.
-\\
\subsubsection{Numerical Stability Criteria}
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