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revision 1.2 by cnh, Thu Sep 27 00:58:17 2001 UTC revision 1.4 by cnh, Wed Oct 24 19:43:07 2001 UTC
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2  % $Name$  % $Name$
3    
4  \section{Example: Four layer Baroclinic Ocean Gyre In Spherical Coordinates}  \section{Example: Four layer Baroclinic Ocean Gyre In Spherical Coordinates}
5    \label{sec:eg-fourlayer}
6    
7  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
8    
# Line 15  Line 16 
16  %{\large May 2001}  %{\large May 2001}
17  %\end{center}  %\end{center}
18    
19  \subsection{Introduction}  This document describes an example experiment using MITgcm
20    to simulate a baroclinic ocean gyre in spherical
21  This document describes the second example MITgcm experiment. The first  polar coordinates. The barotropic
22  example experiment ilustrated how to configure the code for a single layer  example experiment in section \ref{sec:eg-baro}
23    ilustrated how to configure the code for a single layer
24  simulation in a cartesian grid. In this example a similar physical problem  simulation in a cartesian grid. In this example a similar physical problem
25  is simulated, but the code is now configured  is simulated, but the code is now configured
26  for four layers and in a spherical polar coordinate system.  for four layers and in a spherical polar coordinate system.
# Line 38  lateral extent. The fluid is $2$~km deep Line 40  lateral extent. The fluid is $2$~km deep
40  by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally  by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally
41  in the north-south direction. Topologically the simulated  in the north-south direction. Topologically the simulated
42  domain is a sector on a sphere and the coriolis parameter, $f$, is defined  domain is a sector on a sphere and the coriolis parameter, $f$, is defined
43  according to latitude, $\phi$  according to latitude, $\varphi$
44    
45  \begin{equation}  \begin{equation}
46  \label{EQ:fcori}  \label{EQ:fcori}
47  f(\phi) = 2 \Omega \sin( \phi )  f(\varphi) = 2 \Omega \sin( \varphi )
48  \end{equation}  \end{equation}
49    
50  \noindent with the rotation rate, $\Omega$ set to $\frac{2 \pi}{86400s}$.  \noindent with the rotation rate, $\Omega$ set to $\frac{2 \pi}{86400s}$.
# Line 52  f(\phi) = 2 \Omega \sin( \phi ) Line 54  f(\phi) = 2 \Omega \sin( \phi )
54    
55  \begin{equation}  \begin{equation}
56  \label{EQ:taux}  \label{EQ:taux}
57  \tau_x(\phi) = \tau_{0}\sin(\pi \frac{\phi}{L_{\phi}})  \tau_x(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}})
58  \end{equation}  \end{equation}
59    
60  \noindent where $L_{\phi}$ is the lateral domain extent ($60^{\circ}$) and  \noindent where $L_{\varphi}$ is the lateral domain extent ($60^{\circ}$) and
61  $\tau_0$ is set to $0.1N m^{-2}$.  $\tau_0$ is set to $0.1N m^{-2}$.
62  \\  \\
63    
64  Figure \ref{FIG:simulation_config}  Figure \ref{FIG:simulation_config}
65  summarises the configuration simulated.  summarises the configuration simulated.
66  In contrast to example (1) \cite{baro_gyre_case_study}, the current  In contrast to the example in section \ref{sec:eg-baro}, the
67  experiment simulates a spherical polar domain. However, as indicated  current experiment simulates a spherical polar domain. However, as indicated
68  by the axes in the lower left of the figure the model code works internally  by the axes in the lower left of the figure the model code works internally
69  in a locally orthoganal coordinate $(x,y,z)$. In the remainder of this  in a locally orthoganal coordinate $(x,y,z)$. For this experiment description
70  document the local coordinate $(x,y,z)$ will be adopted.  of this document the local orthogonal model coordinate $(x,y,z)$ is synonomous
71    with the spherical polar coordinate shown in figure
72    \ref{fig:spherical-polar-coord}
73  \\  \\
74    
75  The experiment has four levels in the vertical, each of equal thickness,  The experiment has four levels in the vertical, each of equal thickness,
# Line 113  The vertical spacing, $\Delta z$, is con Line 117  The vertical spacing, $\Delta z$, is con
117  \label{FIG:simulation_config}  \label{FIG:simulation_config}
118  \end{figure}  \end{figure}
119    
120  \subsection{Discrete Numerical Configuration}  \subsection{Equations solved}
121    
  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.  
122  The implicit free surface form of the  The implicit free surface form of the
123  pressure equation described in Marshall et. al \cite{Marshall97a} is  pressure equation described in Marshall et. al \cite{Marshall97a} is
124  employed.  employed.
# Line 128  A horizontal laplacian operator $\nabla_ Line 126  A horizontal laplacian operator $\nabla_
126  dissipation. The wind-stress momentum input is added to the momentum equation  dissipation. The wind-stress momentum input is added to the momentum equation
127  for the ``zonal flow'', $u$. Other terms in the model  for the ``zonal flow'', $u$. Other terms in the model
128  are explicitly switched off for this experiement configuration (see section  are explicitly switched off for this experiement configuration (see section
129  \ref{SEC:code_config} ), yielding an active set of equations solved in this  \ref{SEC:code_config} ). This yields an active set of equations in
130  configuration as follows  solved in this configuration, written in spherical polar coordinates as
131    follows
132    
133  \begin{eqnarray}  \begin{eqnarray}
134  \label{EQ:model_equations}  \label{EQ:model_equations}
135  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
136    \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -
137    A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}    A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}
138  & = &  & = &
139  \cal{F}  \cal{F}
140  \\  \\
141  \frac{Dv}{Dt} + fu +  \frac{Dv}{Dt} + fu +
142    \frac{1}{\rho}\frac{\partial p^{'}}{\partial y} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \varphi} -
143    A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}    A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}
144  & = &  & = &
145  0  0
146  \\  \\
147  \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}  \frac{\partial \eta}{\partial t} + \frac{\partial H \hat{u}}{\partial \lambda} +
148    \frac{\partial H \hat{v}}{\partial \varphi}
149  &=&  &=&
150  0  0
151  \\  \\
# Line 154  configuration as follows Line 154  configuration as follows
154  & = &  & = &
155  0  0
156  \\  \\
157  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz & = & p^{'}  p^{\prime} & = &
158    g\rho_{0} \eta + \int^{0}_{-z}\rho^{\prime} dz
159    \\
160    \rho^{\prime} & = &- \alpha_{\theta}\rho_{0}\theta^{\prime}
161  \\  \\
162  {\cal F} |_{s} & = & \frac{\tau_{x}}{\rho_{0}\Delta z_{s}}  {\cal F} |_{s} & = & \frac{\tau_{x}}{\rho_{0}\Delta z_{s}}
163  \\  \\
164  {\cal F} |_{i} & = & 0  {\cal F} |_{i} & = & 0
165  \end{eqnarray}  \end{eqnarray}
166    
167  \noindent where $u$ and $v$ are the $x$ and $y$ components of the  \noindent where $u$ and $v$ are the components of the horizontal
168  flow vector $\vec{u}$. The suffices ${s},{i}$ indicate surface and  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).
169  interior model levels respectively. As described in  The terms $H\hat{u}$ and $H\hat{v}$ are the components of the term
170  MITgcm Numerical Solution Procedure \cite{MITgcm_Numerical_Scheme}, the time  integrated in eqaution \ref{eq:free-surface}, as descirbed in section
171  evolution of potential temperature, $\theta$, equation is solved prognostically.  
172  The total pressure, $p$, is diagnosed by summing pressure due to surface  The suffices ${s},{i}$ indicate surface and interior of the domain.
173    The forcing $\cal F$ is only applied at the surface.
174    The pressure field, $p^{\prime}$, is separated into a barotropic part
175    due to variations in sea-surface height, $\eta$, and a hydrostatic
176    part due to variations in density, $\rho^{\prime}$, over the water column.
177    
178    \subsection{Discrete Numerical Configuration}
179    
180     The model is configured in hydrostatic form.  The domain is discretised with
181    a uniform grid spacing in latitude and longitude
182     $\Delta \lambda=\Delta \varphi=1^{\circ}$, so
183    that there are sixty grid cells in the zonal and meridional directions.
184    Vertically the
185    model is configured with four layers with constant depth,
186    $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate
187    variables $x$ and $y$ are initialised from the values of
188    $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in
189    radians according to
190    
191    \begin{eqnarray}
192    x=r\cos(\varphi)\lambda,~\Delta x & = &r\cos(\varphi)\Delta \lambda \\
193    y=r\varphi,~\Delta y &= &r\Delta \varphi
194    \end{eqnarray}
195    
196    The procedure for generating a set of internal grid variables from a
197    spherical polar grid specification is discussed in section
198    \ref{sec:spatial_discrete_horizontal_grid}.
199    
200    \noindent\fbox{ \begin{minipage}{5.5in}
201    {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em
202    model/src/ini\_spherical\_polar\_grid.F})
203    
204    $A_c$, $A_\zeta$, $A_w$, $A_s$: {\bf rAc}, {\bf rAz}, {\bf rAw}, {\bf rAs}
205    ({\em GRID.h})
206    
207    $\Delta x_g$, $\Delta y_g$: {\bf DXg}, {\bf DYg} ({\em GRID.h})
208    
209    $\Delta x_c$, $\Delta y_c$: {\bf DXc}, {\bf DYc} ({\em GRID.h})
210    
211    $\Delta x_f$, $\Delta y_f$: {\bf DXf}, {\bf DYf} ({\em GRID.h})
212    
213    $\Delta x_v$, $\Delta y_u$: {\bf DXv}, {\bf DYu} ({\em GRID.h})
214    
215    \end{minipage} }\\
216    
217    
218    
219    As described in \ref{sec:tracer_equations}, the time evolution of potential
220    temperature,
221    $\theta$, equation is solved prognostically.
222    The pressure forces that drive the fluid motions, (
223    $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface
224  elevation $\eta$ and the hydrostatic pressure.  elevation $\eta$ and the hydrostatic pressure.
 \\  
225    
226  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
227    
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