/[MITgcm]/manual/s_examples/baroclinic_gyre/fourlayer.tex
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revision 1.9 by cnh, Thu Oct 25 18:36:55 2001 UTC revision 1.12 by cnh, Thu Feb 28 19:32:19 2002 UTC
# Line 1  Line 1 
1  % $Header$  % $Header$
2  % $Name$  % $Name$
3    
4  \section{Example: Four layer Baroclinic Ocean Gyre In Spherical Coordinates}  \section{Four Layer Baroclinic Ocean Gyre In Spherical Coordinates}
5  \label{sec:eg-fourlayer}  \label{sect:eg-fourlayer}
6    
7  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
8    
# Line 19  Line 19 
19  This document describes an example experiment using MITgcm  This document describes an example experiment using MITgcm
20  to simulate a baroclinic ocean gyre in spherical  to simulate a baroclinic ocean gyre in spherical
21  polar coordinates. The barotropic  polar coordinates. The barotropic
22  example experiment in section \ref{sec:eg-baro}  example experiment in section \ref{sect:eg-baro}
23  illustrated how to configure the code for a single layer  illustrated 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
# Line 43  domain is a sector on a sphere and the c Line 43  domain is a sector on a sphere and the c
43  according to latitude, $\varphi$  according to latitude, $\varphi$
44    
45  \begin{equation}  \begin{equation}
46  \label{EQ:fcori}  \label{EQ:eg-fourlayer-fcori}
47  f(\varphi) = 2 \Omega \sin( \varphi )  f(\varphi) = 2 \Omega \sin( \varphi )
48  \end{equation}  \end{equation}
49    
# Line 61  f(\varphi) = 2 \Omega \sin( \varphi ) Line 61  f(\varphi) = 2 \Omega \sin( \varphi )
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:eg-fourlayer-simulation_config}
65  summarizes the configuration simulated.  summarizes the configuration simulated.
66  In contrast to the example in section \ref{sec:eg-baro}, the  In contrast to the example in section \ref{sect:eg-baro}, the
67  current experiment simulates a spherical polar domain. As indicated  current experiment simulates a spherical polar domain. 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 orthogonal coordinate $(x,y,z)$. For this experiment description  in a locally orthogonal coordinate $(x,y,z)$. For this experiment description
# Line 82  $\theta_{1750}=6^{\circ}$~C. The equatio Line 82  $\theta_{1750}=6^{\circ}$~C. The equatio
82  linear  linear
83    
84  \begin{equation}  \begin{equation}
85  \label{EQ:linear1_eos}  \label{EQ:eg-fourlayer-linear1_eos}
86  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )
87  \end{equation}  \end{equation}
88    
89  \noindent which is implemented in the model as a density anomaly equation  \noindent which is implemented in the model as a density anomaly equation
90    
91  \begin{equation}  \begin{equation}
92  \label{EQ:linear1_eos_pert}  \label{EQ:eg-fourlayer-linear1_eos_pert}
93  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}
94  \end{equation}  \end{equation}
95    
# Line 114  An initial stratification is Line 114  An initial stratification is
114  imposed by setting the potential temperature, $\theta$, in each layer.  imposed by setting the potential temperature, $\theta$, in each layer.
115  The vertical spacing, $\Delta z$, is constant and equal to $500$m.  The vertical spacing, $\Delta z$, is constant and equal to $500$m.
116  }  }
117  \label{FIG:simulation_config}  \label{FIG:eg-fourlayer-simulation_config}
118  \end{figure}  \end{figure}
119    
120  \subsection{Equations solved}  \subsection{Equations solved}
121  For this problem  For this problem
122  the implicit free surface, {\bf HPE} (see section \ref{sec:hydrostatic_and_quasi-hydrostatic_forms}) form of the  the implicit free surface, {\bf HPE} (see section \ref{sect:hydrostatic_and_quasi-hydrostatic_forms}) form of the
123  equations described in Marshall et. al \cite{Marshall97a} are  equations described in Marshall et. al \cite{marshall:97a} are
124  employed. The flow is three-dimensional with just temperature, $\theta$, as  employed. The flow is three-dimensional with just temperature, $\theta$, as
125  an active tracer.  The equation of state is linear.  an active tracer.  The equation of state is linear.
126  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous
# Line 133  solved in this configuration, written in Line 133  solved in this configuration, written in
133  follows  follows
134    
135  \begin{eqnarray}  \begin{eqnarray}
136  \label{EQ:model_equations}  \label{EQ:eg-fourlayer-model_equations}
137  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
138    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -
139    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}}
# Line 221  y=r\varphi,~\Delta y &= &r\Delta \varphi Line 221  y=r\varphi,~\Delta y &= &r\Delta \varphi
221    
222  The procedure for generating a set of internal grid variables from a  The procedure for generating a set of internal grid variables from a
223  spherical polar grid specification is discussed in section  spherical polar grid specification is discussed in section
224  \ref{sec:spatial_discrete_horizontal_grid}.  \ref{sect:spatial_discrete_horizontal_grid}.
225    
226  \noindent\fbox{ \begin{minipage}{5.5in}  \noindent\fbox{ \begin{minipage}{5.5in}
227  {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em  {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em
# Line 242  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, { Line 242  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, {
242    
243    
244    
245  As described in \ref{sec:tracer_equations}, the time evolution of potential  As described in \ref{sect:tracer_equations}, the time evolution of potential
246  temperature,  temperature,
247  $\theta$, (equation \ref{eq:eg_fourl_theta})  $\theta$, (equation \ref{eq:eg_fourl_theta})
248  is evaluated prognostically. The centered second-order scheme with  is evaluated prognostically. The centered second-order scheme with
249  Adams-Bashforth time stepping described in section  Adams-Bashforth time stepping described in section
250  \ref{sec:tracer_equations_abII} is used to step forward the temperature  \ref{sect:tracer_equations_abII} is used to step forward the temperature
251  equation. Prognostic terms in  equation. Prognostic terms in
252  the momentum equations are solved using flux form as  the momentum equations are solved using flux form as
253  described in section \ref{sec:flux-form_momentum_eqautions}.  described in section \ref{sect:flux-form_momentum_eqautions}.
254  The pressure forces that drive the fluid motions, (  The pressure forces that drive the fluid motions, (
255  $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface  $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface
256  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the
# Line 258  pressure is diagnosed explicitly by inte Line 258  pressure is diagnosed explicitly by inte
258  height, $\eta$, is diagnosed using an implicit scheme. The pressure  height, $\eta$, is diagnosed using an implicit scheme. The pressure
259  field solution method is described in sections  field solution method is described in sections
260  \ref{sect:pressure-method-linear-backward} and  \ref{sect:pressure-method-linear-backward} and
261  \ref{sec:finding_the_pressure_field}.  \ref{sect:finding_the_pressure_field}.
262    
263  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
264    
# Line 266  The Laplacian viscosity coefficient, $A_ Line 266  The Laplacian viscosity coefficient, $A_
266  This value is chosen to yield a Munk layer width,  This value is chosen to yield a Munk layer width,
267    
268  \begin{eqnarray}  \begin{eqnarray}
269  \label{EQ:munk_layer}  \label{EQ:eg-fourlayer-munk_layer}
270  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
271  \end{eqnarray}  \end{eqnarray}
272    
# Line 282  time step $\delta t=1200$secs. With this Line 282  time step $\delta t=1200$secs. With this
282  parameter to the horizontal Laplacian friction  parameter to the horizontal Laplacian friction
283    
284  \begin{eqnarray}  \begin{eqnarray}
285  \label{EQ:laplacian_stability}  \label{EQ:eg-fourlayer-laplacian_stability}
286  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
287  \end{eqnarray}  \end{eqnarray}
288    
# Line 294  for stability for this term under ABII t Line 294  for stability for this term under ABII t
294  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
295    
296  \begin{eqnarray}  \begin{eqnarray}
297  \label{EQ:laplacian_stability_z}  \label{EQ:eg-fourlayer-laplacian_stability_z}
298  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}
299  \end{eqnarray}  \end{eqnarray}
300    
# Line 307  and vertical ($K_{z}$) diffusion coeffic Line 307  and vertical ($K_{z}$) diffusion coeffic
307  \noindent The numerical stability for inertial oscillations  \noindent The numerical stability for inertial oscillations
308    
309  \begin{eqnarray}  \begin{eqnarray}
310  \label{EQ:inertial_stability}  \label{EQ:eg-fourlayer-inertial_stability}
311  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
312  \end{eqnarray}  \end{eqnarray}
313    
# Line 320  horizontal flow Line 320  horizontal flow
320  speed of $ | \vec{u} | = 2 ms^{-1}$  speed of $ | \vec{u} | = 2 ms^{-1}$
321    
322  \begin{eqnarray}  \begin{eqnarray}
323  \label{EQ:cfl_stability}  \label{EQ:eg-fourlayer-cfl_stability}
324  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
325  \end{eqnarray}  \end{eqnarray}
326    
# Line 332  limit of 0.5. Line 332  limit of 0.5.
332  propagating at $2~{\rm m}~{\rm s}^{-1}$  propagating at $2~{\rm m}~{\rm s}^{-1}$
333    
334  \begin{eqnarray}  \begin{eqnarray}
335  \label{EQ:igw_stability}  \label{EQ:eg-fourlayer-igw_stability}
336  S_{c} = \frac{c_{g} \delta t}{ \Delta x}  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
337  \end{eqnarray}  \end{eqnarray}
338    
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