/[MITgcm]/manual/s_examples/baroclinic_gyre/fourlayer.tex
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revision 1.11 by adcroft, Tue Nov 13 20:13:54 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{sect:eg-fourlayer}  \label{sect:eg-fourlayer}
6    
7  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
# 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{sect: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
# 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}
# 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 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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