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revision 1.4 by adcroft, Tue Nov 13 18:22:24 2001 UTC revision 1.6 by cnh, Thu Feb 28 19:32:19 2002 UTC
# Line 1  Line 1 
1  % $Header$  % $Header$
2  % $Name$  % $Name$
3    
4  \section{Example: Centennial Time Scale Sensitivities}  \section{Centennial Time Scale Tracer Injection}
5    \label{sect:eg-simple-tracer}
6    
7  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
8    
# Line 48  Altogether, this yields the following fo Line 49  Altogether, this yields the following fo
49  in the model surface layer.  in the model surface layer.
50    
51  \begin{eqnarray}  \begin{eqnarray}
52  \label{EQ:global_forcing}  \label{EQ:eg-simple-tracer-global_forcing}
53  \label{EQ:global_forcing_fu}  \label{EQ:eg-simple-tracer-global_forcing_fu}
54  {\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}}  {\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}}
55  \\  \\
56  \label{EQ:global_forcing_fv}  \label{EQ:eg-simple-tracer-global_forcing_fv}
57  {\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}}  {\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}}
58  \\  \\
59  \label{EQ:global_forcing_ft}  \label{EQ:eg-simple-tracer-global_forcing_ft}
60  {\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} )  {\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} )
61   - \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q}   - \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q}
62  \\  \\
63  \label{EQ:global_forcing_fs}  \label{EQ:eg-simple-tracer-global_forcing_fs}
64  {\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} )  {\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} )
65   + \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R})   + \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R})
66  \end{eqnarray}  \end{eqnarray}
# Line 118  $ Line 119  $
119   \Delta z_{20}=815\,{\rm m}   \Delta z_{20}=815\,{\rm m}
120  $ (here the numeric subscript indicates the model level index number, ${\tt k}$).  $ (here the numeric subscript indicates the model level index number, ${\tt k}$).
121  The implicit free surface form of the pressure equation described in Marshall et. al  The implicit free surface form of the pressure equation described in Marshall et. al
122  \cite{Marshall97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous  \cite{marshall:97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous
123  dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.  dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.
124  \\  \\
125    
126  Wind-stress momentum inputs are added to the momentum equations for both  Wind-stress momentum inputs are added to the momentum equations for both
127  the zonal flow, $u$ and the meridional flow $v$, according to equations  the zonal flow, $u$ and the meridional flow $v$, according to equations
128  (\ref{EQ:global_forcing_fu}) and (\ref{EQ:global_forcing_fv}).  (\ref{EQ:eg-simple-tracer-global_forcing_fu}) and (\ref{EQ:eg-simple-tracer-global_forcing_fv}).
129  Thermodynamic forcing inputs are added to the equations for  Thermodynamic forcing inputs are added to the equations for
130  potential temperature, $\theta$, and salinity, $S$, according to equations  potential temperature, $\theta$, and salinity, $S$, according to equations
131  (\ref{EQ:global_forcing_ft}) and (\ref{EQ:global_forcing_fs}).  (\ref{EQ:eg-simple-tracer-global_forcing_ft}) and (\ref{EQ:eg-simple-tracer-global_forcing_fs}).
132  This produces a set of equations solved in this configuration as follows:  This produces a set of equations solved in this configuration as follows:
133  % {\fracktur}  % {\fracktur}
134    
135    
136  \begin{eqnarray}  \begin{eqnarray}
137  \label{EQ:model_equations}  \label{EQ:eg-simple-tracer-model_equations}
138  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
139    \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -    \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -
140    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 179  The Laplacian dissipation coefficient, $ Line 180  The Laplacian dissipation coefficient, $
180  This value is chosen to yield a Munk layer width \cite{adcroft:95},  This value is chosen to yield a Munk layer width \cite{adcroft:95},
181    
182  \begin{eqnarray}  \begin{eqnarray}
183  \label{EQ:munk_layer}  \label{EQ:eg-simple-tracer-munk_layer}
184  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
185  \end{eqnarray}  \end{eqnarray}
186    
# Line 193  time step $\delta t=1200$secs. With this Line 194  time step $\delta t=1200$secs. With this
194  parameter to the horizontal Laplacian friction \cite{adcroft:95}  parameter to the horizontal Laplacian friction \cite{adcroft:95}
195    
196  \begin{eqnarray}  \begin{eqnarray}
197  \label{EQ:laplacian_stability}  \label{EQ:eg-simple-tracer-laplacian_stability}
198  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
199  \end{eqnarray}  \end{eqnarray}
200    
# Line 205  for stability. Line 206  for stability.
206  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
207    
208  \begin{eqnarray}  \begin{eqnarray}
209  \label{EQ:laplacian_stability_z}  \label{EQ:eg-simple-tracer-laplacian_stability_z}
210  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}
211  \end{eqnarray}  \end{eqnarray}
212    
# Line 219  and vertical ($K_{z}$) diffusion coeffic Line 220  and vertical ($K_{z}$) diffusion coeffic
220  \cite{adcroft:95}  \cite{adcroft:95}
221    
222  \begin{eqnarray}  \begin{eqnarray}
223  \label{EQ:inertial_stability}  \label{EQ:eg-simple-tracer-inertial_stability}
224  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
225  \end{eqnarray}  \end{eqnarray}
226    
# Line 232  horizontal flow Line 233  horizontal flow
233  speed of $ | \vec{u} | = 2 ms^{-1}$  speed of $ | \vec{u} | = 2 ms^{-1}$
234    
235  \begin{eqnarray}  \begin{eqnarray}
236  \label{EQ:cfl_stability}  \label{EQ:eg-simple-tracer-cfl_stability}
237  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
238  \end{eqnarray}  \end{eqnarray}
239    
# Line 244  limit of 0.5. Line 245  limit of 0.5.
245  \cite{adcroft:95}  \cite{adcroft:95}
246    
247  \begin{eqnarray}  \begin{eqnarray}
248  \label{EQ:cfl_stability}  \label{EQ:eg-simple-tracer-igw_stability}
249  S_{c} = \frac{c_{g} \delta t}{ \Delta x}  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
250  \end{eqnarray}  \end{eqnarray}
251    
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