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revision 1.5 by adcroft, Tue Nov 13 19:01:42 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 124  dissipation. Thermal and haline diffusio Line 125  dissipation. Thermal and haline diffusio
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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