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revision 1.2 by cnh, Thu Oct 25 18:36:54 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 167  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} d Line 168  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} d
168  \noindent where $u$ and $v$ are the $x$ and $y$ components of the  \noindent where $u$ and $v$ are the $x$ and $y$ components of the
169  flow vector $\vec{u}$. The suffices ${s},{i}$ indicate surface and  flow vector $\vec{u}$. The suffices ${s},{i}$ indicate surface and
170  interior model levels respectively. As described in  interior model levels respectively. As described in
171  MITgcm Numerical Solution Procedure \cite{MITgcm_Numerical_Scheme}, the time  MITgcm Numerical Solution Procedure \ref{chap:discretization}, the time
172  evolution of potential temperature, $\theta$, equation is solved prognostically.  evolution of potential temperature, $\theta$, equation is solved prognostically.
173  The total pressure, $p$, is diagnosed by summing pressure due to surface  The total pressure, $p$, is diagnosed by summing pressure due to surface
174  elevation $\eta$ and the hydrostatic pressure.  elevation $\eta$ and the hydrostatic pressure.
# Line 176  elevation $\eta$ and the hydrostatic pre Line 177  elevation $\eta$ and the hydrostatic pre
177  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
178    
179  The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.
180  This value is chosen to yield a Munk layer width \cite{Adcroft_thesis},  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 190  boundary layer is well resolved. Line 191  boundary layer is well resolved.
191    
192  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
193  time step $\delta t=1200$secs. With this time step the stability  time step $\delta t=1200$secs. With this time step the stability
194  parameter to the horizontal Laplacian friction \cite{Adcroft_thesis}  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 216  and vertical ($K_{z}$) diffusion coeffic Line 217  and vertical ($K_{z}$) diffusion coeffic
217  \\  \\
218    
219  \noindent The numerical stability for inertial oscillations  \noindent The numerical stability for inertial oscillations
220  \cite{Adcroft_thesis}  \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 227  S_{i} = f^{2} {\delta t}^2 Line 228  S_{i} = f^{2} {\delta t}^2
228  limit for stability.  limit for stability.
229  \\  \\
230    
231  \noindent The advective CFL \cite{Adcroft_thesis} for a extreme maximum  \noindent The advective CFL \cite{adcroft:95} for a extreme maximum
232  horizontal flow  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 241  limit of 0.5. Line 242  limit of 0.5.
242  \\  \\
243    
244  \noindent The stability parameter for internal gravity waves  \noindent The stability parameter for internal gravity waves
245  \cite{Adcroft_thesis}  \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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