/[MITgcm]/manual/s_examples/baroclinic_gyre/fourlayer.tex
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revision 1.3 by cnh, Wed Oct 24 15:21:27 2001 UTC revision 1.4 by cnh, Wed Oct 24 19:43:07 2001 UTC
# Line 133  follows Line 133  follows
133  \begin{eqnarray}  \begin{eqnarray}
134  \label{EQ:model_equations}  \label{EQ:model_equations}
135  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
136    \frac{1}{\rho}\frac{\partial p^{'}}{\partial \lambda} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -
137    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}}
138  & = &  & = &
139  \cal{F}  \cal{F}
140  \\  \\
141  \frac{Dv}{Dt} + fu +  \frac{Dv}{Dt} + fu +
142    \frac{1}{\rho}\frac{\partial p^{'}}{\partial \varphi} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \varphi} -
143    A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}    A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}
144  & = &  & = &
145  0  0
# Line 154  follows Line 154  follows
154  & = &  & = &
155  0  0
156  \\  \\
157  p^{'} & = &  p^{\prime} & = &
158  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz  g\rho_{0} \eta + \int^{0}_{-z}\rho^{\prime} dz
159  \\  \\
160  \rho^{'} & = &- \alpha_{\theta}\rho_{0}\theta^{'}  \rho^{\prime} & = &- \alpha_{\theta}\rho_{0}\theta^{\prime}
161  \\  \\
162  {\cal F} |_{s} & = & \frac{\tau_{x}}{\rho_{0}\Delta z_{s}}  {\cal F} |_{s} & = & \frac{\tau_{x}}{\rho_{0}\Delta z_{s}}
163  \\  \\
# Line 166  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} d Line 166  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} d
166    
167  \noindent where $u$ and $v$ are the components of the horizontal  \noindent where $u$ and $v$ are the components of the horizontal
168  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).
169    The terms $H\hat{u}$ and $H\hat{v}$ are the components of the term
170    integrated in eqaution \ref{eq:free-surface}, as descirbed in section
171    
172  The suffices ${s},{i}$ indicate surface and interior of the domain.  The suffices ${s},{i}$ indicate surface and interior of the domain.
173  The forcing $\cal F$ is only applied at the surface.  The forcing $\cal F$ is only applied at the surface.
174  The pressure field $p^{'}$ is separated into a barotropic part  The pressure field, $p^{\prime}$, is separated into a barotropic part
175  due to variations in sea-surface height, $\eta$, and a hydrostatic  due to variations in sea-surface height, $\eta$, and a hydrostatic
176  part due to variations in density, $\rho^{'}$, over the water column.  part due to variations in density, $\rho^{\prime}$, over the water column.
177    
178  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
179    
# Line 179  a uniform grid spacing in latitude and l Line 182  a uniform grid spacing in latitude and l
182   $\Delta \lambda=\Delta \varphi=1^{\circ}$, so   $\Delta \lambda=\Delta \varphi=1^{\circ}$, so
183  that there are sixty grid cells in the zonal and meridional directions.  that there are sixty grid cells in the zonal and meridional directions.
184  Vertically the  Vertically the
185  model is configured with a four layers with constant depth,  model is configured with four layers with constant depth,
186  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate
187  variables $x$ and $y$ are initialised from the values of  variables $x$ and $y$ are initialised from the values of
188  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in
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