/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.30 by jmc, Wed May 4 22:42:48 2011 UTC revision 1.31 by jmc, Mon May 9 13:45:05 2011 UTC
# Line 542  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil Line 542  G_{\theta,S}^{(n+1/2)} & = & (3/2+\epsil
542  \nabla \cdot g H \nabla \eta^{n+1} & - & \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}  \nabla \cdot g H \nabla \eta^{n+1} & - & \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2}
543  ~ = ~ - \frac{\eta^*}{\Delta t^2}  ~ = ~ - \frac{\eta^*}{\Delta t^2}
544  \label{eq:elliptic-sync} \\  \label{eq:elliptic-sync} \\
545  \vec{\bf v}^{n+1} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1}  \vec{\bf v}^{n+1} & = & \vec{\bf v}^{**} - \Delta t g \nabla \eta^{n+1}
546  \label{eq:v-n+1-sync}  \label{eq:v-n+1-sync}
547  \end{eqnarray}  \end{eqnarray}
548  Fig.~\ref{fig:adams-bashforth-sync} illustrates the location of  Fig.~\ref{fig:adams-bashforth-sync} illustrates the location of
# Line 637  position in time of variables appropriat Line 637  position in time of variables appropriat
637  \nabla \cdot g H \nabla \eta^{n+1/2} & - & \frac{\epsilon_{fs} \eta^{n+1/2}}{\Delta t^2}  \nabla \cdot g H \nabla \eta^{n+1/2} & - & \frac{\epsilon_{fs} \eta^{n+1/2}}{\Delta t^2}
638  ~ = ~ - \frac{\eta^*}{\Delta t^2}  ~ = ~ - \frac{\eta^*}{\Delta t^2}
639  \label{eq:elliptic-staggered} \\  \label{eq:elliptic-staggered} \\
640  \vec{\bf v}^{n+1/2} & = & \vec{\bf v}^{*} - \Delta t g \nabla \eta^{n+1/2}  \vec{\bf v}^{n+1/2} & = & \vec{\bf v}^{**} - \Delta t g \nabla \eta^{n+1/2}
641  \label{eq:v-n+1-staggered} \\  \label{eq:v-n+1-staggered} \\
642  G_{\theta,S}^{n} & = & G_{\theta,S} ( u^{n+1/2}, \theta^{n}, S^{n} )  G_{\theta,S}^{n} & = & G_{\theta,S} ( u^{n+1/2}, \theta^{n}, S^{n} )
643  \label{eq:Gt-n-staggered} \\  \label{eq:Gt-n-staggered} \\
# Line 759  Substituting into the depth integrated c Line 759  Substituting into the depth integrated c
759  \partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right)  \partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right)
760  +  +
761  \partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right)  \partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right)
762   - \frac{\epsilon_{fs}\eta^*}{\Delta t^2}   - \frac{\epsilon_{fs}\eta^{n+1}}{\Delta t^2}
763  = - \frac{\eta^*}{\Delta t^2}  = - \frac{\eta^*}{\Delta t^2}
764  \end{equation}  \end{equation}
765  which is approximated by equation  which is approximated by equation
# Line 767  which is approximated by equation Line 767  which is approximated by equation
767  $\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh}  $\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh}
768  << g \nabla \eta$. If \ref{eq:elliptic-backward-free-surface} is  << g \nabla \eta$. If \ref{eq:elliptic-backward-free-surface} is
769  solved accurately then the implication is that $\widehat{\phi}_{nh}  solved accurately then the implication is that $\widehat{\phi}_{nh}
770  \approx 0$ so that thet non-hydrostatic pressure field does not drive  \approx 0$ so that the non-hydrostatic pressure field does not drive
771  barotropic motion.  barotropic motion.
772    
773  The flow must satisfy non-divergence  The flow must satisfy non-divergence
# Line 787  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2 Line 787  u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2
787  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\  v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\
788  w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\  w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\
789  \eta^* ~ = ~ \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)  \eta^* ~ = ~ \epsilon_{fs} \left( \eta^{n} + \Delta t (P-E) \right)
790  & - & \Delta t  & - & \Delta t \left( \partial_x H \widehat{u^{*}}
791    \partial_x H \widehat{u^{*}}                      + \partial_y H \widehat{v^{*}} \right)
 + \partial_y H \widehat{v^{*}}  
792  \\  \\
793    \partial_x g H \partial_x \eta^{n+1}    \partial_x g H \partial_x \eta^{n+1}
794  + \partial_y g H \partial_y \eta^{n+1}  + \partial_y g H \partial_y \eta^{n+1}
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