/[MITgcm]/manual/s_algorithm/text/nonlin_frsurf.tex
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revision 1.10 by jmc, Mon Jul 11 13:49:29 2005 UTC revision 1.13 by edhill, Tue Jun 27 22:31:23 2006 UTC
# Line 11  that concern the free surface formulatio Line 11  that concern the free surface formulatio
11    
12    
13  \subsubsection{pressure/geo-potential and free surface}  \subsubsection{pressure/geo-potential and free surface}
14    \label{sect:phi-freesurface}
15    
16  For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$,  For the atmosphere, since $\phi = \phi_{topo} - \int^p_{p_s} \alpha dp$,
17  subtracting the reference state defined in section  subtracting the reference state defined in section
# Line 30  For both fluid, using the isomorphic not Line 31  For both fluid, using the isomorphic not
31  $$  $$
32  \phi' = \int^{r_{surf}}_r b~ dr - \int^{R_o}_r b_o dr  \phi' = \int^{r_{surf}}_r b~ dr - \int^{R_o}_r b_o dr
33  $$  $$
34  \begin{eqnarray}  and re-write as:
35  \mathrm{and~re~write:}\hspace{10mm}  \begin{equation}
36  \phi' = \int^{r_{surf}}_{R_o} b~ dr & + & \int^{R_o}_r (b - b_o) dr  \phi' = \int^{r_{surf}}_{R_o} b~ dr + \int^{R_o}_r (b - b_o) dr
37  \label{eq:split-phi-Ro} \\  \label{eq:split-phi-Ro}
38  \mathrm{or:}\hspace{10mm}  \end{equation}
39  \phi' = \int^{r_{surf}}_{R_o} b_o dr & + & \int^{r_{surf}}_r (b - b_o) dr  or:
40    \begin{equation}
41    \phi' = \int^{r_{surf}}_{R_o} b_o dr + \int^{r_{surf}}_r (b - b_o) dr
42  \label{eq:split-phi-bo}  \label{eq:split-phi-bo}
43  \end{eqnarray}  \end{equation}
44    
45  In section \ref{sec:finding_the_pressure_field}, following eq.\ref{eq:split-phi-Ro},  In section \ref{sec:finding_the_pressure_field}, following eq.\ref{eq:split-phi-Ro},
46  the pressure/geo-potential $\phi'$ has been separated into surface ($\phi_s$),  the pressure/geo-potential $\phi'$ has been separated into surface ($\phi_s$),
# Line 307  $h^n$ and $dh^n$ are the column and grid Line 310  $h^n$ and $dh^n$ are the column and grid
310  \vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \frac{dh^{n-1}}{dh^{n}} \left(  \vec{\bf v}^{*} & = & \vec{\bf v}^{n-1/2} + \Delta t \frac{dh^{n-1}}{dh^{n}} \left(
311  \vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n}/dh^{n-1} \right)  \vec{\bf G}_{\vec{\bf v}}^{(n)} + F_{\vec{\bf v}}^{n}/dh^{n-1} \right)
312  - \Delta t \nabla \phi_{hyd}^{n}  - \Delta t \nabla \phi_{hyd}^{n}
313  \label{eq:vstar-nlfs} \\  \label{eq:vstar-nlfs}
314    \mathrm{update}\hspace{-4mm} & & \hspace{-4mm}\mathrm{  \end{eqnarray}
315    model~geometry~:~{\bf hFac}}(dh^n)\nonumber \\  \hspace{3cm}$\longrightarrow$~~{\it update model~geometry~:~}${\bf hFac}(dh^n)$\\
316    \begin{eqnarray}
317  \eta^{n+1/2} \hspace{-2mm} & = &  \eta^{n+1/2} \hspace{-2mm} & = &
318  \eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t  \eta^{n-1/2} + \Delta t P^{n+1/2} - \Delta t
319    \nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} \nonumber \\    \nabla \cdot \int \vec{\bf v}^{n+1/2} dh^{n} \nonumber \\
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