/[MITgcm]/manual/s_algorithm/text/time_stepping.tex
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revision 1.4 by jmc, Fri Aug 17 18:38:10 2001 UTC revision 1.5 by jmc, Mon Sep 24 19:30:40 2001 UTC
# Line 77  and equation \ref{eq-tCsC-Vmom} vanishes Line 77  and equation \ref{eq-tCsC-Vmom} vanishes
77    
78  The equation for $\eta$ is obtained by integrating the  The equation for $\eta$ is obtained by integrating the
79  continuity equation over the entire depth of the fluid,  continuity equation over the entire depth of the fluid,
80  from $R_{min}(x,y)$ up to $R_o(x,y)$  from $R_{fixed}(x,y)$ up to $R_o(x,y)$
81  (Linear free surface):  (Linear free surface):
82  \begin{eqnarray}  \begin{eqnarray}
83  \epsilon_{fs} \partial_t \eta =  \epsilon_{fs} \partial_t \eta =
84  \left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) =  \left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) =
85  - {\bf \nabla} \cdot \int_{R_{min}}^{R_o} \vec{\bf v} dr  - {\bf \nabla} \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v} dr
86  + \epsilon_{fw} (P-E)  + \epsilon_{fw} (P-E)
87  \label{eq-tCsC-eta}  \label{eq-tCsC-eta}
88  \end{eqnarray}  \end{eqnarray}
# Line 202  S^{n+1} & = & S^* Line 202  S^{n+1} & = & S^*
202  \label{eq-tDsC-Hmom}  \label{eq-tDsC-Hmom}
203  \\  \\
204  \epsilon_{fs} {\eta}^{n+1} + \Delta t  \epsilon_{fs} {\eta}^{n+1} + \Delta t
205  {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^{n+1} dr  {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^{n+1} dr
206  & = &  & = &
207      \epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n}      \epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n}
208  \nonumber  \nonumber
# Line 311  And Line 311  And
311  %\label{eq-tDsC-Hmom}  %\label{eq-tDsC-Hmom}
312  \\  \\
313  \epsilon_{fs} {\eta}^{n+1} + \Delta t  \epsilon_{fs} {\eta}^{n+1} + \Delta t
314  {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^{n+1} dr  {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^{n+1} dr
315  & = &  & = &
316  \epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n}  \epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta_t (P-E)^{n}
317  \\  \\
# Line 343  Substituting \ref{eq-tDsC-Hmom} into \re Line 343  Substituting \ref{eq-tDsC-Hmom} into \re
343  $\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$:  $\epsilon_{nh} = 0$ yields a Helmholtz equation for ${\eta}^{n+1}$:
344  \begin{eqnarray}  \begin{eqnarray}
345  \epsilon_{fs} {\eta}^{n+1} -  \epsilon_{fs} {\eta}^{n+1} -
346  {\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{min})  {\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed})
347  {\bf \nabla}_h b_s {\eta}^{n+1}  {\bf \nabla}_h b_s {\eta}^{n+1}
348  = {\eta}^*  = {\eta}^*
349  \label{eq-solve2D}  \label{eq-solve2D}
# Line 351  $\epsilon_{nh} = 0$ yields a Helmholtz e Line 351  $\epsilon_{nh} = 0$ yields a Helmholtz e
351  where  where
352  \begin{eqnarray}  \begin{eqnarray}
353  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -
354  \Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o} \vec{\bf v}^* dr  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o} \vec{\bf v}^* dr
355  \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}  \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}
356  \label{eq-solve2D_rhs}  \label{eq-solve2D_rhs}
357  \end{eqnarray}  \end{eqnarray}
# Line 440  $$ Line 440  $$
440  $$  $$
441  $$  $$
442  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}  \epsilon_{fs} \frac{ {\eta}^{n+1} - {\eta}^{n} }{ \Delta t}
443  + {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o}  + {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o}
444  [ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr  [ \gamma \vec{\bf v}^{n+1} + (1-\gamma) \vec{\bf v}^{n}] dr
445  = \epsilon_{fw} (P-E)  = \epsilon_{fw} (P-E)
446  $$  $$
# Line 453  where: Line 453  where:
453  \\  \\
454  {\eta}^* & = &  {\eta}^* & = &
455  \epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta t (P-E)  \epsilon_{fs} {\eta}^{n} + \epsilon_{fw} \Delta t (P-E)
456  - \Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o}  - \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o}
457  [ \gamma \vec{\bf v}^* + (1-\gamma) \vec{\bf v}^{n}] dr  [ \gamma \vec{\bf v}^* + (1-\gamma) \vec{\bf v}^{n}] dr
458  \end{eqnarray*}  \end{eqnarray*}
459  \\  \\
# Line 461  In the hydrostatic case ($\epsilon_{nh}= Line 461  In the hydrostatic case ($\epsilon_{nh}=
461  this allow to find ${\eta}^{n+1}$, according to:  this allow to find ${\eta}^{n+1}$, according to:
462  $$  $$
463  \epsilon_{fs} {\eta}^{n+1} -  \epsilon_{fs} {\eta}^{n+1} -
464  {\bf \nabla}_h \cdot \beta\gamma \Delta t^2 b_s (R_o - R_{min})  {\bf \nabla}_h \cdot \beta\gamma \Delta t^2 b_s (R_o - R_{fixed})
465  {\bf \nabla}_h {\eta}^{n+1}  {\bf \nabla}_h {\eta}^{n+1}
466  = {\eta}^*  = {\eta}^*
467  $$  $$
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