/[MITgcm]/manual/s_algorithm/text/nonlin_frsurf.tex
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revision 1.2 by jmc, Fri Aug 17 18:38:10 2001 UTC revision 1.3 by jmc, Mon Sep 24 19:30:40 2001 UTC
# Line 16  The linear relation between Line 16  The linear relation between
16  surface pressure / geo- potential ($\Phi_{surf}$)  surface pressure / geo- potential ($\Phi_{surf}$)
17  and surface displacement ($\eta$)  and surface displacement ($\eta$)
18  has been considered as uniform ($b_s =$ Constant)  has been considered as uniform ($b_s =$ Constant)
19  but is in fact  \marginpar{add a reference to part.1 here}
20  dependent on the position ($x,y,r$)  but is in fact dependent on the position ($x,y,r$)
21  since we linearize:  since we linearize:
22  $$\Phi_{surf}=\int_{R_o}^{R_o+\eta} b dr \simeq b_s \eta  $$\Phi_{surf}=\int_{R_o}^{R_o+\eta} b dr \simeq b_s \eta
23  ~\mathrm{with}~ b_s = b(\theta,S,r)_{r=R_o}  ~\mathrm{with}~ b_s = b(\theta,S,r)_{r=R_o}
# Line 54  $b_s = c_p \kappa (P_o / Pc)^{(\kappa - Line 54  $b_s = c_p \kappa (P_o / Pc)^{(\kappa -
54  (Non-linear free surface)}  (Non-linear free surface)}
55    
56  The total thickness of the fluid column is  The total thickness of the fluid column is
57  $r_{surf} - R_{min} = \eta + R_o - R_{min}$  $r_{surf} - R_{fixed} = \eta + R_o - R_{fixed}$
58  In the linear free surface approximation  In the linear free surface approximation
59  (detailed before), only the fixed part of  (detailed before), only the fixed part of
60  it ($R_o - R_{min})$ is considered when we integrate the  it ($R_o - R_{fixed})$ is considered when we integrate the
61  continuity equation or compute tracer and momentum advection term.  continuity equation or compute tracer and momentum advection term.
62    
63  This approximation is dropped when using  This approximation is dropped when using
# Line 72  part. Line 72  part.
72  The continuous form of the model equations remains  The continuous form of the model equations remains
73  unchanged, except for the 2D continuity equation  unchanged, except for the 2D continuity equation
74  (\ref{eq-tCsC-eta}) that is now integrated  (\ref{eq-tCsC-eta}) that is now integrated
75  from $R_{min}(x,y)$ up to $r_{surf}=R_o+\eta$ :  from $R_{fixed}(x,y)$ up to $r_{surf}=R_o+\eta$ :
76    
77  \begin{displaymath}  \begin{displaymath}
78  \epsilon_{fs} \partial_t \eta =  \epsilon_{fs} \partial_t \eta =
79  \left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) =  \left. \dot{r} \right|_{r=r_{surf}} + \epsilon_{fw} (P-E) =
80  - {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o+\eta} \vec{\bf v} dr  - {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o+\eta} \vec{\bf v} dr
81  + \epsilon_{fw} (P-E)  + \epsilon_{fw} (P-E)
82  \end{displaymath}  \end{displaymath}
83    
# Line 94  equation (\ref{eq-solve2D} \& \ref{eq-so Line 94  equation (\ref{eq-solve2D} \& \ref{eq-so
94  become:  become:
95  \begin{eqnarray*}  \begin{eqnarray*}
96  \epsilon_{fs} {\eta}^{n+1} -  \epsilon_{fs} {\eta}^{n+1} -
97  {\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}+R_o-R_{min})  {\bf \nabla}_h \cdot \Delta t^2 (\eta^{n}+R_o-R_{fixed})
98  {\bf \nabla}_h b_s {\eta}^{n+1}  {\bf \nabla}_h b_s {\eta}^{n+1}
99  = {\eta}^*  = {\eta}^*
100  %\label{solve_2d}  %\label{solve_2d}
# Line 102  become: Line 102  become:
102  where  where
103  \begin{eqnarray*}  \begin{eqnarray*}
104  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -  {\eta}^* = \epsilon_{fs} \: {\eta}^{n} -
105  \Delta t {\bf \nabla}_h \cdot \int_{R_{min}}^{R_o+\eta^n} \vec{\bf v}^* dr  \Delta t {\bf \nabla}_h \cdot \int_{R_{fixed}}^{R_o+\eta^n} \vec{\bf v}^* dr
106  \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}  \: + \: \epsilon_{fw} \Delta_t (P-E)^{n}
107  %\label{solve_2d_rhs}  %\label{solve_2d_rhs}
108  \end{eqnarray*}  \end{eqnarray*}
# Line 112  Alternatively, the non-linear contributi Line 112  Alternatively, the non-linear contributi
112  explicitly:  explicitly:
113  \begin{eqnarray*}  \begin{eqnarray*}
114  \epsilon_{fs} {\eta}^{n+1} -  \epsilon_{fs} {\eta}^{n+1} -
115  {\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{min})  {\bf \nabla}_h \cdot \Delta t^2 (R_o-R_{fixed})
116  {\bf \nabla}_h b_s {\eta}^{n+1}  {\bf \nabla}_h b_s {\eta}^{n+1}
117  = {\eta}^*  = {\eta}^*
118  +{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n})  +{\bf \nabla}_h \cdot \Delta t^2 (\eta^{n})
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