Changeset 994 for trunk/DOC/TexFiles/Chapters/Chap_DYN.tex

Timestamp:

2008-05-28T11:01:09+02:00 (16 years ago)

Author:

gm

Message:

trunk - add steven correction + several other things + rename BETA into TexFiles?

Location:

trunk/DOC/TexFiles

Files:

• . (moved) (moved from trunk/DOC/BETA)
• Chapters/Chap_DYN.tex (copied) (copied from trunk/DOC/BETA/Chapters/Chap_DYN.tex) (8 diffs)

trunk/DOC/TexFiles/Chapters/Chap_DYN.tex

@@ -107 +107 @@
 \left\{ 
 \begin{aligned}
-{-\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} & {\overline{\overline {\left( {e_{1v} e_{3v} v} \right)}} }^{\,i, j+1/2}    \\
-{+\frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}  & {\overline{\overline {\left( {e_{2u} e_{3u} u} \right)}} }^{\,i+1/2, j}  
+{+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} & {\overline{\overline {\left( {e_{1v} e_{3v} v} \right)}} }^{\,i, j+1/2}    \\
+{-\frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}  & {\overline{\overline {\left( {e_{2u} e_{3u} u} \right)}} }^{\,i+1/2, j}  
 \end{aligned} 
  \right.
@@ -124 +124 @@
 \left\{ {
 \begin{aligned}
-{-\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
+{+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
 \;\overline {\left( {e_{1v} e_{3v} v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
-{+\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
+{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
 \;\overline {\left( {e_{2u} e_{3u} u} \right)} ^{\,j+1/2}} }^{\,i} }
 \end{aligned} 
@@ -145 +145 @@
 \left\{ {
 \begin{aligned}
- {-\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
+ {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
  \; {\overline{\overline {\left( {e_{1v} \; e_{3v} \ v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
  \; {\overline {\left( {\frac{f}{e_{3f} }} \right) 
  \;\overline {\left( {e_{1v} \; e_{3v} \ v} \right)} ^{\,i+1/2}} }^{\,j} } \\
-{+\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
+{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
  \; {\overline{\overline {\left( {e_{2u} \; e_{3u} \ u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
  \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
@@ -220 +220 @@
 -q\,e_3 \,u       &\equiv -\frac{1}{e_{2v} }  \left[ 
 {{\begin{array}{*{20}c}
-   {\,\ \ a_{j-1/2}^{i   }  \left( {e_{2u} e_{3v} \ u} \right)_{j+1}^{i+1/2} } 
-   {\,+\,b_{j-1/2}^{i+1}  \left( {e_{2u} e_{3v} \ u} \right)_{j+1/2}^{i+1} } \\
+   {\,\ \ a_{j-1/2}^{i   }  \left( {e_{2u} e_{3u} \ u} \right)_{j+1}^{i+1/2} } 
+   {\,+\,b_{j-1/2}^{i+1}  \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i+1} } \\
  \\
-      {  +\,c_{j+1/2}^{i+1} \left( {e_{2u} e_{3v} \ u} \right)_{j+1/2}^{i+1} } 
-   {\,+\,d_{j+1/2}^{i   }  \left( {e_{2u} e_{3v} \ u} \right)_{j+1/2}^{i   } } \\
+      {  +\,c_{j+1/2}^{i+1} \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i+1} } 
+   {\,+\,d_{j+1/2}^{i   }  \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i   } } \\
 \end{array} }} \right]
 \end{aligned} 
@@ -670 +670 @@
 \right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)} 
 \end{equation}
-
-where EMP is the surface freshwater budget, expressed in $Kg.m^{-2}.s^{-1}$, 
-and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. If river 
-runoff is expressed as a surface freshwater flux, see \S\ref{SBC}, then EMP 
-can be written as the evaporation minus precipitation, minus the river runoff. 
-The sea-surface height is evaluated using a leapfrog scheme in combination 
-with an Asselin time filter, $i.e.$ the velocity appearing in 
-\eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity). 
+where EMP is the surface freshwater budget, expressed in Kg/m$^2$/s 
+(which is equal to mm/s), and $\rho _w$=1,000~Kg/m$^3$ is the volumic 
+mass of pure water. If river runoff is expressed as a surface freshwater flux 
+(see \S\ref{SBC}) then EMP can be written as the evaporation minus 
+precipitation, minus the river runoff. The sea-surface height is evaluated 
+using a leapfrog scheme in combination with an Asselin time filter, $i.e.$ 
+the velocity appearing in \eqref{Eq_dynspg_ssh} is centred in time 
+(\textit{now} velocity). 
 
 The surface pressure gradient, also evaluated using a leap-frog scheme, is 
@@ -683 +683 @@
 \begin{equation} \label{Eq_dynspg_exp}
 \left\{ \begin{aligned}
- - \frac{1}                      {e_{1u}} \; \delta _{i+1/2} \left[  \,\eta\,  \right]    \\
- \\
- - \frac{1}                      {e_{2v}} \; \delta _{j+1/2} \left[  \,\eta\,  \right]  
+ - \frac{1}{e_{1u}} \;  \delta _{i+1/2} \left[  \,\eta\,  \right]    \\
+ - \frac{1}{e_{2v}} \;  \delta _{j+1/2} \left[  \,\eta\,  \right]  
 \end{aligned} \right.
 \end{equation} 
@@ -704 +703 @@
 proposed by \citet{Griffies2004}. The general idea is to solve the free surface 
 equation with a small time step \np{rdtbt}, while the three dimensional 
-prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt} 
-(Fig.\ref {Fig_DYN_dynspg_ts}). 
+prognostic variables are solved with a longer time step that is a multiple of 
+\np{rdtbt} (Fig.\ref {Fig_DYN_dynspg_ts}). 
 
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
@@ -711 +710 @@
 \begin{center}
 \includegraphics[width=0.90\textwidth]{./Figures/Fig_DYN_dynspg_ts.pdf}
-\caption{Schematic of the split-explicit time stepping scheme for the barotropic 
-and baroclinic modes, after \citet{Griffies2004}. Time increases to the right. 
-Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
-The curved line represents a leap-frog time step, and the smaller barotropic time 
-steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. The vertically 
-integrated forcing \textbf{M}(t) computed at the baroclinic time step $t$ 
-represents the interaction between the barotropic and baroclinic motions. 
-While keeping the total depth, tracer, and freshwater forcing fields fixed, a 
-leap-frog integration carries the surface height and vertically integrated velocity 
-from $t$ to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 
-Time averaging the barotropic fields over the N+1 time steps (endpoints 
-included) centers the vertically integrated velocity at the baroclinic timestep 
-$t+\Delta t$. A baroclinic leap-frog time step carries the surface height to 
+\caption{Schematic of the split-explicit time stepping scheme for the external 
+and internal modes. Time increases to the right. 
+Internal mode time steps (which are also the model time steps) are denoted 
+by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
+The curved line represents a leap-frog time step, and the smaller time 
+steps $N \Delta t_e=\frac{3}{2}\Delta t$ are denoted by the zig-zag line. The vertically 
+integrated forcing \textbf{M}(t) computed at the model time step $t$ 
+represents the interaction between the external and internal motions. 
+While keeping \textbf{M} and freshwater forcing field fixed, a 
+leap-frog integration carries the external mode variables (surface height and vertically integrated velocity) from $t$ to $t+\frac{3}{2} \Delta t$ using N external time steps of length $\Delta t_e$. 
+Time averaging the external fields over the $\frac{2}{3}N+1$ time steps (endpoints 
+included) centers the vertically integrated velocity and the sea surface height at the model timestep $t+\Delta t$. These averaged values are used to update \textbf{M}(t) with both the surface pressure gradient and the Coriolis force. 
+A baroclinic leap-frog time step carries the surface height to The model time stepping scheme can then be achieved by 
 $t+\Delta t$ using the convergence of the time averaged vertically integrated 
 velocity taken from baroclinic time step t. }