<p><b>mpetersen@lanl.gov</b> 2012-02-08 12:28:52 -0700 (Wed, 08 Feb 2012)</p><p>Revised ale design doc. I just clarified Stage 2 in section 3.7, writing each predictor and corrector step separately.<br>
</p><hr noshade><pre><font color="gray">Modified: trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.pdf
===================================================================
--- trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.pdf 2012-02-08 19:24:48 UTC (rev 1485)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.pdf 2012-02-08 19:28:52 UTC (rev 1486)
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startxref
-131801
+131456
%%EOF
Modified: trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.tex
===================================================================
--- trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.tex 2012-02-08 19:24:48 UTC (rev 1485)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.tex 2012-02-08 19:28:52 UTC (rev 1486)
@@ -6,7 +6,7 @@
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\evensidemargin}{0in}
-\setlength{\textheight}{8.0in}
+\setlength{\textheight}{9.0in}
\setlength{\topmargin}{0in}
</font>
<font color="gray">ewcommand{\ds}{\displaystyle}
@@ -14,6 +14,7 @@
\setlength{\parindent}{0mm}
</font>
<font color="black">ewcommand{\bea}{\begin{eqnarray}}
</font>
<font color="blue">ewcommand{\eea}{\end{eqnarray}}
+</font>
<font color="black">ewcommand{</font>
<font color="black">n}{</font>
<font color="gray">onumber}
\begin{document}
@@ -416,47 +417,53 @@
Advance $\ubtr$ and $\zeta$ as a coupled system through $2J$ subcycles, ending at time $t+2\Delta t$.
\begin{eqnarray}
\label{u btr 5} &&
-\left\{\begin{array}{l}
-\mbox{{\bf predictor step:}} \\
-\mbox{compute }\ubtr_{n+(j-1)/J}^{\perp}\mbox{ from }\ubtr_{n+(j-1)/J} \\
+\mbox{{\bf velocity predictor step:}} </font>
<font color="blue">n \\ &&
+\mbox{compute }\ubtr_{n+(j-1)/J}^{\perp}\mbox{ from }\ubtr_{n+(j-1)/J} \\ &&
{\tilde \ubtr}_{n+j/J} = \ubtr_{n+(j-1)/J} + \frac{\Delta t}{J} \left(
- f\ubtr_{n+(j-1)/J}^{\perp}
- g</font>
<font color="red">abla \zeta_{n+(j-1)/J} + {\overline {\bf G}_j}
-\right)\\
-\mbox{boundary update on }{\tilde \ubtr}_{n+j/J}\\
-\zeta_{n+(j-1)/J}^{edge} = Interp(\zeta_{n+(j-1)/J}) \\
+\right)\\ &&
+\mbox{boundary update on }{\tilde \ubtr}_{n+j/J}\\ &&
+</font>
<font color="blue">n \\ &&
+\mbox{{\bf SSH predictor step:}}</font>
<font color="red">n \\ &&
+\zeta_{n+(j-1)/J}^{edge} = Interp(\zeta_{n+(j-1)/J}) \\ &&
\tilde{\bf F}_j = \left((1-\gamma_1){\ubtr}_{n+(j-1)/J} + \gamma_1{\tilde \ubtr}_{n+j/J}\right)
-\left(\zeta_{n+(j-1)/J}^{edge} + \sum_{k=1}^{N^{edge}} \Delta z_k \right)\\
+\left(\zeta_{n+(j-1)/J}^{edge} + H^{edge} \right)\\ &&
\tilde\zeta_{n+j/J} = \zeta_{n+(j-1)/J} + \frac{\Delta t}{J} \left( -
- </font>
<font color="red">abla \cdot \tilde{\bf F}_j \right) \\
-\mbox{boundary update on } \tilde\zeta_{n+j/J} \\ \\
-\mbox{{\bf corrector step:}} \\
-\mbox{compute }{\tilde \ubtr}_{n+j/J}^{\perp}\mbox{ from }{\tilde \ubtr}_{n+j/J} \\
+ </font>
<font color="blue">abla \cdot \tilde{\bf F}_j \right) \\ &&
+\mbox{boundary update on } \tilde\zeta_{n+j/J} \\ &&
+</font>
<font color="blue">n \\ &&
+\mbox{{\bf velocity corrector step:}}</font>
<font color="blue">n \\ &&
+\mbox{compute }{\tilde \ubtr}_{n+j/J}^{\perp}\mbox{ from }{\tilde \ubtr}_{n+j/J} \\ &&
\ubtr_{n+j/J} = \ubtr_{n+(j-1)/J} + \frac{\Delta t}{J} \left(
- f{\tilde \ubtr}_{n+j/J}^{\perp}
- g</font>
<font color="red">abla \left((1-\gamma_2)\zeta_{n+(j-1)/J} + \gamma_2\tilde\zeta_{n+j/J}\right) + {\overline {\bf G}_j}
-\right),\\
-\mbox{boundary update on }\ubtr_{n+j/J}\\
+\right),\\ &&
+\mbox{boundary update on }\ubtr_{n+j/J}\\ &&
+</font>
<font color="blue">n\\ &&
+\mbox{{\bf SSH corrector step:}}</font>
<font color="red">n \\ &&
+\tilde\zeta_{n+j/J}^{\;edge}= Interp\left((1-\gamma_2)\zeta_{n+(j-1)/J} + \gamma_2\tilde\zeta_{n+j/J}\right) \\&&
{\bf F}_j = \left((1-\gamma_3){\ubtr}_{n+(j-1)/J} + \gamma_3 \ubtr_{n+j/J}\right)
-\left(\zeta_{n+(j-1)/J}^{edge} + \sum_{k=1}^{N^{edge}} \Delta z_k \right)\\
+\left(\tilde\zeta_{n+j/J}^{\;edge} + H^{edge} \right)\\ &&
\zeta_{n+j/J} = \zeta_{n+(j-1)/J} + \frac{\Delta t}{J} \left( -
- </font>
<font color="red">abla \cdot {\bf F}_j \right) \\
-\mbox{boundary update on } \zeta_{n+j/J} \\
-\end{array}
-\right.
+ </font>
<font color="red">abla \cdot {\bf F}_j \right) \\ &&
+\mbox{boundary update on } \zeta_{n+j/J}
\end{eqnarray}
-Repeat $j=1\ldots 2J$ to step through the barotropic subcycles. There is a predictor and corrector step for each subcycle. At the end of $2J$ subcycles, we have progressed through two baroclinic timesteps, i.e.\ through $2\Delta t$. In the code, {\tt config\_n\_btr\_subcycles} is $J$, {\tt config\_btr\_subcycle\_loop\_factor=2} is the ``2'' coefficient in $2J$. The input flag {\tt config\_btr\_solve\_SSH2=.true.} runs the algorithm as shown, while {\tt .false.} runs the predictor step only, without the corrector step.
+Repeat $j=1\ldots 2J$ to step through the barotropic subcycles. There are two predictor and two corrector steps for each subcycle. At the end of $2J$ subcycles, we have progressed through two baroclinic timesteps, i.e.\ through $2\Delta t$. In the code, {\tt config\_n\_btr\_subcycles} is $J$, while {\tt config\_btr\_subcycle\_loop\_factor=2} is the ``2'' coefficient in $2J$.
-The first three lines (computing $\ubtr^{\perp}$, $\ubtr$, and boundary update) of the predictor and corrector steps may be iterated to update the velocity in the Coriolis term. The number of iterations is controlled by {\tt config\_n\_btr\_cor\_iter}, and is usually set to two.
+The input flag {\tt config\_btr\_solve\_SSH2=.true.} runs the algorithm as shown, while {\tt .false.} does not include the SSH corrector step. Here $H^{edge}$ is the total column depth without SSH perturbations, that is, from the higher cell adjoining an edge to $z=0$.
+The velocity corrector step may be iterated to update the velocity in the Coriolis term. The number of iterations is controlled by {\tt config\_n\_btr\_cor\_iter}, and is usually set to two.
+
+</font>
<font color="red">ewpage
+{\bf Stage 2 continued}\\
+
The coefficients $(\gamma_1,\gamma_2,\gamma_3)$ control weighting between the old and new variables in the predictor velocity, corrector SSH gradient, and corrector velocity, respectively. These are typically set as $(\gamma_1,\gamma_2,\gamma_3)=(0.5,1,1)$, but this is open to investigation. These flags are
{\tt config\_btr\_gam1\_uWt1, config\_btr\_gam2\_SSHWt1, config\_btr\_gam3\_uWt2}.
The baroclinic forcing ${\overline {\bf G}}_j$ may vary over the barotropic subcycles, as long as
$\frac{1}{2J}\sum_{j=1}^{2J}{\overline {\bf G}}_j={\overline {\bf G}}$. This option is not currently implemented in the code.
-</font>
<font color="red">ewpage
-{\bf Stage 2 continued}\\
\begin{eqnarray} &&
\ubtr_{avg} = \frac{1}{2J+1} \sum_{j=0}^{2J} \ubtr_{n+j/J}
\label{ustar5}\\&&
</font>
</pre>