<p><b>mpetersen@lanl.gov</b> 2012-03-26 14:54:38 -0600 (Mon, 26 Mar 2012)</p><p>Update ALE design document to describe placement of uBolusGM in split explicit doc.<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 2012-03-26 19:48:50 UTC (rev 1723)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.pdf 2012-03-26 20:54:38 UTC (rev 1724)
@@ -7,7 +7,7 @@
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trailer
<< /Size 155 /Root 1 0 R /Info 2 0 R
-/ID [<3E664BA4E9D3F04929D3098E1AC1A131><3E664BA4E9D3F04929D3098E1AC1A131>]
+/ID [<598457ED6C39D21265C87FFC835A133F><598457ED6C39D21265C87FFC835A133F>]
>>
startxref
-133077
+133445
%%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 2012-03-26 19:48:50 UTC (rev 1723)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.tex 2012-03-26 20:54:38 UTC (rev 1724)
@@ -364,7 +364,8 @@
p_k^*=p_{k,n}, \;\;
{\varphi}_k^* = {\varphi}_{k,n},\\&&
h_{k,*} = h_{k,n}, \;\; h_{k,*}^{edge} = h_{k,n}^{edge},
-\zeta_{*} = \zeta_n \\&&
+\zeta_{*} = \zeta_n,\;\;
+{\bf u}_{k,*}^{bolus} = {\bf u}_{k,n}^{bolus} \\&&
h^{*hf}_k=h^{hf}_{k,n},\;\;
D^{*lf}_k=D^{lf}_{k,n} \\ &&
\ubtr_n = \left. \textstyle \sum_{k=1}^{N^{edge}}
@@ -470,22 +471,20 @@
{\overline {\bf F}} = \frac{1}{2J} \sum_{j=1}^{2J}{\bf F}_j \\&&
\mbox{boundary update on }{\overline {\bf F}}
\\&&
-{\bf u}_{k,avg} = \ubtr_{avg} + {\bf u}'_{k,n+1/2}
-\\ &&
-\mbox{compute ${\bf u}_{k,bolus}$ from ${\bf u}_{k,avg}$}
-\\ &&
{\bf u}^{corr} = \left( {\overline {\bf F}}
- \sum_{k=1}^{N^{edge}} h_{k,*}^{edge}
- \left( {\bf u}_{k,avg} + {\bf u}_{k,bolus}\right) \right)
+ \left( \ubtr_{avg} + {\bf u}'_{k,n+1/2} + {\bf u}_{k,*}^{bolus}\right) \right)
\left/ \sum_{k=1}^{N^{edge}} h_{k,*}^{edge} \right. \label{ucorr}
\\ &&
-{\bf u}^{tr}_k = {\bf u}_{k,avg} + {\bf u}_{k,bolus} + {\bf u}^{corr} \label{utr}
+{\bf u}^{tr}_k = \ubtr_{avg} + {\bf u}'_{k,n+1/2} + {\bf u}_{k,*}^{bolus} + {\bf u}^{corr} \label{utr}
\end{eqnarray}
+where ${\bf u}_{k,*}^{bolus}$ is the GM bolus velocity computed at the end of stage 3, and ${\bf u}^{tr}$ is the transport velocity used in the advection terms for for thickness and tracers.
+
For unsplit explicit, skip all computations in stage 2. Instead, set:
\bea
&&
\ubtr_{avg} = 0 \\ &&
-{\bf u}^{tr}_{k} = {\bf u}'_{k,n+1/2}
+{\bf u}^{tr}_{k} = {\bf u}'_{k,n+1/2} + {\bf u}_{k,*}^{bolus}
\eea
{\bf Discussion:} The derivation of ${\bf u}^{corr}$ is as follows. For consistency between the barotropic and summed baroclinic thickness flux, we must enforce
@@ -501,6 +500,7 @@
\mbox{compute tendencies }
T^{hhf}, \;\;
T^{Dlf} \\ &&
+\mbox{boundary update on tendencies: } T^{hhf},T^{Dlf}\\&&
\mbox{ALE step: compute vertical transport $w^{top}_*$}
\mbox{ (branch for different grid types)}\\ &&
\mbox{ based on } u^{tr}, h^* \\ &&
@@ -516,7 +516,7 @@
\left( \kappa_v \frac{\partial \varphi_k^*}{\partial z} \right)
\right)
\\ &&
-\mbox{boundary update on all tendencies: } T^h,T^\varphi,T^{hhf},T^{Dlf}\\&&
+\mbox{boundary update on tendencies: } T^h,T^\varphi\\&&
h_{k,n+1} = h_{k,n} + \Delta t T^h_k \\&&
\label{phi3}
{\varphi}_{k,n+1} = \frac{1}{h_{k,n+1}} \left[
@@ -533,8 +533,8 @@
\begin{array}{ll}
\mbox{\bf if iterating} & \mbox{\bf after final iteration}\smallskip\\
{\bf u}'_{k,*} = {\bf u}'_{k,n+1/2} & {\bf u}'_{k,n+1} \mbox{ from stage 1}\smallskip\\
-\ubtr_* = \ubtr_{avg} & \ubtr_{n+1} = \ubtr_{avg} \mbox{ from stage 2}\smallskip\\
-{\bf u}_{k,*} = {\bf u}_{k}^{tr} \mbox{ (could be $u_{k}^{avg}$, doesn't matter)} & {\bf u}_{k,n+1} = \ubtr_{n+1} + {\bf u}'_{k,n+1}\smallskip\\
+\ubtr_{*} = \ubtr_{avg} \mbox{ from stage 2} & \ubtr_{n+1} = \ubtr_{avg} \mbox{ from stage 2}\smallskip\\
+{\bf u}_{k,*} = \ubtr_{*} + {\bf u}'_{k,*} & {\bf u}_{k,n+1} = \ubtr_{n+1} + {\bf u}'_{k,n+1}\smallskip\\
h_{k,*} = \frac{1}{2}\left(h_{k,n} +h_{k,n+1}\right)
& h_{k,n+1} \mbox{ from stage 3} \smallskip\\
\varphi_{k,*} = \frac{1}{2}\left(\varphi_{k,n} +\varphi_{k,n+1}\right)
@@ -557,7 +557,9 @@
h^{edge}_{k,*} = interp(h_{k,*}) &
h^{edge}_{k,n+1} = interp(h_{k,n+1}) \smallskip\\
\zeta_* = \sum_{k=1}^{kmax}h_{k,*} - H &
-\zeta_{n+1} = \sum_{k=1}^{kmax}h_{k,n+1} - H
+\zeta_{n+1} = \sum_{k=1}^{kmax}h_{k,n+1} - H \\
+\mbox{compute }{\bf u}_{k,*}^{bolus} &
+\mbox{compute }{\bf u}_{k,n+1}^{bolus}
\end{array}
\eea
where H is the total column height with zero sea surface height.
</font>
</pre>