<p><b>mpetersen@lanl.gov</b> 2012-02-03 10:02:23 -0700 (Fri, 03 Feb 2012)</p><p>Updating ale vertical coordinate design doc. The algorithm section now reflects exactly what is in the ale branch. Descriptions of the input flags were placed in the split exlicit section.<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 2012-02-03 00:28:54 UTC (rev 1459)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.pdf 2012-02-03 17:02:23 UTC (rev 1460)
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%%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 2012-02-03 00:28:54 UTC (rev 1459)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.tex 2012-02-03 17:02:23 UTC (rev 1460)
@@ -60,7 +60,7 @@
%-----------------------------------------------------------------------
\chapter{Algorithmic Formulations}
-Date last modified: 11/28/2011 \\
+Date last modified: 2/3/2012 \\
Contributors: Mark, Todd \\
\section{ALE formulation}
@@ -378,6 +378,8 @@
{\bf u}_{k,n+1/2}'={\bf u}_{k,n}'
\end{eqnarray}
+The full algorithm, Stages 1--3 are iterated. This is typically done using two iterations, like a predictor--corrector timestep. The flag for the number of these large iterations is {\tt config\_n\_ts\_iter}.
+
{\bf Stage 1: Baroclinic velocity (3D), explicit with long timestep}\\
Iterate on linear Coriolis term only.
\begin{eqnarray} &&
@@ -388,7 +390,7 @@
\left\{\begin{array}{l} \label{u bcl s2}
\mbox{compute }{\bf u}_{k,n+1/2}^{'\;\perp}\mbox{ from }{\bf u}_{k,n+1/2}'\\
{\tilde {\bf u}}'_{k,n+1} = {\bf u}'_{k,n} + \Delta t
-\left( -f{\bf u}_{k,n+1/2}^{'\;\perp} + {\bf T}({\bf u}_k^*,w_k^*,p_k^*)
+\left( -f{\bf u}_{k,n+1/2}^{'\;\perp} + {\bf T}^u({\bf u}_k^*,w_k^*,p_k^*)
+g</font>
<font color="gray">abla \zeta^* \right)
\\
{\overline {\bf G}} =
@@ -405,11 +407,17 @@
\right. \;\;\; l=1\ldots L \\ &&
\end{eqnarray}
+The bracketed computation is iterated $L$ times. The default method is to use $L=1$ on the first time through Stages 1--3, and $L=2$ the second time through. This is set by the flags {\tt
+config\_n\_bcl\_iter\_beg = 1, config\_n\_bcl\_iter\_mid = 2, config\_n\_bcl\_iter\_end = 2}.
+In the case of two iterations through Stages 1--3, the flag {\tt config\_n\_bcl\_iter\_mid} is not used.
+
+</font>
<font color="gray">ewpage
{\bf Stage 2: Barotropic velocity (2D), explicitly subcycled}\\
-Advance $\ubtr$ and $\zeta$ as a coupled system.
+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} \\
{\tilde \ubtr}_{n+j/J} = \ubtr_{n+(j-1)/J} + \frac{\Delta t}{J} \left(
- f\ubtr_{n+(j-1)/J}^{\perp}
@@ -421,7 +429,8 @@
\left(\zeta_{n+(j-1)/J}^{edge} + \sum_{k=1}^{N^{edge}} \Delta z_k \right)\\
\tilde\zeta_{n+j/J} = \zeta_{n+(j-1)/J} + \frac{\Delta t}{J} \left( -
</font>
<font color="gray">abla \cdot \tilde{\bf F}_j \right) \\
-\mbox{boundary update on } \tilde\zeta_{n+j/J} \\
+\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} \\
\ubtr_{n+j/J} = \ubtr_{n+(j-1)/J} + \frac{\Delta t}{J} \left(
- f{\tilde \ubtr}_{n+j/J}^{\perp}
@@ -434,20 +443,26 @@
</font>
<font color="red">abla \cdot {\bf F}_j \right) \\
\mbox{boundary update on } \zeta_{n+j/J} \\
\end{array}
-\right. \;\;\; j=1\ldots 2J \hspace{1cm}
+\right.
\end{eqnarray}
-Note: The last three lines may be iterated to update the velocity in the Coriolis term. ${\overline {\bf G}}_j$ may vary over the barotropic subcycles, as long as
-$\frac{1}{2J}\sum_{j=1}^{2J}{\overline {\bf G}}_j={\overline {\bf G}}$.
+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.
+
+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 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}\\&&
{\overline {\bf F}} = \frac{1}{2J} \sum_{j=1}^{2J}{\bf F}_j \\&&
\mbox{boundary update on }{\overline {\bf F}}
\\&&
-{\overline D}_* = -</font>
<font color="red">abla \cdot {\overline F}
-\\&&
-\zeta_{avg} = \zeta_{n} + \Delta t \left( - </font>
<font color="gray">abla \cdot {\overline {\bf F}} \right)
-\\ &&
{\bf u}_{k,avg} = \ubtr_{avg} + {\bf u}'_{k,n+1/2}
\\ &&
{\bf u}^{corr} = \left( {\overline {\bf F}}
@@ -460,19 +475,18 @@
\bea
&&
\ubtr_{avg} = 0 \\ &&
-{\bf u}^{tr}_{k} = {\bf u}'_{k,n+1/2} \\ &&
-{\overline D}_* = -</font>
<font color="red">abla \cdot \left( h_k^{*\; edge} {\bf u}_k^{tr} \right)
+{\bf u}^{tr}_{k} = {\bf u}'_{k,n+1/2}
\eea
-
-{\bf Stage 3}
+</font>
<font color="red">ewpage
+{\bf Stage 3: update thickness, tracers, density and pressure}
\bea &&
\mbox{compute tendencies }
T^{hhf}, \;\;
T^{Dlf} \\ &&
\mbox{ALE step: compute vertical transport $w^{top}_*$}
\mbox{ (branch for different grid types)}\\ &&
-\mbox{ based on }{\overline D}^*, u^{tr}, h^* \\ &&
+\mbox{ based on } u^{tr}, h^* \\ &&
T^h_k = \left( - </font>
<font color="gray">abla \cdot \left( h_k^{*\; edge} {\bf u}_k^{tr} \right)
- \frac{\partial}{\partial z} \left( h_k^* w_k^* \right) \right)
%-D_k^* + w_{k+1}^* - w_{k}^*
@@ -485,7 +499,7 @@
\left( \kappa_v \frac{\partial \varphi_k^*}{\partial z} \right)
\right)
\\ &&
-\mbox{boundary update on all tendencies: } {\bf T}^u,T^h,T^{hhf},T^{Dlf},T^\varphi\\&&
+\mbox{boundary update on all tendencies: } T^h,T^\varphi,T^{hhf},T^{Dlf}\\&&
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[
@@ -504,8 +518,6 @@
{\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\\
-\zeta_* = \zeta_{avg} &
-\zeta_{n+1} = \zeta_{avg} \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)
@@ -514,6 +526,8 @@
& h^{hf}_{k,n+1} \mbox{ from stage 3} \smallskip\\
D^{lf}_{k,*} = \frac{1}{2}\left(D^{lf}_{k,n} +D^{lf}_{k,n+1}\right) &
D^{lf}_{k,n+1} \mbox{ from stage 3}\smallskip\\ \label{rho3}
+\mbox{diagnostics:}
+& \mbox{diagnostics:} \\
{\rho}_k^* = EOS(T^*_k, S^*_k)
& {\rho}_{k,n+1} = EOS(T_{k,n+1}, S_{k,n+1}) \smallskip\\
\label{p3}
@@ -524,15 +538,16 @@
{\rho}_{k',n+1} h_{k',n+1}
+ \frac{1}{2}g{\rho}_{k,n+1}^* h_{k,n+1} \smallskip \\
h^{edge}_{k,*} = interp(h_{k,*}) &
-h^{edge}_{k,n+1} = interp(h_{k,n+1})
+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
\end{array}
\eea
+where H is the total column height with zero sea surface height.
-
%-----------------------------------------------------------------------
-\section{Other}
-Check \cite{Leclair:2011ud}: eqns 17-18 is conservative vs nonconservative schemes. This is the method of vertical advection for us - does it effect our coordinate scheme?
+Question: Check \cite{Leclair:2011ud}: eqns 17-18 shows conservative vs nonconservative schemes. This is the method of vertical advection for us - does it effect our coordinate scheme?
\chapter{Design and Implementation}
Date last modified: 11/28/2011 \\
</font>
</pre>