<p><b>mpetersen@lanl.gov</b> 2012-03-12 12:26:51 -0600 (Mon, 12 Mar 2012)</p><p>Updating some minor items on ale design 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        2012-03-12 15:54:42 UTC (rev 1618)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.pdf        2012-03-12 18:26:51 UTC (rev 1619)
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 trailer
 &lt;&lt; /Size 155 /Root 1 0 R /Info 2 0 R
-/ID [&lt;A98A4BA39EE2E4CA9178A253049D87FA&gt;&lt;A98A4BA39EE2E4CA9178A253049D87FA&gt;]
+/ID [&lt;3E664BA4E9D3F04929D3098E1AC1A131&gt;&lt;3E664BA4E9D3F04929D3098E1AC1A131&gt;]
 &gt;&gt;
 startxref
-131456
+133077
 %%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        2012-03-12 15:54:42 UTC (rev 1618)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/ale_vert_coord.tex        2012-03-12 18:26:51 UTC (rev 1619)
@@ -157,14 +157,14 @@
 \mbox{barotropic divergence} &amp;&amp; {\overline D} = \sum\limits_{k=1}^{kmax} D_k  \\
 \mbox{barotropic and baroclinic divergence} &amp;&amp; D_k = {\overline D} + D_k' = {\overline D} + D^{hf}_k + D^{lf}_k \label{D}\\
 \mbox{low frequecy baroclinic divergence} &amp;&amp; 
-  \frac{\partial D^{lf}_k}{\partial t} = - \frac{2\pi}{\tau_{\;lf}} \left[ D^{lf}_k - D'_k \right] \label{Dlf}\\
+  \frac{\partial D^{lf}_k}{\partial t} = - \frac{2\pi}{\tau_{\;Dlf}} \left[ D^{lf}_k - D'_k \right] \label{Dlf}\\
 \mbox{decomposed thickness} &amp;&amp; h_k = h^{init}_k + h^{hf}_k + h^{ext}_k \label{h} \\
 \mbox{decomposed thickness tendency} &amp;&amp; 
   \frac{\partial h_k}{\partial t} = \frac{\partial h^{hf}_k}{\partial t} + \frac{\partial h^{ext}_k}{\partial t} 
    \label{dhdt} \\
 \mbox{high frequency thickness tendency} &amp;&amp; 
-  \frac{\partial h^{hf}_k}{\partial t} = - D^{hf}_k - \frac{2\pi}{\tau_{rest}} h^{hf}_k 
-   + </font>
<font color="black">abla_h\cdot \left( \kappa_{hf} </font>
<font color="blue">abla_h h^{hf}_k \right)
+  \frac{\partial h^{hf}_k}{\partial t} = - D^{hf}_k - \frac{2\pi}{\tau_{hhf}} h^{hf}_k 
+   + </font>
<font color="black">abla_h\cdot \left( \kappa_{hhf} </font>
<font color="gray">abla_h h^{hf}_k \right)
    \label{hhf} \\
 \mbox{external mode thickness response} &amp;&amp; 
   \frac{\partial h^{ext}_k}{\partial t} = - \frac{W_k}{\sum\limits_{k=1}^{kmax} W_k}
@@ -178,11 +178,11 @@
 \frac{\partial h_k}{\partial t}  + D_k + \frac{\partial}{\partial z} \left( h_k w_k \right) = 0.
 \eea
 These equations are discrete in the vertical, indexed by layer $k$, but continuous in the horizontal and in time.  The thickness variables $h$, divergence variables $D$, velocity ${\bf u}$,  and SSH $\zeta$ are functions of $({\bf x},t)$, and vertical dependance is explicitly labelled with a subscript $k$.  External thickness weights $W_k$ do not depend on ${\bf x}$ or $t$, and constant parameters include 
-$\tau_{lf}$, $\tau_{rest}$, and $\kappa_{hf}$.
+$\tau_{Dlf}$, $\tau_{hhf}$, and $\kappa_{hhf}$.
 
-The divergence in each cell, $D$, is partitioned (\ref{D}) into its barotropic component ${\overline D}$ and its baroclinic component $D'$, which is further subpartitioned into the low frequency baroclinic component $D^{lf}$ and the remaining high frequency component $D^{hf}$.  The low frequency baroclinic divergence (\ref{Dlf}) is computed using a first-order low-pass filter, as suggested by \cite{Leclair:2011ud}, with timescale $\tau_{lf}$.  The new prognostic variable $D^{lf}$ should be initialized as zero and saved to the restart file.
+The divergence in each cell, $D$, is partitioned (\ref{D}) into its barotropic component ${\overline D}$ and its baroclinic component $D'$, which is further subpartitioned into the low frequency baroclinic component $D^{lf}$ and the remaining high frequency component $D^{hf}$.  The low frequency baroclinic divergence (\ref{Dlf}) is computed using a first-order low-pass filter, as suggested by \cite{Leclair:2011ud}, with timescale $\tau_{Dlf}$.  The new prognostic variable $D^{lf}$ should be initialized as zero and saved to the restart file.
 
-The thickness $h$ (\ref{h}) is initialized with the 3D field $h^{init}$, and then evolves (\ref{dhdt}) with contributions due to high frequency divergence $h^{hf}$ (the ${\tilde z}$-coordinate part) and external mode oscillations $h^{ext}$ (the $z^*$-coordinate part).  The tendency for $h^{hf}$ (\ref{hhf}) is primarily due to the high frequency divergence term $D^{hf}$, but also includes a restoring term with timescale $\tau_{rest}$ to prevent any longterm drift coordinate drift, and a thickness diffusion term with diffusion coefficient $\kappa_{hf}$ to prevent gridscale noise in the surface interfaces, as explained in \cite{Leclair:2011ud}.  The new prognostic variable $h^{hf}$ should be initialized as zero and saved to the restart file.
+The thickness $h$ (\ref{h}) is initialized with the 3D field $h^{init}$, and then evolves (\ref{dhdt}) with contributions due to high frequency divergence $h^{hf}$ (the ${\tilde z}$-coordinate part) and external mode oscillations $h^{ext}$ (the $z^*$-coordinate part).  The tendency for $h^{hf}$ (\ref{hhf}) is primarily due to the high frequency divergence term $D^{hf}$, but also includes a restoring term with timescale $\tau_{hhf}$ to prevent any longterm drift coordinate drift, and a thickness diffusion term with diffusion coefficient $\kappa_{hhf}$ to prevent gridscale noise in the surface interfaces, as explained in \cite{Leclair:2011ud}.  The new prognostic variable $h^{hf}$ should be initialized as zero and saved to the restart file.
 
 The prognostic equation for $h^{ext}_k$ (\ref{hext}) simply changes the layer thickness in proportion to the change in SSH $\zeta$, where the constant of proportionality $W$ may vary by level (this differs from the formulation in \cite{Leclair:2011ud}).  In a pure $z^*$-coordinate system, one could compute the thickness simply as
 \bea
@@ -197,10 +197,10 @@
 \bea 
 \mbox{low freq. baroclinic div.} &amp;&amp; 
   \frac{\partial D^{lf}_k}{\partial t} 
-= - \frac{2\pi}{\tau_{\;lf}} \left[ D^{lf}_k - (D_k-{\overline D}) \right] \label{Dlf2}\\
+= - \frac{2\pi}{\tau_{\;Dlf}} \left[ D^{lf}_k - (D_k-{\overline D}) \right] \label{Dlf2}\\
 \mbox{high freq. thickness tend.} &amp;&amp; 
-  \frac{\partial h^{hf}_k}{\partial t} = - (D_k-{\overline D}-D^{lf}_k) - \frac{2\pi}{\tau_{rest}} h^{hf}_k 
-   + </font>
<font color="black">abla_h\cdot \left( \kappa_{hf} </font>
<font color="blue">abla_h h^{hf}_k \right)
+  \frac{\partial h^{hf}_k}{\partial t} = - (D_k-{\overline D}-D^{lf}_k) - \frac{2\pi}{\tau_{hhf}} h^{hf}_k 
+   + </font>
<font color="black">abla_h\cdot \left( \kappa_{hhf} </font>
<font color="gray">abla_h h^{hf}_k \right)
    \label{hhf2} 
 \eea
 
@@ -267,7 +267,7 @@
 \end{equation}
 where $\alpha=1/\rho$ is the specific volume.  The hydrostatic condition, 
 $\partial p/\partial z = -\rho g$, 
-implies that $M$ is independent of $z$ in a layer of constant density.  For the shallow water equations, fix $z=z_{top}=h+b$ in (\ref{MontPot}) and assume pressure at the top of the fluid has a constant value, so that 
+implies that $M$ is independent of $z$ in a layer of constant density (See \cite{Higdon06xacta} p. 401 for derivation of $M$).  For the shallow water equations, fix $z=z_{top}=h+b$ in (\ref{MontPot}) and assume pressure at the top of the fluid has a constant value, so that 
 $</font>
<font color="black">abla M = g </font>
<font color="black">abla (h+b)$.  Similarly, for the primitive equations $</font>
<font color="black">abla M = \alpha </font>
<font color="gray">abla p$.
 
 Both $\rho$ and $M$ are constant within each layer, but the pressure and z vary continuously in the vertical.  Taking (\ref{MontPot}) in the limit as one approaches a layer interface, $\Delta M =  p_{int} \Delta \alpha$.  In the top layer, evaluating (\ref{MontPot}) at $z=0$ and using the hydrostatic condition, $M=g\eta$ where $\eta$ is the sea surface height.  These two equations allow us to compute the Montgomery Potential in each column:
@@ -472,11 +472,14 @@
 \\&amp;&amp;
 {\bf u}_{k,avg} = \ubtr_{avg} + {\bf u}'_{k,n+1/2} 
 \\ &amp;&amp;
+\mbox{compute ${\bf u}_{k,bolus}$ from ${\bf u}_{k,avg}$}
+\\ &amp;&amp;
 {\bf u}^{corr} = \left( {\overline {\bf F}} 
-  - \sum_{k=1}^{N^{edge}} h_{k,*}^{edge}  {\bf u}_k^{avg} \right)
-\left/ \sum_{k=1}^{N^{edge}} h_{k,*}^{edge}   \right. 
+  - \sum_{k=1}^{N^{edge}} h_{k,*}^{edge} 
+   \left( {\bf u}_{k,avg} + {\bf u}_{k,bolus}\right) \right)
+\left/ \sum_{k=1}^{N^{edge}} h_{k,*}^{edge}   \right. \label{ucorr}
 \\ &amp;&amp;
-{\bf u}^{tr}_k = {\bf u}^{avg}_k + {\bf u}^{corr}
+{\bf u}^{tr}_k = {\bf u}_{k,avg} + {\bf u}_{k,bolus} + {\bf u}^{corr} \label{utr}
 \end{eqnarray}
 For unsplit explicit, skip all computations in stage 2.  Instead, set:
 \bea
@@ -485,6 +488,13 @@
 {\bf u}^{tr}_{k} = {\bf u}'_{k,n+1/2} 
 \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
+\begin{eqnarray}
+{\overline {\bf F}} &amp;=&amp; \sum_{k=1}^N h_{k,*}^{edge} {\bf u}_k^{tr}. \label{flux.condition}
+\label{BclBtrFluxConsistency}
+\end{eqnarray}
+on each edge.  To do that, introduce a velocity correction ${\bf u}^{corr}$ that is vertically constant, and add it to the velocity as a barotropic correction.  The total transport velocity used in the thickness and tracer tendency terms is then given by (\ref{utr}).  Substitute (\ref{utr}) into (\ref{flux.condition}), note that ${\bf u}^{corr}$ moves outside of the vertical sum, and solve for ${\bf u}^{corr}$ to obtain (\ref{ucorr}).
+
 </font>
<font color="gray">ewpage
 {\bf Stage 3: update thickness, tracers, density and pressure}
 \bea &amp;&amp;
@@ -566,11 +576,11 @@
 namelist character vert_grid config_vert_grid_type      isopycnal
  also: 'zlevel', 'zstar', 'ale'
 \end{verbatim}
-$\tau_{lf}$, $\tau_{rest}$, and $\kappa_{hf}$:
+$\tau_{Dlf}$, $\tau_{hhf}$, and $\kappa_{hhf}$:
 \begin{verbatim}
-namelist real      vert_grid config_div_low_freq_time   86400.0
-namelist real      vert_grid config_h_restore_time      86400.0
-namelist real      vert_grid config_h_diff              1.0e-5
+namelist real      grid     config_Dlf_restore_time    432000.0
+namelist real      grid     config_hhf_restore_time    432000.0
+namelist real      grid     config_hhf_diff            100
 \end{verbatim}
 note: should config\_h\_diff be in list hmix or vert\_grid?
 

Modified: trunk/documents/ocean/current_design_doc/ale_vert_coord/mpas_o.bib
===================================================================
--- trunk/documents/ocean/current_design_doc/ale_vert_coord/mpas_o.bib        2012-03-12 15:54:42 UTC (rev 1618)
+++ trunk/documents/ocean/current_design_doc/ale_vert_coord/mpas_o.bib        2012-03-12 18:26:51 UTC (rev 1619)
@@ -65,3 +65,15 @@
 year = {2011}
 }
 
+@ARTICLE{Higdon06xacta,
+   author = {{Higdon}, R.~L.},
+    title = &quot;{Numerical modelling of ocean circulation}&quot;,
+  journal = {Acta Numerica},
+     year = 2006,
+   volume = 15,
+    pages = {385-407}}
+http://www.math.oregonstate.edu/~higdon/ocean/higdon_acta2006.pdf
+R. L. Higdon, Numerical modelling of ocean circulation (PDF, 3.3 Mb), Acta Numerica 2006, pp. 385-470. Acta Numerica is an annual volume, published by Cambridge University Press, that contains invited surveys on topics in numerical analysis and scientific computing. 
+
+p 401-402 gives the formulation of Mont Potential I used in mpas.
+

</font>
</pre>