<p><b>qchen3@fsu.edu</b> 2012-12-21 10:53:00 -0700 (Fri, 21 Dec 2012)</p><p><br>
GM design document: Specified more implementation details concerning new variables and where calculations are carried out.<br>
</p><hr noshade><pre><font color="gray">Modified: trunk/documents/ocean/current_design_doc/gm/gm.tex
===================================================================
--- trunk/documents/ocean/current_design_doc/gm/gm.tex        2012-12-20 21:44:33 UTC (rev 2370)
+++ trunk/documents/ocean/current_design_doc/gm/gm.tex        2012-12-21 17:53:00 UTC (rev 2371)
@@ -1,4 +1,4 @@
-\documentclass[11pt]{report}
+\documentclass[12pt]{report}
\usepackage{epsf,amsmath,amsfonts}
\usepackage{graphicx}
@@ -246,6 +246,92 @@
Here $(\p \rho/\p\mathrm{n})|_z$ refers to the normal derivative of
the density field on constant height surfaces.
+% \section{Formulae}\label{sec:formulae}
+% Date last modified: 2012/09/21 \\
+% Contributors: Qingshan Chen and Todd Ringler\\
+
+% We label the layers from the top down by $k=1,2,3,\cdots$, and the
+% layer interface between layers $k$ and $k+1$ by $k+1/2$. The height
+% of the midpoint of layer $k$ is denoted as $z_k$, and the height of
+% the layer interface $k+1/2$ is denoted as $z_{k+1/2}$.
+A first order
+approximation to $\rho_z$ at layer interface $k+1/2$ is given by
+\begin{equation}
+ \left(\rho_z\right)_{k+\frac{1}{2}} =
+ \dfrac{\rho_k - \rho_{k+1}}{z_k - z_{k+1}},\label{eq:21}
+\end{equation}
+where $k$ is the layer index.
+
+The calculation of $\p\rho/\p\mathrm{n}$ on constant height surfaces is
+a bit more complicated, because the general vertical coordinate
+usually
+does not align with the traditional $z$-level coordinate. We begin by
+deriving a formula for $</font>
<font color="blue">abla_z\rho$ on the general
+coordinate. Projecting $</font>
<font color="black">abla_z\rho$ in the direction of $</font>
<font color="blue">b$ will
+give $\p\rho/\p\mathrm{n}$ on
+constant height surfaces.
+We call the general vertical coordinate $r$. Thus it can be written that
+\begin{align}
+& r = r(x,y,z),\label{eq:22}\\
+& z = z(x,y,r).\label{eq:23}
+\end{align}
+The time variable $t$ has been dropped for now, since it is not relevant
+in the derivation. Applying the horizontal gradient operator $</font>
<font color="blue">abla_z$
+to $\rho(x,y,r(x,y,z))$, and using the chain rule, we obtain
+\begin{equation}
+</font>
<font color="black">abla_z \rho = </font>
<font color="black">abla_r\rho + \dfrac{\p\rho}{\p r}</font>
<font color="blue">abla_z r.
+\label{eq:24}
+\end{equation}
+Applying $</font>
<font color="blue">abla_z$ to \eqref{eq:23}, we obtain
+\begin{equation*}
+0 = </font>
<font color="black">abla_r z + \dfrac{\p z}{\p r}</font>
<font color="blue">abla_z r.
+\end{equation*}
+Hence
+\begin{equation}
+</font>
<font color="black">abla_z r = \frac{-</font>
<font color="blue">abla_r z}{\frac{\p z}{\p r}}.\label{eq:25}
+\end{equation}
+Applying $\p/\p z$ to \eqref{eq:23}, we have
+\begin{equation}
+1 = \dfrac{\p z}{\p r}\dfrac{\p r}{\p z}.\label{eq:26}
+\end{equation}
+Combining \eqref{eq:24}--\eqref{eq:26} yields
+\begin{equation}
+</font>
<font color="black">abla_z \rho = </font>
<font color="blue">abla_r \rho - \dfrac{\p \rho}{\p r}
+\dfrac{\p r}{\p z}</font>
<font color="blue">abla_r z.\label{eq:27}
+\end{equation}
+We notice that
+\begin{equation}
+\dfrac{\p \rho}{\p z} = \dfrac{\p \rho}{\p r}
+\dfrac{\p r}{\p z}\label{eq:28}
+\end{equation}
+by taking the $z$-derivative of $\rho(x,y,r(x,y,z))$. Hence
+\eqref{eq:27} can be written as
+\begin{equation}
+</font>
<font color="black">abla_z \rho = </font>
<font color="black">abla_r \rho - \dfrac{\p\rho}{\p z}</font>
<font color="blue">abla_r z.
+\label{eq:29}
+\end{equation}
+Taking dot-product of \eqref{eq:29} with the outer normal unit vector
+$</font>
<font color="gray">b$ yields
+\begin{equation}
+\left.\dfrac{\p\rho}{\p \mathrm{n}}\right|_z =
+\left.\dfrac{\p\rho}{\p\mathrm{n}}\right|_r - \left.\dfrac{\p\rho}{\p z}
+\dfrac{\p z}{\p\mathrm{n}}\right|_r.
+\label{eq:30}
+\end{equation}
+The discretization of the RHS of \eqref{eq:30} are as follow,
+\begin{align*}
+\left.\dfrac{\p\rho}{\p n}\right|_r &=
+\dfrac{\rho(\textrm{cell2}) - \rho(\textrm{cell1})}{\textrm{dcEdge}},\\
+\left.\dfrac{\p z}{\p n}\right|_r &=
+\dfrac{z(\textrm{cell2}) - z(\textrm{cell1})}{\textrm{dcEdge}}.
+\end{align*}
+$\p\rho/\p z$ has been calculated before at the cell centers and on
+the layer interfaces. The data first needs to be interpolated to the cell
+edges on the layer interfaces using area weighted averaging, and then
+be interpolated to the cell edges in the middle of the layer
+interfaces.
+
+
\section{Redi isopycnal diffusion}\label{sec:redi-isopycn-diff}
Date last modified: 2012/11/19 \\
Contributors: Qingshan Chen, Todd Ringler, and Peter Gent\\
@@ -656,10 +742,10 @@
\end{equation}
with $\tilde S$ and $K_{33}$ taking the forms
\begin{equation*}
- \tilde{\Sb} = \left(\begin{matrix}(1-\epsilon)k_x- l_x + k_y(k_xl_y - k_yl_x) -
+ \tilde{\Sb} = \left(\begin{matrix}k_x- l_x -\epsilon k_x + k_y(k_xl_y - k_yl_x) -
\epsilon k_x(k_xl_x+k_yl_y) \\
\\
- (1-\epsilon)k_y-l_y
+ k_y-l_y - \epsilon k_y
+ k_x(k_yl_x - k_xl_y) - \epsilon
k_y(k_xl_x+k_yl_y)\end{matrix}\right),
\end{equation*}
@@ -672,16 +758,19 @@
\end{multline*}
With the small-angle approximation, $K^r$ takes the same form as in
\eqref{eq:53}, and $\tilde{S}$ and $K_{33}$ are reduced to
-\begin{equation*}
- \tilde{\Sb} = \left(\begin{matrix}(1-\epsilon)k_x- l_x \\
+\begin{equation}
+\label{eq:56}
+ \tilde{\Sb} = \left(\begin{matrix}k_x- l_x - \epsilon k_x \\
\\
- (1-\epsilon)k_y-l_y
- \end{matrix}\right),
-\end{equation*}
+ k_y-l_y - \epsilon k_y
+ \end{matrix}\right) = -(1-\epsilon)\dfrac{</font>
<font color="blue">abla_r\rho}{\rho_z} -
+ \epsilon</font>
<font color="red">abla_r z,
+\end{equation}
and
-\begin{equation*}
-K_{33} = (k_x - l_x)^2 + (k_y - l_y)^2 +\epsilon.
-\end{equation*}
+\begin{equation}
+\label{eq:57}
+K_{33} = \dfrac{|</font>
<font color="blue">abla_r\rho|^2}{\rho_z^2}+\epsilon.
+\end{equation}
</font>
<font color="gray">oindent{\it Implementation}\\
@@ -697,8 +786,8 @@
r}\varphi\right),
\end{multline}
where
-$$</font>
<font color="red">abla_r \equiv \left(\frac{\p}{\p x}|_r,\, \frac{\p}{\p
- y}|_r\right)$$
+$$</font>
<font color="gray">abla_r \equiv \left(\left.\frac{\p}{\p x}\right|_r,\, \left.\frac{\p}{\p
+ y}\right|_r\right)$$
denotes the two-dimensional gradient operator on
constant $r$ surfaces. According to the third equation in
\eqref{eq:32}, for fixed $(x,\,y)$,
@@ -716,19 +805,28 @@
\dfrac{\p}{\p z}\left( K_{33}\dfrac{\p}{\p
z}\varphi\right).
\end{multline}
-The first and second terms can be easily discretized using the flux
-formulation in the horizontal; the fourth term can be discretized
-using the finite difference method in the vertical. To discretize the
+The first is the usual horizontal diffusion. The second terms can be
+discretized using the flux form,
+\begin{equation}
+ \label{eq:58}
+ \left[ </font>
<font color="blue">abla_r\cdot\left(\tilde{\mathbf{S}}\dfrac{\p\varphi}{\p
+ z}\right)\right] =
+\sum_{e\in EC(i)} \dfrac{\p\varphi}{\p z}\tilde{\mathbf{S}}\cdot</font>
<font color="blue">b_e.
+\end{equation}
+ To discretize the
third term using the FDM, we need the values of
$\mathbf{S}\cdot</font>
<font color="red">abla_r\varphi$ at the cell centers, which can be
approximated by
-\begin{equation*}
+\begin{equation}
+\label{eq:59}
\left[\tilde{\mathbf{S}}\cdot</font>
<font color="black">abla_r\varphi\right]_i = 2\sum_{e\in
EC(i)}\tilde{\mathbf{S}}\cdot\mathbf{n}_e {</font>
<font color="gray">abla_r\varphi}\cdot\mathbf{n}_e.
% &= -2\sum_{e\in
% EC(i)}\dfrac{1}{[\rho_z]_e}\left.\dfrac{\p\rho}{\p
% n_e}\right\vert_r \left.\dfrac{\p\varphi}{\p n_e}\right|_r,
-\end{equation*}
+\end{equation}
+The fourth term can be discretized
+using the finite difference method in the vertical.
% with $(\p/\p n_e)|_r$ denoting the derivative in the normal direction
% along constant $r$ surfaces on
% edge $e$.
@@ -742,7 +840,36 @@
\chapter{Design and Implementation}\label{cha:design-impl}
+\section{Support in the overall code structure for the GM parametrization }\label{sec:code-changes}
+</font>
<font color="blue">oindent Namelist:\\
+Add rhox, rhoz, zMidx and k33.
+\vspace{5mm}
+</font>
<font color="blue">oindent In ocn\_diagnostic\_solve:\\
+Compute rhox, rhoz, zMidx and k33.\\
+Call subroutine ocn\_gm\_compute\_uBolus(rhox, rhoz, zMidx,
+s\%uBolusGM\%array)
+
+\vspace{5mm}
+</font>
<font color="blue">oindent In ocn\_tend\_h:\\
+Call subroutine ocn\_thick\_hadv\_tend with uTransport instead of u.
+
+\vspace{5mm}
+</font>
<font color="blue">oindent In ocn\_tend\_scalar:\\
+Calculate uh with h\_edge and uTransport instead of u.\\
+Pass rhox, rhoz and zMidx along with other necessary arguments to
+the subroutine ocn\_tracer\_hmix\_tend for calculation of horizontal
+compoents of the Redi diffusion.\\
+Pass k33 to the subroutine ocn\_tracer\_vmix\_tend\_explicit for inclusion
+of the vertical component of the Redi diffusion.
+
+\vspace{5mm}
+</font>
<font color="gray">oindent In ocn\_time\_integration:\\
+Pass k33 to ocn\_tracer\_tend\_implicit for inclusion of the vertical
+component of the Redi diffusion. (????)
+
+
+
\section{Placement of the discrete variables}\label{sec:plac-discr-vari}
\begin{figure}[h]
\centering
@@ -766,87 +893,22 @@
$s$ and/or $\gamma$ should be specified on the cell edges and at the
layer interfaces.
-\section{Formulae}\label{sec:formulae}
-Date last modified: 2012/09/21 \\
-Contributors: Qingshan Chen and Todd Ringler\\
+% \section{Sequence of actions in the GM module}
+% \label{sec:sequence-actions-gm}
+% \begin{itemize}
+% \item Step 1. Compute $\frac{\p\rho}{\p n}|_r$ at cell edge and layer center.
+% \item Step 2. Compute $\frac{\p\rho}{\p z}$ at cell center and layer
+% interface.
+% \item Step 3. Compute $(\p z/\p n)|_r$ at cell edge and layer center.
+% \item Step 4a. Compute the slope S and then the uBolusGM according to
+% \eqref{eq:12}. Or
+% \item Step 4b. Compute the stream function $\gamma$ according to
+% \eqref{eq:19}--~\eqref{eq:20}, and uBolusGM according to
+% \eqref{eq:13}.
+% \item 5. Compute $K_{33}$ according \eqref{eq:57}.
+% \item 6. Pass out $(\p\rho/\p n)_r$, $\rho_z$, $(\p z/\p n)_r$ and $K_{33}$.
+% \end{itemize}
-We label the layers from the top down by $k=1,2,3,\cdots$, and the
-layer interface between layers $k$ and $k+1$ by $k+1/2$. The height
-of the midpoint of layer $k$ is denoted as $z_k$, and the height of
-the layer interface $k+1/2$ is denoted as $z_{k+1/2}$. A first order
-approximation to $\rho_z$ at layer interface $k+1/2$ is given by
-\begin{equation}
- \left(\rho_z\right)_{k+\frac{1}{2}} =
- \dfrac{\rho_k - \rho_{k+1}}{z_k - z_{k+1}}.\label{eq:21}
-\end{equation}
-The calculation of $\p\rho/\p\mathrm{n}$ on constant height surfaces is
-a bit more complicated, because the general vertical coordinate usually
-does not align with the traditional $z$-level coordinate. We begin by
-deriving a formula for $</font>
<font color="black">abla_z\rho$ on the general coordinate. Projecting $</font>
<font color="black">abla_z\rho$ in the direction of $</font>
<font color="red">b$ will give $\p\rho/\p\mathrm{n}$ on
-constant height surfaces.
-
-We call the general vertical coordinate $r$. Thus it can be written that
-\begin{align}
-& r = r(x,y,z),\label{eq:22}\\
-& z = z(x,y,r).\label{eq:23}
-\end{align}
-The time variable $t$ has been dropped for now, since it is not relevant
-in the derivation. Applying the horizontal gradient operator $</font>
<font color="red">abla_z$
-to $\rho(x,y,r(x,y,z))$, and using the chain rule, we obtain
-\begin{equation}
-</font>
<font color="black">abla_z \rho = </font>
<font color="black">abla_r\rho + \dfrac{\p\rho}{\p r}</font>
<font color="red">abla_z r.
-\label{eq:24}
-\end{equation}
-Applying $</font>
<font color="red">abla_z$ to \eqref{eq:23}, we obtain
-\begin{equation*}
-0 = </font>
<font color="black">abla_r z + \dfrac{\p z}{\p r}</font>
<font color="red">abla_z r.
-\end{equation*}
-Hence
-\begin{equation}
-</font>
<font color="black">abla_z r = \frac{-</font>
<font color="red">abla_r z}{\frac{\p z}{\p r}}.\label{eq:25}
-\end{equation}
-Applying $\p/\p z$ to \eqref{eq:23}, we have
-\begin{equation}
-1 = \dfrac{\p z}{\p r}\dfrac{\p r}{\p z}.\label{eq:26}
-\end{equation}
-Combining \eqref{eq:24}--\eqref{eq:26} yields
-\begin{equation}
-</font>
<font color="black">abla_z \rho = </font>
<font color="red">abla_r \rho - \dfrac{\p p}{\p r}
-\dfrac{\p r}{\p z}</font>
<font color="red">abla_r z.\label{eq:27}
-\end{equation}
-We notice that
-\begin{equation}
-\dfrac{\p \rho}{\p z} = \dfrac{\p \rho}{\p r}
-\dfrac{\p r}{\p z}\label{eq:28}
-\end{equation}
-by taking the $z$-derivative of $\rho(x,y,r(x,y,z))$. Hence
-\eqref{eq:27} can be written as
-\begin{equation}
-</font>
<font color="black">abla_z \rho = </font>
<font color="black">abla_r \rho - \dfrac{\p\rho}{\p z}</font>
<font color="red">abla_r z.
-\label{eq:29}
-\end{equation}
-Taking dot-product of \eqref{eq:29} with the outer normal unit vector
-$</font>
<font color="gray">b$ yields
-\begin{equation}
-\dfrac{\p\rho}{\p \mathrm{n}}\rvert_z =
-\dfrac{\p\rho}{\p\mathrm{n}}\rvert_r - \dfrac{\p\rho}{\p z}
-\dfrac{\p z}{\p\mathrm{n}}\rvert_r.
-\label{eq:30}
-\end{equation}
-The discretization of the RHS of \eqref{eq:30} are as follow,
-\begin{align*}
-\dfrac{\p\rho}{\p n}\rvert_r &=
-\dfrac{\rho(\textrm{cell2}) - \rho(\textrm{cell1})}{\textrm{dcEdge}},\\
-\dfrac{\p z}{\p n}\rvert_r &=
-\dfrac{z(\textrm{cell2}) - z(\textrm{cell1})}{\textrm{dcEdge}}.
-\end{align*}
-$\p\rho/\p z$ has been calculated before at the cell centers and on
-the layer interfaces. The data first needs to be interpolated to the cell
-edges on the layer interfaces using area weighted averaging, and then
-be interpolated to the cell edges in the middle of the layer
-interfaces.
-
-
%-----------------------------------------------------------------------
\chapter{Testing}\label{cha:testing}
Modified: trunk/documents/ocean/current_design_doc/gm/variable-placement.pdf
===================================================================
--- trunk/documents/ocean/current_design_doc/gm/variable-placement.pdf        2012-12-20 21:44:33 UTC (rev 2370)
+++ trunk/documents/ocean/current_design_doc/gm/variable-placement.pdf        2012-12-21 17:53:00 UTC (rev 2371)
@@ -1,10 +1,10 @@
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