<p><b>qchen3@fsu.edu</b> 2012-11-26 14:02:18 -0700 (Mon, 26 Nov 2012)</p><p><br>
Upload GM design document -- initial draft.<br>
</p><hr noshade><pre><font color="gray">Added: trunk/documents/ocean/current_design_doc/gm/coordinates.pdf
===================================================================
--- trunk/documents/ocean/current_design_doc/gm/coordinates.pdf (rev 0)
+++ trunk/documents/ocean/current_design_doc/gm/coordinates.pdf 2012-11-26 21:02:18 UTC (rev 2319)
@@ -0,0 +1,2362 @@
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+ <rdf:li rdf:parseType="Resource">
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+ <rdf:li rdf:parseType="Resource">
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+ <rdf:li rdf:parseType="Resource">
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+ <xmpG:mode>RGB</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:red>255</xmpG:red>
+ <xmpG:green>255</xmpG:green>
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+ <xmpG:mode>RGB</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
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+ <xmpG:mode>RGB</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
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+ <xmpG:green>255</xmpG:green>
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+ <xmpG:mode>RGB</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
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+ <xmpG:blue>255</xmpG:blue>
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+ <xmpG:mode>RGB</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
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+ <xmpG:mode>RGB</xmpG:mode>
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+startxref
+188609
+%%EOF
Added: trunk/documents/ocean/current_design_doc/gm/gm-references.bib
===================================================================
--- trunk/documents/ocean/current_design_doc/gm/gm-references.bib (rev 0)
+++ trunk/documents/ocean/current_design_doc/gm/gm-references.bib 2012-11-26 21:02:18 UTC (rev 2319)
@@ -0,0 +1,43 @@
+%% Created using Papers on Sun, 23 Sep 2012.
+%% http://mekentosj.com/papers/
+
+@article{Ferrari10,
+author = {Ferrari, Raffaele and Griffies, Stephen M and Nurser, A J George and Vallis, Geoffrey K},
+title = {{A boundary-value problem for the parameterized mesoscale eddy transport}},
+journal = {Ocean Modelling},
+year = {2010},
+volume = {32},
+number = {3-4},
+pages = {143--156}
+}
+
+@article{GentMc90,
+author = {Gent, Peter R. and McWilliams, James C.},
+title = {{Isopycnal Mixing in Ocean Circulation Models}},
+journal = {Journal of Physical Oceanography},
+year = {1990},
+volume = {20},
+number = {1},
+pages = {150--155}
+}
+
+@article{GWMMc95,
+author = {Gent, Peter R. and Willebrand, Jurgen and McDougall, Trevor J. and McWilliams, James C.},
+title = {{Parameterizing Eddy-Induced Tracer Transports in Ocean Circulation Models}},
+journal = {Journal of Physical Oceanography},
+year = {1995},
+volume = {25},
+number = {4},
+pages = {463--474}
+}
+
+@article{Gent20,
+author = {Gent, P.R.},
+title = {{The Gent-McWilliams parameterization: 20/20 hindsight}},
+journal = {Ocean Modelling},
+year = {2011},
+volume = {39},
+number = {1-2},
+pages = {2--9}
+}
+
Added: trunk/documents/ocean/current_design_doc/gm/gm.tex
===================================================================
--- trunk/documents/ocean/current_design_doc/gm/gm.tex (rev 0)
+++ trunk/documents/ocean/current_design_doc/gm/gm.tex 2012-11-26 21:02:18 UTC (rev 2319)
@@ -0,0 +1,848 @@
+\documentclass[11pt]{report}
+
+\usepackage{epsf,amsmath,amsfonts}
+\usepackage{graphicx}
+
+
+\setlength{\textwidth}{6.5in}
+\setlength{\oddsidemargin}{0in}
+\setlength{\evensidemargin}{0in}
+\setlength{\textheight}{9.0in}
+\setlength{\topmargin}{0in}
+
+\setlength{\parskip}{1.2ex}
+\setlength{\parindent}{0mm}
+</font>
<font color="blue">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="blue">onumber}
+
+\def\ep{\epsilon}
+\def\f{\frac}
+\def\df{\dfrac}
+\def\p{\partial}
+\def\tilde{\widetilde}
+\def\hat{\widehat}
+\def\bs{\boldsymbol}
+\def\wh{\widehat}
+\def\ud{\,\mathrm{d}}
+\def\grad{</font>
<font color="blue">abla}
+
+\def\ub{\mathbf{u}}
+\def\vb{\bs{\mathrm{v}}}
+\def\Ub{\mathbf{U}}
+\def\kb{\bs{\mathrm{k}}}
+\def</font>
<font color="blue">b{\bs{\mathrm{n}}}
+\def\Kb{\bs{\mathrm{K}}}
+\def\Sb{\bs{\mathrm{S}}}
+
+
+\begin{document}
+
+\title{Mesoscale Eddy Parametrizations \\
+with General Vertical Coordinates: \\
+Requirements and Design}
+\author{MPAS Development Team}
+
+\maketitle
+\tableofcontents
+
+%-----------------------------------------------------------------------
+
+\chapter{Summary}
+
+Mesoscale eddy parametrization is an important process in determining the distribution of temperature, salinity and, as a result, density though out the world ocean (\cite{GentMc90, GWMMc95, Gent20}). The purpose of this design process is to implement a MEP into MPAS-Ocean model. Our first goal is a successful implementation of GM.
+In the current code, a 'gm' module is available that produces
+a bolus velocity $u^\ast$. The current code also has in place
+the mechanism to add such bolus velocity to the mean transport
+velocity. However, the module can only handle the isopycnal model,
+which is a special case of the general vertical
+coordinate. The new implementation of the GM should calculate
+the bolus velocity and diffusion along isopycnal surfaces from the calculation of isopyncal slopes for use with any vertical coordinate.
+It also includes a tapering process by the means of solving a
+vertical boundary value problem. The GM closure has certain well-known
+properties, such as mass conservation, sink of potential energy, etc.
+The new implementation can be considered a success if it can reproduce
+these properties.
+
+%figure template
+%\begin{figure}
+% \center{\includegraphics[width=14cm]{./Figure1.pdf}}
+% \caption{A graphical representation of the discrete boundary.}
+% \label{Figure1}
+%\end{figure}
+
+
+
+
+%-----------------------------------------------------------------------
+
+\chapter{Requirements}
+
+\section{Requirement: 1}
+Date last modified: 2012/09/06 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+This implementation of GM should make it possible for MPAS-O to
+perform low-resolution (say, 125km) simulations and reproduce model
+physics as expected. For example, GM should reduce available potential energy as compared to a run of the same resolution without GM.
+
+\section{Requirement: 2}
+Date last modified: 2012/09/06 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+This implementation of advective portion of GM should be based on a local stream-function formulation so as to make the GM module portable to other ocean models. The stream function formulation guarantees that the bolus velocity is non-divergent.
+
+
+\section{Requirement: 3}
+Date last modified: 2012/09/06 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+This implementation of GM should involve some ``tapering'' process so
+that in regions where the isopycnal surfaces are nearly vertical, the
+GM closure should have small coefficients or are essentially shut down.
+
+\section{Requirement: 4}
+Date last modified: 2012/09/06 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+This implementation of GM should be valid for any specification of $\kappa$, the GM closure parameter, i.e. we need to permit $\kappa=f(x,y,z,t)$.
+
+\section{Requirement: 5}
+Date last modified: 2012/09/06 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+This implementation of GM must include a component that allows for the diffusion of tracers along isopycnal surface in a manner generally referred to as Redi diffusion (reference).
+
+%-----------------------------------------------------------------------
+
+\chapter{Algorithmic Formulations}
+
+\section{The model with GM closure on a general vertical coordinate}
+Date last modified: 2012/09/06 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+The thickness and the passive tracer equations of the MPAS-O model read
+\begin{align}
+\dfrac{\partial h_k}{\partial t}
+ + </font>
<font color="blue">abla \cdot \left( h_k {\bf u}_k \right)
+% + w^{top}_{k+1} - w^{top}_{k} = 0
+ + w_k^\textrm{t} - w_k^\textrm{b} = 0\label{eq:1},
+\\
+\dfrac{\partial h_k\varphi_k}{\partial t}
+ + </font>
<font color="blue">abla \cdot \left( h_k\varphi_k {\bf u}_k \right)
+%+ w^{top}_{k+1}\varphi^{top}_{k+1} - w^{top}_{k}\varphi^{top}_{k}
+ + \varphi_k^\textrm{t} w_k^\textrm{t} - \varphi_k^\textrm{b} w_k^\textrm{b}
+= 0.\label{eq:4}
+\end{align}
+In the above, the superscripts 't' and 'b' denote the top and bottom surfaces,
+respectively, of a fluid layer.
+The Gent-McWilliams closure replaces the mean transport velocity $\ub$
+and $w$ by the effective transport velocity $\Ub$ and $W$,
+\begin{align}
+\dfrac{\partial h_k}{\partial t}
+ + </font>
<font color="blue">abla \cdot \left( h_k {\bf U}_k \right)
+% + w^{top}_{k+1} - w^{top}_{k} = 0
+ + W_k^\textrm{t} - W_k^\textrm{b} = 0,\label{eq:5}
+\\
+\dfrac{\partial h_k\varphi_k}{\partial t}
+ + </font>
<font color="blue">abla \cdot \left( h_k\varphi_k{\bf U}_k \right)
+%+ w^{top}_{k+1}\varphi^{top}_{k+1} - w^{top}_{k}\varphi^{top}_{k}
+ + \varphi_k^\textrm{t} W_k^\textrm{t} - \varphi_k^\textrm{b} W_k^\textrm{b}
+= </font>
<font color="black">abla\cdot\left(h_k \Kb \cdot </font>
<font color="blue">abla\varphi_k \right).\label{eq:6}
+\end{align}
+In the above, the effective transport velocities can be decomposed
+into the mean transport velocity and the GM bolus velocities,
+\begin{align}
+&{\bf U}_k = u_k + u^\ast_k,\label{eq:7}\\
+& W_k = w_k + w^\ast_k,\label{eq:8},
+\end{align}
+and the $3\times 3$ tensor $\Kb$ and the two dimensional slope $\Sb$
+are given by
+\begin{equation}
+ \label{eq:9}
+ \Kb = \left(
+ \begin{aligned}
+ &\mathbb{I} & \Sb\\
+ &\Sb^T & \Sb\cdot\Sb
+ \end{aligned}\right), \quad \Sb = -\dfrac{</font>
<font color="blue">abla_z \rho}{\rho_z},
+\end{equation}
+where $\mathbb{I}$ is the $2\times 2$ identity matrix.
+
+\section{The stream-function formulation of the GM parametrizations}
+Let
+\begin{equation}
+ \label{eq:10}
+ \Psi = \bs{\gamma}\times\kb,
+\end{equation}
+where $\bs{\gamma}$ is a horizontal 2D vector and $\kb$ is the
+vertical unit vector. Let
+\begin{equation}
+ \label{eq:2}
+ \vb \equiv (\ub^\ast,\,w^\ast) = </font>
<font color="blue">abla_3 \times\Psi.
+\end{equation}
+Then we find that
+\begin{align}
+ &\ub^\ast = \p_z\bs{\gamma},\label{eq:13}\\
+ &w^\ast = -</font>
<font color="blue">abla_z\cdot\bs{\gamma}.\label{eq:14}
+\end{align}
+Now let
+\begin{equation}
+ \label{eq:11}
+ \bs{\gamma}_\textrm{GM} = -\kappa\Sb,
+\end{equation}
+where $\Sb$ is a 2D vector representing the slope of the isopycnal surfaces. Then
+\begin{align}
+ &\ub^\ast_\textrm{GM} = -\p_z(\kappa\Sb), \label{eq:12}\\
+ &w^\ast_\textrm{GM} = </font>
<font color="blue">abla_z\cdot(\kappa \Sb).
+\end{align}
+We note that $w^\ast$ can be determined via (\ref{eq:14}) or using (\ref{eq:13}) along with the condition that
+\begin{equation}
+ \label{eq:2}
+ </font>
<font color="blue">abla \cdot \vb \equiv 0.
+\end{equation}
+
+
+\section{Tapering by solving a boundary value problem}
+Following \cite{Ferrari10}, one solves a boundary value
+problem for the horizontal 2D vector
+$\bs{\gamma}$ in \eqref{eq:13} and \eqref{eq:14},
+\begin{align}
+ &\left( c^2\dfrac{\ud^2}{\ud z^2} - N^2\right)\bs{\gamma} =
+ (g/\rho_0)\kappa </font>
<font color="blue">abla_z\rho,\label{eq:15}\\
+ &\bs{\gamma}(\eta) = \bs{\gamma}(-H) = 0.\label{eq:3}
+\end{align}
+In the above, $c$ is a depth independent speed to be specified,
+$\kappa$ is associated with the barotropic eddy stirring, and
+therefore is also depth independent. $N^2$ is the Brunt-Vasalai
+frequencies, and is give by $N^2 = -(g/\rho_0)\p_z\rho$. We note that,
+when $c\longrightarrow 0$, the solution of \eqref{eq:15} and
+\eqref{eq:3} reduces to $\bs{\gamma}_\textrm{GM}$, except in the top
+and bottom thin boundary layers where $\bs{\gamma}$ goes to zero
+while $\bs{\gamma}_\textrm{GM}$ does not.
+
+\section{Normal components}
+Date last modified: 2012/9/18 \\
+Contributors: Qingshan Chen, Todd Ringler \\
+
+MPAS-O utilizes a C-grid staggering technique. Hence it is necessary
+to recast the full continuous GM formulation, given in the previous
+section, in terms of the normal components of the variables. In what
+follows, we denote the normal components of $\ub^\ast$ and
+$\bs{\gamma}$ by $u^\ast$ and $\gamma$, respectively, i.e.
+\begin{align}
+ &u^\ast = \ub^\ast\cdot</font>
<font color="blue">b,\label{eq:16}\\
+ &\gamma = \bs{\gamma}\cdot</font>
<font color="blue">b,\label{eq:17}
+\end{align}
+where $</font>
<font color="blue">b$ denotes the outer normal unit vector on the cell
+edges. The relation between $u^\ast$ and $\gamma$ is derived by
+taking dot-product of \eqref{eq:13} with $\ub$,
+\begin{equation}
+ \label{eq:18}
+ u^\ast = \p_z \gamma.
+\end{equation}
+We obtain a boundry value problem for the normal component $\gamma$ by
+taking dot-product of \eqref{eq:15}--~\eqref{eq:3} with $</font>
<font color="blue">b$,
+\begin{align}
+ &\left( c^2\dfrac{\ud^2}{\ud z^2} - N^2\right){\gamma} =
+ \left.(g/\rho_0)\kappa \dfrac{\p \rho}{\p \mathrm{n}}\right|_z,\label{eq:19}\\
+ &{\gamma}(\eta) = {\gamma}(-H) = 0.\label{eq:20}
+\end{align}
+Here $(\p \rho/\p\mathrm{n})|_z$ refers to the normal derivative of
+the density field on constant height surfaces.
+
+%-----------------------------------------------------------------------
+
+\chapter{Design and Implementation}
+
+
+\section{Placement of the discrete variables}
+\begin{figure}[h]
+ \centering
+ \scalebox{0.8}{\includegraphics{variable-placement.pdf}}
+ \caption{The placement of the discrete variables}
+ \label{fig:1}
+\end{figure}
+We assume that a general vertical coordinate $r$ is in use. In the 3D
+MPAS-O model, the scalar variables $h,\, T,\, S,\, \rho$ etc.~are
+defined at the cell centers and in the middle of the layers. The
+horizontal normal velocity component $u$ is defined at the cell edges
+and in the middle of the layers. The vertical transport velocity $w$
+is defined at the cell centers and at the layer interfaces. The normal
+derivative of the density field is then most conveniently calculated
+on the cell edges and in the middle of the layers. The vertical
+derivative $\rho_z$ is most conveniently calculated at the cell
+centers and at the layer interfaces. The calculations of the normal
+component of the slope $\bs{s}$ of the constant density surfaces or
+the calculations of the normal component of the vector $\bs{\gamma}$
+involve both $\rho_z$ and $\p\rho/\p\mathrm{n}$, which suggests that
+$s$ and/or $\gamma$ should be specified on the cell edges and at the
+layer interfaces.
+
+\section{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}
+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 $</font>
<font color="blue">b$ 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 p}{\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
+$</font>
<font color="blue">b$ 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.
+
+\section{Redi isopycnal diffusion}
+Date last modified: 2012/11/19 \\
+Contributors: Qingshan Chen, Todd Ringler, and Peter Gent\\
+
+
+\begin{figure}[h]
+ \centering
+ \scalebox{0.7}{\includegraphics{coordinates.pdf}}
+ \caption{Coordinates transformation. Both $y^r$ and $y^\rho$ are on
+ the $xy$-plane. The $x^\rho y^\rho\rho$-coordinates can be reached
+ by first rotating the $xyz$-coordinates along the $z$-axis
+ counter-clockwise by angle $\alpha$, and then rotating the
+ intermediate coordinates along $y^\rho$-axis clockwise by angle
+ $\beta$. The coordinates $x^ry^rr$-coordinates can be reached
+ through a similar process. }
+\end{figure}
+The transformation between $(x^\rho, y^\rho, \rho)$ and $(x, y, z)$ is
+an orthonormal rotation determined by the two angular parameters
+$\alpha$ and $\beta$. Specifically,
+\begin{equation}
+ \label{eq:21}
+ \left(\begin{matrix} x\\ \\ y\\ \\ z\end{matrix}\right) =
+ \left(\begin{matrix}\cos\beta\cos\alpha & -\sin\alpha &
+ -\sin\beta\cos\alpha \\ \\
+ \cos\beta\sin\alpha & \cos\alpha & -\sin\beta\sin\alpha \\ \\
+ \sin\beta & 0 & \cos\beta\end{matrix}\right)
+ \left(\begin{matrix}x^\rho \\ \\ y^\rho \\ \\
+ \rho\end{matrix}\right) \equiv M_1 \left(\begin{matrix}x^\rho
+ \\ \\ y^\rho \\ \\ \rho\end{matrix}\right) .
+\end{equation}
+Similarly,
+\begin{equation}
+\label{eq:31}
+ \left(\begin{matrix} x\\ \\ y\\ \\ z\end{matrix}\right) =
+ \left(\begin{matrix}\cos\gamma\cos\sigma & -\sin\sigma &
+ -\sin\gamma\cos\sigma \\ \\
+ \cos\gamma\sin\sigma & \cos\sigma & -\sin\gamma\sin\sigma \\ \\
+ \sin\gamma & 0 & \cos\gamma\end{matrix}\right)
+ \left(\begin{matrix}x^r \\ \\ y^r \\ \\ r\end{matrix}\right) \equiv
+ M_2 \left(\begin{matrix}x^r \\ \\ y^r \\ \\ r\end{matrix}\right).
+\end{equation}
+For orthonormal matrices, their inverses are simply their
+transposes. Hence, from \eqref{eq:31},
+\begin{equation}
+\label{eq:32}
+ \left(\begin{matrix} x^r\\ \\ y^r\\ \\ r\end{matrix}\right) =
+ M_2^\textrm{T} \left(\begin{matrix}x \\ \\ y \\ \\
+ z\end{matrix}\right) =
+ \left(\begin{matrix}\cos\gamma\cos\sigma & \cos\gamma\sin\sigma &
+ \sin\gamma \\ \\
+ -\sin\sigma & \cos\sigma & 0\\ \\
+ -\sin\gamma\cos\sigma & -\sin\gamma\sin\sigma &
+ \cos\gamma\end{matrix}\right)
+ \left(\begin{matrix}x \\ \\ y \\ \\ z\end{matrix}\right).
+\end{equation}
+Combining \eqref{eq:21} and \eqref{eq:32}, one obtains the
+transformation from the density ($\rho$) coordinate system to the
+general ($r$) coordinate system,
+\begin{equation}
+\label{eq:33}
+ \left(\begin{matrix} x^r\\ \\ y^r\\ \\ r\end{matrix}\right) =
+ M_2^\textrm{T} M_1\left(\begin{matrix}x^\rho \\ \\ y^\rho \\ \\
+ \rho\end{matrix}\right) \equiv M_3\left(\begin{matrix}x^\rho \\
+ \\ y^\rho \\ \\ \rho\end{matrix}\right).
+\end{equation}
+The matrix multiplication is carried out with the symbolic algebraic
+system $Mathematica^\textrm{TM}$, and the resulting transformation
+matrix $M_3$ is given below in Mathematica format:
+\begin{multline}
+M_3 = \{\{\text{Cos}[\alpha ] \text{Cos}[\beta ]
+ \text{Cos}[\gamma ] \text{Cos}[\sigma ]+\text{Sin}[\beta ]
+ \text{Sin}[\gamma ]+\text{Cos}[\beta ] \text{Cos}[\gamma ]
+ \text{Sin}[\alpha ] \text{Sin}[\sigma ],\\
+-\text{Cos}[\gamma ]
+ \text{Cos}[\sigma ] \text{Sin}[\alpha ]+\text{Cos}[\alpha ]
+ \text{Cos}[\gamma ] \text{Sin}[\sigma ],\\
+-\text{Cos}[\alpha ]
+ \text{Cos}[\gamma ] \text{Cos}[\sigma ] \text{Sin}[\beta
+ ]+\text{Cos}[\beta ] \text{Sin}[\gamma ]-\text{Cos}[\gamma ]
+ \text{Sin}[\alpha ] \text{Sin}[\beta ] \text{Sin}[\sigma
+ ]\},\\
+\{\text{Cos}[\beta ] \text{Cos}[\sigma ] \text{Sin}[\alpha
+ ]-\text{Cos}[\alpha ] \text{Cos}[\beta ] \text{Sin}[\sigma
+ ],\text{Cos}[\alpha ] \text{Cos}[\sigma ]+\text{Sin}[\alpha ]
+ \text{Sin}[\sigma ],\\
+-\text{Cos}[\sigma ] \text{Sin}[\alpha ]
+ \text{Sin}[\beta ]+\text{Cos}[\alpha ] \text{Sin}[\beta ]
+ \text{Sin}[\sigma ]\},\\
+\{\text{Cos}[\gamma ] \text{Sin}[\beta
+ ]-\text{Cos}[\alpha ] \text{Cos}[\beta ] \text{Cos}[\sigma ]
+ \text{Sin}[\gamma ]-\text{Cos}[\beta ] \text{Sin}[\alpha ]
+ \text{Sin}[\gamma ] \text{Sin}[\sigma ],\\
+\text{Cos}[\sigma ]
+ \text{Sin}[\alpha ] \text{Sin}[\gamma ]-\text{Cos}[\alpha ]
+ \text{Sin}[\gamma ] \text{Sin}[\sigma ],\\
+\text{Cos}[\beta ]
+ \text{Cos}[\gamma ]+\text{Cos}[\alpha ] \text{Cos}[\sigma ]
+ \text{Sin}[\beta ] \text{Sin}[\gamma ]+\text{Sin}[\alpha ]
+ \text{Sin}[\beta ] \text{Sin}[\gamma ] \text{Sin}[\sigma ]\}\}
+\end{multline}
+The transformation $M$ from the density $\rho$ coordinate system to the
+hybrid curvelinear $(x,y,r)$ coordinate system is obtained by
+combining relevant rows from matrices $M_1$ and $M_3$. Specifically,
+using the Matlab conventions,
+\begin{align}
+ \label{eq:34}
+ M = \left(
+ \begin{aligned}
+ &M_1(1,:)\\
+ &M_1(2,:)\\
+ &M_3(3,:)
+ \end{aligned}\right),
+\end{align}
+and
+\begin{equation}
+\label{eq:35}
+ \left(\begin{matrix} x\\ \\ y\\ \\ r\end{matrix}\right) =
+ M\left(\begin{matrix}x^\rho \\
+ \\ y^\rho \\ \\ \rho\end{matrix}\right).
+\end{equation}
+
+Let $\varphi = \varphi(x^\rho, y^\rho, \rho,t) = \varphi(x,y,r,t)$ be
+some tracer. Then, using chain rules, one has
+\begin{equation}
+\label{eq:36}
+ \left(\begin{matrix}\frac{\p\varphi}{\p x^\rho}\\ \\
+ \frac{\p\varphi}{\p y^\rho} \\ \\ \frac{\p\varphi}{\p
+ \rho} \end{matrix}\right)=
+ \left(\begin{matrix}\frac{\p x}{\p x^\rho} & \frac{\p y}{\p x^\rho} &
+ \frac{\p r}{\p x^\rho} \\ \\
+\frac{\p x}{\p y^\rho} & \frac{\p y}{\p y^\rho} &
+ \frac{\p r}{\p y^\rho} \\ \\
+\frac{\p x}{\p \rho} & \frac{\p y}{\p\rho} &
+ \frac{\p r}{\p\rho}
+\end{matrix}\right)
+ \left(\begin{matrix}\frac{\p\varphi}{\p x}\\ \\
+ \frac{\p\varphi}{\p y} \\ \\ \frac{\p\varphi}{\p
+ r} \end{matrix}\right).
+\end{equation}
+Using \eqref{eq:35}, one can write \eqref{eq:36} as
+\begin{equation}
+\label{eq:37}
+ \left(\begin{matrix}\frac{\p\varphi}{\p x^\rho}\\ \\
+ \frac{\p\varphi}{\p y^\rho} \\ \\ \frac{\p\varphi}{\p
+ \rho} \end{matrix}\right)=
+ M^\textrm{T}
+ \left(\begin{matrix}\frac{\p\varphi}{\p x}\\ \\
+ \frac{\p\varphi}{\p y} \\ \\ \frac{\p\varphi}{\p
+ r} \end{matrix}\right).
+\end{equation}
+
+In the density coordinate system $(x^\rho, y^\rho, \rho)$, the
+diffusion of the tracer can be formulated as
+\begin{equation}
+ \label{eq:38}
+ </font>
<font color="black">abla^\rho\cdot(K^\rho</font>
<font color="blue">abla^\rho\varphi),
+\end{equation}
+where $</font>
<font color="blue">abla^\rho = (\p_{x_\rho}, \p_{y_\rho}, \p_\rho)^\textrm{T}$, and $K^\rho$
+denotes a tensor. The assumption that tracer diffusion should be along
+instead of across isopycnal surfaces can be realized by setting the
+diffusion tensor to
+\begin{equation}
+ \label{eq:39}
+ K^\rho = \left(
+ \begin{matrix}
+ 1 & 0 & 0\\
+ 0 & 1 & 0\\
+ 0 & 0 & 0
+ \end{matrix}\right).
+\end{equation}
+Using the relation \eqref{eq:37}, one can formulate the diffusion
+\eqref{eq:38} in the general hybrid coordinate system $(x, y, r)$ as
+\begin{equation}
+ \label{eq:40}
+ </font>
<font color="black">abla^r\cdot(K^r</font>
<font color="black">abla^r\varphi) \equiv </font>
<font color="blue">abla^r\cdot(MK^\rho
+ M^\textrm{T}</font>
<font color="blue">abla^r\varphi),
+\end{equation}
+where $</font>
<font color="blue">abla^r = (\p_x, \p_y, \p_r)^\textrm{T}$. With $K^\rho$ taking
+the form in \eqref{eq:39}, the new tensor $K^r$ can be easily found
+using Mathematica,
+\begin{multline}
+ \label{eq:41}
+ K^r = \{\{\text{Cos}[\alpha ]^2 \text{Cos}[\beta
+ ]^2+\text{Sin}[\alpha ]^2,\\
+ -\text{Cos}[\alpha ] \text{Sin}[\alpha
+ ]+\text{Cos}[\alpha ] \text{Cos}[\beta ]^2 \text{Sin}[\alpha
+ ],\\
+ \text{Cos}[\alpha ] \text{Cos}[\beta ] \text{Cos}[\gamma ]
+ \text{Sin}[\beta ]-\text{Cos}[\alpha ]^2 \text{Cos}[\beta ]^2
+ \text{Cos}[\sigma ] \text{Sin}[\gamma ]-\\
+ \text{Cos}[\sigma ]
+ \text{Sin}[\alpha ]^2 \text{Sin}[\gamma ]+
+ \text{Cos}[\alpha ]
+ \text{Sin}[\alpha ] \text{Sin}[\gamma ] \text{Sin}[\sigma
+ ]-\text{Cos}[\alpha ] \text{Cos}[\beta ]^2 \text{Sin}[\alpha ]
+ \text{Sin}[\gamma ] \text{Sin}[\sigma
+ ]\},\\
+ \{-\text{Cos}[\alpha ] \text{Sin}[\alpha
+ ]+\text{Cos}[\alpha ] \text{Cos}[\beta ]^2 \text{Sin}[\alpha
+ ],\\
+ \text{Cos}[\alpha ]^2+\text{Cos}[\beta ]^2 \text{Sin}[\alpha
+ ]^2,\\
+ \text{Cos}[\beta ] \text{Cos}[\gamma ] \text{Sin}[\alpha ]
+ \text{Sin}[\beta ]+\text{Cos}[\alpha ] \text{Cos}[\sigma ]
+ \text{Sin}[\alpha ] \text{Sin}[\gamma ]-\\
+ \text{Cos}[\alpha ]
+ \text{Cos}[\beta ]^2 \text{Cos}[\sigma ] \text{Sin}[\alpha ]
+ \text{Sin}[\gamma ]-\text{Cos}[\alpha ]^2 \text{Sin}[\gamma ]
+ \text{Sin}[\sigma ]-\text{Cos}[\beta ]^2 \text{Sin}[\alpha ]^2
+ \text{Sin}[\gamma ] \text{Sin}[\sigma
+ ]\},\\
+ \{\text{Cos}[\alpha ] \text{Cos}[\beta ]
+ \text{Cos}[\gamma ] \text{Sin}[\beta ]-\text{Cos}[\alpha ]^2
+ \text{Cos}[\beta ]^2 \text{Cos}[\sigma ] \text{Sin}[\gamma
+ ]-\\
+ \text{Cos}[\sigma ] \text{Sin}[\alpha ]^2 \text{Sin}[\gamma
+ ]+\text{Cos}[\alpha ] \text{Sin}[\alpha ] \text{Sin}[\gamma ]
+ \text{Sin}[\sigma ]-\text{Cos}[\alpha ] \text{Cos}[\beta ]^2
+ \text{Sin}[\alpha ] \text{Sin}[\gamma ] \text{Sin}[\sigma
+ ],\\
+ \text{Cos}[\beta ] \text{Cos}[\gamma ] \text{Sin}[\alpha ]
+ \text{Sin}[\beta ]+\text{Cos}[\alpha ] \text{Cos}[\sigma ]
+ \text{Sin}[\alpha ] \text{Sin}[\gamma ]-\\
+ \text{Cos}[\alpha ]
+ \text{Cos}[\beta ]^2 \text{Cos}[\sigma ] \text{Sin}[\alpha ]
+ \text{Sin}[\gamma ]-\text{Cos}[\alpha ]^2 \text{Sin}[\gamma ]
+ \text{Sin}[\sigma ]-\text{Cos}[\beta ]^2 \text{Sin}[\alpha ]^2
+ \text{Sin}[\gamma ] \text{Sin}[\sigma ],\\
+ \text{Cos}[\gamma ]^2
+ \text{Sin}[\beta ]^2-2 \text{Cos}[\alpha ] \text{Cos}[\beta ]
+ \text{Cos}[\gamma ] \text{Cos}[\sigma ] \text{Sin}[\beta ]
+ \text{Sin}[\gamma ]+\\
+ \text{Cos}[\alpha ]^2 \text{Cos}[\beta ]^2
+ \text{Cos}[\sigma ]^2 \text{Sin}[\gamma ]^2+\text{Cos}[\sigma
+ ]^2 \text{Sin}[\alpha ]^2 \text{Sin}[\gamma ]^2-\\
+ 2\text{Cos}[\beta ] \text{Cos}[\gamma ] \text{Sin}[\alpha ]
+ \text{Sin}[\beta ] \text{Sin}[\gamma ] \text{Sin}[\sigma ]-2
+ \text{Cos}[\alpha ] \text{Cos}[\sigma ] \text{Sin}[\alpha ]
+ \text{Sin}[\gamma ]^2 \text{Sin}[\sigma ]+\\
+ 2 \text{Cos}[\alpha ]
+ \text{Cos}[\beta ]^2 \text{Cos}[\sigma ] \text{Sin}[\alpha ]
+ \text{Sin}[\gamma ]^2 \text{Sin}[\sigma ]+\\
+ \text{Cos}[\alpha ]^2
+ \text{Sin}[\gamma ]^2 \text{Sin}[\sigma ]^2+\text{Cos}[\beta ]^2
+ \text{Sin}[\alpha ]^2 \text{Sin}[\gamma ]^2 \text{Sin}[\sigma
+ ]^2\}\}
+\end{multline}
+We denote by $k_x$ and $k_y$ the slopes of the constant density
+surfaces along the $x$ and $y$ axes, respectively, and by $l_x$ and
+$l_y$ the slopes of the constant $r$ surfaces along the $x$ and $y$
+axes respectively. We also note the following relations,
+\begin{align}\
+ &k_x = \cos\alpha\tan\beta, & &k_y =
+ \sin\alpha\tan\beta, \label{eq:42}\\
+ &l_x = \cos\sigma\tan\gamma, & &l_y =
+ \sin\sigma\tan\gamma \label{eq:43}
+ \end{align}
+Then the tensor matrix $K^r$ can be written as
+\begin{equation}
+ \label{eq:44}
+ K^r = \cos^2\beta\left(
+ \begin{matrix}
+ 1 + k_y^2 & { } & -k_xk_y & { } & { }\\
+ { } & { } & { } & { } & \cos\gamma{\Large\mathbf{S}}\\
+ -k_xk_y & { } & 1 + k_x^2 & { } & { }\\
+ { } & { } & { } & { } & { } \\
+ { } & \cos\gamma\mathbf{S}^\textrm{T} & { } & { } &
+ \cos^2\gamma K_{33}
+ \end{matrix}\right),
+\end{equation}
+with
+\begin{align*}
+& \mathbf{S} = \left(k_x-(1+k_y^2)l_x + k_xk_yl_y,\, k_y-(1+k_x^2)l_y
+ + k_xk_yl_x\right)^\textrm{T},\\
+&K_{33} = l_x^2(1+k_y^2) + l_y^2(1+k_x^2) - 2(k_xl_x + k_yl_y) + k_x^2
++ k_y^2 - 2k_xk_yl_xl_y.
+\end{align*}
+
+</font>
<font color="blue">oindent{\it Sanity checking}\\
+In the case when the $(x^r, y^r, r)$-coordinate system coincides with
+the $(x,y,z)$-coordinate system, ones $\sigma=\gamma=0$, and hence
+$l_x = l_y=0$ according to \eqref{eq:43}. Therefore, $K^r$ reduces to
+\begin{equation*}
+\cos^2\beta\left(
+ \begin{matrix}
+ 1 + k_y^2 & { } & -k_xk_y & { } & {k_x }\\
+ { } & { } & { } & { } & { }\\
+ -k_xk_y & { } & 1 + k_x^2 & { } & {k_y }\\
+ { } & { } & { } & { } & { } \\
+ {k_x} & { } & {k_y } & { } & k_x^2 + k_y^2
+ \end{matrix}\right),
+\end{equation*}
+which is exactly the form of the Redi diffusion tensor on the
+traditional $z$-level coordinate. In the case when the constant $r$
+surface coincides with the constant density $\rho$ surface, one has
+$l_x = k_x$ and $l_y=k_y$. Therefore, $K^r$ reduces to
+\begin{equation*}
+\cos^2\beta\left(
+ \begin{matrix}
+ 1 + k_y^2 & { } & -k_xk_y & { } & { 0 }\\
+ { } & { } & { } & { } & { }\\
+ -k_xk_y & { } & 1 + k_x^2 & { } & {0 }\\
+ { } & { } & { } & { } & { } \\
+ {0} & { } & {0 } & { } & 0
+ \end{matrix}\right).
+\end{equation*}
+In this case, the requirement that there should be no flux across the
+constant density surface carries over to the constant $r$ surface. The
+extra terms in the upper-left $2\times 2$ matrix are due to the fact
+that the horizontal $(x,y)$-, instead of $(x^\rho, y^\rho)$-,
+coordinate system are used.
+
+</font>
<font color="blue">oindent{\it Calculation of the slopes}\\
+We note that
+\begin{align*}
+ (k_x, k_y) &= -\dfrac{</font>
<font color="blue">abla_z\rho}{\rho_z}\\
+ &= -\dfrac{1}{\rho_z}\left(</font>
<font color="blue">abla_r\rho +
+ \dfrac{\p\rho}{\p r}</font>
<font color="blue">abla_z r\right)\\
+ &= -\dfrac{1}{\rho_z}\left(</font>
<font color="blue">abla_r\rho -
+ \dfrac{\p\rho}{\p z}</font>
<font color="blue">abla_r z\right)\\
+ &= -\dfrac{1}{\rho_z}</font>
<font color="black">abla_r\rho + </font>
<font color="blue">abla_r z.
+\end{align*}
+That is,
+\begin{equation}
+ \label{eq:45}
+(k_x,k_y) = -\dfrac{1}{\rho_z}</font>
<font color="black">abla_r\rho + </font>
<font color="blue">abla_r z.
+\end{equation}
+By definition, the slope of the constant $r$ surface is by
+\begin{equation}
+ \label{eq:46}
+ (l_x, l_y) = </font>
<font color="blue">abla_r z
+\end{equation}
+Using \eqref{eq:45} and \eqref{eq:46}, the vector $\mathbf{S}$ can
+rewritten and simplified as
+\begin{equation}
+ \label{eq:47}
+ \mathbf{S} = -\dfrac{</font>
<font color="blue">abla_r\rho}{\rho_z} +
+ \left(-k_y^2l_x + k_xk_yl_y, -k_x^2l_y +
+ k_xk_yl_x\right)^\textrm{T}.
+\end{equation}
+The term $K_{33}$ can be rewritten as simplified as
+\begin{align*}
+ K_{33} &= (k_x-l_x)^2 + (k_y - l_y)^2 + l_x^2k_y^2 + l_y^2k_x^2 -
+ 2k_xk_yl_xl_y\\
+ &= \dfrac{|</font>
<font color="blue">abla_r\rho|^2}{\rho_z^2}+ l_x^2k_y^2 + l_y^2k_x^2 -
+ 2k_xk_yl_xl_y.
+\end{align*}
+Hence,
+\begin{equation}
+ \label{eq:49}
+ K_{33} = \dfrac{|</font>
<font color="blue">abla_r\rho|^2}{\rho_z^2}+ l_x^2k_y^2 +
+ l_y^2k_x^2 - 2k_xk_yl_xl_y
+\end{equation}
+
+</font>
<font color="blue">oindent{\it Small angle approximation}\\
+The slopes of the constant density surfaces are usually small,
+i.e.~$k_x$ and $k_y$ in \eqref{eq:47} and \eqref{eq:49} are usually
+small. The high order terms involving $k_x$ and/or $k_y$ are even
+smaller, and in practice they may be neglected, leading to simpler forms
+of the Redi diffusion tensor. We propose to neglect the second-order
+terms in the upper-left $2\times 2$ sub-matrix, the third-order
+terms of \eqref{eq:47} and the fourth-order terms in \eqref{eq:49},
+that is, we take
+\begin{equation}
+ \label{eq:53}
+ K^r = \left(\begin{matrix}
+ I_2 & { } & \cos\gamma\mathbf{S}\\
+\\
+ \cos\gamma\mathbf{S} & { } & \cos^2\gamma K_{33}\end{matrix}\right),
+\end{equation}
+with
+\begin{align}
+ &\mathbf{S} =
+ -\dfrac{</font>
<font color="blue">abla_r\rho}{\rho_z}, \label{eq:51}\\
+ &K_{33} = \dfrac{|</font>
<font color="blue">abla_r\rho|^2}{\rho_z^2}\label{eq:52}
+ \end{align}
+It can be easily checked that the diffusion tensor $K^r$ with
+$\mathbf{S}$ and $K_{33}$ taking the simplified forms \eqref{eq:51}
+and \eqref{eq:52} has no effect on the density field, that is,
+\begin{displaymath}
+ K^r</font>
<font color="blue">abla^r\rho = 0.
+\end{displaymath}
+
+</font>
<font color="blue">oindent{\it Implementation}\\
+With $K^r$ taking the form of \eqref{eq:53}, the Redi
+diffusion can be rearranged as
+\begin{multline}
+\label{eq:54}
+ </font>
<font color="black">abla^r\cdot(K^r</font>
<font color="black">abla^r\varphi) = </font>
<font color="black">abla_r\cdot(</font>
<font color="blue">abla_r\varphi)
+ + </font>
<font color="blue">abla_r\cdot\left(\cos\gamma\mathbf{S}\dfrac{\p\varphi}{\p
+ r}\right) +\\ \dfrac{\p}{\p
+ r}\left(\cos\gamma\mathbf{S}\cdot</font>
<font color="blue">abla_r\varphi\right) +
+ \dfrac{\p}{\p r}\left(\cos^2\gamma K_{33}\dfrac{\p}{\p
+ r}\varphi\right),
+\end{multline}
+where
+$$</font>
<font color="blue">abla_r \equiv \left(\frac{\p}{\p x}|_r,\, \frac{\p}{\p
+ y}|_r\right)$$
+denotes the two-dimensional gradient operator on
+constant $r$ surfaces. According to the third equation in
+\eqref{eq:32}, for fixed $(x,\,y)$,
+\begin{displaymath}
+ \dfrac{\p r}{\p z} = \cos\gamma.
+\end{displaymath}
+Therefore, the $\cos\gamma$ term in \eqref{eq:54} can be combined with
+$\p/\p r$ to give $\p/\p z$, and \eqref{eq:54} can then be written as
+\begin{multline}
+\label{eq:55}
+ </font>
<font color="black">abla^r\cdot(K^r</font>
<font color="black">abla^r\varphi) = </font>
<font color="black">abla_r\cdot(</font>
<font color="blue">abla_r\varphi)
+ + </font>
<font color="blue">abla_r\cdot\left(\mathbf{S}\dfrac{\p\varphi}{\p
+ z}\right) + \dfrac{\p}{\p
+ z}\left(\mathbf{S}\cdot</font>
<font color="blue">abla_r\varphi\right) +
+ \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
+third term using the FDM, we need the values of
+$\mathbf{S}\cdot</font>
<font color="blue">abla_r\varphi$ at the cell centers, which can be
+approximated by
+\begin{align*}
+ \left[\mathbf{S}\cdot</font>
<font color="gray">abla_r\varphi\right]_i &= 2\sum_{e\in
+ EC(i)}\mathbf{S}\cdot\mathbf{n}_e \dfrac{\p\varphi}{\p n_e}\\
+ &= -2\sum_{e\in
+ EC(i)}\dfrac{1}{[\rho_z]_e}\dfrac{\p\rho}{\p n_e} \dfrac{\p\varphi}{\p n_e},
+\end{align*}
+with $\p/\p n_e$ denoting the derivative in the normal direction on
+edge $e$.
+
+%-----------------------------------------------------------------------
+
+\chapter{Testing}
+The goal of the testing process is to ensure that the implementation
+is working as it is intended by the design laid out in previous
+sections. We particularly point out that the testing is not about
+whether GM is a good mesoscale eddy parametrization.
+
+\section{Testing and Validation: the vertical boundary value problem}
+Date last modified: 2012/09/20 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+The boundary value problem \eqref{eq:19}-~\eqref{eq:20} will also be
+implemented in Matlab. The Matlab will be evaluated based on exact solutions. The solver will then be implemented into MPAS, but run in diagnostic mode only. We will configure an ocean basin such that every column matches the Matlab testing condition and insure that we obtain the same results as the Matlab code.
+
+\section{Testing and Validation: diagnostic isopycnal setting}
+Date last modified: 2012/09/20 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+A three-layer isopycnal model should be configured and run, with GM
+being implemented based on the layer thickness and on the isopycnal
+slopes. These two different implementations should produce close
+results in terms of the output of the bolus velocities and the
+overall model outputs. In these two cases, the bolus velocities will be diagnostics only (i.e. will not feedback into the model solution).
+
+\section{Testing and Validation: isopycnal setting}
+Date last modified: 2012/09/20 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+A three-layer isopycnal model should be configured and run, with GM
+being implemented based on the layer thickness and on the isopycnal
+slopes. These two different implementations should produce close
+results in terms of the output of the bolus velocities and the
+overall model outputs. In these two cases, the bolus velocities will be included into the model transport velocity. This will test the stability of the new bolus velocity computation.
+
+\section{Testing and Validation: the low-res $z$-level or $z^\ast$ settings}
+Date last modified: 2012/09/20 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+A 125km $z$-level model of the ACC and a 125km $z^\ast$ of the same
+model should be run with and without the GM turned on. The GM closure
+is expected to lead to flatter isopycnal slopes and slower jets.
+
+
+\section{Testing and Validation: Redi isopycnal diffusion}
+Date last modified: 2012/09/20 \\
+Contributors: Qingshan Chen and Todd Ringler\\
+
+????
+
+
+
+%-----------------------------------------------------------------------
+\bibliographystyle{plain}
+\bibliography{gm-references}
+
+
+\end{document}
Added: trunk/documents/ocean/current_design_doc/gm/variable-placement.pdf
===================================================================
--- trunk/documents/ocean/current_design_doc/gm/variable-placement.pdf (rev 0)
+++ trunk/documents/ocean/current_design_doc/gm/variable-placement.pdf 2012-11-26 21:02:18 UTC (rev 2319)
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+ <rdf:Seq>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:groupName>Default Swatch Group</xmpG:groupName>
+ <xmpG:groupType>0</xmpG:groupType>
+ <xmpG:Colorants>
+ <rdf:Seq>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>White</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>0.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>Black</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>0.000000</xmpG:yellow>
+ <xmpG:black>100.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>CMYK Red</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>100.000000</xmpG:magenta>
+ <xmpG:yellow>100.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>CMYK Yellow</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>100.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>CMYK Green</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>100.000000</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>100.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>CMYK Cyan</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>100.000000</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>0.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>CMYK Blue</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>100.000000</xmpG:cyan>
+ <xmpG:magenta>100.000000</xmpG:magenta>
+ <xmpG:yellow>0.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>CMYK Magenta</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>100.000000</xmpG:magenta>
+ <xmpG:yellow>0.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=15 M=100 Y=90 K=10</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>14.999998</xmpG:cyan>
+ <xmpG:magenta>100.000000</xmpG:magenta>
+ <xmpG:yellow>90.000000</xmpG:yellow>
+ <xmpG:black>10.000002</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=0 M=90 Y=85 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>90.000000</xmpG:magenta>
+ <xmpG:yellow>85.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=0 M=80 Y=95 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>80.000000</xmpG:magenta>
+ <xmpG:yellow>95.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=0 M=50 Y=100 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>50.000000</xmpG:magenta>
+ <xmpG:yellow>100.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=0 M=35 Y=85 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>0.000000</xmpG:cyan>
+ <xmpG:magenta>35.000004</xmpG:magenta>
+ <xmpG:yellow>85.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=5 M=0 Y=90 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>5.000001</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>90.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=20 M=0 Y=100 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>19.999998</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>100.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=50 M=0 Y=100 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>50.000000</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>100.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=75 M=0 Y=100 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>75.000000</xmpG:cyan>
+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>100.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=85 M=10 Y=100 K=10</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>85.000000</xmpG:cyan>
+ <xmpG:magenta>10.000002</xmpG:magenta>
+ <xmpG:yellow>100.000000</xmpG:yellow>
+ <xmpG:black>10.000002</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=90 M=30 Y=95 K=30</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>90.000000</xmpG:cyan>
+ <xmpG:magenta>30.000002</xmpG:magenta>
+ <xmpG:yellow>95.000000</xmpG:yellow>
+ <xmpG:black>30.000002</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=75 M=0 Y=75 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
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+ <xmpG:magenta>0.000000</xmpG:magenta>
+ <xmpG:yellow>75.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=80 M=10 Y=45 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>80.000000</xmpG:cyan>
+ <xmpG:magenta>10.000002</xmpG:magenta>
+ <xmpG:yellow>45.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
+ </rdf:li>
+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=70 M=15 Y=0 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
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+ <xmpG:magenta>14.999998</xmpG:magenta>
+ <xmpG:yellow>0.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
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+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=85 M=50 Y=0 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
+ <xmpG:cyan>85.000000</xmpG:cyan>
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+ <xmpG:yellow>0.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
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+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=100 M=95 Y=5 K=0</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
+ <xmpG:type>PROCESS</xmpG:type>
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+ <xmpG:yellow>5.000001</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
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+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=100 M=100 Y=25 K=25</xmpG:swatchName>
+ <xmpG:mode>CMYK</xmpG:mode>
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+ <xmpG:magenta>100.000000</xmpG:magenta>
+ <xmpG:yellow>25.000000</xmpG:yellow>
+ <xmpG:black>25.000000</xmpG:black>
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+ <rdf:li rdf:parseType="Resource">
+ <xmpG:swatchName>C=75 M=100 Y=0 K=0</xmpG:swatchName>
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+ <xmpG:yellow>0.000000</xmpG:yellow>
+ <xmpG:black>0.000000</xmpG:black>
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+ <rdf:li rdf:parseType="Resource">
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+ <xmpG:yellow>0.000000</xmpG:yellow>
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