<p><b>qchen3@fsu.edu</b> 2012-11-28 16:07:28 -0700 (Wed, 28 Nov 2012)</p><p>For GM design document: added the epsilon terms representing weak cross-isopycnal diffusions.<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 2012-11-27 19:49:24 UTC (rev 2325)
+++ trunk/documents/ocean/current_design_doc/gm/gm.tex 2012-11-28 23:07:28 UTC (rev 2326)
@@ -628,7 +628,62 @@
K^r</font>
<font color="blue">abla^r\rho = 0.
\end{displaymath}
+</font>
<font color="blue">oindent{\it Weak cross-isopycnal-surface diffusion}\\
+If weak cross-isopycnal-surface diffusion is to be considered, then
+$K^\rho$ takes the form
+\begin{equation}
+\label{eq:48}
+ K^\rho = \left(
+ \begin{matrix}
+ 1 & 0 & 0\\
+ 0 & 1 & 0\\
+ 0 & 0 & \epsilon
+ \end{matrix}\right).
+\end{equation}
+for a small positive number $\epsilon$. Through the same transformation
+given in \eqref{eq:40}, $K^r$ is found to be
+\begin{equation}
+\label{eq:50}
+ K^r = \cos^2\beta\left(
+ \begin{matrix}
+ 1 + k_y^2 + \epsilon k_x^2& { } & -(1-\epsilon)k_xk_y & { } & { }\\
+ { } & { } & { } & { } & \cos\gamma{\tilde{\mathbf{S}}}\\
+ -(1-\epsilon)k_xk_y & { } & 1 + k_x^2+\epsilon k_y^2 & { } & { }\\
+ { } & { } & { } & { } & { } \\
+ { } & \cos\gamma\tilde{\mathbf{S}}^\textrm{T} & { } & { } &
+ \cos^2\gamma K_{33}
+ \end{matrix}\right),
+\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) -
+ \epsilon k_x(k_xl_x+k_yl_y) \\
+ \\
+ (1-\epsilon)k_y-l_y
+ + k_x(k_yl_x - k_xl_y) - \epsilon
+ k_y(k_xl_x+k_yl_y)\end{matrix}\right),
+\end{equation*}
+and
+\begin{multline*}
+K_{33} = (k_x - l_x)^2 + (k_y - l_y)^2 +\epsilon + (k_yl_x - k_xl_y)^2 +
+2\epsilon(k_xl_x+k_yl_y) + \epsilon(k_xl_x+k_yl_y)^2.
+% 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{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 \\
+ \\
+ (1-\epsilon)k_y-l_y
+ \end{matrix}\right),
+\end{equation*}
+and
+\begin{equation*}
+K_{33} = (k_x - l_x)^2 + (k_y - l_y)^2 +\epsilon.
+\end{equation*}
+
</font>
<font color="gray">oindent{\it Implementation}\\
With $K^r$ taking the form of \eqref{eq:53}, the Redi
diffusion can be rearranged as
@@ -667,16 +722,16 @@
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{align*}
- \left[\tilde{\mathbf{S}}\cdot</font>
<font color="red">abla_r\varphi\right]_i &= 2\sum_{e\in
- EC(i)}\tilde{\mathbf{S}}\cdot\mathbf{n}_e {</font>
<font color="red">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{align*}
-with $(\p/\p n_e)|_r$ denoting the derivative in the normal direction
-along constant $r$ surfaces on
-edge $e$.
+\begin{equation*}
+ \left[\tilde{\mathbf{S}}\cdot</font>
<font color="blue">abla_r\varphi\right]_i = 2\sum_{e\in
+ EC(i)}\tilde{\mathbf{S}}\cdot\mathbf{n}_e {</font>
<font color="blue">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*}
+% with $(\p/\p n_e)|_r$ denoting the derivative in the normal direction
+% along constant $r$ surfaces on
+% edge $e$.
% \section{Normal components}\label{sec:normal-components}
% Date last modified: 2012/9/18 \\
</font>
</pre>