source: trunk/NEMO/DOC/BETA/Chapters/Annex_B.tex @ 707

% ================================================================
% Chapter Ñ Appendix B : Diffusive Operators
% ================================================================
\chapter{Appendix B : Diffusive Operators}
\label{Apdx_B}
\minitoc

% ================================================================
% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
% ================================================================
\section{Horizontal/Vertical 2nd Order Tracer Diffusive Operators}
\label{Apdx_B_1}


In $z$-coordinates, the horizontal/vertical second order tracer diffusive 
operator is given by:



\begin{equation} \label{Apdx_B1}
\begin{split}
 D^T&=D^{lT}+D^{vT} \\ 
 &=\frac{1}{e_1 e_2 }\left[ {\left. {\frac{\partial }{\partial i}\left( 
{\frac{e_2 }{e_1 }A^{lT} \;\left. {\frac{\partial T}{\partial i}} \right|_z 
} \right)} \right|_z +\left. {\frac{\partial }{\partial j}\left( {\frac{e_1 
}{e_2 }A^{lT} \;\left. {\frac{\partial T}{\partial j}} \right|_z } \right)} 
\right|_z } \right] +\frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial 
z}} \right) \\ 
\end{split}
\end{equation}


In $s$-coordinates, we defined the slopes of $s-$surfaces, $\sigma_1$ and $\sigma_2$ by (A.1), the vertical/horizontal ratio of diffusive coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusive operator is given by:

\begin{equation} \label{Apdx_B2}
\begin{aligned} 
&D^T=D^{lT}+D^{vT} =\left. \nabla \right|_s \cdot \left[ {A^{lT} \;\Re \cdot \left. \nabla 
\right|_s \left( T \right)} \right] \\
&\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
 {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1 
^2+\sigma _2 ^2} \hfill \\
\end{array} }} \right)
\end{aligned}
\end{equation}

or in expended form:

\begin{multline} \label{Apdx_B3}
D^T=\frac{1}{e_1 e_2 e_3 }\;\left[ {\;\,e_2 e_3 A^{lT} \;\left. 
{\frac{\partial }{\partial i}\left( {\frac{1}{e_1 }\;\left. {\frac{\partial 
T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial 
T}{\partial s}} \right)} \right|_s } \right.  \\
\;\;+e_1 e_3 A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\
 \;\;+e_1 e_2 A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. \\
\shoveright{\;\;\left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;\,} \right]} \\ 
\end{multline}


Equation (\ref{Apdx_B2}) (or equivalently (\ref{Apdx_B3})~) is obtained from (\ref{Apdx_B1}) without any additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, 
we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix A 
\newline
\newline
and use (A.1) and (A.2). Since no cross horizontal derivate $\partial _i \,\partial _j $ appears neither in (B.1) nor in (A.2), there is a decoupling between $(i,z)$ and $(j,z)$ plans as well as 
$(i,s)$ and $(j,s)$ plans. The demonstration can then be done for the $(i,s)\;\to \;(j,s)$ transformation without loss of generality:

\begin{equation*}
D^T=\frac{1}{e_1 \,e_2 }\left. {\frac{\partial }{\partial i}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)
\end{equation*}

\begin{multline*}
 =\frac{1}{e_1 \,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
 \left. { -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right] \\ 
\shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]} \\ 
\end{multline*}


\begin{multline*}
 =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s -\left. {\frac{e_2 }{e_1 
}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
 \left. {-e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( 
{\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\
 \shoveright{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;} \right] }\\ 
\end{multline*}


Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:

\begin{multline*}
 =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-e_3 \frac{\partial }{\partial 
i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
 -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
\shoveright{ \left. { +e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial z}} \right)\;\;\;} \right] }\\ 
\end{multline*}

\begin{multline*}
 =\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-\frac{\partial }{\partial 
i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
 \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}}  \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ 
-e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ 
\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s}} \right) + \frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
\end{multline*}
using the same remark as just above, it becomes:
\begin{multline*}
= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\ 
+\frac{e_1 \,e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma _1 }{\partial s} - \frac {\sigma _1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ 
-e_2 \left( {A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma _1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma _1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }\\ 
\end{multline*}

Since the horizontal scale factor do not depend on the vertical coordinates, the last term of the first line and the first term of the last line cancel, while the second line reduces to a single vertical derivative, so it becomes:
\begin{multline*}
 =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
 \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma _1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma _1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
\end{multline*}

in other words:

\begin{equation*}
D^T = {\frac{1}{e_1\,e_2\,e_3}}
\left( {{\begin{array}{*{20}c}
{\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
{\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
\end{array}}}\right)
\cdot \left[ {A^{lT}
\left( {{\begin{array}{*{20}c}
 {1} \hfill & {-\sigma_1 } \hfill \\
 {-\sigma_1} \hfill & {\varepsilon_1^2} \hfill \\
\end{array} }} \right)
\cdot 
\left( {{\begin{array}{*{20}c}
{\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
{\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
\end{array}}}
\right) \left( T \right)} \right]
\end{equation*}


% ================================================================
% Isopycnal/Vertical Second Order Tracer Diffusive Operators
% ================================================================
\section{Isopycnal/Vertical Second Order Tracer Diffusive Operators}
\label{Apdx_B_2}


The isopycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the curvilinear coordinate system $(i,j,k)$, in which the equations of the ocean circulation model are formulated, takes the following expression [Redi 1982]:

\begin{equation*}
\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
\left[ {{\begin{array}{*{20}c}
 {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
 {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
\end{array} }} \right]
\end{equation*}
where$(a_1$, $a_2)$ are the isopycnal slopes in $(\textbf{i}$, $\textbf{j})$ directions:
\begin{equation*}
a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
\end{equation*}
\begin{equation*}
a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
\end{equation*}
In practice, the isopycnal slopes are generally less than $10${\small $^{-2}$ in the ocean, so $\textbf {A}_{\textbf I}$ can be simplified appreciably (Cox, 1987) :
\begin{equation*}
{\textbf{A}_{\textbf{I}}} =A^{lT}
\left[ {{\begin{array}{*{20}c}
 1 \hfill & 0 \hfill & {-a_1 } \hfill \\
 0 \hfill & 1 \hfill & {-a_2 } \hfill \\
 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
\end{array} }} \right]
\end{equation*}
The resulting isopycnal operator conserves the quantity it diffuses, and dissipates the square of this quantity. The demonstration of the first property is trivial. Let us demonstrate the second one:
\begin{equation*}
\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
\end{equation*}
since
\begin{multline*}
 \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
\right)=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
\frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
{\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
\shoveright{ \left. {-2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial 
k}+\left( {a_1 ^2+a_2 ^2} \right)\left( {\frac{\partial T}{\partial k}} 
\right)^2} \right]} \\
\end{multline*}

\begin{equation*}
=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial j}-a_2 \frac{\partial T}{\partial k}} \right)^2} \right]\quad \geq 0
\end{equation*}
the property becomes obvious.
\\
Note that the resulting tensor is similar to those obtained for geopotential diffusion in $s$-coordinates. The simplification leads to a decoupling between $(i,z)$ and $(j,z)$ plans.

The resulting diffusive operator in $z$-coordinates has the following 
expression :
\begin{multline*}
 D^T=\frac{1}{e_1 e_2 }\left\{ {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2 }{e_1 }\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right.\;\; \\ 
 \;\left. {\;\;\;+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1 }{e_2 }\frac{\partial T}{\partial j}-a_2 \frac{e_1 }{e_3 }\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\
\shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]} \\ 
\end{multline*}


% ================================================================
% Lateral/Vertical Momentum Diffusive Operators
% ================================================================
\section{Lateral/Vertical Momentum Diffusive Operators}
\label{Apdx_B_3}


*{\footnotesize }lateral/vertical{\footnotesize }2{\small nd}{\footnotesize 
}order{\footnotesize }momentum diffusive operator

Following (I.3.6), the Laplacian of the horizontal velocity can be expressed 
in $z$-coordinates:
\begin{equation*}
\Delta {\textbf{U}}_h =\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)
\end{equation*}

\begin{equation*}
=\left( {{\begin{array}{*{20}c}
 {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
 {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
 {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
 {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 
}\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial 
u}{\partial k}} \right)} \hfill \\
 {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 
}\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta 
}{\partial i}} \hfill \\
 {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 
}{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial 
j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} 
\hfill \\
\end{array} }} \right)
\end{equation*}
\begin{equation*}
=\left( {{\begin{array}{*{20}c}
{\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
{\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
0 \\
\end{array} }} \right)
+\frac{1}{e_3 }
\left( {{\begin{array}{*{20}c}
{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
{\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\
\end{array} }} \right)
\end{equation*}
Using (I.3.8), the definition of the horizontal divergence, the third componant of the second vector is obviously zero and thus :
\begin{equation*}
\Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
\end{equation*}


The lateral/vertical second order (Laplacian type) operator used to diffuse 
horizontal momentum in $z$-coordinates therefore takes the following expression :
\begin{equation*}
\begin{split}
 {\textbf{D}}^{\textbf{U}}&={\textbf{D}}^{lm}+{\textbf{D}}^{vm} \\ 
&=\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)+\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
\end{split}
\end{equation*}

\begin{equation*}
{\textbf{D}}^{\textbf{U}}=
\left( {{\begin{array}{*{20}c}
{\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial i}-\frac{1}{e_2 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}} \\
{\frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j}+\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}} \\
\end{array} }} \right)
+\frac{1}{e_3 }\frac{\partial }{\partial k}
\left( {{\begin{array}{*{20}c}
{\frac{A^{vm}}{e_3 }\frac{\partial u}{\partial k}} \\
{\frac{A^{vm}}{e_3 }\frac{\partial v}{\partial k}} \\
\end{array} }} \right)
\end{equation*}

% ================================================================
% References
% ================================================================
\section{References}


{\small Cox, M. D., 1987 : Isopycnal diffusion in a z-coordinate ocean 
model. }{\small \textit{Ocean Modelling}}{\small , 74, 1-9.}

{\small Redi, M. H., 1982 : oceanic isopycnal mixing by coordinate rotation. 
}{\small \textit{J. Phys. Oceanogr., 13}}{\small , 1154-1158.}