\documentclass[../main/NEMO_manual]{subfiles} \begin{document} \chapter{Diffusive Operators} \label{apdx:DIFFOPERS} \thispagestyle{plain} \chaptertoc \paragraph{Changes record} ~\\ {\footnotesize \begin{tabularx}{\textwidth}{l||X|X} Release & Author(s) & Modifications \\ \hline {\em 4.0} & {\em ...} & {\em ...} \\ {\em 3.6} & {\em ...} & {\em ...} \\ {\em 3.4} & {\em ...} & {\em ...} \\ {\em <=3.4} & {\em ...} & {\em ...} \end{tabularx} } \clearpage %% ================================================================================================= \section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators} \label{sec:DIFFOPERS_1} %% ================================================================================================= \subsubsection*{In z-coordinates} In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: \begin{align} \label{eq:DIFFOPERS_1} &D^T = \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 \right. \left. + \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{align} %% ================================================================================================= \subsubsection*{In generalized vertical coordinates} In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{eq:SCOORD_s_slope} and the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by: \begin{equation} \label{eq:DIFFOPERS_2} D^T = \left. \nabla \right|_s \cdot \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \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{equation} or in expanded form: \begin{align*} { \begin{array}{*{20}l} D^T= \frac{1}{e_1\,e_2\,e_3 } & \left\{ \quad \quad \frac{\partial }{\partial i} \left. \left[ e_2\,e_3 \, A^{lT} \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] \right|_s \right. \\ & \quad \ + \ \left. \frac{\partial }{\partial j} \left. \left[ e_1\,e_3 \, A^{lT} \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] \right|_s \right. \\ & \quad \ + \ \left. e_1\,e_2\, \frac{\partial }{\partial s} \left[ A^{lT} \; \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 +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \; \right\} . \end{array} } \end{align*} \autoref{eq:DIFFOPERS_2} is obtained from \autoref{eq:DIFFOPERS_1} 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 \autoref{apdx:SCOORD} and use \autoref{eq:SCOORD_s_slope} and \autoref{eq:SCOORD_s_chain_rule1}. Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{eq:DIFFOPERS_1}, the ($i$,$z$) and ($j$,$z$) planes are independent. The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without any loss of generality: \begin{align*} { \begin{array}{*{20}l} 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) \\ \\ % &=\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. \\ & \qquad \qquad \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]} \qquad \qquad \qquad \\ \\ % &=\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. \\ & \qquad \qquad \quad \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) \\ & \qquad \qquad \quad \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)\quad} \right] }\\ \end{array} } \\ % { \begin{array}{*{20}l} \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, this becomes:} % D^T & =\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. \\ & \qquad \qquad \quad -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) \\ & \qquad \qquad \quad\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 s}} \right)\;\;\;} \right] }\\ \\ &=\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. \\ & \qquad \qquad \quad \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 \\ & \qquad \qquad \quad-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) \\ & \qquad \qquad \quad\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{array} } \\ { \begin{array}{*{20}l} % \intertext{Using the same remark as just above, $D^T$ becomes:} % D^T &= \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.\;\;\; \\ & \qquad \qquad \quad+\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} \\ & \qquad \qquad \quad-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) \\ & \qquad \qquad \quad\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{array} } \\ { \begin{array}{*{20}l} % \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, the two terms on the second line cancel, while the third line reduces to a single vertical derivative, so it becomes:} % D^T & =\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. \\ & \qquad \qquad \quad \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]} \\ % \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} \end{array} } \\ % {\frac{1}{e_1\,e_2\,e_3}} \left( {{ \begin{array}{*{30}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}{*{30}c} {1} \hfill & {-\sigma_1 } \hfill \\ {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ \end{array} }} \right) \cdot \left( {{ \begin{array}{*{30}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{align*} %\addtocounter{equation}{-2} %% ================================================================================================= \section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} \label{sec:DIFFOPERS_2} %% ================================================================================================= \subsubsection*{In z-coordinates} The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$) curvilinear coordinate system in which the equations of the ocean circulation model are formulated, takes the following form \citep{redi_JPO82}: \begin{equation} \label{eq:DIFFOPERS_3} \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} \left[ {{ \begin{array}{*{20}c} {1+a_2 ^2 +\varepsilon a_1 ^2} \hfill & {-a_1 a_2 (1-\varepsilon)} \hfill & {-a_1 (1-\varepsilon) } \hfill \\ {-a_1 a_2 (1-\varepsilon) } \hfill & {1+a_1 ^2 +\varepsilon a_2 ^2} \hfill & {-a_2 (1-\varepsilon)} \hfill \\ {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ \end{array} }} \right] \end{equation} where ($a_1$, $a_2$) are $(-1) \times$ the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials (or equivalently the slopes of the geopotential surfaces in the isopycnal coordinate framework): \[ a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \qquad , \qquad a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \] and, as before, $\epsilon = A^{vT} / A^{lT}$. In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean, so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0) and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}: \begin{subequations} \label{eq:DIFFOPERS_4} \begin{equation} \label{eq:DIFFOPERS_4a} {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = \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} and the iso/dianeutral diffusive operator in $z$-coordinates is then \begin{equation} \label{eq:DIFFOPERS_4b} D^T = \left. \nabla \right|_z \cdot \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T \right]. \\ \end{equation} \end{subequations} Physically, the full tensor \autoref{eq:DIFFOPERS_3} represents strong isoneutral diffusion on a plane parallel to the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. However, the approximate `weak-slope' tensor \autoref{eq:DIFFOPERS_4a} represents strong diffusion along the isoneutral surface, with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal. This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same form, \autoref{eq:DIFFOPERS_4}, as \autoref{eq:DIFFOPERS_2}, the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. Written out explicitly, \begin{multline} \label{eq:DIFFOPERS_ldfiso} 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]} {+\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+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\ \end{multline} The isopycnal diffusion operator \autoref{eq:DIFFOPERS_4}, \autoref{eq:DIFFOPERS_ldfiso} conserves tracer quantity and dissipates its square. As \autoref{eq:DIFFOPERS_4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero (as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one: \[ \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, \] and since \begin{align*} { \begin{array}{*{20}l} \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf 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. \\ &\qquad \qquad \qquad { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\ &=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} +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right] \\ & \geq 0 . \end{array} } \end{align*} %\addtocounter{equation}{-1} the property becomes obvious. %% ================================================================================================= \subsubsection*{In generalized vertical coordinates} Because the weak-slope operator \autoref{eq:DIFFOPERS_4}, \autoref{eq:DIFFOPERS_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into generalized $s$-coordinates in the same way as \autoref{sec:DIFFOPERS_1} was transformed into \autoref{sec:DIFFOPERS_2}. The resulting operator then takes the simple form \begin{equation} \label{eq:DIFFOPERS_ldfiso_s} D^T = \left. \nabla \right|_s \cdot \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\ \;\;\text{where} \;\Re =\left( {{ \begin{array}{*{20}c} 1 \hfill & 0 \hfill & {-r _1 } \hfill \\ 0 \hfill & 1 \hfill & {-r _2 } \hfill \\ {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 ^2+r _2 ^2} \hfill \\ \end{array} }} \right), \end{equation} where ($r_1$, $r_2$) are $(-1)\times$ the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to $s$-coordinate surfaces (or equivalently the slopes of the $s$-coordinate surfaces in the isopycnal coordinate framework): \[ r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} \qquad , \qquad r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. \] To prove \autoref{eq:DIFFOPERS_ldfiso_s} by direct re-expression of \autoref{eq:DIFFOPERS_ldfiso} is straightforward, but laborious. An easier way is first to note (by reversing the derivation of \autoref{sec:DIFFOPERS_2} from \autoref{sec:DIFFOPERS_1} ) that the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as \begin{equation} \label{eq:DIFFOPERS_5} D^T = \left. \nabla \right|_\rho \cdot \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T \right] \\ \;\;\text{where} \;\Re =\left( {{ \begin{array}{*{20}c} 1 \hfill & 0 \hfill &0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & \varepsilon \hfill \\ \end{array} }} \right). \end{equation} Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives \autoref{eq:DIFFOPERS_ldfiso_s} immediately. Note that the weak-slope approximation is only made in transforming from the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates. The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in \autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. %% ================================================================================================= \section{Lateral/Vertical momentum diffusive operators} \label{sec:DIFFOPERS_3} The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by applying \autoref{eq:MB_lap_vector}, the expression for the Laplacian of a vector, to the horizontal velocity vector: \begin{align*} \Delta {\textbf{U}}_h &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\ &=\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) \\ \\ &=\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{align*} Using \autoref{eq:MB_div}, the definition of the horizontal divergence, the third component of the second vector is obviously zero and thus : \[ \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) . \] Note that this operator ensures a full separation between the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}). It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere. The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in the $z$-coordinate therefore takes the following form: \begin{equation} \label{eq:DIFFOPERS_Lap_U} { \textbf{D}}^{\textbf{U}} = \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 {\mathrm {\mathbf U}}_h }{\partial k}} \right) , \\ \end{equation} that is, in expanded form: \begin{align*} D^{\textbf{U}}_u & = \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_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right) , \\ D^{\textbf{U}}_v & = \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} +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right) . \end{align*} Note Bene: introducing a rotation in \autoref{eq:DIFFOPERS_Lap_U} does not lead to a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. Similarly, we did not found an expression of practical use for the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate. Generally, \autoref{eq:DIFFOPERS_Lap_U} is used in both $z$- and $s$-coordinate systems, that is a Laplacian diffusion is applied on momentum along the coordinate directions. \subinc{\input{../../global/epilogue}} \end{document}