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1\documentclass[../main/NEMO_manual]{subfiles}
2
3%% Local cmds
4\newcommand{\rML}[1][i]{\ensuremath{_{\mathrm{ML}\,#1}}}
5\newcommand{\rMLt}[1][i]{\tilde{r}_{\mathrm{ML}\,#1}}
6%% Move to ../../global/new_cmds.tex to avoid error with \listoffigures
7%\newcommand{\triad}[6][]{\ensuremath{{}_{#2}^{#3}{\mathbb{#4}_{#1}}_{#5}^{\,#6}}
8\newcommand{\triadd}[5]{\ensuremath{{}_{#1}^{#2}{\mathbb{#3}}_{#4}^{\,#5}}}
9\newcommand{\triadt}[5]{\ensuremath{{}_{#1}^{#2}{\tilde{\mathbb{#3}}}_{#4}^{\,#5}}}
10\newcommand{\rtriad}[2][]{\ensuremath{\triad[#1]{i}{k}{#2}{i_p}{k_p}}}
11\newcommand{\rtriadt}[1]{\ensuremath{\triadt{i}{k}{#1}{i_p}{k_p}}}
12
13\begin{document}
14% ================================================================
15% Iso-neutral diffusion :
16% ================================================================
17\chapter{Iso-Neutral Diffusion and Eddy Advection using Triads}
18\label{apdx:triad}
19
20\minitoc
21
22\newpage
23
24\section{Choice of \protect\ngn{namtra\_ldf} namelist parameters}
25%-----------------------------------------nam_traldf------------------------------------------------------
26
27\nlst{namtra_ldf}
28%---------------------------------------------------------------------------------------------------------
29
30Two scheme are available to perform the iso-neutral diffusion.
31If the namelist logical \np{ln\_traldf\_triad} is set true,
32\NEMO updates both active and passive tracers using the Griffies triad representation of iso-neutral diffusion and
33the eddy-induced advective skew (GM) fluxes.
34If the namelist logical \np{ln\_traldf\_iso} is set true,
35the filtered version of Cox's original scheme (the Standard scheme) is employed (\autoref{sec:LDF_slp}).
36In the present implementation of the Griffies scheme,
37the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false.
38
39Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}.
40Note that when GM fluxes are used, the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS,
41even though the eddy advection is accomplished by means of the skew fluxes.
42
43The options specific to the Griffies scheme include:
44\begin{description}
45\item[\np{ln\_triad\_iso}]
46  See \autoref{sec:taper}.
47  If this is set false (the default),
48  then `iso-neutral' mixing is accomplished within the surface mixed-layer along slopes linearly decreasing with
49  depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec:lintaper}).
50  This is the same treatment as used in the default implementation
51  \autoref{subsec:LDF_slp_iso}; \autoref{fig:eiv_slp}.
52  Where \np{ln\_triad\_iso} is set true,
53  the vertical skew flux is further reduced to ensure no vertical buoyancy flux,
54  giving an almost pure horizontal diffusive tracer flux within the mixed layer.
55  This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec:Gerdes-taper}
56\item[\np{ln\_botmix\_triad}]
57  See \autoref{sec:iso_bdry}.
58  If this is set false (the default) then the lateral diffusive fluxes
59  associated with triads partly masked by topography are neglected.
60  If it is set true, however, then these lateral diffusive fluxes are applied,
61  giving smoother bottom tracer fields at the cost of introducing diapycnal mixing.
62\item[\np{rn\_sw\_triad}]
63  blah blah to be added....
64\end{description}
65The options shared with the Standard scheme include:
66\begin{description}
67\item[\np{ln\_traldf\_msc}]   blah blah to be added
68\item[\np{rn\_slpmax}]  blah blah to be added
69\end{description}
70
71\section{Triad formulation of iso-neutral diffusion}
72\label{sec:iso}
73
74We have implemented into \NEMO a scheme inspired by \citet{griffies.gnanadesikan.ea_JPO98},
75but formulated within the \NEMO framework, using scale factors rather than grid-sizes.
76
77\subsection{Iso-neutral diffusion operator}
78
79The iso-neutral second order tracer diffusive operator for small angles between
80iso-neutral surfaces and geopotentials is given by \autoref{eq:iso_tensor_1}:
81\begin{subequations}
82  \label{eq:iso_tensor_1}
83  \begin{equation}
84    D^{lT}=-\nabla \cdot\vect{f}^{lT}\equiv
85    -\frac{1}{e_1e_2e_3}\left[\pd{i}\left (f_1^{lT}e_2e_3\right) +
86      \pd{j}\left (f_2^{lT}e_2e_3\right) + \pd{k}\left (f_3^{lT}e_1e_2\right)\right],
87  \end{equation}
88  where the diffusive flux per unit area of physical space
89  \begin{equation}
90    \vect{f}^{lT}=-{A^{lT}}\Re\cdot\nabla T,
91  \end{equation}
92  \begin{equation}
93    \label{eq:iso_tensor_2}
94    \mbox{with}\quad \;\;\Re =
95    \begin{pmatrix}
96      1   &  0   & -r_1           \rule[-.9 em]{0pt}{1.79 em} \\
97      0   &  1   & -r_2           \rule[-.9 em]{0pt}{1.79 em} \\
98      -r_1 & -r_2 &  r_1 ^2+r_2 ^2 \rule[-.9 em]{0pt}{1.79 em}
99    \end{pmatrix}
100    \quad \text{and} \quad\nabla T=
101    \begin{pmatrix}
102      \frac{1}{e_1} \pd[T]{i} \rule[-.9 em]{0pt}{1.79 em} \\
103      \frac{1}{e_2} \pd[T]{j} \rule[-.9 em]{0pt}{1.79 em} \\
104      \frac{1}{e_3} \pd[T]{k} \rule[-.9 em]{0pt}{1.79 em}
105    \end{pmatrix}
106    .
107  \end{equation}
108\end{subequations}
109% \left( {{\begin{array}{*{20}c}
110%  1 \hfill & 0 \hfill & {-r_1 } \hfill \\
111%  0 \hfill & 1 \hfill & {-r_2 } \hfill \\
112%  {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
113% \end{array} }} \right)
114Here \autoref{eq:PE_iso_slopes} 
115\begin{align*}
116  r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i}
117        \right)
118        \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\
119      &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} +
120        \beta\frac{\partial S }{\partial i} \right) \left(
121        -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S
122        }{\partial k} \right)^{-1}
123\end{align*}
124is the $i$-component of the slope of the iso-neutral surface relative to the computational surface,
125and $r_2$ is the $j$-component.
126
127We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write
128\[
129  % \label{eq:Fijk}
130  \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right).
131\]
132Additionally, we will sometimes write the contributions towards the fluxes $\vect{f}$ and
133$\vect{F}_{\mathrm{iso}}$ from the component $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$,
134with $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc.
135
136The off-diagonal terms of the small angle diffusion tensor
137\autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along
138the $i$- and $j$-directions resulting from the vertical tracer gradient:
139\begin{align}
140  \label{eq:i13c}
141  f_{13}=&+{A^{lT}} r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+{A^{lT}} r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\
142  \intertext{and in the k-direction resulting from the lateral tracer gradients}
143  \label{eq:i31c}
144  f_{31}+f_{32}=& {A^{lT}} r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+{A^{lT}} r_2\frac{1}{e_1}\frac{\partial T}{\partial i}
145\end{align}
146
147The vertical diffusive flux associated with the $_{33}$ component of the small angle diffusion tensor is
148\begin{equation}
149  \label{eq:i33c}
150  f_{33}=-{A^{lT}}(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}.
151\end{equation}
152
153Since there are no cross terms involving $r_1$ and $r_2$ in the above,
154we can consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$ planes,
155just adding together the vertical components from each plane.
156The following description will describe the fluxes on the $i$-$k$ plane.
157
158There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq:i13c},
159as although it must be evaluated at $u$-points,
160it involves vertical gradients (both for the tracer and the slope $r_1$), defined at $w$-points.
161Similarly, the vertical skew flux, \autoref{eq:i31c},
162is evaluated at $w$-points but involves horizontal gradients defined at $u$-points.
163
164\subsection{Standard discretization}
165
166The straightforward approach to discretize the lateral skew flux
167\autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA,
168\autoref{eq:tra_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from
169the average of the four surrounding vertical tracer gradients, and multiply this by a mean slope at the $u$-point,
170calculated from the averaged surrounding vertical density gradients.
171The total area-integrated skew-flux (flux per unit area in $ijk$ space) from tracer cell $i,k$ to $i+1,k$,
172noting that the $e_{{3}_{i+1/2}^k}$ in the area $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with
173the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:tra_ldf_iso}
174\[
175  \left(F_u^{13} \right)_{i+\frac{1}{2}}^k = {A}_{i+\frac{1}{2}}^k
176  {e_{2}}_{i+1/2}^k \overline{\overline
177    r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k},
178\]
179where
180\[
181  \overline{\overline
182    r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k}
183  \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}},
184\]
185and here and in the following we drop the $^{lT}$ superscript from ${A^{lT}}$ for simplicity.
186Unfortunately the resulting combination $\overline{\overline{\delta_k\bullet}}^{\,i,k}$ of a $k$ average and
187a $k$ difference of the tracer reduces to $\bullet_{k+1}-\bullet_{k-1}$,
188so two-grid-point oscillations are invisible to this discretization of the iso-neutral operator.
189These \emph{computational modes} will not be damped by this operator, and may even possibly be amplified by it.
190Consequently, applying this operator to a tracer does not guarantee the decrease of its global-average variance.
191To correct this, we introduced a smoothing of the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}).
192This technique works for $T$ and $S$ in so far as they are active tracers
193(\ie they enter the computation of density), but it does not work for a passive tracer.
194
195\subsection{Expression of the skew-flux in terms of triad slopes}
196
197\citep{griffies.gnanadesikan.ea_JPO98} introduce a different discretization of the off-diagonal terms that
198nicely solves the problem.
199% Instead of multiplying the mean slope calculated at the $u$-point by
200% the mean vertical gradient at the $u$-point,
201% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
202\begin{figure}[tb]
203  \begin{center}
204    \includegraphics[width=\textwidth]{Fig_GRIFF_triad_fluxes}
205    \caption{
206      \protect\label{fig:ISO_triad}
207      (a) Arrangement of triads $S_i$ and tracer gradients to
208      give lateral tracer flux from box $i,k$ to $i+1,k$
209      (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from
210      box $i,k$ to $i,k+1$.
211    }
212  \end{center}
213\end{figure}
214% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
215They get the skew flux from the products of the vertical gradients at each $w$-point surrounding the $u$-point with
216the corresponding `triad' slope calculated from the lateral density gradient across the $u$-point divided by
217the vertical density gradient at the same $w$-point as the tracer gradient.
218See \autoref{fig:ISO_triad}a, where the thick lines denote the tracer gradients,
219and the thin lines the corresponding triads, with slopes $s_1, \dotsc s_4$.
220The total area-integrated skew-flux from tracer cell $i,k$ to $i+1,k$
221\begin{multline}
222  \label{eq:i13}
223  \left( F_u^{13}  \right)_{i+\frac{1}{2}}^k = {A}_{i+1}^k a_1 s_1
224  \delta_{k+\frac{1}{2}} \left[ T^{i+1}
225  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  + {A} _i^k a_2 s_2 \delta
226  _{k+\frac{1}{2}} \left[ T^i
227  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\
228  +{A} _{i+1}^k a_3 s_3 \delta_{k-\frac{1}{2}} \left[ T^{i+1}
229  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  +{A} _i^k a_4 s_4 \delta
230  _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}},
231\end{multline}
232where the contributions of the triad fluxes are weighted by areas $a_1, \dotsc a_4$,
233and ${A}$ is now defined at the tracer points rather than the $u$-points.
234This discretization gives a much closer stencil, and disallows the two-point computational modes.
235
236The vertical skew flux \autoref{eq:i31c} from tracer cell $i,k$ to $i,k+1$ at
237the $w$-point $i,k+\frac{1}{2}$ is constructed similarly (\autoref{fig:ISO_triad}b) by
238multiplying lateral tracer gradients from each of the four surrounding $u$-points by the appropriate triad slope:
239\begin{multline}
240  \label{eq:i31}
241  \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} =  {A}_i^{k+1} a_{1}'
242  s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1}
243  +{A}_i^{k+1} a_{2}' s_{2}' \delta_{i+\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i+\frac{1}{2}}^{k+1} \\
244  + {A}_i^k a_{3}' s_{3}' \delta_{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k
245  +{A}_i^k a_{4}' s_{4}' \delta_{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k.
246\end{multline}
247
248We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$
249(appearing in both the vertical and lateral gradient),
250and the $u$- and $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the triad as follows
251(see also \autoref{fig:ISO_triad}):
252\begin{equation}
253  \label{eq:R}
254  _i^k \mathbb{R}_{i_p}^{k_p}
255  =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}}
256  \
257  \frac
258  { \alpha_i^\ \delta_{i+i_p}[T^k] - \beta_i^k \ \delta_{i+i_p}[S^k] }
259  { \alpha_i^\ \delta_{k+k_p}[T^i] - \beta_i^k \ \delta_{k+k_p}[S^i] }.
260\end{equation}
261In calculating the slopes of the local neutral surfaces,
262the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad,
263while the metrics are calculated at the $u$- and $w$-points on the arms.
264
265% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
266\begin{figure}[tb]
267  \begin{center}
268    \includegraphics[width=\textwidth]{Fig_GRIFF_qcells}
269    \caption{
270      \protect\label{fig:qcells}
271      Triad notation for quarter cells. $T$-cells are inside boxes,
272      while the  $i+\fractext{1}{2},k$ $u$-cell is shaded in green and
273      the $i,k+\fractext{1}{2}$ $w$-cell is shaded in pink.
274    }
275  \end{center}
276\end{figure}
277% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
278
279Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:qcells}) with the quarter cell that is
280the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell.
281Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:i13} and \autoref{eq:i31} in this notation,
282we have \eg \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$.
283Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to
284calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$,
285and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$.
286Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used to
287calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms of a unique triad,
288and we notate these areas, similarly to the triad slopes,
289as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$,
290where \eg in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$,
291and in \autoref{eq:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$.
292
293\subsection{Full triad fluxes}
294
295A key property of iso-neutral diffusion is that it should not affect the (locally referenced) density.
296In particular there should be no lateral or vertical density flux.
297The lateral density flux disappears so long as the area-integrated lateral diffusive flux from
298tracer cell $i,k$ to $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the form
299\begin{equation}
300  \label{eq:i11}
301  \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} =
302  - \left( {A}_i^{k+1} a_{1} + {A}_i^{k+1} a_{2} + {A}_i^k
303    a_{3} + {A}_i^k a_{4} \right)
304  \frac{\delta_{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}},
305\end{equation}
306where the areas $a_i$ are as in \autoref{eq:i13}.
307In this case, separating the total lateral flux, the sum of \autoref{eq:i13} and \autoref{eq:i11},
308into triad components, a lateral tracer flux
309\begin{equation}
310  \label{eq:latflux-triad}
311  _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - {A}_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p}
312  \left(
313    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
314    -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \
315    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
316  \right)
317\end{equation}
318can be identified with each triad.
319Then, because the same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and ${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both
320the density gradients in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients,
321the lateral density flux associated with each triad separately disappears.
322\begin{equation}
323  \label{eq:latflux-rho}
324  {\mathbb{F}_u}_{i_p}^{k_p} (\rho)=-\alpha _i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (S)=0
325\end{equation}
326Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} + \left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from
327tracer cell $i,k$ to $i+1,k$ must also vanish since it is a sum of four such triad fluxes.
328
329The squared slope $r_1^2$ in the expression \autoref{eq:i33c} for the $_{33}$ component is also expressed in
330terms of area-weighted squared triad slopes,
331so the area-integrated vertical flux from tracer cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is
332\begin{equation}
333  \label{eq:i33}
334  \left( F_w^{33} \right) _i^{k+\frac{1}{2}} =
335  - \left( {A}_i^{k+1} a_{1}' s_{1}'^2
336    + {A}_i^{k+1} a_{2}' s_{2}'^2
337    + {A}_i^k a_{3}' s_{3}'^2
338    + {A}_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right],
339\end{equation}
340where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq:i31}.
341Then, separating the total vertical flux, the sum of \autoref{eq:i31} and \autoref{eq:i33},
342into triad components, a vertical flux
343\begin{align}
344  \label{eq:vertflux-triad}
345  _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
346  &= {A}_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p}
347    \left(
348    {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
349    -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \
350    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
351    \right) \\
352  &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right)
353    {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2}
354\end{align}
355may be associated with each triad.
356Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ associated with a triad then
357separately disappears (because the lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$ disappears).
358Consequently the total vertical density flux
359$\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} + \left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from
360tracer cell $i,k$ to $i,k+1$ must also vanish since it is a sum of four such triad fluxes.
361
362We can explicitly identify (\autoref{fig:qcells}) the triads associated with the $s_i$, $a_i$,
363and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq:i31},
364\autoref{eq:i13}, \autoref{eq:i11} \autoref{eq:i33} and \autoref{fig:ISO_triad} to write out
365the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces:
366%(\autoref{fig:ISO_triad}):
367\begin{flalign}
368  \label{eq:iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv
369  \sum_{\substack{i_p,\,k_p}}
370  \begin{pmatrix}
371    {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\
372    {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\
373  \end{pmatrix}.
374\end{flalign}
375
376\subsection{Ensuring the scheme does not increase tracer variance}
377\label{subsec:variance}
378
379We now require that this operator should not increase the globally-integrated tracer variance.
380%This changes according to
381% \begin{align*}
382% &\int_D  D_l^T \; T \;dv \equiv  \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\}    \\
383% &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
384%     \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right]
385%       + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]  \ T \right\}    \\
386% &\equiv  - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
387%                 {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T]
388%              + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}}  \ \delta_{k+1/2} [T]   \right\}      \\
389% \end{align*}
390Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across
391the $u$-point $i+i_p,k$ and a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the $w$-point $i,k+k_p$.
392The lateral flux drives a net rate of change of variance,
393summed over the two $T$-points $i+i_p-\fractext{1}{2},k$ and $i+i_p+\fractext{1}{2},k$, of
394\begin{multline}
395  {b_T}_{i+i_p-1/2}^k\left(\frac{\partial T}{\partial t}T\right)_{i+i_p-1/2}^k+
396  \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial
397      t}T\right)_{i+i_p+1/2}^k \\
398  \begin{aligned}
399    &= -T_{i+i_p-1/2}^k{\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \quad + \quad  T_{i+i_p+1/2}^k
400    {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\
401    &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:dvar_iso_i}
402  \end{aligned}
403\end{multline}
404while the vertical flux similarly drives a net rate of change of variance summed over
405the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of
406\begin{equation}
407  \label{eq:dvar_iso_k}
408  _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
409\end{equation}
410The total variance tendency driven by the triad is the sum of these two.
411Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with
412\autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, it is
413\begin{multline*}
414  -{A}_i^k\left \{
415    { } _i^k{\mathbb{A}_u}_{i_p}^{k_p}
416    \left(
417      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
418      - {_i^k\mathbb{R}_{i_p}^{k_p}} \
419      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\
420  - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p}
421    \left(
422      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
423      -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
424      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
425    \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i]
426  \right \}.
427\end{multline*}
428The key point is then that if we require $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$ to
429be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by
430\begin{equation}
431  \label{eq:V-A}
432  _i^k\mathbb{V}_{i_p}^{k_p}
433  ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k}
434  ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p},
435\end{equation}
436the variance tendency reduces to the perfect square
437\begin{equation}
438  \label{eq:perfect-square}
439  -{A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
440  \left(
441    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
442    -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
443    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
444  \right)^2\leq 0.
445\end{equation}
446Thus, the constraint \autoref{eq:V-A} ensures that the fluxes
447(\autoref{eq:latflux-triad}, \autoref{eq:vertflux-triad}) associated with
448a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase the net variance.
449Since the total fluxes are sums of such fluxes from the various triads, this constraint, applied to all triads,
450is sufficient to ensure that the globally integrated variance does not increase.
451
452The expression \autoref{eq:V-A} can be interpreted as a discretization of the global integral
453\begin{equation}
454  \label{eq:cts-var}
455  \frac{\partial}{\partial t}\int\!\fractext{1}{2} T^2\, dV =
456  \int\!\mathbf{F}\cdot\nabla T\, dV,
457\end{equation}
458where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the lateral and vertical fluxes/unit area
459\[
460  \mathbf{F}=\left(
461    \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p},
462    \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p}
463  \right)
464\]
465and the gradient
466\[
467  \nabla T = \left(
468    \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k},
469    \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p}
470  \right)
471\]
472
473\subsection{Triad volumes in Griffes's scheme and in \NEMO}
474
475To complete the discretization we now need only specify the triad volumes $_i^k\mathbb{V}_{i_p}^{k_p}$.
476\citet{griffies.gnanadesikan.ea_JPO98} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells,
477defined in terms of the distances between $T$, $u$,$f$ and $w$-points.
478This is the natural discretization of \autoref{eq:cts-var}.
479The \NEMO model, however, operates with scale factors instead of grid sizes,
480and scale factors for the quarter cells are not defined.
481Instead, therefore we simply choose
482\begin{equation}
483  \label{eq:V-NEMO}
484  _i^k\mathbb{V}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k,
485\end{equation}
486as a quarter of the volume of the $u$-cell inside which the triad quarter-cell lies.
487This has the nice property that when the slopes $\mathbb{R}$ vanish,
488the lateral flux from tracer cell $i,k$ to $i+1,k$ reduces to the classical form
489\begin{equation}
490  \label{eq:lat-normal}
491  -\overline{A}_{\,i+1/2}^k\;
492  \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}}
493  \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}}
494  = -\overline{A}_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}\;\delta_{i+ 1/2}[T^k]}{{e_{1u}}_{\,i + 1/2}^{\,k}}.
495\end{equation}
496In fact if the diffusive coefficient is defined at $u$-points,
497so that we employ ${A}_{i+i_p}^k$ instead of  ${A}_i^k$ in the definitions of the triad fluxes
498\autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad},
499we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above.
500
501\subsection{Summary of the scheme}
502
503The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that
504cross the $u$- and $w$-faces \autoref{eq:iso_flux}:
505\begin{subequations}
506  % \label{eq:alltriadflux}
507  \begin{flalign*}
508    % \label{eq:vect_isoflux}
509    \vect{F}_{\mathrm{iso}}(T) &\equiv
510    \sum_{\substack{i_p,\,k_p}}
511    \begin{pmatrix}
512      {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\
513      {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T)
514    \end{pmatrix},
515  \end{flalign*}
516  where \autoref{eq:latflux-triad}:
517  \begin{align}
518    \label{eq:triadfluxu}
519    _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - {A}_i^k{
520                                          \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}}
521                                          \left(
522                                          \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
523                                          -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \
524                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
525                                          \right),\\
526    \intertext{and}
527    _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
528                                        &= {A}_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}}
529                                          \left(
530                                          {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
531                                          -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \
532                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
533                                          \right),\label{eq:triadfluxw}
534  \end{align}
535  with \autoref{eq:V-NEMO}
536  \[
537    % \label{eq:V-NEMO2}
538    _i^k{\mathbb{V}}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k.
539  \]
540\end{subequations}
541
542The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at
543each tracer point:
544\[
545  % \label{eq:iso_operator}
546  D_l^T = \frac{1}{b_T}
547  \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k
548        {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[
549      {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\}
550\]
551where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.
552The diffusion scheme satisfies the following six properties:
553\begin{description}
554\item[$\bullet$ horizontal diffusion]
555  The discretization of the diffusion operator recovers the traditional five-point Laplacian
556  \autoref{eq:lat-normal} in the limit of flat iso-neutral direction:
557  \[
558    % \label{eq:iso_property0}
559    D_l^T = \frac{1}{b_T} \
560    \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \;
561      \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] \qquad
562    \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0
563  \]
564
565\item[$\bullet$ implicit treatment in the vertical]
566  Only tracer values associated with a single water column appear in the expression \autoref{eq:i33} for
567  the $_{33}$ fluxes, vertical fluxes driven by vertical gradients.
568  This is of paramount importance since it means that a time-implicit algorithm can be used to
569  solve the vertical diffusion equation.
570  This is necessary since the vertical eddy diffusivity associated with this term,
571  \[
572    \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{
573      {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: {A}_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
574    \right\}  =
575    \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{
576      {b_u}_{i+i_p}^k\: {A}_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
577    \right\},
578  \]
579  (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large.
580
581\item[$\bullet$ pure iso-neutral operator]
582  The iso-neutral flux of locally referenced potential density is zero.
583  See \autoref{eq:latflux-rho} and \autoref{eq:vertflux-triad2}.
584
585\item[$\bullet$ conservation of tracer]
586  The iso-neutral diffusion conserves tracer content, \ie
587  \[
588    % \label{eq:iso_property1}
589    \sum_{i,j,k} \left\{ D_l^T \      b_T \right\} = 0
590  \]
591  This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form.
592
593\item[$\bullet$ no increase of tracer variance]
594  The iso-neutral diffusion does not increase the tracer variance, \ie
595  \[
596    % \label{eq:iso_property2}
597    \sum_{i,j,k} \left\{ T \ D_l^T      \ b_T \right\} \leq 0
598  \]
599  The property is demonstrated in \autoref{subsec:variance} above.
600  It is a key property for a diffusion term.
601  It means that it is also a dissipation term,
602  \ie it dissipates the square of the quantity on which it is applied.
603  It therefore ensures that, when the diffusivity coefficient is large enough,
604  the field on which it is applied becomes free of grid-point noise.
605
606\item[$\bullet$ self-adjoint operator]
607  The iso-neutral diffusion operator is self-adjoint, \ie
608  \begin{equation}
609    \label{eq:iso_property3}
610    \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
611  \end{equation}
612  In other word, there is no need to develop a specific routine from the adjoint of this operator.
613  We just have to apply the same routine.
614  This property can be demonstrated similarly to the proof of the `no increase of tracer variance' property.
615  The contribution by a single triad towards the left hand side of \autoref{eq:iso_property3},
616  can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i} and \autoref{eq:dvar_iso_k}.
617  This results in a term similar to \autoref{eq:perfect-square},
618  \[
619    % \label{eq:TScovar}
620    - {A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
621    \left(
622      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
623      -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
624      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
625    \right)
626    \left(
627      \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
628      -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
629      \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
630    \right).
631  \]
632This is symmetrical in $T $ and $S$, so exactly the same term arises from
633the discretization of this triad's contribution towards the RHS of \autoref{eq:iso_property3}.
634\end{description}
635
636\subsection{Treatment of the triads at the boundaries}
637\label{sec:iso_bdry}
638
639The triad slope can only be defined where both the grid boxes centred at the end of the arms exist.
640Triads that would poke up through the upper ocean surface into the atmosphere,
641or down into the ocean floor, must be masked out.
642See \autoref{fig:bdry_triads}.
643Surface layer triads \triad{i}{1}{R}{1/2}{-1/2} (magenta) and \triad{i+1}{1}{R}{-1/2}{-1/2} (blue) that
644require density to be specified above the ocean surface are masked (\autoref{fig:bdry_triads}a):
645this ensures that lateral tracer gradients produce no flux through the ocean surface.
646However, to prevent surface noise, it is customary to retain the $_{11}$ contributions towards
647the lateral triad fluxes \triad[u]{i}{1}{F}{1/2}{-1/2} and \triad[u]{i+1}{1}{F}{-1/2}{-1/2};
648this drives diapycnal tracer fluxes.
649Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:bdry_triads}b).
650Note that both near bottom triad slopes \triad{i}{k}{R}{1/2}{1/2} and \triad{i+1}{k}{R}{-1/2}{1/2} are masked when
651either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie the $i,k+1$ $u$-point is masked.
652The associated lateral fluxes (grey-black dashed line) are masked if \np{ln\_botmix\_triad}\forcode{ = .false.},
653but left unmasked, giving bottom mixing, if \np{ln\_botmix\_triad}\forcode{ = .true.}.
654
655The default option \np{ln\_botmix\_triad}\forcode{ = .false.} is suitable when the bbl mixing option is enabled
656(\key{trabbl}, with \np{nn\_bbl\_ldf}\forcode{ = 1}), or for simple idealized problems.
657For setups with topography without bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary.
658% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
659\begin{figure}[h]
660  \begin{center}
661    \includegraphics[width=\textwidth]{Fig_GRIFF_bdry_triads}
662    \caption{
663      \protect\label{fig:bdry_triads}
664      (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots),
665      and $i+1/2,1$ $u$-point (blue square).
666      Triad slopes \triad{i}{1}{R}{1/2}{-1/2} (magenta) and \triad{i+1}{1}{R}{-1/2}{-1/2} (blue) poking through
667      the ocean surface are masked (faded in figure).
668      However, the lateral $_{11}$ contributions towards \triad[u]{i}{1}{F}{1/2}{-1/2} and
669      \triad[u]{i+1}{1}{F}{-1/2}{-1/2} (yellow line) are still applied,
670      giving diapycnal diffusive fluxes.
671      \newline
672      (b) Both near bottom triad slopes \triad{i}{k}{R}{1/2}{1/2} and
673      \triad{i+1}{k}{R}{-1/2}{1/2} are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked,
674      \ie the $i,k+1$ $u$-point is masked.
675      The associated lateral fluxes (grey-black dashed line) are masked if
676      \protect\np{botmix\_triad}\forcode{ = .false.}, but left unmasked,
677      giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.}
678    }
679  \end{center}
680\end{figure}
681% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
682
683\subsection{ Limiting of the slopes within the interior}
684\label{sec:limit}
685
686As discussed in \autoref{subsec:LDF_slp_iso},
687iso-neutral slopes relative to geopotentials must be bounded everywhere,
688both for consistency with the small-slope approximation and for numerical stability \citep{cox_OM87, griffies_bk04}.
689The bound chosen in \NEMO is applied to each component of the slope separately and
690has a value of $1/100$ in the ocean interior.
691%, ramping linearly down above 70~m depth to zero at the surface
692It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to geopotentials
693(here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials)
694\autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require
695\[
696  |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01.
697\]
698and then recalculate the slopes $r_i$ relative to coordinates.
699Each individual triad slope
700\begin{equation}
701  \label{eq:Rtilde}
702  _i^k\tilde{\mathbb{R}}_{i_p}^{k_p} = {}_i^k\mathbb{R}_{i_p}^{k_p}  + \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
703\end{equation}
704is limited like this and then the corresponding $_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and
705combined to form the fluxes.
706Note that where the slopes have been limited, there is now a non-zero iso-neutral density flux that
707drives dianeutral mixing.
708In particular this iso-neutral density flux is always downwards,
709and so acts to reduce gravitational potential energy.
710
711\subsection{Tapering within the surface mixed layer}
712\label{sec:taper}
713
714Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer.
715When the Griffies triads are used, we offer two options for this.
716
717\subsubsection{Linear slope tapering within the surface mixed layer}
718\label{sec:lintaper}
719
720This is the option activated by the default choice \np{ln\_triad\_iso}\forcode{ = .false.}.
721Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below
722the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:eiv_slp}, to values
723\begin{equation}
724  \label{eq:rmtilde}
725  \rMLt = -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h,
726\end{equation}
727and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to
728\[
729  % \label{eq:rm}
730  \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h.
731\]
732Thus the diffusion operator within the mixed layer is given by:
733\[
734  % \label{eq:iso_tensor_ML}
735  D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
736  \mbox{with}\quad \;\;\Re =\left( {{
737        \begin{array}{*{20}c}
738          1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\
739          0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\
740          {-\rML[1]}\hfill &   {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill
741        \end{array}
742      }} \right)
743\]
744
745This slope tapering gives a natural connection between tracer in the mixed-layer and
746in isopycnal layers immediately below, in the thermocline.
747It is consistent with the way the $\tilde{r}_i$ are tapered within the mixed layer
748(see \autoref{sec:taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer.
749However, it gives a downwards density flux and so acts so as to reduce potential energy in the same way as
750does the slope limiting discussed above in \autoref{sec:limit}.
751 
752As in \autoref{sec:limit} above, the tapering \autoref{eq:rmtilde} is applied separately to
753each triad $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted.
754For clarity, we assume $z$-coordinates in the following;
755the conversion from $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as
756described above by \autoref{eq:Rtilde}.
757\begin{enumerate}
758\item
759  Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in
760  the slope definition.
761  At each $i,j$ (simplified to $i$ in \autoref{fig:MLB_triad}),
762  we define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer,
763  $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that
764  the potential density ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$,
765  where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m.
766  See the left side of \autoref{fig:MLB_triad}.
767  We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems \eg with thin daytime mixed-layers.
768  Currently we use the same $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to
769  output the diagnosed mixed-layer depth $h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$,
770  the depth of the $w$-point above the $i,k_{\mathrm{ML}}$ tracer point.
771\item
772  We define `basal' triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as
773  the slopes of those triads whose vertical `arms' go down from the $i,k_{\mathrm{ML}}$ tracer point to
774  the $i,k_{\mathrm{ML}}-1$ tracer point below.
775  This is to ensure that the vertical density gradients associated with
776  these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline.
777  The four basal triads defined in the bottom part of \autoref{fig:MLB_triad} are then
778  \begin{align*}
779    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &=
780                                                       {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p},
781                                                       % \label{eq:Rbase}
782    \\
783    \intertext{with \eg the green triad}
784    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&=
785                                                     {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}.
786  \end{align*}
787The vertical flux associated with each of these triads passes through
788the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth
789\[
790  % \label{eq:zbase}
791  {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2}
792\]
793one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in
794\autoref{eq:rmtilde}.
795\item
796  Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within
797  the mixed layer, by multiplying the appropriate ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by
798  the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$.
799  For instance the green triad centred on $i,k$
800  \begin{align*}
801    {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &=
802                                                        \frac{{z_w}_{k-1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2} \\
803    \intertext{and more generally}
804    {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &=
805                                                       \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}.
806                                                       % \label{eq:RML}
807  \end{align*}
808\end{enumerate}
809
810% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
811\begin{figure}[h]
812%  \fcapside {
813  \caption{
814    \protect\label{fig:MLB_triad}
815    Definition of mixed-layer depth and calculation of linearly tapered triads.
816    The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top.
817    Tracer points are denoted by bullets, and black lines the edges of the tracer cells;
818    $k$ increases upwards.
819    \newline
820    \hspace{5 em}
821    We define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer,
822    $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that
823    ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$,
824    where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m.
825    We calculate the triad slopes within the mixed layer by linearly tapering them from zero
826    (at the surface) to the `basal' slopes,
827    the slopes of the four triads passing through the $w$-point $i,k_{\mathrm{ML}}-1/2$ (blue square),
828    ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$.
829    Triads with different $i_p,k_p$, denoted by different colours,
830    (\eg the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}
831  % }
832  \includegraphics[width=\textwidth]{Fig_GRIFF_MLB_triads}
833\end{figure}
834% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
835
836\subsubsection{Additional truncation of skew iso-neutral flux components}
837\label{subsec:Gerdes-taper}
838
839The alternative option is activated by setting \np{ln\_triad\_iso} = true.
840This retains the same tapered slope $\rML$  described above for the calculation of the $_{33}$ term of
841the iso-neutral diffusion tensor (the vertical tracer flux driven by vertical tracer gradients),
842but replaces the $\rML$ in the skew term by
843\begin{equation}
844  \label{eq:rm*}
845  \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i,
846\end{equation}
847giving a ML diffusive operator
848\[
849  % \label{eq:iso_tensor_ML2}
850  D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
851  \mbox{with}\quad \;\;\Re =\left( {{
852        \begin{array}{*{20}c}
853          1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\
854          0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\
855          {-\rML[1]^*}\hfill &   {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\
856        \end{array}
857      }} \right).
858\]
859This operator
860\footnote{
861  To ensure good behaviour where horizontal density gradients are weak,
862  we in fact follow \citet{gerdes.koberle.ea_CD91} and
863  set $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.
864}
865then has the property it gives no vertical density flux, and so does not change the potential energy.
866This approach is similar to multiplying the iso-neutral diffusion coefficient by
867$\tilde{r}_{\mathrm{max}}^{-2}\tilde{r}_i^{-2}$ for steep slopes,
868as suggested by \citet{gerdes.koberle.ea_CD91} (see also \citet{griffies_bk04}).
869Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$
870
871In practice, this approach gives weak vertical tracer fluxes through the mixed-layer,
872as well as vanishing density fluxes.
873While it is theoretically advantageous that it does not change the potential energy,
874it may give a discontinuity between the fluxes within the mixed-layer (purely horizontal) and
875just below (along iso-neutral surfaces).
876% This may give strange looking results,
877% particularly where the mixed-layer depth varies strongly laterally.
878% ================================================================
879% Skew flux formulation for Eddy Induced Velocity :
880% ================================================================
881\section{Eddy induced advection formulated as a skew flux}
882\label{sec:skew-flux}
883
884\subsection{Continuous skew flux formulation}
885\label{sec:continuous-skew-flux}
886
887When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added.
888The associated velocity is the so called eddy induced velocity,
889the formulation of which depends on the slopes of iso-neutral surfaces.
890Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces,
891\ie \autoref{eq:ldfslp_geo} is used in $z$-coordinate,
892and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates.
893
894The eddy induced velocity is given by:
895\begin{subequations}
896  % \label{eq:eiv}
897  \begin{equation}
898    \label{eq:eiv_v}
899    \begin{split}
900      u^* & = - \frac{1}{e_{3}}\;          \partial_i\psi_1,  \\
901      v^* & = - \frac{1}{e_{3}}\;          \partial_j\psi_2,    \\
902      w^* & =    \frac{1}{e_{1}e_{2}}\; \left\{ \partial_\left( e_{2} \, \psi_1\right)
903        + \partial_\left( e_{1} \, \psi_2\right) \right\},
904    \end{split}
905  \end{equation}
906  where the streamfunctions $\psi_i$ are given by
907  \begin{equation}
908    \label{eq:eiv_psi}
909    \begin{split}
910      \psi_1 & = A_{e} \; \tilde{r}_1,   \\
911      \psi_2 & = A_{e} \; \tilde{r}_2,
912    \end{split}
913  \end{equation}
914\end{subequations}
915with $A_{e}$ the eddy induced velocity coefficient,
916and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces.
917
918The traditional way to implement this additional advection is to add it to the Eulerian velocity prior to
919computing the tracer advection.
920This is implemented if \key{traldf\_eiv} is set in the default implementation,
921where \np{ln\_traldf\_triad} is set false.
922This allows us to take advantage of all the advection schemes offered for the tracers
923(see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme.
924This is particularly useful for passive tracers where
925\emph{positivity} of the advection scheme is of paramount importance.
926
927However, when \np{ln\_traldf\_triad} is set true,
928\NEMO instead implements eddy induced advection according to the so-called skew form \citep{griffies_JPO98}.
929It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity.
930For example in the (\textbf{i},\textbf{k}) plane,
931the tracer advective fluxes per unit area in $ijk$ space can be transformed as follows:
932\begin{flalign*}
933  \begin{split}
934    \textbf{F}_{\mathrm{eiv}}^T =
935    \begin{pmatrix}
936      {e_{2}\,e_{3}\;  u^*} \\
937      {e_{1}\,e_{2}\; w^*}
938    \end{pmatrix}   \;   T
939    &=
940    \begin{pmatrix}
941      { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\
942      {+ \partial_\left( e_{2} \, \psi_1 \right) \; T \;}
943    \end{pmatrix}          \\
944    &=
945    \begin{pmatrix}
946      { - \partial_k \left( e_{2} \, \psi_\; T \right) \;} \\
947      {+ \partial_\left( e_{2} \,\psi_1 \; T \right) \;}
948    \end{pmatrix}
949    +
950    \begin{pmatrix}
951      {+ e_{2} \, \psi_\; \partial_k T} \\
952      { - e_{2} \, \psi_\; \partial_i  T}
953    \end{pmatrix}
954  \end{split}
955\end{flalign*}
956and since the eddy induced velocity field is non-divergent,
957we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space:
958\begin{equation}
959  \label{eq:eiv_skew_ijk}
960  \textbf{F}_\mathrm{eiv}^T =
961  \begin{pmatrix}
962    {+ e_{2} \, \psi_\; \partial_k T}   \\
963    { - e_{2} \, \psi_\; \partial_i  T}
964  \end{pmatrix}
965\end{equation}
966The total fluxes per unit physical area are then
967\begin{equation}
968  \label{eq:eiv_skew_physical}
969  \begin{split}
970    f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T   \\
971    f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T   \\
972    f^*_3 & =  -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T + e_{1} \psi_2 \partial_j T \right\}.
973\end{split}
974\end{equation}
975Note that \autoref{eq:eiv_skew_physical} takes the same form whatever the vertical coordinate,
976though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:eiv_psi} are relative to
977geopotentials.
978The tendency associated with eddy induced velocity is then simply the convergence of the fluxes
979(\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so
980\[
981  % \label{eq:skew_eiv_conv}
982  \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 }      \left[
983    \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right)
984    + \frac{\partial}{\partial j} \left( e_1  \;
985      \psi_2 \partial_k T\right)
986    -  \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T
987      + e_{1} \psi_2 \partial_j T \right)  \right]
988\]
989It naturally conserves the tracer content, as it is expressed in flux form.
990Since it has the same divergence as the advective form it also preserves the tracer variance.
991
992\subsection{Discrete skew flux formulation}
993
994The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}),
995like the off-diagonal terms (\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor,
996are best expressed in terms of the triad slopes, as in \autoref{fig:ISO_triad} and
997(\autoref{eq:i13}, \autoref{eq:i31});
998but now in terms of the triad slopes $\tilde{\mathbb{R}}$ relative to geopotentials instead of
999the $\mathbb{R}$ relative to coordinate surfaces.
1000The discrete form of \autoref{eq:eiv_skew_ijk} using the slopes \autoref{eq:R} and
1001defining $A_e$ at $T$-points is then given by:
1002
1003\begin{subequations}
1004  % \label{eq:allskewflux}
1005  \begin{flalign*}
1006    % \label{eq:vect_skew_flux}
1007    \vect{F}_{\mathrm{eiv}}(T) &\equiv    \sum_{\substack{i_p,\,k_p}}
1008    \begin{pmatrix}
1009      {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T)      \\      \\
1010      {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T)      \\
1011    \end{pmatrix},
1012  \end{flalign*}
1013  where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:latflux-triad},
1014  \autoref{eq:triadfluxu}):
1015  \begin{align}
1016    \label{eq:skewfluxu}
1017    _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \fractext{1}{4} {A_e}_i^k{
1018                                          \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}}
1019                                          \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \
1020                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, \\
1021    \intertext{
1022    and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign
1023    to be consistent with \autoref{eq:eiv_skew_ijk}:
1024    }
1025    _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T)
1026                                        &= -\fractext{1}{4} {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}}
1027                                          {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw}
1028  \end{align}
1029\end{subequations}
1030
1031Such a discretisation is consistent with the iso-neutral operator as it uses the same definition for the slopes.
1032It also ensures the following two key properties.
1033
1034\subsubsection{No change in tracer variance}
1035
1036The discretization conserves tracer variance, \ie it does not include a diffusive component but is a `pure' advection term.
1037This can be seen %either from Appendix \autoref{apdx:eiv_skew} or
1038by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$.
1039For, following \autoref{subsec:variance} and \autoref{eq:dvar_iso_i},
1040the associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ drives a net rate of change of variance,
1041summed over the two $T$-points $i+i_p-\fractext{1}{2},k$ and $i+i_p+\fractext{1}{2},k$, of
1042\begin{equation}
1043  \label{eq:dvar_eiv_i}
1044  _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k],
1045\end{equation}
1046while the associated vertical skew-flux gives a variance change summed over
1047the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of
1048\begin{equation}
1049  \label{eq:dvar_eiv_k}
1050  _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
1051\end{equation}
1052Inspection of the definitions (\autoref{eq:skewfluxu}, \autoref{eq:skewfluxw}) shows that
1053these two variance changes (\autoref{eq:dvar_eiv_i}, \autoref{eq:dvar_eiv_k}) sum to zero.
1054Hence the two fluxes associated with each triad make no net contribution to the variance budget.
1055
1056\subsubsection{Reduction in gravitational PE}
1057
1058The vertical density flux associated with the vertical skew-flux always has the same sign as
1059the vertical density gradient;
1060thus, so long as the fluid is stable (the vertical density gradient is negative)
1061the vertical density flux is negative (downward) and hence reduces the gravitational PE.
1062
1063For the change in gravitational PE driven by the $k$-flux is
1064\begin{align}
1065  \label{eq:vert_densityPE}
1066  g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho)
1067  &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k
1068    {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k
1069    {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\
1070  \intertext{Substituting  ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:skewfluxw}, gives}
1071  % and separating out
1072  % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$,
1073  % gives two terms. The
1074  % first $\rtriad{R}$ term (the only term for $z$-coordinates) is:
1075  &=-\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}
1076    \frac{ -\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{\,i + i_p}^{\,k} } \notag \\
1077  &=+\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
1078    \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) {_i^k\mathbb{R}_{i_p}^{k_p}}
1079    \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
1080\end{align}
1081using the definition of the triad slope $\rtriad{R}$, \autoref{eq:R} to
1082express $-\alpha _i^k\delta_{i+ i_p}[T^k]+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of
1083$-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$.
1084
1085Where the coordinates slope, the $i$-flux gives a PE change
1086\begin{multline}
1087  \label{eq:lat_densityPE}
1088  g \delta_{i+i_p}[z_T^k]
1089  \left[
1090    -\alpha _i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (S)
1091  \right] \\
1092  = +\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
1093  \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
1094  \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)
1095  \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
1096\end{multline}
1097(using \autoref{eq:skewfluxu}) and so the total PE change \autoref{eq:vert_densityPE} +
1098\autoref{eq:lat_densityPE} associated with the triad fluxes is
1099\begin{multline*}
1100  % \label{eq:tot_densityPE}
1101  g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) +
1102  g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\
1103  = +\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
1104  \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)^2
1105  \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}.
1106\end{multline*}
1107Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+
1108\beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative.
1109
1110\subsection{Treatment of the triads at the boundaries}
1111\label{sec:skew_bdry}
1112
1113Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries
1114in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes,
1115as described in \autoref{sec:iso_bdry} and \autoref{fig:bdry_triads}.
1116Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked,
1117and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when
1118either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie the $i,k+1$ $u$-point is masked.
1119The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes.
1120
1121\subsection{Limiting of the slopes within the interior}
1122\label{sec:limitskew}
1123
1124Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$,
1125exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:limit}.
1126Each individual triad \rtriadt{R} is so limited.
1127
1128\subsection{Tapering within the surface mixed layer}
1129\label{sec:taperskew}
1130
1131The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R})
1132are always tapered linearly from their value immediately below the mixed layer to zero at the surface
1133\autoref{eq:rmtilde}, as described in \autoref{sec:lintaper}.
1134This is option (c) of \autoref{fig:eiv_slp}.
1135This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by
1136the value of \np{ln\_triad\_iso}.
1137
1138The justification for this linear slope tapering is that, for $A_e$ that is constant or varies only in
1139the horizontal (the most commonly used options in \NEMO: see \autoref{sec:LDF_coef}),
1140it is equivalent to a horizontal eiv (eddy-induced velocity) that is uniform within the mixed layer
1141\autoref{eq:eiv_v}.
1142This ensures that the eiv velocities do not restratify the mixed layer \citep{treguier.held.ea_JPO97,danabasoglu.ferrari.ea_JC08}.
1143Equivantly, in terms of the skew-flux formulation we use here,
1144the linear slope tapering within the mixed-layer gives a linearly varying vertical flux,
1145and so a tracer convergence uniform in depth
1146(the horizontal flux convergence is relatively insignificant within the mixed-layer).
1147
1148\subsection{Streamfunction diagnostics}
1149\label{sec:sfdiag}
1150
1151Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.},
1152diagnosed mean eddy-induced velocities are output.
1153Each time step, streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at
1154$uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ (integer $i$, integer +1/2 $j$, integer +1/2 $k$)
1155points (see Table \autoref{tab:cell}) respectively.
1156We follow \citep{griffies_bk04} and calculate the streamfunction at a given $uw$-point from
1157the surrounding four triads according to:
1158\[
1159  % \label{eq:sfdiagi}
1160  {\psi_1}_{i+1/2}^{k+1/2}={\fractext{1}{4}}\sum_{\substack{i_p,\,k_p}}
1161  {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}.
1162\]
1163The streamfunction $\psi_1$ is calculated similarly at $vw$ points.
1164The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:eiv_v}:
1165\[
1166  % \label{eq:eiv_v_discrete}
1167  \begin{split}
1168    {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right),   \\
1169    {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right),   \\
1170    {w^*}_{i,j}^{k+1/2} & =    \frac{1}{e_{1t}e_{2t}}\; \left\{
1171      {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} -
1172      {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\
1173    \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\},
1174  \end{split}
1175\]
1176
1177\biblio
1178
1179\pindex
1180
1181\end{document}
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