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apdx_triads.tex in NEMO/trunk/doc/latex/NEMO/subfiles – NEMO

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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\chapter{Iso-Neutral Diffusion and Eddy Advection using Triads}
15\label{apdx:TRIADS}
16
17\chaptertoc
18
19%% =================================================================================================
20\section[Choice of \forcode{namtra\_ldf} namelist parameters]{Choice of \protect\nam{tra_ldf}{tra\_ldf} namelist parameters}
21
22
23Two scheme are available to perform the iso-neutral diffusion.
24If the namelist logical \np{ln_traldf_triad}{ln\_traldf\_triad} is set true,
25\NEMO\ updates both active and passive tracers using the Griffies triad representation of iso-neutral diffusion and
26the eddy-induced advective skew (GM) fluxes.
27If the namelist logical \np{ln_traldf_iso}{ln\_traldf\_iso} is set true,
28the filtered version of Cox's original scheme (the Standard scheme) is employed (\autoref{sec:LDF_slp}).
29In the present implementation of the Griffies scheme,
30the advective skew fluxes are implemented even if \np{ln_traldf_eiv}{ln\_traldf\_eiv} is false.
31
32Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}.
33Note that when GM fluxes are used, the eddy-advective (GM) velocities are output for diagnostic purposes using XIOS,
34even though the eddy advection is accomplished by means of the skew fluxes.
35
36The options specific to the Griffies scheme include:
37\begin{description}
38\item [{\np{ln_triad_iso}{ln\_triad\_iso}}]
39  See \autoref{sec:TRIADS_taper}.
40  If this is set false (the default),
41  then `iso-neutral' mixing is accomplished within the surface mixed-layer along slopes linearly decreasing with
42  depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec:TRIADS_lintaper}).
43  This is the same treatment as used in the default implementation
44  \autoref{subsec:LDF_slp_iso}; \autoref{fig:LDF_eiv_slp}.
45  Where \np{ln_triad_iso}{ln\_triad\_iso} is set true,
46  the vertical skew flux is further reduced to ensure no vertical buoyancy flux,
47  giving an almost pure horizontal diffusive tracer flux within the mixed layer.
48  This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec:TRIADS_Gerdes-taper}
49\item [{\np{ln_botmix_triad}{ln\_botmix\_triad}}]
50  See \autoref{sec:TRIADS_iso_bdry}.
51  If this is set false (the default) then the lateral diffusive fluxes
52  associated with triads partly masked by topography are neglected.
53  If it is set true, however, then these lateral diffusive fluxes are applied,
54  giving smoother bottom tracer fields at the cost of introducing diapycnal mixing.
55\item [{\np{rn_sw_triad}{rn\_sw\_triad}}]
56  blah blah to be added....
57\end{description}
58The options shared with the Standard scheme include:
59\begin{description}
60\item [{\np{ln_traldf_msc}{ln\_traldf\_msc}}]   blah blah to be added
61\item [{\np{rn_slpmax}{rn\_slpmax}}]  blah blah to be added
62\end{description}
63
64%% =================================================================================================
65\section{Triad formulation of iso-neutral diffusion}
66\label{sec:TRIADS_iso}
67
68We have implemented into \NEMO\ a scheme inspired by \citet{griffies.gnanadesikan.ea_JPO98},
69but formulated within the \NEMO\ framework, using scale factors rather than grid-sizes.
70
71%% =================================================================================================
72\subsection{Iso-neutral diffusion operator}
73
74The iso-neutral second order tracer diffusive operator for small angles between
75iso-neutral surfaces and geopotentials is given by \autoref{eq:TRIADS_iso_tensor_1}:
76\begin{subequations}
77  \label{eq:TRIADS_iso_tensor_1}
78  \begin{equation}
79    D^{lT}=-\nabla \cdot\vect{f}^{lT}\equiv
80    -\frac{1}{e_1e_2e_3}\left[\pd{i}\left (f_1^{lT}e_2e_3\right) +
81      \pd{j}\left (f_2^{lT}e_2e_3\right) + \pd{k}\left (f_3^{lT}e_1e_2\right)\right],
82  \end{equation}
83  where the diffusive flux per unit area of physical space
84  \begin{equation}
85    \vect{f}^{lT}=-{A^{lT}}\Re\cdot\nabla T,
86  \end{equation}
87  \begin{equation}
88    \label{eq:TRIADS_iso_tensor_2}
89    \mbox{with}\quad \;\;\Re =
90    \begin{pmatrix}
91      1   &  0   & -r_1           \rule[-.9 em]{0pt}{1.79 em} \\
92      0   &  1   & -r_2           \rule[-.9 em]{0pt}{1.79 em} \\
93      -r_1 & -r_2 &  r_1 ^2+r_2 ^2 \rule[-.9 em]{0pt}{1.79 em}
94    \end{pmatrix}
95    \quad \text{and} \quad\nabla T=
96    \begin{pmatrix}
97      \frac{1}{e_1} \pd[T]{i} \rule[-.9 em]{0pt}{1.79 em} \\
98      \frac{1}{e_2} \pd[T]{j} \rule[-.9 em]{0pt}{1.79 em} \\
99      \frac{1}{e_3} \pd[T]{k} \rule[-.9 em]{0pt}{1.79 em}
100    \end{pmatrix}
101    .
102  \end{equation}
103\end{subequations}
104% \left( {{\begin{array}{*{20}c}
105%  1 \hfill & 0 \hfill & {-r_1 } \hfill \\
106%  0 \hfill & 1 \hfill & {-r_2 } \hfill \\
107%  {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
108% \end{array} }} \right)
109Here \autoref{eq:MB_iso_slopes}
110\begin{align*}
111  r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i}
112        \right)
113        \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\
114      &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} +
115        \beta\frac{\partial S }{\partial i} \right) \left(
116        -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S
117        }{\partial k} \right)^{-1}
118\end{align*}
119is the $i$-component of the slope of the iso-neutral surface relative to the computational surface,
120and $r_2$ is the $j$-component.
121
122We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write
123\[
124  % \label{eq:TRIADS_Fijk}
125  \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right).
126\]
127Additionally, we will sometimes write the contributions towards the fluxes $\vect{f}$ and
128$\vect{F}_{\mathrm{iso}}$ from the component $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$,
129with $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc.
130
131The off-diagonal terms of the small angle diffusion tensor
132\autoref{eq:TRIADS_iso_tensor_1}, \autoref{eq:TRIADS_iso_tensor_2} produce skew-fluxes along
133the $i$- and $j$-directions resulting from the vertical tracer gradient:
134\begin{align}
135  \label{eq:TRIADS_i13c}
136  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}\\
137  \intertext{and in the k-direction resulting from the lateral tracer gradients}
138  \label{eq:TRIADS_i31c}
139  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}
140\end{align}
141
142The vertical diffusive flux associated with the $_{33}$ component of the small angle diffusion tensor is
143\begin{equation}
144  \label{eq:TRIADS_i33c}
145  f_{33}=-{A^{lT}}(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}.
146\end{equation}
147
148Since there are no cross terms involving $r_1$ and $r_2$ in the above,
149we can consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$ planes,
150just adding together the vertical components from each plane.
151The following description will describe the fluxes on the $i$-$k$ plane.
152
153There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq:TRIADS_i13c},
154as although it must be evaluated at $u$-points,
155it involves vertical gradients (both for the tracer and the slope $r_1$), defined at $w$-points.
156Similarly, the vertical skew flux, \autoref{eq:TRIADS_i31c},
157is evaluated at $w$-points but involves horizontal gradients defined at $u$-points.
158
159%% =================================================================================================
160\subsection{Standard discretization}
161
162The straightforward approach to discretize the lateral skew flux
163\autoref{eq:TRIADS_i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA,
164\autoref{eq:TRA_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from
165the average of the four surrounding vertical tracer gradients, and multiply this by a mean slope at the $u$-point,
166calculated from the averaged surrounding vertical density gradients.
167The total area-integrated skew-flux (flux per unit area in $ijk$ space) from tracer cell $i,k$ to $i+1,k$,
168noting 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
169the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:TRA_ldf_iso}
170\[
171  \left(F_u^{13} \right)_{i+\frac{1}{2}}^k = {A}_{i+\frac{1}{2}}^k
172  {e_{2}}_{i+1/2}^k \overline{\overline
173    r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k},
174\]
175where
176\[
177  \overline{\overline
178    r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k}
179  \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}},
180\]
181and here and in the following we drop the $^{lT}$ superscript from ${A^{lT}}$ for simplicity.
182Unfortunately the resulting combination $\overline{\overline{\delta_k\bullet}}^{\,i,k}$ of a $k$ average and
183a $k$ difference of the tracer reduces to $\bullet_{k+1}-\bullet_{k-1}$,
184so two-grid-point oscillations are invisible to this discretization of the iso-neutral operator.
185These \emph{computational modes} will not be damped by this operator, and may even possibly be amplified by it.
186Consequently, applying this operator to a tracer does not guarantee the decrease of its global-average variance.
187To correct this, we introduced a smoothing of the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}).
188This technique works for $T$ and $S$ in so far as they are active tracers
189(\ie\ they enter the computation of density), but it does not work for a passive tracer.
190
191%% =================================================================================================
192\subsection{Expression of the skew-flux in terms of triad slopes}
193
194\citep{griffies.gnanadesikan.ea_JPO98} introduce a different discretization of the off-diagonal terms that
195nicely solves the problem.
196% Instead of multiplying the mean slope calculated at the $u$-point by
197% the mean vertical gradient at the $u$-point,
198% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
199\begin{figure}[tb]
200  \centering
201  \includegraphics[width=0.66\textwidth]{Fig_GRIFF_triad_fluxes}
202  \caption[Triads arrangement and tracer gradients to give lateral and vertical tracer fluxes]{
203    (a) Arrangement of triads $S_i$ and tracer gradients to
204    give lateral tracer flux from box $i,k$ to $i+1,k$
205    (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from
206    box $i,k$ to $i,k+1$.}
207  \label{fig:TRIADS_ISO_triad}
208\end{figure}
209% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
210They get the skew flux from the products of the vertical gradients at each $w$-point surrounding the $u$-point with
211the corresponding `triad' slope calculated from the lateral density gradient across the $u$-point divided by
212the vertical density gradient at the same $w$-point as the tracer gradient.
213See \autoref{fig:TRIADS_ISO_triad}a, where the thick lines denote the tracer gradients,
214and the thin lines the corresponding triads, with slopes $s_1, \dotsc s_4$.
215The total area-integrated skew-flux from tracer cell $i,k$ to $i+1,k$
216\begin{multline}
217  \label{eq:TRIADS_i13}
218  \left( F_u^{13}  \right)_{i+\frac{1}{2}}^k = {A}_{i+1}^k a_1 s_1
219  \delta_{k+\frac{1}{2}} \left[ T^{i+1}
220  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  + {A} _i^k a_2 s_2 \delta
221  _{k+\frac{1}{2}} \left[ T^i
222  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\
223  +{A} _{i+1}^k a_3 s_3 \delta_{k-\frac{1}{2}} \left[ T^{i+1}
224  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  +{A} _i^k a_4 s_4 \delta
225  _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}},
226\end{multline}
227where the contributions of the triad fluxes are weighted by areas $a_1, \dotsc a_4$,
228and ${A}$ is now defined at the tracer points rather than the $u$-points.
229This discretization gives a much closer stencil, and disallows the two-point computational modes.
230
231The vertical skew flux \autoref{eq:TRIADS_i31c} from tracer cell $i,k$ to $i,k+1$ at
232the $w$-point $i,k+\frac{1}{2}$ is constructed similarly (\autoref{fig:TRIADS_ISO_triad}b) by
233multiplying lateral tracer gradients from each of the four surrounding $u$-points by the appropriate triad slope:
234\begin{multline}
235  \label{eq:TRIADS_i31}
236  \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} =  {A}_i^{k+1} a_{1}'
237  s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1}
238  +{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} \\
239  + {A}_i^k a_{3}' s_{3}' \delta_{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k
240  +{A}_i^k a_{4}' s_{4}' \delta_{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k.
241\end{multline}
242
243We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$
244(appearing in both the vertical and lateral gradient),
245and the $u$- and $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the triad as follows
246(see also \autoref{fig:TRIADS_ISO_triad}):
247\begin{equation}
248  \label{eq:TRIADS_R}
249  _i^k \mathbb{R}_{i_p}^{k_p}
250  =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}}
251  \
252  \frac
253  { \alpha_i^\ \delta_{i+i_p}[T^k] - \beta_i^k \ \delta_{i+i_p}[S^k] }
254  { \alpha_i^\ \delta_{k+k_p}[T^i] - \beta_i^k \ \delta_{k+k_p}[S^i] }.
255\end{equation}
256In calculating the slopes of the local neutral surfaces,
257the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad,
258while the metrics are calculated at the $u$- and $w$-points on the arms.
259
260% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
261\begin{figure}[tb]
262  \centering
263  \includegraphics[width=0.66\textwidth]{Fig_GRIFF_qcells}
264  \caption[Triad notation for quarter cells]{
265    Triad notation for quarter cells.
266    $T$-cells are inside boxes,
267    while the $i+\fractext{1}{2},k$ $u$-cell is shaded in green and
268    the $i,k+\fractext{1}{2}$ $w$-cell is shaded in pink.}
269  \label{fig:TRIADS_qcells}
270\end{figure}
271% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
272
273Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:TRIADS_qcells}) with the quarter cell that is
274the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell.
275Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:TRIADS_i13} and \autoref{eq:TRIADS_i31} in this notation,
276we have \eg\ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$.
277Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to
278calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$,
279and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$.
280Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used to
281calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms of a unique triad,
282and we notate these areas, similarly to the triad slopes,
283as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$,
284where \eg\ in \autoref{eq:TRIADS_i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$,
285and in \autoref{eq:TRIADS_i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$.
286
287%% =================================================================================================
288\subsection{Full triad fluxes}
289
290A key property of iso-neutral diffusion is that it should not affect the (locally referenced) density.
291In particular there should be no lateral or vertical density flux.
292The lateral density flux disappears so long as the area-integrated lateral diffusive flux from
293tracer cell $i,k$ to $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the form
294\begin{equation}
295  \label{eq:TRIADS_i11}
296  \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} =
297  - \left( {A}_i^{k+1} a_{1} + {A}_i^{k+1} a_{2} + {A}_i^k
298    a_{3} + {A}_i^k a_{4} \right)
299  \frac{\delta_{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}},
300\end{equation}
301where the areas $a_i$ are as in \autoref{eq:TRIADS_i13}.
302In this case, separating the total lateral flux, the sum of \autoref{eq:TRIADS_i13} and \autoref{eq:TRIADS_i11},
303into triad components, a lateral tracer flux
304\begin{equation}
305  \label{eq:TRIADS_latflux-triad}
306  _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - {A}_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p}
307  \left(
308    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
309    -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \
310    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
311  \right)
312\end{equation}
313can be identified with each triad.
314Then, because the same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and ${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both
315the density gradients in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients,
316the lateral density flux associated with each triad separately disappears.
317\begin{equation}
318  \label{eq:TRIADS_latflux-rho}
319  {\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
320\end{equation}
321Thus 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
322tracer cell $i,k$ to $i+1,k$ must also vanish since it is a sum of four such triad fluxes.
323
324The squared slope $r_1^2$ in the expression \autoref{eq:TRIADS_i33c} for the $_{33}$ component is also expressed in
325terms of area-weighted squared triad slopes,
326so the area-integrated vertical flux from tracer cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is
327\begin{equation}
328  \label{eq:TRIADS_i33}
329  \left( F_w^{33} \right) _i^{k+\frac{1}{2}} =
330  - \left( {A}_i^{k+1} a_{1}' s_{1}'^2
331    + {A}_i^{k+1} a_{2}' s_{2}'^2
332    + {A}_i^k a_{3}' s_{3}'^2
333    + {A}_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right],
334\end{equation}
335where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq:TRIADS_i31}.
336Then, separating the total vertical flux, the sum of \autoref{eq:TRIADS_i31} and \autoref{eq:TRIADS_i33},
337into triad components, a vertical flux
338\begin{align}
339  \label{eq:TRIADS_vertflux-triad}
340  _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
341  &= {A}_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p}
342    \left(
343    {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
344    -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \
345    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
346    \right) \\
347  &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right)
348    {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:TRIADS_vertflux-triad2}
349\end{align}
350may be associated with each triad.
351Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ associated with a triad then
352separately disappears (because the lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$ disappears).
353Consequently the total vertical density flux
354$\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} + \left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from
355tracer cell $i,k$ to $i,k+1$ must also vanish since it is a sum of four such triad fluxes.
356
357We can explicitly identify (\autoref{fig:TRIADS_qcells}) the triads associated with the $s_i$, $a_i$,
358and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq:TRIADS_i31},
359\autoref{eq:TRIADS_i13}, \autoref{eq:TRIADS_i11} \autoref{eq:TRIADS_i33} and \autoref{fig:TRIADS_ISO_triad} to write out
360the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces:
361%(\autoref{fig:TRIADS_ISO_triad}):
362\begin{flalign}
363  \label{eq:TRIADS_iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv
364  \sum_{\substack{i_p,\,k_p}}
365  \begin{pmatrix}
366    {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\
367    {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\
368  \end{pmatrix}.
369\end{flalign}
370
371%% =================================================================================================
372\subsection{Ensuring the scheme does not increase tracer variance}
373\label{subsec:TRIADS_variance}
374
375We now require that this operator should not increase the globally-integrated tracer variance.
376%This changes according to
377% \begin{align*}
378% &\int_D  D_l^T \; T \;dv \equiv  \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\}    \\
379% &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
380%     \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right]
381%       + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]  \ T \right\}    \\
382% &\equiv  - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
383%                 {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T]
384%              + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}}  \ \delta_{k+1/2} [T]   \right\}      \\
385% \end{align*}
386Each 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
387the $u$-point $i+i_p,k$ and a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the $w$-point $i,k+k_p$.
388The lateral flux drives a net rate of change of variance,
389summed over the two $T$-points $i+i_p-\fractext{1}{2},k$ and $i+i_p+\fractext{1}{2},k$, of
390\begin{multline}
391  {b_T}_{i+i_p-1/2}^k\left(\frac{\partial T}{\partial t}T\right)_{i+i_p-1/2}^k+
392  \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial
393      t}T\right)_{i+i_p+1/2}^k \\
394  \begin{aligned}
395    &= -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
396    {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\
397    &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:TRIADS_dvar_iso_i}
398  \end{aligned}
399\end{multline}
400while the vertical flux similarly drives a net rate of change of variance summed over
401the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of
402\begin{equation}
403  \label{eq:TRIADS_dvar_iso_k}
404  _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
405\end{equation}
406The total variance tendency driven by the triad is the sum of these two.
407Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with
408\autoref{eq:TRIADS_latflux-triad} and \autoref{eq:TRIADS_vertflux-triad}, it is
409\begin{multline*}
410  -{A}_i^k\left \{
411    { } _i^k{\mathbb{A}_u}_{i_p}^{k_p}
412    \left(
413      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
414      - {_i^k\mathbb{R}_{i_p}^{k_p}} \
415      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\
416  - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p}
417    \left(
418      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
419      -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
420      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
421    \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i]
422  \right \}.
423\end{multline*}
424The 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
425be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by
426\begin{equation}
427  \label{eq:TRIADS_V-A}
428  _i^k\mathbb{V}_{i_p}^{k_p}
429  ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k}
430  ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p},
431\end{equation}
432the variance tendency reduces to the perfect square
433\begin{equation}
434  \label{eq:TRIADS_perfect-square}
435  -{A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
436  \left(
437    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
438    -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
439    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
440  \right)^2\leq 0.
441\end{equation}
442Thus, the constraint \autoref{eq:TRIADS_V-A} ensures that the fluxes
443(\autoref{eq:TRIADS_latflux-triad}, \autoref{eq:TRIADS_vertflux-triad}) associated with
444a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase the net variance.
445Since the total fluxes are sums of such fluxes from the various triads, this constraint, applied to all triads,
446is sufficient to ensure that the globally integrated variance does not increase.
447
448The expression \autoref{eq:TRIADS_V-A} can be interpreted as a discretization of the global integral
449\begin{equation}
450  \label{eq:TRIADS_cts-var}
451  \frac{\partial}{\partial t}\int\!\fractext{1}{2} T^2\, dV =
452  \int\!\mathbf{F}\cdot\nabla T\, dV,
453\end{equation}
454where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the lateral and vertical fluxes/unit area
455\[
456  \mathbf{F}=\left(
457    \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p},
458    \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p}
459  \right)
460\]
461and the gradient
462\[
463  \nabla T = \left(
464    \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k},
465    \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p}
466  \right)
467\]
468
469%% =================================================================================================
470\subsection{Triad volumes in Griffes's scheme and in \NEMO}
471
472To complete the discretization we now need only specify the triad volumes $_i^k\mathbb{V}_{i_p}^{k_p}$.
473\citet{griffies.gnanadesikan.ea_JPO98} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells,
474defined in terms of the distances between $T$, $u$,$f$ and $w$-points.
475This is the natural discretization of \autoref{eq:TRIADS_cts-var}.
476The \NEMO\ model, however, operates with scale factors instead of grid sizes,
477and scale factors for the quarter cells are not defined.
478Instead, therefore we simply choose
479\begin{equation}
480  \label{eq:TRIADS_V-NEMO}
481  _i^k\mathbb{V}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k,
482\end{equation}
483as a quarter of the volume of the $u$-cell inside which the triad quarter-cell lies.
484This has the nice property that when the slopes $\mathbb{R}$ vanish,
485the lateral flux from tracer cell $i,k$ to $i+1,k$ reduces to the classical form
486\begin{equation}
487  \label{eq:TRIADS_lat-normal}
488  -\overline{A}_{\,i+1/2}^k\;
489  \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}}
490  \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}}
491  = -\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}}.
492\end{equation}
493In fact if the diffusive coefficient is defined at $u$-points,
494so that we employ ${A}_{i+i_p}^k$ instead of  ${A}_i^k$ in the definitions of the triad fluxes
495\autoref{eq:TRIADS_latflux-triad} and \autoref{eq:TRIADS_vertflux-triad},
496we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above.
497
498%% =================================================================================================
499\subsection{Summary of the scheme}
500
501The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that
502cross the $u$- and $w$-faces \autoref{eq:TRIADS_iso_flux}:
503\begin{subequations}
504  % \label{eq:TRIADS_alltriadflux}
505  \begin{flalign*}
506    % \label{eq:TRIADS_vect_isoflux}
507    \vect{F}_{\mathrm{iso}}(T) &\equiv
508    \sum_{\substack{i_p,\,k_p}}
509    \begin{pmatrix}
510      {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\
511      {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T)
512    \end{pmatrix},
513  \end{flalign*}
514  where \autoref{eq:TRIADS_latflux-triad}:
515  \begin{align}
516    \label{eq:TRIADS_triadfluxu}
517    _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - {A}_i^k{
518                                          \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}}
519                                          \left(
520                                          \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
521                                          -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \
522                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
523                                          \right),\\
524    \intertext{and}
525    _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
526                                        &= {A}_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}}
527                                          \left(
528                                          {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
529                                          -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \
530                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
531                                          \right),\label{eq:TRIADS_triadfluxw}
532  \end{align}
533  with \autoref{eq:TRIADS_V-NEMO}
534  \[
535    % \label{eq:TRIADS_V-NEMO2}
536    _i^k{\mathbb{V}}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k.
537  \]
538\end{subequations}
539
540The divergence of the expression \autoref{eq:TRIADS_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at
541each tracer point:
542\[
543  % \label{eq:TRIADS_iso_operator}
544  D_l^T = \frac{1}{b_T}
545  \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k
546        {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[
547      {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\}
548\]
549where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.
550The diffusion scheme satisfies the following six properties:
551\begin{description}
552\item [$\bullet$ horizontal diffusion]
553  The discretization of the diffusion operator recovers the traditional five-point Laplacian
554  \autoref{eq:TRIADS_lat-normal} in the limit of flat iso-neutral direction:
555  \[
556    % \label{eq:TRIADS_iso_property0}
557    D_l^T = \frac{1}{b_T} \
558    \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \;
559      \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] \qquad
560    \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0
561  \]
562
563\item [$\bullet$ implicit treatment in the vertical]
564  Only tracer values associated with a single water column appear in the expression \autoref{eq:TRIADS_i33} for
565  the $_{33}$ fluxes, vertical fluxes driven by vertical gradients.
566  This is of paramount importance since it means that a time-implicit algorithm can be used to
567  solve the vertical diffusion equation.
568  This is necessary since the vertical eddy diffusivity associated with this term,
569  \[
570    \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{
571      {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: {A}_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
572    \right\}  =
573    \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{
574      {b_u}_{i+i_p}^k\: {A}_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
575    \right\},
576  \]
577  (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large.
578
579\item [$\bullet$ pure iso-neutral operator]
580  The iso-neutral flux of locally referenced potential density is zero.
581  See \autoref{eq:TRIADS_latflux-rho} and \autoref{eq:TRIADS_vertflux-triad2}.
582
583\item [$\bullet$ conservation of tracer]
584  The iso-neutral diffusion conserves tracer content, \ie
585  \[
586    % \label{eq:TRIADS_iso_property1}
587    \sum_{i,j,k} \left\{ D_l^T \      b_T \right\} = 0
588  \]
589  This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form.
590
591\item [$\bullet$ no increase of tracer variance]
592  The iso-neutral diffusion does not increase the tracer variance, \ie
593  \[
594    % \label{eq:TRIADS_iso_property2}
595    \sum_{i,j,k} \left\{ T \ D_l^T      \ b_T \right\} \leq 0
596  \]
597  The property is demonstrated in \autoref{subsec:TRIADS_variance} above.
598  It is a key property for a diffusion term.
599  It means that it is also a dissipation term,
600  \ie\ it dissipates the square of the quantity on which it is applied.
601  It therefore ensures that, when the diffusivity coefficient is large enough,
602  the field on which it is applied becomes free of grid-point noise.
603
604\item [$\bullet$ self-adjoint operator]
605  The iso-neutral diffusion operator is self-adjoint, \ie
606  \begin{equation}
607    \label{eq:TRIADS_iso_property3}
608    \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
609  \end{equation}
610  In other word, there is no need to develop a specific routine from the adjoint of this operator.
611  We just have to apply the same routine.
612  This property can be demonstrated similarly to the proof of the `no increase of tracer variance' property.
613  The contribution by a single triad towards the left hand side of \autoref{eq:TRIADS_iso_property3},
614  can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:TRIADS_dvar_iso_i} and \autoref{eq:TRIADS_dvar_iso_k}.
615  This results in a term similar to \autoref{eq:TRIADS_perfect-square},
616  \[
617    % \label{eq:TRIADS_TScovar}
618    - {A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
619    \left(
620      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
621      -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
622      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
623    \right)
624    \left(
625      \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
626      -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
627      \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
628    \right).
629  \]
630This is symmetrical in $T $ and $S$, so exactly the same term arises from
631the discretization of this triad's contribution towards the RHS of \autoref{eq:TRIADS_iso_property3}.
632\end{description}
633
634%% =================================================================================================
635\subsection{Treatment of the triads at the boundaries}
636\label{sec:TRIADS_iso_bdry}
637
638The triad slope can only be defined where both the grid boxes centred at the end of the arms exist.
639Triads that would poke up through the upper ocean surface into the atmosphere,
640or down into the ocean floor, must be masked out.
641See \autoref{fig:TRIADS_bdry_triads}.
642Surface layer triads \triad{i}{1}{R}{1/2}{-1/2} (magenta) and \triad{i+1}{1}{R}{-1/2}{-1/2} (blue) that
643require density to be specified above the ocean surface are masked (\autoref{fig:TRIADS_bdry_triads}a):
644this ensures that lateral tracer gradients produce no flux through the ocean surface.
645However, to prevent surface noise, it is customary to retain the $_{11}$ contributions towards
646the lateral triad fluxes \triad[u]{i}{1}{F}{1/2}{-1/2} and \triad[u]{i+1}{1}{F}{-1/2}{-1/2};
647this drives diapycnal tracer fluxes.
648Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:TRIADS_bdry_triads}b).
649Note 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
650either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked.
651The associated lateral fluxes (grey-black dashed line) are masked if \np[=.false.]{ln_botmix_triad}{ln\_botmix\_triad},
652but left unmasked, giving bottom mixing, if \np[=.true.]{ln_botmix_triad}{ln\_botmix\_triad}.
653
654The default option \np[=.false.]{ln_botmix_triad}{ln\_botmix\_triad} is suitable when the bbl mixing option is enabled
655(\np[=.true.]{ln_trabbl}{ln\_trabbl}, with \np[=1]{nn_bbl_ldf}{nn\_bbl\_ldf}), or for simple idealized problems.
656For setups with topography without bbl mixing, \np[=.true.]{ln_botmix_triad}{ln\_botmix\_triad} may be necessary.
657% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
658\begin{figure}[h]
659  \centering
660  \includegraphics[width=0.66\textwidth]{Fig_GRIFF_bdry_triads}
661  \caption[Boundary triads]{
662    (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots),
663    and $i+1/2,1$ $u$-point (blue square).
664    Triad slopes \triad{i}{1}{R}{1/2}{-1/2} (magenta) and
665    \triad{i+1}{1}{R}{-1/2}{-1/2} (blue) poking through the ocean surface are masked
666    (faded in figure).
667    However,
668    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
674    either of the $i,k+1$ or $i+1,k+1$ tracer points is masked,
675    \ie\ the $i,k+1$ $u$-point is masked.
676    The associated lateral fluxes (grey-black dashed line) are masked if
677    \protect\np[=.false.]{ln_botmix_triad}{ln\_botmix\_triad}, but left unmasked,
678    giving bottom mixing, if \protect\np[=.true.]{ln_botmix_triad}{ln\_botmix\_triad}}
679  \label{fig:TRIADS_bdry_triads}
680\end{figure}
681% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
682
683%% =================================================================================================
684\subsection{ Limiting of the slopes within the interior}
685\label{sec:TRIADS_limit}
686
687As discussed in \autoref{subsec:LDF_slp_iso},
688iso-neutral slopes relative to geopotentials must be bounded everywhere,
689both for consistency with the small-slope approximation and for numerical stability \citep{cox_OM87, griffies_bk04}.
690The bound chosen in \NEMO\ is applied to each component of the slope separately and
691has a value of $1/100$ in the ocean interior.
692%, ramping linearly down above 70~m depth to zero at the surface
693It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to geopotentials
694(here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials)
695\autoref{eq:MB_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require
696\[
697  |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01.
698\]
699and then recalculate the slopes $r_i$ relative to coordinates.
700Each individual triad slope
701\begin{equation}
702  \label{eq:TRIADS_Rtilde}
703  _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}}
704\end{equation}
705is limited like this and then the corresponding $_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and
706combined to form the fluxes.
707Note that where the slopes have been limited, there is now a non-zero iso-neutral density flux that
708drives dianeutral mixing.
709In particular this iso-neutral density flux is always downwards,
710and so acts to reduce gravitational potential energy.
711
712%% =================================================================================================
713\subsection{Tapering within the surface mixed layer}
714\label{sec:TRIADS_taper}
715
716Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer.
717When the Griffies triads are used, we offer two options for this.
718
719%% =================================================================================================
720\subsubsection{Linear slope tapering within the surface mixed layer}
721\label{sec:TRIADS_lintaper}
722
723This is the option activated by the default choice \np[=.false.]{ln_triad_iso}{ln\_triad\_iso}.
724Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below
725the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:LDF_eiv_slp}, to values
726\begin{equation}
727  \label{eq:TRIADS_rmtilde}
728  \rMLt = -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h,
729\end{equation}
730and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to
731\[
732  % \label{eq:TRIADS_rm}
733  \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h.
734\]
735Thus the diffusion operator within the mixed layer is given by:
736\[
737  % \label{eq:TRIADS_iso_tensor_ML}
738  D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
739  \mbox{with}\quad \;\;\Re =\left( {{
740        \begin{array}{*{20}c}
741          1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\
742          0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\
743          {-\rML[1]}\hfill &   {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill
744        \end{array}
745      }} \right)
746\]
747
748This slope tapering gives a natural connection between tracer in the mixed-layer and
749in isopycnal layers immediately below, in the thermocline.
750It is consistent with the way the $\tilde{r}_i$ are tapered within the mixed layer
751(see \autoref{sec:TRIADS_taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer.
752However, it gives a downwards density flux and so acts so as to reduce potential energy in the same way as
753does the slope limiting discussed above in \autoref{sec:TRIADS_limit}.
754
755As in \autoref{sec:TRIADS_limit} above, the tapering \autoref{eq:TRIADS_rmtilde} is applied separately to
756each triad $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted.
757For clarity, we assume $z$-coordinates in the following;
758the conversion from $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as
759described above by \autoref{eq:TRIADS_Rtilde}.
760\begin{enumerate}
761\item Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in
762  the slope definition.
763  At each $i,j$ (simplified to $i$ in \autoref{fig:TRIADS_MLB_triad}),
764  we define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer,
765  $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that
766  the potential density ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$,
767  where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m.
768  See the left side of \autoref{fig:TRIADS_MLB_triad}.
769  We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems \eg\ with thin daytime mixed-layers.
770  Currently we use the same $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to
771  output the diagnosed mixed-layer depth $h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$,
772  the depth of the $w$-point above the $i,k_{\mathrm{ML}}$ tracer point.
773\item We define `basal' triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as
774  the slopes of those triads whose vertical `arms' go down from the $i,k_{\mathrm{ML}}$ tracer point to
775  the $i,k_{\mathrm{ML}}-1$ tracer point below.
776  This is to ensure that the vertical density gradients associated with
777  these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline.
778  The four basal triads defined in the bottom part of \autoref{fig:TRIADS_MLB_triad} are then
779  \begin{align*}
780    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &=
781                                                       {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p},
782                                                       % \label{eq:TRIADS_Rbase}
783    \\
784    \intertext{with \eg\ the green triad}
785    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&=
786                                                     {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}.
787  \end{align*}
788The vertical flux associated with each of these triads passes through
789the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth
790\[
791  % \label{eq:TRIADS_zbase}
792  {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2}
793\]
794one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in
795\autoref{eq:TRIADS_rmtilde}.
796\item 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:TRIADS_RML}
807  \end{align*}
808\end{enumerate}
809
810% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
811\begin{figure}[h]
812  \centering
813  \includegraphics[width=0.66\textwidth]{Fig_GRIFF_MLB_triads}
814  \caption[Definition of mixed-layer depth and calculation of linearly tapered triads]{
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 $i$),
817    with the ocean surface at the top.
818    Tracer points are denoted by bullets, and black lines the edges of the tracer cells;
819    $k$ increases upwards.
820    \newline
821    We define the mixed-layer by setting the vertical index of the tracer point immediately below
822    the mixed layer, $k_{\mathrm{ML}}$, 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  \label{fig:TRIADS_MLB_triad}
832\end{figure}
833% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
834
835%% =================================================================================================
836\subsubsection{Additional truncation of skew iso-neutral flux components}
837\label{subsec:TRIADS_Gerdes-taper}
838
839The alternative option is activated by setting \np{ln_triad_iso}{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:TRIADS_rm*}
845  \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i,
846\end{equation}
847giving a ML diffusive operator
848\[
849  % \label{eq:TRIADS_iso_tensor_ML2}
850  D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
851  \mbox{with}\quad \;\;\Re =\left( {{
852        \begin{array}{*{20}c}
853          1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\
854          0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\
855          {-\rML[1]^*}\hfill &   {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\
856        \end{array}
857      }} \right).
858\]
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\section{Eddy induced advection formulated as a skew flux}
880\label{sec:TRIADS_skew-flux}
881
882%% =================================================================================================
883\subsection{Continuous skew flux formulation}
884\label{sec:TRIADS_continuous-skew-flux}
885
886When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added.
887The associated velocity is the so called eddy induced velocity,
888the formulation of which depends on the slopes of iso-neutral surfaces.
889Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces,
890\ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinate,
891and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $z^*$ or $s$-coordinates.
892
893The eddy induced velocity is given by:
894\begin{subequations}
895  % \label{eq:TRIADS_eiv}
896  \begin{equation}
897    \label{eq:TRIADS_eiv_v}
898    \begin{split}
899      u^* & = - \frac{1}{e_{3}}\;          \partial_i\psi_1,  \\
900      v^* & = - \frac{1}{e_{3}}\;          \partial_j\psi_2,    \\
901      w^* & =    \frac{1}{e_{1}e_{2}}\; \left\{ \partial_\left( e_{2} \, \psi_1\right)
902        + \partial_\left( e_{1} \, \psi_2\right) \right\},
903    \end{split}
904  \end{equation}
905  where the streamfunctions $\psi_i$ are given by
906  \begin{equation}
907    \label{eq:TRIADS_eiv_psi}
908    \begin{split}
909      \psi_1 & = A_{e} \; \tilde{r}_1,   \\
910      \psi_2 & = A_{e} \; \tilde{r}_2,
911    \end{split}
912  \end{equation}
913\end{subequations}
914with $A_{e}$ the eddy induced velocity coefficient,
915and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces.
916
917The traditional way to implement this additional advection is to add it to the Eulerian velocity prior to
918computing the tracer advection.
919This is implemented if \texttt{traldf\_eiv?} is set in the default implementation,
920where \np{ln_traldf_triad}{ln\_traldf\_triad} is set false.
921This allows us to take advantage of all the advection schemes offered for the tracers
922(see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme.
923This is particularly useful for passive tracers where
924\emph{positivity} of the advection scheme is of paramount importance.
925
926However, when \np{ln_traldf_triad}{ln\_traldf\_triad} is set true,
927\NEMO\ instead implements eddy induced advection according to the so-called skew form \citep{griffies_JPO98}.
928It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity.
929For example in the (\textbf{i},\textbf{k}) plane,
930the tracer advective fluxes per unit area in $ijk$ space can be transformed as follows:
931\begin{flalign*}
932  \begin{split}
933    \textbf{F}_{\mathrm{eiv}}^T =
934    \begin{pmatrix}
935      {e_{2}\,e_{3}\;  u^*} \\
936      {e_{1}\,e_{2}\; w^*}
937    \end{pmatrix}   \;   T
938    &=
939    \begin{pmatrix}
940      { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\
941      {+ \partial_\left( e_{2} \, \psi_1 \right) \; T \;}
942    \end{pmatrix}          \\
943    &=
944    \begin{pmatrix}
945      { - \partial_k \left( e_{2} \, \psi_\; T \right) \;} \\
946      {+ \partial_\left( e_{2} \,\psi_1 \; T \right) \;}
947    \end{pmatrix}
948    +
949    \begin{pmatrix}
950      {+ e_{2} \, \psi_\; \partial_k T} \\
951      { - e_{2} \, \psi_\; \partial_i  T}
952    \end{pmatrix}
953  \end{split}
954\end{flalign*}
955and since the eddy induced velocity field is non-divergent,
956we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space:
957\begin{equation}
958  \label{eq:TRIADS_eiv_skew_ijk}
959  \textbf{F}_\mathrm{eiv}^T =
960  \begin{pmatrix}
961    {+ e_{2} \, \psi_\; \partial_k T}   \\
962    { - e_{2} \, \psi_\; \partial_i  T}
963  \end{pmatrix}
964\end{equation}
965The total fluxes per unit physical area are then
966\begin{equation}
967  \label{eq:TRIADS_eiv_skew_physical}
968  \begin{split}
969    f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T   \\
970    f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T   \\
971    f^*_3 & =  -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T + e_{1} \psi_2 \partial_j T \right\}.
972\end{split}
973\end{equation}
974Note that \autoref{eq:TRIADS_eiv_skew_physical} takes the same form whatever the vertical coordinate,
975though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:TRIADS_eiv_psi} are relative to
976geopotentials.
977The tendency associated with eddy induced velocity is then simply the convergence of the fluxes
978(\autoref{eq:TRIADS_eiv_skew_ijk}, \autoref{eq:TRIADS_eiv_skew_physical}), so
979\[
980  % \label{eq:TRIADS_skew_eiv_conv}
981  \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 }      \left[
982    \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right)
983    + \frac{\partial}{\partial j} \left( e_1  \;
984      \psi_2 \partial_k T\right)
985    -  \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T
986      + e_{1} \psi_2 \partial_j T \right)  \right]
987\]
988It naturally conserves the tracer content, as it is expressed in flux form.
989Since it has the same divergence as the advective form it also preserves the tracer variance.
990
991%% =================================================================================================
992\subsection{Discrete skew flux formulation}
993
994The skew fluxes in (\autoref{eq:TRIADS_eiv_skew_physical}, \autoref{eq:TRIADS_eiv_skew_ijk}),
995like the off-diagonal terms (\autoref{eq:TRIADS_i13c}, \autoref{eq:TRIADS_i31c}) of the small angle diffusion tensor,
996are best expressed in terms of the triad slopes, as in \autoref{fig:TRIADS_ISO_triad} and
997(\autoref{eq:TRIADS_i13}, \autoref{eq:TRIADS_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:TRIADS_eiv_skew_ijk} using the slopes \autoref{eq:TRIADS_R} and
1001defining $A_e$ at $T$-points is then given by:
1002
1003\begin{subequations}
1004  % \label{eq:TRIADS_allskewflux}
1005  \begin{flalign*}
1006    % \label{eq:TRIADS_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:TRIADS_latflux-triad},
1014  \autoref{eq:TRIADS_triadfluxu}):
1015  \begin{align}
1016    \label{eq:TRIADS_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:TRIADS_triadfluxw} in the $k$-direction, changing the sign
1023    to be consistent with \autoref{eq:TRIADS_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:TRIADS_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%% =================================================================================================
1035\subsubsection{No change in tracer variance}
1036
1037The discretization conserves tracer variance, \ie\ it does not include a diffusive component but is a `pure' advection term.
1038This can be seen %either from Appendix \autoref{apdx:eiv_skew} or
1039by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$.
1040For, following \autoref{subsec:TRIADS_variance} and \autoref{eq:TRIADS_dvar_iso_i},
1041the associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ drives a net rate of change of variance,
1042summed over the two $T$-points $i+i_p-\fractext{1}{2},k$ and $i+i_p+\fractext{1}{2},k$, of
1043\begin{equation}
1044  \label{eq:TRIADS_dvar_eiv_i}
1045  _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k],
1046\end{equation}
1047while the associated vertical skew-flux gives a variance change summed over
1048the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of
1049\begin{equation}
1050  \label{eq:TRIADS_dvar_eiv_k}
1051  _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
1052\end{equation}
1053Inspection of the definitions (\autoref{eq:TRIADS_skewfluxu}, \autoref{eq:TRIADS_skewfluxw}) shows that
1054these two variance changes (\autoref{eq:TRIADS_dvar_eiv_i}, \autoref{eq:TRIADS_dvar_eiv_k}) sum to zero.
1055Hence the two fluxes associated with each triad make no net contribution to the variance budget.
1056
1057%% =================================================================================================
1058\subsubsection{Reduction in gravitational PE}
1059
1060The vertical density flux associated with the vertical skew-flux always has the same sign as
1061the vertical density gradient;
1062thus, so long as the fluid is stable (the vertical density gradient is negative)
1063the vertical density flux is negative (downward) and hence reduces the gravitational PE.
1064
1065For the change in gravitational PE driven by the $k$-flux is
1066\begin{align}
1067  \label{eq:TRIADS_vert_densityPE}
1068  g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho)
1069  &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k
1070    {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k
1071    {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\
1072  \intertext{Substituting  ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:TRIADS_skewfluxw}, gives}
1073  % and separating out
1074  % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$,
1075  % gives two terms. The
1076  % first $\rtriad{R}$ term (the only term for $z$-coordinates) is:
1077  &=-\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}
1078    \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 \\
1079  &=+\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
1080    \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}}
1081    \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}},
1082\end{align}
1083using the definition of the triad slope $\rtriad{R}$, \autoref{eq:TRIADS_R} to
1084express $-\alpha _i^k\delta_{i+ i_p}[T^k]+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of
1085$-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$.
1086
1087Where the coordinates slope, the $i$-flux gives a PE change
1088\begin{multline}
1089  \label{eq:TRIADS_lat_densityPE}
1090  g \delta_{i+i_p}[z_T^k]
1091  \left[
1092    -\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)
1093  \right] \\
1094  = +\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
1095  \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
1096  \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)
1097  \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}},
1098\end{multline}
1099(using \autoref{eq:TRIADS_skewfluxu}) and so the total PE change \autoref{eq:TRIADS_vert_densityPE} +
1100\autoref{eq:TRIADS_lat_densityPE} associated with the triad fluxes is
1101\begin{multline*}
1102  % \label{eq:TRIADS_tot_densityPE}
1103  g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) +
1104  g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\
1105  = +\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
1106  \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
1107  \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}}.
1108\end{multline*}
1109Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+
1110\beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative.
1111
1112%% =================================================================================================
1113\subsection{Treatment of the triads at the boundaries}
1114\label{sec:TRIADS_skew_bdry}
1115
1116Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries
1117in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes,
1118as described in \autoref{sec:TRIADS_iso_bdry} and \autoref{fig:TRIADS_bdry_triads}.
1119Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked,
1120and 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
1121either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked.
1122The namelist parameter \np{ln_botmix_triad}{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes.
1123
1124%% =================================================================================================
1125\subsection{Limiting of the slopes within the interior}
1126\label{sec:TRIADS_limitskew}
1127
1128Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$,
1129exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:TRIADS_limit}.
1130Each individual triad \rtriadt{R} is so limited.
1131
1132%% =================================================================================================
1133\subsection{Tapering within the surface mixed layer}
1134\label{sec:TRIADS_taperskew}
1135
1136The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R})
1137are always tapered linearly from their value immediately below the mixed layer to zero at the surface
1138\autoref{eq:TRIADS_rmtilde}, as described in \autoref{sec:TRIADS_lintaper}.
1139This is option (c) of \autoref{fig:LDF_eiv_slp}.
1140This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by
1141the value of \np{ln_triad_iso}{ln\_triad\_iso}.
1142
1143The justification for this linear slope tapering is that, for $A_e$ that is constant or varies only in
1144the horizontal (the most commonly used options in \NEMO: see \autoref{sec:LDF_coef}),
1145it is equivalent to a horizontal eiv (eddy-induced velocity) that is uniform within the mixed layer
1146\autoref{eq:TRIADS_eiv_v}.
1147This ensures that the eiv velocities do not restratify the mixed layer \citep{treguier.held.ea_JPO97,danabasoglu.ferrari.ea_JC08}.
1148Equivantly, in terms of the skew-flux formulation we use here,
1149the linear slope tapering within the mixed-layer gives a linearly varying vertical flux,
1150and so a tracer convergence uniform in depth
1151(the horizontal flux convergence is relatively insignificant within the mixed-layer).
1152
1153%% =================================================================================================
1154\subsection{Streamfunction diagnostics}
1155\label{sec:TRIADS_sfdiag}
1156
1157Where the namelist parameter \np[=.true.]{ln_traldf_gdia}{ln\_traldf\_gdia},
1158diagnosed mean eddy-induced velocities are output.
1159Each time step, streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at
1160$uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ (integer $i$, integer +1/2 $j$, integer +1/2 $k$)
1161points (see Table \autoref{tab:DOM_cell}) respectively.
1162We follow \citep{griffies_bk04} and calculate the streamfunction at a given $uw$-point from
1163the surrounding four triads according to:
1164\[
1165  % \label{eq:TRIADS_sfdiagi}
1166  {\psi_1}_{i+1/2}^{k+1/2}={\fractext{1}{4}}\sum_{\substack{i_p,\,k_p}}
1167  {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}.
1168\]
1169The streamfunction $\psi_1$ is calculated similarly at $vw$ points.
1170The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:TRIADS_eiv_v}:
1171\[
1172  % \label{eq:TRIADS_eiv_v_discrete}
1173  \begin{split}
1174    {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right),   \\
1175    {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right),   \\
1176    {w^*}_{i,j}^{k+1/2} & =    \frac{1}{e_{1t}e_{2t}}\; \left\{
1177      {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} -
1178      {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\
1179    \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\},
1180  \end{split}
1181\]
1182
1183\onlyinsubfile{\input{../../global/epilogue}}
1184
1185\end{document}
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