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1\documentclass[../main/NEMO_manual]{subfiles}
2
3%% Custom aliases
4\newcommand{\cf}{\ensuremath{C\kern-0.14em f}}
5
6\begin{document}
7% ================================================================
8% Chapter  Vertical Ocean Physics (ZDF)
9% ================================================================
10\chapter{Vertical Ocean Physics (ZDF)}
11\label{chap:ZDF}
12
13\chaptertoc
14
15%gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN.
16
17\newpage
18
19% ================================================================
20% Vertical Mixing
21% ================================================================
22\section{Vertical mixing}
23\label{sec:ZDF}
24
25The discrete form of the ocean subgrid scale physics has been presented in
26\autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}.
27At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt have to be defined.
28At the surface they are prescribed from the surface forcing (see \autoref{chap:SBC}),
29while at the bottom they are set to zero for heat and salt,
30unless a geothermal flux forcing is prescribed as a bottom boundary condition (\ie\ \np{ln\_trabbc} defined,
31see \autoref{subsec:TRA_bbc}), and specified through a bottom friction parameterisation for momentum
32(see \autoref{sec:ZDF_drg}).
33
34In this section we briefly discuss the various choices offered to compute the vertical eddy viscosity and
35diffusivity coefficients, $A_u^{vm}$ , $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- points,
36respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}).
37These coefficients can be assumed to be either constant, or a function of the local Richardson number,
38or computed from a turbulent closure model (either TKE or GLS or OSMOSIS formulation).
39The computation of these coefficients is initialized in the \mdl{zdfphy} module and performed in
40the \mdl{zdfric}, \mdl{zdftke} or \mdl{zdfgls} or \mdl{zdfosm} modules.
41The trends due to the vertical momentum and tracer diffusion, including the surface forcing,
42are computed and added to the general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively.
43%These trends can be computed using either a forward time stepping scheme
44%(namelist parameter \np{ln\_zdfexp}\forcode{=.true.}) or a backward time stepping scheme
45%(\np{ln\_zdfexp}\forcode{=.false.}) depending on the magnitude of the mixing coefficients,
46%and thus of the formulation used (see \autoref{chap:TD}).
47
48%--------------------------------------------namzdf--------------------------------------------------------
49
50\begin{listing}
51  \nlst{namzdf}
52  \caption{\texttt{namzdf}}
53  \label{lst:namzdf}
54\end{listing}
55%--------------------------------------------------------------------------------------------------------------
56
57% -------------------------------------------------------------------------------------------------------------
58%        Constant
59% -------------------------------------------------------------------------------------------------------------
60\subsection[Constant (\forcode{ln_zdfcst=.true.})]
61{Constant (\protect\np{ln\_zdfcst}\forcode{=.true.})}
62\label{subsec:ZDF_cst}
63
64Options are defined through the \nam{zdf} namelist variables.
65When \np{ln\_zdfcst} is defined, the momentum and tracer vertical eddy coefficients are set to
66constant values over the whole ocean.
67This is the crudest way to define the vertical ocean physics.
68It is recommended to use this option only in process studies, not in basin scale simulations.
69Typical values used in this case are:
70\begin{align*}
71  A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}   \\
72  A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1}
73\end{align*}
74
75These values are set through the \np{rn\_avm0} and \np{rn\_avt0} namelist parameters.
76In all cases, do not use values smaller that those associated with the molecular viscosity and diffusivity,
77that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and
78$\sim10^{-9}~m^2.s^{-1}$ for salinity.
79
80% -------------------------------------------------------------------------------------------------------------
81%        Richardson Number Dependent
82% -------------------------------------------------------------------------------------------------------------
83\subsection[Richardson number dependent (\forcode{ln_zdfric=.true.})]
84{Richardson number dependent (\protect\np{ln\_zdfric}\forcode{=.true.})}
85\label{subsec:ZDF_ric}
86
87%--------------------------------------------namric---------------------------------------------------------
88
89\begin{listing}
90  \nlst{namzdf_ric}
91  \caption{\texttt{namzdf\_ric}}
92  \label{lst:namzdf_ric}
93\end{listing}
94%--------------------------------------------------------------------------------------------------------------
95
96When \np{ln\_zdfric}\forcode{=.true.}, a local Richardson number dependent formulation for the vertical momentum and
97tracer eddy coefficients is set through the \nam{zdf\_ric} namelist variables.
98The vertical mixing coefficients are diagnosed from the large scale variables computed by the model.
99\textit{In situ} measurements have been used to link vertical turbulent activity to large scale ocean structures.
100The hypothesis of a mixing mainly maintained by the growth of Kelvin-Helmholtz like instabilities leads to
101a dependency between the vertical eddy coefficients and the local Richardson number
102(\ie\ the ratio of stratification to vertical shear).
103Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented:
104\[
105  % \label{eq:ZDF_ric}
106  \left\{
107    \begin{aligned}
108      A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\
109      A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm}
110    \end{aligned}
111  \right.
112\]
113where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number,
114$N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
115$A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the constant case
116(see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ is the maximum value that
117can be reached by the coefficient when $Ri\leq 0$, $a=5$ and $n=2$.
118The last three values can be modified by setting the \np{rn\_avmri}, \np{rn\_alp} and
119\np{nn\_ric} namelist parameters, respectively.
120
121A simple mixing-layer model to transfer and dissipate the atmospheric forcings
122(wind-stress and buoyancy fluxes) can be activated setting the \np{ln\_mldw}\forcode{=.true.} in the namelist.
123
124In this case, the local depth of turbulent wind-mixing or "Ekman depth" $h_{e}(x,y,t)$ is evaluated and
125the vertical eddy coefficients prescribed within this layer.
126
127This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation:
128\[
129  h_{e} = Ek \frac {u^{*}} {f_{0}}
130\]
131where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter.
132
133In this similarity height relationship, the turbulent friction velocity:
134\[
135  u^{*} = \sqrt \frac {|\tau|} {\rho_o}
136\]
137is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$.
138The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}.
139Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to
140the empirical values \np{rn\_wtmix} and \np{rn\_wvmix} \citep{lermusiaux_JMS01}.
141
142% -------------------------------------------------------------------------------------------------------------
143%        TKE Turbulent Closure Scheme
144% -------------------------------------------------------------------------------------------------------------
145\subsection[TKE turbulent closure scheme (\forcode{ln_zdftke=.true.})]
146{TKE turbulent closure scheme (\protect\np{ln\_zdftke}\forcode{=.true.})}
147\label{subsec:ZDF_tke}
148%--------------------------------------------namzdf_tke--------------------------------------------------
149
150\begin{listing}
151  \nlst{namzdf_tke}
152  \caption{\texttt{namzdf\_tke}}
153  \label{lst:namzdf_tke}
154\end{listing}
155%--------------------------------------------------------------------------------------------------------------
156
157The vertical eddy viscosity and diffusivity coefficients are computed from a TKE turbulent closure model based on
158a prognostic equation for $\bar{e}$, the turbulent kinetic energy,
159and a closure assumption for the turbulent length scales.
160This turbulent closure model has been developed by \citet{bougeault.lacarrere_MWR89} in the atmospheric case,
161adapted by \citet{gaspar.gregoris.ea_JGR90} for the oceanic case, and embedded in OPA, the ancestor of \NEMO,
162by \citet{blanke.delecluse_JPO93} for equatorial Atlantic simulations.
163Since then, significant modifications have been introduced by \citet{madec.delecluse.ea_NPM98} in both the implementation and
164the formulation of the mixing length scale.
165The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear,
166its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type:
167\begin{equation}
168  \label{eq:ZDF_tke_e}
169  \frac{\partial \bar{e}}{\partial t} =
170  \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
171      +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
172  -K_\rho\,N^2
173  +\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
174      \;\frac{\partial \bar{e}}{\partial k}} \right]
175  - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
176\end{equation}
177\[
178  % \label{eq:ZDF_tke_kz}
179  \begin{split}
180    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }    \\
181    K_\rho &= A^{vm} / P_{rt}
182  \end{split}
183\]
184where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
185$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,
186$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity and diffusivity coefficients.
187The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$ $\approx 0.7$ are designed to deal with
188vertical mixing at any depth \citep{gaspar.gregoris.ea_JGR90}.
189They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}.
190$P_{rt}$ can be set to unity or, following \citet{blanke.delecluse_JPO93}, be a function of the local Richardson number, $R_i$:
191\begin{align*}
192  % \label{eq:ZDF_prt}
193  P_{rt} =
194  \begin{cases}
195    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$}   \\
196    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$}   \\
197    \ \ 10 &      \text{if $\ 2 \leq R_i$}
198  \end{cases}
199\end{align*}
200The choice of $P_{rt}$ is controlled by the \np{nn\_pdl} namelist variable.
201
202At the sea surface, the value of $\bar{e}$ is prescribed from the wind stress field as
203$\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb} namelist parameter.
204The default value of $e_{bb}$ is 3.75. \citep{gaspar.gregoris.ea_JGR90}), however a much larger value can be used when
205taking into account the surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}).
206The bottom value of TKE is assumed to be equal to the value of the level just above.
207The time integration of the $\bar{e}$ equation may formally lead to negative values because
208the numerical scheme does not ensure its positivity.
209To overcome this problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} namelist parameter).
210Following \citet{gaspar.gregoris.ea_JGR90}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$.
211This allows the subsequent formulations to match that of \citet{gargett_JMR84} for the diffusion in
212the thermocline and deep ocean :  $K_\rho = 10^{-3} / N$.
213In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical instabilities associated with
214too weak vertical diffusion.
215They must be specified at least larger than the molecular values, and are set through \np{rn\_avm0} and
216\np{rn\_avt0} (\nam{zdf} namelist, see \autoref{subsec:ZDF_cst}).
217
218\subsubsection{Turbulent length scale}
219
220For computational efficiency, the original formulation of the turbulent length scales proposed by
221\citet{gaspar.gregoris.ea_JGR90} has been simplified.
222Four formulations are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist parameter.
223The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}:
224\begin{equation}
225  \label{eq:ZDF_tke_mxl0_1}
226  l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N
227\end{equation}
228which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency.
229The resulting length scale is bounded by the distance to the surface or to the bottom
230(\np{nn\_mxl}\forcode{=0}) or by the local vertical scale factor (\np{nn\_mxl}\forcode{=1}).
231\citet{blanke.delecluse_JPO93} notice that this simplification has two major drawbacks:
232it makes no sense for locally unstable stratification and the computation no longer uses all
233the information contained in the vertical density profile.
234To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np{nn\_mxl}\forcode{=2, 3} cases,
235which add an extra assumption concerning the vertical gradient of the computed length scale.
236So, the length scales are first evaluated as in \autoref{eq:ZDF_tke_mxl0_1} and then bounded such that:
237\begin{equation}
238  \label{eq:ZDF_tke_mxl_constraint}
239  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
240  \qquad \text{with }\  l =  l_k = l_\epsilon
241\end{equation}
242\autoref{eq:ZDF_tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than
243the variations of depth.
244It provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less
245time consuming.
246In particular, it allows the length scale to be limited not only by the distance to the surface or
247to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as
248the thermocline (\autoref{fig:ZDF_mixing_length}).
249In order to impose the \autoref{eq:ZDF_tke_mxl_constraint} constraint, we introduce two additional length scales:
250$l_{up}$ and $l_{dwn}$, the upward and downward length scales, and
251evaluate the dissipation and mixing length scales as
252(and note that here we use numerical indexing):
253%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
254\begin{figure}[!t]
255  \centering
256  \includegraphics[width=0.66\textwidth]{Fig_mixing_length}
257  \caption[Mixing length computation]{Illustration of the mixing length computation}
258  \label{fig:ZDF_mixing_length}
259\end{figure}
260%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
261\[
262  % \label{eq:ZDF_tke_mxl2}
263  \begin{aligned}
264    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
265    \quad &\text{ from $k=1$ to $jpk$ }\ \\
266    l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)\right)
267    \quad &\text{ from $k=jpk$ to $1$ }\ \\
268  \end{aligned}
269\]
270where $l^{(k)}$ is computed using \autoref{eq:ZDF_tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
271
272In the \np{nn\_mxl}\forcode{=2} case, the dissipation and mixing length scales take the same value:
273$ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the \np{nn\_mxl}\forcode{=3} case,
274the dissipation and mixing turbulent length scales are give as in \citet{gaspar.gregoris.ea_JGR90}:
275\[
276  % \label{eq:ZDF_tke_mxl_gaspar}
277  \begin{aligned}
278    & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }   \\
279    & l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)
280  \end{aligned}
281\]
282
283At the ocean surface, a non zero length scale is set through the  \np{rn\_mxl0} namelist parameter.
284Usually the surface scale is given by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant and
285$z_o$ the roughness parameter of the surface.
286Assuming $z_o=0.1$~m \citep{craig.banner_JPO94} leads to a 0.04~m, the default value of \np{rn\_mxl0}.
287In the ocean interior a minimum length scale is set to recover the molecular viscosity when
288$\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).
289
290\subsubsection{Surface wave breaking parameterization}
291%-----------------------------------------------------------------------%
292
293Following \citet{mellor.blumberg_JPO04}, the TKE turbulence closure model has been modified to
294include the effect of surface wave breaking energetics.
295This results in a reduction of summertime surface temperature when the mixed layer is relatively shallow.
296The \citet{mellor.blumberg_JPO04} modifications acts on surface length scale and TKE values and
297air-sea drag coefficient.
298The latter concerns the bulk formulae and is not discussed here.
299
300Following \citet{craig.banner_JPO94}, the boundary condition on surface TKE value is :
301\begin{equation}
302  \label{eq:ZDF_Esbc}
303  \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
304\end{equation}
305where $\alpha_{CB}$ is the \citet{craig.banner_JPO94} constant of proportionality which depends on the ''wave age'',
306ranging from 57 for mature waves to 146 for younger waves \citep{mellor.blumberg_JPO04}.
307The boundary condition on the turbulent length scale follows the Charnock's relation:
308\begin{equation}
309  \label{eq:ZDF_Lsbc}
310  l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o}
311\end{equation}
312where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant.
313\citet{mellor.blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by
314\citet{stacey_JPO99} citing observation evidence, and
315$\alpha_{CB} = 100$ the Craig and Banner's value.
316As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$,
317with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}\forcode{ = 67.83} corresponds
318to $\alpha_{CB} = 100$.
319Further setting  \np{ln\_mxl0}\forcode{ =.true.},  applies \autoref{eq:ZDF_Lsbc} as the surface boundary condition on the length scale,
320with $\beta$ hard coded to the Stacey's value.
321Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) is applied on the
322surface $\bar{e}$ value.
323
324\subsubsection{Langmuir cells}
325%--------------------------------------%
326
327Langmuir circulations (LC) can be described as ordered large-scale vertical motions in
328the surface layer of the oceans.
329Although LC have nothing to do with convection, the circulation pattern is rather similar to
330so-called convective rolls in the atmospheric boundary layer.
331The detailed physics behind LC is described in, for example, \citet{craik.leibovich_JFM76}.
332The prevailing explanation is that LC arise from a nonlinear interaction between the Stokes drift and
333wind drift currents.
334
335Here we introduced in the TKE turbulent closure the simple parameterization of Langmuir circulations proposed by
336\citep{axell_JGR02} for a $k-\epsilon$ turbulent closure.
337The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in
338an extra source term of TKE, $P_{LC}$.
339The presence of $P_{LC}$ in \autoref{eq:ZDF_tke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to
340\forcode{.true.} in the \nam{zdf\_tke} namelist.
341
342By making an analogy with the characteristic convective velocity scale (\eg, \citet{dalessio.abdella.ea_JPO98}),
343$P_{LC}$ is assumed to be :
344\[
345P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}}
346\]
347where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth.
348With no information about the wave field, $w_{LC}$ is assumed to be proportional to
349the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module
350\footnote{Following \citet{li.garrett_JMR93}, the surface Stoke drift velocity may be expressed as
351  $u_s =  0.016 \,|U_{10m}|$.
352  Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of
353  $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress
354}.
355For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at
356a finite depth $H_{LC}$ (which is often close to the mixed layer depth),
357and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures).
358The resulting expression for $w_{LC}$ is :
359\[
360  w_{LC}  =
361  \begin{cases}
362    c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$}    \\
363    0                             &      \text{otherwise}
364  \end{cases}
365\]
366where $c_{LC} = 0.15$ has been chosen by \citep{axell_JGR02} as a good compromise to fit LES data.
367The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second.
368The value of $c_{LC}$ is set through the \np{rn\_lc} namelist parameter,
369having in mind that it should stay between 0.15 and 0.54 \citep{axell_JGR02}.
370
371The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations:
372$H_{LC}$ is the depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by
373converting its kinetic energy to potential energy, according to
374\[
375- \int_{-H_{LC}}^0 { N^2\;\;dz} = \frac{1}{2} u_s^2
376\]
377
378\subsubsection{Mixing just below the mixed layer}
379%--------------------------------------------------------------%
380
381Vertical mixing parameterizations commonly used in ocean general circulation models tend to
382produce mixed-layer depths that are too shallow during summer months and windy conditions.
383This bias is particularly acute over the Southern Ocean.
384To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{rodgers.aumont.ea_B14}.
385The parameterization is an empirical one, \ie\ not derived from theoretical considerations,
386but rather is meant to account for observed processes that affect the density structure of
387the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme
388(\ie\ near-inertial oscillations and ocean swells and waves).
389
390When using this parameterization (\ie\ when \np{nn\_etau}\forcode{=1}),
391the TKE input to the ocean ($S$) imposed by the winds in the form of near-inertial oscillations,
392swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition,
393plus a depth depend one given by:
394\begin{equation}
395  \label{eq:ZDF_Ehtau}
396  S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau}
397\end{equation}
398where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that
399penetrates in the ocean, $h_\tau$ is a vertical mixing length scale that controls exponential shape of
400the penetration, and $f_i$ is the ice concentration
401(no penetration if $f_i=1$, \ie\ if the ocean is entirely covered by sea-ice).
402The value of $f_r$, usually a few percents, is specified through \np{rn\_efr} namelist parameter.
403The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np{nn\_etau}\forcode{=0}) or
404a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m at high latitudes
405(\np{nn\_etau}\forcode{=1}).
406
407Note that two other option exist, \np{nn\_etau}\forcode{=2, 3}.
408They correspond to applying \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer,
409or to using the high frequency part of the stress to evaluate the fraction of TKE that penetrates the ocean.
410Those two options are obsolescent features introduced for test purposes.
411They will be removed in the next release.
412
413% This should be explain better below what this rn_eice parameter is meant for:
414In presence of Sea Ice, the value of this mixing can be modulated by the \np{rn\_eice} namelist parameter.
415This parameter varies from \forcode{0} for no effect to \forcode{4} to suppress the TKE input into the ocean when Sea Ice concentration
416is greater than 25\%.
417
418% from Burchard et al OM 2008 :
419% the most critical process not reproduced by statistical turbulence models is the activity of
420% internal waves and their interaction with turbulence. After the Reynolds decomposition,
421% internal waves are in principle included in the RANS equations, but later partially
422% excluded by the hydrostatic assumption and the model resolution.
423% Thus far, the representation of internal wave mixing in ocean models has been relatively crude
424% (\eg\ Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002).
425
426% -------------------------------------------------------------------------------------------------------------
427%        GLS Generic Length Scale Scheme
428% -------------------------------------------------------------------------------------------------------------
429\subsection[GLS: Generic Length Scale (\forcode{ln_zdfgls=.true.})]
430{GLS: Generic Length Scale (\protect\np{ln\_zdfgls}\forcode{=.true.})}
431\label{subsec:ZDF_gls}
432
433%--------------------------------------------namzdf_gls---------------------------------------------------------
434
435\begin{listing}
436  \nlst{namzdf_gls}
437  \caption{\texttt{namzdf\_gls}}
438  \label{lst:namzdf_gls}
439\end{listing}
440%--------------------------------------------------------------------------------------------------------------
441
442The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on two prognostic equations:
443one for the turbulent kinetic energy $\bar {e}$, and another for the generic length scale,
444$\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}.
445This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,
446where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:ZDF_GLS} allows to recover a number of
447well-known turbulent closures ($k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87},
448$k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}).
449The GLS scheme is given by the following set of equations:
450\begin{equation}
451  \label{eq:ZDF_gls_e}
452  \frac{\partial \bar{e}}{\partial t} =
453  \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
454      +\left( \frac{\partial v}{\partial k} \right)^2} \right]
455  -K_\rho \,N^2
456  +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right]
457  - \epsilon
458\end{equation}
459
460\[
461  % \label{eq:ZDF_gls_psi}
462  \begin{split}
463    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{
464      \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
465          +\left( \frac{\partial v}{\partial k} \right)^2} \right]
466      - C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\
467    &+\frac{1}{e_3\;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 }
468        \;\frac{\partial \psi}{\partial k}} \right]\;
469  \end{split}
470\]
471
472\[
473  % \label{eq:ZDF_gls_kz}
474  \begin{split}
475    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
476    K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l
477  \end{split}
478\]
479
480\[
481  % \label{eq:ZDF_gls_eps}
482  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
483\]
484where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and
485$\epsilon$ the dissipation rate.
486The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of
487the choice of the turbulence model.
488Four different turbulent models are pre-defined (\autoref{tab:ZDF_GLS}).
489They are made available through the \np{nn\_clo} namelist parameter.
490
491%--------------------------------------------------TABLE--------------------------------------------------
492\begin{table}[htbp]
493  \centering
494  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c}
495  \begin{tabular}{ccccc}
496    &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\
497    % & \citep{mellor.yamada_RG82} &  \citep{rodi_JGR87}       & \citep{wilcox_AJ88} &                 \\
498    \hline
499    \hline
500    \np{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\
501    \hline
502    $( p , n , m )$         &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\
503    $\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\
504    $\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\
505    $C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\
506    $C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\
507    $C_3$              &      1.           &     1.              &      1.                &       1.           \\
508    $F_{wall}$        &      Yes        &       --             &     --                  &      --          \\
509    \hline
510    \hline
511  \end{tabular}
512  \caption[Set of predefined GLS parameters or equivalently predefined turbulence models available]{
513    Set of predefined GLS parameters, or equivalently predefined turbulence models available with
514    \protect\np{ln\_zdfgls}\forcode{=.true.} and controlled by
515    the \protect\np{nn\_clos} namelist variable in \protect\nam{zdf\_gls}.}
516  \label{tab:ZDF_GLS}
517\end{table}
518%--------------------------------------------------------------------------------------------------------------
519
520In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force the convergence of
521the mixing length towards $\kappa z_b$ ($\kappa$ is the Von Karman constant and $z_b$ the rugosity length scale) value near physical boundaries
522(logarithmic boundary layer law).
523$C_{\mu}$ and $C_{\mu'}$ are calculated from stability function proposed by \citet{galperin.kantha.ea_JAS88},
524or by \citet{kantha.clayson_JGR94} or one of the two functions suggested by \citet{canuto.howard.ea_JPO01}
525(\np{nn\_stab\_func}\forcode{=0, 3}, resp.).
526The value of $C_{0\mu}$ depends on the choice of the stability function.
527
528The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated thanks to Dirichlet or
529Neumann condition through \np{nn\_bc\_surf} and \np{nn\_bc\_bot}, resp.
530As for TKE closure, the wave effect on the mixing is considered when
531\np{rn\_crban}\forcode{ > 0.} \citep{craig.banner_JPO94, mellor.blumberg_JPO04}.
532The \np{rn\_crban} namelist parameter is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and
533\np{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}.
534
535The $\psi$ equation is known to fail in stably stratified flows, and for this reason
536almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.
537With this clipping, the maximum permissible length scale is determined by $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$.
538A value of $c_{lim} = 0.53$ is often used \citep{galperin.kantha.ea_JAS88}.
539\cite{umlauf.burchard_CSR05} show that the value of the clipping factor is of crucial importance for
540the entrainment depth predicted in stably stratified situations,
541and that its value has to be chosen in accordance with the algebraic model for the turbulent fluxes.
542The clipping is only activated if \np{ln\_length\_lim}\forcode{=.true.},
543and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value.
544
545The time and space discretization of the GLS equations follows the same energetic consideration as for
546the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{burchard_OM02}.
547Evaluation of the 4 GLS turbulent closure schemes can be found in \citet{warner.sherwood.ea_OM05} in ROMS model and
548 in \citet{reffray.guillaume.ea_GMD15} for the \NEMO\ model.
549
550
551% -------------------------------------------------------------------------------------------------------------
552%        OSM OSMOSIS BL Scheme
553% -------------------------------------------------------------------------------------------------------------
554\subsection[OSM: OSMosis boundary layer scheme (\forcode{ln_zdfosm=.true.})]
555{OSM: OSMosis boundary layer scheme (\protect\np{ln\_zdfosm}\forcode{=.true.})}
556\label{subsec:ZDF_osm}
557%--------------------------------------------namzdf_osm---------------------------------------------------------
558
559\begin{listing}
560  \nlst{namzdf_osm}
561  \caption{\texttt{namzdf\_osm}}
562  \label{lst:namzdf_osm}
563\end{listing}
564%--------------------------------------------------------------------------------------------------------------
565
566The OSMOSIS turbulent closure scheme is based on......   TBC
567
568% -------------------------------------------------------------------------------------------------------------
569%        TKE and GLS discretization considerations
570% -------------------------------------------------------------------------------------------------------------
571\subsection[ Discrete energy conservation for TKE and GLS schemes]
572{Discrete energy conservation for TKE and GLS schemes}
573\label{subsec:ZDF_tke_ene}
574
575%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
576\begin{figure}[!t]
577  \centering
578  \includegraphics[width=0.66\textwidth]{Fig_ZDF_TKE_time_scheme}
579  \caption[Subgrid kinetic energy integration in GLS and TKE schemes]{
580    Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and
581    its links to the momentum and tracer time integration.}
582  \label{fig:ZDF_TKE_time_scheme}
583\end{figure}
584%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
585
586The production of turbulence by vertical shear (the first term of the right hand side of
587\autoref{eq:ZDF_tke_e}) and  \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion
588(first line in \autoref{eq:MB_zdf}).
589To do so a special care has to be taken for both the time and space discretization of
590the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}.
591
592Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how
593the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with
594the one-level forward time stepping of the equation for $\bar{e}$.
595With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to
596the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and
597summing the result vertically:
598\begin{equation}
599  \label{eq:ZDF_energ1}
600  \begin{split}
601    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\
602    &= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}
603    - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz }
604  \end{split}
605\end{equation}
606Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$,
607known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme
608(see \autoref{sec:TD_forward_imp}).
609The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at
610the surface (atmospheric forcing) and at the bottom (friction effect).
611The second term is always negative.
612It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$.
613\autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
614the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
615${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$
616(and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
617
618A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
619(second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}).
620This term must balance the input of potential energy resulting from vertical mixing.
621The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by
622multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically:
623\begin{equation}
624  \label{eq:ZDF_energ2}
625  \begin{split}
626    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\
627    &= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}
628    - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\
629    &= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta}
630    + \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  }
631  \end{split}
632\end{equation}
633where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.
634The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because
635there is no diffusive flux through the ocean surface and bottom).
636The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
637Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
638the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and  \autoref{eq:ZDF_gls_e}.
639
640Let us now address the space discretization issue.
641The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in
642the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}).
643A space averaging is thus required to obtain the shear TKE production term.
644By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by
645the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
646Furthermore, the time variation of $e_3$ has be taken into account.
647
648The above energetic considerations leads to the following final discrete form for the TKE equation:
649\begin{equation}
650  \label{eq:ZDF_tke_ene}
651  \begin{split}
652    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv
653    \Biggl\{ \Biggr.
654    &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} }
655        \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\
656    +&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} }
657        \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j}
658    \Biggr. \Biggr\}   \\
659    %
660    - &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\
661    %
662    +&\frac{1}{{e_3w}^{t+\rdt}}  \;\delta_{k+1/2} \left[   {K_m}^{t-\rdt} \,\frac{\delta_{k}[(\bar{e})^{t+\rdt}]} {{e_3w}^{t+\rdt}}   \right]   \\
663    %
664    - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt}
665  \end{split}
666\end{equation}
667where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation)
668are time stepped using a backward scheme (see\autoref{sec:TD_forward_imp}).
669Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible.
670%The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as
671%they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}.
672%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.
673
674% ================================================================
675% Convection
676% ================================================================
677\section{Convection}
678\label{sec:ZDF_conv}
679
680Static instabilities (\ie\ light potential densities under heavy ones) may occur at particular ocean grid points.
681In nature, convective processes quickly re-establish the static stability of the water column.
682These processes have been removed from the model via the hydrostatic assumption so they must be parameterized.
683Three parameterisations are available to deal with convective processes:
684a non-penetrative convective adjustment or an enhanced vertical diffusion,
685or/and the use of a turbulent closure scheme.
686
687% -------------------------------------------------------------------------------------------------------------
688%       Non-Penetrative Convective Adjustment
689% -------------------------------------------------------------------------------------------------------------
690\subsection[Non-penetrative convective adjustment (\forcode{ln_tranpc=.true.})]
691{Non-penetrative convective adjustment (\protect\np{ln\_tranpc}\forcode{=.true.})}
692\label{subsec:ZDF_npc}
693
694%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
695\begin{figure}[!htb]
696  \centering
697  \includegraphics[width=0.66\textwidth]{Fig_npc}
698  \caption[Unstable density profile treated by the non penetrative convective adjustment algorithm]{
699    Example of an unstable density profile treated by
700    the non penetrative convective adjustment algorithm.
701    $1^{st}$ step: the initial profile is checked from the surface to the bottom.
702    It is found to be unstable between levels 3 and 4.
703    They are mixed.
704    The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
705    The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
706    The $1^{st}$ step ends since the density profile is then stable below the level 3.
707    $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
708    levels 2 to 5 are mixed.
709    The new density profile is checked.
710    It is found stable: end of algorithm.}
711  \label{fig:ZDF_npc}
712\end{figure}
713%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
714
715Options are defined through the \nam{zdf} namelist variables.
716The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}\forcode{=.true.}.
717It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of
718the water column, but only until the density structure becomes neutrally stable
719(\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below)
720\citep{madec.delecluse.ea_JPO91}.
721The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}):
722starting from the top of the ocean, the first instability is found.
723Assume in the following that the instability is located between levels $k$ and $k+1$.
724The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of
725the water column.
726The new density is then computed by a linear approximation.
727If the new density profile is still unstable between levels $k+1$ and $k+2$,
728levels $k$, $k+1$ and $k+2$ are then mixed.
729This process is repeated until stability is established below the level $k$
730(the mixing process can go down to the ocean bottom).
731The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or
732if there is no deeper instability.
733
734This algorithm is significantly different from mixing statically unstable levels two by two.
735The latter procedure cannot converge with a finite number of iterations for some vertical profiles while
736the algorithm used in \NEMO\ converges for any profile in a number of iterations which is less than
737the number of vertical levels.
738This property is of paramount importance as pointed out by \citet{killworth_iprc89}:
739it avoids the existence of permanent and unrealistic static instabilities at the sea surface.
740This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in
741the north-western Mediterranean Sea \citep{madec.delecluse.ea_JPO91, madec.chartier.ea_DAO91, madec.crepon_iprc91}.
742
743The current implementation has been modified in order to deal with any non linear equation of seawater
744(L. Brodeau, personnal communication).
745Two main differences have been introduced compared to the original algorithm:
746$(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency
747(not the difference in potential density);
748$(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in
749the same way their temperature and salinity has been mixed.
750These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without
751having to recompute the expansion coefficients at each mixing iteration.
752
753% -------------------------------------------------------------------------------------------------------------
754%       Enhanced Vertical Diffusion
755% -------------------------------------------------------------------------------------------------------------
756\subsection[Enhanced vertical diffusion (\forcode{ln_zdfevd=.true.})]
757{Enhanced vertical diffusion (\protect\np{ln\_zdfevd}\forcode{=.true.})}
758\label{subsec:ZDF_evd}
759
760Options are defined through the  \nam{zdf} namelist variables.
761The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}\forcode{=.true.}.
762In this case, the vertical eddy mixing coefficients are assigned very large values
763in regions where the stratification is unstable
764(\ie\ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{lazar_phd97, lazar.madec.ea_JPO99}.
765This is done either on tracers only (\np{nn\_evdm}\forcode{=0}) or
766on both momentum and tracers (\np{nn\_evdm}\forcode{=1}).
767
768In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np{nn\_evdm}\forcode{=1},
769the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to
770the namelist parameter \np{rn\_avevd}.
771A typical value for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$.
772This parameterisation of convective processes is less time consuming than
773the convective adjustment algorithm presented above when mixing both tracers and
774momentum in the case of static instabilities.
775
776Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$.
777This removes a potential source of divergence of odd and even time step in
778a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}).
779
780% -------------------------------------------------------------------------------------------------------------
781%       Turbulent Closure Scheme
782% -------------------------------------------------------------------------------------------------------------
783\subsection{Handling convection with turbulent closure schemes (\forcode{ln_zdf{tke,gls,osm}=.true.})}
784\label{subsec:ZDF_tcs}
785
786
787The turbulent closure schemes presented in \autoref{subsec:ZDF_tke}, \autoref{subsec:ZDF_gls} and
788\autoref{subsec:ZDF_osm} (\ie\ \np{ln\_zdftke} or \np{ln\_zdfgls} or \np{ln\_zdfosm} defined) deal, in theory,
789with statically unstable density profiles.
790In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in
791\autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative.
792It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also of the four neighboring values at
793velocity points $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1}$).
794These large values restore the static stability of the water column in a way similar to that of
795the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}).
796However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by
797the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$,
798because the mixing length scale is bounded by the distance to the sea surface.
799It can thus be useful to combine the enhanced vertical diffusion with the turbulent closure scheme,
800\ie\ setting the \np{ln\_zdfnpc} namelist parameter to true and
801defining the turbulent closure (\np{ln\_zdftke} or \np{ln\_zdfgls} = \forcode{.true.}) all together.
802
803The OSMOSIS turbulent closure scheme already includes enhanced vertical diffusion in the case of convection,
804%as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp},
805therefore \np{ln\_zdfevd}\forcode{=.false.} should be used with the OSMOSIS scheme.
806% gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
807
808% ================================================================
809% Double Diffusion Mixing
810% ================================================================
811\section[Double diffusion mixing (\forcode{ln_zdfddm=.true.})]
812{Double diffusion mixing (\protect\np{ln\_zdfddm}\forcode{=.true.})}
813\label{subsec:ZDF_ddm}
814
815
816%-------------------------------------------namzdf_ddm-------------------------------------------------
817%
818%\nlst{namzdf_ddm}
819%--------------------------------------------------------------------------------------------------------------
820
821This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the namelist parameter
822\np{ln\_zdfddm} in \nam{zdf}.
823Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa.
824The former condition leads to salt fingering and the latter to diffusive convection.
825Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean.
826\citet{merryfield.holloway.ea_JPO99} include a parameterisation of such phenomena in a global ocean model and show that
827it leads to relatively minor changes in circulation but exerts significant regional influences on
828temperature and salinity.
829
830
831Diapycnal mixing of S and T are described by diapycnal diffusion coefficients
832\begin{align*}
833  % \label{eq:ZDF_ddm_Kz}
834  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\
835  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
836\end{align*}
837where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
838and $o$ by processes other than double diffusion.
839The rates of double-diffusive mixing depend on the buoyancy ratio
840$R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of
841thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}).
842To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt
843(1981):
844\begin{align}
845  \label{eq:ZDF_ddm_f}
846  A_f^{vS} &=
847             \begin{cases}
848               \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
849               0                              &\text{otherwise}
850             \end{cases}
851  \\         \label{eq:ZDF_ddm_f_T}
852  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho
853\end{align}
854
855%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
856\begin{figure}[!t]
857  \centering
858  \includegraphics[width=0.66\textwidth]{Fig_zdfddm}
859  \caption[Diapycnal diffusivities for temperature and salt in regions of salt fingering and
860  diffusive convection]{
861    From \citet{merryfield.holloway.ea_JPO99}:
862    (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in
863    regions of salt fingering.
864    Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and
865    thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
866    (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in
867    regions of diffusive convection.
868    Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
869    The latter is not implemented in \NEMO.}
870  \label{fig:ZDF_ddm}
871\end{figure}
872%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
873
874The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of
875buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}).
876Following  \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
877
878To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by
879Federov (1988) is used:
880\begin{align}
881  % \label{eq:ZDF_ddm_d}
882  A_d^{vT} &=
883             \begin{cases}
884               1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
885               &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
886               0                       &\text{otherwise}
887             \end{cases}
888                                       \nonumber \\
889  \label{eq:ZDF_ddm_d_S}
890  A_d^{vS} &=
891             \begin{cases}
892               A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
893               A_d^{vT} \ 0.15 \ R_\rho               &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
894               0                       &\text{otherwise}
895             \end{cases}
896\end{align}
897
898The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in
899\autoref{fig:ZDF_ddm}.
900Implementing this requires computing $R_\rho$ at each grid point on every time step.
901This is done in \mdl{eosbn2} at the same time as $N^2$ is computed.
902This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive).
903
904% ================================================================
905% Bottom Friction
906% ================================================================
907 \section[Bottom and top friction (\textit{zdfdrg.F90})]
908 {Bottom and top friction (\protect\mdl{zdfdrg})}
909 \label{sec:ZDF_drg}
910
911%--------------------------------------------namdrg--------------------------------------------------------
912%
913\begin{listing}
914  \nlst{namdrg}
915  \caption{\texttt{namdrg}}
916  \label{lst:namdrg}
917\end{listing}
918\begin{listing}
919  \nlst{namdrg_top}
920  \caption{\texttt{namdrg\_top}}
921  \label{lst:namdrg_top}
922\end{listing}
923\begin{listing}
924  \nlst{namdrg_bot}
925  \caption{\texttt{namdrg\_bot}}
926  \label{lst:namdrg_bot}
927\end{listing}
928
929%--------------------------------------------------------------------------------------------------------------
930
931Options to define the top and bottom friction are defined through the \nam{drg} namelist variables.
932The bottom friction represents the friction generated by the bathymetry.
933The top friction represents the friction generated by the ice shelf/ocean interface.
934As the friction processes at the top and the bottom are treated in and identical way,
935the description below considers mostly the bottom friction case, if not stated otherwise.
936
937
938Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as
939a condition on the vertical diffusive flux.
940For the bottom boundary layer, one has:
941 \[
942   % \label{eq:ZDF_bfr_flux}
943   A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
944 \]
945where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside
946the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean).
947How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near
948the bottom relative to the Ekman layer depth.
949For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m,
950one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$
951(for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$).
952With a background diffusion coefficient $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
953When the vertical mixing coefficient is this small, using a flux condition is equivalent to
954entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or
955bottom model layer.
956To illustrate this, consider the equation for $u$ at $k$, the last ocean level:
957\begin{equation}
958  \label{eq:ZDF_drg_flux2}
959  \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}}
960\end{equation}
961If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth.
962On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used,
963the turbulent Ekman layer will be represented explicitly by the model.
964However, the logarithmic layer is never represented in current primitive equation model applications:
965it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
966Two choices are available in \NEMO: a linear and a quadratic bottom friction.
967Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in
968the present release of \NEMO.
969
970In the code, the bottom friction is imposed by adding the trend due to the bottom friction to
971 the general momentum trend in \mdl{dynzdf}.
972For the time-split surface pressure gradient algorithm, the momentum trend due to
973the barotropic component needs to be handled separately.
974For this purpose it is convenient to compute and store coefficients which can be simply combined with
975bottom velocities and geometric values to provide the momentum trend due to bottom friction.
976 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where:
977\begin{equation}
978  \label{eq:ZDF_bfr_bdef}
979  \frac{\partial {\textbf U_h}}{\partial t} =
980  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b
981\end{equation}
982where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity.
983Note than from \NEMO\ 4.0, drag coefficients are only computed at cell centers (\ie\ at T-points) and refer to as $c_b^T$ in the following. These are then linearly interpolated in space to get $c_b^\textbf{U}$ at velocity points.
984
985% -------------------------------------------------------------------------------------------------------------
986%       Linear Bottom Friction
987% -------------------------------------------------------------------------------------------------------------
988 \subsection[Linear top/bottom friction (\forcode{ln_lin=.true.})]
989 {Linear top/bottom friction (\protect\np{ln\_lin}\forcode{=.true.)}}
990 \label{subsec:ZDF_drg_linear}
991
992The linear friction parameterisation (including the special case of a free-slip condition) assumes that
993the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level):
994\[
995  % \label{eq:ZDF_bfr_linear}
996  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
997\]
998where $r$ is a friction coefficient expressed in $m s^{-1}$.
999This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean,
1000and setting $r = H / \tau$, where $H$ is the ocean depth.
1001Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{weatherly_JMR84}.
1002A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models.
1003One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$
1004(\citet{gill_bk82}, Eq. 9.6.6).
1005For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$,
1006and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.
1007This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days.
1008It can be changed by specifying \np{rn\_Uc0} (namelist parameter).
1009
1010 For the linear friction case the drag coefficient used in the general expression \autoref{eq:ZDF_bfr_bdef} is:
1011\[
1012  % \label{eq:ZDF_bfr_linbfr_b}
1013    c_b^T = - r
1014\]
1015When \np{ln\_lin} \forcode{= .true.}, the value of $r$ used is \np{rn\_Uc0}*\np{rn\_Cd0}.
1016Setting \np{ln\_OFF} \forcode{= .true.} (and \forcode{ln_lin=.true.}) is equivalent to setting $r=0$ and leads to a free-slip boundary condition.
1017
1018These values are assigned in \mdl{zdfdrg}.
1019Note that there is support for local enhancement of these values via an externally defined 2D mask array
1020(\np{ln\_boost}\forcode{=.true.}) given in the \ifile{bfr\_coef} input NetCDF file.
1021The mask values should vary from 0 to 1.
1022Locations with a non-zero mask value will have the friction coefficient increased by
1023$mask\_value$ * \np{rn\_boost} * \np{rn\_Cd0}.
1024
1025% -------------------------------------------------------------------------------------------------------------
1026%       Non-Linear Bottom Friction
1027% -------------------------------------------------------------------------------------------------------------
1028 \subsection[Non-linear top/bottom friction (\forcode{ln_non_lin=.true.})]
1029 {Non-linear top/bottom friction (\protect\np{ln\_non\_lin}\forcode{=.true.})}
1030 \label{subsec:ZDF_drg_nonlinear}
1031
1032The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic:
1033\[
1034  % \label{eq:ZDF_drg_nonlinear}
1035  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
1036  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
1037\]
1038where $C_D$ is a drag coefficient, and $e_b $ a top/bottom turbulent kinetic energy due to tides,
1039internal waves breaking and other short time scale currents.
1040A typical value of the drag coefficient is $C_D = 10^{-3} $.
1041As an example, the CME experiment \citep{treguier_JGR92} uses $C_D = 10^{-3}$ and
1042$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{killworth_JPO92} uses $C_D = 1.4\;10^{-3}$ and
1043$e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.
1044The CME choices have been set as default values (\np{rn\_Cd0} and \np{rn\_ke0} namelist parameters).
1045
1046As for the linear case, the friction is imposed in the code by adding the trend due to
1047the friction to the general momentum trend in \mdl{dynzdf}.
1048For the non-linear friction case the term computed in \mdl{zdfdrg} is:
1049\[
1050  % \label{eq:ZDF_drg_nonlinbfr}
1051    c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2}
1052\]
1053
1054The coefficients that control the strength of the non-linear friction are initialised as namelist parameters:
1055$C_D$= \np{rn\_Cd0}, and $e_b$ =\np{rn\_bfeb2}.
1056Note that for applications which consider tides explicitly, a low or even zero value of \np{rn\_bfeb2} is recommended. A local enhancement of $C_D$ is again possible via an externally defined 2D mask array
1057(\np{ln\_boost}\forcode{=.true.}).
1058This works in the same way as for the linear friction case with non-zero masked locations increased by
1059$mask\_value$ * \np{rn\_boost} * \np{rn\_Cd0}.
1060
1061% -------------------------------------------------------------------------------------------------------------
1062%       Bottom Friction Log-layer
1063% -------------------------------------------------------------------------------------------------------------
1064 \subsection[Log-layer top/bottom friction (\forcode{ln_loglayer=.true.})]
1065 {Log-layer top/bottom friction (\protect\np{ln\_loglayer}\forcode{=.true.})}
1066 \label{subsec:ZDF_drg_loglayer}
1067
1068In the non-linear friction case, the drag coefficient, $C_D$, can be optionally enhanced using
1069a "law of the wall" scaling. This assumes that the model vertical resolution can capture the logarithmic layer which typically occur for layers thinner than 1 m or so.
1070If  \np{ln\_loglayer} \forcode{= .true.}, $C_D$ is no longer constant but is related to the distance to the wall (or equivalently to the half of the top/bottom layer thickness):
1071\[
1072  C_D = \left ( {\kappa \over {\mathrm log}\left ( 0.5 \; e_{3b} / rn\_{z0} \right ) } \right )^2
1073\]
1074
1075\noindent where $\kappa$ is the von-Karman constant and \np{rn\_z0} is a roughness length provided via the namelist.
1076
1077The drag coefficient is bounded such that it is kept greater or equal to
1078the base \np{rn\_Cd0} value which occurs where layer thicknesses become large and presumably logarithmic layers are not resolved at all. For stability reason, it is also not allowed to exceed the value of an additional namelist parameter:
1079\np{rn\_Cdmax}, \ie
1080\[
1081  rn\_Cd0 \leq C_D \leq rn\_Cdmax
1082\]
1083
1084\noindent The log-layer enhancement can also be applied to the top boundary friction if
1085under ice-shelf cavities are activated (\np{ln\_isfcav}\forcode{=.true.}).
1086%In this case, the relevant namelist parameters are \np{rn\_tfrz0}, \np{rn\_tfri2} and \np{rn\_tfri2\_max}.
1087
1088% -------------------------------------------------------------------------------------------------------------
1089%       Explicit bottom Friction
1090% -------------------------------------------------------------------------------------------------------------
1091 \subsection{Explicit top/bottom friction (\forcode{ln_drgimp=.false.})}
1092 \label{subsec:ZDF_drg_stability}
1093
1094Setting \np{ln\_drgimp} \forcode{= .false.} means that bottom friction is treated explicitly in time, which has the advantage of simplifying the interaction with the split-explicit free surface (see \autoref{subsec:ZDF_drg_ts}). The latter does indeed require the knowledge of bottom stresses in the course of the barotropic sub-iteration, which becomes less straightforward in the implicit case. In the explicit case, top/bottom stresses can be computed using \textit{before} velocities and inserted in the overall momentum tendency budget. This reads:
1095
1096At the top (below an ice shelf cavity):
1097\[
1098  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1099  = c_{t}^{\textbf{U}}\textbf{u}^{n-1}_{t}
1100\]
1101
1102At the bottom (above the sea floor):
1103\[
1104  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1105  = c_{b}^{\textbf{U}}\textbf{u}^{n-1}_{b}
1106\]
1107
1108Since this is conditionally stable, some care needs to exercised over the choice of parameters to ensure that the implementation of explicit top/bottom friction does not induce numerical instability.
1109For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is:
1110\begin{equation}
1111  \label{eq:ZDF_Eqn_drgstab}
1112  \begin{split}
1113    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
1114    &= -\frac{ru}{e_{3u}}\;2\rdt\\
1115  \end{split}
1116\end{equation}
1117\noindent where linear friction and a leapfrog timestep have been assumed.
1118To ensure that the friction cannot reverse the direction of flow it is necessary to have:
1119\[
1120  |\Delta u| < \;|u|
1121\]
1122\noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives:
1123\[
1124  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
1125\]
1126This same inequality can also be derived in the non-linear bottom friction case if
1127a velocity of 1 m.s$^{-1}$ is assumed.
1128Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability:
1129\[
1130  e_{3u} > 2\;r\;\rdt
1131\]
1132\noindent which it may be necessary to impose if partial steps are being used.
1133For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m.
1134For most applications, with physically sensible parameters these restrictions should not be of concern.
1135But caution may be necessary if attempts are made to locally enhance the bottom friction parameters.
1136To ensure stability limits are imposed on the top/bottom friction coefficients both
1137during initialisation and at each time step.
1138Checks at initialisation are made in \mdl{zdfdrg} (assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
1139The number of breaches of the stability criterion are reported as well as
1140the minimum and maximum values that have been set.
1141The criterion is also checked at each time step, using the actual velocity, in \mdl{dynzdf}.
1142Values of the friction coefficient are reduced as necessary to ensure stability;
1143these changes are not reported.
1144
1145Limits on the top/bottom friction coefficient are not imposed if the user has elected to
1146handle the friction implicitly (see \autoref{subsec:ZDF_drg_imp}).
1147The number of potential breaches of the explicit stability criterion are still reported for information purposes.
1148
1149% -------------------------------------------------------------------------------------------------------------
1150%       Implicit Bottom Friction
1151% -------------------------------------------------------------------------------------------------------------
1152 \subsection[Implicit top/bottom friction (\forcode{ln_drgimp=.true.})]
1153 {Implicit top/bottom friction (\protect\np{ln\_drgimp}\forcode{=.true.})}
1154 \label{subsec:ZDF_drg_imp}
1155
1156An optional implicit form of bottom friction has been implemented to improve model stability.
1157We recommend this option for shelf sea and coastal ocean applications. %, especially for split-explicit time splitting.
1158This option can be invoked by setting \np{ln\_drgimp} to \forcode{.true.} in the \nam{drg} namelist.
1159%This option requires \np{ln\_zdfexp} to be \forcode{.false.} in the \nam{zdf} namelist.
1160
1161This implementation is performed in \mdl{dynzdf} where the following boundary conditions are set while solving the fully implicit diffusion step:
1162
1163At the top (below an ice shelf cavity):
1164\[
1165  % \label{eq:ZDF_dynZDF__drg_top}
1166  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1167  = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t}
1168\]
1169
1170At the bottom (above the sea floor):
1171\[
1172  % \label{eq:ZDF_dynZDF__drg_bot}
1173  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1174  = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b}
1175\]
1176
1177where $t$ and $b$ refers to top and bottom layers respectively.
1178Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so it is implicit.
1179
1180% -------------------------------------------------------------------------------------------------------------
1181%       Bottom Friction with split-explicit free surface
1182% -------------------------------------------------------------------------------------------------------------
1183 \subsection[Bottom friction with split-explicit free surface]
1184 {Bottom friction with split-explicit free surface}
1185 \label{subsec:ZDF_drg_ts}
1186
1187With split-explicit free surface, the sub-stepping of barotropic equations needs the knowledge of top/bottom stresses. An obvious way to satisfy this is to take them as constant over the course of the barotropic integration and equal to the value used to update the baroclinic momentum trend. Provided \np{ln\_drgimp}\forcode{= .false.} and a centred or \textit{leap-frog} like integration of barotropic equations is used (\ie\ \forcode{ln_bt_fw=.false.}, cf \autoref{subsec:DYN_spg_ts}), this does ensure that barotropic and baroclinic dynamics feel the same stresses during one leapfrog time step. However, if \np{ln\_drgimp}\forcode{= .true.},  stresses depend on the \textit{after} value of the velocities which themselves depend on the barotropic iteration result. This cyclic dependency makes difficult obtaining consistent stresses in 2d and 3d dynamics. Part of this mismatch is then removed when setting the final barotropic component of 3d velocities to the time splitting estimate. This last step can be seen as a necessary evil but should be minimized since it interferes with the adjustment to the boundary conditions.
1188
1189The strategy to handle top/bottom stresses with split-explicit free surface in \NEMO\ is as follows:
1190\begin{enumerate}
1191\item To extend the stability of the barotropic sub-stepping, bottom stresses are refreshed at each sub-iteration. The baroclinic part of the flow entering the stresses is frozen at the initial time of the barotropic iteration. In case of non-linear friction, the drag coefficient is also constant.
1192\item In case of an implicit drag, specific computations are performed in \mdl{dynzdf} which renders the overall scheme mixed explicit/implicit: the barotropic components of 3d velocities are removed before seeking for the implicit vertical diffusion result. Top/bottom stresses due to the barotropic components are explicitly accounted for thanks to the updated values of barotropic velocities. Then the implicit solution of 3d velocities is obtained. Lastly, the residual barotropic component is replaced by the time split estimate.
1193\end{enumerate}
1194
1195Note that other strategies are possible, like considering vertical diffusion step in advance, \ie\ prior barotropic integration.
1196
1197
1198% ================================================================
1199% Internal wave-driven mixing
1200% ================================================================
1201\section[Internal wave-driven mixing (\forcode{ln_zdfiwm=.true.})]
1202{Internal wave-driven mixing (\protect\np{ln\_zdfiwm}\forcode{=.true.})}
1203\label{subsec:ZDF_tmx_new}
1204
1205%--------------------------------------------namzdf_iwm------------------------------------------
1206%
1207\begin{listing}
1208  \nlst{namzdf_iwm}
1209  \caption{\texttt{namzdf\_iwm}}
1210  \label{lst:namzdf_iwm}
1211\end{listing}
1212%--------------------------------------------------------------------------------------------------------------
1213
1214The parameterization of mixing induced by breaking internal waves is a generalization of
1215the approach originally proposed by \citet{st-laurent.simmons.ea_GRL02}.
1216A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed,
1217and the resulting diffusivity is obtained as
1218\[
1219  % \label{eq:ZDF_Kwave}
1220  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
1221\]
1222where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of
1223the energy available for mixing.
1224If the \np{ln\_mevar} namelist parameter is set to \forcode{.false.}, the mixing efficiency is taken as constant and
1225equal to 1/6 \citep{osborn_JPO80}.
1226In the opposite (recommended) case, $R_f$ is instead a function of
1227the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$,
1228with $\nu$ the molecular viscosity of seawater, following the model of \cite{bouffard.boegman_DAO13} and
1229the implementation of \cite{de-lavergne.madec.ea_JPO16}.
1230Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when
1231the mixing efficiency is constant.
1232
1233In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary
1234as a function of $Re_b$ by setting the \np{ln\_tsdiff} parameter to \forcode{.true.}, a recommended choice.
1235This parameterization of differential mixing, due to \cite{jackson.rehmann_JPO14},
1236is implemented as in \cite{de-lavergne.madec.ea_JPO16}.
1237
1238The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$,
1239is constructed from three static maps of column-integrated internal wave energy dissipation,
1240$E_{cri}(i,j)$, $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures:
1241
1242\begin{align*}
1243  F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\
1244  F_{pyc}(i,j,k) &\propto N^{n_p}\\
1245  F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} }
1246\end{align*}
1247In the above formula, $h_{ab}$ denotes the height above bottom,
1248$h_{wkb}$ denotes the WKB-stretched height above bottom, defined by
1249\[
1250  h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; ,
1251\]
1252The $n_p$ parameter (given by \np{nn\_zpyc} in \nam{zdf\_iwm} namelist)
1253controls the stratification-dependence of the pycnocline-intensified dissipation.
1254It can take values of $1$ (recommended) or $2$.
1255Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of
1256the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps.
1257$h_{cri}$ is related to the large-scale topography of the ocean (etopo2) and
1258$h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of
1259the abyssal hill topography \citep{goff_JGR10} and the latitude.
1260%
1261% Jc: input files names ?
1262
1263% ================================================================
1264% surface wave-induced mixing
1265% ================================================================
1266\section[Surface wave-induced mixing (\forcode{ln_zdfswm=.true.})]
1267{Surface wave-induced mixing (\protect\np{ln\_zdfswm}\forcode{=.true.})}
1268\label{subsec:ZDF_swm}
1269
1270Surface waves produce an enhanced mixing through wave-turbulence interaction.
1271In addition to breaking waves induced turbulence (\autoref{subsec:ZDF_tke}),
1272the influence of non-breaking waves can be accounted introducing
1273wave-induced viscosity and diffusivity as a function of the wave number spectrum.
1274Following \citet{qiao.yuan.ea_OD10}, a formulation of wave-induced mixing coefficient
1275is provided  as a function of wave amplitude, Stokes Drift and wave-number:
1276
1277\begin{equation}
1278  \label{eq:ZDF_Bv}
1279  B_{v} = \alpha {A} {U}_{st} {exp(3kz)}
1280\end{equation}
1281
1282Where $B_{v}$ is the wave-induced mixing coefficient, $A$ is the wave amplitude,
1283${U}_{st}$ is the Stokes Drift velocity, $k$ is the wave number and $\alpha$
1284is a constant which should be determined by observations or
1285numerical experiments and is set to be 1.
1286
1287The coefficient $B_{v}$ is then directly added to the vertical viscosity
1288and diffusivity coefficients.
1289
1290In order to account for this contribution set: \forcode{ln_zdfswm=.true.},
1291then wave interaction has to be activated through \forcode{ln_wave=.true.},
1292the Stokes Drift can be evaluated by setting \forcode{ln_sdw=.true.}
1293(see \autoref{subsec:SBC_wave_sdw})
1294and the needed wave fields can be provided either in forcing or coupled mode
1295(for more information on wave parameters and settings see \autoref{sec:SBC_wave})
1296
1297% ================================================================
1298% Adaptive-implicit vertical advection
1299% ================================================================
1300\section[Adaptive-implicit vertical advection (\forcode{ln_zad_Aimp=.true.})]
1301{Adaptive-implicit vertical advection(\protect\np{ln\_zad\_Aimp}\forcode{=.true.})}
1302\label{subsec:ZDF_aimp}
1303
1304The adaptive-implicit vertical advection option in NEMO is based on the work of
1305\citep{shchepetkin_OM15}.  In common with most ocean models, the timestep used with NEMO
1306needs to satisfy multiple criteria associated with different physical processes in order
1307to maintain numerical stability. \citep{shchepetkin_OM15} pointed out that the vertical
1308CFL criterion is commonly the most limiting. \citep{lemarie.debreu.ea_OM15} examined the
1309constraints for a range of time and space discretizations and provide the CFL stability
1310criteria for a range of advection schemes. The values for the Leap-Frog with Robert
1311asselin filter time-stepping (as used in NEMO) are reproduced in
1312\autoref{tab:ZDF_zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these
1313restrictions but at the cost of large dispersive errors and, possibly, large numerical
1314viscosity. The adaptive-implicit vertical advection option provides a targetted use of the
1315implicit scheme only when and where potential breaches of the vertical CFL condition
1316occur. In many practical applications these events may occur remote from the main area of
1317interest or due to short-lived conditions such that the extra numerical diffusion or
1318viscosity does not greatly affect the overall solution. With such applications, setting:
1319\forcode{ln_zad_Aimp=.true.} should allow much longer model timesteps to be used whilst
1320retaining the accuracy of the high order explicit schemes over most of the domain.
1321
1322\begin{table}[htbp]
1323  \centering
1324  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}}
1325  \begin{tabular}{r|ccc}
1326    \hline
1327    spatial discretization  & 2$^nd$ order centered & 3$^rd$ order upwind & 4$^th$ order compact \\
1328    advective CFL criterion &                 0.904 &              0.472  &                0.522 \\
1329    \hline
1330  \end{tabular}
1331  \caption[Advective CFL criteria for the leapfrog with Robert Asselin filter time-stepping]{
1332    The advective CFL criteria for a range of spatial discretizations for
1333    the leapfrog with Robert Asselin filter time-stepping
1334    ($\nu=0.1$) as given in \citep{lemarie.debreu.ea_OM15}.}
1335  \label{tab:ZDF_zad_Aimp_CFLcrit}
1336\end{table}
1337
1338In particular, the advection scheme remains explicit everywhere except where and when
1339local vertical velocities exceed a threshold set just below the explicit stability limit.
1340Once the threshold is reached a tapered transition towards an implicit scheme is used by
1341partitioning the vertical velocity into a part that can be treated explicitly and any
1342excess that must be treated implicitly. The partitioning is achieved via a Courant-number
1343dependent weighting algorithm as described in \citep{shchepetkin_OM15}.
1344
1345The local cell Courant number ($Cu$) used for this partitioning is:
1346
1347\begin{equation}
1348  \label{eq:ZDF_Eqn_zad_Aimp_Courant}
1349  \begin{split}
1350    Cu &= {2 \rdt \over e^n_{3t_{ijk}}} \bigg (\big [ \texttt{Max}(w^n_{ijk},0.0) - \texttt{Min}(w^n_{ijk+1},0.0) \big ]    \\
1351       &\phantom{=} +\big [ \texttt{Max}(e_{{2_u}ij}e^n_{{3_{u}}ijk}u^n_{ijk},0.0) - \texttt{Min}(e_{{2_u}i-1j}e^n_{{3_{u}}i-1jk}u^n_{i-1jk},0.0) \big ]
1352                     \big / e_{{1_t}ij}e_{{2_t}ij}            \\
1353       &\phantom{=} +\big [ \texttt{Max}(e_{{1_v}ij}e^n_{{3_{v}}ijk}v^n_{ijk},0.0) - \texttt{Min}(e_{{1_v}ij-1}e^n_{{3_{v}}ij-1k}v^n_{ij-1k},0.0) \big ]
1354                     \big / e_{{1_t}ij}e_{{2_t}ij} \bigg )    \\
1355  \end{split}
1356\end{equation}
1357
1358\noindent and the tapering algorithm follows \citep{shchepetkin_OM15} as:
1359
1360\begin{align}
1361  \label{eq:ZDF_Eqn_zad_Aimp_partition}
1362Cu_{min} &= 0.15 \nonumber \\
1363Cu_{max} &= 0.3  \nonumber \\
1364Cu_{cut} &= 2Cu_{max} - Cu_{min} \nonumber \\
1365Fcu    &= 4Cu_{max}*(Cu_{max}-Cu_{min}) \nonumber \\
1366\cf &=
1367     \begin{cases}
1368        0.0                                                        &\text{if $Cu \leq Cu_{min}$} \\
1369        (Cu - Cu_{min})^2 / (Fcu +  (Cu - Cu_{min})^2)             &\text{else if $Cu < Cu_{cut}$} \\
1370        (Cu - Cu_{max}) / Cu                                       &\text{else}
1371     \end{cases}
1372\end{align}
1373
1374\begin{figure}[!t]
1375  \centering
1376  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_coeff}
1377  \caption[Partitioning coefficient used to partition vertical velocities into parts]{
1378    The value of the partitioning coefficient (\cf) used to partition vertical velocities into
1379    parts to be treated implicitly and explicitly for a range of typical Courant numbers
1380    (\forcode{ln_zad_Aimp=.true.}).}
1381  \label{fig:ZDF_zad_Aimp_coeff}
1382\end{figure}
1383
1384\noindent The partitioning coefficient is used to determine the part of the vertical
1385velocity that must be handled implicitly ($w_i$) and to subtract this from the total
1386vertical velocity ($w_n$) to leave that which can continue to be handled explicitly:
1387
1388\begin{align}
1389  \label{eq:ZDF_Eqn_zad_Aimp_partition2}
1390    w_{i_{ijk}} &= \cf_{ijk} w_{n_{ijk}}     \nonumber \\
1391    w_{n_{ijk}} &= (1-\cf_{ijk}) w_{n_{ijk}}
1392\end{align}
1393
1394\noindent Note that the coefficient is such that the treatment is never fully implicit;
1395the three cases from \autoref{eq:ZDF_Eqn_zad_Aimp_partition} can be considered as:
1396fully-explicit; mixed explicit/implicit and mostly-implicit.  With the settings shown the
1397coefficient (\cf) varies as shown in \autoref{fig:ZDF_zad_Aimp_coeff}. Note with these values
1398the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly
1399implicit' is 0.45 which is just below the stability limited given in
1400\autoref{tab:ZDF_zad_Aimp_CFLcrit}  for a 3rd order scheme.
1401
1402The $w_i$ component is added to the implicit solvers for the vertical mixing in
1403\mdl{dynzdf} and \mdl{trazdf} in a similar way to \citep{shchepetkin_OM15}.  This is
1404sufficient for the flux-limited advection scheme (\forcode{ln_traadv_mus}) but further
1405intervention is required when using the flux-corrected scheme (\forcode{ln_traadv_fct}).
1406For these schemes the implicit upstream fluxes must be added to both the monotonic guess
1407and to the higher order solution when calculating the antidiffusive fluxes. The implicit
1408vertical fluxes are then removed since they are added by the implicit solver later on.
1409
1410The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be
1411used in a wide range of simulations. The following test simulation, however, does illustrate
1412the potential benefits and will hopefully encourage further testing and feedback from users:
1413
1414\begin{figure}[!t]
1415  \centering
1416  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames}
1417  \caption[OVERFLOW: time-series of temperature vertical cross-sections]{
1418    A time-series of temperature vertical cross-sections for the OVERFLOW test case.
1419    These results are for the default settings with \forcode{nn_rdt=10.0} and
1420    without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}).}
1421  \label{fig:ZDF_zad_Aimp_overflow_frames}
1422\end{figure}
1423
1424\subsection{Adaptive-implicit vertical advection in the OVERFLOW test-case}
1425
1426The \href{https://forge.ipsl.jussieu.fr/nemo/chrome/site/doc/NEMO/guide/html/test\_cases.html\#overflow}{OVERFLOW test case}
1427provides a simple illustration of the adaptive-implicit advection in action. The example here differs from the basic test case
1428by only a few extra physics choices namely:
1429
1430\begin{verbatim}
1431     ln_dynldf_OFF = .false.
1432     ln_dynldf_lap = .true.
1433     ln_dynldf_hor = .true.
1434     ln_zdfnpc     = .true.
1435     ln_traadv_fct = .true.
1436        nn_fct_h   =  2
1437        nn_fct_v   =  2
1438\end{verbatim}
1439
1440\noindent which were chosen to provide a slightly more stable and less noisy solution. The
1441result when using the default value of \forcode{nn_rdt=10.} without adaptive-implicit
1442vertical velocity is illustrated in \autoref{fig:ZDF_zad_Aimp_overflow_frames}. The mass of
1443cold water, initially sitting on the shelf, moves down the slope and forms a
1444bottom-trapped, dense plume. Even with these extra physics choices the model is close to
1445stability limits and attempts with \forcode{nn_rdt=30.} will fail after about 5.5 hours
1446with excessively high horizontal velocities. This time-scale corresponds with the time the
1447plume reaches the steepest part of the topography and, although detected as a horizontal
1448CFL breach, the instability originates from a breach of the vertical CFL limit. This is a good
1449candidate, therefore, for use of the adaptive-implicit vertical advection scheme.
1450
1451The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps
1452are shown in \autoref{fig:ZDF_zad_Aimp_overflow_all_rdt} (together with the equivalent
1453frames from the base run).  In this simple example the use of the adaptive-implicit
1454vertcal advection scheme has enabled a 12x increase in the model timestep without
1455significantly altering the solution (although at this extreme the plume is more diffuse
1456and has not travelled so far).  Notably, the solution with and without the scheme is
1457slightly different even with \forcode{nn_rdt=10.}; suggesting that the base run was
1458close enough to instability to trigger the scheme despite completing successfully.
1459To assist in diagnosing how active the scheme is, in both location and time, the 3D
1460implicit and explicit components of the vertical velocity are available via XIOS as
1461\texttt{wimp} and \texttt{wexp} respectively.  Likewise, the partitioning coefficient
1462(\cf) is also available as \texttt{wi\_cff}. For a quick oversight of
1463the schemes activity the global maximum values of the absolute implicit component
1464of the vertical velocity and the partitioning coefficient are written to the netCDF
1465version of the run statistics file (\texttt{run.stat.nc}) if this is active (see
1466\autoref{sec:MISC_opt} for activation details).
1467
1468\autoref{fig:ZDF_zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for
1469the various overflow tests.  Note that the adaptive-implicit vertical advection scheme is
1470active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the
1471test case is close to stability limits even with this value. At the larger timesteps, the
1472vertical velocity is treated mostly implicitly at some location throughout the run. The
1473oscillatory nature of this measure appears to be linked to the progress of the plume front
1474as each cusp is associated with the location of the maximum shifting to the adjacent cell.
1475This is illustrated in \autoref{fig:ZDF_zad_Aimp_maxCf_loc} where the i- and k- locations of the
1476maximum have been overlaid for the base run case.
1477
1478\medskip
1479\noindent Only limited tests have been performed in more realistic configurations. In the
1480ORCA2\_ICE\_PISCES reference configuration the scheme does activate and passes
1481restartability and reproducibility tests but it is unable to improve the model's stability
1482enough to allow an increase in the model time-step. A view of the time-series of maximum
1483partitioning coefficient (not shown here)  suggests that the default time-step of 5400s is
1484already pushing at stability limits, especially in the initial start-up phase. The
1485time-series does not, however, exhibit any of the 'cuspiness' found with the overflow
1486tests.
1487
1488\medskip
1489\noindent A short test with an eORCA1 configuration promises more since a test using a
1490time-step of 3600s remains stable with \forcode{ln_zad_Aimp=.true.} whereas the
1491time-step is limited to 2700s without.
1492
1493\begin{figure}[!t]
1494  \centering
1495  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_overflow_all_rdt}
1496  \caption[OVERFLOW: sample temperature vertical cross-sections from mid- and end-run]{
1497    Sample temperature vertical cross-sections from mid- and end-run using
1498    different values for \forcode{nn_rdt} and with or without adaptive implicit vertical advection.
1499    Without the adaptive implicit vertical advection
1500    only the run with the shortest timestep is able to run to completion.
1501    Note also that the colour-scale has been chosen to confirm that
1502    temperatures remain within the original range of 10$^o$ to 20$^o$.}
1503  \label{fig:ZDF_zad_Aimp_overflow_all_rdt}
1504\end{figure}
1505
1506\begin{figure}[!t]
1507  \centering
1508  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_maxCf}
1509  \caption[OVERFLOW: maximum partitioning coefficient during a series of test runs]{
1510    The maximum partitioning coefficient during a series of test runs with
1511    increasing model timestep length.
1512    At the larger timesteps,
1513    the vertical velocity is treated mostly implicitly at some location throughout the run.}
1514  \label{fig:ZDF_zad_Aimp_maxCf}
1515\end{figure}
1516
1517\begin{figure}[!t]
1518  \centering
1519  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_maxCf_loc}
1520  \caption[OVERFLOW: maximum partitioning coefficient for the case overlaid]{
1521    The maximum partitioning coefficient for the \forcode{nn_rdt=10.0} case overlaid with
1522    information on the gridcell i- and k-locations of the maximum value.}
1523  \label{fig:ZDF_zad_Aimp_maxCf_loc}
1524\end{figure}
1525
1526% ================================================================
1527
1528\biblio
1529
1530\pindex
1531
1532\end{document}
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