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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\label{chap:ZDF}
10
11\thispagestyle{plain}
12
13\chaptertoc
14
15\paragraph{Changes record} ~\\
16
17{\footnotesize
18  \begin{tabularx}{\textwidth}{l||X|X}
19    Release & Author(s) & Modifications \\
20    \hline
21    {\em   4.0} & {\em ...} & {\em ...} \\
22    {\em   3.6} & {\em ...} & {\em ...} \\
23    {\em   3.4} & {\em ...} & {\em ...} \\
24    {\em <=3.4} & {\em ...} & {\em ...}
25  \end{tabularx}
26}
27
28\clearpage
29
30%gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN.
31
32%% =================================================================================================
33\section{Vertical mixing}
34\label{sec:ZDF}
35
36The discrete form of the ocean subgrid scale physics has been presented in
37\autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}.
38At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt have to be defined.
39At the surface they are prescribed from the surface forcing (see \autoref{chap:SBC}),
40while at the bottom they are set to zero for heat and salt,
41unless a geothermal flux forcing is prescribed as a bottom boundary condition (\ie\ \np{ln_trabbc}{ln\_trabbc} defined,
42see \autoref{subsec:TRA_bbc}), and specified through a bottom friction parameterisation for momentum
43(see \autoref{sec:ZDF_drg}).
44
45In this section we briefly discuss the various choices offered to compute the vertical eddy viscosity and
46diffusivity coefficients, $A_u^{vm}$ , $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- points,
47respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}).
48These coefficients can be assumed to be either constant, or a function of the local Richardson number,
49or computed from a turbulent closure model (either TKE or GLS or OSMOSIS formulation).
50The computation of these coefficients is initialized in the \mdl{zdfphy} module and performed in
51the \mdl{zdfric}, \mdl{zdftke} or \mdl{zdfgls} or \mdl{zdfosm} modules.
52The trends due to the vertical momentum and tracer diffusion, including the surface forcing,
53are computed and added to the general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively.
54%These trends can be computed using either a forward time stepping scheme
55%(namelist parameter \np[=.true.]{ln_zdfexp}{ln\_zdfexp}) or a backward time stepping scheme
56%(\np[=.false.]{ln_zdfexp}{ln\_zdfexp}) depending on the magnitude of the mixing coefficients,
57%and thus of the formulation used (see \autoref{chap:TD}).
58
59\begin{listing}
60  \nlst{namzdf}
61  \caption{\forcode{&namzdf}}
62  \label{lst:namzdf}
63\end{listing}
64
65%% =================================================================================================
66\subsection[Constant (\forcode{ln_zdfcst})]{Constant (\protect\np{ln_zdfcst}{ln\_zdfcst})}
67\label{subsec:ZDF_cst}
68
69Options are defined through the \nam{zdf}{zdf} namelist variables.
70When \np{ln_zdfcst}{ln\_zdfcst} is defined, the momentum and tracer vertical eddy coefficients are set to
71constant values over the whole ocean.
72This is the crudest way to define the vertical ocean physics.
73It is recommended to use this option only in process studies, not in basin scale simulations.
74Typical values used in this case are:
75\begin{align*}
76  A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}   \\
77  A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1}
78\end{align*}
79
80These values are set through the \np{rn_avm0}{rn\_avm0} and \np{rn_avt0}{rn\_avt0} namelist parameters.
81In all cases, do not use values smaller that those associated with the molecular viscosity and diffusivity,
82that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and
83$\sim10^{-9}~m^2.s^{-1}$ for salinity.
84
85%% =================================================================================================
86\subsection[Richardson number dependent (\forcode{ln_zdfric})]{Richardson number dependent (\protect\np{ln_zdfric}{ln\_zdfric})}
87\label{subsec:ZDF_ric}
88
89\begin{listing}
90  \nlst{namzdf_ric}
91  \caption{\forcode{&namzdf_ric}}
92  \label{lst:namzdf_ric}
93\end{listing}
94
95When \np[=.true.]{ln_zdfric}{ln\_zdfric}, a local Richardson number dependent formulation for the vertical momentum and
96tracer eddy coefficients is set through the \nam{zdf_ric}{zdf\_ric} namelist variables.
97The vertical mixing coefficients are diagnosed from the large scale variables computed by the model.
98\textit{In situ} measurements have been used to link vertical turbulent activity to large scale ocean structures.
99The hypothesis of a mixing mainly maintained by the growth of Kelvin-Helmholtz like instabilities leads to
100a dependency between the vertical eddy coefficients and the local Richardson number
101(\ie\ the ratio of stratification to vertical shear).
102Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented:
103\[
104  % \label{eq:ZDF_ric}
105  \left\{
106    \begin{aligned}
107      A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\
108      A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm}
109    \end{aligned}
110  \right.
111\]
112where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number,
113$N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
114$A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the constant case
115(see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ is the maximum value that
116can be reached by the coefficient when $Ri\leq 0$, $a=5$ and $n=2$.
117The last three values can be modified by setting the \np{rn_avmri}{rn\_avmri}, \np{rn_alp}{rn\_alp} and
118\np{nn_ric}{nn\_ric} namelist parameters, respectively.
119
120A simple mixing-layer model to transfer and dissipate the atmospheric forcings
121(wind-stress and buoyancy fluxes) can be activated setting the \np[=.true.]{ln_mldw}{ln\_mldw} in the namelist.
122
123In this case, the local depth of turbulent wind-mixing or "Ekman depth" $h_{e}(x,y,t)$ is evaluated and
124the vertical eddy coefficients prescribed within this layer.
125
126This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation:
127\[
128  h_{e} = Ek \frac {u^{*}} {f_{0}}
129\]
130where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter.
131
132In this similarity height relationship, the turbulent friction velocity:
133\[
134  u^{*} = \sqrt \frac {|\tau|} {\rho_o}
135\]
136is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$.
137The final $h_{e}$ is further constrained by the adjustable bounds \np{rn_mldmin}{rn\_mldmin} and \np{rn_mldmax}{rn\_mldmax}.
138Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to
139the empirical values \np{rn_wtmix}{rn\_wtmix} and \np{rn_wvmix}{rn\_wvmix} \citep{lermusiaux_JMS01}.
140
141%% =================================================================================================
142\subsection[TKE turbulent closure scheme (\forcode{ln_zdftke})]{TKE turbulent closure scheme (\protect\np{ln_zdftke}{ln\_zdftke})}
143\label{subsec:ZDF_tke}
144
145\begin{listing}
146  \nlst{namzdf_tke}
147  \caption{\forcode{&namzdf_tke}}
148  \label{lst:namzdf_tke}
149\end{listing}
150
151The vertical eddy viscosity and diffusivity coefficients are computed from a TKE turbulent closure model based on
152a prognostic equation for $\bar{e}$, the turbulent kinetic energy,
153and a closure assumption for the turbulent length scales.
154This turbulent closure model has been developed by \citet{bougeault.lacarrere_MWR89} in the atmospheric case,
155adapted by \citet{gaspar.gregoris.ea_JGR90} for the oceanic case, and embedded in OPA, the ancestor of \NEMO,
156by \citet{blanke.delecluse_JPO93} for equatorial Atlantic simulations.
157Since then, significant modifications have been introduced by \citet{madec.delecluse.ea_NPM98} in both the implementation and
158the formulation of the mixing length scale.
159The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear,
160its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type:
161\begin{equation}
162  \label{eq:ZDF_tke_e}
163  \frac{\partial \bar{e}}{\partial t} =
164  \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
165      +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
166  -K_\rho\,N^2
167  +\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
168      \;\frac{\partial \bar{e}}{\partial k}} \right]
169  - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
170\end{equation}
171\[
172  % \label{eq:ZDF_tke_kz}
173  \begin{split}
174    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }    \\
175    K_\rho &= A^{vm} / P_{rt}
176  \end{split}
177\]
178where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
179$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,
180$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity and diffusivity coefficients.
181The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$ $\approx 0.7$ are designed to deal with
182vertical mixing at any depth \citep{gaspar.gregoris.ea_JGR90}.
183They are set through namelist parameters \np{nn_ediff}{nn\_ediff} and \np{nn_ediss}{nn\_ediss}.
184$P_{rt}$ can be set to unity or, following \citet{blanke.delecluse_JPO93}, be a function of the local Richardson number, $R_i$:
185\begin{align*}
186  % \label{eq:ZDF_prt}
187  P_{rt} =
188  \begin{cases}
189    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$}   \\
190    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$}   \\
191    \ \ 10 &      \text{if $\ 2 \leq R_i$}
192  \end{cases}
193\end{align*}
194The choice of $P_{rt}$ is controlled by the \np{nn_pdl}{nn\_pdl} namelist variable.
195
196At the sea surface, the value of $\bar{e}$ is prescribed from the wind stress field as
197$\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn_ebb}{rn\_ebb} namelist parameter.
198The default value of $e_{bb}$ is 3.75. \citep{gaspar.gregoris.ea_JGR90}), however a much larger value can be used when
199taking into account the surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}).
200The bottom value of TKE is assumed to be equal to the value of the level just above.
201The time integration of the $\bar{e}$ equation may formally lead to negative values because
202the numerical scheme does not ensure its positivity.
203To overcome this problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn_emin}{rn\_emin} namelist parameter).
204Following \citet{gaspar.gregoris.ea_JGR90}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$.
205This allows the subsequent formulations to match that of \citet{gargett_JMR84} for the diffusion in
206the thermocline and deep ocean :  $K_\rho = 10^{-3} / N$.
207In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical instabilities associated with
208too weak vertical diffusion.
209They must be specified at least larger than the molecular values, and are set through \np{rn_avm0}{rn\_avm0} and
210\np{rn_avt0}{rn\_avt0} (\nam{zdf}{zdf} namelist, see \autoref{subsec:ZDF_cst}).
211
212%% =================================================================================================
213\subsubsection{Turbulent length scale}
214
215For computational efficiency, the original formulation of the turbulent length scales proposed by
216\citet{gaspar.gregoris.ea_JGR90} has been simplified.
217Four formulations are proposed, the choice of which is controlled by the \np{nn_mxl}{nn\_mxl} namelist parameter.
218The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}:
219\begin{equation}
220  \label{eq:ZDF_tke_mxl0_1}
221  l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N
222\end{equation}
223which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency.
224The resulting length scale is bounded by the distance to the surface or to the bottom
225(\np[=0]{nn_mxl}{nn\_mxl}) or by the local vertical scale factor (\np[=1]{nn_mxl}{nn\_mxl}).
226\citet{blanke.delecluse_JPO93} notice that this simplification has two major drawbacks:
227it makes no sense for locally unstable stratification and the computation no longer uses all
228the information contained in the vertical density profile.
229To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np[=2, 3]{nn_mxl}{nn\_mxl} cases,
230which add an extra assumption concerning the vertical gradient of the computed length scale.
231So, the length scales are first evaluated as in \autoref{eq:ZDF_tke_mxl0_1} and then bounded such that:
232\begin{equation}
233  \label{eq:ZDF_tke_mxl_constraint}
234  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
235  \qquad \text{with }\  l =  l_k = l_\epsilon
236\end{equation}
237\autoref{eq:ZDF_tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than
238the variations of depth.
239It provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less
240time consuming.
241In particular, it allows the length scale to be limited not only by the distance to the surface or
242to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as
243the thermocline (\autoref{fig:ZDF_mixing_length}).
244In order to impose the \autoref{eq:ZDF_tke_mxl_constraint} constraint, we introduce two additional length scales:
245$l_{up}$ and $l_{dwn}$, the upward and downward length scales, and
246evaluate the dissipation and mixing length scales as
247(and note that here we use numerical indexing):
248\begin{figure}[!t]
249  \centering
250  \includegraphics[width=0.66\textwidth]{Fig_mixing_length}
251  \caption[Mixing length computation]{Illustration of the mixing length computation}
252  \label{fig:ZDF_mixing_length}
253\end{figure}
254\[
255  % \label{eq:ZDF_tke_mxl2}
256  \begin{aligned}
257    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
258    \quad &\text{ from $k=1$ to $jpk$ }\ \\
259    l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)\right)
260    \quad &\text{ from $k=jpk$ to $1$ }\ \\
261  \end{aligned}
262\]
263where $l^{(k)}$ is computed using \autoref{eq:ZDF_tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
264
265In the \np[=2]{nn_mxl}{nn\_mxl} case, the dissipation and mixing length scales take the same value:
266$ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the \np[=3]{nn_mxl}{nn\_mxl} case,
267the dissipation and mixing turbulent length scales are give as in \citet{gaspar.gregoris.ea_JGR90}:
268\[
269  % \label{eq:ZDF_tke_mxl_gaspar}
270  \begin{aligned}
271    & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }   \\
272    & l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)
273  \end{aligned}
274\]
275
276At the ocean surface, a non zero length scale is set through the  \np{rn_mxl0}{rn\_mxl0} namelist parameter.
277Usually the surface scale is given by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant and
278$z_o$ the roughness parameter of the surface.
279Assuming $z_o=0.1$~m \citep{craig.banner_JPO94} leads to a 0.04~m, the default value of \np{rn_mxl0}{rn\_mxl0}.
280In the ocean interior a minimum length scale is set to recover the molecular viscosity when
281$\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).
282
283%% =================================================================================================
284\subsubsection{Surface wave breaking parameterization}
285
286Following \citet{mellor.blumberg_JPO04}, the TKE turbulence closure model has been modified to
287include the effect of surface wave breaking energetics.
288This results in a reduction of summertime surface temperature when the mixed layer is relatively shallow.
289The \citet{mellor.blumberg_JPO04} modifications acts on surface length scale and TKE values and
290air-sea drag coefficient.
291The latter concerns the bulk formulae and is not discussed here.
292
293Following \citet{craig.banner_JPO94}, the boundary condition on surface TKE value is :
294\begin{equation}
295  \label{eq:ZDF_Esbc}
296  \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
297\end{equation}
298where $\alpha_{CB}$ is the \citet{craig.banner_JPO94} constant of proportionality which depends on the ''wave age'',
299ranging from 57 for mature waves to 146 for younger waves \citep{mellor.blumberg_JPO04}.
300The boundary condition on the turbulent length scale follows the Charnock's relation:
301\begin{equation}
302  \label{eq:ZDF_Lsbc}
303  l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o}
304\end{equation}
305where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant.
306\citet{mellor.blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by
307\citet{stacey_JPO99} citing observation evidence, and
308$\alpha_{CB} = 100$ the Craig and Banner's value.
309As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$,
310with $e_{bb}$ the \np{rn_ebb}{rn\_ebb} namelist parameter, setting \np[=67.83]{rn_ebb}{rn\_ebb} corresponds
311to $\alpha_{CB} = 100$.
312Further setting  \np[=.true.]{ln_mxl0}{ln\_mxl0},  applies \autoref{eq:ZDF_Lsbc} as the surface boundary condition on the length scale,
313with $\beta$ hard coded to the Stacey's value.
314Note that a minimal threshold of \np{rn_emin0}{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) is applied on the
315surface $\bar{e}$ value.
316
317%% =================================================================================================
318\subsubsection{Langmuir cells}
319
320Langmuir circulations (LC) can be described as ordered large-scale vertical motions in
321the surface layer of the oceans.
322Although LC have nothing to do with convection, the circulation pattern is rather similar to
323so-called convective rolls in the atmospheric boundary layer.
324The detailed physics behind LC is described in, for example, \citet{craik.leibovich_JFM76}.
325The prevailing explanation is that LC arise from a nonlinear interaction between the Stokes drift and
326wind drift currents.
327
328Here we introduced in the TKE turbulent closure the simple parameterization of Langmuir circulations proposed by
329\citep{axell_JGR02} for a $k-\epsilon$ turbulent closure.
330The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in
331an extra source term of TKE, $P_{LC}$.
332The presence of $P_{LC}$ in \autoref{eq:ZDF_tke_e}, the TKE equation, is controlled by setting \np{ln_lc}{ln\_lc} to
333\forcode{.true.} in the \nam{zdf_tke}{zdf\_tke} namelist.
334
335By making an analogy with the characteristic convective velocity scale (\eg, \citet{dalessio.abdella.ea_JPO98}),
336$P_{LC}$ is assumed to be :
337\[
338P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}}
339\]
340where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth.
341With no information about the wave field, $w_{LC}$ is assumed to be proportional to
342the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module
343\footnote{Following \citet{li.garrett_JMR93}, the surface Stoke drift velocity may be expressed as
344  $u_s =  0.016 \,|U_{10m}|$.
345  Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of
346  $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress
347}.
348For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at
349a finite depth $H_{LC}$ (which is often close to the mixed layer depth),
350and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures).
351The resulting expression for $w_{LC}$ is :
352\[
353  w_{LC}  =
354  \begin{cases}
355    c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$}    \\
356    0                             &      \text{otherwise}
357  \end{cases}
358\]
359where $c_{LC} = 0.15$ has been chosen by \citep{axell_JGR02} as a good compromise to fit LES data.
360The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second.
361The value of $c_{LC}$ is set through the \np{rn_lc}{rn\_lc} namelist parameter,
362having in mind that it should stay between 0.15 and 0.54 \citep{axell_JGR02}.
363
364The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations:
365$H_{LC}$ is the depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by
366converting its kinetic energy to potential energy, according to
367\[
368- \int_{-H_{LC}}^0 { N^2\;\;dz} = \frac{1}{2} u_s^2
369\]
370
371%% =================================================================================================
372\subsubsection{Mixing just below the mixed layer}
373
374Vertical mixing parameterizations commonly used in ocean general circulation models tend to
375produce mixed-layer depths that are too shallow during summer months and windy conditions.
376This bias is particularly acute over the Southern Ocean.
377To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{rodgers.aumont.ea_B14}.
378The parameterization is an empirical one, \ie\ not derived from theoretical considerations,
379but rather is meant to account for observed processes that affect the density structure of
380the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme
381(\ie\ near-inertial oscillations and ocean swells and waves).
382
383When using this parameterization (\ie\ when \np[=1]{nn_etau}{nn\_etau}),
384the TKE input to the ocean ($S$) imposed by the winds in the form of near-inertial oscillations,
385swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition,
386plus a depth depend one given by:
387\begin{equation}
388  \label{eq:ZDF_Ehtau}
389  S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau}
390\end{equation}
391where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that
392penetrates in the ocean, $h_\tau$ is a vertical mixing length scale that controls exponential shape of
393the penetration, and $f_i$ is the ice concentration
394(no penetration if $f_i=1$, \ie\ if the ocean is entirely covered by sea-ice).
395The value of $f_r$, usually a few percents, is specified through \np{rn_efr}{rn\_efr} namelist parameter.
396The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np[=0]{nn_etau}{nn\_etau}) or
397a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m at high latitudes
398(\np[=1]{nn_etau}{nn\_etau}).
399
400Note that two other option exist, \np[=2, 3]{nn_etau}{nn\_etau}.
401They correspond to applying \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer,
402or to using the high frequency part of the stress to evaluate the fraction of TKE that penetrates the ocean.
403Those two options are obsolescent features introduced for test purposes.
404They will be removed in the next release.
405
406% This should be explain better below what this rn_eice parameter is meant for:
407In presence of Sea Ice, the value of this mixing can be modulated by the \np{rn_eice}{rn\_eice} namelist parameter.
408This parameter varies from \forcode{0} for no effect to \forcode{4} to suppress the TKE input into the ocean when Sea Ice concentration
409is greater than 25\%.
410
411% from Burchard et al OM 2008 :
412% the most critical process not reproduced by statistical turbulence models is the activity of
413% internal waves and their interaction with turbulence. After the Reynolds decomposition,
414% internal waves are in principle included in the RANS equations, but later partially
415% excluded by the hydrostatic assumption and the model resolution.
416% Thus far, the representation of internal wave mixing in ocean models has been relatively crude
417% (\eg\ Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002).
418
419%% =================================================================================================
420\subsection[GLS: Generic Length Scale (\forcode{ln_zdfgls})]{GLS: Generic Length Scale (\protect\np{ln_zdfgls}{ln\_zdfgls})}
421\label{subsec:ZDF_gls}
422
423\begin{listing}
424  \nlst{namzdf_gls}
425  \caption{\forcode{&namzdf_gls}}
426  \label{lst:namzdf_gls}
427\end{listing}
428
429The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on two prognostic equations:
430one for the turbulent kinetic energy $\bar {e}$, and another for the generic length scale,
431$\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}.
432This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,
433where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:ZDF_GLS} allows to recover a number of
434well-known turbulent closures ($k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87},
435$k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}).
436The GLS scheme is given by the following set of equations:
437\begin{equation}
438  \label{eq:ZDF_gls_e}
439  \frac{\partial \bar{e}}{\partial t} =
440  \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
441      +\left( \frac{\partial v}{\partial k} \right)^2} \right]
442  -K_\rho \,N^2
443  +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right]
444  - \epsilon
445\end{equation}
446
447\[
448  % \label{eq:ZDF_gls_psi}
449  \begin{split}
450    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{
451      \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
452          +\left( \frac{\partial v}{\partial k} \right)^2} \right]
453      - C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\
454    &+\frac{1}{e_3\;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 }
455        \;\frac{\partial \psi}{\partial k}} \right]\;
456  \end{split}
457\]
458
459\[
460  % \label{eq:ZDF_gls_kz}
461  \begin{split}
462    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
463    K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l
464  \end{split}
465\]
466
467\[
468  % \label{eq:ZDF_gls_eps}
469  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
470\]
471where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and
472$\epsilon$ the dissipation rate.
473The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of
474the choice of the turbulence model.
475Four different turbulent models are pre-defined (\autoref{tab:ZDF_GLS}).
476They are made available through the \np{nn_clo}{nn\_clo} namelist parameter.
477
478\begin{table}[htbp]
479  \centering
480  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c}
481  \begin{tabular}{ccccc}
482    &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\
483    % & \citep{mellor.yamada_RG82} &  \citep{rodi_JGR87}       & \citep{wilcox_AJ88} &                 \\
484    \hline
485    \hline
486    \np{nn_clo}{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\
487    \hline
488    $( p , n , m )$         &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\
489    $\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\
490    $\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\
491    $C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\
492    $C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\
493    $C_3$              &      1.           &     1.              &      1.                &       1.           \\
494    $F_{wall}$        &      Yes        &       --             &     --                  &      --          \\
495    \hline
496    \hline
497  \end{tabular}
498  \caption[Set of predefined GLS parameters or equivalently predefined turbulence models available]{
499    Set of predefined GLS parameters, or equivalently predefined turbulence models available with
500    \protect\np[=.true.]{ln_zdfgls}{ln\_zdfgls} and controlled by
501    the \protect\np{nn_clos}{nn\_clos} namelist variable in \protect\nam{zdf_gls}{zdf\_gls}.}
502  \label{tab:ZDF_GLS}
503\end{table}
504
505In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force the convergence of
506the mixing length towards $\kappa z_b$ ($\kappa$ is the Von Karman constant and $z_b$ the rugosity length scale) value near physical boundaries
507(logarithmic boundary layer law).
508$C_{\mu}$ and $C_{\mu'}$ are calculated from stability function proposed by \citet{galperin.kantha.ea_JAS88},
509or by \citet{kantha.clayson_JGR94} or one of the two functions suggested by \citet{canuto.howard.ea_JPO01}
510(\np[=0, 3]{nn_stab_func}{nn\_stab\_func}, resp.).
511The value of $C_{0\mu}$ depends on the choice of the stability function.
512
513The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated thanks to Dirichlet or
514Neumann condition through \np{nn_bc_surf}{nn\_bc\_surf} and \np{nn_bc_bot}{nn\_bc\_bot}, resp.
515As for TKE closure, the wave effect on the mixing is considered when
516\np[ > 0.]{rn_crban}{rn\_crban} \citep{craig.banner_JPO94, mellor.blumberg_JPO04}.
517The \np{rn_crban}{rn\_crban} namelist parameter is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and
518\np{rn_charn}{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}.
519
520The $\psi$ equation is known to fail in stably stratified flows, and for this reason
521almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.
522With this clipping, the maximum permissible length scale is determined by $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$.
523A value of $c_{lim} = 0.53$ is often used \citep{galperin.kantha.ea_JAS88}.
524\cite{umlauf.burchard_CSR05} show that the value of the clipping factor is of crucial importance for
525the entrainment depth predicted in stably stratified situations,
526and that its value has to be chosen in accordance with the algebraic model for the turbulent fluxes.
527The clipping is only activated if \np[=.true.]{ln_length_lim}{ln\_length\_lim},
528and the $c_{lim}$ is set to the \np{rn_clim_galp}{rn\_clim\_galp} value.
529
530The time and space discretization of the GLS equations follows the same energetic consideration as for
531the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{burchard_OM02}.
532Evaluation of the 4 GLS turbulent closure schemes can be found in \citet{warner.sherwood.ea_OM05} in ROMS model and
533 in \citet{reffray.guillaume.ea_GMD15} for the \NEMO\ model.
534
535% -------------------------------------------------------------------------------------------------------------
536%        OSM OSMOSIS BL Scheme
537% -------------------------------------------------------------------------------------------------------------
538\subsection[OSM: OSMOSIS boundary layer scheme (\forcode{ln_zdfosm = .true.})]
539{OSM: OSMOSIS boundary layer scheme (\protect\np{ln\_zdfosm}\forcode{ = .true.})}
540\label{subsec:ZDF_osm}
541
542\begin{listing}
543  \nlst{namzdf_osm}
544  \caption{\forcode{&namzdf_osm}}
545  \label{lst:namzdf_osm}
546\end{listing}
547
548Much of the time the turbulent motions in the ocean surface boundary
549layer (OSBL) are not given by
550classical shear turbulence. Instead they are in a regime dominated by an
551interaction between the currents and the Stokes drift of the surface waves known as
552`Langmuir turbulence' \citep[e.g.][]{mcwilliams.ea_JFM97}.
553The OSMOSIS model is fundamentally based on results of Large Eddy
554Simulations (LES) of Langmuir turbulence and aims to fully describe
555this Langmuir regime.
556
557The OSMOSIS turbulent closure scheme is a similarity-scale scheme in
558the same spirit as the K-profile
559parameterization (KPP) scheme of \citet{large.ea_RG97}.
560A specified shape of diffusivity, scaled by the (OSBL) depth
561$h_{\mathrm{BL}}$ and a turbulent velocity scale, is imposed throughout the
562boundary layer
563$-h_{\mathrm{BL}}<z<\eta$.
564However, rather than the OSBL
565depth being diagnosed in terms of a bulk Richardson number criterion,
566as in KPP, it is set by a prognostic equation that is informed by
567energy budget considerations reminiscent of the classical mixed layer
568models of \citet{kraus.turner_tellus67}.
569The model also includes an explicit parametrization of the structure
570of the pycnocline (the stratified region at the bottom of the OSBL).
571
572\subsubsection{The flux gradient model}
573
574The turbulent closure model
575also includes ``non-local'' (independent of the local property gradient)
576fluxes of tracers and momentum.
577
578%% =================================================================================================
579\subsection[ Discrete energy conservation for TKE and GLS schemes]{Discrete energy conservation for TKE and GLS schemes}
580\label{subsec:ZDF_tke_ene}
581
582\begin{figure}[!t]
583  \centering
584  \includegraphics[width=0.66\textwidth]{Fig_ZDF_TKE_time_scheme}
585  \caption[Subgrid kinetic energy integration in GLS and TKE schemes]{
586    Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and
587    its links to the momentum and tracer time integration.}
588  \label{fig:ZDF_TKE_time_scheme}
589\end{figure}
590
591The production of turbulence by vertical shear (the first term of the right hand side of
592\autoref{eq:ZDF_tke_e}) and  \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion
593(first line in \autoref{eq:MB_zdf}).
594To do so a special care has to be taken for both the time and space discretization of
595the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}.
596
597Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how
598the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with
599the one-level forward time stepping of the equation for $\bar{e}$.
600With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to
601the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and
602summing the result vertically:
603\begin{equation}
604  \label{eq:ZDF_energ1}
605  \begin{split}
606    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\
607    &= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}
608    - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz }
609  \end{split}
610\end{equation}
611Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$,
612known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme
613(see \autoref{sec:TD_forward_imp}).
614The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at
615the surface (atmospheric forcing) and at the bottom (friction effect).
616The second term is always negative.
617It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$.
618\autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
619the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
620${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$
621(and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
622
623A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
624(second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}).
625This term must balance the input of potential energy resulting from vertical mixing.
626The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by
627multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically:
628\begin{equation}
629  \label{eq:ZDF_energ2}
630  \begin{split}
631    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\
632    &= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}
633    - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\
634    &= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta}
635    + \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  }
636  \end{split}
637\end{equation}
638where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.
639The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because
640there is no diffusive flux through the ocean surface and bottom).
641The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
642Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
643the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and  \autoref{eq:ZDF_gls_e}.
644
645Let us now address the space discretization issue.
646The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in
647the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}).
648A space averaging is thus required to obtain the shear TKE production term.
649By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by
650the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
651Furthermore, the time variation of $e_3$ has be taken into account.
652
653The above energetic considerations leads to the following final discrete form for the TKE equation:
654\begin{equation}
655  \label{eq:ZDF_tke_ene}
656  \begin{split}
657    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv
658    \Biggl\{ \Biggr.
659    &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} }
660        \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\
661    +&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} }
662        \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j}
663    \Biggr. \Biggr\}   \\
664    %
665    - &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\
666    %
667    +&\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]   \\
668    %
669    - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt}
670  \end{split}
671\end{equation}
672where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation)
673are time stepped using a backward scheme (see\autoref{sec:TD_forward_imp}).
674Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible.
675%The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as
676%they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}.
677%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.
678
679%% =================================================================================================
680\section{Convection}
681\label{sec:ZDF_conv}
682
683Static instabilities (\ie\ light potential densities under heavy ones) may occur at particular ocean grid points.
684In nature, convective processes quickly re-establish the static stability of the water column.
685These processes have been removed from the model via the hydrostatic assumption so they must be parameterized.
686Three parameterisations are available to deal with convective processes:
687a non-penetrative convective adjustment or an enhanced vertical diffusion,
688or/and the use of a turbulent closure scheme.
689
690%% =================================================================================================
691\subsection[Non-penetrative convective adjustment (\forcode{ln_tranpc})]{Non-penetrative convective adjustment (\protect\np{ln_tranpc}{ln\_tranpc})}
692\label{subsec:ZDF_npc}
693
694\begin{figure}[!htb]
695  \centering
696  \includegraphics[width=0.66\textwidth]{Fig_npc}
697  \caption[Unstable density profile treated by the non penetrative convective adjustment algorithm]{
698    Example of an unstable density profile treated by
699    the non penetrative convective adjustment algorithm.
700    $1^{st}$ step: the initial profile is checked from the surface to the bottom.
701    It is found to be unstable between levels 3 and 4.
702    They are mixed.
703    The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
704    The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
705    The $1^{st}$ step ends since the density profile is then stable below the level 3.
706    $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
707    levels 2 to 5 are mixed.
708    The new density profile is checked.
709    It is found stable: end of algorithm.}
710  \label{fig:ZDF_npc}
711\end{figure}
712
713Options are defined through the \nam{zdf}{zdf} namelist variables.
714The non-penetrative convective adjustment is used when \np[=.true.]{ln_zdfnpc}{ln\_zdfnpc}.
715It is applied at each \np{nn_npc}{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of
716the water column, but only until the density structure becomes neutrally stable
717(\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below)
718\citep{madec.delecluse.ea_JPO91}.
719The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}):
720starting from the top of the ocean, the first instability is found.
721Assume in the following that the instability is located between levels $k$ and $k+1$.
722The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of
723the water column.
724The new density is then computed by a linear approximation.
725If the new density profile is still unstable between levels $k+1$ and $k+2$,
726levels $k$, $k+1$ and $k+2$ are then mixed.
727This process is repeated until stability is established below the level $k$
728(the mixing process can go down to the ocean bottom).
729The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or
730if there is no deeper instability.
731
732This algorithm is significantly different from mixing statically unstable levels two by two.
733The latter procedure cannot converge with a finite number of iterations for some vertical profiles while
734the algorithm used in \NEMO\ converges for any profile in a number of iterations which is less than
735the number of vertical levels.
736This property is of paramount importance as pointed out by \citet{killworth_iprc89}:
737it avoids the existence of permanent and unrealistic static instabilities at the sea surface.
738This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in
739the north-western Mediterranean Sea \citep{madec.delecluse.ea_JPO91, madec.chartier.ea_DAO91, madec.crepon_iprc91}.
740
741The current implementation has been modified in order to deal with any non linear equation of seawater
742(L. Brodeau, personnal communication).
743Two main differences have been introduced compared to the original algorithm:
744$(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency
745(not the difference in potential density);
746$(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in
747the same way their temperature and salinity has been mixed.
748These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without
749having to recompute the expansion coefficients at each mixing iteration.
750
751%% =================================================================================================
752\subsection[Enhanced vertical diffusion (\forcode{ln_zdfevd})]{Enhanced vertical diffusion (\protect\np{ln_zdfevd}{ln\_zdfevd})}
753\label{subsec:ZDF_evd}
754
755Options are defined through the  \nam{zdf}{zdf} namelist variables.
756The enhanced vertical diffusion parameterisation is used when \np[=.true.]{ln_zdfevd}{ln\_zdfevd}.
757In this case, the vertical eddy mixing coefficients are assigned very large values
758in regions where the stratification is unstable
759(\ie\ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{lazar_phd97, lazar.madec.ea_JPO99}.
760This is done either on tracers only (\np[=0]{nn_evdm}{nn\_evdm}) or
761on both momentum and tracers (\np[=1]{nn_evdm}{nn\_evdm}).
762
763In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np[=1]{nn_evdm}{nn\_evdm},
764the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to
765the namelist parameter \np{rn_avevd}{rn\_avevd}.
766A typical value for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$.
767This parameterisation of convective processes is less time consuming than
768the convective adjustment algorithm presented above when mixing both tracers and
769momentum in the case of static instabilities.
770
771Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$.
772This removes a potential source of divergence of odd and even time step in
773a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}).
774
775%% =================================================================================================
776\subsection[Handling convection with turbulent closure schemes (\forcode{ln_zdf_}\{\forcode{tke,gls,osm}\})]{Handling convection with turbulent closure schemes (\forcode{ln_zdf{tke,gls,osm}})}
777\label{subsec:ZDF_tcs}
778
779The turbulent closure schemes presented in \autoref{subsec:ZDF_tke}, \autoref{subsec:ZDF_gls} and
780\autoref{subsec:ZDF_osm} (\ie\ \np{ln_zdftke}{ln\_zdftke} or \np{ln_zdfgls}{ln\_zdfgls} or \np{ln_zdfosm}{ln\_zdfosm} defined) deal, in theory,
781with statically unstable density profiles.
782In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in
783\autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative.
784It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also of the four neighboring values at
785velocity points $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1}$).
786These large values restore the static stability of the water column in a way similar to that of
787the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}).
788However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by
789the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$,
790because the mixing length scale is bounded by the distance to the sea surface.
791It can thus be useful to combine the enhanced vertical diffusion with the turbulent closure scheme,
792\ie\ setting the \np{ln_zdfnpc}{ln\_zdfnpc} namelist parameter to true and
793defining the turbulent closure (\np{ln_zdftke}{ln\_zdftke} or \np{ln_zdfgls}{ln\_zdfgls} = \forcode{.true.}) all together.
794
795The OSMOSIS turbulent closure scheme already includes enhanced vertical diffusion in the case of convection,
796%as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp},
797therefore \np[=.false.]{ln_zdfevd}{ln\_zdfevd} should be used with the OSMOSIS scheme.
798% gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
799
800%% =================================================================================================
801\section[Double diffusion mixing (\forcode{ln_zdfddm})]{Double diffusion mixing (\protect\np{ln_zdfddm}{ln\_zdfddm})}
802\label{subsec:ZDF_ddm}
803
804%\nlst{namzdf_ddm}
805
806This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the namelist parameter
807\np{ln_zdfddm}{ln\_zdfddm} in \nam{zdf}{zdf}.
808Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa.
809The former condition leads to salt fingering and the latter to diffusive convection.
810Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean.
811\citet{merryfield.holloway.ea_JPO99} include a parameterisation of such phenomena in a global ocean model and show that
812it leads to relatively minor changes in circulation but exerts significant regional influences on
813temperature and salinity.
814
815Diapycnal mixing of S and T are described by diapycnal diffusion coefficients
816\begin{align*}
817  % \label{eq:ZDF_ddm_Kz}
818  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\
819  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
820\end{align*}
821where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
822and $o$ by processes other than double diffusion.
823The rates of double-diffusive mixing depend on the buoyancy ratio
824$R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of
825thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}).
826To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt
827(1981):
828\begin{align}
829  \label{eq:ZDF_ddm_f}
830  A_f^{vS} &=
831             \begin{cases}
832               \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
833               0                              &\text{otherwise}
834             \end{cases}
835  \\         \label{eq:ZDF_ddm_f_T}
836  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho
837\end{align}
838
839\begin{figure}[!t]
840  \centering
841  \includegraphics[width=0.66\textwidth]{Fig_zdfddm}
842  \caption[Diapycnal diffusivities for temperature and salt in regions of salt fingering and
843  diffusive convection]{
844    From \citet{merryfield.holloway.ea_JPO99}:
845    (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in
846    regions of salt fingering.
847    Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and
848    thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
849    (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in
850    regions of diffusive convection.
851    Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
852    The latter is not implemented in \NEMO.}
853  \label{fig:ZDF_ddm}
854\end{figure}
855
856The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of
857buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}).
858Following  \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
859
860To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by
861Federov (1988) is used:
862\begin{align}
863  % \label{eq:ZDF_ddm_d}
864  A_d^{vT} &=
865             \begin{cases}
866               1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
867               &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
868               0                       &\text{otherwise}
869             \end{cases}
870                                       \nonumber \\
871  \label{eq:ZDF_ddm_d_S}
872  A_d^{vS} &=
873             \begin{cases}
874               A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
875               A_d^{vT} \ 0.15 \ R_\rho               &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
876               0                       &\text{otherwise}
877             \end{cases}
878\end{align}
879
880The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in
881\autoref{fig:ZDF_ddm}.
882Implementing this requires computing $R_\rho$ at each grid point on every time step.
883This is done in \mdl{eosbn2} at the same time as $N^2$ is computed.
884This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive).
885
886%% =================================================================================================
887\section[Bottom and top friction (\textit{zdfdrg.F90})]{Bottom and top friction (\protect\mdl{zdfdrg})}
888\label{sec:ZDF_drg}
889
890\begin{listing}
891  \nlst{namdrg}
892  \caption{\forcode{&namdrg}}
893  \label{lst:namdrg}
894\end{listing}
895\begin{listing}
896  \nlst{namdrg_top}
897  \caption{\forcode{&namdrg_top}}
898  \label{lst:namdrg_top}
899\end{listing}
900\begin{listing}
901  \nlst{namdrg_bot}
902  \caption{\forcode{&namdrg_bot}}
903  \label{lst:namdrg_bot}
904\end{listing}
905
906Options to define the top and bottom friction are defined through the \nam{drg}{drg} namelist variables.
907The bottom friction represents the friction generated by the bathymetry.
908The top friction represents the friction generated by the ice shelf/ocean interface.
909As the friction processes at the top and the bottom are treated in and identical way,
910the description below considers mostly the bottom friction case, if not stated otherwise.
911
912Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as
913a condition on the vertical diffusive flux.
914For the bottom boundary layer, one has:
915 \[
916   % \label{eq:ZDF_bfr_flux}
917   A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
918 \]
919where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside
920the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean).
921How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near
922the bottom relative to the Ekman layer depth.
923For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m,
924one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$
925(for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$).
926With a background diffusion coefficient $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
927When the vertical mixing coefficient is this small, using a flux condition is equivalent to
928entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or
929bottom model layer.
930To illustrate this, consider the equation for $u$ at $k$, the last ocean level:
931\begin{equation}
932  \label{eq:ZDF_drg_flux2}
933  \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}}
934\end{equation}
935If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth.
936On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used,
937the turbulent Ekman layer will be represented explicitly by the model.
938However, the logarithmic layer is never represented in current primitive equation model applications:
939it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
940Two choices are available in \NEMO: a linear and a quadratic bottom friction.
941Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in
942the present release of \NEMO.
943
944In the code, the bottom friction is imposed by adding the trend due to the bottom friction to
945 the general momentum trend in \mdl{dynzdf}.
946For the time-split surface pressure gradient algorithm, the momentum trend due to
947the barotropic component needs to be handled separately.
948For this purpose it is convenient to compute and store coefficients which can be simply combined with
949bottom velocities and geometric values to provide the momentum trend due to bottom friction.
950 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where:
951\begin{equation}
952  \label{eq:ZDF_bfr_bdef}
953  \frac{\partial {\textbf U_h}}{\partial t} =
954  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b
955\end{equation}
956where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity.
957Note 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.
958
959%% =================================================================================================
960\subsection[Linear top/bottom friction (\forcode{ln_lin})]{Linear top/bottom friction (\protect\np{ln_lin}{ln\_lin})}
961\label{subsec:ZDF_drg_linear}
962
963The linear friction parameterisation (including the special case of a free-slip condition) assumes that
964the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level):
965\[
966  % \label{eq:ZDF_bfr_linear}
967  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
968\]
969where $r$ is a friction coefficient expressed in $m s^{-1}$.
970This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean,
971and setting $r = H / \tau$, where $H$ is the ocean depth.
972Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{weatherly_JMR84}.
973A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models.
974One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$
975(\citet{gill_bk82}, Eq. 9.6.6).
976For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$,
977and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.
978This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days.
979It can be changed by specifying \np{rn_Uc0}{rn\_Uc0} (namelist parameter).
980
981 For the linear friction case the drag coefficient used in the general expression \autoref{eq:ZDF_bfr_bdef} is:
982\[
983  % \label{eq:ZDF_bfr_linbfr_b}
984    c_b^T = - r
985\]
986When \np[=.true.]{ln_lin}{ln\_lin}, the value of $r$ used is \np{rn_Uc0}{rn\_Uc0}*\np{rn_Cd0}{rn\_Cd0}.
987Setting \np[=.true.]{ln_OFF}{ln\_OFF} (and \forcode{ln_lin=.true.}) is equivalent to setting $r=0$ and leads to a free-slip boundary condition.
988
989These values are assigned in \mdl{zdfdrg}.
990Note that there is support for local enhancement of these values via an externally defined 2D mask array
991(\np[=.true.]{ln_boost}{ln\_boost}) given in the \ifile{bfr\_coef} input NetCDF file.
992The mask values should vary from 0 to 1.
993Locations with a non-zero mask value will have the friction coefficient increased by
994$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}.
995
996%% =================================================================================================
997\subsection[Non-linear top/bottom friction (\forcode{ln_non_lin})]{Non-linear top/bottom friction (\protect\np{ln_non_lin}{ln\_non\_lin})}
998\label{subsec:ZDF_drg_nonlinear}
999
1000The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic:
1001\[
1002  % \label{eq:ZDF_drg_nonlinear}
1003  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
1004  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
1005\]
1006where $C_D$ is a drag coefficient, and $e_b $ a top/bottom turbulent kinetic energy due to tides,
1007internal waves breaking and other short time scale currents.
1008A typical value of the drag coefficient is $C_D = 10^{-3} $.
1009As an example, the CME experiment \citep{treguier_JGR92} uses $C_D = 10^{-3}$ and
1010$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{killworth_JPO92} uses $C_D = 1.4\;10^{-3}$ and
1011$e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.
1012The CME choices have been set as default values (\np{rn_Cd0}{rn\_Cd0} and \np{rn_ke0}{rn\_ke0} namelist parameters).
1013
1014As for the linear case, the friction is imposed in the code by adding the trend due to
1015the friction to the general momentum trend in \mdl{dynzdf}.
1016For the non-linear friction case the term computed in \mdl{zdfdrg} is:
1017\[
1018  % \label{eq:ZDF_drg_nonlinbfr}
1019    c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2}
1020\]
1021
1022The coefficients that control the strength of the non-linear friction are initialised as namelist parameters:
1023$C_D$= \np{rn_Cd0}{rn\_Cd0}, and $e_b$ =\np{rn_bfeb2}{rn\_bfeb2}.
1024Note that for applications which consider tides explicitly, a low or even zero value of \np{rn_bfeb2}{rn\_bfeb2} is recommended. A local enhancement of $C_D$ is again possible via an externally defined 2D mask array
1025(\np[=.true.]{ln_boost}{ln\_boost}).
1026This works in the same way as for the linear friction case with non-zero masked locations increased by
1027$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}.
1028
1029%% =================================================================================================
1030\subsection[Log-layer top/bottom friction (\forcode{ln_loglayer})]{Log-layer top/bottom friction (\protect\np{ln_loglayer}{ln\_loglayer})}
1031\label{subsec:ZDF_drg_loglayer}
1032
1033In the non-linear friction case, the drag coefficient, $C_D$, can be optionally enhanced using
1034a "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.
1035If  \np[=.true.]{ln_loglayer}{ln\_loglayer}, $C_D$ is no longer constant but is related to the distance to the wall (or equivalently to the half of the top/bottom layer thickness):
1036\[
1037  C_D = \left ( {\kappa \over {\mathrm log}\left ( 0.5 \; e_{3b} / rn\_{z0} \right ) } \right )^2
1038\]
1039
1040\noindent where $\kappa$ is the von-Karman constant and \np{rn_z0}{rn\_z0} is a roughness length provided via the namelist.
1041
1042The drag coefficient is bounded such that it is kept greater or equal to
1043the base \np{rn_Cd0}{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:
1044\np{rn_Cdmax}{rn\_Cdmax}, \ie
1045\[
1046  rn\_Cd0 \leq C_D \leq rn\_Cdmax
1047\]
1048
1049\noindent The log-layer enhancement can also be applied to the top boundary friction if
1050under ice-shelf cavities are activated (\np[=.true.]{ln_isfcav}{ln\_isfcav}).
1051%In this case, the relevant namelist parameters are \np{rn_tfrz0}{rn\_tfrz0}, \np{rn_tfri2}{rn\_tfri2} and \np{rn_tfri2_max}{rn\_tfri2\_max}.
1052
1053%% =================================================================================================
1054\subsection[Explicit top/bottom friction (\forcode{ln_drgimp=.false.})]{Explicit top/bottom friction (\protect\np[=.false.]{ln_drgimp}{ln\_drgimp})}
1055\label{subsec:ZDF_drg_stability}
1056
1057Setting \np[=.false.]{ln_drgimp}{ln\_drgimp} 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:
1058
1059At the top (below an ice shelf cavity):
1060\[
1061  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1062  = c_{t}^{\textbf{U}}\textbf{u}^{n-1}_{t}
1063\]
1064
1065At the bottom (above the sea floor):
1066\[
1067  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1068  = c_{b}^{\textbf{U}}\textbf{u}^{n-1}_{b}
1069\]
1070
1071Since 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.
1072For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is:
1073\begin{equation}
1074  \label{eq:ZDF_Eqn_drgstab}
1075  \begin{split}
1076    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
1077    &= -\frac{ru}{e_{3u}}\;2\rdt\\
1078  \end{split}
1079\end{equation}
1080\noindent where linear friction and a leapfrog timestep have been assumed.
1081To ensure that the friction cannot reverse the direction of flow it is necessary to have:
1082\[
1083  |\Delta u| < \;|u|
1084\]
1085\noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives:
1086\[
1087  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
1088\]
1089This same inequality can also be derived in the non-linear bottom friction case if
1090a velocity of 1 m.s$^{-1}$ is assumed.
1091Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability:
1092\[
1093  e_{3u} > 2\;r\;\rdt
1094\]
1095\noindent which it may be necessary to impose if partial steps are being used.
1096For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m.
1097For most applications, with physically sensible parameters these restrictions should not be of concern.
1098But caution may be necessary if attempts are made to locally enhance the bottom friction parameters.
1099To ensure stability limits are imposed on the top/bottom friction coefficients both
1100during initialisation and at each time step.
1101Checks at initialisation are made in \mdl{zdfdrg} (assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
1102The number of breaches of the stability criterion are reported as well as
1103the minimum and maximum values that have been set.
1104The criterion is also checked at each time step, using the actual velocity, in \mdl{dynzdf}.
1105Values of the friction coefficient are reduced as necessary to ensure stability;
1106these changes are not reported.
1107
1108Limits on the top/bottom friction coefficient are not imposed if the user has elected to
1109handle the friction implicitly (see \autoref{subsec:ZDF_drg_imp}).
1110The number of potential breaches of the explicit stability criterion are still reported for information purposes.
1111
1112%% =================================================================================================
1113\subsection[Implicit top/bottom friction (\forcode{ln_drgimp=.true.})]{Implicit top/bottom friction (\protect\np[=.true.]{ln_drgimp}{ln\_drgimp})}
1114\label{subsec:ZDF_drg_imp}
1115
1116An optional implicit form of bottom friction has been implemented to improve model stability.
1117We recommend this option for shelf sea and coastal ocean applications. %, especially for split-explicit time splitting.
1118This option can be invoked by setting \np{ln_drgimp}{ln\_drgimp} to \forcode{.true.} in the \nam{drg}{drg} namelist.
1119%This option requires \np{ln_zdfexp}{ln\_zdfexp} to be \forcode{.false.} in the \nam{zdf}{zdf} namelist.
1120
1121This implementation is performed in \mdl{dynzdf} where the following boundary conditions are set while solving the fully implicit diffusion step:
1122
1123At the top (below an ice shelf cavity):
1124\[
1125  % \label{eq:ZDF_dynZDF__drg_top}
1126  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1127  = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t}
1128\]
1129
1130At the bottom (above the sea floor):
1131\[
1132  % \label{eq:ZDF_dynZDF__drg_bot}
1133  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1134  = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b}
1135\]
1136
1137where $t$ and $b$ refers to top and bottom layers respectively.
1138Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so it is implicit.
1139
1140%% =================================================================================================
1141\subsection[Bottom friction with split-explicit free surface]{Bottom friction with split-explicit free surface}
1142\label{subsec:ZDF_drg_ts}
1143
1144With 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[=.false.]{ln_drgimp}{ln\_drgimp} 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[=.true.]{ln_drgimp}{ln\_drgimp},  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.
1145
1146The strategy to handle top/bottom stresses with split-explicit free surface in \NEMO\ is as follows:
1147\begin{enumerate}
1148\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.
1149\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.
1150\end{enumerate}
1151
1152Note that other strategies are possible, like considering vertical diffusion step in advance, \ie\ prior barotropic integration.
1153
1154%% =================================================================================================
1155\section[Internal wave-driven mixing (\forcode{ln_zdfiwm})]{Internal wave-driven mixing (\protect\np{ln_zdfiwm}{ln\_zdfiwm})}
1156\label{subsec:ZDF_tmx_new}
1157
1158\begin{listing}
1159  \nlst{namzdf_iwm}
1160  \caption{\forcode{&namzdf_iwm}}
1161  \label{lst:namzdf_iwm}
1162\end{listing}
1163
1164The parameterization of mixing induced by breaking internal waves is a generalization of
1165the approach originally proposed by \citet{st-laurent.simmons.ea_GRL02}.
1166A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed,
1167and the resulting diffusivity is obtained as
1168\[
1169  % \label{eq:ZDF_Kwave}
1170  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
1171\]
1172where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of
1173the energy available for mixing.
1174If the \np{ln_mevar}{ln\_mevar} namelist parameter is set to \forcode{.false.}, the mixing efficiency is taken as constant and
1175equal to 1/6 \citep{osborn_JPO80}.
1176In the opposite (recommended) case, $R_f$ is instead a function of
1177the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$,
1178with $\nu$ the molecular viscosity of seawater, following the model of \cite{bouffard.boegman_DAO13} and
1179the implementation of \cite{de-lavergne.madec.ea_JPO16}.
1180Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when
1181the mixing efficiency is constant.
1182
1183In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary
1184as a function of $Re_b$ by setting the \np{ln_tsdiff}{ln\_tsdiff} parameter to \forcode{.true.}, a recommended choice.
1185This parameterization of differential mixing, due to \cite{jackson.rehmann_JPO14},
1186is implemented as in \cite{de-lavergne.madec.ea_JPO16}.
1187
1188The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$,
1189is constructed from three static maps of column-integrated internal wave energy dissipation,
1190$E_{cri}(i,j)$, $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures:
1191
1192\begin{align*}
1193  F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\
1194  F_{pyc}(i,j,k) &\propto N^{n_p}\\
1195  F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} }
1196\end{align*}
1197In the above formula, $h_{ab}$ denotes the height above bottom,
1198$h_{wkb}$ denotes the WKB-stretched height above bottom, defined by
1199\[
1200  h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; ,
1201\]
1202The $n_p$ parameter (given by \np{nn_zpyc}{nn\_zpyc} in \nam{zdf_iwm}{zdf\_iwm} namelist)
1203controls the stratification-dependence of the pycnocline-intensified dissipation.
1204It can take values of $1$ (recommended) or $2$.
1205Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of
1206the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps.
1207$h_{cri}$ is related to the large-scale topography of the ocean (etopo2) and
1208$h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of
1209the abyssal hill topography \citep{goff_JGR10} and the latitude.
1210% Jc: input files names ?
1211
1212%% =================================================================================================
1213\section[Surface wave-induced mixing (\forcode{ln_zdfswm})]{Surface wave-induced mixing (\protect\np{ln_zdfswm}{ln\_zdfswm})}
1214\label{subsec:ZDF_swm}
1215
1216Surface waves produce an enhanced mixing through wave-turbulence interaction.
1217In addition to breaking waves induced turbulence (\autoref{subsec:ZDF_tke}),
1218the influence of non-breaking waves can be accounted introducing
1219wave-induced viscosity and diffusivity as a function of the wave number spectrum.
1220Following \citet{qiao.yuan.ea_OD10}, a formulation of wave-induced mixing coefficient
1221is provided  as a function of wave amplitude, Stokes Drift and wave-number:
1222
1223\begin{equation}
1224  \label{eq:ZDF_Bv}
1225  B_{v} = \alpha {A} {U}_{st} {exp(3kz)}
1226\end{equation}
1227
1228Where $B_{v}$ is the wave-induced mixing coefficient, $A$ is the wave amplitude,
1229${U}_{st}$ is the Stokes Drift velocity, $k$ is the wave number and $\alpha$
1230is a constant which should be determined by observations or
1231numerical experiments and is set to be 1.
1232
1233The coefficient $B_{v}$ is then directly added to the vertical viscosity
1234and diffusivity coefficients.
1235
1236In order to account for this contribution set: \forcode{ln_zdfswm=.true.},
1237then wave interaction has to be activated through \forcode{ln_wave=.true.},
1238the Stokes Drift can be evaluated by setting \forcode{ln_sdw=.true.}
1239(see \autoref{subsec:SBC_wave_sdw})
1240and the needed wave fields can be provided either in forcing or coupled mode
1241(for more information on wave parameters and settings see \autoref{sec:SBC_wave})
1242
1243%% =================================================================================================
1244\section[Adaptive-implicit vertical advection (\forcode{ln_zad_Aimp})]{Adaptive-implicit vertical advection(\protect\np{ln_zad_Aimp}{ln\_zad\_Aimp})}
1245\label{subsec:ZDF_aimp}
1246
1247The adaptive-implicit vertical advection option in NEMO is based on the work of
1248\citep{shchepetkin_OM15}.  In common with most ocean models, the timestep used with NEMO
1249needs to satisfy multiple criteria associated with different physical processes in order
1250to maintain numerical stability. \citep{shchepetkin_OM15} pointed out that the vertical
1251CFL criterion is commonly the most limiting. \citep{lemarie.debreu.ea_OM15} examined the
1252constraints for a range of time and space discretizations and provide the CFL stability
1253criteria for a range of advection schemes. The values for the Leap-Frog with Robert
1254asselin filter time-stepping (as used in NEMO) are reproduced in
1255\autoref{tab:ZDF_zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these
1256restrictions but at the cost of large dispersive errors and, possibly, large numerical
1257viscosity. The adaptive-implicit vertical advection option provides a targetted use of the
1258implicit scheme only when and where potential breaches of the vertical CFL condition
1259occur. In many practical applications these events may occur remote from the main area of
1260interest or due to short-lived conditions such that the extra numerical diffusion or
1261viscosity does not greatly affect the overall solution. With such applications, setting:
1262\forcode{ln_zad_Aimp=.true.} should allow much longer model timesteps to be used whilst
1263retaining the accuracy of the high order explicit schemes over most of the domain.
1264
1265\begin{table}[htbp]
1266  \centering
1267  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}}
1268  \begin{tabular}{r|ccc}
1269    \hline
1270    spatial discretization  & 2$^nd$ order centered & 3$^rd$ order upwind & 4$^th$ order compact \\
1271    advective CFL criterion &                 0.904 &              0.472  &                0.522 \\
1272    \hline
1273  \end{tabular}
1274  \caption[Advective CFL criteria for the leapfrog with Robert Asselin filter time-stepping]{
1275    The advective CFL criteria for a range of spatial discretizations for
1276    the leapfrog with Robert Asselin filter time-stepping
1277    ($\nu=0.1$) as given in \citep{lemarie.debreu.ea_OM15}.}
1278  \label{tab:ZDF_zad_Aimp_CFLcrit}
1279\end{table}
1280
1281In particular, the advection scheme remains explicit everywhere except where and when
1282local vertical velocities exceed a threshold set just below the explicit stability limit.
1283Once the threshold is reached a tapered transition towards an implicit scheme is used by
1284partitioning the vertical velocity into a part that can be treated explicitly and any
1285excess that must be treated implicitly. The partitioning is achieved via a Courant-number
1286dependent weighting algorithm as described in \citep{shchepetkin_OM15}.
1287
1288The local cell Courant number ($Cu$) used for this partitioning is:
1289
1290\begin{equation}
1291  \label{eq:ZDF_Eqn_zad_Aimp_Courant}
1292  \begin{split}
1293    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 ]    \\
1294       &\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 ]
1295                     \big / e_{{1_t}ij}e_{{2_t}ij}            \\
1296       &\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 ]
1297                     \big / e_{{1_t}ij}e_{{2_t}ij} \bigg )    \\
1298  \end{split}
1299\end{equation}
1300
1301\noindent and the tapering algorithm follows \citep{shchepetkin_OM15} as:
1302
1303\begin{align}
1304  \label{eq:ZDF_Eqn_zad_Aimp_partition}
1305Cu_{min} &= 0.15 \nonumber \\
1306Cu_{max} &= 0.3  \nonumber \\
1307Cu_{cut} &= 2Cu_{max} - Cu_{min} \nonumber \\
1308Fcu    &= 4Cu_{max}*(Cu_{max}-Cu_{min}) \nonumber \\
1309\cf &=
1310     \begin{cases}
1311        0.0                                                        &\text{if $Cu \leq Cu_{min}$} \\
1312        (Cu - Cu_{min})^2 / (Fcu +  (Cu - Cu_{min})^2)             &\text{else if $Cu < Cu_{cut}$} \\
1313        (Cu - Cu_{max}) / Cu                                       &\text{else}
1314     \end{cases}
1315\end{align}
1316
1317\begin{figure}[!t]
1318  \centering
1319  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_coeff}
1320  \caption[Partitioning coefficient used to partition vertical velocities into parts]{
1321    The value of the partitioning coefficient (\cf) used to partition vertical velocities into
1322    parts to be treated implicitly and explicitly for a range of typical Courant numbers
1323    (\forcode{ln_zad_Aimp=.true.}).}
1324  \label{fig:ZDF_zad_Aimp_coeff}
1325\end{figure}
1326
1327\noindent The partitioning coefficient is used to determine the part of the vertical
1328velocity that must be handled implicitly ($w_i$) and to subtract this from the total
1329vertical velocity ($w_n$) to leave that which can continue to be handled explicitly:
1330
1331\begin{align}
1332  \label{eq:ZDF_Eqn_zad_Aimp_partition2}
1333    w_{i_{ijk}} &= \cf_{ijk} w_{n_{ijk}}     \nonumber \\
1334    w_{n_{ijk}} &= (1-\cf_{ijk}) w_{n_{ijk}}
1335\end{align}
1336
1337\noindent Note that the coefficient is such that the treatment is never fully implicit;
1338the three cases from \autoref{eq:ZDF_Eqn_zad_Aimp_partition} can be considered as:
1339fully-explicit; mixed explicit/implicit and mostly-implicit.  With the settings shown the
1340coefficient (\cf) varies as shown in \autoref{fig:ZDF_zad_Aimp_coeff}. Note with these values
1341the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly
1342implicit' is 0.45 which is just below the stability limited given in
1343\autoref{tab:ZDF_zad_Aimp_CFLcrit}  for a 3rd order scheme.
1344
1345The $w_i$ component is added to the implicit solvers for the vertical mixing in
1346\mdl{dynzdf} and \mdl{trazdf} in a similar way to \citep{shchepetkin_OM15}.  This is
1347sufficient for the flux-limited advection scheme (\forcode{ln_traadv_mus}) but further
1348intervention is required when using the flux-corrected scheme (\forcode{ln_traadv_fct}).
1349For these schemes the implicit upstream fluxes must be added to both the monotonic guess
1350and to the higher order solution when calculating the antidiffusive fluxes. The implicit
1351vertical fluxes are then removed since they are added by the implicit solver later on.
1352
1353The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be
1354used in a wide range of simulations. The following test simulation, however, does illustrate
1355the potential benefits and will hopefully encourage further testing and feedback from users:
1356
1357\begin{figure}[!t]
1358  \centering
1359  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames}
1360  \caption[OVERFLOW: time-series of temperature vertical cross-sections]{
1361    A time-series of temperature vertical cross-sections for the OVERFLOW test case.
1362    These results are for the default settings with \forcode{nn_rdt=10.0} and
1363    without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}).}
1364  \label{fig:ZDF_zad_Aimp_overflow_frames}
1365\end{figure}
1366
1367%% =================================================================================================
1368\subsection{Adaptive-implicit vertical advection in the OVERFLOW test-case}
1369
1370The \href{https://forge.ipsl.jussieu.fr/nemo/chrome/site/doc/NEMO/guide/html/test\_cases.html\#overflow}{OVERFLOW test case}
1371provides a simple illustration of the adaptive-implicit advection in action. The example here differs from the basic test case
1372by only a few extra physics choices namely:
1373
1374\begin{verbatim}
1375     ln_dynldf_OFF = .false.
1376     ln_dynldf_lap = .true.
1377     ln_dynldf_hor = .true.
1378     ln_zdfnpc     = .true.
1379     ln_traadv_fct = .true.
1380        nn_fct_h   =  2
1381        nn_fct_v   =  2
1382\end{verbatim}
1383
1384\noindent which were chosen to provide a slightly more stable and less noisy solution. The
1385result when using the default value of \forcode{nn_rdt=10.} without adaptive-implicit
1386vertical velocity is illustrated in \autoref{fig:ZDF_zad_Aimp_overflow_frames}. The mass of
1387cold water, initially sitting on the shelf, moves down the slope and forms a
1388bottom-trapped, dense plume. Even with these extra physics choices the model is close to
1389stability limits and attempts with \forcode{nn_rdt=30.} will fail after about 5.5 hours
1390with excessively high horizontal velocities. This time-scale corresponds with the time the
1391plume reaches the steepest part of the topography and, although detected as a horizontal
1392CFL breach, the instability originates from a breach of the vertical CFL limit. This is a good
1393candidate, therefore, for use of the adaptive-implicit vertical advection scheme.
1394
1395The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps
1396are shown in \autoref{fig:ZDF_zad_Aimp_overflow_all_rdt} (together with the equivalent
1397frames from the base run).  In this simple example the use of the adaptive-implicit
1398vertcal advection scheme has enabled a 12x increase in the model timestep without
1399significantly altering the solution (although at this extreme the plume is more diffuse
1400and has not travelled so far).  Notably, the solution with and without the scheme is
1401slightly different even with \forcode{nn_rdt=10.}; suggesting that the base run was
1402close enough to instability to trigger the scheme despite completing successfully.
1403To assist in diagnosing how active the scheme is, in both location and time, the 3D
1404implicit and explicit components of the vertical velocity are available via XIOS as
1405\texttt{wimp} and \texttt{wexp} respectively.  Likewise, the partitioning coefficient
1406(\cf) is also available as \texttt{wi\_cff}. For a quick oversight of
1407the schemes activity the global maximum values of the absolute implicit component
1408of the vertical velocity and the partitioning coefficient are written to the netCDF
1409version of the run statistics file (\texttt{run.stat.nc}) if this is active (see
1410\autoref{sec:MISC_opt} for activation details).
1411
1412\autoref{fig:ZDF_zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for
1413the various overflow tests.  Note that the adaptive-implicit vertical advection scheme is
1414active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the
1415test case is close to stability limits even with this value. At the larger timesteps, the
1416vertical velocity is treated mostly implicitly at some location throughout the run. The
1417oscillatory nature of this measure appears to be linked to the progress of the plume front
1418as each cusp is associated with the location of the maximum shifting to the adjacent cell.
1419This is illustrated in \autoref{fig:ZDF_zad_Aimp_maxCf_loc} where the i- and k- locations of the
1420maximum have been overlaid for the base run case.
1421
1422\medskip
1423\noindent Only limited tests have been performed in more realistic configurations. In the
1424ORCA2\_ICE\_PISCES reference configuration the scheme does activate and passes
1425restartability and reproducibility tests but it is unable to improve the model's stability
1426enough to allow an increase in the model time-step. A view of the time-series of maximum
1427partitioning coefficient (not shown here)  suggests that the default time-step of 5400s is
1428already pushing at stability limits, especially in the initial start-up phase. The
1429time-series does not, however, exhibit any of the 'cuspiness' found with the overflow
1430tests.
1431
1432\medskip
1433\noindent A short test with an eORCA1 configuration promises more since a test using a
1434time-step of 3600s remains stable with \forcode{ln_zad_Aimp=.true.} whereas the
1435time-step is limited to 2700s without.
1436
1437\begin{figure}[!t]
1438  \centering
1439  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_overflow_all_rdt}
1440  \caption[OVERFLOW: sample temperature vertical cross-sections from mid- and end-run]{
1441    Sample temperature vertical cross-sections from mid- and end-run using
1442    different values for \forcode{nn_rdt} and with or without adaptive implicit vertical advection.
1443    Without the adaptive implicit vertical advection
1444    only the run with the shortest timestep is able to run to completion.
1445    Note also that the colour-scale has been chosen to confirm that
1446    temperatures remain within the original range of 10$^o$ to 20$^o$.}
1447  \label{fig:ZDF_zad_Aimp_overflow_all_rdt}
1448\end{figure}
1449
1450\begin{figure}[!t]
1451  \centering
1452  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_maxCf}
1453  \caption[OVERFLOW: maximum partitioning coefficient during a series of test runs]{
1454    The maximum partitioning coefficient during a series of test runs with
1455    increasing model timestep length.
1456    At the larger timesteps,
1457    the vertical velocity is treated mostly implicitly at some location throughout the run.}
1458  \label{fig:ZDF_zad_Aimp_maxCf}
1459\end{figure}
1460
1461\begin{figure}[!t]
1462  \centering
1463  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_maxCf_loc}
1464  \caption[OVERFLOW: maximum partitioning coefficient for the case overlaid]{
1465    The maximum partitioning coefficient for the \forcode{nn_rdt=10.0} case overlaid with
1466    information on the gridcell i- and k-locations of the maximum value.}
1467  \label{fig:ZDF_zad_Aimp_maxCf_loc}
1468\end{figure}
1469
1470\onlyinsubfile{\input{../../global/epilogue}}
1471
1472\end{document}
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