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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}{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
548%--------------------------------------------------------------------------------------------------------------
549\paragraph{Namelist choices}
550Most of the namelist options refer to how to specify the Stokes
551surface drift and penetration depth. There are three options:
552\begin{description}
553  \item \protect\np[=0]{nn_osm_wave}{nn\_osm\_wave} Default value in \texttt{namelist\_ref}. In this case the Stokes drift is
554      assumed to be parallel to the surface wind stress, with
555      magnitude consistent with a constant turbulent Langmuir number
556    $\mathrm{La}_t=$ \protect\np{rn_m_la} {rn\_m\_la} i.e.\
557    $u_{s0}=\tau/(\mathrm{La}_t^2\rho_0)$.  Default value of
558    \protect\np{rn_m_la}{rn\_m\_la} is 0.3. The Stokes penetration
559      depth $\delta = $ \protect\np{rn_osm_dstokes}{rn\_osm\_dstokes}; this has default value
560      of \SI{5}{m}.
561 
562  \item \protect\np[=1]{nn_osm_wave}{nn\_osm\_wave} In this case the Stokes drift is
563      assumed to be parallel to the surface wind stress, with
564      magnitude as in the classical Pierson-Moskowitz wind-sea
565      spectrum.  Significant wave height and
566      wave-mean period taken from this spectrum are used to calculate the Stokes penetration
567      depth, following the approach set out in Breivik(XXxx)
568 
569    \item \protect\np[=2]{nn\_osm_wave}{nn\_osm\_wave} In this case the Stokes drift is
570      taken from  ECMWF wave model output, though only the component parallel
571      to the wind stress is retained. Significant wave height and
572      wave-mean period from ECMWF wave model output are used to calculate the Stokes penetration
573      depth, following the approach set out in Breivik(XXxx).
574
575    \end{description}
576
577    Others refer to the treatment of diffusion and viscosity beneath
578    the surface boundary layer:
579\begin{description}
580   \item \protect\np{ln_kpprimix} {ln\_kpprimix}  Default is \np{.true.}. Switches on KPP-style Ri \#-dependent
581     mixing below the surface boundary layer. If this is set
582     \np{.true.}  the following variable settings are honoured:
583    \item \protect\np{rn_riinfty}{rn\_riinfty} Critical value of local Ri \# below which
584      shear instability increases vertical mixing from background value.
585    \item \protect\np{rn_difri} {rn\_difri} Maximum value of Ri \#-dependent mixing at $\mathrm{Ri}=0$.
586    \item \protect\np{ln_convmix}{ln\_convmix} If \texttt{.true.} then, where water column is unstable, specify
587       diffusivity equal to \protect\np{rn_dif_conv}{rn\_dif\_conv} (default value is 1 ms$^{-2}$).
588 \end{description}
589 Diagnostic output is controlled by:
590  \begin{description}
591    \item \protect\np{ln_dia_osm}{ln\_dia\_osm} Default is \np{.false.}; allows XIOS output of OSMOSIS internal fields.
592  \end{description}
593Obsolete namelist parameters include:
594\begin{description}
595   \item \protect\np{ln_use_osm_la}\np{ln\_use\_osm\_la} With \protect\np[=0]{nn_osm_wave}{nn\_osm\_wave},
596      \protect\np{rn_osm_dstokes} {rn\_osm\_dstokes} is always used to specify the Stokes
597      penetration depth.
598   \item \protect\np{nn_ave} {nn\_ave} Choice of averaging method for KPP-style Ri \#
599      mixing. Not taken account of.
600   \item \protect\np{rn_osm_hbl0} {rn\_osm\_hbl0} Depth of initial boundary layer is now set
601     by a density criterion similar to that used in calculating hmlp (output as mldr10_1) in zdfmxl.F90.
602\end{description}
603
604\subsubsection{Summary}
605Much of the time the turbulent motions in the ocean surface boundary
606layer (OSBL) are not given by
607classical shear turbulence. Instead they are in a regime known as
608`Langmuir turbulence',  dominated by an
609interaction between the currents and the Stokes drift of the surface waves \citep[e.g.][]{mcwilliams.ea_JFM97}.
610This regime is characterised by strong vertical turbulent motion, and appears when the surface Stokes drift $u_{s0}$ is much greater than the friction velocity $u_{\ast}$. More specifically Langmuir turbulence is thought to be crucial where the turbulent Langmuir number $\mathrm{La}_{t}=(u_{\ast}/u_{s0}) > 0.4$.
611
612The OSMOSIS model is fundamentally based on results of Large Eddy
613Simulations (LES) of Langmuir turbulence and aims to fully describe
614this Langmuir regime. The description in this section is of necessity incomplete and further details are available in the manuscript ``The OSMOSIS scheme'', Grant. A (2019); in prep.
615
616The OSMOSIS turbulent closure scheme is a similarity-scale scheme in
617the same spirit as the K-profile
618parameterization (KPP) scheme of \citet{large.ea_RG97}.
619A specified shape of diffusivity, scaled by the (OSBL) depth
620$h_{\mathrm{BL}}$ and a turbulent velocity scale, is imposed throughout the
621boundary layer
622$-h_{\mathrm{BL}}<z<\eta$. The turbulent closure model
623also includes fluxes of tracers and momentum that are``non-local'' (independent of the local property gradient).
624
625Rather than the OSBL
626depth being diagnosed in terms of a bulk Richardson number criterion,
627as in KPP, it is set by a prognostic equation that is informed by
628energy budget considerations reminiscent of the classical mixed layer
629models of \citet{kraus.turner_tellus67}.
630The model also includes an explicit parametrization of the structure
631of the pycnocline (the stratified region at the bottom of the OSBL).
632
633Presently, mixing below the OSBL is handled by the Richardson
634number-dependent mixing scheme used in \citet{large.ea_RG97}.
635
636Convective parameterizations such as described in \ref{sec:ZDF_conv}
637below should not be used with the OSMOSIS-OBL model: instabilities
638within the OSBL are part of the model, while instabilities below the
639ML are handled by the Ri \# dependent scheme.
640
641\subsubsection{Depth and velocity scales}
642The model supposes a boundary layer of thickness $h_{\mathrm{bl}}$ enclosing a well-mixed layer of thickness $h_{\mathrm{ml}}$ and a relatively thin pycnocline at the base of thickness $\Delta h$; Fig.~\ref{fig: OSBL_structure} shows typical (a) buoyancy structure and (b) turbulent buoyancy flux profile for the unstable boundary layer (losing buoyancy at the surface; e.g.\ cooling).
643\begin{figure}[!t]
644  \begin{center}
645    \includegraphics[width=0.7\textwidth]{Fig_ZDF_OSM_structure_of_OSBL}
646    \caption{
647      \protect\label{fig: OSBL_structure}
648     The structure of the entraining  boundary layer. (a) Mean buoyancy profile. (b) Profile of the buoyancy flux.
649    }
650  \end{center}
651\end{figure}
652The pycnocline in the OSMOSIS scheme is assumed to have a finite thickness, and may include a number of model levels. This means that the OSMOSIS scheme must parametrize both the thickness of the pycnocline, and the turbulent fluxes within the pycnocline.
653
654Consideration of the power input by wind acting on the Stokes drift suggests that the Langmuir turbulence has velocity scale:
655\begin{equation}\label{eq:w_La}
656w_{*L}= \left(u_*^2 u_{s\,0}\right)^{1/3};
657\end{equation} 
658but at times the Stokes drift may be weak due to e.g.\ ice cover, short fetch, misalignment with the surface stress, etc.\ so  a composite velocity scale is assumed for the stable (warming) boundary layer:
659\begin{equation}\label{eq:composite-nu}
660  \nu_{\ast}= \left\{ u_*^3 \left[1-\exp(-.5 \mathrm{La}_t^2)\right]+w_{*L}^3\right\}^{1/3}.
661\end{equation}
662For the unstable boundary layer this is merged with the standard convective velocity scale $w_{*C}=\left(\overline{w^\prime b^\prime}_0 \,h_\mathrm{ml}\right)^{1/3}$, where $\overline{w^\prime b^\prime}_0$ is the upwards surface buoyancy flux, to give:
663\begin{equation}\label{eq:vel-scale-unstable}
664\omega_* = \left(\nu_*^3 + 0.5 w_{*C}^3\right)^{1/3}.
665\end{equation}
666
667\subsubsection{The flux gradient model}
668The flux-gradient relationships used in the OSMOSIS scheme take the form:
669%
670\begin{equation}\label{eq:flux-grad-gen}
671\overline{w^\prime\chi^\prime}=-K\frac{\partial\overline{\chi}}{\partial z} + N_{\chi,s} +N_{\chi,b} +N_{\chi,t},
672\end{equation}
673%
674where $\chi$ is a general variable and $N_{\chi,s}, N_{\chi,b} \mathrm{and} N_{\chi,t}$  are the non-gradient terms, and represent the effects of the different terms in the turbulent flux-budget on the transport of $\chi$. $N_{\chi,s}$ represents the effects that the Stokes shear has on the transport of $\chi$, $N_{\chi,b}$  the effect of buoyancy, and $N_{\chi,t}$ the effect of the turbulent transport.  The same general form for the flux-gradient relationship is used to parametrize the transports of momentum, heat and salinity.
675
676In terms of the non-dimensionalized depth variables
677%
678\begin{equation}\label{eq:sigma}
679\sigma_{\mathrm{ml}}= -z/h_{\mathrm{ml}}; \;\sigma_{\mathrm{bl}}= -z/h_{\mathrm{bl}},
680\end{equation}
681%
682in unstable conditions the eddy diffusivity ($K_d$) and eddy viscosity ($K_\nu$) profiles are parametrized as:
683%
684\begin{align}\label{eq:diff-unstable}
685K_d=&0.8\, \omega_*\, h_{\mathrm{ml}} \, \sigma_{\mathrm{ml}} \left(1-\beta_d \sigma_{\mathrm{ml}}\right)^{3/2}
686\\\label{eq:visc-unstable}
687K_\nu =& 0.3\, \omega_* \,h_{\mathrm{ml}}\, \sigma_{\mathrm{ml}} \left(1-\beta_\nu \sigma_{\mathrm{ml}}\right)\left(1-\tfrac{1}{2}\sigma_{\mathrm{ml}}^2\right)
688\end{align}
689%
690where $\beta_d$ and $\beta_\nu$ are parameters that are determined by matching Eqs \ref{eq:diff-unstable} and \ref{eq:visc-unstable} to the eddy diffusivity and viscosity at the base of the well-mixed layer, given by
691%
692\begin{equation}\label{eq:diff-wml-base} 
693K_{d,\mathrm{ml}}=K_{\nu,\mathrm{ml}}=\,0.16\,\omega_* \Delta h.
694\end{equation}
695%
696For stable conditions the eddy diffusivity/viscosity profiles are given by:
697%
698\begin{align}\label{diff-stable}
699K_d= & 0.75\,\, \nu_*\, h_{\mathrm{ml}}\,\,  \exp\left[-2.8 \left(h_{\mathrm{bl}}/L_L\right)^2\right]\sigma_{\mathrm{ml}} \left(1-\sigma_{\mathrm{ml}}\right)^{3/2} \\\label{eq:visc-stable}
700K_\nu = & 0.375\,\,  \nu_*\, h_{\mathrm{ml}} \,\, \exp\left[-2.8 \left(h_{\mathrm{bl}}/L_L\right)^2\right] \sigma_{\mathrm{ml}} \left(1-\sigma_{\mathrm{ml}}\right)\left(1-\tfrac{1}{2}\sigma_{\mathrm{ml}}^2\right).
701\end{align}
702%
703The shape of the eddy viscosity and diffusivity profiles is the same as the shape in the unstable OSBL. The eddy diffusivity/viscosity depends on the stability parameter $h_{\mathrm{bl}}/{L_L}$ where $ L_L$ is analogous to the Obukhov length, but for Langmuir turbulence:
704\begin{equation}\label{eq:L_L} 
705  L_L=-w_{*L}^3/\left<\overline{w^\prime b^\prime}\right>_L,
706\end{equation}
707with the mean turbulent buoyancy flux averaged over the boundary layer given in terms of its surface value $\overline{w^\prime b^\prime}_0$ and (downwards) )solar irradiance $I(z)$ by
708\begin{equation} \label{eq:stable-av-buoy-flux}
709\left<\overline{w^\prime b^\prime}\right>_L = \tfrac{1}{2} {\overline{w^\prime b^\prime}}_0-g\alpha_E\left[\tfrac{1}{2}(I(0)+I(-h))-\left<I\right>\right].
710\end{equation}
711%
712In unstable conditions the eddy diffusivity and viscosity depend on stability through the velocity scale $\omega_*$, which depends on the two velocity scales $\nu_*$ and $w_{*C}$.
713
714Details of the non-gradient terms in \eqref{eq:flux-grad-gen} and of the fluxes within the pycnocline $-h_{\mathrm{bl}}<z<h_{\mathrm{ml}}$ can be found in Grant (2019).
715
716\subsubsection{Evolution of the boundary layer depth}
717
718The prognostic equation for the depth of the neutral/unstable boundary layer is given by \citep{grant+etal18},
719
720\begin{equation} \label{eq:dhdt-unstable}
721%\frac{\partial h_\mathrm{bl}}{\partial t} + \mathbf{U}_b\cdot\nabla h_\mathrm{bl}= W_b - \frac{{\overline{w^\prime b^\prime}}_\mathrm{ent}}{\Delta B_\mathrm{bl}}
722\frac{\partial h_\mathrm{bl}}{\partial t} = W_b - \frac{{\overline{w^\prime b^\prime}}_\mathrm{ent}}{\Delta B_\mathrm{bl}}
723\end{equation}
724where $h_\mathrm{bl}$ is the horizontally-varying depth of the OSBL, $\mathbf{U}_b$ and $W_b$ are the mean horizontal and vertical velocities at the base of the OSBL, ${\overline{w^\prime b^\prime}}_\mathrm{ent}$ is the buoyancy flux due to entrainment and $\Delta B_\mathrm{bl}$ is the difference between the buoyancy averaged over the depth of the OSBL (i.e.\ including the ML and pycnocline) and the buoyancy just below the base of the OSBL. This equation for the case when the pycnocline has a finite thickness, based on the potential energy budget of the OSBL, is the leading term \citep{grant+etal18} of a generalization of that used in mixed-layer models \citet[e.g.][]{kraus.turner_tellus67}, in which the thickness of the pycnocline is taken to be zero.
725
726The entrainment flux for the combination of convective and Langmuir turbulence is given by
727\begin{equation} \label{eq:entrain-flux}
728  {\overline{w^\prime b^\prime}}_\mathrm{ent} = -\alpha_{\mathrm{B}} {\overline{w^\prime b^\prime}}_0 - \alpha_{\mathrm{S}} \frac{u_*^3}{h_{\mathrm{ml}}}
729  + G\left(\delta/h_{\mathrm{ml}} \right)\left[\alpha_{\mathrm{S}}e^{-1.5\, \mathrm{La}_t}-\alpha_{\mathrm{L}} \frac{w_{\mathrm{*L}}^3}{h_{\mathrm{ml}}}\right]
730\end{equation}
731where the factor $G\equiv 1 - \mathrm{e}^ {-25\delta/h_{\mathrm{bl}}}(1-4\delta/h_{\mathrm{bl}})$ models the lesser efficiency of Langmuir mixing when the boundary-layer depth is much greater than the Stokes depth, and $\alpha_{\mathrm{B}}$, $\alpha_{S}$  and $\alpha_{\mathrm{L}}$ depend on the ratio of the appropriate eddy turnover time to the inertial timescale $f^{-1}$. Results from the LES suggest $\alpha_{\mathrm{B}}=0.18 F(fh_{\mathrm{bl}}/w_{*C})$, $\alpha_{S}=0.15 F(fh_{\mathrm{bl}}/u_*$  and $\alpha_{\mathrm{L}}=0.035 F(fh_{\mathrm{bl}}/u_{*L})$, where $F(x)\equiv\tanh(x^{-1})^{0.69}$.
732
733For the stable boundary layer, the equation for the depth of the OSBL is:
734
735\begin{equation}\label{eq:dhdt-stable}
736\max\left(\Delta B_{bl},\frac{w_{*L}^2}{h_\mathrm{bl}}\right)\frac{\partial h_\mathrm{bl}}{\partial t} = \left(0.06 + 0.52\,\frac{ h_\mathrm{bl}}{L_L}\right) \frac{w_{*L}^3}{h_\mathrm{bl}} +\left<\overline{w^\prime b^\prime}\right>_L.
737\end{equation} 
738
739Equation. \ref{eq:dhdt-unstable} always leads to the depth of the entraining OSBL increasing (ignoring the effect of the mean vertical motion), but the change in the thickness of the stable OSBL given by Eq. \ref{eq:dhdt-stable} can be positive or negative, depending on the magnitudes of $\left<\overline{w^\prime b^\prime}\right>_L$ and $h_\mathrm{bl}/L_L$. The rate at which the depth of the OSBL can decrease is limited by choosing an effective buoyancy $w_{*L}^2/h_\mathrm{bl}$, in place of $\Delta B_{bl}$ which will be $\approx 0$ for the collapsing OSBL.
740
741
742%% =================================================================================================
743\subsection[ Discrete energy conservation for TKE and GLS schemes]{Discrete energy conservation for TKE and GLS schemes}
744\label{subsec:ZDF_tke_ene}
745
746\begin{figure}[!t]
747  \centering
748  \includegraphics[width=0.66\textwidth]{Fig_ZDF_TKE_time_scheme}
749  \caption[Subgrid kinetic energy integration in GLS and TKE schemes]{
750    Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and
751    its links to the momentum and tracer time integration.}
752  \label{fig:ZDF_TKE_time_scheme}
753\end{figure}
754
755The production of turbulence by vertical shear (the first term of the right hand side of
756\autoref{eq:ZDF_tke_e}) and  \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion
757(first line in \autoref{eq:MB_zdf}).
758To do so a special care has to be taken for both the time and space discretization of
759the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}.
760
761Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how
762the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with
763the one-level forward time stepping of the equation for $\bar{e}$.
764With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to
765the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and
766summing the result vertically:
767\begin{equation}
768  \label{eq:ZDF_energ1}
769  \begin{split}
770    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\
771    &= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}
772    - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz }
773  \end{split}
774\end{equation}
775Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$,
776known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme
777(see \autoref{sec:TD_forward_imp}).
778The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at
779the surface (atmospheric forcing) and at the bottom (friction effect).
780The second term is always negative.
781It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$.
782\autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
783the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
784${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$
785(and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
786
787A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
788(second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}).
789This term must balance the input of potential energy resulting from vertical mixing.
790The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by
791multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically:
792\begin{equation}
793  \label{eq:ZDF_energ2}
794  \begin{split}
795    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\
796    &= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}
797    - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\
798    &= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta}
799    + \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  }
800  \end{split}
801\end{equation}
802where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.
803The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because
804there is no diffusive flux through the ocean surface and bottom).
805The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
806Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
807the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and  \autoref{eq:ZDF_gls_e}.
808
809Let us now address the space discretization issue.
810The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in
811the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}).
812A space averaging is thus required to obtain the shear TKE production term.
813By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by
814the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
815Furthermore, the time variation of $e_3$ has be taken into account.
816
817The above energetic considerations leads to the following final discrete form for the TKE equation:
818\begin{equation}
819  \label{eq:ZDF_tke_ene}
820  \begin{split}
821    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv
822    \Biggl\{ \Biggr.
823    &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} }
824        \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\
825    +&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} }
826        \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j}
827    \Biggr. \Biggr\}   \\
828    %
829    - &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\
830    %
831    +&\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]   \\
832    %
833    - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt}
834  \end{split}
835\end{equation}
836where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation)
837are time stepped using a backward scheme (see\autoref{sec:TD_forward_imp}).
838Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible.
839%The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as
840%they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}.
841%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.
842
843%% =================================================================================================
844\section{Convection}
845\label{sec:ZDF_conv}
846
847Static instabilities (\ie\ light potential densities under heavy ones) may occur at particular ocean grid points.
848In nature, convective processes quickly re-establish the static stability of the water column.
849These processes have been removed from the model via the hydrostatic assumption so they must be parameterized.
850Three parameterisations are available to deal with convective processes:
851a non-penetrative convective adjustment or an enhanced vertical diffusion,
852or/and the use of a turbulent closure scheme.
853
854%% =================================================================================================
855\subsection[Non-penetrative convective adjustment (\forcode{ln_tranpc})]{Non-penetrative convective adjustment (\protect\np{ln_tranpc}{ln\_tranpc})}
856\label{subsec:ZDF_npc}
857
858\begin{figure}[!htb]
859  \centering
860  \includegraphics[width=0.66\textwidth]{Fig_npc}
861  \caption[Unstable density profile treated by the non penetrative convective adjustment algorithm]{
862    Example of an unstable density profile treated by
863    the non penetrative convective adjustment algorithm.
864    $1^{st}$ step: the initial profile is checked from the surface to the bottom.
865    It is found to be unstable between levels 3 and 4.
866    They are mixed.
867    The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
868    The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
869    The $1^{st}$ step ends since the density profile is then stable below the level 3.
870    $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
871    levels 2 to 5 are mixed.
872    The new density profile is checked.
873    It is found stable: end of algorithm.}
874  \label{fig:ZDF_npc}
875\end{figure}
876
877Options are defined through the \nam{zdf}{zdf} namelist variables.
878The non-penetrative convective adjustment is used when \np[=.true.]{ln_zdfnpc}{ln\_zdfnpc}.
879It is applied at each \np{nn_npc}{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of
880the water column, but only until the density structure becomes neutrally stable
881(\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below)
882\citep{madec.delecluse.ea_JPO91}.
883The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}):
884starting from the top of the ocean, the first instability is found.
885Assume in the following that the instability is located between levels $k$ and $k+1$.
886The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of
887the water column.
888The new density is then computed by a linear approximation.
889If the new density profile is still unstable between levels $k+1$ and $k+2$,
890levels $k$, $k+1$ and $k+2$ are then mixed.
891This process is repeated until stability is established below the level $k$
892(the mixing process can go down to the ocean bottom).
893The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or
894if there is no deeper instability.
895
896This algorithm is significantly different from mixing statically unstable levels two by two.
897The latter procedure cannot converge with a finite number of iterations for some vertical profiles while
898the algorithm used in \NEMO\ converges for any profile in a number of iterations which is less than
899the number of vertical levels.
900This property is of paramount importance as pointed out by \citet{killworth_iprc89}:
901it avoids the existence of permanent and unrealistic static instabilities at the sea surface.
902This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in
903the north-western Mediterranean Sea \citep{madec.delecluse.ea_JPO91, madec.chartier.ea_DAO91, madec.crepon_iprc91}.
904
905The current implementation has been modified in order to deal with any non linear equation of seawater
906(L. Brodeau, personnal communication).
907Two main differences have been introduced compared to the original algorithm:
908$(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency
909(not the difference in potential density);
910$(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in
911the same way their temperature and salinity has been mixed.
912These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without
913having to recompute the expansion coefficients at each mixing iteration.
914
915%% =================================================================================================
916\subsection[Enhanced vertical diffusion (\forcode{ln_zdfevd})]{Enhanced vertical diffusion (\protect\np{ln_zdfevd}{ln\_zdfevd})}
917\label{subsec:ZDF_evd}
918
919Options are defined through the  \nam{zdf}{zdf} namelist variables.
920The enhanced vertical diffusion parameterisation is used when \np[=.true.]{ln_zdfevd}{ln\_zdfevd}.
921In this case, the vertical eddy mixing coefficients are assigned very large values
922in regions where the stratification is unstable
923(\ie\ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{lazar_phd97, lazar.madec.ea_JPO99}.
924This is done either on tracers only (\np[=0]{nn_evdm}{nn\_evdm}) or
925on both momentum and tracers (\np[=1]{nn_evdm}{nn\_evdm}).
926
927In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np[=1]{nn_evdm}{nn\_evdm},
928the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to
929the namelist parameter \np{rn_avevd}{rn\_avevd}.
930A typical value for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$.
931This parameterisation of convective processes is less time consuming than
932the convective adjustment algorithm presented above when mixing both tracers and
933momentum in the case of static instabilities.
934
935Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$.
936This removes a potential source of divergence of odd and even time step in
937a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}).
938
939%% =================================================================================================
940\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}})}
941\label{subsec:ZDF_tcs}
942
943The turbulent closure schemes presented in \autoref{subsec:ZDF_tke}, \autoref{subsec:ZDF_gls} and
944\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,
945with statically unstable density profiles.
946In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in
947\autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative.
948It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also of the four neighboring values at
949velocity points $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1}$).
950These large values restore the static stability of the water column in a way similar to that of
951the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}).
952However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by
953the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$,
954because the mixing length scale is bounded by the distance to the sea surface.
955It can thus be useful to combine the enhanced vertical diffusion with the turbulent closure scheme,
956\ie\ setting the \np{ln_zdfnpc}{ln\_zdfnpc} namelist parameter to true and
957defining the turbulent closure (\np{ln_zdftke}{ln\_zdftke} or \np{ln_zdfgls}{ln\_zdfgls} = \forcode{.true.}) all together.
958
959The OSMOSIS turbulent closure scheme already includes enhanced vertical diffusion in the case of convection,
960%as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp},
961therefore \np[=.false.]{ln_zdfevd}{ln\_zdfevd} should be used with the OSMOSIS scheme.
962% gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
963
964%% =================================================================================================
965\section[Double diffusion mixing (\forcode{ln_zdfddm})]{Double diffusion mixing (\protect\np{ln_zdfddm}{ln\_zdfddm})}
966\label{subsec:ZDF_ddm}
967
968%\nlst{namzdf_ddm}
969
970This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the namelist parameter
971\np{ln_zdfddm}{ln\_zdfddm} in \nam{zdf}{zdf}.
972Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa.
973The former condition leads to salt fingering and the latter to diffusive convection.
974Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean.
975\citet{merryfield.holloway.ea_JPO99} include a parameterisation of such phenomena in a global ocean model and show that
976it leads to relatively minor changes in circulation but exerts significant regional influences on
977temperature and salinity.
978
979Diapycnal mixing of S and T are described by diapycnal diffusion coefficients
980\begin{align*}
981  % \label{eq:ZDF_ddm_Kz}
982  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\
983  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
984\end{align*}
985where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
986and $o$ by processes other than double diffusion.
987The rates of double-diffusive mixing depend on the buoyancy ratio
988$R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of
989thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}).
990To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt
991(1981):
992\begin{align}
993  \label{eq:ZDF_ddm_f}
994  A_f^{vS} &=
995             \begin{cases}
996               \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
997               0                              &\text{otherwise}
998             \end{cases}
999  \\         \label{eq:ZDF_ddm_f_T}
1000  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho
1001\end{align}
1002
1003\begin{figure}[!t]
1004  \centering
1005  \includegraphics[width=0.66\textwidth]{Fig_zdfddm}
1006  \caption[Diapycnal diffusivities for temperature and salt in regions of salt fingering and
1007  diffusive convection]{
1008    From \citet{merryfield.holloway.ea_JPO99}:
1009    (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in
1010    regions of salt fingering.
1011    Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and
1012    thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
1013    (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in
1014    regions of diffusive convection.
1015    Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
1016    The latter is not implemented in \NEMO.}
1017  \label{fig:ZDF_ddm}
1018\end{figure}
1019
1020The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of
1021buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}).
1022Following  \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
1023
1024To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by
1025Federov (1988) is used:
1026\begin{align}
1027  % \label{eq:ZDF_ddm_d}
1028  A_d^{vT} &=
1029             \begin{cases}
1030               1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
1031               &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
1032               0                       &\text{otherwise}
1033             \end{cases}
1034                                       \nonumber \\
1035  \label{eq:ZDF_ddm_d_S}
1036  A_d^{vS} &=
1037             \begin{cases}
1038               A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
1039               A_d^{vT} \ 0.15 \ R_\rho               &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
1040               0                       &\text{otherwise}
1041             \end{cases}
1042\end{align}
1043
1044The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in
1045\autoref{fig:ZDF_ddm}.
1046Implementing this requires computing $R_\rho$ at each grid point on every time step.
1047This is done in \mdl{eosbn2} at the same time as $N^2$ is computed.
1048This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive).
1049
1050%% =================================================================================================
1051\section[Bottom and top friction (\textit{zdfdrg.F90})]{Bottom and top friction (\protect\mdl{zdfdrg})}
1052\label{sec:ZDF_drg}
1053
1054\begin{listing}
1055  \nlst{namdrg}
1056  \caption{\forcode{&namdrg}}
1057  \label{lst:namdrg}
1058\end{listing}
1059\begin{listing}
1060  \nlst{namdrg_top}
1061  \caption{\forcode{&namdrg_top}}
1062  \label{lst:namdrg_top}
1063\end{listing}
1064\begin{listing}
1065  \nlst{namdrg_bot}
1066  \caption{\forcode{&namdrg_bot}}
1067  \label{lst:namdrg_bot}
1068\end{listing}
1069
1070Options to define the top and bottom friction are defined through the \nam{drg}{drg} namelist variables.
1071The bottom friction represents the friction generated by the bathymetry.
1072The top friction represents the friction generated by the ice shelf/ocean interface.
1073As the friction processes at the top and the bottom are treated in and identical way,
1074the description below considers mostly the bottom friction case, if not stated otherwise.
1075
1076Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as
1077a condition on the vertical diffusive flux.
1078For the bottom boundary layer, one has:
1079 \[
1080   % \label{eq:ZDF_bfr_flux}
1081   A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
1082 \]
1083where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside
1084the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean).
1085How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near
1086the bottom relative to the Ekman layer depth.
1087For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m,
1088one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$
1089(for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$).
1090With a background diffusion coefficient $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
1091When the vertical mixing coefficient is this small, using a flux condition is equivalent to
1092entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or
1093bottom model layer.
1094To illustrate this, consider the equation for $u$ at $k$, the last ocean level:
1095\begin{equation}
1096  \label{eq:ZDF_drg_flux2}
1097  \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}}
1098\end{equation}
1099If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth.
1100On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used,
1101the turbulent Ekman layer will be represented explicitly by the model.
1102However, the logarithmic layer is never represented in current primitive equation model applications:
1103it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
1104Two choices are available in \NEMO: a linear and a quadratic bottom friction.
1105Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in
1106the present release of \NEMO.
1107
1108In the code, the bottom friction is imposed by adding the trend due to the bottom friction to
1109 the general momentum trend in \mdl{dynzdf}.
1110For the time-split surface pressure gradient algorithm, the momentum trend due to
1111the barotropic component needs to be handled separately.
1112For this purpose it is convenient to compute and store coefficients which can be simply combined with
1113bottom velocities and geometric values to provide the momentum trend due to bottom friction.
1114 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where:
1115\begin{equation}
1116  \label{eq:ZDF_bfr_bdef}
1117  \frac{\partial {\textbf U_h}}{\partial t} =
1118  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b
1119\end{equation}
1120where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity.
1121Note 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.
1122
1123%% =================================================================================================
1124\subsection[Linear top/bottom friction (\forcode{ln_lin})]{Linear top/bottom friction (\protect\np{ln_lin}{ln\_lin})}
1125\label{subsec:ZDF_drg_linear}
1126
1127The linear friction parameterisation (including the special case of a free-slip condition) assumes that
1128the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level):
1129\[
1130  % \label{eq:ZDF_bfr_linear}
1131  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
1132\]
1133where $r$ is a friction coefficient expressed in $m s^{-1}$.
1134This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean,
1135and setting $r = H / \tau$, where $H$ is the ocean depth.
1136Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{weatherly_JMR84}.
1137A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models.
1138One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$
1139(\citet{gill_bk82}, Eq. 9.6.6).
1140For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$,
1141and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.
1142This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days.
1143It can be changed by specifying \np{rn_Uc0}{rn\_Uc0} (namelist parameter).
1144
1145 For the linear friction case the drag coefficient used in the general expression \autoref{eq:ZDF_bfr_bdef} is:
1146\[
1147  % \label{eq:ZDF_bfr_linbfr_b}
1148    c_b^T = - r
1149\]
1150When \np[=.true.]{ln_lin}{ln\_lin}, the value of $r$ used is \np{rn_Uc0}{rn\_Uc0}*\np{rn_Cd0}{rn\_Cd0}.
1151Setting \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.
1152
1153These values are assigned in \mdl{zdfdrg}.
1154Note that there is support for local enhancement of these values via an externally defined 2D mask array
1155(\np[=.true.]{ln_boost}{ln\_boost}) given in the \ifile{bfr\_coef} input NetCDF file.
1156The mask values should vary from 0 to 1.
1157Locations with a non-zero mask value will have the friction coefficient increased by
1158$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}.
1159
1160%% =================================================================================================
1161\subsection[Non-linear top/bottom friction (\forcode{ln_non_lin})]{Non-linear top/bottom friction (\protect\np{ln_non_lin}{ln\_non\_lin})}
1162\label{subsec:ZDF_drg_nonlinear}
1163
1164The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic:
1165\[
1166  % \label{eq:ZDF_drg_nonlinear}
1167  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
1168  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
1169\]
1170where $C_D$ is a drag coefficient, and $e_b $ a top/bottom turbulent kinetic energy due to tides,
1171internal waves breaking and other short time scale currents.
1172A typical value of the drag coefficient is $C_D = 10^{-3} $.
1173As an example, the CME experiment \citep{treguier_JGR92} uses $C_D = 10^{-3}$ and
1174$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{killworth_JPO92} uses $C_D = 1.4\;10^{-3}$ and
1175$e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.
1176The CME choices have been set as default values (\np{rn_Cd0}{rn\_Cd0} and \np{rn_ke0}{rn\_ke0} namelist parameters).
1177
1178As for the linear case, the friction is imposed in the code by adding the trend due to
1179the friction to the general momentum trend in \mdl{dynzdf}.
1180For the non-linear friction case the term computed in \mdl{zdfdrg} is:
1181\[
1182  % \label{eq:ZDF_drg_nonlinbfr}
1183    c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2}
1184\]
1185
1186The coefficients that control the strength of the non-linear friction are initialised as namelist parameters:
1187$C_D$= \np{rn_Cd0}{rn\_Cd0}, and $e_b$ =\np{rn_bfeb2}{rn\_bfeb2}.
1188Note 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
1189(\np[=.true.]{ln_boost}{ln\_boost}).
1190This works in the same way as for the linear friction case with non-zero masked locations increased by
1191$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}.
1192
1193%% =================================================================================================
1194\subsection[Log-layer top/bottom friction (\forcode{ln_loglayer})]{Log-layer top/bottom friction (\protect\np{ln_loglayer}{ln\_loglayer})}
1195\label{subsec:ZDF_drg_loglayer}
1196
1197In the non-linear friction case, the drag coefficient, $C_D$, can be optionally enhanced using
1198a "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.
1199If  \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):
1200\[
1201  C_D = \left ( {\kappa \over {\mathrm log}\left ( 0.5 \; e_{3b} / rn\_{z0} \right ) } \right )^2
1202\]
1203
1204\noindent where $\kappa$ is the von-Karman constant and \np{rn_z0}{rn\_z0} is a roughness length provided via the namelist.
1205
1206The drag coefficient is bounded such that it is kept greater or equal to
1207the 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:
1208\np{rn_Cdmax}{rn\_Cdmax}, \ie
1209\[
1210  rn\_Cd0 \leq C_D \leq rn\_Cdmax
1211\]
1212
1213\noindent The log-layer enhancement can also be applied to the top boundary friction if
1214under ice-shelf cavities are activated (\np[=.true.]{ln_isfcav}{ln\_isfcav}).
1215%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}.
1216
1217%% =================================================================================================
1218\subsection[Explicit top/bottom friction (\forcode{ln_drgimp=.false.})]{Explicit top/bottom friction (\protect\np[=.false.]{ln_drgimp}{ln\_drgimp})}
1219\label{subsec:ZDF_drg_stability}
1220
1221Setting \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:
1222
1223At the top (below an ice shelf cavity):
1224\[
1225  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1226  = c_{t}^{\textbf{U}}\textbf{u}^{n-1}_{t}
1227\]
1228
1229At the bottom (above the sea floor):
1230\[
1231  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1232  = c_{b}^{\textbf{U}}\textbf{u}^{n-1}_{b}
1233\]
1234
1235Since 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.
1236For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is:
1237\begin{equation}
1238  \label{eq:ZDF_Eqn_drgstab}
1239  \begin{split}
1240    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
1241    &= -\frac{ru}{e_{3u}}\;2\rdt\\
1242  \end{split}
1243\end{equation}
1244\noindent where linear friction and a leapfrog timestep have been assumed.
1245To ensure that the friction cannot reverse the direction of flow it is necessary to have:
1246\[
1247  |\Delta u| < \;|u|
1248\]
1249\noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives:
1250\[
1251  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
1252\]
1253This same inequality can also be derived in the non-linear bottom friction case if
1254a velocity of 1 m.s$^{-1}$ is assumed.
1255Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability:
1256\[
1257  e_{3u} > 2\;r\;\rdt
1258\]
1259\noindent which it may be necessary to impose if partial steps are being used.
1260For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m.
1261For most applications, with physically sensible parameters these restrictions should not be of concern.
1262But caution may be necessary if attempts are made to locally enhance the bottom friction parameters.
1263To ensure stability limits are imposed on the top/bottom friction coefficients both
1264during initialisation and at each time step.
1265Checks at initialisation are made in \mdl{zdfdrg} (assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
1266The number of breaches of the stability criterion are reported as well as
1267the minimum and maximum values that have been set.
1268The criterion is also checked at each time step, using the actual velocity, in \mdl{dynzdf}.
1269Values of the friction coefficient are reduced as necessary to ensure stability;
1270these changes are not reported.
1271
1272Limits on the top/bottom friction coefficient are not imposed if the user has elected to
1273handle the friction implicitly (see \autoref{subsec:ZDF_drg_imp}).
1274The number of potential breaches of the explicit stability criterion are still reported for information purposes.
1275
1276%% =================================================================================================
1277\subsection[Implicit top/bottom friction (\forcode{ln_drgimp=.true.})]{Implicit top/bottom friction (\protect\np[=.true.]{ln_drgimp}{ln\_drgimp})}
1278\label{subsec:ZDF_drg_imp}
1279
1280An optional implicit form of bottom friction has been implemented to improve model stability.
1281We recommend this option for shelf sea and coastal ocean applications. %, especially for split-explicit time splitting.
1282This option can be invoked by setting \np{ln_drgimp}{ln\_drgimp} to \forcode{.true.} in the \nam{drg}{drg} namelist.
1283%This option requires \np{ln_zdfexp}{ln\_zdfexp} to be \forcode{.false.} in the \nam{zdf}{zdf} namelist.
1284
1285This implementation is performed in \mdl{dynzdf} where the following boundary conditions are set while solving the fully implicit diffusion step:
1286
1287At the top (below an ice shelf cavity):
1288\[
1289  % \label{eq:ZDF_dynZDF__drg_top}
1290  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1291  = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t}
1292\]
1293
1294At the bottom (above the sea floor):
1295\[
1296  % \label{eq:ZDF_dynZDF__drg_bot}
1297  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1298  = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b}
1299\]
1300
1301where $t$ and $b$ refers to top and bottom layers respectively.
1302Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so it is implicit.
1303
1304%% =================================================================================================
1305\subsection[Bottom friction with split-explicit free surface]{Bottom friction with split-explicit free surface}
1306\label{subsec:ZDF_drg_ts}
1307
1308With 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.
1309
1310The strategy to handle top/bottom stresses with split-explicit free surface in \NEMO\ is as follows:
1311\begin{enumerate}
1312\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.
1313\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.
1314\end{enumerate}
1315
1316Note that other strategies are possible, like considering vertical diffusion step in advance, \ie\ prior barotropic integration.
1317
1318%% =================================================================================================
1319\section[Internal wave-driven mixing (\forcode{ln_zdfiwm})]{Internal wave-driven mixing (\protect\np{ln_zdfiwm}{ln\_zdfiwm})}
1320\label{subsec:ZDF_tmx_new}
1321
1322\begin{listing}
1323  \nlst{namzdf_iwm}
1324  \caption{\forcode{&namzdf_iwm}}
1325  \label{lst:namzdf_iwm}
1326\end{listing}
1327
1328The parameterization of mixing induced by breaking internal waves is a generalization of
1329the approach originally proposed by \citet{st-laurent.simmons.ea_GRL02}.
1330A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed,
1331and the resulting diffusivity is obtained as
1332\[
1333  % \label{eq:ZDF_Kwave}
1334  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
1335\]
1336where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of
1337the energy available for mixing.
1338If the \np{ln_mevar}{ln\_mevar} namelist parameter is set to \forcode{.false.}, the mixing efficiency is taken as constant and
1339equal to 1/6 \citep{osborn_JPO80}.
1340In the opposite (recommended) case, $R_f$ is instead a function of
1341the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$,
1342with $\nu$ the molecular viscosity of seawater, following the model of \cite{bouffard.boegman_DAO13} and
1343the implementation of \cite{de-lavergne.madec.ea_JPO16}.
1344Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when
1345the mixing efficiency is constant.
1346
1347In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary
1348as a function of $Re_b$ by setting the \np{ln_tsdiff}{ln\_tsdiff} parameter to \forcode{.true.}, a recommended choice.
1349This parameterization of differential mixing, due to \cite{jackson.rehmann_JPO14},
1350is implemented as in \cite{de-lavergne.madec.ea_JPO16}.
1351
1352The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$,
1353is constructed from three static maps of column-integrated internal wave energy dissipation,
1354$E_{cri}(i,j)$, $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures:
1355
1356\begin{align*}
1357  F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\
1358  F_{pyc}(i,j,k) &\propto N^{n_p}\\
1359  F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} }
1360\end{align*}
1361In the above formula, $h_{ab}$ denotes the height above bottom,
1362$h_{wkb}$ denotes the WKB-stretched height above bottom, defined by
1363\[
1364  h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; ,
1365\]
1366The $n_p$ parameter (given by \np{nn_zpyc}{nn\_zpyc} in \nam{zdf_iwm}{zdf\_iwm} namelist)
1367controls the stratification-dependence of the pycnocline-intensified dissipation.
1368It can take values of $1$ (recommended) or $2$.
1369Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of
1370the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps.
1371$h_{cri}$ is related to the large-scale topography of the ocean (etopo2) and
1372$h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of
1373the abyssal hill topography \citep{goff_JGR10} and the latitude.
1374% Jc: input files names ?
1375
1376%% =================================================================================================
1377\section[Surface wave-induced mixing (\forcode{ln_zdfswm})]{Surface wave-induced mixing (\protect\np{ln_zdfswm}{ln\_zdfswm})}
1378\label{subsec:ZDF_swm}
1379
1380Surface waves produce an enhanced mixing through wave-turbulence interaction.
1381In addition to breaking waves induced turbulence (\autoref{subsec:ZDF_tke}),
1382the influence of non-breaking waves can be accounted introducing
1383wave-induced viscosity and diffusivity as a function of the wave number spectrum.
1384Following \citet{qiao.yuan.ea_OD10}, a formulation of wave-induced mixing coefficient
1385is provided  as a function of wave amplitude, Stokes Drift and wave-number:
1386
1387\begin{equation}
1388  \label{eq:ZDF_Bv}
1389  B_{v} = \alpha {A} {U}_{st} {exp(3kz)}
1390\end{equation}
1391
1392Where $B_{v}$ is the wave-induced mixing coefficient, $A$ is the wave amplitude,
1393${U}_{st}$ is the Stokes Drift velocity, $k$ is the wave number and $\alpha$
1394is a constant which should be determined by observations or
1395numerical experiments and is set to be 1.
1396
1397The coefficient $B_{v}$ is then directly added to the vertical viscosity
1398and diffusivity coefficients.
1399
1400In order to account for this contribution set: \forcode{ln_zdfswm=.true.},
1401then wave interaction has to be activated through \forcode{ln_wave=.true.},
1402the Stokes Drift can be evaluated by setting \forcode{ln_sdw=.true.}
1403(see \autoref{subsec:SBC_wave_sdw})
1404and the needed wave fields can be provided either in forcing or coupled mode
1405(for more information on wave parameters and settings see \autoref{sec:SBC_wave})
1406
1407%% =================================================================================================
1408\section[Adaptive-implicit vertical advection (\forcode{ln_zad_Aimp})]{Adaptive-implicit vertical advection(\protect\np{ln_zad_Aimp}{ln\_zad\_Aimp})}
1409\label{subsec:ZDF_aimp}
1410
1411The adaptive-implicit vertical advection option in NEMO is based on the work of
1412\citep{shchepetkin_OM15}.  In common with most ocean models, the timestep used with NEMO
1413needs to satisfy multiple criteria associated with different physical processes in order
1414to maintain numerical stability. \citep{shchepetkin_OM15} pointed out that the vertical
1415CFL criterion is commonly the most limiting. \citep{lemarie.debreu.ea_OM15} examined the
1416constraints for a range of time and space discretizations and provide the CFL stability
1417criteria for a range of advection schemes. The values for the Leap-Frog with Robert
1418asselin filter time-stepping (as used in NEMO) are reproduced in
1419\autoref{tab:ZDF_zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these
1420restrictions but at the cost of large dispersive errors and, possibly, large numerical
1421viscosity. The adaptive-implicit vertical advection option provides a targetted use of the
1422implicit scheme only when and where potential breaches of the vertical CFL condition
1423occur. In many practical applications these events may occur remote from the main area of
1424interest or due to short-lived conditions such that the extra numerical diffusion or
1425viscosity does not greatly affect the overall solution. With such applications, setting:
1426\forcode{ln_zad_Aimp=.true.} should allow much longer model timesteps to be used whilst
1427retaining the accuracy of the high order explicit schemes over most of the domain.
1428
1429\begin{table}[htbp]
1430  \centering
1431  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}}
1432  \begin{tabular}{r|ccc}
1433    \hline
1434    spatial discretization  & 2$^nd$ order centered & 3$^rd$ order upwind & 4$^th$ order compact \\
1435    advective CFL criterion &                 0.904 &              0.472  &                0.522 \\
1436    \hline
1437  \end{tabular}
1438  \caption[Advective CFL criteria for the leapfrog with Robert Asselin filter time-stepping]{
1439    The advective CFL criteria for a range of spatial discretizations for
1440    the leapfrog with Robert Asselin filter time-stepping
1441    ($\nu=0.1$) as given in \citep{lemarie.debreu.ea_OM15}.}
1442  \label{tab:ZDF_zad_Aimp_CFLcrit}
1443\end{table}
1444
1445In particular, the advection scheme remains explicit everywhere except where and when
1446local vertical velocities exceed a threshold set just below the explicit stability limit.
1447Once the threshold is reached a tapered transition towards an implicit scheme is used by
1448partitioning the vertical velocity into a part that can be treated explicitly and any
1449excess that must be treated implicitly. The partitioning is achieved via a Courant-number
1450dependent weighting algorithm as described in \citep{shchepetkin_OM15}.
1451
1452The local cell Courant number ($Cu$) used for this partitioning is:
1453
1454\begin{equation}
1455  \label{eq:ZDF_Eqn_zad_Aimp_Courant}
1456  \begin{split}
1457    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 ]    \\
1458       &\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 ]
1459                     \big / e_{{1_t}ij}e_{{2_t}ij}            \\
1460       &\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 ]
1461                     \big / e_{{1_t}ij}e_{{2_t}ij} \bigg )    \\
1462  \end{split}
1463\end{equation}
1464
1465\noindent and the tapering algorithm follows \citep{shchepetkin_OM15} as:
1466
1467\begin{align}
1468  \label{eq:ZDF_Eqn_zad_Aimp_partition}
1469Cu_{min} &= 0.15 \nonumber \\
1470Cu_{max} &= 0.3  \nonumber \\
1471Cu_{cut} &= 2Cu_{max} - Cu_{min} \nonumber \\
1472Fcu    &= 4Cu_{max}*(Cu_{max}-Cu_{min}) \nonumber \\
1473\cf &=
1474     \begin{cases}
1475        0.0                                                        &\text{if $Cu \leq Cu_{min}$} \\
1476        (Cu - Cu_{min})^2 / (Fcu +  (Cu - Cu_{min})^2)             &\text{else if $Cu < Cu_{cut}$} \\
1477        (Cu - Cu_{max}) / Cu                                       &\text{else}
1478     \end{cases}
1479\end{align}
1480
1481\begin{figure}[!t]
1482  \centering
1483  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_coeff}
1484  \caption[Partitioning coefficient used to partition vertical velocities into parts]{
1485    The value of the partitioning coefficient (\cf) used to partition vertical velocities into
1486    parts to be treated implicitly and explicitly for a range of typical Courant numbers
1487    (\forcode{ln_zad_Aimp=.true.}).}
1488  \label{fig:ZDF_zad_Aimp_coeff}
1489\end{figure}
1490
1491\noindent The partitioning coefficient is used to determine the part of the vertical
1492velocity that must be handled implicitly ($w_i$) and to subtract this from the total
1493vertical velocity ($w_n$) to leave that which can continue to be handled explicitly:
1494
1495\begin{align}
1496  \label{eq:ZDF_Eqn_zad_Aimp_partition2}
1497    w_{i_{ijk}} &= \cf_{ijk} w_{n_{ijk}}     \nonumber \\
1498    w_{n_{ijk}} &= (1-\cf_{ijk}) w_{n_{ijk}}
1499\end{align}
1500
1501\noindent Note that the coefficient is such that the treatment is never fully implicit;
1502the three cases from \autoref{eq:ZDF_Eqn_zad_Aimp_partition} can be considered as:
1503fully-explicit; mixed explicit/implicit and mostly-implicit.  With the settings shown the
1504coefficient (\cf) varies as shown in \autoref{fig:ZDF_zad_Aimp_coeff}. Note with these values
1505the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly
1506implicit' is 0.45 which is just below the stability limited given in
1507\autoref{tab:ZDF_zad_Aimp_CFLcrit}  for a 3rd order scheme.
1508
1509The $w_i$ component is added to the implicit solvers for the vertical mixing in
1510\mdl{dynzdf} and \mdl{trazdf} in a similar way to \citep{shchepetkin_OM15}.  This is
1511sufficient for the flux-limited advection scheme (\forcode{ln_traadv_mus}) but further
1512intervention is required when using the flux-corrected scheme (\forcode{ln_traadv_fct}).
1513For these schemes the implicit upstream fluxes must be added to both the monotonic guess
1514and to the higher order solution when calculating the antidiffusive fluxes. The implicit
1515vertical fluxes are then removed since they are added by the implicit solver later on.
1516
1517The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be
1518used in a wide range of simulations. The following test simulation, however, does illustrate
1519the potential benefits and will hopefully encourage further testing and feedback from users:
1520
1521\begin{figure}[!t]
1522  \centering
1523  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames}
1524  \caption[OVERFLOW: time-series of temperature vertical cross-sections]{
1525    A time-series of temperature vertical cross-sections for the OVERFLOW test case.
1526    These results are for the default settings with \forcode{nn_rdt=10.0} and
1527    without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}).}
1528  \label{fig:ZDF_zad_Aimp_overflow_frames}
1529\end{figure}
1530
1531%% =================================================================================================
1532\subsection{Adaptive-implicit vertical advection in the OVERFLOW test-case}
1533
1534The \href{https://forge.ipsl.jussieu.fr/nemo/chrome/site/doc/NEMO/guide/html/test\_cases.html\#overflow}{OVERFLOW test case}
1535provides a simple illustration of the adaptive-implicit advection in action. The example here differs from the basic test case
1536by only a few extra physics choices namely:
1537
1538\begin{verbatim}
1539     ln_dynldf_OFF = .false.
1540     ln_dynldf_lap = .true.
1541     ln_dynldf_hor = .true.
1542     ln_zdfnpc     = .true.
1543     ln_traadv_fct = .true.
1544        nn_fct_h   =  2
1545        nn_fct_v   =  2
1546\end{verbatim}
1547
1548\noindent which were chosen to provide a slightly more stable and less noisy solution. The
1549result when using the default value of \forcode{nn_rdt=10.} without adaptive-implicit
1550vertical velocity is illustrated in \autoref{fig:ZDF_zad_Aimp_overflow_frames}. The mass of
1551cold water, initially sitting on the shelf, moves down the slope and forms a
1552bottom-trapped, dense plume. Even with these extra physics choices the model is close to
1553stability limits and attempts with \forcode{nn_rdt=30.} will fail after about 5.5 hours
1554with excessively high horizontal velocities. This time-scale corresponds with the time the
1555plume reaches the steepest part of the topography and, although detected as a horizontal
1556CFL breach, the instability originates from a breach of the vertical CFL limit. This is a good
1557candidate, therefore, for use of the adaptive-implicit vertical advection scheme.
1558
1559The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps
1560are shown in \autoref{fig:ZDF_zad_Aimp_overflow_all_rdt} (together with the equivalent
1561frames from the base run).  In this simple example the use of the adaptive-implicit
1562vertcal advection scheme has enabled a 12x increase in the model timestep without
1563significantly altering the solution (although at this extreme the plume is more diffuse
1564and has not travelled so far).  Notably, the solution with and without the scheme is
1565slightly different even with \forcode{nn_rdt=10.}; suggesting that the base run was
1566close enough to instability to trigger the scheme despite completing successfully.
1567To assist in diagnosing how active the scheme is, in both location and time, the 3D
1568implicit and explicit components of the vertical velocity are available via XIOS as
1569\texttt{wimp} and \texttt{wexp} respectively.  Likewise, the partitioning coefficient
1570(\cf) is also available as \texttt{wi\_cff}. For a quick oversight of
1571the schemes activity the global maximum values of the absolute implicit component
1572of the vertical velocity and the partitioning coefficient are written to the netCDF
1573version of the run statistics file (\texttt{run.stat.nc}) if this is active (see
1574\autoref{sec:MISC_opt} for activation details).
1575
1576\autoref{fig:ZDF_zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for
1577the various overflow tests.  Note that the adaptive-implicit vertical advection scheme is
1578active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the
1579test case is close to stability limits even with this value. At the larger timesteps, the
1580vertical velocity is treated mostly implicitly at some location throughout the run. The
1581oscillatory nature of this measure appears to be linked to the progress of the plume front
1582as each cusp is associated with the location of the maximum shifting to the adjacent cell.
1583This is illustrated in \autoref{fig:ZDF_zad_Aimp_maxCf_loc} where the i- and k- locations of the
1584maximum have been overlaid for the base run case.
1585
1586\medskip
1587\noindent Only limited tests have been performed in more realistic configurations. In the
1588ORCA2\_ICE\_PISCES reference configuration the scheme does activate and passes
1589restartability and reproducibility tests but it is unable to improve the model's stability
1590enough to allow an increase in the model time-step. A view of the time-series of maximum
1591partitioning coefficient (not shown here)  suggests that the default time-step of 5400s is
1592already pushing at stability limits, especially in the initial start-up phase. The
1593time-series does not, however, exhibit any of the 'cuspiness' found with the overflow
1594tests.
1595
1596\medskip
1597\noindent A short test with an eORCA1 configuration promises more since a test using a
1598time-step of 3600s remains stable with \forcode{ln_zad_Aimp=.true.} whereas the
1599time-step is limited to 2700s without.
1600
1601\begin{figure}[!t]
1602  \centering
1603  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_overflow_all_rdt}
1604  \caption[OVERFLOW: sample temperature vertical cross-sections from mid- and end-run]{
1605    Sample temperature vertical cross-sections from mid- and end-run using
1606    different values for \forcode{nn_rdt} and with or without adaptive implicit vertical advection.
1607    Without the adaptive implicit vertical advection
1608    only the run with the shortest timestep is able to run to completion.
1609    Note also that the colour-scale has been chosen to confirm that
1610    temperatures remain within the original range of 10$^o$ to 20$^o$.}
1611  \label{fig:ZDF_zad_Aimp_overflow_all_rdt}
1612\end{figure}
1613
1614\begin{figure}[!t]
1615  \centering
1616  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_maxCf}
1617  \caption[OVERFLOW: maximum partitioning coefficient during a series of test runs]{
1618    The maximum partitioning coefficient during a series of test runs with
1619    increasing model timestep length.
1620    At the larger timesteps,
1621    the vertical velocity is treated mostly implicitly at some location throughout the run.}
1622  \label{fig:ZDF_zad_Aimp_maxCf}
1623\end{figure}
1624
1625\begin{figure}[!t]
1626  \centering
1627  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_maxCf_loc}
1628  \caption[OVERFLOW: maximum partitioning coefficient for the case overlaid]{
1629    The maximum partitioning coefficient for the \forcode{nn_rdt=10.0} case overlaid with
1630    information on the gridcell i- and k-locations of the maximum value.}
1631  \label{fig:ZDF_zad_Aimp_maxCf_loc}
1632\end{figure}
1633
1634\onlyinsubfile{\input{../../global/epilogue}}
1635
1636\end{document}
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