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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\cmtgm{ Add here a small introduction to ZDF and naming of the different physics
31(similar to what have been written for TRA and DYN).}
32
33%% =================================================================================================
34\section{Vertical mixing}
35\label{sec:ZDF}
36
37The discrete form of the ocean subgrid scale physics has been presented in
38\autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}.
39At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt have to be defined.
40At the surface they are prescribed from the surface forcing (see \autoref{chap:SBC}),
41while at the bottom they are set to zero for heat and salt,
42unless a geothermal flux forcing is prescribed as a bottom boundary condition (\ie\ \np{ln_trabbc}{ln\_trabbc} defined,
43see \autoref{subsec:TRA_bbc}), and specified through a bottom friction parameterisation for momentum
44(see \autoref{sec:ZDF_drg}).
45
46In this section we briefly discuss the various choices offered to compute the vertical eddy viscosity and
47diffusivity coefficients, $A_u^{vm}$ , $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- points,
48respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}).
49These coefficients can be assumed to be either constant, or a function of the local Richardson number,
50or computed from a turbulent closure model (either TKE or GLS or OSMOSIS formulation).
51The computation of these coefficients is initialized in the \mdl{zdfphy} module and performed in
52the \mdl{zdfric}, \mdl{zdftke} or \mdl{zdfgls} or \mdl{zdfosm} modules.
53The trends due to the vertical momentum and tracer diffusion, including the surface forcing,
54are computed and added to the general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively.
55%These trends can be computed using either a forward time stepping scheme
56%(namelist parameter \np[=.true.]{ln_zdfexp}{ln\_zdfexp}) or a backward time stepping scheme
57%(\np[=.false.]{ln_zdfexp}{ln\_zdfexp}) depending on the magnitude of the mixing coefficients,
58%and thus of the formulation used (see \autoref{chap:TD}).
59
60\begin{listing}
61  \nlst{namzdf}
62  \caption{\forcode{&namzdf}}
63  \label{lst:namzdf}
64\end{listing}
65
66%% =================================================================================================
67\subsection[Constant (\forcode{ln_zdfcst})]{Constant (\protect\np{ln_zdfcst}{ln\_zdfcst})}
68\label{subsec:ZDF_cst}
69
70Options are defined through the \nam{zdf}{zdf} namelist variables.
71When \np{ln_zdfcst}{ln\_zdfcst} is defined, the momentum and tracer vertical eddy coefficients are set to
72constant values over the whole ocean.
73This is the crudest way to define the vertical ocean physics.
74It is recommended to use this option only in process studies, not in basin scale simulations.
75Typical values used in this case are:
76\begin{align*}
77  A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}   \\
78  A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1}
79\end{align*}
80
81These values are set through the \np{rn_avm0}{rn\_avm0} and \np{rn_avt0}{rn\_avt0} namelist parameters.
82In all cases, do not use values smaller that those associated with the molecular viscosity and diffusivity,
83that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and
84$\sim10^{-9}~m^2.s^{-1}$ for salinity.
85
86%% =================================================================================================
87\subsection[Richardson number dependent (\forcode{ln_zdfric})]{Richardson number dependent (\protect\np{ln_zdfric}{ln\_zdfric})}
88\label{subsec:ZDF_ric}
89
90\begin{listing}
91  \nlst{namzdf_ric}
92  \caption{\forcode{&namzdf_ric}}
93  \label{lst:namzdf_ric}
94\end{listing}
95
96When \np[=.true.]{ln_zdfric}{ln\_zdfric}, a local Richardson number dependent formulation for the vertical momentum and
97tracer eddy coefficients is set through the \nam{zdf_ric}{zdf\_ric} namelist variables.
98The vertical mixing coefficients are diagnosed from the large scale variables computed by the model.
99\textit{In situ} measurements have been used to link vertical turbulent activity to large scale ocean structures.
100The hypothesis of a mixing mainly maintained by the growth of Kelvin-Helmholtz like instabilities leads to
101a dependency between the vertical eddy coefficients and the local Richardson number
102(\ie\ the ratio of stratification to vertical shear).
103Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented:
104\[
105  % \label{eq:ZDF_ric}
106  \left\{
107    \begin{aligned}
108      A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\
109      A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm}
110    \end{aligned}
111  \right.
112\]
113where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number,
114$N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
115$A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the constant case
116(see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ is the maximum value that
117can be reached by the coefficient when $Ri\leq 0$, $a=5$ and $n=2$.
118The last three values can be modified by setting the \np{rn_avmri}{rn\_avmri}, \np{rn_alp}{rn\_alp} and
119\np{nn_ric}{nn\_ric} namelist parameters, respectively.
120
121A simple mixing-layer model to transfer and dissipate the atmospheric forcings
122(wind-stress and buoyancy fluxes) can be activated setting the \np[=.true.]{ln_mldw}{ln\_mldw} in the namelist.
123
124In this case, the local depth of turbulent wind-mixing or "Ekman depth" $h_{e}(x,y,t)$ is evaluated and
125the vertical eddy coefficients prescribed within this layer.
126
127This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation:
128\[
129  h_{e} = Ek \frac {u^{*}} {f_{0}}
130\]
131where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter.
132
133In this similarity height relationship, the turbulent friction velocity:
134\[
135  u^{*} = \sqrt \frac {|\tau|} {\rho_o}
136\]
137is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$.
138The final $h_{e}$ is further constrained by the adjustable bounds \np{rn_mldmin}{rn\_mldmin} and \np{rn_mldmax}{rn\_mldmax}.
139Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to
140the empirical values \np{rn_wtmix}{rn\_wtmix} and \np{rn_wvmix}{rn\_wvmix} \citep{lermusiaux_JMS01}.
141
142%% =================================================================================================
143\subsection[TKE turbulent closure scheme (\forcode{ln_zdftke})]{TKE turbulent closure scheme (\protect\np{ln_zdftke}{ln\_zdftke})}
144\label{subsec:ZDF_tke}
145
146\begin{listing}
147  \nlst{namzdf_tke}
148  \caption{\forcode{&namzdf_tke}}
149  \label{lst:namzdf_tke}
150\end{listing}
151
152The vertical eddy viscosity and diffusivity coefficients are computed from a TKE turbulent closure model based on
153a prognostic equation for $\bar{e}$, the turbulent kinetic energy,
154and a closure assumption for the turbulent length scales.
155This turbulent closure model has been developed by \citet{bougeault.lacarrere_MWR89} in the atmospheric case,
156adapted by \citet{gaspar.gregoris.ea_JGR90} for the oceanic case, and embedded in OPA, the ancestor of \NEMO,
157by \citet{blanke.delecluse_JPO93} for equatorial Atlantic simulations.
158Since then, significant modifications have been introduced by \citet{madec.delecluse.ea_NPM98} in both the implementation and
159the formulation of the mixing length scale.
160The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear,
161its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type:
162\begin{equation}
163  \label{eq:ZDF_tke_e}
164  \frac{\partial \bar{e}}{\partial t} =
165  \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
166      +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
167  -K_\rho\,N^2
168  +\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
169      \;\frac{\partial \bar{e}}{\partial k}} \right]
170  - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
171\end{equation}
172\[
173  % \label{eq:ZDF_tke_kz}
174  \begin{split}
175    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }    \\
176    K_\rho &= A^{vm} / P_{rt}
177  \end{split}
178\]
179where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
180$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,
181$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity and diffusivity coefficients.
182The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$ $\approx 0.7$ are designed to deal with
183vertical mixing at any depth \citep{gaspar.gregoris.ea_JGR90}.
184They are set through namelist parameters \np{nn_ediff}{nn\_ediff} and \np{nn_ediss}{nn\_ediss}.
185$P_{rt}$ can be set to unity or, following \citet{blanke.delecluse_JPO93}, be a function of the local Richardson number, $R_i$:
186\begin{align*}
187  % \label{eq:ZDF_prt}
188  P_{rt} =
189  \begin{cases}
190    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$}   \\
191    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$}   \\
192    \ \ 10 &      \text{if $\ 2 \leq R_i$}
193  \end{cases}
194\end{align*}
195The choice of $P_{rt}$ is controlled by the \np{nn_pdl}{nn\_pdl} namelist variable.
196
197At the sea surface, the value of $\bar{e}$ is prescribed from the wind stress field as
198$\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn_ebb}{rn\_ebb} namelist parameter.
199The default value of $e_{bb}$ is 3.75. \citep{gaspar.gregoris.ea_JGR90}), however a much larger value can be used when
200taking into account the surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}).
201The bottom value of TKE is assumed to be equal to the value of the level just above.
202The time integration of the $\bar{e}$ equation may formally lead to negative values because
203the numerical scheme does not ensure its positivity.
204To overcome this problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn_emin}{rn\_emin} namelist parameter).
205Following \citet{gaspar.gregoris.ea_JGR90}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$.
206This allows the subsequent formulations to match that of \citet{gargett_JMR84} for the diffusion in
207the thermocline and deep ocean :  $K_\rho = 10^{-3} / N$.
208In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical instabilities associated with
209too weak vertical diffusion.
210They must be specified at least larger than the molecular values, and are set through \np{rn_avm0}{rn\_avm0} and
211\np{rn_avt0}{rn\_avt0} (\nam{zdf}{zdf} namelist, see \autoref{subsec:ZDF_cst}).
212
213%% =================================================================================================
214\subsubsection{Turbulent length scale}
215
216For computational efficiency, the original formulation of the turbulent length scales proposed by
217\citet{gaspar.gregoris.ea_JGR90} has been simplified.
218Four formulations are proposed, the choice of which is controlled by the \np{nn_mxl}{nn\_mxl} namelist parameter.
219The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}:
220\begin{equation}
221  \label{eq:ZDF_tke_mxl0_1}
222  l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N
223\end{equation}
224which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency.
225The resulting length scale is bounded by the distance to the surface or to the bottom
226(\np[=0]{nn_mxl}{nn\_mxl}) or by the local vertical scale factor (\np[=1]{nn_mxl}{nn\_mxl}).
227\citet{blanke.delecluse_JPO93} notice that this simplification has two major drawbacks:
228it makes no sense for locally unstable stratification and the computation no longer uses all
229the information contained in the vertical density profile.
230To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np[=2, 3]{nn_mxl}{nn\_mxl} cases,
231which add an extra assumption concerning the vertical gradient of the computed length scale.
232So, the length scales are first evaluated as in \autoref{eq:ZDF_tke_mxl0_1} and then bounded such that:
233\begin{equation}
234  \label{eq:ZDF_tke_mxl_constraint}
235  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
236  \qquad \text{with }\  l =  l_k = l_\epsilon
237\end{equation}
238\autoref{eq:ZDF_tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than
239the variations of depth.
240It provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less
241time consuming.
242In particular, it allows the length scale to be limited not only by the distance to the surface or
243to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as
244the thermocline (\autoref{fig:ZDF_mixing_length}).
245In order to impose the \autoref{eq:ZDF_tke_mxl_constraint} constraint, we introduce two additional length scales:
246$l_{up}$ and $l_{dwn}$, the upward and downward length scales, and
247evaluate the dissipation and mixing length scales as
248(and note that here we use numerical indexing):
249\begin{figure}[!t]
250  \centering
251  \includegraphics[width=0.66\textwidth]{ZDF_mixing_length}
252  \caption[Mixing length computation]{Illustration of the mixing length computation}
253  \label{fig:ZDF_mixing_length}
254\end{figure}
255\[
256  % \label{eq:ZDF_tke_mxl2}
257  \begin{aligned}
258    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
259    \quad &\text{ from $k=1$ to $jpk$ }\ \\
260    l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)\right)
261    \quad &\text{ from $k=jpk$ to $1$ }\ \\
262  \end{aligned}
263\]
264where $l^{(k)}$ is computed using \autoref{eq:ZDF_tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
265
266In the \np[=2]{nn_mxl}{nn\_mxl} case, the dissipation and mixing length scales take the same value:
267$ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the \np[=3]{nn_mxl}{nn\_mxl} case,
268the dissipation and mixing turbulent length scales are give as in \citet{gaspar.gregoris.ea_JGR90}:
269\[
270  % \label{eq:ZDF_tke_mxl_gaspar}
271  \begin{aligned}
272    & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }   \\
273    & l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)
274  \end{aligned}
275\]
276
277At the ocean surface, a non zero length scale is set through the  \np{rn_mxl0}{rn\_mxl0} namelist parameter.
278Usually the surface scale is given by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant and
279$z_o$ the roughness parameter of the surface.
280Assuming $z_o=0.1$~m \citep{craig.banner_JPO94} leads to a 0.04~m, the default value of \np{rn_mxl0}{rn\_mxl0}.
281In the ocean interior a minimum length scale is set to recover the molecular viscosity when
282$\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).
283
284%% =================================================================================================
285\subsubsection{Surface wave breaking parameterization}
286
287Following \citet{mellor.blumberg_JPO04}, the TKE turbulence closure model has been modified to
288include the effect of surface wave breaking energetics.
289This results in a reduction of summertime surface temperature when the mixed layer is relatively shallow.
290The \citet{mellor.blumberg_JPO04} modifications acts on surface length scale and TKE values and
291air-sea drag coefficient.
292The latter concerns the bulk formulae and is not discussed here.
293
294Following \citet{craig.banner_JPO94}, the boundary condition on surface TKE value is :
295\begin{equation}
296  \label{eq:ZDF_Esbc}
297  \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
298\end{equation}
299where $\alpha_{CB}$ is the \citet{craig.banner_JPO94} constant of proportionality which depends on the ''wave age'',
300ranging from 57 for mature waves to 146 for younger waves \citep{mellor.blumberg_JPO04}.
301The boundary condition on the turbulent length scale follows the Charnock's relation:
302\begin{equation}
303  \label{eq:ZDF_Lsbc}
304  l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o}
305\end{equation}
306where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant.
307\citet{mellor.blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by
308\citet{stacey_JPO99} citing observation evidence, and
309$\alpha_{CB} = 100$ the Craig and Banner's value.
310As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$,
311with $e_{bb}$ the \np{rn_ebb}{rn\_ebb} namelist parameter, setting \np[=67.83]{rn_ebb}{rn\_ebb} corresponds
312to $\alpha_{CB} = 100$.
313Further setting  \np[=.true.]{ln_mxl0}{ln\_mxl0},  applies \autoref{eq:ZDF_Lsbc} as the surface boundary condition on the length scale,
314with $\beta$ hard coded to the Stacey's value.
315Note that a minimal threshold of \np{rn_emin0}{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) is applied on the
316surface $\bar{e}$ value.
317
318%% =================================================================================================
319\subsubsection{Langmuir cells}
320
321Langmuir circulations (LC) can be described as ordered large-scale vertical motions in
322the surface layer of the oceans.
323Although LC have nothing to do with convection, the circulation pattern is rather similar to
324so-called convective rolls in the atmospheric boundary layer.
325The detailed physics behind LC is described in, for example, \citet{craik.leibovich_JFM76}.
326The prevailing explanation is that LC arise from a nonlinear interaction between the Stokes drift and
327wind drift currents.
328
329Here we introduced in the TKE turbulent closure the simple parameterization of Langmuir circulations proposed by
330\citep{axell_JGR02} for a $k-\epsilon$ turbulent closure.
331The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in
332an extra source term of TKE, $P_{LC}$.
333The presence of $P_{LC}$ in \autoref{eq:ZDF_tke_e}, the TKE equation, is controlled by setting \np{ln_lc}{ln\_lc} to
334\forcode{.true.} in the \nam{zdf_tke}{zdf\_tke} namelist.
335
336By making an analogy with the characteristic convective velocity scale (\eg, \citet{dalessio.abdella.ea_JPO98}),
337$P_{LC}$ is assumed to be :
338\[
339P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}}
340\]
341where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth.
342With no information about the wave field, $w_{LC}$ is assumed to be proportional to
343the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module
344\footnote{Following \citet{li.garrett_JMR93}, the surface Stoke drift velocity may be expressed as
345  $u_s =  0.016 \,|U_{10m}|$.
346  Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of
347  $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress
348}.
349For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at
350a finite depth $H_{LC}$ (which is often close to the mixed layer depth),
351and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures).
352The resulting expression for $w_{LC}$ is :
353\[
354  w_{LC}  =
355  \begin{cases}
356    c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$}    \\
357    0                             &      \text{otherwise}
358  \end{cases}
359\]
360where $c_{LC} = 0.15$ has been chosen by \citep{axell_JGR02} as a good compromise to fit LES data.
361The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second.
362The value of $c_{LC}$ is set through the \np{rn_lc}{rn\_lc} namelist parameter,
363having in mind that it should stay between 0.15 and 0.54 \citep{axell_JGR02}.
364
365The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations:
366$H_{LC}$ is the depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by
367converting its kinetic energy to potential energy, according to
368\[
369- \int_{-H_{LC}}^0 { N^2\;\;dz} = \frac{1}{2} u_s^2
370\]
371
372%% =================================================================================================
373\subsubsection{Mixing just below the mixed layer}
374
375Vertical mixing parameterizations commonly used in ocean general circulation models tend to
376produce mixed-layer depths that are too shallow during summer months and windy conditions.
377This bias is particularly acute over the Southern Ocean.
378To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{rodgers.aumont.ea_B14}.
379The parameterization is an empirical one, \ie\ not derived from theoretical considerations,
380but rather is meant to account for observed processes that affect the density structure of
381the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme
382(\ie\ near-inertial oscillations and ocean swells and waves).
383
384When using this parameterization (\ie\ when \np[=1]{nn_etau}{nn\_etau}),
385the TKE input to the ocean ($S$) imposed by the winds in the form of near-inertial oscillations,
386swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition,
387plus a depth depend one given by:
388\begin{equation}
389  \label{eq:ZDF_Ehtau}
390  S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau}
391\end{equation}
392where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that
393penetrates in the ocean, $h_\tau$ is a vertical mixing length scale that controls exponential shape of
394the penetration, and $f_i$ is the ice concentration
395(no penetration if $f_i=1$, \ie\ if the ocean is entirely covered by sea-ice).
396The value of $f_r$, usually a few percents, is specified through \np{rn_efr}{rn\_efr} namelist parameter.
397The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np[=0]{nn_etau}{nn\_etau}) or
398a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m at high latitudes
399(\np[=1]{nn_etau}{nn\_etau}).
400
401Note that two other option exist, \np[=2, 3]{nn_etau}{nn\_etau}.
402They correspond to applying \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer,
403or to using the high frequency part of the stress to evaluate the fraction of TKE that penetrates the ocean.
404Those two options are obsolescent features introduced for test purposes.
405They will be removed in the next release.
406
407% This should be explain better below what this rn_eice parameter is meant for:
408In presence of Sea Ice, the value of this mixing can be modulated by the \np{rn_eice}{rn\_eice} namelist parameter.
409This parameter varies from \forcode{0} for no effect to \forcode{4} to suppress the TKE input into the ocean when Sea Ice concentration
410is greater than 25\%.
411
412% from Burchard et al OM 2008 :
413% the most critical process not reproduced by statistical turbulence models is the activity of
414% internal waves and their interaction with turbulence. After the Reynolds decomposition,
415% internal waves are in principle included in the RANS equations, but later partially
416% excluded by the hydrostatic assumption and the model resolution.
417% Thus far, the representation of internal wave mixing in ocean models has been relatively crude
418% (\eg\ Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002).
419
420%% =================================================================================================
421\subsection[GLS: Generic Length Scale (\forcode{ln_zdfgls})]{GLS: Generic Length Scale (\protect\np{ln_zdfgls}{ln\_zdfgls})}
422\label{subsec:ZDF_gls}
423
424\begin{listing}
425  \nlst{namzdf_gls}
426  \caption{\forcode{&namzdf_gls}}
427  \label{lst:namzdf_gls}
428\end{listing}
429
430The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on two prognostic equations:
431one for the turbulent kinetic energy $\bar {e}$, and another for the generic length scale,
432$\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}.
433This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,
434where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:ZDF_GLS} allows to recover a number of
435well-known turbulent closures ($k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87},
436$k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}).
437The GLS scheme is given by the following set of equations:
438\begin{equation}
439  \label{eq:ZDF_gls_e}
440  \frac{\partial \bar{e}}{\partial t} =
441  \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
442      +\left( \frac{\partial v}{\partial k} \right)^2} \right]
443  -K_\rho \,N^2
444  +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right]
445  - \epsilon
446\end{equation}
447
448\[
449  % \label{eq:ZDF_gls_psi}
450  \begin{split}
451    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{
452      \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
453          +\left( \frac{\partial v}{\partial k} \right)^2} \right]
454      - C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\
455    &+\frac{1}{e_3\;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 }
456        \;\frac{\partial \psi}{\partial k}} \right]\;
457  \end{split}
458\]
459
460\[
461  % \label{eq:ZDF_gls_kz}
462  \begin{split}
463    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
464    K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l
465  \end{split}
466\]
467
468\[
469  % \label{eq:ZDF_gls_eps}
470  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
471\]
472where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and
473$\epsilon$ the dissipation rate.
474The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of
475the choice of the turbulence model.
476Four different turbulent models are pre-defined (\autoref{tab:ZDF_GLS}).
477They are made available through the \np{nn_clo}{nn\_clo} namelist parameter.
478
479\begin{table}[htbp]
480  \centering
481  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c}
482  \begin{tabular}{ccccc}
483    &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\
484    % & \citep{mellor.yamada_RG82} &  \citep{rodi_JGR87}       & \citep{wilcox_AJ88} &                 \\
485    \hline
486    \hline
487    \np{nn_clo}{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\
488    \hline
489    $( p , n , m )$         &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\
490    $\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\
491    $\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\
492    $C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\
493    $C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\
494    $C_3$              &      1.           &     1.              &      1.                &       1.           \\
495    $F_{wall}$        &      Yes        &       --             &     --                  &      --          \\
496    \hline
497    \hline
498  \end{tabular}
499  \caption[Set of predefined GLS parameters or equivalently predefined turbulence models available]{
500    Set of predefined GLS parameters, or equivalently predefined turbulence models available with
501    \protect\np[=.true.]{ln_zdfgls}{ln\_zdfgls} and controlled by
502    the \protect\np{nn_clos}{nn\_clos} namelist variable in \protect\nam{zdf_gls}{zdf\_gls}.}
503  \label{tab:ZDF_GLS}
504\end{table}
505
506In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force the convergence of
507the mixing length towards $\kappa z_b$ ($\kappa$ is the Von Karman constant and $z_b$ the rugosity length scale) value near physical boundaries
508(logarithmic boundary layer law).
509$C_{\mu}$ and $C_{\mu'}$ are calculated from stability function proposed by \citet{galperin.kantha.ea_JAS88},
510or by \citet{kantha.clayson_JGR94} or one of the two functions suggested by \citet{canuto.howard.ea_JPO01}
511(\np[=0, 3]{nn_stab_func}{nn\_stab\_func}, resp.).
512The value of $C_{0\mu}$ depends on the choice of the stability function.
513
514The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated thanks to Dirichlet or
515Neumann condition through \np{nn_bc_surf}{nn\_bc\_surf} and \np{nn_bc_bot}{nn\_bc\_bot}, resp.
516As for TKE closure, the wave effect on the mixing is considered when
517\np[ > 0.]{rn_crban}{rn\_crban} \citep{craig.banner_JPO94, mellor.blumberg_JPO04}.
518The \np{rn_crban}{rn\_crban} namelist parameter is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and
519\np{rn_charn}{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}.
520
521The $\psi$ equation is known to fail in stably stratified flows, and for this reason
522almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.
523With this clipping, the maximum permissible length scale is determined by $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$.
524A value of $c_{lim} = 0.53$ is often used \citep{galperin.kantha.ea_JAS88}.
525\cite{umlauf.burchard_CSR05} show that the value of the clipping factor is of crucial importance for
526the entrainment depth predicted in stably stratified situations,
527and that its value has to be chosen in accordance with the algebraic model for the turbulent fluxes.
528The clipping is only activated if \np[=.true.]{ln_length_lim}{ln\_length\_lim},
529and the $c_{lim}$ is set to the \np{rn_clim_galp}{rn\_clim\_galp} value.
530
531The time and space discretization of the GLS equations follows the same energetic consideration as for
532the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{burchard_OM02}.
533Evaluation of the 4 GLS turbulent closure schemes can be found in \citet{warner.sherwood.ea_OM05} in ROMS model and
534 in \citet{reffray.guillaume.ea_GMD15} for the \NEMO\ model.
535
536% -------------------------------------------------------------------------------------------------------------
537%        OSM OSMOSIS BL Scheme
538% -------------------------------------------------------------------------------------------------------------
539\subsection[OSM: OSMOSIS boundary layer scheme (\forcode{ln_zdfosm = .true.})]
540{OSM: OSMOSIS boundary layer scheme (\protect\np{ln_zdfosm}{ln\_zdfosm})}
541\label{subsec:ZDF_osm}
542
543\begin{listing}
544  \nlst{namzdf_osm}
545  \caption{\forcode{&namzdf_osm}}
546  \label{lst:namzdf_osm}
547\end{listing}
548
549%--------------------------------------------------------------------------------------------------------------
550\paragraph{Namelist choices}
551Most of the namelist options refer to how to specify the Stokes
552surface drift and penetration depth. There are three options:
553\begin{description}
554  \item \protect\np[=0]{nn_osm_wave}{nn\_osm\_wave} Default value in \texttt{namelist\_ref}. In this case the Stokes drift is
555      assumed to be parallel to the surface wind stress, with
556      magnitude consistent with a constant turbulent Langmuir number
557    $\mathrm{La}_t=$ \protect\np{rn_m_la} {rn\_m\_la} i.e.\
558    $u_{s0}=\tau/(\mathrm{La}_t^2\rho_0)$.  Default value of
559    \protect\np{rn_m_la}{rn\_m\_la} is 0.3. The Stokes penetration
560      depth $\delta = $ \protect\np{rn_osm_dstokes}{rn\_osm\_dstokes}; this has default value
561      of 5~m.
562
563  \item \protect\np[=1]{nn_osm_wave}{nn\_osm\_wave} In this case the Stokes drift is
564      assumed to be parallel to the surface wind stress, with
565      magnitude as in the classical Pierson-Moskowitz wind-sea
566      spectrum.  Significant wave height and
567      wave-mean period taken from this spectrum are used to calculate the Stokes penetration
568      depth, following the approach set out in  \citet{breivik.janssen.ea_JPO14}.
569
570    \item \protect\np[=2]{nn_osm_wave}{nn\_osm\_wave} In this case the Stokes drift is
571      taken from  ECMWF wave model output, though only the component parallel
572      to the wind stress is retained. Significant wave height and
573      wave-mean period from ECMWF wave model output are used to calculate the Stokes penetration
574      depth, again following \citet{breivik.janssen.ea_JPO14}.
575
576    \end{description}
577
578    Others refer to the treatment of diffusion and viscosity beneath
579    the surface boundary layer:
580\begin{description}
581   \item \protect\np{ln_kpprimix} {ln\_kpprimix}  Default is \np{.true.}. Switches on KPP-style Ri \#-dependent
582     mixing below the surface boundary layer. If this is set
583     \texttt{.true.}  the following variable settings are honoured:
584    \item \protect\np{rn_riinfty}{rn\_riinfty} Critical value of local Ri \# below which
585      shear instability increases vertical mixing from background value.
586    \item \protect\np{rn_difri} {rn\_difri} Maximum value of Ri \#-dependent mixing at $\mathrm{Ri}=0$.
587    \item \protect\np{ln_convmix}{ln\_convmix} If \texttt{.true.} then, where water column is unstable, specify
588       diffusivity equal to \protect\np{rn_dif_conv}{rn\_dif\_conv} (default value is 1 m~s$^{-2}$).
589 \end{description}
590 Diagnostic output is controlled by:
591  \begin{description}
592    \item \protect\np{ln_dia_osm}{ln\_dia\_osm} Default is \np{.false.}; allows XIOS output of OSMOSIS internal fields.
593  \end{description}
594Obsolete namelist parameters include:
595\begin{description}
596   \item \protect\np{ln_use_osm_la}\np{ln\_use\_osm\_la} With \protect\np[=0]{nn_osm_wave}{nn\_osm\_wave},
597      \protect\np{rn_osm_dstokes} {rn\_osm\_dstokes} is always used to specify the Stokes
598      penetration depth.
599   \item \protect\np{nn_ave} {nn\_ave} Choice of averaging method for KPP-style Ri \#
600      mixing. Not taken account of.
601   \item \protect\np{rn_osm_hbl0} {rn\_osm\_hbl0} Depth of initial boundary layer is now set
602     by a density criterion similar to that used in calculating \emph{hmlp} (output as \texttt{mldr10\_1}) in \mdl{zdfmxl}.
603\end{description}
604
605\subsubsection{Summary}
606Much of the time the turbulent motions in the ocean surface boundary
607layer (OSBL) are not given by
608classical shear turbulence. Instead they are in a regime known as
609`Langmuir turbulence',  dominated by an
610interaction between the currents and the Stokes drift of the surface waves \citep[e.g.][]{mcwilliams.ea_JFM97}.
611This 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 $\mathrm{La}_{t}=(u_{\ast}/u_{s0}) > 0.4$.
612
613The OSMOSIS model is fundamentally based on results of Large Eddy
614Simulations (LES) of Langmuir turbulence and aims to fully describe
615this Langmuir regime. The description in this section is of necessity incomplete and further details are available in Grant. A (2019); in prep.
616
617The OSMOSIS turbulent closure scheme is a similarity-scale scheme in
618the same spirit as the K-profile
619parameterization (KPP) scheme of \citet{large.ea_RG97}.
620A specified shape of diffusivity, scaled by the (OSBL) depth
621$h_{\mathrm{BL}}$ and a turbulent velocity scale, is imposed throughout the
622boundary layer
623$-h_{\mathrm{BL}}<z<\eta$. The turbulent closure model
624also includes fluxes of tracers and momentum that are``non-local'' (independent of the local property gradient).
625
626Rather than the OSBL
627depth being diagnosed in terms of a bulk Richardson number criterion,
628as in KPP, it is set by a prognostic equation that is informed by
629energy budget considerations reminiscent of the classical mixed layer
630models of \citet{kraus.turner_tellus67}.
631The model also includes an explicit parametrization of the structure
632of the pycnocline (the stratified region at the bottom of the OSBL).
633
634Presently, mixing below the OSBL is handled by the Richardson
635number-dependent mixing scheme used in \citet{large.ea_RG97}.
636
637Convective parameterizations such as described in \ref{sec:ZDF_conv}
638below should not be used with the OSMOSIS-OBL model: instabilities
639within the OSBL are part of the model, while instabilities below the
640ML are handled by the Ri \# dependent scheme.
641
642\subsubsection{Depth and velocity scales}
643The 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).
644\begin{figure}[!t]
645  \begin{center}
646    %\includegraphics[width=0.7\textwidth]{ZDF_OSM_structure_of_OSBL}
647    \caption{
648      \protect\label{fig: OSBL_structure}
649     The structure of the entraining  boundary layer. (a) Mean buoyancy profile. (b) Profile of the buoyancy flux.
650    }
651  \end{center}
652\end{figure}
653The 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.
654
655Consideration of the power input by wind acting on the Stokes drift suggests that the Langmuir turbulence has velocity scale:
656\begin{equation}\label{eq:w_La}
657w_{*L}= \left(u_*^2 u_{s\,0}\right)^{1/3};
658\end{equation}
659but 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:
660\begin{equation}\label{eq:composite-nu}
661  \nu_{\ast}= \left\{ u_*^3 \left[1-\exp(-.5 \mathrm{La}_t^2)\right]+w_{*L}^3\right\}^{1/3}.
662\end{equation}
663For 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:
664\begin{equation}\label{eq:vel-scale-unstable}
665\omega_* = \left(\nu_*^3 + 0.5 w_{*C}^3\right)^{1/3}.
666\end{equation}
667
668\subsubsection{The flux gradient model}
669The flux-gradient relationships used in the OSMOSIS scheme take the form:
670%
671\begin{equation}\label{eq:flux-grad-gen}
672\overline{w^\prime\chi^\prime}=-K\frac{\partial\overline{\chi}}{\partial z} + N_{\chi,s} +N_{\chi,b} +N_{\chi,t},
673\end{equation}
674%
675where $\chi$ 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.
676
677In terms of the non-dimensionalized depth variables
678%
679\begin{equation}\label{eq:sigma}
680\sigma_{\mathrm{ml}}= -z/h_{\mathrm{ml}}; \;\sigma_{\mathrm{bl}}= -z/h_{\mathrm{bl}},
681\end{equation}
682%
683in unstable conditions the eddy diffusivity ($K_d$) and eddy viscosity ($K_\nu$) profiles are parametrized as:
684%
685\begin{align}\label{eq:diff-unstable}
686K_d=&0.8\, \omega_*\, h_{\mathrm{ml}} \, \sigma_{\mathrm{ml}} \left(1-\beta_d \sigma_{\mathrm{ml}}\right)^{3/2}
687\\\label{eq:visc-unstable}
688K_\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)
689\end{align}
690%
691where $\beta_d$ 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
692%
693\begin{equation}\label{eq:diff-wml-base}
694K_{d,\mathrm{ml}}=K_{\nu,\mathrm{ml}}=\,0.16\,\omega_* \Delta h.
695\end{equation}
696%
697For stable conditions the eddy diffusivity/viscosity profiles are given by:
698%
699\begin{align}\label{diff-stable}
700K_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}
701K_\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).
702\end{align}
703%
704The 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:
705\begin{equation}\label{eq:L_L}
706  L_L=-w_{*L}^3/\left<\overline{w^\prime b^\prime}\right>_L,
707\end{equation}
708with 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
709\begin{equation} \label{eq:stable-av-buoy-flux}
710\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].
711\end{equation}
712%
713In unstable conditions the eddy diffusivity and viscosity depend on stability through the velocity scale $\omega_*$, which depends on the two velocity scales $\nu_*$ and $w_{*C}$.
714
715Details 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).
716
717\subsubsection{Evolution of the boundary layer depth}
718
719The prognostic equation for the depth of the neutral/unstable boundary layer is given by \citep{grant+etal18},
720
721\begin{equation} \label{eq:dhdt-unstable}
722%\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}}
723\frac{\partial h_\mathrm{bl}}{\partial t} = W_b - \frac{{\overline{w^\prime b^\prime}}_\mathrm{ent}}{\Delta B_\mathrm{bl}}
724\end{equation}
725where $h_\mathrm{bl}$ is the horizontally-varying depth of the OSBL,
726$\mathbf{U}_b$ and $W_b$ are the mean horizontal and vertical
727velocities at the base of the OSBL, ${\overline{w^\prime
728    b^\prime}}_\mathrm{ent}$ is the buoyancy flux due to entrainment
729and $\Delta B_\mathrm{bl}$ is the difference between the buoyancy
730averaged over the depth of the OSBL (i.e.\ including the ML and
731pycnocline) and the buoyancy just below the base of the OSBL. This
732equation for the case when the pycnocline has a finite thickness,
733based on the potential energy budget of the OSBL, is the leading term
734\citep{grant+etal18} of a generalization of that used in mixed-layer
735models e.g.\ \citet{kraus.turner_tellus67}, in which the thickness of the pycnocline is taken to be zero.
736
737The entrainment flux for the combination of convective and Langmuir turbulence is given by
738\begin{equation} \label{eq:entrain-flux}
739  {\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}}}
740  + 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]
741\end{equation}
742where 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 $F(x)\equiv\tanh(x^{-1})^{0.69}$.
743
744For the stable boundary layer, the equation for the depth of the OSBL is:
745
746\begin{equation}\label{eq:dhdt-stable}
747\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.
748\end{equation}
749
750Equation. \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.
751
752
753%% =================================================================================================
754\subsection[ Discrete energy conservation for TKE and GLS schemes]{Discrete energy conservation for TKE and GLS schemes}
755\label{subsec:ZDF_tke_ene}
756
757\begin{figure}[!t]
758  \centering
759  \includegraphics[width=0.66\textwidth]{ZDF_TKE_time_scheme}
760  \caption[Subgrid kinetic energy integration in GLS and TKE schemes]{
761    Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and
762    its links to the momentum and tracer time integration.}
763  \label{fig:ZDF_TKE_time_scheme}
764\end{figure}
765
766The production of turbulence by vertical shear (the first term of the right hand side of
767\autoref{eq:ZDF_tke_e}) and  \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion
768(first line in \autoref{eq:MB_zdf}).
769To do so a special care has to be taken for both the time and space discretization of
770the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}.
771
772Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how
773the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with
774the one-level forward time stepping of the equation for $\bar{e}$.
775With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to
776the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and
777summing the result vertically:
778\begin{equation}
779  \label{eq:ZDF_energ1}
780  \begin{split}
781    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\
782    &= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}
783    - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz }
784  \end{split}
785\end{equation}
786Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$,
787known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme
788(see \autoref{sec:TD_forward_imp}).
789The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at
790the surface (atmospheric forcing) and at the bottom (friction effect).
791The second term is always negative.
792It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$.
793\autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
794the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
795${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$
796(and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
797
798A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
799(second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}).
800This term must balance the input of potential energy resulting from vertical mixing.
801The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by
802multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically:
803\begin{equation}
804  \label{eq:ZDF_energ2}
805  \begin{split}
806    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\
807    &= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}
808    - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\
809    &= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta}
810    + \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  }
811  \end{split}
812\end{equation}
813where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.
814The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because
815there is no diffusive flux through the ocean surface and bottom).
816The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
817Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
818the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and  \autoref{eq:ZDF_gls_e}.
819
820Let us now address the space discretization issue.
821The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in
822the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}).
823A space averaging is thus required to obtain the shear TKE production term.
824By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by
825the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
826Furthermore, the time variation of $e_3$ has be taken into account.
827
828The above energetic considerations leads to the following final discrete form for the TKE equation:
829\begin{equation}
830  \label{eq:ZDF_tke_ene}
831  \begin{split}
832    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv
833    \Biggl\{ \Biggr.
834    &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} }
835        \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\
836    +&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} }
837        \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j}
838    \Biggr. \Biggr\}   \\
839    %
840    - &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\
841    %
842    +&\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]   \\
843    %
844    - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt}
845  \end{split}
846\end{equation}
847where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation)
848are time stepped using a backward scheme (see\autoref{sec:TD_forward_imp}).
849Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible.
850%The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as
851%they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}.
852%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.
853
854%% =================================================================================================
855\section{Convection}
856\label{sec:ZDF_conv}
857
858Static instabilities (\ie\ light potential densities under heavy ones) may occur at particular ocean grid points.
859In nature, convective processes quickly re-establish the static stability of the water column.
860These processes have been removed from the model via the hydrostatic assumption so they must be parameterized.
861Three parameterisations are available to deal with convective processes:
862a non-penetrative convective adjustment or an enhanced vertical diffusion,
863or/and the use of a turbulent closure scheme.
864
865%% =================================================================================================
866\subsection[Non-penetrative convective adjustment (\forcode{ln_tranpc})]{Non-penetrative convective adjustment (\protect\np{ln_tranpc}{ln\_tranpc})}
867\label{subsec:ZDF_npc}
868
869\begin{figure}[!htb]
870  \centering
871  \includegraphics[width=0.66\textwidth]{ZDF_npc}
872  \caption[Unstable density profile treated by the non penetrative convective adjustment algorithm]{
873    Example of an unstable density profile treated by
874    the non penetrative convective adjustment algorithm.
875    $1^{st}$ step: the initial profile is checked from the surface to the bottom.
876    It is found to be unstable between levels 3 and 4.
877    They are mixed.
878    The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
879    The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
880    The $1^{st}$ step ends since the density profile is then stable below the level 3.
881    $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
882    levels 2 to 5 are mixed.
883    The new density profile is checked.
884    It is found stable: end of algorithm.}
885  \label{fig:ZDF_npc}
886\end{figure}
887
888Options are defined through the \nam{zdf}{zdf} namelist variables.
889The non-penetrative convective adjustment is used when \np[=.true.]{ln_zdfnpc}{ln\_zdfnpc}.
890It is applied at each \np{nn_npc}{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of
891the water column, but only until the density structure becomes neutrally stable
892(\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below)
893\citep{madec.delecluse.ea_JPO91}.
894The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}):
895starting from the top of the ocean, the first instability is found.
896Assume in the following that the instability is located between levels $k$ and $k+1$.
897The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of
898the water column.
899The new density is then computed by a linear approximation.
900If the new density profile is still unstable between levels $k+1$ and $k+2$,
901levels $k$, $k+1$ and $k+2$ are then mixed.
902This process is repeated until stability is established below the level $k$
903(the mixing process can go down to the ocean bottom).
904The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or
905if there is no deeper instability.
906
907This algorithm is significantly different from mixing statically unstable levels two by two.
908The latter procedure cannot converge with a finite number of iterations for some vertical profiles while
909the algorithm used in \NEMO\ converges for any profile in a number of iterations which is less than
910the number of vertical levels.
911This property is of paramount importance as pointed out by \citet{killworth_iprc89}:
912it avoids the existence of permanent and unrealistic static instabilities at the sea surface.
913This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in
914the north-western Mediterranean Sea \citep{madec.delecluse.ea_JPO91, madec.chartier.ea_DAO91, madec.crepon_iprc91}.
915
916The current implementation has been modified in order to deal with any non linear equation of seawater
917(L. Brodeau, personnal communication).
918Two main differences have been introduced compared to the original algorithm:
919$(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency
920(not the difference in potential density);
921$(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in
922the same way their temperature and salinity has been mixed.
923These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without
924having to recompute the expansion coefficients at each mixing iteration.
925
926%% =================================================================================================
927\subsection[Enhanced vertical diffusion (\forcode{ln_zdfevd})]{Enhanced vertical diffusion (\protect\np{ln_zdfevd}{ln\_zdfevd})}
928\label{subsec:ZDF_evd}
929
930Options are defined through the  \nam{zdf}{zdf} namelist variables.
931The enhanced vertical diffusion parameterisation is used when \np[=.true.]{ln_zdfevd}{ln\_zdfevd}.
932In this case, the vertical eddy mixing coefficients are assigned very large values
933in regions where the stratification is unstable
934(\ie\ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{lazar_phd97, lazar.madec.ea_JPO99}.
935This is done either on tracers only (\np[=0]{nn_evdm}{nn\_evdm}) or
936on both momentum and tracers (\np[=1]{nn_evdm}{nn\_evdm}).
937
938In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np[=1]{nn_evdm}{nn\_evdm},
939the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to
940the namelist parameter \np{rn_avevd}{rn\_avevd}.
941A typical value for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$.
942This parameterisation of convective processes is less time consuming than
943the convective adjustment algorithm presented above when mixing both tracers and
944momentum in the case of static instabilities.
945
946Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$.
947This removes a potential source of divergence of odd and even time step in
948a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}).
949
950%% =================================================================================================
951\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}})}
952\label{subsec:ZDF_tcs}
953
954The turbulent closure schemes presented in \autoref{subsec:ZDF_tke}, \autoref{subsec:ZDF_gls} and
955\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,
956with statically unstable density profiles.
957In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in
958\autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative.
959It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also of the four neighboring values at
960velocity points $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1}$).
961These large values restore the static stability of the water column in a way similar to that of
962the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}).
963However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by
964the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$,
965because the mixing length scale is bounded by the distance to the sea surface.
966It can thus be useful to combine the enhanced vertical diffusion with the turbulent closure scheme,
967\ie\ setting the \np{ln_zdfnpc}{ln\_zdfnpc} namelist parameter to true and
968defining the turbulent closure (\np{ln_zdftke}{ln\_zdftke} or \np{ln_zdfgls}{ln\_zdfgls} = \forcode{.true.}) all together.
969
970The OSMOSIS turbulent closure scheme already includes enhanced vertical diffusion in the case of convection,
971%as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp},
972therefore \np[=.false.]{ln_zdfevd}{ln\_zdfevd} should be used with the OSMOSIS scheme.
973% gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
974
975%% =================================================================================================
976\section[Double diffusion mixing (\forcode{ln_zdfddm})]{Double diffusion mixing (\protect\np{ln_zdfddm}{ln\_zdfddm})}
977\label{subsec:ZDF_ddm}
978
979%\nlst{namzdf_ddm}
980
981This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the namelist parameter
982\np{ln_zdfddm}{ln\_zdfddm} in \nam{zdf}{zdf}.
983Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa.
984The former condition leads to salt fingering and the latter to diffusive convection.
985Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean.
986\citet{merryfield.holloway.ea_JPO99} include a parameterisation of such phenomena in a global ocean model and show that
987it leads to relatively minor changes in circulation but exerts significant regional influences on
988temperature and salinity.
989
990Diapycnal mixing of S and T are described by diapycnal diffusion coefficients
991\begin{align*}
992  % \label{eq:ZDF_ddm_Kz}
993  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\
994  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
995\end{align*}
996where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
997and $o$ by processes other than double diffusion.
998The rates of double-diffusive mixing depend on the buoyancy ratio
999$R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of
1000thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}).
1001To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt
1002(1981):
1003\begin{align}
1004  \label{eq:ZDF_ddm_f}
1005  A_f^{vS} &=
1006             \begin{cases}
1007               \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
1008               0                              &\text{otherwise}
1009             \end{cases}
1010  \\         \label{eq:ZDF_ddm_f_T}
1011  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho
1012\end{align}
1013
1014\begin{figure}[!t]
1015  \centering
1016  \includegraphics[width=0.66\textwidth]{ZDF_ddm}
1017  \caption[Diapycnal diffusivities for temperature and salt in regions of salt fingering and
1018  diffusive convection]{
1019    From \citet{merryfield.holloway.ea_JPO99}:
1020    (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in
1021    regions of salt fingering.
1022    Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and
1023    thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
1024    (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in
1025    regions of diffusive convection.
1026    Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
1027    The latter is not implemented in \NEMO.}
1028  \label{fig:ZDF_ddm}
1029\end{figure}
1030
1031The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of
1032buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}).
1033Following  \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
1034
1035To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by
1036Federov (1988) is used:
1037\begin{align}
1038  % \label{eq:ZDF_ddm_d}
1039  A_d^{vT} &=
1040             \begin{cases}
1041               1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
1042               &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
1043               0                       &\text{otherwise}
1044             \end{cases}
1045                                       \nonumber \\
1046  \label{eq:ZDF_ddm_d_S}
1047  A_d^{vS} &=
1048             \begin{cases}
1049               A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
1050               A_d^{vT} \ 0.15 \ R_\rho               &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
1051               0                       &\text{otherwise}
1052             \end{cases}
1053\end{align}
1054
1055The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in
1056\autoref{fig:ZDF_ddm}.
1057Implementing this requires computing $R_\rho$ at each grid point on every time step.
1058This is done in \mdl{eosbn2} at the same time as $N^2$ is computed.
1059This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive).
1060
1061%% =================================================================================================
1062\section[Bottom and top friction (\textit{zdfdrg.F90})]{Bottom and top friction (\protect\mdl{zdfdrg})}
1063\label{sec:ZDF_drg}
1064
1065\begin{listing}
1066  \nlst{namdrg}
1067  \caption{\forcode{&namdrg}}
1068  \label{lst:namdrg}
1069\end{listing}
1070\begin{listing}
1071  \nlst{namdrg_top}
1072  \caption{\forcode{&namdrg_top}}
1073  \label{lst:namdrg_top}
1074\end{listing}
1075\begin{listing}
1076  \nlst{namdrg_bot}
1077  \caption{\forcode{&namdrg_bot}}
1078  \label{lst:namdrg_bot}
1079\end{listing}
1080
1081Options to define the top and bottom friction are defined through the \nam{drg}{drg} namelist variables.
1082The bottom friction represents the friction generated by the bathymetry.
1083The top friction represents the friction generated by the ice shelf/ocean interface.
1084As the friction processes at the top and the bottom are treated in and identical way,
1085the description below considers mostly the bottom friction case, if not stated otherwise.
1086
1087Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as
1088a condition on the vertical diffusive flux.
1089For the bottom boundary layer, one has:
1090 \[
1091   % \label{eq:ZDF_bfr_flux}
1092   A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
1093 \]
1094where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside
1095the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean).
1096How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near
1097the bottom relative to the Ekman layer depth.
1098For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m,
1099one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$
1100(for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$).
1101With a background diffusion coefficient $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
1102When the vertical mixing coefficient is this small, using a flux condition is equivalent to
1103entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or
1104bottom model layer.
1105To illustrate this, consider the equation for $u$ at $k$, the last ocean level:
1106\begin{equation}
1107  \label{eq:ZDF_drg_flux2}
1108  \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}}
1109\end{equation}
1110If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth.
1111On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used,
1112the turbulent Ekman layer will be represented explicitly by the model.
1113However, the logarithmic layer is never represented in current primitive equation model applications:
1114it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
1115Two choices are available in \NEMO: a linear and a quadratic bottom friction.
1116Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in
1117the present release of \NEMO.
1118
1119In the code, the bottom friction is imposed by adding the trend due to the bottom friction to
1120 the general momentum trend in \mdl{dynzdf}.
1121For the time-split surface pressure gradient algorithm, the momentum trend due to
1122the barotropic component needs to be handled separately.
1123For this purpose it is convenient to compute and store coefficients which can be simply combined with
1124bottom velocities and geometric values to provide the momentum trend due to bottom friction.
1125 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where:
1126\begin{equation}
1127  \label{eq:ZDF_bfr_bdef}
1128  \frac{\partial {\textbf U_h}}{\partial t} =
1129  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b
1130\end{equation}
1131where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity.
1132Note 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.
1133
1134%% =================================================================================================
1135\subsection[Linear top/bottom friction (\forcode{ln_lin})]{Linear top/bottom friction (\protect\np{ln_lin}{ln\_lin})}
1136\label{subsec:ZDF_drg_linear}
1137
1138The linear friction parameterisation (including the special case of a free-slip condition) assumes that
1139the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level):
1140\[
1141  % \label{eq:ZDF_bfr_linear}
1142  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
1143\]
1144where $r$ is a friction coefficient expressed in $m s^{-1}$.
1145This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean,
1146and setting $r = H / \tau$, where $H$ is the ocean depth.
1147Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{weatherly_JMR84}.
1148A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models.
1149One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$
1150(\citet{gill_bk82}, Eq. 9.6.6).
1151For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$,
1152and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.
1153This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days.
1154It can be changed by specifying \np{rn_Uc0}{rn\_Uc0} (namelist parameter).
1155
1156 For the linear friction case the drag coefficient used in the general expression \autoref{eq:ZDF_bfr_bdef} is:
1157\[
1158  % \label{eq:ZDF_bfr_linbfr_b}
1159    c_b^T = - r
1160\]
1161When \np[=.true.]{ln_lin}{ln\_lin}, the value of $r$ used is \np{rn_Uc0}{rn\_Uc0}*\np{rn_Cd0}{rn\_Cd0}.
1162Setting \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.
1163
1164These values are assigned in \mdl{zdfdrg}.
1165Note that there is support for local enhancement of these values via an externally defined 2D mask array
1166(\np[=.true.]{ln_boost}{ln\_boost}) given in the \ifile{bfr\_coef} input NetCDF file.
1167The mask values should vary from 0 to 1.
1168Locations with a non-zero mask value will have the friction coefficient increased by
1169$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}.
1170
1171%% =================================================================================================
1172\subsection[Non-linear top/bottom friction (\forcode{ln_non_lin})]{Non-linear top/bottom friction (\protect\np{ln_non_lin}{ln\_non\_lin})}
1173\label{subsec:ZDF_drg_nonlinear}
1174
1175The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic:
1176\[
1177  % \label{eq:ZDF_drg_nonlinear}
1178  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
1179  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
1180\]
1181where $C_D$ is a drag coefficient, and $e_b $ a top/bottom turbulent kinetic energy due to tides,
1182internal waves breaking and other short time scale currents.
1183A typical value of the drag coefficient is $C_D = 10^{-3} $.
1184As an example, the CME experiment \citep{treguier_JGR92} uses $C_D = 10^{-3}$ and
1185$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{killworth_JPO92} uses $C_D = 1.4\;10^{-3}$ and
1186$e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.
1187The CME choices have been set as default values (\np{rn_Cd0}{rn\_Cd0} and \np{rn_ke0}{rn\_ke0} namelist parameters).
1188
1189As for the linear case, the friction is imposed in the code by adding the trend due to
1190the friction to the general momentum trend in \mdl{dynzdf}.
1191For the non-linear friction case the term computed in \mdl{zdfdrg} is:
1192\[
1193  % \label{eq:ZDF_drg_nonlinbfr}
1194    c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2}
1195\]
1196
1197The coefficients that control the strength of the non-linear friction are initialised as namelist parameters:
1198$C_D$= \np{rn_Cd0}{rn\_Cd0}, and $e_b$ =\np{rn_bfeb2}{rn\_bfeb2}.
1199Note 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
1200(\np[=.true.]{ln_boost}{ln\_boost}).
1201This works in the same way as for the linear friction case with non-zero masked locations increased by
1202$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}.
1203
1204%% =================================================================================================
1205\subsection[Log-layer top/bottom friction (\forcode{ln_loglayer})]{Log-layer top/bottom friction (\protect\np{ln_loglayer}{ln\_loglayer})}
1206\label{subsec:ZDF_drg_loglayer}
1207
1208In the non-linear friction case, the drag coefficient, $C_D$, can be optionally enhanced using
1209a "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.
1210If  \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):
1211\[
1212  C_D = \left ( {\kappa \over {\mathrm log}\left ( 0.5 \; e_{3b} / rn\_{z0} \right ) } \right )^2
1213\]
1214
1215\noindent where $\kappa$ is the von-Karman constant and \np{rn_z0}{rn\_z0} is a roughness length provided via the namelist.
1216
1217The drag coefficient is bounded such that it is kept greater or equal to
1218the 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:
1219\np{rn_Cdmax}{rn\_Cdmax}, \ie
1220\[
1221  rn\_Cd0 \leq C_D \leq rn\_Cdmax
1222\]
1223
1224\noindent The log-layer enhancement can also be applied to the top boundary friction if
1225under ice-shelf cavities are activated (\np[=.true.]{ln_isfcav}{ln\_isfcav}).
1226%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}.
1227
1228%% =================================================================================================
1229\subsection[Explicit top/bottom friction (\forcode{ln_drgimp=.false.})]{Explicit top/bottom friction (\protect\np[=.false.]{ln_drgimp}{ln\_drgimp})}
1230\label{subsec:ZDF_drg_stability}
1231
1232Setting \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:
1233
1234At the top (below an ice shelf cavity):
1235\[
1236  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1237  = c_{t}^{\textbf{U}}\textbf{u}^{n-1}_{t}
1238\]
1239
1240At the bottom (above the sea floor):
1241\[
1242  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1243  = c_{b}^{\textbf{U}}\textbf{u}^{n-1}_{b}
1244\]
1245
1246Since 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.
1247For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is:
1248\begin{equation}
1249  \label{eq:ZDF_Eqn_drgstab}
1250  \begin{split}
1251    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
1252    &= -\frac{ru}{e_{3u}}\;2\rdt\\
1253  \end{split}
1254\end{equation}
1255\noindent where linear friction and a leapfrog timestep have been assumed.
1256To ensure that the friction cannot reverse the direction of flow it is necessary to have:
1257\[
1258  |\Delta u| < \;|u|
1259\]
1260\noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives:
1261\[
1262  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
1263\]
1264This same inequality can also be derived in the non-linear bottom friction case if
1265a velocity of 1 m.s$^{-1}$ is assumed.
1266Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability:
1267\[
1268  e_{3u} > 2\;r\;\rdt
1269\]
1270\noindent which it may be necessary to impose if partial steps are being used.
1271For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m.
1272For most applications, with physically sensible parameters these restrictions should not be of concern.
1273But caution may be necessary if attempts are made to locally enhance the bottom friction parameters.
1274To ensure stability limits are imposed on the top/bottom friction coefficients both
1275during initialisation and at each time step.
1276Checks at initialisation are made in \mdl{zdfdrg} (assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
1277The number of breaches of the stability criterion are reported as well as
1278the minimum and maximum values that have been set.
1279The criterion is also checked at each time step, using the actual velocity, in \mdl{dynzdf}.
1280Values of the friction coefficient are reduced as necessary to ensure stability;
1281these changes are not reported.
1282
1283Limits on the top/bottom friction coefficient are not imposed if the user has elected to
1284handle the friction implicitly (see \autoref{subsec:ZDF_drg_imp}).
1285The number of potential breaches of the explicit stability criterion are still reported for information purposes.
1286
1287%% =================================================================================================
1288\subsection[Implicit top/bottom friction (\forcode{ln_drgimp=.true.})]{Implicit top/bottom friction (\protect\np[=.true.]{ln_drgimp}{ln\_drgimp})}
1289\label{subsec:ZDF_drg_imp}
1290
1291An optional implicit form of bottom friction has been implemented to improve model stability.
1292We recommend this option for shelf sea and coastal ocean applications. %, especially for split-explicit time splitting.
1293This option can be invoked by setting \np{ln_drgimp}{ln\_drgimp} to \forcode{.true.} in the \nam{drg}{drg} namelist.
1294%This option requires \np{ln_zdfexp}{ln\_zdfexp} to be \forcode{.false.} in the \nam{zdf}{zdf} namelist.
1295
1296This implementation is performed in \mdl{dynzdf} where the following boundary conditions are set while solving the fully implicit diffusion step:
1297
1298At the top (below an ice shelf cavity):
1299\[
1300  % \label{eq:ZDF_dynZDF__drg_top}
1301  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1302  = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t}
1303\]
1304
1305At the bottom (above the sea floor):
1306\[
1307  % \label{eq:ZDF_dynZDF__drg_bot}
1308  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1309  = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b}
1310\]
1311
1312where $t$ and $b$ refers to top and bottom layers respectively.
1313Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so it is implicit.
1314
1315%% =================================================================================================
1316\subsection[Bottom friction with split-explicit free surface]{Bottom friction with split-explicit free surface}
1317\label{subsec:ZDF_drg_ts}
1318
1319With 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.
1320
1321The strategy to handle top/bottom stresses with split-explicit free surface in \NEMO\ is as follows:
1322\begin{enumerate}
1323\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.
1324\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.
1325\end{enumerate}
1326
1327Note that other strategies are possible, like considering vertical diffusion step in advance, \ie\ prior barotropic integration.
1328
1329%% =================================================================================================
1330\section[Internal wave-driven mixing (\forcode{ln_zdfiwm})]{Internal wave-driven mixing (\protect\np{ln_zdfiwm}{ln\_zdfiwm})}
1331\label{subsec:ZDF_tmx_new}
1332
1333\begin{listing}
1334  \nlst{namzdf_iwm}
1335  \caption{\forcode{&namzdf_iwm}}
1336  \label{lst:namzdf_iwm}
1337\end{listing}
1338
1339The parameterization of mixing induced by breaking internal waves is a generalization of
1340the approach originally proposed by \citet{st-laurent.simmons.ea_GRL02}.
1341A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed,
1342and the resulting diffusivity is obtained as
1343\[
1344  % \label{eq:ZDF_Kwave}
1345  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
1346\]
1347where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of
1348the energy available for mixing.
1349If the \np{ln_mevar}{ln\_mevar} namelist parameter is set to \forcode{.false.}, the mixing efficiency is taken as constant and
1350equal to 1/6 \citep{osborn_JPO80}.
1351In the opposite (recommended) case, $R_f$ is instead a function of
1352the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$,
1353with $\nu$ the molecular viscosity of seawater, following the model of \cite{bouffard.boegman_DAO13} and
1354the implementation of \cite{de-lavergne.madec.ea_JPO16}.
1355Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when
1356the mixing efficiency is constant.
1357
1358In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary
1359as a function of $Re_b$ by setting the \np{ln_tsdiff}{ln\_tsdiff} parameter to \forcode{.true.}, a recommended choice.
1360This parameterization of differential mixing, due to \cite{jackson.rehmann_JPO14},
1361is implemented as in \cite{de-lavergne.madec.ea_JPO16}.
1362
1363The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$,
1364is constructed from three static maps of column-integrated internal wave energy dissipation,
1365$E_{cri}(i,j)$, $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures:
1366
1367\begin{align*}
1368  F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\
1369  F_{pyc}(i,j,k) &\propto N^{n_p}\\
1370  F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} }
1371\end{align*}
1372In the above formula, $h_{ab}$ denotes the height above bottom,
1373$h_{wkb}$ denotes the WKB-stretched height above bottom, defined by
1374\[
1375  h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; ,
1376\]
1377The $n_p$ parameter (given by \np{nn_zpyc}{nn\_zpyc} in \nam{zdf_iwm}{zdf\_iwm} namelist)
1378controls the stratification-dependence of the pycnocline-intensified dissipation.
1379It can take values of $1$ (recommended) or $2$.
1380Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of
1381the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps.
1382$h_{cri}$ is related to the large-scale topography of the ocean (etopo2) and
1383$h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of
1384the abyssal hill topography \citep{goff_JGR10} and the latitude.
1385% Jc: input files names ?
1386
1387%% =================================================================================================
1388\section[Surface wave-induced mixing (\forcode{ln_zdfswm})]{Surface wave-induced mixing (\protect\np{ln_zdfswm}{ln\_zdfswm})}
1389\label{subsec:ZDF_swm}
1390
1391Surface waves produce an enhanced mixing through wave-turbulence interaction.
1392In addition to breaking waves induced turbulence (\autoref{subsec:ZDF_tke}),
1393the influence of non-breaking waves can be accounted introducing
1394wave-induced viscosity and diffusivity as a function of the wave number spectrum.
1395Following \citet{qiao.yuan.ea_OD10}, a formulation of wave-induced mixing coefficient
1396is provided  as a function of wave amplitude, Stokes Drift and wave-number:
1397
1398\begin{equation}
1399  \label{eq:ZDF_Bv}
1400  B_{v} = \alpha {A} {U}_{st} {exp(3kz)}
1401\end{equation}
1402
1403Where $B_{v}$ is the wave-induced mixing coefficient, $A$ is the wave amplitude,
1404${U}_{st}$ is the Stokes Drift velocity, $k$ is the wave number and $\alpha$
1405is a constant which should be determined by observations or
1406numerical experiments and is set to be 1.
1407
1408The coefficient $B_{v}$ is then directly added to the vertical viscosity
1409and diffusivity coefficients.
1410
1411In order to account for this contribution set: \forcode{ln_zdfswm=.true.},
1412then wave interaction has to be activated through \forcode{ln_wave=.true.},
1413the Stokes Drift can be evaluated by setting \forcode{ln_sdw=.true.}
1414(see \autoref{subsec:SBC_wave_sdw})
1415and the needed wave fields can be provided either in forcing or coupled mode
1416(for more information on wave parameters and settings see \autoref{sec:SBC_wave})
1417
1418%% =================================================================================================
1419\section[Adaptive-implicit vertical advection (\forcode{ln_zad_Aimp})]{Adaptive-implicit vertical advection(\protect\np{ln_zad_Aimp}{ln\_zad\_Aimp})}
1420\label{subsec:ZDF_aimp}
1421
1422The adaptive-implicit vertical advection option in NEMO is based on the work of
1423\citep{shchepetkin_OM15}.  In common with most ocean models, the timestep used with NEMO
1424needs to satisfy multiple criteria associated with different physical processes in order
1425to maintain numerical stability. \citep{shchepetkin_OM15} pointed out that the vertical
1426CFL criterion is commonly the most limiting. \citep{lemarie.debreu.ea_OM15} examined the
1427constraints for a range of time and space discretizations and provide the CFL stability
1428criteria for a range of advection schemes. The values for the Leap-Frog with Robert
1429asselin filter time-stepping (as used in NEMO) are reproduced in
1430\autoref{tab:ZDF_zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these
1431restrictions but at the cost of large dispersive errors and, possibly, large numerical
1432viscosity. The adaptive-implicit vertical advection option provides a targetted use of the
1433implicit scheme only when and where potential breaches of the vertical CFL condition
1434occur. In many practical applications these events may occur remote from the main area of
1435interest or due to short-lived conditions such that the extra numerical diffusion or
1436viscosity does not greatly affect the overall solution. With such applications, setting:
1437\forcode{ln_zad_Aimp=.true.} should allow much longer model timesteps to be used whilst
1438retaining the accuracy of the high order explicit schemes over most of the domain.
1439
1440\begin{table}[htbp]
1441  \centering
1442  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}}
1443  \begin{tabular}{r|ccc}
1444    \hline
1445    spatial discretization  & 2$^nd$ order centered & 3$^rd$ order upwind & 4$^th$ order compact \\
1446    advective CFL criterion &                 0.904 &              0.472  &                0.522 \\
1447    \hline
1448  \end{tabular}
1449  \caption[Advective CFL criteria for the leapfrog with Robert Asselin filter time-stepping]{
1450    The advective CFL criteria for a range of spatial discretizations for
1451    the leapfrog with Robert Asselin filter time-stepping
1452    ($\nu=0.1$) as given in \citep{lemarie.debreu.ea_OM15}.}
1453  \label{tab:ZDF_zad_Aimp_CFLcrit}
1454\end{table}
1455
1456In particular, the advection scheme remains explicit everywhere except where and when
1457local vertical velocities exceed a threshold set just below the explicit stability limit.
1458Once the threshold is reached a tapered transition towards an implicit scheme is used by
1459partitioning the vertical velocity into a part that can be treated explicitly and any
1460excess that must be treated implicitly. The partitioning is achieved via a Courant-number
1461dependent weighting algorithm as described in \citep{shchepetkin_OM15}.
1462
1463The local cell Courant number ($Cu$) used for this partitioning is:
1464
1465\begin{equation}
1466  \label{eq:ZDF_Eqn_zad_Aimp_Courant}
1467  \begin{split}
1468    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 ]    \\
1469       &\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 ]
1470                     \big / e_{{1_t}ij}e_{{2_t}ij}            \\
1471       &\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 ]
1472                     \big / e_{{1_t}ij}e_{{2_t}ij} \bigg )    \\
1473  \end{split}
1474\end{equation}
1475
1476\noindent and the tapering algorithm follows \citep{shchepetkin_OM15} as:
1477
1478\begin{align}
1479  \label{eq:ZDF_Eqn_zad_Aimp_partition}
1480Cu_{min} &= 0.15 \nonumber \\
1481Cu_{max} &= 0.3  \nonumber \\
1482Cu_{cut} &= 2Cu_{max} - Cu_{min} \nonumber \\
1483Fcu    &= 4Cu_{max}*(Cu_{max}-Cu_{min}) \nonumber \\
1484\cf &=
1485     \begin{cases}
1486        0.0                                                        &\text{if $Cu \leq Cu_{min}$} \\
1487        (Cu - Cu_{min})^2 / (Fcu +  (Cu - Cu_{min})^2)             &\text{else if $Cu < Cu_{cut}$} \\
1488        (Cu - Cu_{max}) / Cu                                       &\text{else}
1489     \end{cases}
1490\end{align}
1491
1492\begin{figure}[!t]
1493  \centering
1494  \includegraphics[width=0.66\textwidth]{ZDF_zad_Aimp_coeff}
1495  \caption[Partitioning coefficient used to partition vertical velocities into parts]{
1496    The value of the partitioning coefficient (\cf) used to partition vertical velocities into
1497    parts to be treated implicitly and explicitly for a range of typical Courant numbers
1498    (\forcode{ln_zad_Aimp=.true.}).}
1499  \label{fig:ZDF_zad_Aimp_coeff}
1500\end{figure}
1501
1502\noindent The partitioning coefficient is used to determine the part of the vertical
1503velocity that must be handled implicitly ($w_i$) and to subtract this from the total
1504vertical velocity ($w_n$) to leave that which can continue to be handled explicitly:
1505
1506\begin{align}
1507  \label{eq:ZDF_Eqn_zad_Aimp_partition2}
1508    w_{i_{ijk}} &= \cf_{ijk} w_{n_{ijk}}     \nonumber \\
1509    w_{n_{ijk}} &= (1-\cf_{ijk}) w_{n_{ijk}}
1510\end{align}
1511
1512\noindent Note that the coefficient is such that the treatment is never fully implicit;
1513the three cases from \autoref{eq:ZDF_Eqn_zad_Aimp_partition} can be considered as:
1514fully-explicit; mixed explicit/implicit and mostly-implicit.  With the settings shown the
1515coefficient (\cf) varies as shown in \autoref{fig:ZDF_zad_Aimp_coeff}. Note with these values
1516the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly
1517implicit' is 0.45 which is just below the stability limited given in
1518\autoref{tab:ZDF_zad_Aimp_CFLcrit}  for a 3rd order scheme.
1519
1520The $w_i$ component is added to the implicit solvers for the vertical mixing in
1521\mdl{dynzdf} and \mdl{trazdf} in a similar way to \citep{shchepetkin_OM15}.  This is
1522sufficient for the flux-limited advection scheme (\forcode{ln_traadv_mus}) but further
1523intervention is required when using the flux-corrected scheme (\forcode{ln_traadv_fct}).
1524For these schemes the implicit upstream fluxes must be added to both the monotonic guess
1525and to the higher order solution when calculating the antidiffusive fluxes. The implicit
1526vertical fluxes are then removed since they are added by the implicit solver later on.
1527
1528The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be
1529used in a wide range of simulations. The following test simulation, however, does illustrate
1530the potential benefits and will hopefully encourage further testing and feedback from users:
1531
1532\begin{figure}[!t]
1533  \centering
1534  \includegraphics[width=0.66\textwidth]{ZDF_zad_Aimp_overflow_frames}
1535  \caption[OVERFLOW: time-series of temperature vertical cross-sections]{
1536    A time-series of temperature vertical cross-sections for the OVERFLOW test case.
1537    These results are for the default settings with \forcode{nn_rdt=10.0} and
1538    without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}).}
1539  \label{fig:ZDF_zad_Aimp_overflow_frames}
1540\end{figure}
1541
1542%% =================================================================================================
1543\subsection{Adaptive-implicit vertical advection in the OVERFLOW test-case}
1544
1545The \href{https://forge.ipsl.jussieu.fr/nemo/chrome/site/doc/NEMO/guide/html/test\_cases.html\#overflow}{OVERFLOW test case}
1546provides a simple illustration of the adaptive-implicit advection in action. The example here differs from the basic test case
1547by only a few extra physics choices namely:
1548
1549\begin{verbatim}
1550     ln_dynldf_OFF = .false.
1551     ln_dynldf_lap = .true.
1552     ln_dynldf_hor = .true.
1553     ln_zdfnpc     = .true.
1554     ln_traadv_fct = .true.
1555        nn_fct_h   =  2
1556        nn_fct_v   =  2
1557\end{verbatim}
1558
1559\noindent which were chosen to provide a slightly more stable and less noisy solution. The
1560result when using the default value of \forcode{nn_rdt=10.} without adaptive-implicit
1561vertical velocity is illustrated in \autoref{fig:ZDF_zad_Aimp_overflow_frames}. The mass of
1562cold water, initially sitting on the shelf, moves down the slope and forms a
1563bottom-trapped, dense plume. Even with these extra physics choices the model is close to
1564stability limits and attempts with \forcode{nn_rdt=30.} will fail after about 5.5 hours
1565with excessively high horizontal velocities. This time-scale corresponds with the time the
1566plume reaches the steepest part of the topography and, although detected as a horizontal
1567CFL breach, the instability originates from a breach of the vertical CFL limit. This is a good
1568candidate, therefore, for use of the adaptive-implicit vertical advection scheme.
1569
1570The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps
1571are shown in \autoref{fig:ZDF_zad_Aimp_overflow_all_rdt} (together with the equivalent
1572frames from the base run).  In this simple example the use of the adaptive-implicit
1573vertcal advection scheme has enabled a 12x increase in the model timestep without
1574significantly altering the solution (although at this extreme the plume is more diffuse
1575and has not travelled so far).  Notably, the solution with and without the scheme is
1576slightly different even with \forcode{nn_rdt=10.}; suggesting that the base run was
1577close enough to instability to trigger the scheme despite completing successfully.
1578To assist in diagnosing how active the scheme is, in both location and time, the 3D
1579implicit and explicit components of the vertical velocity are available via XIOS as
1580\texttt{wimp} and \texttt{wexp} respectively.  Likewise, the partitioning coefficient
1581(\cf) is also available as \texttt{wi\_cff}. For a quick oversight of
1582the schemes activity the global maximum values of the absolute implicit component
1583of the vertical velocity and the partitioning coefficient are written to the netCDF
1584version of the run statistics file (\texttt{run.stat.nc}) if this is active (see
1585\autoref{sec:MISC_opt} for activation details).
1586
1587\autoref{fig:ZDF_zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for
1588the various overflow tests.  Note that the adaptive-implicit vertical advection scheme is
1589active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the
1590test case is close to stability limits even with this value. At the larger timesteps, the
1591vertical velocity is treated mostly implicitly at some location throughout the run. The
1592oscillatory nature of this measure appears to be linked to the progress of the plume front
1593as each cusp is associated with the location of the maximum shifting to the adjacent cell.
1594This is illustrated in \autoref{fig:ZDF_zad_Aimp_maxCf_loc} where the i- and k- locations of the
1595maximum have been overlaid for the base run case.
1596
1597\medskip
1598\noindent Only limited tests have been performed in more realistic configurations. In the
1599ORCA2\_ICE\_PISCES reference configuration the scheme does activate and passes
1600restartability and reproducibility tests but it is unable to improve the model's stability
1601enough to allow an increase in the model time-step. A view of the time-series of maximum
1602partitioning coefficient (not shown here)  suggests that the default time-step of 5400s is
1603already pushing at stability limits, especially in the initial start-up phase. The
1604time-series does not, however, exhibit any of the 'cuspiness' found with the overflow
1605tests.
1606
1607\medskip
1608\noindent A short test with an eORCA1 configuration promises more since a test using a
1609time-step of 3600s remains stable with \forcode{ln_zad_Aimp=.true.} whereas the
1610time-step is limited to 2700s without.
1611
1612\begin{figure}[!t]
1613  \centering
1614  \includegraphics[width=0.66\textwidth]{ZDF_zad_Aimp_overflow_all_rdt}
1615  \caption[OVERFLOW: sample temperature vertical cross-sections from mid- and end-run]{
1616    Sample temperature vertical cross-sections from mid- and end-run using
1617    different values for \forcode{nn_rdt} and with or without adaptive implicit vertical advection.
1618    Without the adaptive implicit vertical advection
1619    only the run with the shortest timestep is able to run to completion.
1620    Note also that the colour-scale has been chosen to confirm that
1621    temperatures remain within the original range of 10$^o$ to 20$^o$.}
1622  \label{fig:ZDF_zad_Aimp_overflow_all_rdt}
1623\end{figure}
1624
1625\begin{figure}[!t]
1626  \centering
1627  \includegraphics[width=0.66\textwidth]{ZDF_zad_Aimp_maxCf}
1628  \caption[OVERFLOW: maximum partitioning coefficient during a series of test runs]{
1629    The maximum partitioning coefficient during a series of test runs with
1630    increasing model timestep length.
1631    At the larger timesteps,
1632    the vertical velocity is treated mostly implicitly at some location throughout the run.}
1633  \label{fig:ZDF_zad_Aimp_maxCf}
1634\end{figure}
1635
1636\begin{figure}[!t]
1637  \centering
1638  \includegraphics[width=0.66\textwidth]{ZDF_zad_Aimp_maxCf_loc}
1639  \caption[OVERFLOW: maximum partitioning coefficient for the case overlaid]{
1640    The maximum partitioning coefficient for the \forcode{nn_rdt=10.0} case overlaid with
1641    information on the gridcell i- and k-locations of the maximum value.}
1642  \label{fig:ZDF_zad_Aimp_maxCf_loc}
1643\end{figure}
1644
1645\subinc{\input{../../global/epilogue}}
1646
1647\end{document}
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