\documentclass[../main/NEMO_manual]{subfiles}

\begin{document}

\chapter{Vertical Ocean Physics (ZDF)}
\label{chap:ZDF}

\chaptertoc

\paragraph{Changes record} ~\\

{\footnotesize
  \begin{tabularx}{\textwidth}{l||X|X}
    Release & Author(s) & Modifications \\
    \hline
    {\em   5.0}                                                       &
      {\em D. Calvert \newline
           C. de Lavergne, K. Hutchinson \newline
           R. Bourdalle-Badie, S. Techene \newline
           A. Moulin, E. Clementi}                                    &
        {\em General updates \newline
             Update Internal Wave Mixing section \newline
             Add Mass Flux Correction scheme section \newline
             Update of \autoref{subsec:ZDF_tke} in for wave coupling} \\[2mm]
    {\em   4.0} & {\em ...} & {\em ...} \\
    {\em   3.6} & {\em ...} & {\em ...} \\
    {\em   3.4} & {\em ...} & {\em ...} \\
    {\em <=3.4} & {\em ...} & {\em ...}
  \end{tabularx}
}

\clearpage

\cmtgm{ Add here a small introduction to ZDF and naming of the different physics
(similar to what have been written for TRA and DYN).}

%% =================================================================================================
\section{Vertical mixing}
\label{sec:ZDF}

The discrete form of the ocean subgrid scale physics has been presented in
\autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}.
At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt have to be defined.
At the surface they are prescribed from the surface forcing (see \autoref{chap:SBC}),
while at the bottom they are set to zero for heat and salt,
unless a geothermal flux forcing is prescribed as a bottom boundary condition (\ie\ \np{ln_trabbc}{ln\_trabbc} defined,
see \autoref{subsec:TRA_bbc}), and specified through a bottom friction parameterisation for momentum
(see \autoref{sec:ZDF_drg}).

In this section we briefly discuss the various choices offered to compute the vertical eddy viscosity and
diffusivity coefficients, $A_u^{vm}$ , $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- points,
respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}).
These coefficients can be defined as constant, or a function of the local Richardson number,
or computed from a turbulent closure model (either TKE or GLS or OSMOSIS formulation).
This choice is specified via the appropriate namelist parameter in \nam{zdf}{zdf}.

The computation of these coefficients is initialized in the \mdl{zdfphy} module and performed in
the \mdl{zdfric}, \mdl{zdftke} or \mdl{zdfgls} or \mdl{zdfosm} modules.
The trends due to the vertical momentum and tracer diffusion, including the surface forcing,
are computed and added to the general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively.
%These trends can be computed using either a forward time stepping scheme
%(namelist parameter \np[=.true.]{ln_zdfexp}{ln\_zdfexp}) or a backward time stepping scheme
%(\np[=.false.]{ln_zdfexp}{ln\_zdfexp}) depending on the magnitude of the mixing coefficients,
%and thus of the formulation used (see \autoref{chap:TD}).

\begin{listing}
  \nlst{namzdf}
  \caption{\forcode{&namzdf}}
  \label{lst:namzdf}
\end{listing}

%% =================================================================================================
\subsection[Background values]{Background values}
\label{subsec:ZDF_bkg}

To avoid numerical instabilities associated with weak vertical diffusion, all methods of calculating
the eddy viscosity and diffusivity coefficients will enforce a minimum background value
on their final values: $A_b^{vm}$ and $A_b^{vT}$ respectively. These background values are set by the
namelist parameters \np{rn_avm0}{rn\_avm0} and \np{rn_avt0}{rn\_avt0} respectively, which
should be at least as large as molecular values (see \autoref{subsec:ZDF_cst}).

Vertical and horizontal profiles may be applied to $A_b^{vT}$ via the \np{nn_avb}{nn\_avb} and
\np{nn_havtb}{nn\_havtb} namelist parameters respectively.
When these parameters are set to \forcode{0}, no profile is applied and $A_b^{vT}$ is constant everywhere.
When setting \np[=1]{nn_avb}{nn\_avb}, a theoretical vertical profile will be applied to $A_b^{vT}$ \citep{kraus1990diapycnal}.
When setting \np[=1]{nn_havtb}{nn\_havtb}, $A_b^{vT}$ will be reduced in the tropics, decreasing linearly
from $\pm 15^o$ latitude to 10\% of its nominal value (\np{rn_avt0}{rn\_avt0}) at $\pm 5^o$ latitude.

%% =================================================================================================
\subsection[Constant (\forcode{ln_zdfcst})]{Constant (\protect\np{ln_zdfcst}{ln\_zdfcst})}
\label{subsec:ZDF_cst}

When \np[=.true.]{ln_zdfcst}{ln\_zdfcst}, the momentum and tracer vertical eddy coefficients are set to
constant values over the whole ocean.
This is the crudest way to define the vertical ocean physics.
It is recommended to use this option only in process studies, not in basin scale simulations.
Typical values used in this case are:
\begin{align*}
  A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1} 	\\
  A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1}
\end{align*}

The coefficient values are set to their background values; see \autoref{subsec:ZDF_bkg} and the parameters described within.
In all cases, do not use values smaller that those associated with the molecular viscosity and diffusivity,
that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and
$\sim10^{-9}~m^2.s^{-1}$ for salinity.

%% =================================================================================================
\subsection[Richardson number dependent (\forcode{ln_zdfric})]{Richardson number dependent (\protect\np{ln_zdfric}{ln\_zdfric})}
\label{subsec:ZDF_ric}

\begin{listing}
  \nlst{namzdf_ric}
  \caption{\forcode{&namzdf_ric}}
  \label{lst:namzdf_ric}
\end{listing}

When \np[=.true.]{ln_zdfric}{ln\_zdfric}, a local Richardson number dependent formulation for the vertical momentum and
tracer eddy coefficients is set through the \nam{zdf_ric}{zdf\_ric} namelist variables.
The vertical mixing coefficients are diagnosed from the large scale variables computed by the model.
\textit{In situ} measurements have been used to link vertical turbulent activity to large scale ocean structures.
The hypothesis of a mixing mainly maintained by the growth of Kelvin-Helmholtz like instabilities leads to
a dependency between the vertical eddy coefficients and the local Richardson number
(\ie\ the ratio of stratification to vertical shear).
Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented:
\[
  % \label{eq:ZDF_ric}
  \left\{
    \begin{aligned}
      A^{vm} &= \frac {A_{ric}^{vm}}{\left( 1+a \; Ri \right)^n} + A_b^{vm}       \\
      A^{vT} &= \frac{A^{vm}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vT}
    \end{aligned}
  \right.
\]
where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number,
$N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
$A_b^{vT} $ and $A_b^{vm}$ are the constant background values
(see \autoref{subsec:ZDF_bkg}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ is the maximum value that
can be reached by the coefficient when $Ri\leq 0$, $a=5$ and $n=2$.
The last three values can be modified by setting the \np{rn_avmri}{rn\_avmri}, \np{rn_alp}{rn\_alp} and
\np{nn_ric}{nn\_ric} namelist parameters, respectively.

A simple mixing-layer model to transfer and dissipate the atmospheric forcings
(wind-stress and buoyancy fluxes) can be activated setting \np[=.true.]{ln_mldw}{ln\_mldw} in the namelist.
In this case, the local depth of turbulent wind-mixing or "Ekman depth" ($h_{e}$) is evaluated and
the vertical eddy coefficients prescribed within this layer.

This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation:
\[
  h_{e} = Ek \frac {u^{*}} {f_{0}}
\]
where $Ek$ is an empirical parameter set by the namelist parameter \np{rn_ekmfc}{rn\_ekmfc},
$u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter.

In this similarity height relationship, the turbulent friction velocity:
\[
  u^{*} = \sqrt \frac {|\tau|} {\rho_o}
\]
is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$.
The minimum and maximum value of $h_{e}$ is constrained by the namelist parameters
\np{rn_mldmin}{rn\_mldmin} and \np{rn_mldmax}{rn\_mldmax} respectively.
Once $h_{e}$ is computed, the minimum values of the vertical eddy coefficients $A^{vT}$ and $A^{vm}$
within $h_{e}$ are constrained by the namelist parameters \np{rn_wtmix}{rn\_wtmix} and
\np{rn_wvmix}{rn\_wvmix} respectively \citep{lermusiaux_JMS01}.

%% =================================================================================================
\subsection[TKE turbulent closure scheme (\forcode{ln_zdftke})]{TKE turbulent closure scheme (\protect\np{ln_zdftke}{ln\_zdftke})}
\label{subsec:ZDF_tke}

\begin{listing}
  \nlst{namzdf_tke}
  \caption{\forcode{&namzdf_tke}}
  \label{lst:namzdf_tke}
\end{listing}

When \np[=.true.]{ln_zdftke}{ln\_zdftke}, the vertical eddy viscosity and diffusivity coefficients
are computed from a TKE turbulent closure model based on
a prognostic equation for $\bar{e}$, the turbulent kinetic energy,
and a closure assumption for the turbulent length scales.
This turbulent closure model has been developed by \citet{bougeault.lacarrere_MWR89} in the atmospheric case,
adapted by \citet{gaspar.gregoris.ea_JGR90} for the oceanic case, and embedded in OPA, the ancestor of \NEMO,
by \citet{blanke.delecluse_JPO93} for equatorial Atlantic simulations.
Since then, significant modifications have been introduced by \citet{madec.delecluse.ea_NPM98} in both the implementation and
the formulation of the mixing length scale.

The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear,
its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type:
\begin{equation}
  \label{eq:ZDF_tke_e}
  \frac{\partial \bar{e}}{\partial t} =
  \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
      +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
  -K_\rho\,N^2
  +\frac{1}{e_3}	\;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
      \;\frac{\partial \bar{e}}{\partial k}} \right]
  - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
\end{equation}
\[
  % \label{eq:ZDF_tke_kz}
  \begin{split}
    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }  	\\
    K_\rho &= A^{vm} / P_{rt}
  \end{split}
\]
where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,
$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity and diffusivity coefficients.

The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$ $\approx 0.7$ are designed to deal with
vertical mixing at any depth \citep{gaspar.gregoris.ea_JGR90}.
They are set through namelist parameters \np{nn_ediff}{nn\_ediff} and \np{nn_ediss}{nn\_ediss}.

The definition used for $P_{rt}$ is controlled by the \np{nn_pdl}{nn\_pdl} namelist parameter.
If \np[=0]{nn_pdl}{nn\_pdl}, then $P_{rt}=1$.
If \np[=1]{nn_pdl}{nn\_pdl}, then $P_{rt}$ is a function of the local Richardson number ($R_i$)
following \citet{blanke.delecluse_JPO93}:
\begin{align*}
  % \label{eq:ZDF_prt}
  P_{rt} =
  \begin{cases}
    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$} 	\\
    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$}
  \end{cases}
\end{align*}

At the sea surface, the value of $\bar{e}$ is prescribed from the wind stress field as
$\bar{e}_o = e_{bb} |\tau| / \rho_o$, where $e_{bb}$ is set by the \np{rn_ebb}{rn\_ebb} namelist parameter.
The default value of $e_{bb}$ is 3.75 \citep{gaspar.gregoris.ea_JGR90}, however a much larger value can be used when
taking into account the surface wave breaking (see below \autoref{eq:ZDF_Esbc}).
The bottom value of $\bar{e}$ is assumed to be equal to the value of the level just above.

The time integration of the $\bar{e}$ equation may formally lead to negative values because
the numerical scheme does not ensure its positivity.
To overcome this problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn_emin}{rn\_emin} namelist parameter).
Following \citet{gaspar.gregoris.ea_JGR90}, the cut-off value is set to $10^{-6}~m^{2}s^{-2}$.
This allows the subsequent formulations to match that of \citet{gargett_JMR84} for the diffusion in
the thermocline and deep ocean: $K_\rho = 10^{-3} / N$.

A separate minimum value is applied to the surface value of $\bar{e}$, set by the \np{rn_emin0}{rn\_emin0} namelist parameter.
This is typically larger; here it is set to $10^{-4}~m^{2}s^{-2}$ by default.

%% =================================================================================================
\subsubsection{Turbulent length scale}

For computational efficiency, the original formulation of the turbulent length scales proposed by
\citet{gaspar.gregoris.ea_JGR90} has been simplified.
Four formulations are proposed, the choice of which is controlled by the \np{nn_mxl}{nn\_mxl} namelist parameter.

The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}:
\begin{equation}
  \label{eq:ZDF_tke_mxl0_1}
  l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N
\end{equation}
which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency.
The resulting length scale is bounded by the distance to the surface or to the bottom
(\np[=0]{nn_mxl}{nn\_mxl}) or by the local vertical scale factor (\np[=1]{nn_mxl}{nn\_mxl}).
\citet{blanke.delecluse_JPO93} notice that this simplification has two major drawbacks:
it makes no sense for locally unstable stratification and the computation no longer uses all
the information contained in the vertical density profile.

To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np[=2, 3]{nn_mxl}{nn\_mxl} cases,
which add an extra assumption concerning the vertical gradient of the computed length scale.
The length scales are first evaluated as in \autoref{eq:ZDF_tke_mxl0_1} and then bounded such that
the vertical variations of the length scale cannot be larger than the variations of depth:
\begin{equation}
  \label{eq:ZDF_tke_mxl_constraint}
  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
  \qquad \text{with }\  l =  l_k = l_\epsilon
\end{equation}
This provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less
time consuming.
In particular, it allows the length scale to be limited not only by the distance to the surface or
to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as
the thermocline (\autoref{fig:ZDF_mixing_length}).

In order to impose the \autoref{eq:ZDF_tke_mxl_constraint} constraint, we introduce two additional length scales:
$l_{up}$ and $l_{dwn}$, the upward and downward length scales, and
evaluate the dissipation and mixing length scales as
(and note that here we use numerical indexing):
\begin{figure}[!t]
  \centering
  \includegraphics[width=0.66\textwidth]{ZDF_mixing_length}
  \caption[Mixing length computation]{Illustration of the mixing length computation}
  \label{fig:ZDF_mixing_length}
\end{figure}
\[
  % \label{eq:ZDF_tke_mxl2}
  \begin{aligned}
    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
    \quad &\text{ from $k=1$ to $jpk$ }\ \\
    l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)}  \right)
    \quad &\text{ from $k=jpk$ to $1$ }\ \\
  \end{aligned}
\]
where $l^{(k)}$ is computed using \autoref{eq:ZDF_tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.

In the \np[=2]{nn_mxl}{nn\_mxl} case, the dissipation and mixing length scales take the same value:
$ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the \np[=3]{nn_mxl}{nn\_mxl} case,
the dissipation and mixing turbulent length scales are as given in \citet{gaspar.gregoris.ea_JGR90}:
\[
  % \label{eq:ZDF_tke_mxl_gaspar}
  \begin{aligned}
    & l_\epsilon  = \sqrt{\  l_{up} \ \ l_{dwn}\ }  	\\
    & l_k             = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)
  \end{aligned}
\]

At the ocean surface, a non zero length scale is set through the  \np{rn_mxl0}{rn\_mxl0} namelist parameter.
Usually the surface scale is given by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant and
$z_o$ the roughness parameter of the surface.
Assuming $z_o=0.1$~m \citep{craig.banner_JPO94} leads to a default value for \np{rn_mxl0}{rn\_mxl0} of 0.04~m.
In the ocean interior, a minimum length scale is set to recover the molecular viscosity when
$\bar{e}$ reaches its minimum value (such that $1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).

%% =================================================================================================
\subsubsection{Surface wave breaking parameterization (no information from an external wave model)}
\label{subsubsec:ZDF_tke_wave} 

Following \citet{mellor.blumberg_JPO04}, the TKE turbulence closure model has been modified to
include the effect of surface wave breaking energetics.
This results in a reduction of summertime surface temperature when the mixed layer is relatively shallow.
The \citet{mellor.blumberg_JPO04} modifications act on values of the surface length scale, TKE and the
air-sea drag coefficient.
The latter concerns the bulk formulae and is not discussed here.

Following \citet{craig.banner_JPO94}, the boundary condition on surface TKE value is :
\begin{equation}
  \label{eq:ZDF_Esbc}
  \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
\end{equation}
where $\alpha_{CB}$ is the \citet{craig.banner_JPO94} constant of proportionality which depends on the ''wave age'',
ranging from 57 for mature waves to 146 for younger waves \citep{mellor.blumberg_JPO04}.
\citet{mellor.blumberg_JPO04} suggest $\alpha_{CB} = 100$ which, as the surface boundary condition on TKE in \NEMO\
is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, corresponds to setting \np[=67.83]{rn_ebb}{rn\_ebb}.

The namelist parameter \np{ln_mxl0}{ln\_mxl0} determines the surface boundary condition on the
turbulent length scale, $l_o$.
When \np[=.false.]{ln_mxl0}{ln\_mxl0}, $l_o$ is set to the value specified by the namelist parameter
\np{rn_mxl0}{rn\_mxl0} (see previous subsection).
When \np[=.true.]{ln_mxl0}{ln\_mxl0}, $l_o$ follows Charnock's relation:

\begin{equation}
  \label{eq:ZDF_Lsbc}
  l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o}
\end{equation}

where $\kappa=0.40$ is the von Karman constant, and $\beta = 2.10^{5}$ is Charnock's constant set to the
value chosen by \citet{stacey_JPO99}.

\subsubsection{Surface wave breaking parameterization (using information from an external wave model)}
\label{subsubsec:ZDF_tke_waveco} 

Surface boundary conditions for the turbulent kinetic energy, the mixing length scale and the
dissipative length scale can be defined using wave fields provided from an external wave model
(see \autoref{chap:SBC}, \autoref{sec:SBC_wave}).
The injection of turbulent kinetic energy at the surface can be given by the dissipation of the wave
field usually dominated by wave breaking. In coupled mode, the wave to ocean energy flux term ($\Phi_o$)
from an external wave model can be provided and then converted into an ocean turbulence source by setting
\np[=.true.]{ln_phioc}{ln\_phioc}.

The surface TKE can be defined by a Dirichlet boundary condition by setting \np[=0]{nn_bc_surf}{nn\_bc\_surf}
in the \nam{zdf}{tke} namelist:
\begin{equation}
  \bar{e}_o  = \frac{1}{2}\,\left( 15.8 \, \frac{\Phi_o}{\rho_o}\right) ^{2/3}
\end{equation}

Due to the definition of the computational grid, the TKE flux is not applied at the free surface but
at the centre of the topmost grid cell ($z = z1$).

To be more accurate, a Neumann boundary condition equivalent to interpreting the half-grid cell
at the top as a constant flux layer (consistent with the surface layer
Monin–Obukhov theory) can be applied by instead setting \np[=1]{nn_bc_surf}{nn\_bc\_surf} \citep{couvelard_2020}:

\begin{equation}
  \left(\frac{Km}{e_3}\,\partial_k e \right)_{z=z1} = \frac{\Phi_o}{\rho_o}
\end{equation}

The mixing length scale surface value $l_0$ can be estimated from the surface roughness length $z0$:
\begin{equation}
  l_o = \kappa \, \frac{ \left( C_k\,C_\epsilon \right) ^{1/4}}{C_k}\, z0
\end{equation}
where $z0$ is directly estimated from the significant wave height ($Hs$) provided by the external wave model as
$z0=1.6Hs$. To use this option, \np{ln_mxhsw}{ln\_mxhsw}, \np{ln_wave}{ln\_wave} and \np{ln_sdw}{ln\_sdw}
have to be set to \forcode{.true.}.

%% =================================================================================================
\subsubsection{Langmuir cells}
\label{subsubsec:ZDF_tke_langmuir}

Langmuir circulations (LC) can be described as organised large-scale vertical motions in
the surface layer of the oceans.
Although LC have nothing to do with convection, the circulation pattern is rather similar to
so-called convective rolls in the atmospheric boundary layer.
The detailed physics behind LC are described in, for example, \citet{craik.leibovich_JFM76}.
The prevailing explanation is that LC arise from a nonlinear interaction between the Stokes drift and
wind drift currents.

Here we introduced in the TKE turbulent closure the simple parameterization of Langmuir circulations proposed by
\citep{axell_JGR02} for a $k-\epsilon$ turbulent closure.
The parameterization, tuned against large-eddy simulations, includes the whole effect of LC in
an extra source term of TKE, $P_{LC}$.
The presence of $P_{LC}$ in \autoref{eq:ZDF_tke_e}, the TKE equation, is controlled by setting
\np[=.true.]{ln_lc}{ln\_lc} in the \nam{zdf_tke}{zdf\_tke} namelist.

By making an analogy with the characteristic convective velocity scale (\eg, \citet{dalessio.abdella.ea_JPO98}),
$P_{LC}$ is assumed to be :
\[
P_{LC}(z) = (1-F_i) \frac{w_{LC}^3(z)}{H_{LC}}
\]
where $w_{LC}(z)$ is the vertical velocity profile of LC, $H_{LC}$ is the LC depth and $F_i$ is a function of
sea ice concentration ($f_i$) representing the attenuation of wind-driven mixing under sea ice.

$F_i$ has several possible definitions, chosen via the \np{nn_eice}{nn\_eice} namelist parameter:

\begin{description}
\item [{\np[=0]{nn_eice}{nn\_eice}}] \hfill \\
  No attenuation of mixing under sea ice ($F_i = 0$)
\item [{\np[=1]{nn_eice}{nn\_eice}}] \hfill \\
  TANH profile with no mixing at 100\% ice concentration ($F_i = \tanh \lt( 10 f_i \rt)$)
\item [{\np[=2]{nn_eice}{nn\_eice}}] \hfill \\
  Linear profile with no mixing at 100\% ice concentration ($F_i = f_i$)
\item [{\np[=3]{nn_eice}{nn\_eice}}] \hfill \\
  Linear profile with no mixing at 25\% ice concentration ($F_i = \min \lt( 4 f_i, 1 \rt)$)
\end{description}

$w_{LC}$ is assumed to be zero at the surface and at a finite depth $H_{LC}$
(which is often close to the mixed layer depth), and simply varies as a sine function in between
(a first-order profile for the Langmuir cell structures).
The resulting expression for $w_{LC}$ is :
\[
  w_{LC}  =
  \begin{cases}
    c_{LC} \,\|u_s^{LC}\| \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$} 	\\
    0                 				 &      \text{otherwise}
  \end{cases}
\]

In the absence of information about the wave field, $w_{LC}$ is assumed to be proportional to
the surface Stokes drift ($u_s^{LC}=u_{s0} $) empirically estimated by $ u_{s0} = 0.377\,\,|\tau|^{1/2}$,
where $|\tau|$ is the surface wind stress module
\footnote{Following \citet{li.garrett_JMR93}, the surface Stoke drift velocity may be expressed as
  $u_{s0} =  0.016 \,|U_{10m}|$.
  Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of
  $1.5~10^{-3}$ allows $u_{s0}$ to be expressed as a function of the module of surface stress
}.

In the case of online coupling with an external wave model (see \autoref{chap:SBC}, \autoref{sec:SBC_wave}),
$w_{LC}$ is proportional to the component of the Stokes drift aligned with the wind \citep{couvelard_2020}
and $ u_s^{LC}  = \max(u_{s0}.e_\tau,0)$ where $e_\tau$ is the unit vector in the wind stress direction and
$u_{s0}$ is the surface Stokes drift provided by the external wave model.

$c_{LC} = 0.15$ has been chosen by \citet{axell_JGR02} as a good compromise to fit large-eddy simulation data
and yields maximum vertical velocities $w_{LC}$ of the order of a few centimetres per second.
The value of $c_{LC}$ is set through the \np{rn_lc}{rn\_lc} namelist parameter and should have a value of
between 0.15 and 0.54 \citep{axell_JGR02}.

$H_{LC}$ is estimated in a similar way to the turbulent length scale of TKE equations:
it is the depth to which a water parcel with kinetic energy due to Stokes drift can reach on its own by
converting its kinetic energy to potential energy, according to
\[
- \int_{-H_{LC}}^0 { N^2\;z  \;dz} = \frac{1}{2} \|u_s^{LC}\|^2
\]

%% =================================================================================================
\subsubsection{Mixing just below the mixed layer}

Vertical mixing parameterizations commonly used in ocean general circulation models tend to
produce mixed-layer depths that are too shallow during summer months and windy conditions.
This bias is particularly acute over the Southern Ocean.
To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \citep{rodgers.aumont.ea_B14}.
The parameterization is an empirical one, \ie\ not derived from theoretical considerations,
but rather is meant to account for observed processes that affect the density structure of
the ocean’s planetary boundary layer that are not explicitly captured by the TKE scheme
(\ie\ near-inertial oscillations and ocean swells and waves).

When using this parameterization (\ie\ when \np[=1]{nn_etau}{nn\_etau}),
the TKE input to the ocean ($S$) imposed by the winds in the form of near-inertial oscillations,
swell and waves is parameterized by \autoref{eq:ZDF_Esbc}, the standard TKE surface boundary condition,
plus a depth dependance given by:
\begin{equation}
  \label{eq:ZDF_Ehtau}
  S = (1-F_i) \; f_r \; e_s \; e^{-z / h_\tau}
\end{equation}
where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that
penetrates into the ocean, $h_\tau$ is a vertical mixing length scale that controls the exponential shape of
the penetration, and $F_i$ is a function of
sea ice concentration ($f_i$) representing the attenuation of wind-driven mixing under sea ice.

The value of $f_r$, usually a few percent, is specified through the \np{rn_efr}{rn\_efr} namelist parameter.
The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np[=0]{nn_htau}{nn\_htau}) or
a latitude dependent value varying from 0.5~m at the Equator to a maximum value of 30~m at high latitudes
(\np[=1]{nn_htau}{nn\_htau}).
As for the parameterisation of Langmuir Circulations, $F_i$ has several possible definitions chosen via the
\np{nn_eice}{nn\_eice} namelist parameter (see \autoref{subsubsec:ZDF_tke_langmuir}).

Note that two other options exist, \np[=2, 3]{nn_etau}{nn\_etau}.
They correspond to applying \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer,
or to using the high frequency part of the stress to evaluate the fraction of TKE that penetrates the ocean.
Those two options are obsolescent features introduced for test purposes.
They will be removed in the next release.

% from Burchard et al OM 2008 :
% the most critical process not reproduced by statistical turbulence models is the activity of
% internal waves and their interaction with turbulence. After the Reynolds decomposition,
% internal waves are in principle included in the RANS equations, but later partially
% excluded by the hydrostatic assumption and the model resolution.
% Thus far, the representation of internal wave mixing in ocean models has been relatively crude
% (\eg\ Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002).

%% =================================================================================================
\subsection[GLS: Generic Length Scale (\forcode{ln_zdfgls})]{GLS: Generic Length Scale (\protect\np{ln_zdfgls}{ln\_zdfgls})}
\label{subsec:ZDF_gls}

\begin{listing}
  \nlst{namzdf_gls}
  \caption{\forcode{&namzdf_gls}}
  \label{lst:namzdf_gls}
\end{listing}

When \np[=.true.]{ln_zdfgls}{ln\_zdfgls}, the vertical eddy viscosity and diffusivity coefficients
are computed using the Generic Length Scale (GLS) scheme.
The GLS scheme is a turbulent closure model based on two prognostic equations:
one for the turbulent kinetic energy $\bar {e}$, and another for the generic length scale,
$\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}.
This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,
where the triplet $(p, m, n)$ value given in \autoref{tab:ZDF_GLS} allows to recover a number of
well-known turbulent closures including $k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87} and
$k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}.

The GLS scheme is given by the following set of equations:
\begin{equation}
  \label{eq:ZDF_gls_e}
  \frac{\partial \bar{e}}{\partial t} =
  \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
      +\left( \frac{\partial v}{\partial k} \right)^2} \right]
  -K_\rho \,N^2
  +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right]
  - \epsilon
\end{equation}

\[
  % \label{eq:ZDF_gls_psi}
  \begin{split}
    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{
      \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
          +\left( \frac{\partial v}{\partial k} \right)^2} \right]
      - C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\
    &+\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 }
        \;\frac{\partial \psi}{\partial k}} \right]\;
  \end{split}
\]

\[
  % \label{eq:ZDF_gls_kz}
  \begin{split}
    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
    K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l
  \end{split}
\]

\[
  % \label{eq:ZDF_gls_eps}
  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
\]
where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and
$\epsilon$ the dissipation rate.

The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depend on
the choice of the turbulence model.
Four different turbulent models are pre-defined (\autoref{tab:ZDF_GLS}).
They are made available through the \np{nn_clo}{nn\_clo} namelist parameter.

\begin{table}[htbp]
  \centering
  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c}
  \begin{tabular}{ccccc}
    &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\
    % & \citep{mellor.yamada_RG82} &  \citep{rodi_JGR87}       & \citep{wilcox_AJ88} &                 \\
    \hline
    \hline
    \np{nn_clo}{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\
    \hline
    $( p , n , m )$	       &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\
    $\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\
    $\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\
    $C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\
    $C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\
    $C_3$              &      1.           &     1.              &      1.                &       1.           \\
    $F_{wall}$        &      Yes        &       --             &     --                  &      --          \\
    \hline
    \hline
  \end{tabular}
  \caption[Set of predefined GLS parameters or equivalently predefined turbulence models available]{
    Set of predefined GLS parameters, or equivalently predefined turbulence models available with
    \protect\np[=.true.]{ln_zdfgls}{ln\_zdfgls} and controlled by
    the \protect\np{nn_clo}{nn\_clo} namelist variable in \protect\nam{zdf_gls}{zdf\_gls}.}
  \label{tab:ZDF_GLS}
\end{table}

In the Mellor-Yamada model, the negativity of $n$ requires the use of a wall function to force the convergence of
the mixing length towards $\kappa z_b$ (where $\kappa$ is the Von Karman constant and $z_b$ the rugosity length scale)
value near physical boundaries (logarithmic boundary layer law).

The stability functions $C_{\mu}$ and $C_{\mu'}$ have several definitions, chosen via the namelist parameter
\np{nn_stab_func}{nn\_stab\_func}:

\begin{description}
\item [{\np[=0]{nn_stab_func}{nn\_stab\_func}}] \hfill \\
  \citet{galperin.kantha.ea_JAS88} functions
\item [{\np[=1]{nn_stab_func}{nn\_stab\_func}}] \hfill \\
  \citet{kantha.clayson_JGR94} functions
\item [{\np[=2]{nn_stab_func}{nn\_stab\_func}}] \hfill \\
  \citet{canuto.howard.ea_JPO01} "model A" functions
\item [{\np[=3]{nn_stab_func}{nn\_stab\_func}}] \hfill \\
  \citet{canuto.howard.ea_JPO01} "model B" functions
\end{description}

The value of $C_{0\mu}$ depends on this choice of the stability function.

The surface and bottom boundary conditions on both $\bar{e}$ and $\psi$ are chosen via
the \np{nn_bc_surf}{nn\_bc\_surf} and \np{nn_bc_bot}{nn\_bc\_bot} namelist parameters respectively.
They can be calculated using a Dirichlet (\forcode{= 0}) or Neumann (\forcode{= 1}) condition.

As for the TKE turbulent closure scheme (\autoref{subsec:ZDF_tke}), the wave effect on the mixing
is parameterised following \citet{mellor.blumberg_JPO04} and \citet{craig.banner_JPO94}.
The namelist parameters \np{rn_crban}{rn\_crban} and \np{rn_charn}{rn\_charn} correspond to
$\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and $\beta$ in \autoref{eq:ZDF_Lsbc}.
Setting \np[=0.]{rn_crban}{rn\_crban} will disable this parameterisation.

The $\psi$ equation is known to fail in stably stratified flows, and for this reason
almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.
With this clipping, the maximum permissible length scale is determined by $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$
where a value of $c_{lim} = 0.53$ is often used \citep{galperin.kantha.ea_JAS88}.
\cite{umlauf.burchard_CSR05} show that the value of the clipping factor is of crucial importance for
the entrainment depth predicted in stably stratified situations,
and that its value has to be chosen in accordance with the algebraic model for the turbulent fluxes.
This clipping is activated by setting \np[=.true.]{ln_length_lim}{ln\_length\_lim} and $c_{lim}$ is set to
the value of \np{rn_clim_galp}{rn\_clim\_galp}.

The time and space discretization of the GLS equations follows the same energetic consideration as for
the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{burchard_OM02}.
An evaluation of the 4 GLS turbulent closure schemes can be found in \citet{warner.sherwood.ea_OM05} for the
ROMS model and in \citet{reffray.bourdalle-badie.ea_GMD15} for the \NEMO\ model.

% -------------------------------------------------------------------------------------------------------------
%        OSM OSMOSIS BL Scheme
% -------------------------------------------------------------------------------------------------------------
\subsection[OSMOSIS boundary layer scheme (\forcode{ln_zdfosm = .true.})]
{OSMOSIS boundary layer scheme (\protect\np{ln_zdfosm}{ln\_zdfosm})}
\label{subsec:ZDF_osm}

\begin{listing}
  \nlst{namzdf_osm}
  \caption{\forcode{&namzdf_osm}}
  \label{lst:namzdf_osm}
\end{listing}

%--------------------------------------------------------------------------------------------------------------
When \np[=.true.]{ln_zdfosm}{ln\_zdfosm}, the vertical eddy viscosity and diffusivity coefficients
are computed using the OSMOSIS scheme.
\paragraph{Namelist choices}
Most of the namelist options refer to how to specify the Stokes
surface drift and penetration depth. There are three options:
\begin{description}
  \item \protect\np[=0]{nn_osm_wave}{nn\_osm\_wave} Default value in \texttt{namelist\_ref}. In this case the Stokes drift is
      assumed to be parallel to the surface wind stress, with
      magnitude consistent with a constant turbulent Langmuir number
    $\mathrm{La}_t=$ \protect\np{rn_m_la}{rn\_m\_la} i.e.\
    $u_{s0}=\tau/(\mathrm{La}_t^2\rho_0)$.  Default value of
    \protect\np{rn_m_la}{rn\_m\_la} is 0.3. The Stokes penetration
      depth $\delta = $ \protect\np{rn_osm_dstokes}{rn\_osm\_dstokes}; this has default value
      of 5~m.

  \item \protect\np[=1]{nn_osm_wave}{nn\_osm\_wave} In this case the Stokes drift is
      assumed to be parallel to the surface wind stress, with
      magnitude as in the classical Pierson-Moskowitz wind-sea
      spectrum.  Significant wave height and
      wave-mean period taken from this spectrum are used to calculate the Stokes penetration
      depth, following the approach set out in  \citet{breivik.janssen.ea_JPO14}.

  \item \protect\np[=2]{nn_osm_wave}{nn\_osm\_wave} In this case the Stokes drift is
      calculated by the \NEMO\ surface wave module (see \autoref{sec:SBC_wave}),
      though only the component parallel to the wind stress is retained. Significant wave height and
      wave-mean period are used to calculate the Stokes penetration depth, again following \citet{breivik.janssen.ea_JPO14}.
\end{description}

    Others refer to the treatment of diffusion and viscosity beneath
    the surface boundary layer:
\begin{description}
   \item \protect\np{ln_kpprimix}{ln\_kpprimix}  Default is \texttt{.true.}. Switches on KPP-style Ri \#-dependent
     mixing below the surface boundary layer. If this is set
     \texttt{.true.}  the following variable settings are honoured:
    \item \protect\np{rn_riinfty}{rn\_riinfty} Critical value of local Ri \# below which
      shear instability increases vertical mixing from background value.
    \item \protect\np{rn_difri}{rn\_difri} Maximum value of Ri \#-dependent mixing at $\mathrm{Ri}=0$.
    \item \protect\np{ln_convmix}{ln\_convmix} If \texttt{.true.} then, where water column is unstable, specify
       diffusivity equal to \protect\np{rn_dif_conv}{rn\_dif\_conv} (default value is 1 m~s$^{-2}$).
 \end{description}
 Diagnostic output is controlled by:
  \begin{description}
    \item \protect\np{ln_dia_osm}{ln\_dia\_osm} Default is \texttt{.false.}; allows XIOS output of OSMOSIS internal fields.
  \end{description}
Obsolete namelist parameters include:
\begin{description}
\item \protect\np{ln_use_osm_la}{ln\_use\_osm\_la} With \protect\np[=0]{nn_osm_wave}{nn\_osm\_wave},
  \protect\np{rn_osm_dstokes}{rn\_osm\_dstokes} is always used to specify the Stokes
  penetration depth.
\item \protect\np{nn_ave}{nn\_ave} Choice of averaging method for KPP-style Ri \#
  mixing. Not taken account of.
\item \protect\np{rn_osm_hbl0}{rn\_osm\_hbl0} Depth of initial boundary layer is now set
  by a density criterion similar to that used in calculating \emph{hmlp} (output as \texttt{mldr10\_1}) in \mdl{zdfmxl}.
\end{description}

\subsubsection{Summary}
Much of the time the turbulent motions in the ocean surface boundary
layer (OSBL) are not given by
classical shear turbulence. Instead they are in a regime known as
`Langmuir turbulence',  dominated by an
interaction between the currents and the Stokes drift of the surface waves \citep[e.g.][]{mcwilliams.sullivan.ea_JFM97}.
This 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 $\mathrm{La}_{t}=(u_{\ast}/u_{s0}) > 0.4$.

The OSMOSIS model is fundamentally based on results of Large Eddy
Simulations (LES) of Langmuir turbulence and aims to fully describe
this Langmuir regime. The description in this section is of necessity incomplete and further details are available in Grant. A (2019); in prep.

The OSMOSIS turbulent closure scheme is a similarity-scale scheme in
the same spirit as the K-profile
parameterization (KPP) scheme of \citet{large.mcwilliams.ea_RG94}.
A specified shape of diffusivity, scaled by the (OSBL) depth
$h_{\mathrm{BL}}$ and a turbulent velocity scale, is imposed throughout the
boundary layer
$-h_{\mathrm{BL}}<z<\eta$. The turbulent closure model
also includes fluxes of tracers and momentum that are ``non-local'' (independent of the local property gradient).

Rather than the OSBL
depth being diagnosed in terms of a bulk Richardson number criterion,
as in KPP, it is set by a prognostic equation that is informed by
energy budget considerations reminiscent of the classical mixed layer
models of \citet{kraus.turner_T67}.
The model also includes an explicit parametrization of the structure
of the pycnocline (the stratified region at the bottom of the OSBL).

Presently, mixing below the OSBL is handled by the Richardson
number-dependent mixing scheme used in \citet{large.mcwilliams.ea_RG94}.

Convective parameterizations such as described in \autoref{sec:ZDF_conv}
below should not be used with the OSMOSIS-OBL model: instabilities
within the OSBL are part of the model, while instabilities below the
ML are handled by the Ri \# dependent scheme.

\subsubsection{Depth and velocity scales}

The 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 $\Delta h$; \autoref{fig:OSBL_structure} shows typical (a) buoyancy structure
and (b) turbulent buoyancy flux profiles for the unstable boundary layer (losing buoyancy
at the surface; e.g.\ cooling).

\begin{figure}[!t]
  \begin{center}
    \includegraphics[width=0.7\textwidth]{ZDF_OSM_structure_of_OSBL}
    \caption{
      \protect\label{fig:OSBL_structure}
     The structure of the entraining  boundary layer. (a) Mean buoyancy profile. (b) Profile of the buoyancy flux.
    }
  \end{center}
\end{figure}

The 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.

Consideration of the power input by wind acting on the Stokes drift suggests that the Langmuir turbulence has velocity scale:
\begin{equation}
  \label{eq:ZDF_w_La}
  w_{*L}= \left(u_*^2 u_{s\,0}\right)^{1/3};
\end{equation}
but 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:
\begin{equation}
  \label{eq:ZDF_composite-nu}
  \nu_{\ast}= \left\{ u_*^3 \left[1-\exp(-.5 \mathrm{La}_t^2)\right]+w_{*L}^3\right\}^{1/3}.
\end{equation}
For 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:
\begin{equation}
  \label{eq:ZDF_vel-scale-unstable}
  \omega_* = \left(\nu_*^3 + 0.5 w_{*C}^3\right)^{1/3}.
\end{equation}

\subsubsection{The flux gradient model}
The flux-gradient relationships used in the OSMOSIS scheme take the form:

\begin{equation}
  \label{eq:ZDF_flux-grad-gen}
  \overline{w^\prime\chi^\prime}=-K\frac{\partial\overline{\chi}}{\partial z} + N_{\chi,s} +N_{\chi,b} +N_{\chi,t},
\end{equation}

where $\chi$ is a general variable and $N_{\chi,s}$, $N_{\chi,b}$ 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.

In terms of the non-dimensionalized depth variables

\begin{equation}
  \label{eq:ZDF_sigma}
  \sigma_{\mathrm{ml}}= -z/h_{\mathrm{ml}}; \;\sigma_{\mathrm{bl}}= -z/h_{\mathrm{bl}},
\end{equation}

in unstable conditions the eddy diffusivity ($K_d$) and eddy viscosity ($K_\nu$) profiles are parametrized as:

\begin{align}
  \label{eq:ZDF_diff-unstable}
  K_d=&0.8\, \omega_*\, h_{\mathrm{ml}} \, \sigma_{\mathrm{ml}} \left(1-\beta_d \sigma_{\mathrm{ml}}\right)^{3/2}
  \\
  \label{eq:ZDF_visc-unstable}
  K_\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)
\end{align}

where $\beta_d$ and $\beta_\nu$ are parameters that are determined by matching \autoref{eq:ZDF_diff-unstable} and \autoref{eq:ZDF_visc-unstable} to the eddy diffusivity and viscosity at the base of the well-mixed layer, given by

\begin{equation}
  \label{eq:ZDF_diff-wml-base}
  K_{d,\mathrm{ml}}=K_{\nu,\mathrm{ml}}=\,0.16\,\omega_* \Delta h.
\end{equation}

For stable conditions the eddy diffusivity/viscosity profiles are given by:

\begin{align}
  \label{eq:ZDF_diff-stable}
  K_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:ZDF_visc-stable}
  K_\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).
\end{align}

The 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:
\begin{equation}
  \label{eq:ZDF_L_L}
  L_L=-w_{*L}^3/\left<\overline{w^\prime b^\prime}\right>_L,
\end{equation}
with 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
\begin{equation}
  \label{eq:ZDF_stable-av-buoy-flux}
  \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].
\end{equation}

In unstable conditions the eddy diffusivity and viscosity depend on stability through the velocity scale $\omega_*$, which depends on the two velocity scales $\nu_*$ and $w_{*C}$.

Details of the non-gradient terms in \autoref{eq:ZDF_flux-grad-gen} and of the fluxes within the pycnocline $-h_{\mathrm{bl}}<z<h_{\mathrm{ml}}$ can be found in Grant (2019).

\subsubsection{Evolution of the boundary layer depth}

The prognostic equation for the depth of the neutral/unstable boundary layer is given by \iffalse \citep{grant+etal18?}, \fi

\begin{equation}
  \label{eq:ZDF_dhdt-unstable}
%\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}}
   \frac{\partial h_\mathrm{bl}}{\partial t} = W_b - \frac{{\overline{w^\prime b^\prime}}_\mathrm{ent}}{\Delta B_\mathrm{bl}}
\end{equation}
where $h_\mathrm{bl}$ is the horizontally-varying depth of the OSBL,
$\mathbf{U}_b$ and $W_b$ are the mean horizontal and vertical
velocities at the base of the OSBL, ${\overline{w^\prime
    b^\prime}}_\mathrm{ent}$ is the buoyancy flux due to entrainment
and $\Delta B_\mathrm{bl}$ is the difference between the buoyancy
averaged over the depth of the OSBL (i.e.\ including the ML and
pycnocline) and the buoyancy just below the base of the OSBL. This
equation for the case when the pycnocline has a finite thickness,
based on the potential energy budget of the OSBL, is the leading term
\iffalse \citep{grant+etal18?} \fi of a generalization of that used in mixed-layer
models e.g.\ \citet{kraus.turner_T67}, in which the thickness of the pycnocline is taken to be zero.

The entrainment flux for the combination of convective and Langmuir turbulence is given by
\begin{equation}
  \label{eq:ZDF_entrain-flux}
  {\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}}}
  + 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]
\end{equation}
where 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 $F(x)\equiv\tanh(x^{-1})^{0.69}$.

For the stable boundary layer, the equation for the depth of the OSBL is:

\begin{equation}
  \label{eq:ZDF_dhdt-stable}
\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.
\end{equation}

\autoref{eq:ZDF_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 \autoref{eq:ZDF_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.


%% =================================================================================================
\subsection[ Discrete energy conservation for TKE and GLS schemes]{Discrete energy conservation for TKE and GLS schemes}
\label{subsec:ZDF_tke_ene}

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.66\textwidth]{ZDF_TKE_time_scheme}
  \caption[Subgrid kinetic energy integration in GLS and TKE schemes]{
    Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and
    its links to the momentum and tracer time integration.}
  \label{fig:ZDF_TKE_time_scheme}
\end{figure}

The production of turbulence by vertical shear (the first term of the right hand side of
\autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy
associated with the vertical momentum diffusion (first line in \autoref{eq:MB_zdf}).
To do so, a special care has to be taken for both the time and space discretization of
the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}.

Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how
the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with
the one-level forward time stepping of the equation for $\bar{e}$.
With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to
the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and
summing the result vertically:
\begin{equation}
  \label{eq:ZDF_energ1}
  \begin{split}
    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\
    &= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}
    - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz }
  \end{split}
\end{equation}
Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$,
known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme
(see \autoref{sec:TD_forward_imp}).
The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at
the surface (atmospheric forcing) and at the bottom (friction effect).
The second term is always negative.
It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$.
\autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$
(and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).

A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
(second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}).
This term must balance the input of potential energy resulting from vertical mixing.
The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by
multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically:
\begin{equation}
  \label{eq:ZDF_energ2}
  \begin{split}
    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\
    &= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}
    - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\
    &= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta}
    + \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  }
  \end{split}
\end{equation}
where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.
The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because
there is no diffusive flux through the ocean surface and bottom.
The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and  \autoref{eq:ZDF_gls_e}.

Let us now address the space discretization issue.
The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in
the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}).
A space averaging is thus required to obtain the shear TKE production term.
By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by
the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
Furthermore, the time variation of $e_3$ has be taken into account.

The above energetic considerations lead to the following final discrete form for the TKE equation:
\begin{equation}
  \label{eq:ZDF_tke_ene}
  \begin{split}
    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv
    \Biggl\{ \Biggr.
    &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} }
        \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\
    +&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} }
        \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j}
    \Biggr. \Biggr\}   \\
    %
    - &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\
    %
    +&\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]   \\
    %
    - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt}
  \end{split}
\end{equation}
where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation)
are time stepped using a backward scheme (see \autoref{sec:TD_forward_imp}).
Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible.
%The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as
%they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}.
%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.

%% =================================================================================================
\section{Convection}
\label{sec:ZDF_conv}

Static instabilities (\ie\ light potential densities under heavy ones) may occur at particular ocean grid points.
In nature, convective processes quickly re-establish the static stability of the water column.
These processes have been removed from the model via the hydrostatic assumption, so they must be parameterized.

Two parameterisations are available to deal specifically with convective processes:
a non-penetrative convective adjustment (\autoref{subsec:ZDF_npc}) or an enhanced
vertical diffusion (\autoref{subsec:ZDF_evd}).
It is recommended that one of these parameterisations be enabled when using either the TKE or GLS turbulent
closure scheme, but not when using the OSMOSIS turbulent closure scheme (see \autoref{subsec:ZDF_tcs}).

%% =================================================================================================
\subsection[Non-penetrative convective adjustment (\forcode{ln_tranpc})]{Non-penetrative convective adjustment (\protect\np{ln_tranpc}{ln\_tranpc})}
\label{subsec:ZDF_npc}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=0.66\textwidth]{ZDF_npc}
  \caption[Unstable density profile treated by the non penetrative convective adjustment algorithm]{
    Example of an unstable density profile treated by
    the non penetrative convective adjustment algorithm.
    $1^{st}$ step: the initial profile is checked from the surface to the bottom.
    It is found to be unstable between levels 3 and 4.
    They are mixed.
    The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
    The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
    The $1^{st}$ step ends since the density profile is then stable below the level 3.
    $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
    levels 2 to 5 are mixed.
    The new density profile is checked.
    It is found stable: end of algorithm.}
  \label{fig:ZDF_npc}
\end{figure}

Options are defined through the \nam{zdf}{zdf} namelist variables.
The non-penetrative convective adjustment is used when \np[=.true.]{ln_zdfnpc}{ln\_zdfnpc}.
It is applied at each \np{nn_npc}{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of
the water column, but only until the density structure becomes neutrally stable
(\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below)
\citep{madec.delecluse.ea_JPO91}.
The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}):
starting from the top of the ocean, the first instability is found.
Assume in the following that the instability is located between levels $k$ and $k+1$.
The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of
the water column.
The new density is then computed by a linear approximation.
If the new density profile is still unstable between levels $k+1$ and $k+2$,
levels $k$, $k+1$ and $k+2$ are then mixed.
This process is repeated until stability is established below the level $k$
(the mixing process can go down to the ocean bottom).
The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or
if there is no deeper instability.

This algorithm is significantly different from mixing statically unstable levels two by two.
The latter procedure cannot converge with a finite number of iterations for some vertical profiles while
the algorithm used in \NEMO\ converges for any profile in a number of iterations which is less than
the number of vertical levels.
This property is of paramount importance as pointed out by \citet{killworth_iprc89}:
it avoids the existence of permanent and unrealistic static instabilities at the sea surface.
This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in
the north-western Mediterranean Sea \citep{madec.delecluse.ea_JPO91, madec.chartier.ea_DAO91, madec.crepon_iprc91}.

The current implementation has been modified in order to deal with any non linear equation of seawater
(L. Brodeau, personnal communication).
Two main differences have been introduced compared to the original algorithm:
$(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency
(not the difference in potential density);
$(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in
the same way their temperature and salinity has been mixed.
These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without
having to recompute the expansion coefficients at each mixing iteration.

%% =================================================================================================
\subsection[Enhanced vertical diffusion (\forcode{ln_zdfevd})]{Enhanced vertical diffusion (\protect\np{ln_zdfevd}{ln\_zdfevd})}
\label{subsec:ZDF_evd}

Options are defined through the  \nam{zdf}{zdf} namelist variables.
The enhanced vertical diffusion parameterisation is used when \np[=.true.]{ln_zdfevd}{ln\_zdfevd}.
In this case, the vertical eddy mixing coefficients are assigned very large values
in regions where the stratification is unstable
(\ie\ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{lazar_phd97, lazar.madec.ea_JPO99}.
This is done either on tracers only (\np[=0]{nn_evdm}{nn\_evdm}) or
on both momentum and tracers (\np[=1]{nn_evdm}{nn\_evdm}).

In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np[=1]{nn_evdm}{nn\_evdm},
the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to
the namelist parameter \np{rn_evd}{rn\_evd}.
A typical value for \np{rn_evd}{rn\_evd} is between 1 and $100~m^{2}s^{-1}$.
This parameterisation of convective processes is less time consuming than
the convective adjustment algorithm presented above when mixing both tracers and
momentum in the case of static instabilities.

Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$.
This removes a potential source of divergence of odd and even time step in
a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}).

%% =================================================================================================
\subsection[Mass Flux Convection (\forcode{ln_zdfmfc})]{Mass Flux Convection (\protect\np{ln_zdfmfc}{ln\_zdfmfc})}
\label{subsec:ZDF_mfc}

The \np{ln_zdfmfc}{ln\_zdfmfc} option offers a new, coherent way to simultaneously parameterize 
local and non-local transport within the oceanic convective mixing layer \citep{giordani.ea_JAMES2020}. 
This approach, initially developed for atmospheric models (\cite{grant_QJRMS01}; \cite{soares.ea_QJRMS04}; \cite{pergaud.ea_BLM09}), 
assumes that subgrid turbulent fluxes in the convective boundary layer result from two distinct mixing scales: 
the local scale of small eddies, represented by vertical diffusion (TKE, GLS, etc.), 
and the scale of large eddies or convective thermals, represented by a non-local mass flux approach. 
The combination of both diffusive and convective schemes operating simultaneously is called as Eddy Diffusivity Mass Flux (EDMF).

The mass flux scheme is designed to represent all convection regimes, i.e., from moderate to strong, and from intermediate to deep convection. 
Its goal is to calculate the effects of a population of convective plumes occupying a fraction of an ocean model grid cell on its prognostic variables. 
This scheme thus parameterizes the subgrid-scale effects of very fine-scale convective plumes on the resolved variables at the model grid scale. 
This phenomenon is a one-dimensional vertical process. Since convection is a non-local phenomenon, 
this scheme complements mixing schemes such as the TKE and Richardson schemes. 
It serves as an alternative to the "Enhanced Vertical Diffusion" (\np{ln_zdfevd}{ln\_zdfevd}) or
"Non-Penetrative Convection" (\np{ln_zdfnpc}{ln\_zdfnpc}) schemes currently implemented in NEMO. 
The scheme is active at every model time step, alongside diffusion. 
It is broken down into several steps, corresponding to the evolution equations 
for tracers within convective plumes and the mass flux resulting from the work of
buoyancy forces in the vertical direction and the conservation of energy. 
The system of equations, along with its implementation in NEMO, is described in
\citep{giordani.ea_JAMES2020}, specifically in equations 12 and 13.

%% =================================================================================================
\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}})}
\label{subsec:ZDF_tcs}

The TKE and GLS turbulent closure schemes presented in \autoref{subsec:ZDF_tke} and \autoref{subsec:ZDF_gls}
are, in theory, able to handle statically unstable density profiles.
In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in
\autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative.
It results in large values of $A_T^{vT}$ and $A_T^{vT}$, and also of the four neighboring values at
velocity points $A_u^{vm}$ and $A_v^{vm}$ (up to $1\;m^2s^{-1}$).
These large values restore the static stability of the water column in a way similar to that of
the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}).
However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by
the turbulent closure scheme do not usually exceed $10^{-2}~ms^{-1}$,
because the mixing length scale is bounded by the distance to the sea surface.

When using either the TKE (\np[=.true.]{ln_zdftke}{ln\_zdftke}) or GLS (\np[=.true.]{ln_zdfgls}{ln\_zdfgls})
turbulent closure scheme, it is therefore recommended to also enable the enhanced vertical diffusion
parameterisation (\np[=.true.]{ln_zdfevd}{ln\_zdfevd}).
This should not be done when using the OSMOSIS turbulent closure scheme (\np[=.true.]{ln_zdfosm}{ln\_zdfosm}), as this
already includes enhanced vertical diffusion in the case of convection (see \autoref{subsec:ZDF_osm}).
%as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp},
% gm%  + one word on non local flux with KPP scheme trakpp.F90 module...

%% =================================================================================================
\section[Double diffusion mixing (\forcode{ln_zdfddm})]{Double diffusion mixing (\protect\np{ln_zdfddm}{ln\_zdfddm})}
\label{subsec:ZDF_ddm}

%\nlst{namzdf_ddm}

This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the namelist parameter
\np{ln_zdfddm}{ln\_zdfddm} in \nam{zdf}{zdf}.
Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa.
The former condition leads to salt fingering and the latter to diffusive convection.
Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean.
\citet{merryfield.holloway.ea_JPO99} include a parameterisation of such phenomena in a global ocean model and show that
it leads to relatively minor changes in circulation but exerts significant regional influences on
temperature and salinity.

Diapycnal mixing of S and T are described by diapycnal diffusion coefficients
\begin{align*}
  % \label{eq:ZDF_ddm_Kz}
  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT}	\\
  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
\end{align*}
where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
and $o$ by processes other than double diffusion.
The rates of double-diffusive mixing depend on the buoyancy ratio
$R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of
thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}).
To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt
(1981):
\begin{align}
  \label{eq:ZDF_ddm_f}
  A_f^{vS} &=
             \begin{cases}
               \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
               0 				  					    &\text{otherwise}
             \end{cases}
  \\ 		    \label{eq:ZDF_ddm_f_T}
  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho
\end{align}

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.66\textwidth]{ZDF_ddm}
  \caption[Diapycnal diffusivities for temperature and salt in regions of salt fingering and
  diffusive convection]{
    From \citet{merryfield.holloway.ea_JPO99}:
    (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in
    regions of salt fingering.
    Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and
    thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
    (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in
    regions of diffusive convection.
    Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
    The latter is not implemented in \NEMO.}
  \label{fig:ZDF_ddm}
\end{figure}

The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of
buoyancy flux of heat to buoyancy flux of salt, \eg\ \citet{mcdougall.taylor_JMR84}.
Following  \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^{2}s^{-1}$.

To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested by
\citet{federov1988_ddm} are used:
\begin{align}
  % \label{eq:ZDF_ddm_d}
  A_d^{vT} &=
             \begin{cases}
               1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
               &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
               0 								&\text{otherwise}
             \end{cases}
                                       \nonumber \\
  \label{eq:ZDF_ddm_d_S}
  A_d^{vS} &=
             \begin{cases}
               A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
               A_d^{vT} \ 0.15 \ R_\rho               &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
               0 								&\text{otherwise}
             \end{cases}
\end{align}

The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in
\autoref{fig:ZDF_ddm}.
Implementing this requires computing $R_\rho$ at each grid point on every time step.
This is done in \mdl{eosbn2} (\autoref{sec:TRA_eosbn2}) at the same time as $N^2$ is computed,
avoiding duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive).

%% =================================================================================================
\section[Bottom and top friction (\textit{zdfdrg.F90})]{Bottom and top friction (\protect\mdl{zdfdrg})}
\label{sec:ZDF_drg}

\begin{listing}
  \nlst{namdrg}
  \caption{\forcode{&namdrg}}
  \label{lst:namdrg}
\end{listing}

\begin{listing}
  \nlst{namdrg_top}
  \caption{\forcode{&namdrg_top}}
  \label{lst:namdrg_top}
\end{listing}

\begin{listing}
  \nlst{namdrg_bot}
  \caption{\forcode{&namdrg_bot}}
  \label{lst:namdrg_bot}
\end{listing}

Options to define the top and bottom friction are defined via parameters in the \nam{drg}{drg} namelist,
and, for top and bottom friction specifically, in the \nam{drg_top}{drg\_top} and \nam{drg_bot}{drg\_bot}
namelists respectively.
The bottom friction represents the friction generated by the bathymetry.
The top friction represents the friction generated by the ice shelf/ocean interface.
As the friction processes at the top and the bottom are treated in an identical way,
the description below considers mostly the bottom friction case, if not stated otherwise.

Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as
a condition on the vertical diffusive flux.
For the bottom boundary layer, one has:
 \[
   % \label{eq:ZDF_bfr_flux}
   A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
 \]
where ${\cal F}_h^{\textbf U}$ represents the downward flux of horizontal momentum outside
the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean).

How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near
the bottom relative to the Ekman layer depth.
For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m,
one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$
(for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$).
With a background value, $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
When the vertical mixing coefficient is this small, using a flux condition is equivalent to
entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or
bottom model layer.

To illustrate this, consider the equation for $u$ at $k$, the last ocean level:
\begin{equation}
  \label{eq:ZDF_drg_flux2}
  \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}}
\end{equation}
If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth.
On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used,
the turbulent Ekman layer will be represented explicitly by the model.
However, the logarithmic layer is never represented in current primitive equation model applications:
it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
Two choices are available in \NEMO: a linear and a quadratic bottom friction.
Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in
the present release of \NEMO.

In the code, the bottom friction is imposed by adding the trend due to the bottom friction to
the general momentum trend in \mdl{dynzdf} (\autoref{sec:DYN_zdf}).
For the time-split surface pressure gradient algorithm, the momentum trend due to
the barotropic component needs to be handled separately.
For this purpose it is convenient to compute and store coefficients which can be simply combined with
bottom velocities and geometric values to provide the momentum trend due to bottom friction.
 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where:
\begin{equation}
  \label{eq:ZDF_bfr_bdef}
  \frac{\partial {\textbf U_h}}{\partial t} =
  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b
\end{equation}
where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity.

Note that from \NEMO\ 4.0, drag coefficients are only computed at cell centers (\ie\ at T-points) and are
referred to as $c_b^T$ in the following. These are then linearly interpolated in space to get $c_b^\textbf{U}$
at velocity points.

%% =================================================================================================
\subsection[Free-slip boundary conditions (\forcode{ln_drg_OFF})]{Free-slip boundary conditions (\protect\np{ln_drg_OFF}{ln\_drg\_OFF})}
\label{subsec:ZDF_drg_free}

When setting \np[=.true.]{ln_drg_OFF}{ln\_drg\_OFF} free-slip conditions are used at the top and bottom boundaries,
\ie\ the drag coefficient used in \autoref{eq:ZDF_bfr_bdef} is set to $c_b^T = 0$.

%% =================================================================================================
\subsection[Linear top/bottom friction (\forcode{ln_lin})]{Linear top/bottom friction (\protect\np{ln_lin}{ln\_lin})}
\label{subsec:ZDF_drg_linear}

The linear friction parameterisation (including the special case of a free-slip condition, \autoref{subsec:ZDF_drg_free})
assumes that the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level):
\[
  % \label{eq:ZDF_bfr_linear}
  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
\]
where $r$ is a friction coefficient expressed in $m s^{-1}$.
The drag coefficient used in the general expression (\autoref{eq:ZDF_bfr_bdef}) is therefore:

\[
  % \label{eq:ZDF_bfr_linbfr_b}
    c_b^T = - r
\]

$r$ is generally estimated as $H / \tau$, where $\tau$ is a typical decay time in the deep ocean and $H$ is the ocean depth.
Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{weatherly_JMR84}.
A value of $\tau^{-1} = 10^{-7}$~s$^{-1}$, equivalent to 115 days, is usually used in quasi-geostrophic models.
One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$
\citep[][Eq. 9.6.6]{gill_bk82}. \\

In \NEMO, linear friction is enabled by setting \np[=.true.]{ln_lin}{ln\_lin}.
$c_b^T$ is calculated in \mdl{zdfdrg} and the trend due to the friction is added to the general momentum trend
in \mdl{dynzdf}.
$r$ is calculated as \forcode{rn_Cd0 * rn_Uc0}, where \np{rn_Cd0}{rn\_Cd0} and \np{rn_Uc0}{rn\_Uc0} are
namelist parameters corresponding to the drag coefficient $C_D$ and velocity scale $U_{av}$ respectively.
Their default values (0.001 and $0.4$~m\;s$^{-1}$ respectively) result in a friction coefficient of
$r = 4\;10^{-4}$~m\;s$^{-1}$, corresponding to a decay time scale of 115~days when assuming an ocean
depth of $H = 4000$~m.

Local enhancements may be applied to the values of $c_b^T$ by setting \np[=.true.]{ln_boost}{ln\_boost}
and providing a 2D mask array (with values $0 \leq M_b \leq 1$) via a NetCDF file.
For bottom (top) friction, the array and file are named \forcode{bfr_coef} and \textit{bfr\_coef.nc}
(\forcode{tfr_coef} and \textit{tfr\_coef.nc}) respectively.
Locations with a non-zero mask value will have the friction coefficient increased by
$M_b * \mathrm{rn\_boost} * \mathrm{rn\_Cd0}$.

%% =================================================================================================
\subsection[Non-linear top/bottom friction (\forcode{ln_non_lin})]{Non-linear top/bottom friction (\protect\np{ln_non_lin}{ln\_non\_lin})}
\label{subsec:ZDF_drg_nonlinear}

The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic:
\[
  % \label{eq:ZDF_drg_nonlinear}
  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
\]
where $C_D$ is a drag coefficient, and $e_b$ a top/bottom turbulent kinetic energy due to tides,
internal waves breaking and other short time scale currents.
The drag coefficient used in the general expression (\autoref{eq:ZDF_bfr_bdef}) is therefore:
\[
  % \label{eq:ZDF_drg_nonlinbfr}
    c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2}
\]

A typical value of the drag coefficient is $C_D = 10^{-3}$.
As an example, the CME experiment \citep{treguier_JGR92} uses $C_D = 10^{-3}$ and
$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{killworth_JPO92} uses $C_D = 1.4\;10^{-3}$ and
$e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$. \\

In \NEMO, non-linear friction is enabled by setting \np[=.true.]{ln_non_lin}{ln\_non\_lin}.
As for linear friction, $c_b^T$ is calculated in \mdl{zdfdrg} and the trend due to the friction is
added to the general momentum trend in \mdl{dynzdf}.
$C_D$ and $e_b$ correspond to the namelist parameters \np{rn_Cd0}{rn\_Cd0} and \np{rn_ke0}{rn\_ke0} respectively,
with their default values set to those of the CME experiment.
Note that for applications which consider tides explicitly, a low or even zero value of \np{rn_ke0}{rn\_ke0} is recommended.

As for linear friction, local enhancements may be applied to the values of $c_b^T$ by setting
\np[=.true.]{ln_boost}{ln\_boost} (see \autoref{subsec:ZDF_drg_linear}).

%% =================================================================================================
\subsection[Log-layer top/bottom friction (\forcode{ln_loglayer})]{Log-layer top/bottom friction (\protect\np{ln_loglayer}{ln\_loglayer})}
\label{subsec:ZDF_drg_loglayer}

In the non-linear friction case, the drag coefficient, $C_D$, can be optionally enhanced using
a "law of the wall" scaling.
$C_D$ is then no longer constant, but instead related to the distance to the wall (or equivalently, to
the half of the top/bottom layer thickness):
\[
  C_D = \left ( {\kappa \over {\mathrm log}\left ( 0.5 \; e_{3b} / z_0 \right ) } \right )^2
\]

\noindent where $\kappa$ is the von-Karman constant and $z_0$ is a roughness length.
This assumes that the model vertical resolution can capture the logarithmic layer,
which typically occurs for layers thinner than 1 m or so.

This special case of the non-linear friction is enabled by setting
\np[=.true.]{ln_loglayer}{ln\_loglayer} instead of \np[=.true.]{ln_non_lin}{ln\_non\_lin}.
$z_0$ corresponds to the namelist parameter \np{rn_z0}{rn\_z0}, while $C_D$ is bounded by
the namelist parameters \np{rn_Cd0}{rn\_Cd0} and \np{rn_Cdmax}{rn\_Cdmax} such that
$\mathrm{rn\_Cd0} \leq C_D \leq \mathrm{rn\_Cdmax}$. The lower bound of \np{rn_Cd0}{rn\_Cd0}
covers large layer thicknesses where logarithmic layers are presumably not resolved, while the upper bound
of \np{rn_Cdmax}{rn\_Cdmax} is applied for stability reasons.

\noindent The log-layer enhancement can also be applied to the top boundary friction if
ice-shelf cavities are activated (\np[=.true.]{ln_isfcav}{ln\_isfcav}).
%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}.

%% =================================================================================================
\subsection[Explicit top/bottom friction (\forcode{ln_drgimp=.false.})]{Explicit top/bottom friction (\protect\np[=.false.]{ln_drgimp}{ln\_drgimp})}
\label{subsec:ZDF_drg_stability}

Setting \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:

At the top (below an ice shelf cavity):
\[
  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
  = c_{t}^{\textbf{U}}\textbf{u}^{n-1}_{t}
\]

At the bottom (above the sea floor):
\[
  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
  = c_{b}^{\textbf{U}}\textbf{u}^{n-1}_{b}
\]

Since 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.
For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is:
\begin{equation}
  \label{eq:ZDF_Eqn_drgstab}
  \begin{split}
    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
    &= -\frac{ru}{e_{3u}}\;2\rdt\\
  \end{split}
\end{equation}
\noindent where linear friction and a leapfrog timestep have been assumed.
To ensure that the friction cannot reverse the direction of flow it is necessary to have:
\[
  |\Delta u| < \;|u|
\]
\noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives:
\[
  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
\]
This same inequality can also be derived in the non-linear bottom friction case if
a velocity of 1 m.s$^{-1}$ is assumed.
Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability:
\[
  e_{3u} > 2\;r\;\rdt
\]
\noindent which it may be necessary to impose if partial steps are being used.
For example, if $|u| = 1~\mathrm{ms}^{-1}$, $\rdt = 1800~\mathrm{s}$, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m.
For most applications, with physically sensible parameters these restrictions should not be of concern.
But caution may be necessary if attempts are made to locally enhance the bottom friction parameters.
To ensure stability limits are imposed on the top/bottom friction coefficients both
during initialisation and at each time step, checks at initialisation are made in
\mdl{zdfdrg} (assuming a $1~\mathrm{ms}^{-1}$ velocity in the non-linear case).
The number of breaches of the stability criterion are reported as well as
the minimum and maximum values that have been set.
The criterion is also checked at each time step, using the actual velocity, in \mdl{dynzdf}.
Values of the friction coefficient are reduced as necessary to ensure stability;
these changes are not reported.

Limits on the top/bottom friction coefficient are not imposed if the user has elected to
handle the friction implicitly (see \autoref{subsec:ZDF_drg_imp}).
The number of potential breaches of the explicit stability criterion are still reported for information purposes.

%% =================================================================================================
\subsection[Implicit top/bottom friction (\forcode{ln_drgimp=.true.})]{Implicit top/bottom friction (\protect\np[=.true.]{ln_drgimp}{ln\_drgimp})}
\label{subsec:ZDF_drg_imp}

An optional implicit form of bottom friction has been implemented to improve model stability.
We recommend this option for shelf sea and coastal ocean applications. %, especially for split-explicit time splitting.
This option can be invoked by setting \np{ln_drgimp}{ln\_drgimp} to \forcode{.true.} in the \nam{drg}{drg} namelist.
%This option requires \np{ln_zdfexp}{ln\_zdfexp} to be \forcode{.false.} in the \nam{zdf}{zdf} namelist.

This implementation is performed in \mdl{dynzdf} where the following boundary conditions are set while solving the fully implicit diffusion step:

At the top (below an ice shelf cavity):
\[
  % \label{eq:ZDF_dynZDF__drg_top}
  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
  = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t}
\]

At the bottom (above the sea floor):
\[
  % \label{eq:ZDF_dynZDF__drg_bot}
  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
  = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b}
\]

where $t$ and $b$ refers to top and bottom layers respectively.
Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so it is implicit.

%% =================================================================================================
\subsection[Bottom friction with split-explicit free surface]{Bottom friction with split-explicit free surface}
\label{subsec:ZDF_drg_ts}

With 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:D2D_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 it difficult to obtain 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.

The strategy to handle top/bottom stresses with split-explicit free surface in \NEMO\ is as follows:
\begin{enumerate}
\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.
\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 obtaining the implicit
      vertical diffusion result. Top/bottom stresses due to the barotropic components are explicitly accounted for
      due 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.
\end{enumerate}

Note that other strategies are possible, like considering vertical diffusion step in advance, \ie\ prior barotropic integration.

%% =================================================================================================
\section[Internal wave-driven mixing (\forcode{ln_zdfiwm})]{Internal wave-driven mixing (\protect\np{ln_zdfiwm}{ln\_zdfiwm})}
\label{subsec:ZDF_tmx_new}

\begin{listing}
  \nlst{namzdf_iwm}
  \caption{\forcode{&namzdf_iwm}}
  \label{lst:namzdf_iwm}
\end{listing}

The parameterization of mixing induced by breaking internal waves \citep{de-lavergne.vic.ea_JAMES20} is a generalization of the approach originally proposed by \citet{st-laurent.simmons.ea_GRL02}.
A three-dimensional field of turbulence production by breaking internal waves $\epsilon(i,j,k)$ is first constructed,
and the resulting diffusivity is obtained as
\[
  % \label{eq:ZDF_Kwave}
  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
\]
where $R_f$ is the mixing efficiency \citep{osborn_JPO80}.
If the \np{ln_mevar}{ln\_mevar} namelist parameter is set to \forcode{.false.}, the mixing efficiency is taken as constant and equal to 1/6 \citep{osborn_JPO80}.
In the opposite case, $R_f$ is instead a function of the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$, with $\nu$ the molecular viscosity of seawater, following the model of \cite{bouffard.boegman_DAO13} and the implementation of \cite{de-lavergne.madec.ea_JPO16}.
Note that $A^{vT}_{wave}$ is bounded by $10^{-2}$ m$^2$ s$^{-1}$, a limit that is often reached in weak stratification when the mixing efficiency is constant. 
In addition to the mixing efficiency, the ratio of salt to heat diffusivities can be chosen to vary as a function of $Re_b$ by setting the \np{ln_tsdiff}{ln\_tsdiff} parameter to \forcode{.true.}, a recommended choice. 
This parameterization of differential mixing, due to \cite{jackson.rehmann_JPO14}, is implemented as in \cite{de-lavergne.madec.ea_JPO16}.

This parameterization of internal wave-driven mixing is energetically constrained because vertical diffusivities derive from a known and globally constant power supply \citep{de-lavergne.vic.ea_JAMES20}. In the routine \mdl{zdfiwm}, the background diffusivity (\forcode{avtb}) is forced to a very small value (10$^{-10}$ m$^2$ s$^{-1}$), irrespective of namelist choices for background mixing. In addition, in the \mdl{zdftke} routine, when \np{ln_zdfiwm}{ln\_zdfiwm} is set to \forcode{.true.}, the background level of turbulent kinetic energy and the minimum mixing length are set to small values avoiding background diffusivities higher than molecular rates of diffusion. Hence, when the parameterization of internal wave-driven mixing is activated, vertical mixing in the deep ocean is truly energy constrained.

The three-dimensional distribution of turbulence production, $\epsilon(i,j,k)$ [W kg$^{-1}$],
is constructed from four static maps of column-integrated internal wave energy dissipation [W m$^{-2}$] (\autoref{fig:ZDF_iwm}):
\begin{itemize}
\item $E_{nsq}(i,j)$: dissipation scaling with the squared buoyancy frequency $N^2$;
\item $E_{sho}(i,j)$: dissipation due to shoaling (scales with $N$ in the vertical);
\item $E_{cri}(i,j)$: bottom-intensified dissipation at critical slopes;
\item $E_{bot}(i,j)$: bottom-intensified dissipation above abyssal hills.
\end{itemize}
These power fields are read in a NetCDF forcing file whose name is set in the namelist section \nam{zdf}{zdf}. This forcing file also contains two maps of decay scales [m] necessary for the vertical distribution of $E_{cri}$ and $E_{bot}$:
\begin{itemize}
\item $H_{cri}$(i,j): related to the height difference of the critical slope.
\item $H_{bot}$(i,j): related to the wavelength of abyssal hills and to the energy flux $E_{bot}$.
\end{itemize}
Each power field goes with a specific vertical structure. The local turbulence production $\epsilon(i,j,k)$ is thus obtained as the sum of four contributions,
$$ \epsilon = \epsilon_{nsq} + \epsilon_{sho} + \epsilon_{cri} + \epsilon_{bot} ,$$
where, within each $(i,j)$ water column,
\begin{itemize}
\item $\epsilon_{nsq} \propto N^2$
\item $\epsilon_{sho} \propto N$
\item $\epsilon_{cri} \propto \exp(-h_{ab}/H_{cri})$
\item $\epsilon_{bot} \propto 1/(1+h_{ab}/H_{bot})^2$
\end{itemize}
with $h_{ab}$ the height above bottom and $N$ the simulated buoyancy frequency. These vertical structures are interactive in the sense that they depend on the simulated stratification and sea surface height. Vertical distributions are thus computed every time step and are normalized so that
$$\int \rho\,\epsilon_{nsq}\,dz = E_{nsq}\,,\; \int \rho\,\epsilon_{sho}\,dz = E_{sho}\,,\; \int \rho\,\epsilon_{cri}\,dz = E_{cri}\,,\; \int \rho\,\epsilon_{bot}\,dz = E_{bot}\;.$$

\begin{figure}
  \centering
  \includegraphics[width=0.66\textwidth]{ZDF_iwm}
  \caption[Internal tide energy dissipation]{
    Four power maps (in log$_{10}$(W m$^{-2}$)), as estimated by \cite{de-lavergne.rathore.ea_ESS24}, that enter the parameterization of internal wave-driven mixing. The globally integrated power (in GW) within each field is indicated at the top left of each panel. The overall power amounts to 1.40 TW.}
  \label{fig:ZDF_iwm}
\end{figure}

As of now, the parameterization only accounts for mixing powered by internal tides, thought to be the dominant supply of small-scale turbulence in the ocean interior \citep{de-lavergne.vic.ea_JAMES20}. It is recommended to use the forcing fields available at https://doi.org/10.17882/103233. These fields are state-of-the-art estimates of internal tide energy dissipation, including subinertial tides \citep{de-lavergne.rathore.ea_ESS24}. Details on the implementation of this parameterization and impacts in long NEMO eORCA1 experiments can be found in \citep{de-lavergne.rathore.ea_ESS24}.


%% =================================================================================================
\section[Surface wave-induced mixing (\forcode{ln_zdfswm})]{Surface wave-induced mixing (\protect\np{ln_zdfswm}{ln\_zdfswm})}
\label{subsec:ZDF_swm}

Surface waves produce an enhanced mixing through wave-turbulence interaction.
In addition to breaking waves induced turbulence (\autoref{subsec:ZDF_tke}),
the influence of non-breaking waves can be accounted for by introducing
wave-induced viscosity and diffusivity as a function of the wave number spectrum.

Following \citet{qiao.yuan.ea_OD10}, a formulation of wave-induced mixing coefficient
is provided as a function of wave amplitude, Stokes Drift and wave-number:

\begin{equation}
  \label{eq:ZDF_Bv}
  B_{v} = \alpha {A} {U}_{st} {\exp(3kz)}
\end{equation}

Where $B_{v}$ is the wave-induced mixing coefficient, $A$ is the wave amplitude,
${U}_{st}$ is the Stokes Drift velocity, $k$ is the wave number and $\alpha$
is a constant which should be determined by observations or
numerical experiments and is set to be 1.

The coefficient $B_{v}$ is then directly added to the vertical viscosity
and diffusivity coefficients.

This parameterisation is enabled by setting \np[=.true.]{ln_zdfswm}{ln\_zdfswm}.
Additionally, both wave interaction (\np[=.true.]{ln_wave}{ln\_wave}) and calculation of the Stokes Drift
(\np[=.true.]{ln_sdw}{ln\_sdw}) must be enabled.
The required wave fields (significant wave height and mean wave number) can be provided either in forced or coupled mode.
For more information on wave parameters and settings, see \autoref{sec:SBC_wave}.


\subinc{\input{../../global/epilogue}}

\end{document}
