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1% ================================================================
2% Chapter  Vertical Ocean Physics (ZDF)
3% ================================================================
4\chapter{Vertical Ocean Physics (ZDF)}
5\label{ZDF}
6\minitoc
7
8%gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN.
9
10
11\newpage
12$\ $\newline    % force a new ligne
13
14
15% ================================================================
16% Vertical Mixing
17% ================================================================
18\section{Vertical Mixing}
19\label{ZDF_zdf}
20
21The discrete form of the ocean subgrid scale physics has been presented in
22\S\ref{TRA_zdf} and \S\ref{DYN_zdf}. At the surface and bottom boundaries,
23the turbulent fluxes of momentum, heat and salt have to be defined. At the
24surface they are prescribed from the surface forcing (see Chap.~\ref{SBC}),
25while at the bottom they are set to zero for heat and salt, unless a geothermal
26flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} 
27defined, see \S\ref{TRA_bbc}), and specified through a bottom friction
28parameterisation for momentum (see \S\ref{ZDF_bfr}).
29
30In this section we briefly discuss the various choices offered to compute
31the vertical eddy viscosity and diffusivity coefficients, $A_u^{vm}$ ,
32$A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$-
33points, respectively (see \S\ref{TRA_zdf} and \S\ref{DYN_zdf}). These
34coefficients can be assumed to be either constant, or a function of the local
35Richardson number, or computed from a turbulent closure model (either
36TKE or KPP formulation). The computation of these coefficients is initialized
37in the \mdl{zdfini} module and performed in the \mdl{zdfric}, \mdl{zdftke} or
38\mdl{zdfkpp} modules. The trends due to the vertical momentum and tracer
39diffusion, including the surface forcing, are computed and added to the
40general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively.
41These trends can be computed using either a forward time stepping scheme
42(namelist parameter \np{ln\_zdfexp}=true) or a backward time stepping
43scheme (\np{ln\_zdfexp}=false) depending on the magnitude of the mixing
44coefficients, and thus of the formulation used (see \S\ref{STP}).
45
46% -------------------------------------------------------------------------------------------------------------
47%        Constant
48% -------------------------------------------------------------------------------------------------------------
49\subsection{Constant (\key{zdfcst})}
50\label{ZDF_cst}
51%--------------------------------------------namzdf---------------------------------------------------------
52\namdisplay{namzdf}
53%--------------------------------------------------------------------------------------------------------------
54
55Options are defined through the  \ngn{namzdf} namelist variables.
56When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients
57are set to constant values over the whole ocean. This is the crudest way to define
58the vertical ocean physics. It is recommended that this option is only used in
59process studies, not in basin scale simulations. Typical values used in this case are:
60\begin{align*} 
61A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}  \\
62A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1}
63\end{align*}
64
65These values are set through the \np{rn\_avm0} and \np{rn\_avt0} namelist parameters.
66In all cases, do not use values smaller that those associated with the molecular
67viscosity and diffusivity, that is $\sim10^{-6}~m^2.s^{-1}$ for momentum,
68$\sim10^{-7}~m^2.s^{-1}$ for temperature and $\sim10^{-9}~m^2.s^{-1}$ for salinity.
69
70
71% -------------------------------------------------------------------------------------------------------------
72%        Richardson Number Dependent
73% -------------------------------------------------------------------------------------------------------------
74\subsection{Richardson Number Dependent (\key{zdfric})}
75\label{ZDF_ric}
76
77%--------------------------------------------namric---------------------------------------------------------
78\namdisplay{namzdf_ric}
79%--------------------------------------------------------------------------------------------------------------
80
81When \key{zdfric} is defined, a local Richardson number dependent formulation
82for the vertical momentum and tracer eddy coefficients is set through the  \ngn{namzdf\_ric} 
83namelist variables.The vertical mixing
84coefficients are diagnosed from the large scale variables computed by the model.
85\textit{In situ} measurements have been used to link vertical turbulent activity to
86large scale ocean structures. The hypothesis of a mixing mainly maintained by the
87growth of Kelvin-Helmholtz like instabilities leads to a dependency between the
88vertical eddy coefficients and the local Richardson number ($i.e.$ the
89ratio of stratification to vertical shear). Following \citet{Pacanowski_Philander_JPO81}, the following
90formulation has been implemented:
91\begin{equation} \label{Eq_zdfric}
92   \left\{      \begin{aligned}
93         A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\
94         A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm}
95   \end{aligned}    \right.
96\end{equation}
97where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson
98number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}),
99$A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the
100constant case (see \S\ref{ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ 
101is the maximum value that can be reached by the coefficient when $Ri\leq 0$,
102$a=5$ and $n=2$. The last three values can be modified by setting the
103\np{rn\_avmri}, \np{rn\_alp} and \np{nn\_ric} namelist parameters, respectively.
104
105A simple mixing-layer model to transfer and dissipate the atmospheric
106 forcings (wind-stress and buoyancy fluxes) can be activated setting
107the \np{ln\_mldw} =.true. in the namelist.
108
109In this case, the local depth of turbulent wind-mixing or "Ekman depth"
110 $h_{e}(x,y,t)$ is evaluated and the vertical eddy coefficients prescribed within this layer.
111
112This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation:
113\begin{equation}
114         h_{e} = Ek \frac {u^{*}} {f_{0}}    \\
115\end{equation}
116where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis
117parameter.
118
119In this similarity height relationship, the turbulent friction velocity:
120\begin{equation}
121         u^{*} = \sqrt \frac {|\tau|} {\rho_o}     \\
122\end{equation}
123
124is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$.
125The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}.
126Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to
127the empirical values \np{rn\_wtmix} and \np{rn\_wvmix} \citep{Lermusiaux2001}.
128
129% -------------------------------------------------------------------------------------------------------------
130%        TKE Turbulent Closure Scheme
131% -------------------------------------------------------------------------------------------------------------
132\subsection{TKE Turbulent Closure Scheme (\key{zdftke})}
133\label{ZDF_tke}
134
135%--------------------------------------------namzdf_tke--------------------------------------------------
136\namdisplay{namzdf_tke}
137%--------------------------------------------------------------------------------------------------------------
138
139The vertical eddy viscosity and diffusivity coefficients are computed from a TKE
140turbulent closure model based on a prognostic equation for $\bar{e}$, the turbulent
141kinetic energy, and a closure assumption for the turbulent length scales. This
142turbulent closure model has been developed by \citet{Bougeault1989} in the
143atmospheric case, adapted by \citet{Gaspar1990} for the oceanic case, and
144embedded in OPA, the ancestor of NEMO, by \citet{Blanke1993} for equatorial Atlantic
145simulations. Since then, significant modifications have been introduced by
146\citet{Madec1998} in both the implementation and the formulation of the mixing
147length scale. The time evolution of $\bar{e}$ is the result of the production of
148$\bar{e}$ through vertical shear, its destruction through stratification, its vertical
149diffusion, and its dissipation of \citet{Kolmogorov1942} type:
150\begin{equation} \label{Eq_zdftke_e}
151\frac{\partial \bar{e}}{\partial t} =
152\frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
153                    +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
154-K_\rho\,N^2
155+\frac{1}{e_3} \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
156            \;\frac{\partial \bar{e}}{\partial k}} \right]
157- c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
158\end{equation}
159\begin{equation} \label{Eq_zdftke_kz}
160   \begin{split}
161         K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }     \\
162         K_\rho &= A^{vm} / P_{rt}
163   \end{split}
164\end{equation}
165where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}),
166$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,
167$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity
168and diffusivity coefficients. The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$ 
169$\approx 0.7$ are designed to deal with vertical mixing at any depth \citep{Gaspar1990}.
170They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}.
171$P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function
172of the local Richardson number, $R_i$:
173\begin{align*} \label{Eq_prt}
174P_{rt} = \begin{cases}
175                    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$}  \\
176                    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$}  \\
177                    \ \ 10 &      \text{if $\ 2 \leq R_i$} 
178            \end{cases}
179\end{align*}
180Options are defined through the  \ngn{namzdfy\_tke} namelist variables.
181The choice of $P_{rt}$ is controlled by the \np{nn\_pdl} namelist variable.
182
183At the sea surface, the value of $\bar{e}$ is prescribed from the wind
184stress field as $\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb} 
185namelist parameter. The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}),
186however a much larger value can be used when taking into account the
187surface wave breaking (see below Eq. \eqref{ZDF_Esbc}).
188The bottom value of TKE is assumed to be equal to the value of the level just above.
189The time integration of the $\bar{e}$ equation may formally lead to negative values
190because the numerical scheme does not ensure its positivity. To overcome this
191problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} 
192namelist parameter). Following \citet{Gaspar1990}, the cut-off value is set
193to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. This allows the subsequent formulations
194to match that of \citet{Gargett1984} for the diffusion in the thermocline and
195deep ocean :  $K_\rho = 10^{-3} / N$.
196In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical
197instabilities associated with too weak vertical diffusion. They must be
198specified at least larger than the molecular values, and are set through
199\np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \S\ref{ZDF_cst}).
200
201\subsubsection{Turbulent length scale}
202For computational efficiency, the original formulation of the turbulent length
203scales proposed by \citet{Gaspar1990} has been simplified. Four formulations
204are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist
205parameter. The first two are based on the following first order approximation
206\citep{Blanke1993}:
207\begin{equation} \label{Eq_tke_mxl0_1}
208l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N
209\end{equation}
210which is valid in a stable stratified region with constant values of the Brunt-
211Vais\"{a}l\"{a} frequency. The resulting length scale is bounded by the distance
212to the surface or to the bottom (\np{nn\_mxl} = 0) or by the local vertical scale factor
213(\np{nn\_mxl} = 1). \citet{Blanke1993} notice that this simplification has two major
214drawbacks: it makes no sense for locally unstable stratification and the
215computation no longer uses all the information contained in the vertical density
216profile. To overcome these drawbacks, \citet{Madec1998} introduces the
217\np{nn\_mxl} = 2 or 3 cases, which add an extra assumption concerning the vertical
218gradient of the computed length scale. So, the length scales are first evaluated
219as in \eqref{Eq_tke_mxl0_1} and then bounded such that:
220\begin{equation} \label{Eq_tke_mxl_constraint}
221\frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
222\qquad \text{with }\  l =  l_k = l_\epsilon
223\end{equation}
224\eqref{Eq_tke_mxl_constraint} means that the vertical variations of the length
225scale cannot be larger than the variations of depth. It provides a better
226approximation of the \citet{Gaspar1990} formulation while being much less
227time consuming. In particular, it allows the length scale to be limited not only
228by the distance to the surface or to the ocean bottom but also by the distance
229to a strongly stratified portion of the water column such as the thermocline
230(Fig.~\ref{Fig_mixing_length}). In order to impose the \eqref{Eq_tke_mxl_constraint} 
231constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$,
232the upward and downward length scales, and evaluate the dissipation and
233mixing length scales as (and note that here we use numerical indexing):
234%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
235\begin{figure}[!t] \begin{center}
236\includegraphics[width=1.00\textwidth]{./TexFiles/Figures/Fig_mixing_length.pdf}
237\caption{ \label{Fig_mixing_length} 
238Illustration of the mixing length computation. }
239\end{center} 
240\end{figure}
241%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
242\begin{equation} \label{Eq_tke_mxl2}
243\begin{aligned}
244 l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
245    \quad &\text{ from $k=1$ to $jpk$ }\ \\
246 l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)}  \right)   
247    \quad &\text{ from $k=jpk$ to $1$ }\ \\
248\end{aligned}
249\end{equation}
250where $l^{(k)}$ is computed using \eqref{Eq_tke_mxl0_1},
251$i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
252
253In the \np{nn\_mxl}~=~2 case, the dissipation and mixing length scales take the same
254value: $ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the
255\np{nn\_mxl}~=~3 case, the dissipation and mixing turbulent length scales are give
256as in \citet{Gaspar1990}:
257\begin{equation} \label{Eq_tke_mxl_gaspar}
258\begin{aligned}
259& l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }    \\
260& l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)
261\end{aligned}
262\end{equation}
263
264At the ocean surface, a non zero length scale is set through the  \np{rn\_lmin0} namelist
265parameter. Usually the surface scale is given by $l_o = \kappa \,z_o$ 
266where $\kappa = 0.4$ is von Karman's constant and $z_o$ the roughness
267parameter of the surface. Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94} 
268leads to a 0.04~m, the default value of \np{rn\_lsurf}. In the ocean interior
269a minimum length scale is set to recover the molecular viscosity when $\bar{e}$ 
270reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).
271
272
273\subsubsection{Surface wave breaking parameterization}
274%-----------------------------------------------------------------------%
275Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified
276to include the effect of surface wave breaking energetics. This results in a reduction of summertime
277surface temperature when the mixed layer is relatively shallow. The \citet{Mellor_Blumberg_JPO04} 
278modifications acts on surface length scale and TKE values and air-sea drag coefficient.
279The latter concerns the bulk formulea and is not discussed here.
280
281Following \citet{Craig_Banner_JPO94}, the boundary condition on surface TKE value is :
282\begin{equation}  \label{ZDF_Esbc}
283\bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
284\end{equation}
285where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality
286which depends on the ''wave age'', ranging from 57 for mature waves to 146 for
287younger waves \citep{Mellor_Blumberg_JPO04}.
288The boundary condition on the turbulent length scale follows the Charnock's relation:
289\begin{equation} \label{ZDF_Lsbc}
290l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o}
291\end{equation}
292where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant.
293\citet{Mellor_Blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by \citet{Stacey_JPO99}
294citing observation evidence, and $\alpha_{CB} = 100$ the Craig and Banner's value.
295As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$,
296with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}~=~67.83 corresponds
297to $\alpha_{CB} = 100$. further setting  \np{ln\_lsurf} to true applies \eqref{ZDF_Lsbc} 
298as surface boundary condition on length scale, with $\beta$ hard coded to the Stacet's value.
299Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters)
300is applied on surface $\bar{e}$ value.
301
302
303\subsubsection{Langmuir cells}
304%--------------------------------------%
305Langmuir circulations (LC) can be described as ordered large-scale vertical motions
306in the surface layer of the oceans. Although LC have nothing to do with convection,
307the circulation pattern is rather similar to so-called convective rolls in the atmospheric
308boundary layer. The detailed physics behind LC is described in, for example,
309\citet{Craik_Leibovich_JFM76}. The prevailing explanation is that LC arise from
310a nonlinear interaction between the Stokes drift and wind drift currents.
311
312Here we introduced in the TKE turbulent closure the simple parameterization of
313Langmuir circulations proposed by \citep{Axell_JGR02} for a $k-\epsilon$ turbulent closure.
314The parameterization, tuned against large-eddy simulation, includes the whole effect
315of LC in an extra source terms of TKE, $P_{LC}$.
316The presence of $P_{LC}$ in \eqref{Eq_zdftke_e}, the TKE equation, is controlled
317by setting \np{ln\_lc} to \textit{true} in the namtke namelist.
318 
319By making an analogy with the characteristic convective velocity scale
320($e.g.$, \citet{D'Alessio_al_JPO98}), $P_{LC}$ is assumed to be :
321\begin{equation}
322P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}}
323\end{equation}
324where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth.
325With no information about the wave field, $w_{LC}$ is assumed to be proportional to
326the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module
327\footnote{Following \citet{Li_Garrett_JMR93}, the surface Stoke drift velocity
328may be expressed as $u_s =  0.016 \,|U_{10m}|$. Assuming an air density of
329$\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of $1.5~10^{-3}$ give the expression
330used of $u_s$ as a function of the module of surface stress}.
331For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as
332at a finite depth $H_{LC}$ (which is often close to the mixed layer depth), and simply
333varies as a sine function in between (a first-order profile for the Langmuir cell structures).
334The resulting expression for $w_{LC}$ is :
335\begin{equation}
336w_{LC}  = \begin{cases}
337                   c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$}    \\
338                   0                             &      \text{otherwise} 
339                 \end{cases}
340\end{equation}
341where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise
342to fit LES data. The chosen value yields maximum vertical velocities $w_{LC}$ of the order
343of a few centimeters per second. The value of $c_{LC}$ is set through the \np{rn\_lc} 
344namelist parameter, having in mind that it should stay between 0.15 and 0.54 \citep{Axell_JGR02}.
345
346The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations:
347$H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift
348can reach on its own by converting its kinetic energy to potential energy, according to
349\begin{equation}
350- \int_{-H_{LC}}^0 { N^2\;\;dz} = \frac{1}{2} u_s^2
351\end{equation}
352
353
354\subsubsection{Mixing just below the mixed layer}
355%--------------------------------------------------------------%
356
357To be add here a description of "penetration of TKE" and the associated namelist parameters
358 \np{nn\_etau}, \np{rn\_efr} and \np{nn\_htau}.
359
360% from Burchard et al OM 2008 :
361% 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 (e.g. Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002).
362
363
364
365% -------------------------------------------------------------------------------------------------------------
366%        TKE discretization considerations
367% -------------------------------------------------------------------------------------------------------------
368\subsection{TKE discretization considerations (\key{zdftke})}
369\label{ZDF_tke_ene}
370
371%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
372\begin{figure}[!t]   \begin{center}
373\includegraphics[width=1.00\textwidth]{./TexFiles/Figures/Fig_ZDF_TKE_time_scheme.pdf}
374\caption{ \label{Fig_TKE_time_scheme} 
375Illustration of the TKE time integration and its links to the momentum and tracer time integration. }
376\end{center} 
377\end{figure}
378%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
379
380The production of turbulence by vertical shear (the first term of the right hand side
381of \eqref{Eq_zdftke_e}) should balance the loss of kinetic energy associated with
382the vertical momentum diffusion (first line in \eqref{Eq_PE_zdf}). To do so a special care
383have to be taken for both the time and space discretization of the TKE equation
384\citep{Burchard_OM02,Marsaleix_al_OM08}.
385
386Let us first address the time stepping issue. Fig.~\ref{Fig_TKE_time_scheme} shows
387how the two-level Leap-Frog time stepping of the momentum and tracer equations interplays
388with the one-level forward time stepping of TKE equation. With this framework, the total loss
389of kinetic energy (in 1D for the demonstration) due to the vertical momentum diffusion is
390obtained by multiplying this quantity by $u^t$ and summing the result vertically:   
391\begin{equation} \label{Eq_energ1}
392\begin{split}
393\int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\
394&= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}         
395 - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz }
396\end{split}
397\end{equation}
398Here, the vertical diffusion of momentum is discretized backward in time
399with a coefficient, $K_m$, known at time $t$ (Fig.~\ref{Fig_TKE_time_scheme}),
400as it is required when using the TKE scheme (see \S\ref{STP_forward_imp}).
401The first term of the right hand side of \eqref{Eq_energ1} represents the kinetic energy
402transfer at the surface (atmospheric forcing) and at the bottom (friction effect).
403The second term is always negative. It is the dissipation rate of kinetic energy,
404and thus minus the shear production rate of $\bar{e}$. \eqref{Eq_energ1} 
405implies that, to be energetically consistent, the production rate of $\bar{e}$ 
406used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
407${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ (and not by the more straightforward
408$K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
409
410A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
411(second term of the right hand side of \eqref{Eq_zdftke_e}). This term
412must balance the input of potential energy resulting from vertical mixing.
413The rate of change of potential energy (in 1D for the demonstration) due vertical
414mixing is obtained by multiplying vertical density diffusion
415tendency by $g\,z$ and and summing the result vertically:
416\begin{equation} \label{Eq_energ2}
417\begin{split}
418\int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\
419&= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta} 
420   - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\
421&= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta}
422+ \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  }
423\end{split}
424\end{equation}
425where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.
426The first term of the right hand side of \eqref{Eq_energ2}  is always zero
427because there is no diffusive flux through the ocean surface and bottom).
428The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
429Therefore \eqref{Eq_energ1} implies that, to be energetically consistent, the product
430${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \eqref{Eq_zdftke_e}, the TKE equation.
431
432Let us now address the space discretization issue.
433The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity
434components are in the centre of the side faces of a $t$-box in staggered C-grid
435(Fig.\ref{Fig_cell}). A space averaging is thus required to obtain the shear TKE production term.
436By redoing the \eqref{Eq_energ1} in the 3D case, it can be shown that the product of
437eddy coefficient by the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
438Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into
439account.
440
441The above energetic considerations leads to
442the following final discrete form for the TKE equation:
443\begin{equation} \label{Eq_zdftke_ene}
444\begin{split}
445\frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv 
446\Biggl\{ \Biggr.
447  &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} } 
448                                                                              \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\
449+&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} } 
450                                                                               \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j} 
451\Biggr. \Biggr\}   \\
452%
453- &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\
454%
455+&\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]   \\
456%
457- &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt}
458\end{split}
459\end{equation}
460where the last two terms in \eqref{Eq_zdftke_ene} (vertical diffusion and Kolmogorov dissipation)
461are time stepped using a backward scheme (see\S\ref{STP_forward_imp}).
462Note that the Kolmogorov term has been linearized in time in order to render
463the implicit computation possible. The restart of the TKE scheme
464requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as they all appear in
465the right hand side of \eqref{Eq_zdftke_ene}. For the latter, it is in fact
466the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.
467
468% -------------------------------------------------------------------------------------------------------------
469%        GLS Generic Length Scale Scheme
470% -------------------------------------------------------------------------------------------------------------
471\subsection{GLS Generic Length Scale (\key{zdfgls})}
472\label{ZDF_gls}
473
474%--------------------------------------------namzdf_gls---------------------------------------------------------
475\namdisplay{namzdf_gls}
476%--------------------------------------------------------------------------------------------------------------
477
478The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on
479two prognostic equations: one for the turbulent kinetic energy $\bar {e}$, and another
480for the generic length scale, $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}.
481This later variable is defined as : $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,
482where the triplet $(p, m, n)$ value given in Tab.\ref{Tab_GLS} allows to recover
483a number of well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982},
484$k$-$\epsilon$ \citep{Rodi_1987}, $k$-$\omega$ \citep{Wilcox_1988} 
485among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}).
486The GLS scheme is given by the following set of equations:
487\begin{equation} \label{Eq_zdfgls_e}
488\frac{\partial \bar{e}}{\partial t} =
489\frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
490                                                   +\left( \frac{\partial v}{\partial k} \right)^2} \right]
491-K_\rho \,N^2
492+\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right]
493- \epsilon
494\end{equation}
495
496\begin{equation} \label{Eq_zdfgls_psi}
497   \begin{split}
498\frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{
499\frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
500                                                                   +\left( \frac{\partial v}{\partial k} \right)^2} \right]
501- C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\
502&+\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 }
503                                  \;\frac{\partial \psi}{\partial k}} \right]\;
504   \end{split}
505\end{equation}
506
507\begin{equation} \label{Eq_zdfgls_kz}
508   \begin{split}
509         K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
510         K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l
511   \end{split}
512\end{equation}
513
514\begin{equation} \label{Eq_zdfgls_eps}
515{\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
516\end{equation}
517where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2})
518and $\epsilon$ the dissipation rate.
519The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$)
520depends of the choice of the turbulence model. Four different turbulent models are pre-defined
521(Tab.\ref{Tab_GLS}). They are made available through the \np{nn\_clo} namelist parameter.
522
523%--------------------------------------------------TABLE--------------------------------------------------
524\begin{table}[htbp]  \begin{center}
525%\begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c}
526\begin{tabular}{ccccc}
527                         &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\ 
528%                        & \citep{Mellor_Yamada_1982} &  \citep{Rodi_1987}       & \citep{Wilcox_1988} &                 \\ 
529\hline  \hline 
530\np{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\ 
531\hline 
532$( p , n , m )$          &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\
533$\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\
534$\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\
535$C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\
536$C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\
537$C_3$              &      1.           &     1.              &      1.                &       1.           \\
538$F_{wall}$        &      Yes        &       --             &     --                  &      --          \\
539\hline
540\hline
541\end{tabular}
542\caption{   \label{Tab_GLS} 
543Set of predefined GLS parameters, or equivalently predefined turbulence models available
544with \key{zdfgls} and controlled by the \np{nn\_clos} namelist variable in \ngn{namzdf\_gls} .}
545\end{center}   \end{table}
546%--------------------------------------------------------------------------------------------------------------
547
548In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force
549the convergence of the mixing length towards $K z_b$ ($K$: Kappa and $z_b$: rugosity length)
550value near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$ 
551are calculated from stability function proposed by \citet{Galperin_al_JAS88}, or by \citet{Kantha_Clayson_1994} 
552or one of the two functions suggested by \citet{Canuto_2001}  (\np{nn\_stab\_func} = 0, 1, 2 or 3, resp.}).
553The value of $C_{0\mu}$ depends of the choice of the stability function.
554
555The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated
556thanks to Dirichlet or Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp.
557As for TKE closure , the wave effect on the mixing is considered when \np{ln\_crban}~=~true
558\citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}. The \np{rn\_crban} namelist parameter
559is $\alpha_{CB}$ in \eqref{ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \eqref{ZDF_Lsbc}.
560
561The $\psi$ equation is known to fail in stably stratified flows, and for this reason
562almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.
563With this clipping, the maximum permissible length scale is determined by
564$l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$. A value of $c_{lim} = 0.53$ is often used
565\citep{Galperin_al_JAS88}. \cite{Umlauf_Burchard_CSR05} show that the value of
566the clipping factor is of crucial importance for the entrainment depth predicted in
567stably stratified situations, and that its value has to be chosen in accordance
568with the algebraic model for the turbulent fluxes. The clipping is only activated
569if \np{ln\_length\_lim}=true, and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value.
570
571The time and space discretization of the GLS equations follows the same energetic
572consideration as for the TKE case described in \S\ref{ZDF_tke_ene}  \citep{Burchard_OM02}.
573Examples of performance of the 4 turbulent closure scheme can be found in \citet{Warner_al_OM05}.
574
575% -------------------------------------------------------------------------------------------------------------
576%        K Profile Parametrisation (KPP)
577% -------------------------------------------------------------------------------------------------------------
578\subsection{K Profile Parametrisation (KPP) (\key{zdfkpp}) }
579\label{ZDF_kpp}
580
581%--------------------------------------------namkpp--------------------------------------------------------
582\namdisplay{namzdf_kpp}
583%--------------------------------------------------------------------------------------------------------------
584
585The KKP scheme has been implemented by J. Chanut ...
586Options are defined through the  \ngn{namzdf\_kpp} namelist variables.
587
588\colorbox{yellow}{Add a description of KPP here.}
589
590
591% ================================================================
592% Convection
593% ================================================================
594\section{Convection}
595\label{ZDF_conv}
596
597%--------------------------------------------namzdf--------------------------------------------------------
598\namdisplay{namzdf}
599%--------------------------------------------------------------------------------------------------------------
600
601Static instabilities (i.e. light potential densities under heavy ones) may
602occur at particular ocean grid points. In nature, convective processes
603quickly re-establish the static stability of the water column. These
604processes have been removed from the model via the hydrostatic
605assumption so they must be parameterized. Three parameterisations
606are available to deal with convective processes: a non-penetrative
607convective adjustment or an enhanced vertical diffusion, or/and the
608use of a turbulent closure scheme.
609
610% -------------------------------------------------------------------------------------------------------------
611%       Non-Penetrative Convective Adjustment
612% -------------------------------------------------------------------------------------------------------------
613\subsection   [Non-Penetrative Convective Adjustment (\np{ln\_tranpc}) ]
614         {Non-Penetrative Convective Adjustment (\np{ln\_tranpc}=.true.) }
615\label{ZDF_npc}
616
617%--------------------------------------------namzdf--------------------------------------------------------
618\namdisplay{namzdf}
619%--------------------------------------------------------------------------------------------------------------
620
621%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
622\begin{figure}[!htb]    \begin{center}
623\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_npc.pdf}
624\caption{  \label{Fig_npc} 
625Example of an unstable density profile treated by the non penetrative
626convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from
627the surface to the bottom. It is found to be unstable between levels 3 and 4.
628They are mixed. The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5
629are mixed. The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are
630mixed. The $1^{st}$ step ends since the density profile is then stable below
631the level 3. $2^{nd}$ step: the new $\rho$ profile is checked following the same
632procedure as in $1^{st}$ step: levels 2 to 5 are mixed. The new density profile
633is checked. It is found stable: end of algorithm.}
634\end{center}   \end{figure}
635%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
636
637Options are defined through the  \ngn{namzdf} namelist variables.
638The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}=true.
639It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously
640the statically unstable portion of the water column, but only until the density
641structure becomes neutrally stable ($i.e.$ until the mixed portion of the water
642column has \textit{exactly} the density of the water just below) \citep{Madec_al_JPO91}.
643The associated algorithm is an iterative process used in the following way
644(Fig. \ref{Fig_npc}): starting from the top of the ocean, the first instability is
645found. Assume in the following that the instability is located between levels
646$k$ and $k+1$. The potential temperature and salinity in the two levels are
647vertically mixed, conserving the heat and salt contents of the water column.
648The new density is then computed by a linear approximation. If the new
649density profile is still unstable between levels $k+1$ and $k+2$, levels $k$,
650$k+1$ and $k+2$ are then mixed. This process is repeated until stability is
651established below the level $k$ (the mixing process can go down to the
652ocean bottom). The algorithm is repeated to check if the density profile
653between level $k-1$ and $k$ is unstable and/or if there is no deeper instability.
654
655This algorithm is significantly different from mixing statically unstable levels
656two by two. The latter procedure cannot converge with a finite number
657of iterations for some vertical profiles while the algorithm used in \NEMO 
658converges for any profile in a number of iterations which is less than the
659number of vertical levels. This property is of paramount importance as
660pointed out by \citet{Killworth1989}: it avoids the existence of permanent
661and unrealistic static instabilities at the sea surface. This non-penetrative
662convective algorithm has been proved successful in studies of the deep
663water formation in the north-western Mediterranean Sea
664\citep{Madec_al_JPO91, Madec_al_DAO91, Madec_Crepon_Bk91}.
665
666Note that in the current implementation of this algorithm presents several
667limitations. First, potential density referenced to the sea surface is used to
668check whether the density profile is stable or not. This is a strong
669simplification which leads to large errors for realistic ocean simulations.
670Indeed, many water masses of the world ocean, especially Antarctic Bottom
671Water, are unstable when represented in surface-referenced potential density.
672The scheme will erroneously mix them up. Second, the mixing of potential
673density is assumed to be linear. This assures the convergence of the algorithm
674even when the equation of state is non-linear. Small static instabilities can thus
675persist due to cabbeling: they will be treated at the next time step.
676Third, temperature and salinity, and thus density, are mixed, but the
677corresponding velocity fields remain unchanged. When using a Richardson
678Number dependent eddy viscosity, the mixing of momentum is done through
679the vertical diffusion: after a static adjustment, the Richardson Number is zero
680and thus the eddy viscosity coefficient is at a maximum. When this convective
681adjustment algorithm is used with constant vertical eddy viscosity, spurious
682solutions can occur since the vertical momentum diffusion remains small even
683after a static adjustment. In that case, we recommend the addition of momentum
684mixing in a manner that mimics the mixing in temperature and salinity
685\citep{Speich_PhD92, Speich_al_JPO96}.
686
687% -------------------------------------------------------------------------------------------------------------
688%       Enhanced Vertical Diffusion
689% -------------------------------------------------------------------------------------------------------------
690\subsection   [Enhanced Vertical Diffusion (\np{ln\_zdfevd})]
691         {Enhanced Vertical Diffusion (\np{ln\_zdfevd}=true)}
692\label{ZDF_evd}
693
694%--------------------------------------------namzdf--------------------------------------------------------
695\namdisplay{namzdf}
696%--------------------------------------------------------------------------------------------------------------
697
698Options are defined through the  \ngn{namzdf} namelist variables.
699The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}=true.
700In this case, the vertical eddy mixing coefficients are assigned very large values
701(a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable
702($i.e.$ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative)
703\citep{Lazar_PhD97, Lazar_al_JPO99}. This is done either on tracers only
704(\np{nn\_evdm}=0) or on both momentum and tracers (\np{nn\_evdm}=1).
705
706In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and
707if \np{nn\_evdm}=1, the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ 
708values also, are set equal to the namelist parameter \np{rn\_avevd}. A typical value
709for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$. This parameterisation of
710convective processes is less time consuming than the convective adjustment
711algorithm presented above when mixing both tracers and momentum in the
712case of static instabilities. It requires the use of an implicit time stepping on
713vertical diffusion terms (i.e. \np{ln\_zdfexp}=false).
714
715Note that the stability test is performed on both \textit{before} and \textit{now} 
716values of $N^2$. This removes a potential source of divergence of odd and
717even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \S\ref{STP_mLF}).
718
719% -------------------------------------------------------------------------------------------------------------
720%       Turbulent Closure Scheme
721% -------------------------------------------------------------------------------------------------------------
722\subsection{Turbulent Closure Scheme (\key{zdftke} or \key{zdfgls})}
723\label{ZDF_tcs}
724
725The turbulent closure scheme presented in \S\ref{ZDF_tke} and \S\ref{ZDF_gls} 
726(\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically
727unstable density profiles. In such a case, the term corresponding to the
728destruction of turbulent kinetic energy through stratification in \eqref{Eq_zdftke_e} 
729or \eqref{Eq_zdfgls_e} becomes a source term, since $N^2$ is negative.
730It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also the four neighbouring
731$A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1})$. These large values
732restore the static stability of the water column in a way similar to that of the
733enhanced vertical diffusion parameterisation (\S\ref{ZDF_evd}). However,
734in the vicinity of the sea surface (first ocean layer), the eddy coefficients
735computed by the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$,
736because the mixing length scale is bounded by the distance to the sea surface.
737It can thus be useful to combine the enhanced vertical
738diffusion with the turbulent closure scheme, $i.e.$ setting the \np{ln\_zdfnpc} 
739namelist parameter to true and defining the turbulent closure CPP key all together.
740
741The KPP turbulent closure scheme already includes enhanced vertical diffusion
742in the case of convection, as governed by the variables $bvsqcon$ and $difcon$ 
743found in \mdl{zdfkpp}, therefore \np{ln\_zdfevd}=false should be used with the KPP
744scheme. %gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
745
746% ================================================================
747% Double Diffusion Mixing
748% ================================================================
749\section  [Double Diffusion Mixing (\key{zdfddm})]
750      {Double Diffusion Mixing (\key{zdfddm})}
751\label{ZDF_ddm}
752
753%-------------------------------------------namzdf_ddm-------------------------------------------------
754\namdisplay{namzdf_ddm}
755%--------------------------------------------------------------------------------------------------------------
756
757Options are defined through the  \ngn{namzdf\_ddm} namelist variables.
758Double diffusion occurs when relatively warm, salty water overlies cooler, fresher
759water, or vice versa. The former condition leads to salt fingering and the latter
760to diffusive convection. Double-diffusive phenomena contribute to diapycnal
761mixing in extensive regions of the ocean.  \citet{Merryfield1999} include a
762parameterisation of such phenomena in a global ocean model and show that
763it leads to relatively minor changes in circulation but exerts significant regional
764influences on temperature and salinity. This parameterisation has been
765introduced in \mdl{zdfddm} module and is controlled by the \key{zdfddm} CPP key.
766
767Diapycnal mixing of S and T are described by diapycnal diffusion coefficients
768\begin{align*} % \label{Eq_zdfddm_Kz}
769    &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT}  \\
770    &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
771\end{align*}
772where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
773and $o$ by processes other than double diffusion. The rates of double-diffusive
774mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$,
775where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline
776contraction (see \S\ref{TRA_eos}). To represent mixing of $S$ and $T$ by salt
777fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981):
778\begin{align} \label{Eq_zdfddm_f}
779A_f^{vS} &=    \begin{cases}
780   \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
781   0                              &\text{otherwise} 
782            \end{cases}   
783\\           \label{Eq_zdfddm_f_T}
784A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 
785\end{align}
786
787%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
788\begin{figure}[!t]   \begin{center}
789\includegraphics[width=0.99\textwidth]{./TexFiles/Figures/Fig_zdfddm.pdf}
790\caption{  \label{Fig_zdfddm}
791From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ 
792and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy
793curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves
794$A^{\ast v} = 10^{-4}~m^2.s^{-1}$ ; (b) diapycnal diffusivities $A_d^{vT}$ and
795$A_d^{vS}$ for temperature and salt in regions of diffusive convection. Heavy
796curves denote the Federov parameterisation and thin curves the Kelley
797parameterisation. The latter is not implemented in \NEMO. }
798\end{center}    \end{figure}
799%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
800
801The factor 0.7 in \eqref{Eq_zdfddm_f_T} reflects the measured ratio
802$\alpha F_T /\beta F_S \approx  0.7$ of buoyancy flux of heat to buoyancy
803flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). Following  \citet{Merryfield1999},
804we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
805
806To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by Federov (1988) is used:
807\begin{align}  \label{Eq_zdfddm_d}
808A_d^{vT} &=    \begin{cases}
809   1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
810                           &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
811   0                       &\text{otherwise} 
812            \end{cases}   
813\\          \label{Eq_zdfddm_d_S}
814A_d^{vS} &=    \begin{cases}
815   A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right)
816                           &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
817   A_d^{vT} \ 0.15 \ R_\rho
818                           &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
819   0                       &\text{otherwise} 
820            \end{cases}   
821\end{align}
822
823The dependencies of \eqref{Eq_zdfddm_f} to \eqref{Eq_zdfddm_d_S} on $R_\rho$ 
824are illustrated in Fig.~\ref{Fig_zdfddm}. Implementing this requires computing
825$R_\rho$ at each grid point on every time step. This is done in \mdl{eosbn2} at the
826same time as $N^2$ is computed. This avoids duplication in the computation of
827$\alpha$ and $\beta$ (which is usually quite expensive).
828
829% ================================================================
830% Bottom Friction
831% ================================================================
832\section  [Bottom and top Friction (\textit{zdfbfr})]   {Bottom Friction (\mdl{zdfbfr} module)}
833\label{ZDF_bfr}
834
835%--------------------------------------------nambfr--------------------------------------------------------
836\namdisplay{nambfr}
837%--------------------------------------------------------------------------------------------------------------
838
839Options to defined the top and bottom friction are defined through the  \ngn{nambfr} namelist variables.
840The top friction is activated only if the ice shelf cavities are opened (\np{ln\_isfcav}~=~true).
841As the friction processes at the top and bottom are the same, only the bottom friction is described in details.
842
843Both the surface momentum flux (wind stress) and the bottom momentum
844flux (bottom friction) enter the equations as a condition on the vertical
845diffusive flux. For the bottom boundary layer, one has:
846\begin{equation} \label{Eq_zdfbfr_flux}
847A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
848\end{equation}
849where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum
850outside the logarithmic turbulent boundary layer (thickness of the order of
8511~m in the ocean). How ${\cal F}_h^{\textbf U}$ influences the interior depends on the
852vertical resolution of the model near the bottom relative to the Ekman layer
853depth. For example, in order to obtain an Ekman layer depth
854$d = \sqrt{2\;A^{vm}} / f = 50$~m, one needs a vertical diffusion coefficient
855$A^{vm} = 0.125$~m$^2$s$^{-1}$ (for a Coriolis frequency
856$f = 10^{-4}$~m$^2$s$^{-1}$). With a background diffusion coefficient
857$A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
858When the vertical mixing coefficient is this small, using a flux condition is
859equivalent to entering the viscous forces (either wind stress or bottom friction)
860as a body force over the depth of the top or bottom model layer. To illustrate
861this, consider the equation for $u$ at $k$, the last ocean level:
862\begin{equation} \label{Eq_zdfbfr_flux2}
863\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}}
864\end{equation}
865If the bottom layer thickness is 200~m, the Ekman transport will
866be distributed over that depth. On the other hand, if the vertical resolution
867is high (1~m or less) and a turbulent closure model is used, the turbulent
868Ekman layer will be represented explicitly by the model. However, the
869logarithmic layer is never represented in current primitive equation model
870applications: it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
871Two choices are available in \NEMO: a linear and a quadratic bottom friction.
872Note that in both cases, the rotation between the interior velocity and the
873bottom friction is neglected in the present release of \NEMO.
874
875In the code, the bottom friction is imposed by adding the trend due to the bottom
876friction to the general momentum trend in \mdl{dynbfr}. For the time-split surface
877pressure gradient algorithm, the momentum trend due to the barotropic component
878needs to be handled separately. For this purpose it is convenient to compute and
879store coefficients which can be simply combined with bottom velocities and geometric
880values to provide the momentum trend due to bottom friction.
881These coefficients are computed in \mdl{zdfbfr} and generally take the form
882$c_b^{\textbf U}$ where:
883\begin{equation} \label{Eq_zdfbfr_bdef}
884\frac{\partial {\textbf U_h}}{\partial t} =
885  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b
886\end{equation}
887where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity.
888
889% -------------------------------------------------------------------------------------------------------------
890%       Linear Bottom Friction
891% -------------------------------------------------------------------------------------------------------------
892\subsection{Linear Bottom Friction (\np{nn\_botfr} = 0 or 1) }
893\label{ZDF_bfr_linear}
894
895The linear bottom friction parameterisation (including the special case
896of a free-slip condition) assumes that the bottom friction
897is proportional to the interior velocity (i.e. the velocity of the last
898model level):
899\begin{equation} \label{Eq_zdfbfr_linear}
900{\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
901\end{equation}
902where $r$ is a friction coefficient expressed in ms$^{-1}$.
903This coefficient is generally estimated by setting a typical decay time
904$\tau$ in the deep ocean,
905and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted
906values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}.
907A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used
908in quasi-geostrophic models. One may consider the linear friction as an
909approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982},
910Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed
911of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, and assuming an ocean depth
912$H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.
913This is the default value used in \NEMO. It corresponds to a decay time scale
914of 115~days. It can be changed by specifying \np{rn\_bfric1} (namelist parameter).
915
916For the linear friction case the coefficients defined in the general
917expression \eqref{Eq_zdfbfr_bdef} are:
918\begin{equation} \label{Eq_zdfbfr_linbfr_b}
919\begin{split}
920 c_b^u &= - r\\
921 c_b^v &= - r\\
922\end{split}
923\end{equation}
924When \np{nn\_botfr}=1, the value of $r$ used is \np{rn\_bfric1}.
925Setting \np{nn\_botfr}=0 is equivalent to setting $r=0$ and leads to a free-slip
926bottom boundary condition. These values are assigned in \mdl{zdfbfr}.
927From v3.2 onwards there is support for local enhancement of these values
928via an externally defined 2D mask array (\np{ln\_bfr2d}=true) given
929in the \ifile{bfr\_coef} input NetCDF file. The mask values should vary from 0 to 1.
930Locations with a non-zero mask value will have the friction coefficient increased
931by $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfric1}.
932
933% -------------------------------------------------------------------------------------------------------------
934%       Non-Linear Bottom Friction
935% -------------------------------------------------------------------------------------------------------------
936\subsection{Non-Linear Bottom Friction (\np{nn\_botfr} = 2)}
937\label{ZDF_bfr_nonlinear}
938
939The non-linear bottom friction parameterisation assumes that the bottom
940friction is quadratic:
941\begin{equation} \label{Eq_zdfbfr_nonlinear}
942{\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
943}{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
944\end{equation}
945where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy
946due to tides, internal waves breaking and other short time scale currents.
947A typical value of the drag coefficient is $C_D = 10^{-3} $. As an example,
948the CME experiment \citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and
949$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992} 
950uses $C_D = 1.4\;10^{-3}$ and $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.
951The CME choices have been set as default values (\np{rn\_bfric2} and \np{rn\_bfeb2} 
952namelist parameters).
953
954As for the linear case, the bottom friction is imposed in the code by
955adding the trend due to the bottom friction to the general momentum trend
956in \mdl{dynbfr}.
957For the non-linear friction case the terms
958computed in \mdl{zdfbfr}  are:
959\begin{equation} \label{Eq_zdfbfr_nonlinbfr}
960\begin{split}
961 c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\
962 c_b^v &= - \; C_D\;\left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\
963\end{split}
964\end{equation}
965
966The coefficients that control the strength of the non-linear bottom friction are
967initialised as namelist parameters: $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}.
968Note for applications which treat tides explicitly a low or even zero value of
969\np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$ 
970is possible via an externally defined 2D mask array (\np{ln\_bfr2d}=true).
971See previous section for details.
972
973% -------------------------------------------------------------------------------------------------------------
974%       Bottom Friction stability
975% -------------------------------------------------------------------------------------------------------------
976\subsection{Bottom Friction stability considerations}
977\label{ZDF_bfr_stability}
978
979Some care needs to exercised over the choice of parameters to ensure that the
980implementation of bottom friction does not induce numerical instability. For
981the purposes of stability analysis, an approximation to \eqref{Eq_zdfbfr_flux2}
982is:
983\begin{equation} \label{Eqn_bfrstab}
984\begin{split}
985 \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
986               &= -\frac{ru}{e_{3u}}\;2\rdt\\
987\end{split}
988\end{equation}
989\noindent where linear bottom friction and a leapfrog timestep have been assumed.
990To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have:
991\begin{equation}
992 |\Delta u| < \;|u|
993\end{equation}
994\noindent which, using \eqref{Eqn_bfrstab}, gives:
995\begin{equation}
996r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
997\end{equation}
998This same inequality can also be derived in the non-linear bottom friction case
999if a velocity of 1 m.s$^{-1}$ is assumed. Alternatively, this criterion can be
1000rearranged to suggest a minimum bottom box thickness to ensure stability:
1001\begin{equation}
1002e_{3u} > 2\;r\;\rdt
1003\end{equation}
1004\noindent which it may be necessary to impose if partial steps are being used.
1005For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then
1006$e_{3u}$ should be greater than 3.6 m. For most applications, with physically
1007sensible parameters these restrictions should not be of concern. But
1008caution may be necessary if attempts are made to locally enhance the bottom
1009friction parameters.
1010To ensure stability limits are imposed on the bottom friction coefficients both during
1011initialisation and at each time step. Checks at initialisation are made in \mdl{zdfbfr} 
1012(assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
1013The number of breaches of the stability criterion are reported as well as the minimum
1014and maximum values that have been set. The criterion is also checked at each time step,
1015using the actual velocity, in \mdl{dynbfr}. Values of the bottom friction coefficient are
1016reduced as necessary to ensure stability; these changes are not reported.
1017
1018Limits on the bottom friction coefficient are not imposed if the user has elected to
1019handle the bottom friction implicitly (see \S\ref{ZDF_bfr_imp}). The number of potential
1020breaches of the explicit stability criterion are still reported for information purposes.
1021
1022% -------------------------------------------------------------------------------------------------------------
1023%       Implicit Bottom Friction
1024% -------------------------------------------------------------------------------------------------------------
1025\subsection{Implicit Bottom Friction (\np{ln\_bfrimp}$=$\textit{T})}
1026\label{ZDF_bfr_imp}
1027
1028An optional implicit form of bottom friction has been implemented to improve
1029model stability. We recommend this option for shelf sea and coastal ocean applications, especially
1030for split-explicit time splitting. This option can be invoked by setting \np{ln\_bfrimp} 
1031to \textit{true} in the \textit{nambfr} namelist. This option requires \np{ln\_zdfexp} to be \textit{false} 
1032in the \textit{namzdf} namelist.
1033
1034This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, the
1035bottom boundary condition is implemented implicitly.
1036
1037\begin{equation} \label{Eq_dynzdf_bfr}
1038\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk}
1039    = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}}
1040\end{equation}
1041
1042where $mbk$ is the layer number of the bottom wet layer. superscript $n+1$ means the velocity used in the
1043friction formula is to be calculated, so, it is implicit.
1044
1045If split-explicit time splitting is used, care must be taken to avoid the double counting of
1046the bottom friction in the 2-D barotropic momentum equations. As NEMO only updates the barotropic
1047pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, we need to remove
1048the bottom friction induced by these two terms which has been included in the 3-D momentum trend
1049and update it with the latest value. On the other hand, the bottom friction contributed by the
1050other terms (e.g. the advection term, viscosity term) has been included in the 3-D momentum equations
1051and should not be added in the 2-D barotropic mode.
1052
1053The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the
1054following:
1055
1056\begin{equation} \label{Eq_dynspg_ts_bfr1}
1057\frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b}
1058\left(\textbf{U}_{med}-\textbf{U}^{m-1}\right)
1059\end{equation}
1060\begin{equation} \label{Eq_dynspg_ts_bfr2}
1061\frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+
1062\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)-
10632\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right)
1064\end{equation}
1065
1066where $\textbf{T}$ is the vertical integrated 3-D momentum trend. We assume the leap-frog time-stepping
1067is used here. $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step.
1068 $c_{b}$ is the friction coefficient. $\eta$ is the sea surface level calculated in the barotropic loops
1069while $\eta^{'}$ is the sea surface level used in the 3-D baroclinic mode. $\textbf{u}_{b}$ is the bottom
1070layer horizontal velocity.
1071
1072
1073
1074
1075% -------------------------------------------------------------------------------------------------------------
1076%       Bottom Friction with split-explicit time splitting
1077% -------------------------------------------------------------------------------------------------------------
1078\subsection{Bottom Friction with split-explicit time splitting (\np{ln\_bfrimp}$=$\textit{F})}
1079\label{ZDF_bfr_ts}
1080
1081When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the
1082bottom velocity at the before time step is used. This velocity includes both the
1083baroclinic and barotropic components which is appropriate when using either the
1084explicit or filtered surface pressure gradient algorithms (\key{dynspg\_exp} or
1085{\key{dynspg\_flt}). Extra attention is required, however, when using
1086split-explicit time stepping (\key{dynspg\_ts}). In this case the free surface
1087equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, while the three
1088dimensional prognostic variables are solved with the longer time step
1089of \np{rn\_rdt} seconds. The trend in the barotropic momentum due to bottom
1090friction appropriate to this method is that given by the selected parameterisation
1091($i.e.$ linear or non-linear bottom friction) computed with the evolving velocities
1092at each barotropic timestep.
1093
1094In the case of non-linear bottom friction, we have elected to partially linearise
1095the problem by keeping the coefficients fixed throughout the barotropic
1096time-stepping to those computed in \mdl{zdfbfr} using the now timestep.
1097This decision allows an efficient use of the $c_b^{\vect{U}}$ coefficients to:
1098
1099\begin{enumerate}
1100\item On entry to \rou{dyn\_spg\_ts}, remove the contribution of the before
1101barotropic velocity to the bottom friction component of the vertically
1102integrated momentum trend. Note the same stability check that is carried out
1103on the bottom friction coefficient in \mdl{dynbfr} has to be applied here to
1104ensure that the trend removed matches that which was added in \mdl{dynbfr}.
1105\item At each barotropic step, compute the contribution of the current barotropic
1106velocity to the trend due to bottom friction. Add this contribution to the
1107vertically integrated momentum trend. This contribution is handled implicitly which
1108eliminates the need to impose a stability criteria on the values of the bottom friction
1109coefficient within the barotropic loop.
1110\end{enumerate}
1111
1112Note that the use of an implicit formulation within the barotropic loop
1113for the bottom friction trend means that any limiting of the bottom friction coefficient
1114in \mdl{dynbfr} does not adversely affect the solution when using split-explicit time
1115splitting. This is because the major contribution to bottom friction is likely to come from
1116the barotropic component which uses the unrestricted value of the coefficient. However, if the
1117limiting is thought to be having a major effect (a more likely prospect in coastal and shelf seas
1118applications) then the fully implicit form of the bottom friction should be used (see \S\ref{ZDF_bfr_imp} )
1119which can be selected by setting \np{ln\_bfrimp} $=$ \textit{true}.
1120
1121Otherwise, the implicit formulation takes the form:
1122\begin{equation} \label{Eq_zdfbfr_implicitts}
1123 \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ] 
1124\end{equation}
1125where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height),
1126$c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and $RHS$ represents
1127all the components to the vertically integrated momentum trend except for that due to bottom friction.
1128
1129
1130
1131
1132% ================================================================
1133% Tidal Mixing
1134% ================================================================
1135\section{Tidal Mixing (\key{zdftmx})}
1136\label{ZDF_tmx}
1137
1138%--------------------------------------------namzdf_tmx--------------------------------------------------
1139\namdisplay{namzdf_tmx}
1140%--------------------------------------------------------------------------------------------------------------
1141
1142
1143% -------------------------------------------------------------------------------------------------------------
1144%        Bottom intensified tidal mixing
1145% -------------------------------------------------------------------------------------------------------------
1146\subsection{Bottom intensified tidal mixing}
1147\label{ZDF_tmx_bottom}
1148
1149Options are defined through the  \ngn{namzdf\_tmx} namelist variables.
1150The parameterization of tidal mixing follows the general formulation for
1151the vertical eddy diffusivity proposed by \citet{St_Laurent_al_GRL02} and
1152first introduced in an OGCM by \citep{Simmons_al_OM04}.
1153In this formulation an additional vertical diffusivity resulting from internal tide breaking,
1154$A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, the energy transfer from barotropic
1155tides to baroclinic tides :
1156\begin{equation} \label{Eq_Ktides}
1157A^{vT}_{tides} =  q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 }
1158\end{equation}
1159where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency
1160(see \S\ref{TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency,
1161and $F(z)$ the vertical structure function.
1162
1163The mixing efficiency of turbulence is set by $\Gamma$ (\np{rn\_me} namelist parameter)
1164and is usually taken to be the canonical value of $\Gamma = 0.2$ (Osborn 1980).
1165The tidal dissipation efficiency is given by the parameter $q$ (\np{rn\_tfe} namelist parameter)
1166represents the part of the internal wave energy flux $E(x, y)$ that is dissipated locally,
1167with the remaining $1-q$ radiating away as low mode internal waves and
1168contributing to the background internal wave field. A value of $q=1/3$ is typically used 
1169\citet{St_Laurent_al_GRL02}.
1170The vertical structure function $F(z)$ models the distribution of the turbulent mixing in the vertical.
1171It is implemented as a simple exponential decaying upward away from the bottom,
1172with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04},
1173\begin{equation} \label{Eq_Fz}
1174F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) }
1175\end{equation}
1176and is normalized so that vertical integral over the water column is unity.
1177
1178The associated vertical viscosity is calculated from the vertical
1179diffusivity assuming a Prandtl number of 1, $i.e.$ $A^{vm}_{tides}=A^{vT}_{tides}$.
1180In the limit of $N \rightarrow 0$ (or becoming negative), the vertical diffusivity
1181is capped at $300\,cm^2/s$ and impose a lower limit on $N^2$ of \np{rn\_n2min} 
1182usually set to $10^{-8} s^{-2}$. These bounds are usually rarely encountered.
1183
1184The internal wave energy map, $E(x, y)$ in \eqref{Eq_Ktides}, is derived
1185from a barotropic model of the tides utilizing a parameterization of the
1186conversion of barotropic tidal energy into internal waves.
1187The essential goal of the parameterization is to represent the momentum
1188exchange between the barotropic tides and the unrepresented internal waves
1189induced by the tidal flow over rough topography in a stratified ocean.
1190In the current version of \NEMO, the map is built from the output of
1191the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}.
1192This model provides the dissipation associated with internal wave energy for the M2 and K1
1193tides component (Fig.~\ref{Fig_ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated
1194as being $1/4$ of the M2 one. The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$.
1195Its global mean value is $1.1$ TW, in agreement with independent estimates
1196\citep{Egbert_Ray_Nat00, Egbert_Ray_JGR01}.
1197
1198%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1199\begin{figure}[!t]   \begin{center}
1200\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_ZDF_M2_K1_tmx.pdf}
1201\caption{  \label{Fig_ZDF_M2_K1_tmx} 
1202(a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). }
1203\end{center}   \end{figure}
1204%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1205 
1206% -------------------------------------------------------------------------------------------------------------
1207%        Indonesian area specific treatment
1208% -------------------------------------------------------------------------------------------------------------
1209\subsection{Indonesian area specific treatment (\np{ln\_zdftmx\_itf})}
1210\label{ZDF_tmx_itf}
1211
1212When the Indonesian Through Flow (ITF) area is included in the model domain,
1213a specific treatment of tidal induced mixing in this area can be used.
1214It is activated through the namelist logical \np{ln\_tmx\_itf}, and the user must provide
1215an input NetCDF file, \ifile{mask\_itf}, which contains a mask array defining the ITF area
1216where the specific treatment is applied.
1217
1218When \np{ln\_tmx\_itf}=true, the two key parameters $q$ and $F(z)$ are adjusted following
1219the parameterisation developed by \citet{Koch-Larrouy_al_GRL07}:
1220
1221First, the Indonesian archipelago is a complex geographic region
1222with a series of large, deep, semi-enclosed basins connected via
1223numerous narrow straits. Once generated, internal tides remain
1224confined within this semi-enclosed area and hardly radiate away.
1225Therefore all the internal tides energy is consumed within this area.
1226So it is assumed that $q = 1$, $i.e.$ all the energy generated is available for mixing.
1227Note that for test purposed, the ITF tidal dissipation efficiency is a
1228namelist parameter (\np{rn\_tfe\_itf}). A value of $1$ or close to is
1229this recommended for this parameter.
1230
1231Second, the vertical structure function, $F(z)$, is no more associated
1232with a bottom intensification of the mixing, but with a maximum of
1233energy available within the thermocline. \citet{Koch-Larrouy_al_GRL07} 
1234have suggested that the vertical distribution of the energy dissipation
1235proportional to $N^2$ below the core of the thermocline and to $N$ above.
1236The resulting $F(z)$ is:
1237\begin{equation} \label{Eq_Fz_itf}
1238F(i,j,k) \sim     \left\{ \begin{aligned}
1239\frac{q\,\Gamma E(i,j) } {\rho N \, \int N     dz}    \qquad \text{when $\partial_z N < 0$} \\
1240\frac{q\,\Gamma E(i,j) } {\rho     \, \int N^2 dz}    \qquad \text{when $\partial_z N > 0$}
1241                      \end{aligned} \right.
1242\end{equation}
1243
1244Averaged over the ITF area, the resulting tidal mixing coefficient is $1.5\,cm^2/s$,
1245which agrees with the independent estimates inferred from observations.
1246Introduced in a regional OGCM, the parameterization improves the water mass
1247characteristics in the different Indonesian seas, suggesting that the horizontal
1248and vertical distributions of the mixing are adequately prescribed
1249\citep{Koch-Larrouy_al_GRL07, Koch-Larrouy_al_OD08a, Koch-Larrouy_al_OD08b}.
1250Note also that such a parameterisation has a significant impact on the behaviour
1251of global coupled GCMs \citep{Koch-Larrouy_al_CD10}.
1252
1253
1254% ================================================================
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