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Last change on this file since 11582 was 11582, checked in by nicolasmartin, 5 years ago

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1\documentclass[../main/NEMO_manual]{subfiles}
2
3%% Custom aliases
4\newcommand{\cf}{\ensuremath{C\kern-0.14em f}}
5
6\onlyinsubfile{\makeindex}
7
8\begin{document}
9% ================================================================
10% Chapter  Vertical Ocean Physics (ZDF)
11% ================================================================
12\chapter{Vertical Ocean Physics (ZDF)}
13\label{chap:ZDF}
14
15\chaptertoc
16
17%gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN.
18
19\newpage
20
21% ================================================================
22% Vertical Mixing
23% ================================================================
24\section{Vertical mixing}
25\label{sec:ZDF}
26
27The discrete form of the ocean subgrid scale physics has been presented in
28\autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}.
29At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt have to be defined.
30At the surface they are prescribed from the surface forcing (see \autoref{chap:SBC}),
31while at the bottom they are set to zero for heat and salt,
32unless a geothermal flux forcing is prescribed as a bottom boundary condition (\ie\ \np{ln_trabbc}{ln\_trabbc} defined,
33see \autoref{subsec:TRA_bbc}), and specified through a bottom friction parameterisation for momentum
34(see \autoref{sec:ZDF_drg}).
35
36In this section we briefly discuss the various choices offered to compute the vertical eddy viscosity and
37diffusivity coefficients, $A_u^{vm}$ , $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- points,
38respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}).
39These coefficients can be assumed to be either constant, or a function of the local Richardson number,
40or computed from a turbulent closure model (either TKE or GLS or OSMOSIS formulation).
41The computation of these coefficients is initialized in the \mdl{zdfphy} module and performed in
42the \mdl{zdfric}, \mdl{zdftke} or \mdl{zdfgls} or \mdl{zdfosm} modules.
43The trends due to the vertical momentum and tracer diffusion, including the surface forcing,
44are computed and added to the general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively.
45%These trends can be computed using either a forward time stepping scheme
46%(namelist parameter \np[=.true.]{ln_zdfexp}{ln\_zdfexp}) or a backward time stepping scheme
47%(\np[=.false.]{ln_zdfexp}{ln\_zdfexp}) depending on the magnitude of the mixing coefficients,
48%and thus of the formulation used (see \autoref{chap:TD}).
49
50%--------------------------------------------namzdf--------------------------------------------------------
51
52\begin{listing}
53  \nlst{namzdf}
54  \caption{\forcode{&namzdf}}
55  \label{lst:namzdf}
56\end{listing}
57%--------------------------------------------------------------------------------------------------------------
58
59% -------------------------------------------------------------------------------------------------------------
60%        Constant
61% -------------------------------------------------------------------------------------------------------------
62\subsection[Constant (\forcode{ln_zdfcst})]{Constant (\protect\np{ln_zdfcst}{ln\_zdfcst})}
63\label{subsec:ZDF_cst}
64
65Options are defined through the \nam{zdf}{zdf} namelist variables.
66When \np{ln_zdfcst}{ln\_zdfcst} is defined, the momentum and tracer vertical eddy coefficients are set to
67constant values over the whole ocean.
68This is the crudest way to define the vertical ocean physics.
69It is recommended to use this option only in process studies, not in basin scale simulations.
70Typical values used in this case are:
71\begin{align*}
72  A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}   \\
73  A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1}
74\end{align*}
75
76These values are set through the \np{rn_avm0}{rn\_avm0} and \np{rn_avt0}{rn\_avt0} namelist parameters.
77In all cases, do not use values smaller that those associated with the molecular viscosity and diffusivity,
78that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and
79$\sim10^{-9}~m^2.s^{-1}$ for salinity.
80
81% -------------------------------------------------------------------------------------------------------------
82%        Richardson Number Dependent
83% -------------------------------------------------------------------------------------------------------------
84\subsection[Richardson number dependent (\forcode{ln_zdfric})]{Richardson number dependent (\protect\np{ln_zdfric}{ln\_zdfric})}
85\label{subsec:ZDF_ric}
86
87%--------------------------------------------namric---------------------------------------------------------
88
89\begin{listing}
90  \nlst{namzdf_ric}
91  \caption{\forcode{&namzdf_ric}}
92  \label{lst:namzdf_ric}
93\end{listing}
94%--------------------------------------------------------------------------------------------------------------
95
96When \np[=.true.]{ln_zdfric}{ln\_zdfric}, a local Richardson number dependent formulation for the vertical momentum and
97tracer eddy coefficients is set through the \nam{zdf_ric}{zdf\_ric} namelist variables.
98The vertical mixing coefficients are diagnosed from the large scale variables computed by the model.
99\textit{In situ} measurements have been used to link vertical turbulent activity to large scale ocean structures.
100The hypothesis of a mixing mainly maintained by the growth of Kelvin-Helmholtz like instabilities leads to
101a dependency between the vertical eddy coefficients and the local Richardson number
102(\ie\ the ratio of stratification to vertical shear).
103Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented:
104\[
105  % \label{eq:ZDF_ric}
106  \left\{
107    \begin{aligned}
108      A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\
109      A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm}
110    \end{aligned}
111  \right.
112\]
113where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number,
114$N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
115$A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the constant case
116(see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ is the maximum value that
117can be reached by the coefficient when $Ri\leq 0$, $a=5$ and $n=2$.
118The last three values can be modified by setting the \np{rn_avmri}{rn\_avmri}, \np{rn_alp}{rn\_alp} and
119\np{nn_ric}{nn\_ric} namelist parameters, respectively.
120
121A simple mixing-layer model to transfer and dissipate the atmospheric forcings
122(wind-stress and buoyancy fluxes) can be activated setting the \np[=.true.]{ln_mldw}{ln\_mldw} in the namelist.
123
124In this case, the local depth of turbulent wind-mixing or "Ekman depth" $h_{e}(x,y,t)$ is evaluated and
125the vertical eddy coefficients prescribed within this layer.
126
127This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation:
128\[
129  h_{e} = Ek \frac {u^{*}} {f_{0}}
130\]
131where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter.
132
133In this similarity height relationship, the turbulent friction velocity:
134\[
135  u^{*} = \sqrt \frac {|\tau|} {\rho_o}
136\]
137is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$.
138The final $h_{e}$ is further constrained by the adjustable bounds \np{rn_mldmin}{rn\_mldmin} and \np{rn_mldmax}{rn\_mldmax}.
139Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to
140the empirical values \np{rn_wtmix}{rn\_wtmix} and \np{rn_wvmix}{rn\_wvmix} \citep{lermusiaux_JMS01}.
141
142% -------------------------------------------------------------------------------------------------------------
143%        TKE Turbulent Closure Scheme
144% -------------------------------------------------------------------------------------------------------------
145\subsection[TKE turbulent closure scheme (\forcode{ln_zdftke})]{TKE turbulent closure scheme (\protect\np{ln_zdftke}{ln\_zdftke})}
146\label{subsec:ZDF_tke}
147%--------------------------------------------namzdf_tke--------------------------------------------------
148
149\begin{listing}
150  \nlst{namzdf_tke}
151  \caption{\forcode{&namzdf_tke}}
152  \label{lst:namzdf_tke}
153\end{listing}
154%--------------------------------------------------------------------------------------------------------------
155
156The vertical eddy viscosity and diffusivity coefficients are computed from a TKE turbulent closure model based on
157a prognostic equation for $\bar{e}$, the turbulent kinetic energy,
158and a closure assumption for the turbulent length scales.
159This turbulent closure model has been developed by \citet{bougeault.lacarrere_MWR89} in the atmospheric case,
160adapted by \citet{gaspar.gregoris.ea_JGR90} for the oceanic case, and embedded in OPA, the ancestor of \NEMO,
161by \citet{blanke.delecluse_JPO93} for equatorial Atlantic simulations.
162Since then, significant modifications have been introduced by \citet{madec.delecluse.ea_NPM98} in both the implementation and
163the formulation of the mixing length scale.
164The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear,
165its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type:
166\begin{equation}
167  \label{eq:ZDF_tke_e}
168  \frac{\partial \bar{e}}{\partial t} =
169  \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
170      +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
171  -K_\rho\,N^2
172  +\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
173      \;\frac{\partial \bar{e}}{\partial k}} \right]
174  - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
175\end{equation}
176\[
177  % \label{eq:ZDF_tke_kz}
178  \begin{split}
179    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }    \\
180    K_\rho &= A^{vm} / P_{rt}
181  \end{split}
182\]
183where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),
184$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,
185$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity and diffusivity coefficients.
186The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$ $\approx 0.7$ are designed to deal with
187vertical mixing at any depth \citep{gaspar.gregoris.ea_JGR90}.
188They are set through namelist parameters \np{nn_ediff}{nn\_ediff} and \np{nn_ediss}{nn\_ediss}.
189$P_{rt}$ can be set to unity or, following \citet{blanke.delecluse_JPO93}, be a function of the local Richardson number, $R_i$:
190\begin{align*}
191  % \label{eq:ZDF_prt}
192  P_{rt} =
193  \begin{cases}
194    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$}   \\
195    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$}   \\
196    \ \ 10 &      \text{if $\ 2 \leq R_i$}
197  \end{cases}
198\end{align*}
199The choice of $P_{rt}$ is controlled by the \np{nn_pdl}{nn\_pdl} namelist variable.
200
201At the sea surface, the value of $\bar{e}$ is prescribed from the wind stress field as
202$\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn_ebb}{rn\_ebb} namelist parameter.
203The default value of $e_{bb}$ is 3.75. \citep{gaspar.gregoris.ea_JGR90}), however a much larger value can be used when
204taking into account the surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}).
205The bottom value of TKE is assumed to be equal to the value of the level just above.
206The time integration of the $\bar{e}$ equation may formally lead to negative values because
207the numerical scheme does not ensure its positivity.
208To overcome this problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn_emin}{rn\_emin} namelist parameter).
209Following \citet{gaspar.gregoris.ea_JGR90}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$.
210This allows the subsequent formulations to match that of \citet{gargett_JMR84} for the diffusion in
211the thermocline and deep ocean :  $K_\rho = 10^{-3} / N$.
212In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical instabilities associated with
213too weak vertical diffusion.
214They must be specified at least larger than the molecular values, and are set through \np{rn_avm0}{rn\_avm0} and
215\np{rn_avt0}{rn\_avt0} (\nam{zdf}{zdf} namelist, see \autoref{subsec:ZDF_cst}).
216
217\subsubsection{Turbulent length scale}
218
219For computational efficiency, the original formulation of the turbulent length scales proposed by
220\citet{gaspar.gregoris.ea_JGR90} has been simplified.
221Four formulations are proposed, the choice of which is controlled by the \np{nn_mxl}{nn\_mxl} namelist parameter.
222The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}:
223\begin{equation}
224  \label{eq:ZDF_tke_mxl0_1}
225  l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N
226\end{equation}
227which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency.
228The resulting length scale is bounded by the distance to the surface or to the bottom
229(\np[=0]{nn_mxl}{nn\_mxl}) or by the local vertical scale factor (\np[=1]{nn_mxl}{nn\_mxl}).
230\citet{blanke.delecluse_JPO93} notice that this simplification has two major drawbacks:
231it makes no sense for locally unstable stratification and the computation no longer uses all
232the information contained in the vertical density profile.
233To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np[=2, 3]{nn_mxl}{nn\_mxl} cases,
234which add an extra assumption concerning the vertical gradient of the computed length scale.
235So, the length scales are first evaluated as in \autoref{eq:ZDF_tke_mxl0_1} and then bounded such that:
236\begin{equation}
237  \label{eq:ZDF_tke_mxl_constraint}
238  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
239  \qquad \text{with }\  l =  l_k = l_\epsilon
240\end{equation}
241\autoref{eq:ZDF_tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than
242the variations of depth.
243It provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less
244time consuming.
245In particular, it allows the length scale to be limited not only by the distance to the surface or
246to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as
247the thermocline (\autoref{fig:ZDF_mixing_length}).
248In order to impose the \autoref{eq:ZDF_tke_mxl_constraint} constraint, we introduce two additional length scales:
249$l_{up}$ and $l_{dwn}$, the upward and downward length scales, and
250evaluate the dissipation and mixing length scales as
251(and note that here we use numerical indexing):
252%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
253\begin{figure}[!t]
254  \centering
255  \includegraphics[width=0.66\textwidth]{Fig_mixing_length}
256  \caption[Mixing length computation]{Illustration of the mixing length computation}
257  \label{fig:ZDF_mixing_length}
258\end{figure}
259%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
260\[
261  % \label{eq:ZDF_tke_mxl2}
262  \begin{aligned}
263    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
264    \quad &\text{ from $k=1$ to $jpk$ }\ \\
265    l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)\right)
266    \quad &\text{ from $k=jpk$ to $1$ }\ \\
267  \end{aligned}
268\]
269where $l^{(k)}$ is computed using \autoref{eq:ZDF_tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
270
271In the \np[=2]{nn_mxl}{nn\_mxl} case, the dissipation and mixing length scales take the same value:
272$ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the \np[=3]{nn_mxl}{nn\_mxl} case,
273the dissipation and mixing turbulent length scales are give as in \citet{gaspar.gregoris.ea_JGR90}:
274\[
275  % \label{eq:ZDF_tke_mxl_gaspar}
276  \begin{aligned}
277    & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }   \\
278    & l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)
279  \end{aligned}
280\]
281
282At the ocean surface, a non zero length scale is set through the  \np{rn_mxl0}{rn\_mxl0} namelist parameter.
283Usually the surface scale is given by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant and
284$z_o$ the roughness parameter of the surface.
285Assuming $z_o=0.1$~m \citep{craig.banner_JPO94} leads to a 0.04~m, the default value of \np{rn_mxl0}{rn\_mxl0}.
286In the ocean interior a minimum length scale is set to recover the molecular viscosity when
287$\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).
288
289\subsubsection{Surface wave breaking parameterization}
290%-----------------------------------------------------------------------%
291
292Following \citet{mellor.blumberg_JPO04}, the TKE turbulence closure model has been modified to
293include the effect of surface wave breaking energetics.
294This results in a reduction of summertime surface temperature when the mixed layer is relatively shallow.
295The \citet{mellor.blumberg_JPO04} modifications acts on surface length scale and TKE values and
296air-sea drag coefficient.
297The latter concerns the bulk formulae and is not discussed here.
298
299Following \citet{craig.banner_JPO94}, the boundary condition on surface TKE value is :
300\begin{equation}
301  \label{eq:ZDF_Esbc}
302  \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
303\end{equation}
304where $\alpha_{CB}$ is the \citet{craig.banner_JPO94} constant of proportionality which depends on the ''wave age'',
305ranging from 57 for mature waves to 146 for younger waves \citep{mellor.blumberg_JPO04}.
306The boundary condition on the turbulent length scale follows the Charnock's relation:
307\begin{equation}
308  \label{eq:ZDF_Lsbc}
309  l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o}
310\end{equation}
311where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant.
312\citet{mellor.blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by
313\citet{stacey_JPO99} citing observation evidence, and
314$\alpha_{CB} = 100$ the Craig and Banner's value.
315As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$,
316with $e_{bb}$ the \np{rn_ebb}{rn\_ebb} namelist parameter, setting \np[=67.83]{rn_ebb}{rn\_ebb} corresponds
317to $\alpha_{CB} = 100$.
318Further setting  \np[=.true.]{ln_mxl0}{ln\_mxl0},  applies \autoref{eq:ZDF_Lsbc} as the surface boundary condition on the length scale,
319with $\beta$ hard coded to the Stacey's value.
320Note that a minimal threshold of \np{rn_emin0}{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) is applied on the
321surface $\bar{e}$ value.
322
323\subsubsection{Langmuir cells}
324%--------------------------------------%
325
326Langmuir circulations (LC) can be described as ordered large-scale vertical motions in
327the surface layer of the oceans.
328Although LC have nothing to do with convection, the circulation pattern is rather similar to
329so-called convective rolls in the atmospheric boundary layer.
330The detailed physics behind LC is described in, for example, \citet{craik.leibovich_JFM76}.
331The prevailing explanation is that LC arise from a nonlinear interaction between the Stokes drift and
332wind drift currents.
333
334Here we introduced in the TKE turbulent closure the simple parameterization of Langmuir circulations proposed by
335\citep{axell_JGR02} for a $k-\epsilon$ turbulent closure.
336The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in
337an extra source term of TKE, $P_{LC}$.
338The presence of $P_{LC}$ in \autoref{eq:ZDF_tke_e}, the TKE equation, is controlled by setting \np{ln_lc}{ln\_lc} to
339\forcode{.true.} in the \nam{zdf_tke}{zdf\_tke} namelist.
340
341By making an analogy with the characteristic convective velocity scale (\eg, \citet{dalessio.abdella.ea_JPO98}),
342$P_{LC}$ is assumed to be :
343\[
344P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}}
345\]
346where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth.
347With no information about the wave field, $w_{LC}$ is assumed to be proportional to
348the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module
349\footnote{Following \citet{li.garrett_JMR93}, the surface Stoke drift velocity may be expressed as
350  $u_s =  0.016 \,|U_{10m}|$.
351  Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of
352  $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress
353}.
354For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at
355a finite depth $H_{LC}$ (which is often close to the mixed layer depth),
356and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures).
357The resulting expression for $w_{LC}$ is :
358\[
359  w_{LC}  =
360  \begin{cases}
361    c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$}    \\
362    0                             &      \text{otherwise}
363  \end{cases}
364\]
365where $c_{LC} = 0.15$ has been chosen by \citep{axell_JGR02} as a good compromise to fit LES data.
366The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second.
367The value of $c_{LC}$ is set through the \np{rn_lc}{rn\_lc} namelist parameter,
368having in mind that it should stay between 0.15 and 0.54 \citep{axell_JGR02}.
369
370The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations:
371$H_{LC}$ is the depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by
372converting its kinetic energy to potential energy, according to
373\[
374- \int_{-H_{LC}}^0 { N^2\;\;dz} = \frac{1}{2} u_s^2
375\]
376
377\subsubsection{Mixing just below the mixed layer}
378%--------------------------------------------------------------%
379
380Vertical mixing parameterizations commonly used in ocean general circulation models tend to
381produce mixed-layer depths that are too shallow during summer months and windy conditions.
382This bias is particularly acute over the Southern Ocean.
383To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{rodgers.aumont.ea_B14}.
384The parameterization is an empirical one, \ie\ not derived from theoretical considerations,
385but rather is meant to account for observed processes that affect the density structure of
386the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme
387(\ie\ near-inertial oscillations and ocean swells and waves).
388
389When using this parameterization (\ie\ when \np[=1]{nn_etau}{nn\_etau}),
390the TKE input to the ocean ($S$) imposed by the winds in the form of near-inertial oscillations,
391swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition,
392plus a depth depend one given by:
393\begin{equation}
394  \label{eq:ZDF_Ehtau}
395  S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau}
396\end{equation}
397where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that
398penetrates in the ocean, $h_\tau$ is a vertical mixing length scale that controls exponential shape of
399the penetration, and $f_i$ is the ice concentration
400(no penetration if $f_i=1$, \ie\ if the ocean is entirely covered by sea-ice).
401The value of $f_r$, usually a few percents, is specified through \np{rn_efr}{rn\_efr} namelist parameter.
402The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np[=0]{nn_etau}{nn\_etau}) or
403a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m at high latitudes
404(\np[=1]{nn_etau}{nn\_etau}).
405
406Note that two other option exist, \np[=2, 3]{nn_etau}{nn\_etau}.
407They correspond to applying \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer,
408or to using the high frequency part of the stress to evaluate the fraction of TKE that penetrates the ocean.
409Those two options are obsolescent features introduced for test purposes.
410They will be removed in the next release.
411
412% This should be explain better below what this rn_eice parameter is meant for:
413In presence of Sea Ice, the value of this mixing can be modulated by the \np{rn_eice}{rn\_eice} namelist parameter.
414This parameter varies from \forcode{0} for no effect to \forcode{4} to suppress the TKE input into the ocean when Sea Ice concentration
415is greater than 25\%.
416
417% from Burchard et al OM 2008 :
418% the most critical process not reproduced by statistical turbulence models is the activity of
419% internal waves and their interaction with turbulence. After the Reynolds decomposition,
420% internal waves are in principle included in the RANS equations, but later partially
421% excluded by the hydrostatic assumption and the model resolution.
422% Thus far, the representation of internal wave mixing in ocean models has been relatively crude
423% (\eg\ Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002).
424
425% -------------------------------------------------------------------------------------------------------------
426%        GLS Generic Length Scale Scheme
427% -------------------------------------------------------------------------------------------------------------
428\subsection[GLS: Generic Length Scale (\forcode{ln_zdfgls})]{GLS: Generic Length Scale (\protect\np{ln_zdfgls}{ln\_zdfgls})}
429\label{subsec:ZDF_gls}
430
431%--------------------------------------------namzdf_gls---------------------------------------------------------
432
433\begin{listing}
434  \nlst{namzdf_gls}
435  \caption{\forcode{&namzdf_gls}}
436  \label{lst:namzdf_gls}
437\end{listing}
438%--------------------------------------------------------------------------------------------------------------
439
440The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on two prognostic equations:
441one for the turbulent kinetic energy $\bar {e}$, and another for the generic length scale,
442$\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}.
443This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,
444where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:ZDF_GLS} allows to recover a number of
445well-known turbulent closures ($k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87},
446$k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}).
447The GLS scheme is given by the following set of equations:
448\begin{equation}
449  \label{eq:ZDF_gls_e}
450  \frac{\partial \bar{e}}{\partial t} =
451  \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
452      +\left( \frac{\partial v}{\partial k} \right)^2} \right]
453  -K_\rho \,N^2
454  +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right]
455  - \epsilon
456\end{equation}
457
458\[
459  % \label{eq:ZDF_gls_psi}
460  \begin{split}
461    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{
462      \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
463          +\left( \frac{\partial v}{\partial k} \right)^2} \right]
464      - C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\
465    &+\frac{1}{e_3\;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 }
466        \;\frac{\partial \psi}{\partial k}} \right]\;
467  \end{split}
468\]
469
470\[
471  % \label{eq:ZDF_gls_kz}
472  \begin{split}
473    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
474    K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l
475  \end{split}
476\]
477
478\[
479  % \label{eq:ZDF_gls_eps}
480  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
481\]
482where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and
483$\epsilon$ the dissipation rate.
484The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of
485the choice of the turbulence model.
486Four different turbulent models are pre-defined (\autoref{tab:ZDF_GLS}).
487They are made available through the \np{nn_clo}{nn\_clo} namelist parameter.
488
489%--------------------------------------------------TABLE--------------------------------------------------
490\begin{table}[htbp]
491  \centering
492  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c}
493  \begin{tabular}{ccccc}
494    &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\
495    % & \citep{mellor.yamada_RG82} &  \citep{rodi_JGR87}       & \citep{wilcox_AJ88} &                 \\
496    \hline
497    \hline
498    \np{nn_clo}{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\
499    \hline
500    $( p , n , m )$         &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\
501    $\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\
502    $\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\
503    $C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\
504    $C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\
505    $C_3$              &      1.           &     1.              &      1.                &       1.           \\
506    $F_{wall}$        &      Yes        &       --             &     --                  &      --          \\
507    \hline
508    \hline
509  \end{tabular}
510  \caption[Set of predefined GLS parameters or equivalently predefined turbulence models available]{
511    Set of predefined GLS parameters, or equivalently predefined turbulence models available with
512    \protect\np[=.true.]{ln_zdfgls}{ln\_zdfgls} and controlled by
513    the \protect\np{nn_clos}{nn\_clos} namelist variable in \protect\nam{zdf_gls}{zdf\_gls}.}
514  \label{tab:ZDF_GLS}
515\end{table}
516%--------------------------------------------------------------------------------------------------------------
517
518In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force the convergence of
519the mixing length towards $\kappa z_b$ ($\kappa$ is the Von Karman constant and $z_b$ the rugosity length scale) value near physical boundaries
520(logarithmic boundary layer law).
521$C_{\mu}$ and $C_{\mu'}$ are calculated from stability function proposed by \citet{galperin.kantha.ea_JAS88},
522or by \citet{kantha.clayson_JGR94} or one of the two functions suggested by \citet{canuto.howard.ea_JPO01}
523(\np[=0, 3]{nn_stab_func}{nn\_stab\_func}, resp.).
524The value of $C_{0\mu}$ depends on the choice of the stability function.
525
526The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated thanks to Dirichlet or
527Neumann condition through \np{nn_bc_surf}{nn\_bc\_surf} and \np{nn_bc_bot}{nn\_bc\_bot}, resp.
528As for TKE closure, the wave effect on the mixing is considered when
529\np[ > 0.]{rn_crban}{rn\_crban} \citep{craig.banner_JPO94, mellor.blumberg_JPO04}.
530The \np{rn_crban}{rn\_crban} namelist parameter is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and
531\np{rn_charn}{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}.
532
533The $\psi$ equation is known to fail in stably stratified flows, and for this reason
534almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.
535With this clipping, the maximum permissible length scale is determined by $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$.
536A value of $c_{lim} = 0.53$ is often used \citep{galperin.kantha.ea_JAS88}.
537\cite{umlauf.burchard_CSR05} show that the value of the clipping factor is of crucial importance for
538the entrainment depth predicted in stably stratified situations,
539and that its value has to be chosen in accordance with the algebraic model for the turbulent fluxes.
540The clipping is only activated if \np[=.true.]{ln_length_lim}{ln\_length\_lim},
541and the $c_{lim}$ is set to the \np{rn_clim_galp}{rn\_clim\_galp} value.
542
543The time and space discretization of the GLS equations follows the same energetic consideration as for
544the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{burchard_OM02}.
545Evaluation of the 4 GLS turbulent closure schemes can be found in \citet{warner.sherwood.ea_OM05} in ROMS model and
546 in \citet{reffray.guillaume.ea_GMD15} for the \NEMO\ model.
547
548
549% -------------------------------------------------------------------------------------------------------------
550%        OSM OSMOSIS BL Scheme
551% -------------------------------------------------------------------------------------------------------------
552\subsection[OSM: OSMosis boundary layer scheme (\forcode{ln_zdfosm})]{OSM: OSMosis boundary layer scheme (\protect\np{ln_zdfosm}{ln\_zdfosm})}
553\label{subsec:ZDF_osm}
554%--------------------------------------------namzdf_osm---------------------------------------------------------
555
556\begin{listing}
557  \nlst{namzdf_osm}
558  \caption{\forcode{&namzdf_osm}}
559  \label{lst:namzdf_osm}
560\end{listing}
561%--------------------------------------------------------------------------------------------------------------
562
563The OSMOSIS turbulent closure scheme is based on......   TBC
564
565% -------------------------------------------------------------------------------------------------------------
566%        TKE and GLS discretization considerations
567% -------------------------------------------------------------------------------------------------------------
568\subsection[ Discrete energy conservation for TKE and GLS schemes]{Discrete energy conservation for TKE and GLS schemes}
569\label{subsec:ZDF_tke_ene}
570
571%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
572\begin{figure}[!t]
573  \centering
574  \includegraphics[width=0.66\textwidth]{Fig_ZDF_TKE_time_scheme}
575  \caption[Subgrid kinetic energy integration in GLS and TKE schemes]{
576    Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and
577    its links to the momentum and tracer time integration.}
578  \label{fig:ZDF_TKE_time_scheme}
579\end{figure}
580%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
581
582The production of turbulence by vertical shear (the first term of the right hand side of
583\autoref{eq:ZDF_tke_e}) and  \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion
584(first line in \autoref{eq:MB_zdf}).
585To do so a special care has to be taken for both the time and space discretization of
586the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}.
587
588Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how
589the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with
590the one-level forward time stepping of the equation for $\bar{e}$.
591With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to
592the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and
593summing the result vertically:
594\begin{equation}
595  \label{eq:ZDF_energ1}
596  \begin{split}
597    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\
598    &= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}
599    - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz }
600  \end{split}
601\end{equation}
602Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$,
603known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme
604(see \autoref{sec:TD_forward_imp}).
605The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at
606the surface (atmospheric forcing) and at the bottom (friction effect).
607The second term is always negative.
608It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$.
609\autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
610the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as
611${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$
612(and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
613
614A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification
615(second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}).
616This term must balance the input of potential energy resulting from vertical mixing.
617The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by
618multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically:
619\begin{equation}
620  \label{eq:ZDF_energ2}
621  \begin{split}
622    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\
623    &= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}
624    - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\
625    &= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta}
626    + \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  }
627  \end{split}
628\end{equation}
629where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.
630The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because
631there is no diffusive flux through the ocean surface and bottom).
632The second term is minus the destruction rate of  $\bar{e}$ due to stratification.
633Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent,
634the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and  \autoref{eq:ZDF_gls_e}.
635
636Let us now address the space discretization issue.
637The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in
638the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}).
639A space averaging is thus required to obtain the shear TKE production term.
640By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by
641the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
642Furthermore, the time variation of $e_3$ has be taken into account.
643
644The above energetic considerations leads to the following final discrete form for the TKE equation:
645\begin{equation}
646  \label{eq:ZDF_tke_ene}
647  \begin{split}
648    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv
649    \Biggl\{ \Biggr.
650    &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} }
651        \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\
652    +&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} }
653        \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j}
654    \Biggr. \Biggr\}   \\
655    %
656    - &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\
657    %
658    +&\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]   \\
659    %
660    - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt}
661  \end{split}
662\end{equation}
663where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation)
664are time stepped using a backward scheme (see\autoref{sec:TD_forward_imp}).
665Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible.
666%The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as
667%they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}.
668%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.
669
670% ================================================================
671% Convection
672% ================================================================
673\section{Convection}
674\label{sec:ZDF_conv}
675
676Static instabilities (\ie\ light potential densities under heavy ones) may occur at particular ocean grid points.
677In nature, convective processes quickly re-establish the static stability of the water column.
678These processes have been removed from the model via the hydrostatic assumption so they must be parameterized.
679Three parameterisations are available to deal with convective processes:
680a non-penetrative convective adjustment or an enhanced vertical diffusion,
681or/and the use of a turbulent closure scheme.
682
683% -------------------------------------------------------------------------------------------------------------
684%       Non-Penetrative Convective Adjustment
685% -------------------------------------------------------------------------------------------------------------
686\subsection[Non-penetrative convective adjustment (\forcode{ln_tranpc})]{Non-penetrative convective adjustment (\protect\np{ln_tranpc}{ln\_tranpc})}
687\label{subsec:ZDF_npc}
688
689%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
690\begin{figure}[!htb]
691  \centering
692  \includegraphics[width=0.66\textwidth]{Fig_npc}
693  \caption[Unstable density profile treated by the non penetrative convective adjustment algorithm]{
694    Example of an unstable density profile treated by
695    the non penetrative convective adjustment algorithm.
696    $1^{st}$ step: the initial profile is checked from the surface to the bottom.
697    It is found to be unstable between levels 3 and 4.
698    They are mixed.
699    The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
700    The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
701    The $1^{st}$ step ends since the density profile is then stable below the level 3.
702    $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
703    levels 2 to 5 are mixed.
704    The new density profile is checked.
705    It is found stable: end of algorithm.}
706  \label{fig:ZDF_npc}
707\end{figure}
708%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
709
710Options are defined through the \nam{zdf}{zdf} namelist variables.
711The non-penetrative convective adjustment is used when \np[=.true.]{ln_zdfnpc}{ln\_zdfnpc}.
712It is applied at each \np{nn_npc}{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of
713the water column, but only until the density structure becomes neutrally stable
714(\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below)
715\citep{madec.delecluse.ea_JPO91}.
716The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}):
717starting from the top of the ocean, the first instability is found.
718Assume in the following that the instability is located between levels $k$ and $k+1$.
719The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of
720the water column.
721The new density is then computed by a linear approximation.
722If the new density profile is still unstable between levels $k+1$ and $k+2$,
723levels $k$, $k+1$ and $k+2$ are then mixed.
724This process is repeated until stability is established below the level $k$
725(the mixing process can go down to the ocean bottom).
726The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or
727if there is no deeper instability.
728
729This algorithm is significantly different from mixing statically unstable levels two by two.
730The latter procedure cannot converge with a finite number of iterations for some vertical profiles while
731the algorithm used in \NEMO\ converges for any profile in a number of iterations which is less than
732the number of vertical levels.
733This property is of paramount importance as pointed out by \citet{killworth_iprc89}:
734it avoids the existence of permanent and unrealistic static instabilities at the sea surface.
735This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in
736the north-western Mediterranean Sea \citep{madec.delecluse.ea_JPO91, madec.chartier.ea_DAO91, madec.crepon_iprc91}.
737
738The current implementation has been modified in order to deal with any non linear equation of seawater
739(L. Brodeau, personnal communication).
740Two main differences have been introduced compared to the original algorithm:
741$(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency
742(not the difference in potential density);
743$(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in
744the same way their temperature and salinity has been mixed.
745These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without
746having to recompute the expansion coefficients at each mixing iteration.
747
748% -------------------------------------------------------------------------------------------------------------
749%       Enhanced Vertical Diffusion
750% -------------------------------------------------------------------------------------------------------------
751\subsection[Enhanced vertical diffusion (\forcode{ln_zdfevd})]{Enhanced vertical diffusion (\protect\np{ln_zdfevd}{ln\_zdfevd})}
752\label{subsec:ZDF_evd}
753
754Options are defined through the  \nam{zdf}{zdf} namelist variables.
755The enhanced vertical diffusion parameterisation is used when \np[=.true.]{ln_zdfevd}{ln\_zdfevd}.
756In this case, the vertical eddy mixing coefficients are assigned very large values
757in regions where the stratification is unstable
758(\ie\ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{lazar_phd97, lazar.madec.ea_JPO99}.
759This is done either on tracers only (\np[=0]{nn_evdm}{nn\_evdm}) or
760on both momentum and tracers (\np[=1]{nn_evdm}{nn\_evdm}).
761
762In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np[=1]{nn_evdm}{nn\_evdm},
763the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to
764the namelist parameter \np{rn_avevd}{rn\_avevd}.
765A typical value for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$.
766This parameterisation of convective processes is less time consuming than
767the convective adjustment algorithm presented above when mixing both tracers and
768momentum in the case of static instabilities.
769
770Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$.
771This removes a potential source of divergence of odd and even time step in
772a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}).
773
774% -------------------------------------------------------------------------------------------------------------
775%       Turbulent Closure Scheme
776% -------------------------------------------------------------------------------------------------------------
777\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}})}
778\label{subsec:ZDF_tcs}
779
780
781The turbulent closure schemes presented in \autoref{subsec:ZDF_tke}, \autoref{subsec:ZDF_gls} and
782\autoref{subsec:ZDF_osm} (\ie\ \np{ln_zdftke}{ln\_zdftke} or \np{ln_zdfgls}{ln\_zdfgls} or \np{ln_zdfosm}{ln\_zdfosm} defined) deal, in theory,
783with statically unstable density profiles.
784In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in
785\autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative.
786It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also of the four neighboring values at
787velocity points $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1}$).
788These large values restore the static stability of the water column in a way similar to that of
789the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}).
790However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by
791the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$,
792because the mixing length scale is bounded by the distance to the sea surface.
793It can thus be useful to combine the enhanced vertical diffusion with the turbulent closure scheme,
794\ie\ setting the \np{ln_zdfnpc}{ln\_zdfnpc} namelist parameter to true and
795defining the turbulent closure (\np{ln_zdftke}{ln\_zdftke} or \np{ln_zdfgls}{ln\_zdfgls} = \forcode{.true.}) all together.
796
797The OSMOSIS turbulent closure scheme already includes enhanced vertical diffusion in the case of convection,
798%as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp},
799therefore \np[=.false.]{ln_zdfevd}{ln\_zdfevd} should be used with the OSMOSIS scheme.
800% gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
801
802% ================================================================
803% Double Diffusion Mixing
804% ================================================================
805\section[Double diffusion mixing (\forcode{ln_zdfddm})]{Double diffusion mixing (\protect\np{ln_zdfddm}{ln\_zdfddm})}
806\label{subsec:ZDF_ddm}
807
808
809%-------------------------------------------namzdf_ddm-------------------------------------------------
810%
811%\nlst{namzdf_ddm}
812%--------------------------------------------------------------------------------------------------------------
813
814This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the namelist parameter
815\np{ln_zdfddm}{ln\_zdfddm} in \nam{zdf}{zdf}.
816Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa.
817The former condition leads to salt fingering and the latter to diffusive convection.
818Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean.
819\citet{merryfield.holloway.ea_JPO99} include a parameterisation of such phenomena in a global ocean model and show that
820it leads to relatively minor changes in circulation but exerts significant regional influences on
821temperature and salinity.
822
823
824Diapycnal mixing of S and T are described by diapycnal diffusion coefficients
825\begin{align*}
826  % \label{eq:ZDF_ddm_Kz}
827  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\
828  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
829\end{align*}
830where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
831and $o$ by processes other than double diffusion.
832The rates of double-diffusive mixing depend on the buoyancy ratio
833$R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of
834thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}).
835To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt
836(1981):
837\begin{align}
838  \label{eq:ZDF_ddm_f}
839  A_f^{vS} &=
840             \begin{cases}
841               \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
842               0                              &\text{otherwise}
843             \end{cases}
844  \\         \label{eq:ZDF_ddm_f_T}
845  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho
846\end{align}
847
848%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
849\begin{figure}[!t]
850  \centering
851  \includegraphics[width=0.66\textwidth]{Fig_zdfddm}
852  \caption[Diapycnal diffusivities for temperature and salt in regions of salt fingering and
853  diffusive convection]{
854    From \citet{merryfield.holloway.ea_JPO99}:
855    (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in
856    regions of salt fingering.
857    Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and
858    thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
859    (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in
860    regions of diffusive convection.
861    Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
862    The latter is not implemented in \NEMO.}
863  \label{fig:ZDF_ddm}
864\end{figure}
865%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
866
867The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of
868buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}).
869Following  \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
870
871To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by
872Federov (1988) is used:
873\begin{align}
874  % \label{eq:ZDF_ddm_d}
875  A_d^{vT} &=
876             \begin{cases}
877               1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
878               &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
879               0                       &\text{otherwise}
880             \end{cases}
881                                       \nonumber \\
882  \label{eq:ZDF_ddm_d_S}
883  A_d^{vS} &=
884             \begin{cases}
885               A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
886               A_d^{vT} \ 0.15 \ R_\rho               &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
887               0                       &\text{otherwise}
888             \end{cases}
889\end{align}
890
891The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in
892\autoref{fig:ZDF_ddm}.
893Implementing this requires computing $R_\rho$ at each grid point on every time step.
894This is done in \mdl{eosbn2} at the same time as $N^2$ is computed.
895This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive).
896
897% ================================================================
898% Bottom Friction
899% ================================================================
900\section[Bottom and top friction (\textit{zdfdrg.F90})]{Bottom and top friction (\protect\mdl{zdfdrg})}
901\label{sec:ZDF_drg}
902
903%--------------------------------------------namdrg--------------------------------------------------------
904%
905\begin{listing}
906  \nlst{namdrg}
907  \caption{\forcode{&namdrg}}
908  \label{lst:namdrg}
909\end{listing}
910\begin{listing}
911  \nlst{namdrg_top}
912  \caption{\forcode{&namdrg_top}}
913  \label{lst:namdrg_top}
914\end{listing}
915\begin{listing}
916  \nlst{namdrg_bot}
917  \caption{\forcode{&namdrg_bot}}
918  \label{lst:namdrg_bot}
919\end{listing}
920
921%--------------------------------------------------------------------------------------------------------------
922
923Options to define the top and bottom friction are defined through the \nam{drg}{drg} namelist variables.
924The bottom friction represents the friction generated by the bathymetry.
925The top friction represents the friction generated by the ice shelf/ocean interface.
926As the friction processes at the top and the bottom are treated in and identical way,
927the description below considers mostly the bottom friction case, if not stated otherwise.
928
929
930Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as
931a condition on the vertical diffusive flux.
932For the bottom boundary layer, one has:
933 \[
934   % \label{eq:ZDF_bfr_flux}
935   A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
936 \]
937where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside
938the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean).
939How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near
940the bottom relative to the Ekman layer depth.
941For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m,
942one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$
943(for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$).
944With a background diffusion coefficient $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m.
945When the vertical mixing coefficient is this small, using a flux condition is equivalent to
946entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or
947bottom model layer.
948To illustrate this, consider the equation for $u$ at $k$, the last ocean level:
949\begin{equation}
950  \label{eq:ZDF_drg_flux2}
951  \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}}
952\end{equation}
953If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth.
954On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used,
955the turbulent Ekman layer will be represented explicitly by the model.
956However, the logarithmic layer is never represented in current primitive equation model applications:
957it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.
958Two choices are available in \NEMO: a linear and a quadratic bottom friction.
959Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in
960the present release of \NEMO.
961
962In the code, the bottom friction is imposed by adding the trend due to the bottom friction to
963 the general momentum trend in \mdl{dynzdf}.
964For the time-split surface pressure gradient algorithm, the momentum trend due to
965the barotropic component needs to be handled separately.
966For this purpose it is convenient to compute and store coefficients which can be simply combined with
967bottom velocities and geometric values to provide the momentum trend due to bottom friction.
968 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where:
969\begin{equation}
970  \label{eq:ZDF_bfr_bdef}
971  \frac{\partial {\textbf U_h}}{\partial t} =
972  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b
973\end{equation}
974where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity.
975Note than from \NEMO\ 4.0, drag coefficients are only computed at cell centers (\ie\ at T-points) and refer to as $c_b^T$ in the following. These are then linearly interpolated in space to get $c_b^\textbf{U}$ at velocity points.
976
977% -------------------------------------------------------------------------------------------------------------
978%       Linear Bottom Friction
979% -------------------------------------------------------------------------------------------------------------
980\subsection[Linear top/bottom friction (\forcode{ln_lin})]{Linear top/bottom friction (\protect\np{ln_lin}{ln\_lin})}
981\label{subsec:ZDF_drg_linear}
982
983The linear friction parameterisation (including the special case of a free-slip condition) assumes that
984the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level):
985\[
986  % \label{eq:ZDF_bfr_linear}
987  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
988\]
989where $r$ is a friction coefficient expressed in $m s^{-1}$.
990This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean,
991and setting $r = H / \tau$, where $H$ is the ocean depth.
992Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{weatherly_JMR84}.
993A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models.
994One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$
995(\citet{gill_bk82}, Eq. 9.6.6).
996For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$,
997and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.
998This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days.
999It can be changed by specifying \np{rn_Uc0}{rn\_Uc0} (namelist parameter).
1000
1001 For the linear friction case the drag coefficient used in the general expression \autoref{eq:ZDF_bfr_bdef} is:
1002\[
1003  % \label{eq:ZDF_bfr_linbfr_b}
1004    c_b^T = - r
1005\]
1006When \np[=.true.]{ln_lin}{ln\_lin}, the value of $r$ used is \np{rn_Uc0}{rn\_Uc0}*\np{rn_Cd0}{rn\_Cd0}.
1007Setting \np[=.true.]{ln_OFF}{ln\_OFF} (and \forcode{ln_lin=.true.}) is equivalent to setting $r=0$ and leads to a free-slip boundary condition.
1008
1009These values are assigned in \mdl{zdfdrg}.
1010Note that there is support for local enhancement of these values via an externally defined 2D mask array
1011(\np[=.true.]{ln_boost}{ln\_boost}) given in the \ifile{bfr\_coef} input NetCDF file.
1012The mask values should vary from 0 to 1.
1013Locations with a non-zero mask value will have the friction coefficient increased by
1014$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}.
1015
1016% -------------------------------------------------------------------------------------------------------------
1017%       Non-Linear Bottom Friction
1018% -------------------------------------------------------------------------------------------------------------
1019\subsection[Non-linear top/bottom friction (\forcode{ln_non_lin})]{Non-linear top/bottom friction (\protect\np{ln_non_lin}{ln\_non\_lin})}
1020\label{subsec:ZDF_drg_nonlinear}
1021
1022The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic:
1023\[
1024  % \label{eq:ZDF_drg_nonlinear}
1025  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
1026  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
1027\]
1028where $C_D$ is a drag coefficient, and $e_b $ a top/bottom turbulent kinetic energy due to tides,
1029internal waves breaking and other short time scale currents.
1030A typical value of the drag coefficient is $C_D = 10^{-3} $.
1031As an example, the CME experiment \citep{treguier_JGR92} uses $C_D = 10^{-3}$ and
1032$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{killworth_JPO92} uses $C_D = 1.4\;10^{-3}$ and
1033$e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.
1034The CME choices have been set as default values (\np{rn_Cd0}{rn\_Cd0} and \np{rn_ke0}{rn\_ke0} namelist parameters).
1035
1036As for the linear case, the friction is imposed in the code by adding the trend due to
1037the friction to the general momentum trend in \mdl{dynzdf}.
1038For the non-linear friction case the term computed in \mdl{zdfdrg} is:
1039\[
1040  % \label{eq:ZDF_drg_nonlinbfr}
1041    c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2}
1042\]
1043
1044The coefficients that control the strength of the non-linear friction are initialised as namelist parameters:
1045$C_D$= \np{rn_Cd0}{rn\_Cd0}, and $e_b$ =\np{rn_bfeb2}{rn\_bfeb2}.
1046Note that for applications which consider tides explicitly, a low or even zero value of \np{rn_bfeb2}{rn\_bfeb2} is recommended. A local enhancement of $C_D$ is again possible via an externally defined 2D mask array
1047(\np[=.true.]{ln_boost}{ln\_boost}).
1048This works in the same way as for the linear friction case with non-zero masked locations increased by
1049$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}.
1050
1051% -------------------------------------------------------------------------------------------------------------
1052%       Bottom Friction Log-layer
1053% -------------------------------------------------------------------------------------------------------------
1054\subsection[Log-layer top/bottom friction (\forcode{ln_loglayer})]{Log-layer top/bottom friction (\protect\np{ln_loglayer}{ln\_loglayer})}
1055\label{subsec:ZDF_drg_loglayer}
1056
1057In the non-linear friction case, the drag coefficient, $C_D$, can be optionally enhanced using
1058a "law of the wall" scaling. This assumes that the model vertical resolution can capture the logarithmic layer which typically occur for layers thinner than 1 m or so.
1059If  \np[=.true.]{ln_loglayer}{ln\_loglayer}, $C_D$ is no longer constant but is related to the distance to the wall (or equivalently to the half of the top/bottom layer thickness):
1060\[
1061  C_D = \left ( {\kappa \over {\mathrm log}\left ( 0.5 \; e_{3b} / rn\_{z0} \right ) } \right )^2
1062\]
1063
1064\noindent where $\kappa$ is the von-Karman constant and \np{rn_z0}{rn\_z0} is a roughness length provided via the namelist.
1065
1066The drag coefficient is bounded such that it is kept greater or equal to
1067the base \np{rn_Cd0}{rn\_Cd0} value which occurs where layer thicknesses become large and presumably logarithmic layers are not resolved at all. For stability reason, it is also not allowed to exceed the value of an additional namelist parameter:
1068\np{rn_Cdmax}{rn\_Cdmax}, \ie
1069\[
1070  rn\_Cd0 \leq C_D \leq rn\_Cdmax
1071\]
1072
1073\noindent The log-layer enhancement can also be applied to the top boundary friction if
1074under ice-shelf cavities are activated (\np[=.true.]{ln_isfcav}{ln\_isfcav}).
1075%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}.
1076
1077% -------------------------------------------------------------------------------------------------------------
1078%       Explicit bottom Friction
1079% -------------------------------------------------------------------------------------------------------------
1080\subsection[Explicit top/bottom friction (\forcode{ln_drgimp=.false.})]{Explicit top/bottom friction (\protect\np[=.false.]{ln_drgimp}{ln\_drgimp})}
1081\label{subsec:ZDF_drg_stability}
1082
1083Setting \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:
1084
1085At the top (below an ice shelf cavity):
1086\[
1087  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1088  = c_{t}^{\textbf{U}}\textbf{u}^{n-1}_{t}
1089\]
1090
1091At the bottom (above the sea floor):
1092\[
1093  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1094  = c_{b}^{\textbf{U}}\textbf{u}^{n-1}_{b}
1095\]
1096
1097Since 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.
1098For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is:
1099\begin{equation}
1100  \label{eq:ZDF_Eqn_drgstab}
1101  \begin{split}
1102    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
1103    &= -\frac{ru}{e_{3u}}\;2\rdt\\
1104  \end{split}
1105\end{equation}
1106\noindent where linear friction and a leapfrog timestep have been assumed.
1107To ensure that the friction cannot reverse the direction of flow it is necessary to have:
1108\[
1109  |\Delta u| < \;|u|
1110\]
1111\noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives:
1112\[
1113  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
1114\]
1115This same inequality can also be derived in the non-linear bottom friction case if
1116a velocity of 1 m.s$^{-1}$ is assumed.
1117Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability:
1118\[
1119  e_{3u} > 2\;r\;\rdt
1120\]
1121\noindent which it may be necessary to impose if partial steps are being used.
1122For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m.
1123For most applications, with physically sensible parameters these restrictions should not be of concern.
1124But caution may be necessary if attempts are made to locally enhance the bottom friction parameters.
1125To ensure stability limits are imposed on the top/bottom friction coefficients both
1126during initialisation and at each time step.
1127Checks at initialisation are made in \mdl{zdfdrg} (assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
1128The number of breaches of the stability criterion are reported as well as
1129the minimum and maximum values that have been set.
1130The criterion is also checked at each time step, using the actual velocity, in \mdl{dynzdf}.
1131Values of the friction coefficient are reduced as necessary to ensure stability;
1132these changes are not reported.
1133
1134Limits on the top/bottom friction coefficient are not imposed if the user has elected to
1135handle the friction implicitly (see \autoref{subsec:ZDF_drg_imp}).
1136The number of potential breaches of the explicit stability criterion are still reported for information purposes.
1137
1138% -------------------------------------------------------------------------------------------------------------
1139%       Implicit Bottom Friction
1140% -------------------------------------------------------------------------------------------------------------
1141\subsection[Implicit top/bottom friction (\forcode{ln_drgimp=.true.})]{Implicit top/bottom friction (\protect\np[=.true.]{ln_drgimp}{ln\_drgimp})}
1142\label{subsec:ZDF_drg_imp}
1143
1144An optional implicit form of bottom friction has been implemented to improve model stability.
1145We recommend this option for shelf sea and coastal ocean applications. %, especially for split-explicit time splitting.
1146This option can be invoked by setting \np{ln_drgimp}{ln\_drgimp} to \forcode{.true.} in the \nam{drg}{drg} namelist.
1147%This option requires \np{ln_zdfexp}{ln\_zdfexp} to be \forcode{.false.} in the \nam{zdf}{zdf} namelist.
1148
1149This implementation is performed in \mdl{dynzdf} where the following boundary conditions are set while solving the fully implicit diffusion step:
1150
1151At the top (below an ice shelf cavity):
1152\[
1153  % \label{eq:ZDF_dynZDF__drg_top}
1154  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t}
1155  = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t}
1156\]
1157
1158At the bottom (above the sea floor):
1159\[
1160  % \label{eq:ZDF_dynZDF__drg_bot}
1161  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b}
1162  = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b}
1163\]
1164
1165where $t$ and $b$ refers to top and bottom layers respectively.
1166Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so it is implicit.
1167
1168% -------------------------------------------------------------------------------------------------------------
1169%       Bottom Friction with split-explicit free surface
1170% -------------------------------------------------------------------------------------------------------------
1171\subsection[Bottom friction with split-explicit free surface]{Bottom friction with split-explicit free surface}
1172\label{subsec:ZDF_drg_ts}
1173
1174With split-explicit free surface, the sub-stepping of barotropic equations needs the knowledge of top/bottom stresses. An obvious way to satisfy this is to take them as constant over the course of the barotropic integration and equal to the value used to update the baroclinic momentum trend. Provided \np[=.false.]{ln_drgimp}{ln\_drgimp} and a centred or \textit{leap-frog} like integration of barotropic equations is used (\ie\ \forcode{ln_bt_fw=.false.}, cf \autoref{subsec:DYN_spg_ts}), this does ensure that barotropic and baroclinic dynamics feel the same stresses during one leapfrog time step. However, if \np[=.true.]{ln_drgimp}{ln\_drgimp},  stresses depend on the \textit{after} value of the velocities which themselves depend on the barotropic iteration result. This cyclic dependency makes difficult obtaining consistent stresses in 2d and 3d dynamics. Part of this mismatch is then removed when setting the final barotropic component of 3d velocities to the time splitting estimate. This last step can be seen as a necessary evil but should be minimized since it interferes with the adjustment to the boundary conditions.
1175
1176The strategy to handle top/bottom stresses with split-explicit free surface in \NEMO\ is as follows:
1177\begin{enumerate}
1178\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.
1179\item In case of an implicit drag, specific computations are performed in \mdl{dynzdf} which renders the overall scheme mixed explicit/implicit: the barotropic components of 3d velocities are removed before seeking for the implicit vertical diffusion result. Top/bottom stresses due to the barotropic components are explicitly accounted for thanks to the updated values of barotropic velocities. Then the implicit solution of 3d velocities is obtained. Lastly, the residual barotropic component is replaced by the time split estimate.
1180\end{enumerate}
1181
1182Note that other strategies are possible, like considering vertical diffusion step in advance, \ie\ prior barotropic integration.
1183
1184
1185% ================================================================
1186% Internal wave-driven mixing
1187% ================================================================
1188\section[Internal wave-driven mixing (\forcode{ln_zdfiwm})]{Internal wave-driven mixing (\protect\np{ln_zdfiwm}{ln\_zdfiwm})}
1189\label{subsec:ZDF_tmx_new}
1190
1191%--------------------------------------------namzdf_iwm------------------------------------------
1192%
1193\begin{listing}
1194  \nlst{namzdf_iwm}
1195  \caption{\forcode{&namzdf_iwm}}
1196  \label{lst:namzdf_iwm}
1197\end{listing}
1198%--------------------------------------------------------------------------------------------------------------
1199
1200The parameterization of mixing induced by breaking internal waves is a generalization of
1201the approach originally proposed by \citet{st-laurent.simmons.ea_GRL02}.
1202A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed,
1203and the resulting diffusivity is obtained as
1204\[
1205  % \label{eq:ZDF_Kwave}
1206  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
1207\]
1208where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of
1209the energy available for mixing.
1210If the \np{ln_mevar}{ln\_mevar} namelist parameter is set to \forcode{.false.}, the mixing efficiency is taken as constant and
1211equal to 1/6 \citep{osborn_JPO80}.
1212In the opposite (recommended) case, $R_f$ is instead a function of
1213the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$,
1214with $\nu$ the molecular viscosity of seawater, following the model of \cite{bouffard.boegman_DAO13} and
1215the implementation of \cite{de-lavergne.madec.ea_JPO16}.
1216Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when
1217the mixing efficiency is constant.
1218
1219In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary
1220as a function of $Re_b$ by setting the \np{ln_tsdiff}{ln\_tsdiff} parameter to \forcode{.true.}, a recommended choice.
1221This parameterization of differential mixing, due to \cite{jackson.rehmann_JPO14},
1222is implemented as in \cite{de-lavergne.madec.ea_JPO16}.
1223
1224The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$,
1225is constructed from three static maps of column-integrated internal wave energy dissipation,
1226$E_{cri}(i,j)$, $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures:
1227
1228\begin{align*}
1229  F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\
1230  F_{pyc}(i,j,k) &\propto N^{n_p}\\
1231  F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} }
1232\end{align*}
1233In the above formula, $h_{ab}$ denotes the height above bottom,
1234$h_{wkb}$ denotes the WKB-stretched height above bottom, defined by
1235\[
1236  h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; ,
1237\]
1238The $n_p$ parameter (given by \np{nn_zpyc}{nn\_zpyc} in \nam{zdf_iwm}{zdf\_iwm} namelist)
1239controls the stratification-dependence of the pycnocline-intensified dissipation.
1240It can take values of $1$ (recommended) or $2$.
1241Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of
1242the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps.
1243$h_{cri}$ is related to the large-scale topography of the ocean (etopo2) and
1244$h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of
1245the abyssal hill topography \citep{goff_JGR10} and the latitude.
1246%
1247% Jc: input files names ?
1248
1249% ================================================================
1250% surface wave-induced mixing
1251% ================================================================
1252\section[Surface wave-induced mixing (\forcode{ln_zdfswm})]{Surface wave-induced mixing (\protect\np{ln_zdfswm}{ln\_zdfswm})}
1253\label{subsec:ZDF_swm}
1254
1255Surface waves produce an enhanced mixing through wave-turbulence interaction.
1256In addition to breaking waves induced turbulence (\autoref{subsec:ZDF_tke}),
1257the influence of non-breaking waves can be accounted introducing
1258wave-induced viscosity and diffusivity as a function of the wave number spectrum.
1259Following \citet{qiao.yuan.ea_OD10}, a formulation of wave-induced mixing coefficient
1260is provided  as a function of wave amplitude, Stokes Drift and wave-number:
1261
1262\begin{equation}
1263  \label{eq:ZDF_Bv}
1264  B_{v} = \alpha {A} {U}_{st} {exp(3kz)}
1265\end{equation}
1266
1267Where $B_{v}$ is the wave-induced mixing coefficient, $A$ is the wave amplitude,
1268${U}_{st}$ is the Stokes Drift velocity, $k$ is the wave number and $\alpha$
1269is a constant which should be determined by observations or
1270numerical experiments and is set to be 1.
1271
1272The coefficient $B_{v}$ is then directly added to the vertical viscosity
1273and diffusivity coefficients.
1274
1275In order to account for this contribution set: \forcode{ln_zdfswm=.true.},
1276then wave interaction has to be activated through \forcode{ln_wave=.true.},
1277the Stokes Drift can be evaluated by setting \forcode{ln_sdw=.true.}
1278(see \autoref{subsec:SBC_wave_sdw})
1279and the needed wave fields can be provided either in forcing or coupled mode
1280(for more information on wave parameters and settings see \autoref{sec:SBC_wave})
1281
1282% ================================================================
1283% Adaptive-implicit vertical advection
1284% ================================================================
1285\section[Adaptive-implicit vertical advection (\forcode{ln_zad_Aimp})]{Adaptive-implicit vertical advection(\protect\np{ln_zad_Aimp}{ln\_zad\_Aimp})}
1286\label{subsec:ZDF_aimp}
1287
1288The adaptive-implicit vertical advection option in NEMO is based on the work of
1289\citep{shchepetkin_OM15}.  In common with most ocean models, the timestep used with NEMO
1290needs to satisfy multiple criteria associated with different physical processes in order
1291to maintain numerical stability. \citep{shchepetkin_OM15} pointed out that the vertical
1292CFL criterion is commonly the most limiting. \citep{lemarie.debreu.ea_OM15} examined the
1293constraints for a range of time and space discretizations and provide the CFL stability
1294criteria for a range of advection schemes. The values for the Leap-Frog with Robert
1295asselin filter time-stepping (as used in NEMO) are reproduced in
1296\autoref{tab:ZDF_zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these
1297restrictions but at the cost of large dispersive errors and, possibly, large numerical
1298viscosity. The adaptive-implicit vertical advection option provides a targetted use of the
1299implicit scheme only when and where potential breaches of the vertical CFL condition
1300occur. In many practical applications these events may occur remote from the main area of
1301interest or due to short-lived conditions such that the extra numerical diffusion or
1302viscosity does not greatly affect the overall solution. With such applications, setting:
1303\forcode{ln_zad_Aimp=.true.} should allow much longer model timesteps to be used whilst
1304retaining the accuracy of the high order explicit schemes over most of the domain.
1305
1306\begin{table}[htbp]
1307  \centering
1308  % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}}
1309  \begin{tabular}{r|ccc}
1310    \hline
1311    spatial discretization  & 2$^nd$ order centered & 3$^rd$ order upwind & 4$^th$ order compact \\
1312    advective CFL criterion &                 0.904 &              0.472  &                0.522 \\
1313    \hline
1314  \end{tabular}
1315  \caption[Advective CFL criteria for the leapfrog with Robert Asselin filter time-stepping]{
1316    The advective CFL criteria for a range of spatial discretizations for
1317    the leapfrog with Robert Asselin filter time-stepping
1318    ($\nu=0.1$) as given in \citep{lemarie.debreu.ea_OM15}.}
1319  \label{tab:ZDF_zad_Aimp_CFLcrit}
1320\end{table}
1321
1322In particular, the advection scheme remains explicit everywhere except where and when
1323local vertical velocities exceed a threshold set just below the explicit stability limit.
1324Once the threshold is reached a tapered transition towards an implicit scheme is used by
1325partitioning the vertical velocity into a part that can be treated explicitly and any
1326excess that must be treated implicitly. The partitioning is achieved via a Courant-number
1327dependent weighting algorithm as described in \citep{shchepetkin_OM15}.
1328
1329The local cell Courant number ($Cu$) used for this partitioning is:
1330
1331\begin{equation}
1332  \label{eq:ZDF_Eqn_zad_Aimp_Courant}
1333  \begin{split}
1334    Cu &= {2 \rdt \over e^n_{3t_{ijk}}} \bigg (\big [ \texttt{Max}(w^n_{ijk},0.0) - \texttt{Min}(w^n_{ijk+1},0.0) \big ]    \\
1335       &\phantom{=} +\big [ \texttt{Max}(e_{{2_u}ij}e^n_{{3_{u}}ijk}u^n_{ijk},0.0) - \texttt{Min}(e_{{2_u}i-1j}e^n_{{3_{u}}i-1jk}u^n_{i-1jk},0.0) \big ]
1336                     \big / e_{{1_t}ij}e_{{2_t}ij}            \\
1337       &\phantom{=} +\big [ \texttt{Max}(e_{{1_v}ij}e^n_{{3_{v}}ijk}v^n_{ijk},0.0) - \texttt{Min}(e_{{1_v}ij-1}e^n_{{3_{v}}ij-1k}v^n_{ij-1k},0.0) \big ]
1338                     \big / e_{{1_t}ij}e_{{2_t}ij} \bigg )    \\
1339  \end{split}
1340\end{equation}
1341
1342\noindent and the tapering algorithm follows \citep{shchepetkin_OM15} as:
1343
1344\begin{align}
1345  \label{eq:ZDF_Eqn_zad_Aimp_partition}
1346Cu_{min} &= 0.15 \nonumber \\
1347Cu_{max} &= 0.3  \nonumber \\
1348Cu_{cut} &= 2Cu_{max} - Cu_{min} \nonumber \\
1349Fcu    &= 4Cu_{max}*(Cu_{max}-Cu_{min}) \nonumber \\
1350\cf &=
1351     \begin{cases}
1352        0.0                                                        &\text{if $Cu \leq Cu_{min}$} \\
1353        (Cu - Cu_{min})^2 / (Fcu +  (Cu - Cu_{min})^2)             &\text{else if $Cu < Cu_{cut}$} \\
1354        (Cu - Cu_{max}) / Cu                                       &\text{else}
1355     \end{cases}
1356\end{align}
1357
1358\begin{figure}[!t]
1359  \centering
1360  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_coeff}
1361  \caption[Partitioning coefficient used to partition vertical velocities into parts]{
1362    The value of the partitioning coefficient (\cf) used to partition vertical velocities into
1363    parts to be treated implicitly and explicitly for a range of typical Courant numbers
1364    (\forcode{ln_zad_Aimp=.true.}).}
1365  \label{fig:ZDF_zad_Aimp_coeff}
1366\end{figure}
1367
1368\noindent The partitioning coefficient is used to determine the part of the vertical
1369velocity that must be handled implicitly ($w_i$) and to subtract this from the total
1370vertical velocity ($w_n$) to leave that which can continue to be handled explicitly:
1371
1372\begin{align}
1373  \label{eq:ZDF_Eqn_zad_Aimp_partition2}
1374    w_{i_{ijk}} &= \cf_{ijk} w_{n_{ijk}}     \nonumber \\
1375    w_{n_{ijk}} &= (1-\cf_{ijk}) w_{n_{ijk}}
1376\end{align}
1377
1378\noindent Note that the coefficient is such that the treatment is never fully implicit;
1379the three cases from \autoref{eq:ZDF_Eqn_zad_Aimp_partition} can be considered as:
1380fully-explicit; mixed explicit/implicit and mostly-implicit.  With the settings shown the
1381coefficient (\cf) varies as shown in \autoref{fig:ZDF_zad_Aimp_coeff}. Note with these values
1382the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly
1383implicit' is 0.45 which is just below the stability limited given in
1384\autoref{tab:ZDF_zad_Aimp_CFLcrit}  for a 3rd order scheme.
1385
1386The $w_i$ component is added to the implicit solvers for the vertical mixing in
1387\mdl{dynzdf} and \mdl{trazdf} in a similar way to \citep{shchepetkin_OM15}.  This is
1388sufficient for the flux-limited advection scheme (\forcode{ln_traadv_mus}) but further
1389intervention is required when using the flux-corrected scheme (\forcode{ln_traadv_fct}).
1390For these schemes the implicit upstream fluxes must be added to both the monotonic guess
1391and to the higher order solution when calculating the antidiffusive fluxes. The implicit
1392vertical fluxes are then removed since they are added by the implicit solver later on.
1393
1394The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be
1395used in a wide range of simulations. The following test simulation, however, does illustrate
1396the potential benefits and will hopefully encourage further testing and feedback from users:
1397
1398\begin{figure}[!t]
1399  \centering
1400  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames}
1401  \caption[OVERFLOW: time-series of temperature vertical cross-sections]{
1402    A time-series of temperature vertical cross-sections for the OVERFLOW test case.
1403    These results are for the default settings with \forcode{nn_rdt=10.0} and
1404    without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}).}
1405  \label{fig:ZDF_zad_Aimp_overflow_frames}
1406\end{figure}
1407
1408\subsection{Adaptive-implicit vertical advection in the OVERFLOW test-case}
1409
1410The \href{https://forge.ipsl.jussieu.fr/nemo/chrome/site/doc/NEMO/guide/html/test\_cases.html\#overflow}{OVERFLOW test case}
1411provides a simple illustration of the adaptive-implicit advection in action. The example here differs from the basic test case
1412by only a few extra physics choices namely:
1413
1414\begin{verbatim}
1415     ln_dynldf_OFF = .false.
1416     ln_dynldf_lap = .true.
1417     ln_dynldf_hor = .true.
1418     ln_zdfnpc     = .true.
1419     ln_traadv_fct = .true.
1420        nn_fct_h   =  2
1421        nn_fct_v   =  2
1422\end{verbatim}
1423
1424\noindent which were chosen to provide a slightly more stable and less noisy solution. The
1425result when using the default value of \forcode{nn_rdt=10.} without adaptive-implicit
1426vertical velocity is illustrated in \autoref{fig:ZDF_zad_Aimp_overflow_frames}. The mass of
1427cold water, initially sitting on the shelf, moves down the slope and forms a
1428bottom-trapped, dense plume. Even with these extra physics choices the model is close to
1429stability limits and attempts with \forcode{nn_rdt=30.} will fail after about 5.5 hours
1430with excessively high horizontal velocities. This time-scale corresponds with the time the
1431plume reaches the steepest part of the topography and, although detected as a horizontal
1432CFL breach, the instability originates from a breach of the vertical CFL limit. This is a good
1433candidate, therefore, for use of the adaptive-implicit vertical advection scheme.
1434
1435The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps
1436are shown in \autoref{fig:ZDF_zad_Aimp_overflow_all_rdt} (together with the equivalent
1437frames from the base run).  In this simple example the use of the adaptive-implicit
1438vertcal advection scheme has enabled a 12x increase in the model timestep without
1439significantly altering the solution (although at this extreme the plume is more diffuse
1440and has not travelled so far).  Notably, the solution with and without the scheme is
1441slightly different even with \forcode{nn_rdt=10.}; suggesting that the base run was
1442close enough to instability to trigger the scheme despite completing successfully.
1443To assist in diagnosing how active the scheme is, in both location and time, the 3D
1444implicit and explicit components of the vertical velocity are available via XIOS as
1445\texttt{wimp} and \texttt{wexp} respectively.  Likewise, the partitioning coefficient
1446(\cf) is also available as \texttt{wi\_cff}. For a quick oversight of
1447the schemes activity the global maximum values of the absolute implicit component
1448of the vertical velocity and the partitioning coefficient are written to the netCDF
1449version of the run statistics file (\texttt{run.stat.nc}) if this is active (see
1450\autoref{sec:MISC_opt} for activation details).
1451
1452\autoref{fig:ZDF_zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for
1453the various overflow tests.  Note that the adaptive-implicit vertical advection scheme is
1454active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the
1455test case is close to stability limits even with this value. At the larger timesteps, the
1456vertical velocity is treated mostly implicitly at some location throughout the run. The
1457oscillatory nature of this measure appears to be linked to the progress of the plume front
1458as each cusp is associated with the location of the maximum shifting to the adjacent cell.
1459This is illustrated in \autoref{fig:ZDF_zad_Aimp_maxCf_loc} where the i- and k- locations of the
1460maximum have been overlaid for the base run case.
1461
1462\medskip
1463\noindent Only limited tests have been performed in more realistic configurations. In the
1464ORCA2\_ICE\_PISCES reference configuration the scheme does activate and passes
1465restartability and reproducibility tests but it is unable to improve the model's stability
1466enough to allow an increase in the model time-step. A view of the time-series of maximum
1467partitioning coefficient (not shown here)  suggests that the default time-step of 5400s is
1468already pushing at stability limits, especially in the initial start-up phase. The
1469time-series does not, however, exhibit any of the 'cuspiness' found with the overflow
1470tests.
1471
1472\medskip
1473\noindent A short test with an eORCA1 configuration promises more since a test using a
1474time-step of 3600s remains stable with \forcode{ln_zad_Aimp=.true.} whereas the
1475time-step is limited to 2700s without.
1476
1477\begin{figure}[!t]
1478  \centering
1479  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_overflow_all_rdt}
1480  \caption[OVERFLOW: sample temperature vertical cross-sections from mid- and end-run]{
1481    Sample temperature vertical cross-sections from mid- and end-run using
1482    different values for \forcode{nn_rdt} and with or without adaptive implicit vertical advection.
1483    Without the adaptive implicit vertical advection
1484    only the run with the shortest timestep is able to run to completion.
1485    Note also that the colour-scale has been chosen to confirm that
1486    temperatures remain within the original range of 10$^o$ to 20$^o$.}
1487  \label{fig:ZDF_zad_Aimp_overflow_all_rdt}
1488\end{figure}
1489
1490\begin{figure}[!t]
1491  \centering
1492  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_maxCf}
1493  \caption[OVERFLOW: maximum partitioning coefficient during a series of test runs]{
1494    The maximum partitioning coefficient during a series of test runs with
1495    increasing model timestep length.
1496    At the larger timesteps,
1497    the vertical velocity is treated mostly implicitly at some location throughout the run.}
1498  \label{fig:ZDF_zad_Aimp_maxCf}
1499\end{figure}
1500
1501\begin{figure}[!t]
1502  \centering
1503  \includegraphics[width=0.66\textwidth]{Fig_ZDF_zad_Aimp_maxCf_loc}
1504  \caption[OVERFLOW: maximum partitioning coefficient for the case overlaid]{
1505    The maximum partitioning coefficient for the \forcode{nn_rdt=10.0} case overlaid with
1506    information on the gridcell i- and k-locations of the maximum value.}
1507  \label{fig:ZDF_zad_Aimp_maxCf_loc}
1508\end{figure}
1509
1510% ================================================================
1511
1512\onlyinsubfile{\bibliography{../main/bibliography}}
1513
1514\onlyinsubfile{\printindex}
1515
1516\end{document}
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