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Changeset 2226 for branches/DEV_r1826_DOC/DOC – NEMO

Ignore:
Timestamp:
2010-10-12T16:23:34+02:00 (14 years ago)
Author:
gm
Message:

ticket:#658 update ZDF with TKE time-stepping and GLS scheme

Location:
branches/DEV_r1826_DOC/DOC/TexFiles
Files:
1 added
2 edited

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  • branches/DEV_r1826_DOC/DOC/TexFiles/Biblio/Biblio.bib

    r2213 r2226  
    1212@STRING{AW = {Addison-Wesley}} 
    1313 
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  • branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_ZDF.tex

    r2164 r2226  
    5959\begin{align*}  
    6060A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}  \\ 
    61 \\ 
    6261A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1} 
    6362\end{align*} 
    … …  
    9190   \left\{      \begin{aligned} 
    9291         A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\ 
    93          \\ 
    9492         A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm} 
    9593   \end{aligned}    \right. 
    … …  
    126124\begin{equation} \label{Eq_zdftke_e} 
    127125\frac{\partial \bar{e}}{\partial t} =  
    128 \frac{A^{vm}}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 
    129                      +\left( {\frac{\partial v}{\partial k}} \right)^2} \right] 
    130 -A^{vT}\,N^2 
     126\frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 
     127                    +\left( {\frac{\partial v}{\partial k}} \right)^2} \right] 
     128-K_\rho\,N^2 
    131129+\frac{1}{e_3} \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 } 
    132130            \;\frac{\partial \bar{e}}{\partial k}} \right] 
    … …  
    135133\begin{equation} \label{Eq_zdftke_kz} 
    136134   \begin{split} 
    137          A^{vm} &= C_k\  l_k\  \sqrt {\bar{e}\; }     \\ 
    138          A^{vT} &= A^{vm} / P_{rt} 
     135         K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }     \\ 
     136         K_\rho &= A^{vm} / P_{rt} 
    139137   \end{split} 
    140138\end{equation} 
    141139where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}),  
    142140$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,  
    143 $P_{rt}$ is the Prandtl number. The constants $C_k =  0.1$ and  
    144 $C_\epsilon = \sqrt {2} /2$  $\approx 0.7$ are designed to deal with vertical mixing  
    145 at any depth \citep{Gaspar1990}. They are set through namelist parameters  
    146 \np{nn\_ediff} and \np{nn\_ediss}. $P_{rt}$ can be set to unity or, following  
    147 \citet{Blanke1993}, be a function of the local Richardson number, $R_i$: 
     141$P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity  
     142and diffusivity coefficients. The constants $C_k =  0.1$ and $C_\epsilon = \sqrt {2} /2$   
     143$\approx 0.7$ are designed to deal with vertical mixing at any depth \citep{Gaspar1990}.  
     144They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}.  
     145$P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function  
     146of the local Richardson number, $R_i$: 
    148147\begin{align*} \label{Eq_prt} 
    149148P_{rt} = \begin{cases} 
    … …  
    165164which is valid in a stable stratified region with constant values of the Brunt- 
    166165Vais\"{a}l\"{a} frequency. The resulting length scale is bounded by the distance  
    167 to the surface or to the bottom (\np{nn\_mxl}=0) or by the local vertical scale factor  
    168 (\np{nn\_mxl}=1). \citet{Blanke1993} notice that this simplification has two major  
     166to the surface or to the bottom (\np{nn\_mxl} = 0) or by the local vertical scale factor  
     167(\np{nn\_mxl} = 1). \citet{Blanke1993} notice that this simplification has two major  
    169168drawbacks: it makes no sense for locally unstable stratification and the  
    170169computation no longer uses all the information contained in the vertical density  
    171170profile. To overcome these drawbacks, \citet{Madec1998} introduces the  
    172 \np{nn\_mxl}=2 or 3 cases, which add an extra assumption concerning the vertical  
     171\np{nn\_mxl} = 2 or 3 cases, which add an extra assumption concerning the vertical  
    173172gradient of the computed length scale. So, the length scales are first evaluated  
    174173as in \eqref{Eq_tke_mxl0_1} and then bounded such that: 
    … …  
    227226to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. This allows the subsequent formulations  
    228227to match that of \citet{Gargett1984} for the diffusion in the thermocline and  
    229 deep ocean :  $(A^{vT} = 10^{-3} / N)$.  
    230 In addition, a cut-off is applied on $A^{vm}$ and $A^{vT}$ to avoid numerical  
     228deep ocean :  $K_\rho = 10^{-3} / N$.  
     229In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical  
    231230instabilities associated with too weak vertical diffusion. They must be  
    232231specified at least larger than the molecular values, and are set through  
    … …  
    236235%        TKE Turbulent Closure Scheme : new organization to energetic considerations 
    237236% ------------------------------------------------------------------------------------------------------------- 
    238 \subsection{TKE Turbulent Closure Scheme - time integration (\key{zdftke})} 
     237\subsection{TKE discretization considerations (\key{zdftke})} 
    239238\label{ZDF_tke_ene} 
    240239 
    … …  
    249248The production of turbulence by vertical shear (the first term of the right hand side  
    250249of \eqref{Eq_zdftke_e}) should balance the loss of kinetic energy associated with 
    251 the vertical momentum diffusion (first line in \eqref{Eq_PE_zdf}).  
    252 The total loss of kinetic energy (in 1D for the demonstration)  
    253 due to this term is obtained by multiply this quantity by $u^n$ and verticaly integrating:    
    254  
     250the vertical momentum diffusion (first line in \eqref{Eq_PE_zdf}). To do so a special care  
     251have to be taken for both the time and space discretization of the TKE equation \citep{Burchard_OM02}. 
     252 
     253Let us first address the time stepping issue. Fig.~\ref{Fig_TKE_time_scheme} shows  
     254how the two-level Leap-Frog time stepping of the momentum and tracer equations interplays  
     255with the one-level forward time stepping of TKE equation. With this framework, the total loss  
     256of kinetic energy (in 1D for the demonstration) due to the vertical momentum diffusion is  
     257obtained by multiplying this quantity by $u^t$ and summing the result vertically:    
    255258\begin{equation} \label{Eq_energ1} 
    256 \int_{k_b}^{k_s} {u^t \frac{1}{e_3} 
    257                                  \frac{\partial   }  
    258                                         {\partial k} \left( \frac{A^{vm}}{e_3}  
    259                                                                  \frac{\partial{u^{t+1}}} 
    260                                                                         {\partial k           }     \right) \; e_3 \; dk } 
    261 = \left[  \frac{u^t}{e_3}   A^{vm} \frac{\partial{u^{t+1}}}{\partial k} \right]_{k_b}^{k_s} 
    262  - \int_{k_b}^{k_s}{\frac{A^{vm}}{{e_3}}\frac{\partial{u^t}}{\partial k}\frac{\partial{u^{t+1}}}{\partial k}} \ dk 
    263 \end{equation} 
    264  
    265 The first term of the right hand side of \eqref{Eq_energ1} represents the kinetic  
    266 energy transfer at the surface (atmospheric forcing) and at the bottom (friction effect).  
    267 The second term is always negative and have to balance the term of \eqref{Eq_zdftke_e}  
    268 previously identified. 
    269  
    270 The sink term (possibly a source term in statically unstable situations) of turbulence  
    271 by buoyancy (second term of the right hand side of \eqref{Eq_zdftke_e}) must balance 
    272 the source of potential energy associated with the vertical diffusion  
    273 in the density equation (second line in \eqref{Eq_PE_zdf}). The source of potential  
    274 energy (in 1D for the demonstration) due to this term is obtained by multiply this quantity  
    275 by $gz{\rho_r}^{-1}$ and verticaly integrating: 
    276  
     259\begin{split} 
     260\int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\ 
     261&= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}           
     262 - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz } 
     263\end{split} 
     264\end{equation} 
     265Here, the vertical diffusion of momentum is discretized backward in time  
     266with a coefficient, $K_m$, known at time $t$ (Fig.~\ref{Fig_TKE_time_scheme}),  
     267as it is required when using the TKE scheme (see \S\ref{STP_forward_imp}).  
     268The first term of the right hand side of \eqref{Eq_energ1} represents the kinetic energy  
     269transfer at the surface (atmospheric forcing) and at the bottom (friction effect).  
     270The second term is always negative. It is the dissipation rate of kinetic energy,  
     271and thus minus the shear production rate of $\bar{e}$. \eqref{Eq_energ1}  
     272implies that, to be energetically consistent, the production rate of $\bar{e}$  
     273used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as  
     274${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ (and not by the more straightforward  
     275$K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$). 
     276 
     277A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification  
     278(second term of the right hand side of \eqref{Eq_zdftke_e}). This term  
     279must balance the input of potential energy resulting from vertical mixing.  
     280The rate of change of potential energy (in 1D for the demonstration) due vertical  
     281mixing is obtained by multiplying vertical density diffusion  
     282tendency by $g\,z$ and and summing the result vertically: 
    277283\begin{equation} \label{Eq_energ2} 
    278 \begin{aligned} 
    279 \int_{k_b}^{k_s}{\frac{g\;z}{e_3} \frac{\partial }{\partial k} 
    280                         \left( \frac{A^{vT}}{e_3}  
    281                         \frac{\partial{\rho^{t+1}}}{\partial k}   \right)} \; e_3 \; dk 
    282 =\left[  g\;z \frac{A^{vT}}{e_3} 
    283                  \frac{\partial{\rho^{t+1}}}{\partial k} \right]_{k_b}^{k_s}   
    284 - \int_{k_b}^{k_s}{  \frac{A^{vT}}{e_3} g \frac{\partial{\rho^{t+1}}}{\partial k}} \; dk\\ 
    285 \\ 
    286 = - \left[  z\,A^{vT} {N_{t+1}}^2 \right]_{k_b}^{k_s} 
    287 + \int_{k_b}^{k_s}{  \rho^{t+1} \; A^{vT}{N_{t+1}}^2 \; e_3 \; dk  }\\ 
    288 \end{aligned} 
    289 \end{equation} 
    290 where $N^2_{t+1}$ is  the Brunt-Vais\"{a}l\"{a} frequency at $t+1$  
    291 and noting that $\frac{\partial z}{\partial k} = e_3$. 
    292  
    293 The first term is always zero because theBrunt-Vais\"{a}l\"{a} frequency is set to zero at the  
    294 surface and the bottom. The second term is of opposite sign than the buoyancy term 
    295 identified previously. 
    296  
    297 Under these energetic considerations, \eqref{Eq_zdftke_e} have to be written like this  
    298 to be consistant:  
    299  
     284\begin{split} 
     285\int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\ 
     286&= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}   
     287   - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\ 
     288&= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta} 
     289+ \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  } 
     290\end{split} 
     291\end{equation} 
     292where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.  
     293The first term of the right hand side of \eqref{Eq_energ2}  is always zero  
     294because there is no diffusive flux through the ocean surface and bottom).  
     295The second term is minus the destruction rate of  $\bar{e}$ due to stratification.  
     296Therefore \eqref{Eq_energ1} implies that, to be energetically consistent, the product  
     297${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \eqref{Eq_zdftke_e}, the TKE equation. 
     298 
     299Let us now address the space discretization issue.  
     300The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity  
     301components are in the centre of the side faces of a $t$-box in staggered C-grid  
     302(Fig.\ref{Fig_cell}). A space averaging is thus required to obtain the shear TKE production term. 
     303By redoing the \eqref{Eq_energ1} in the 3D case, it can be shown that the product of  
     304eddy coefficient by the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. 
     305Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into  
     306account. 
     307 
     308The above energetic considerations leads to  
     309the following final discrete form for the TKE equation: 
    300310\begin{equation} \label{Eq_zdftke_ene} 
    301 \frac{\partial \bar{e}}{\partial t} = 
    302 \frac{A^{vm}}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u^a}{\partial k}} \right)\left( {\frac{\partial u^n}{\partial k}} \right) 
    303                                                         +\left( {\frac{\partial v^a}{\partial k}} \right)\left( {\frac{\partial v^n}{\partial k}} \right) 
    304 } \right]-A^{vT}\,{N_n}^2+... 
    305 \end{equation} 
    306  
    307 Note that during a time step, the equation \eqref{Eq_zdftke_e} is resolved before those  
    308 of momentum and density. So, the indice "a" (after) become "n" (now) and the indice "n"  
    309 (now) become "b" (before). 
    310  
    311 Moreover, the vertical shear have to be multiply by the appropriate viscosity for numerical  
    312 stability. Thus, the vertical shear at U-point have to be multiply by the viscosity avmu and  
    313 the vertical shear at V-point have to be multiply by the viscosity avmv. Next, these two  
    314 quantities are averaged to obtain a production term by vertical shear at W-point :  
    315  
    316 \begin{equation} \label{Eq_zdftke_ene2} 
    317 \frac{\partial \bar{e}}{\partial t} = 
    318 \frac{1}{{e_3}^2 }\;\left[ {   
    319     A^{vmu}\left({\frac{\partial u^a}{\partial k}} \right)\left( {\frac{\partial u^n}{\partial k}} \right) 
    320    +A^{vmv}\left({\frac{\partial v^a}{\partial k}} \right)\left( {\frac{\partial v^n}{\partial k}} \right) 
    321                            } \right]-A^{vT}\,{N_n}^2+... 
    322 \end{equation}  
    323  
    324 The TKE equation is resolved before the mixing length, the viscosity and diffusivity. Two tabs  
    325 are then declared : dissl (dissipation length) (Remark : it's not only the dissipation lenght,  
    326 it's the root of the TKE divided by the dissipation lenght) and avmt (viscosity at the points T)  
    327 used for the vertical diffusion of the TKE.  
    328  
    329 This new organization needs also a reorganization of the routine step.F90.  
    330 The bigger change is the estimation of the Brunt-Vais\"{a}l\"{a} 
    331 frequency at "n" instead of "b". Moreover for energetic considerations, the call of tranxt.F90  
    332 is done after the density update. On the contrary, the density is updated with scalars fields  
    333 filtered by the Asselin filter.  
    334  
    335 This new organisation of the routine zdftke force to save five three dimensionnal tabs in  
    336 the restart : avmu, avmv, avt, avmt and dissl are needed to calculate $e_n$. At the end  
    337 of the run (time step = nitend), the alternative is to save only $e_n$ estimated at the  
    338 following time step (nitend+1). The next run using this restart file, the mixing length  
    339 and turbulents coefficients are directly calculated using $e_n$. It is the same thing  
    340 for the intermediate restart. 
    341  
    342  
    343 %%GM for figure of the time scheme: 
    344 \begin{equation}  
    345   \rho^{t+1} \; A^{vT}{N_{t+1}}^2 \; dk  \\ 
    346 \end{equation} 
    347  
    348 %%end GM  
    349  
     311\begin{split} 
     312\frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv   
     313\Biggl\{ \Biggr. 
     314  &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} }  
     315                                                                              \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\ 
     316+&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} }  
     317                                                                               \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j}  
     318\Biggr. \Biggr\}   \\ 
     319% 
     320- &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\ 
     321% 
     322+&\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]   \\ 
     323% 
     324- &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt} 
     325\end{split} 
     326\end{equation} 
     327where the last two terms in \eqref{Eq_zdftke_ene} (vertical diffusion and Kolmogorov dissipation)  
     328are time stepped using a backward scheme (see\S\ref{STP_forward_imp}).  
     329Note that the Kolmogorov term has been linearized in time in order to render  
     330the implicit computation possible. The restart of the TKE scheme  
     331requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as they all appear in  
     332the right hand side of \eqref{Eq_zdftke_ene}. For the latter, it is in fact  
     333the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored.  
     334 
     335% ------------------------------------------------------------------------------------------------------------- 
     336%        GLS Generic Length Scale Scheme  
     337% ------------------------------------------------------------------------------------------------------------- 
     338\subsection{GLS Generic Length Scale (\key{zdfgls})} 
     339\label{ZDF_gls} 
     340 
     341%--------------------------------------------namgls--------------------------------------------------------- 
     342\namdisplay{namgls} 
     343%-------------------------------------------------------------------------------------------------------------- 
     344 
     345The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on  
     346two prognostic equations: one for the turbulent kinetic energy $\bar {e}$, and another  
     347for the generic length scale, $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}.  
     348This later variable is defined as : $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$,  
     349where the triplet $(p, m, n)$ value given in Tab.\ref{Tab_GLS} allows to recover  
     350a number of well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982},  
     351$k$-$\epsilon$ \citep{Rodi_1987}, $k$-$\omega$ \citep{Wilcox_1988}  
     352among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}).  
     353The GLS scheme is given by the following set of equations: 
     354\begin{equation} \label{Eq_zdfgls_e} 
     355\frac{\partial \bar{e}}{\partial t} =  
     356\frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 
     357                                                   +\left( \frac{\partial v}{\partial k} \right)^2} \right] 
     358-K_\rho \,N^2 
     359+\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right] 
     360- \epsilon 
     361\end{equation} 
     362 
     363\begin{equation} \label{Eq_zdfgls_psi} 
     364   \begin{split} 
     365\frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ 
     366\frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 
     367                                                                   +\left( \frac{\partial v}{\partial k} \right)^2} \right] 
     368- C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\ 
     369&+\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 } 
     370                                  \;\frac{\partial \psi}{\partial k}} \right]\; 
     371   \end{split} 
     372\end{equation} 
     373 
     374\begin{equation} \label{Eq_zdfgls_kz} 
     375   \begin{split} 
     376         K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\ 
     377         K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l 
     378   \end{split} 
     379\end{equation} 
     380 
     381\begin{equation} \label{Eq_zdfgls_eps} 
     382{\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 
     383\end{equation} 
     384where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2})  
     385and $\epsilon$ the dissipation rate.  
     386The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$)  
     387depends of the choice of the turbulence model. Four different turbulent models are pre-defined  
     388(Tab.\ref{Tab_GLS}). They are made available through th \np{gls} namelist parameter.  
     389 
     390%--------------------------------------------------TABLE-------------------------------------------------- 
     391\begin{table}[htbp]  \label{Tab_GLS} 
     392\begin{center} 
     393%\begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c} 
     394\begin{tabular}{ccccc} 
     395                         &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\   
     396%                        & \citep{Mellor_Yamada_1982} &  \citep{Rodi_1987}       & \citep{Wilcox_1988} &                 \\   
     397\hline  \hline  
     398\np{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\   
     399\hline  
     400$( p , n , m )$          &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\ 
     401$\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\ 
     402$\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\ 
     403$C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\ 
     404$C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\ 
     405$C_3$              &      1.           &     1.              &      1.                &       1.           \\ 
     406$F_{wall}$        &      Yes        &       --             &     --                  &      --          \\ 
     407\hline 
     408\hline 
     409\end{tabular} 
     410\caption {Set of predefined GLS parameters, or equivalently predefined turbulence models available with \key{gls} and controlled by the \np{nn\_clos} namelist parameter.} 
     411\end{center} 
     412\end{table} 
     413%-------------------------------------------------------------------------------------------------------------- 
     414 
     415In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force 
     416the convergence of the mixing length towards $K\,z_b$ ($K$: Kappa and $z_b$: rugosity length)  
     417value near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$  
     418are calculated from stability function proposed by \citet{Galperin_al_JAS88}, or by \citet{Kantha_Clayson_1994}  
     419or one of the two functions suggested by \citet{Canuto_2001}  (\np{nn\_stab\_func} = 0, 1, 2 or 3, resp.}). The value of $C_{0\mu}$ depends of the choice of the stability function. 
     420 
     421The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated  
     422thanks to Dirichlet or Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp.  
     423The wave effect on the mixing could be also being considered \citep{Craig_Banner_1994}. 
     424 
     425The $\psi$ equation is known to fail in stably stratified flows, and for this reason  
     426almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy.  
     427With this clipping, the maximum permissible length scale is determined by  
     428$l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$. A value of $c_{lim} = 0.53$ is often used  
     429\citep{Galperin_al_JAS88}. \cite{Umlauf_Burchard_CSR05} show that the value of  
     430the clipping factor is of crucial importance for the entrainment depth predicted in  
     431stably stratified situations, and that its value has to be chosen in accordance  
     432with the algebraic model for the turbulent ßuxes. The clipping is only activated  
     433if \np{ln\_length\_lim}=true, and the $c_{lim}$ is set to the \np{clim\_galp} value. 
    350434 
    351435% ------------------------------------------------------------------------------------------------------------- 
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