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branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_ZDF.tex
r2349 r2376 154 154 The choice of $P_{rt}$ is controlled by the \np{nn\_pdl} namelist parameter. 155 155 156 At the sea surface, the value of $\bar{e}$ is prescribed from the wind 157 stress field as $\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb} 158 namelist parameter. The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}), 159 however a much larger value can be used when taking into account the 160 surface wave breaking (see below Eq. \eqref{ZDF_Esbc}). 161 The bottom value of TKE is assumed to be equal to the value of the level just above. 162 The time integration of the $\bar{e}$ equation may formally lead to negative values 163 because the numerical scheme does not ensure its positivity. To overcome this 164 problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} 165 namelist parameter). Following \citet{Gaspar1990}, the cut-off value is set 166 to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. This allows the subsequent formulations 167 to match that of \citet{Gargett1984} for the diffusion in the thermocline and 168 deep ocean : $K_\rho = 10^{-3} / N$. 169 In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical 170 instabilities associated with too weak vertical diffusion. They must be 171 specified at least larger than the molecular values, and are set through 172 \np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \S\ref{ZDF_cst}). 173 174 \subsubsection{Turbulent length scale} 156 175 For computational efficiency, the original formulation of the turbulent length 157 176 scales proposed by \citet{Gaspar1990} has been simplified. Four formulations … … 187 206 mixing length scales as (and note that here we use numerical indexing): 188 207 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 189 \begin{figure}[!t] \ label{Fig_mixing_length} \begin{center}208 \begin{figure}[!t] \begin{center} 190 209 \includegraphics[width=1.00\textwidth]{./TexFiles/Figures/Fig_mixing_length.pdf} 191 \caption {Illustration of the mixing length computation. } 210 \caption{ \label{Fig_mixing_length} 211 Illustration of the mixing length computation. } 192 212 \end{center} 193 213 \end{figure} … … 204 224 $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 205 225 206 In the \np{nn\_mxl} =2 case, the dissipation and mixing length scales take the same226 In the \np{nn\_mxl}~=~2 case, the dissipation and mixing length scales take the same 207 227 value: $ l_k= l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right)$, while in the 208 \np{nn\_mxl} =2case, the dissipation and mixing turbulent length scales are give228 \np{nn\_mxl}~=~3 case, the dissipation and mixing turbulent length scales are give 209 229 as in \citet{Gaspar1990}: 210 230 \begin{equation} \label{Eq_tke_mxl_gaspar} … … 215 235 \end{equation} 216 236 217 At the sea surface the value of $\bar{e}$ is prescribed from the wind 218 stress field: $\bar{e}=rn\_ebb\;\left| \tau \right|$ (\np{rn\_ebb}=60 by default) 219 with a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist 220 parameters). Its value at the bottom of the ocean is assumed to be 221 equal to the value of the level just above. The time integration of the 222 $\bar{e}$ equation may formally lead to negative values because the 223 numerical scheme does not ensure its positivity. To overcome this 224 problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} 225 namelist parameter). Following \citet{Gaspar1990}, the cut-off value is set 226 to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. This allows the subsequent formulations 227 to match that of \citet{Gargett1984} for the diffusion in the thermocline and 228 deep ocean : $K_\rho = 10^{-3} / N$. 229 In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical 230 instabilities associated with too weak vertical diffusion. They must be 231 specified at least larger than the molecular values, and are set through 232 \np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \S\ref{ZDF_cst}). 233 234 % ------------------------------------------------------------------------------------------------------------- 235 % TKE Turbulent Closure Scheme : new organization to energetic considerations 237 At the ocean surface, a non zero length scale is set through the \np{rn\_lmin0} namelist 238 parameter. Usually the surface scale is given by $l_o = \kappa \,z_o$ 239 where $\kappa = 0.4$ is von Karman's constant and $z_o$ the roughness 240 parameter of the surface. Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94} 241 leads to a 0.04~m, the default value of \np{rn\_lsurf}. In the ocean interior 242 a minimum length scale is set to recover the molecular viscosity when $\bar{e}$ 243 reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ). 244 245 246 \subsubsection{Surface wave breaking parameterization} 247 %-----------------------------------------------------------------------% 248 249 Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified 250 to include the effect of surface wave breaking energetics. This results in a reduction of summertime 251 surface temperature when the mixed layer is relatively shallow. The \citet{Mellor_Blumberg_JPO04} 252 modifications acts on surface length scale and TKE values and air-sea drag coefficient. 253 The latter concerns the bulk formulea and is not discussed here. 254 255 Following \citet{Craig_Banner_JPO94}, the boundary condition on surface TKE value is : 256 \begin{equation} \label{ZDF_Esbc} 257 \bar{e}_o = \frac{1}{2}\,\left( 15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} 258 \end{equation} 259 where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality 260 which depends on the ''wave age'', ranging from 57 for mature waves to 146 for 261 younger waves \citep{Mellor_Blumberg_JPO04}. 262 The boundary condition on the turbulent length scale follows the Charnock's relation: 263 \begin{equation} \label{ZDF_Lsbc} 264 l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o} 265 \end{equation} 266 where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant. 267 \citet{Mellor_Blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by \citet{Stacey_JPO99} 268 citing observation evidence, and $\alpha_{CB} = 100$ the Craig and Banner's value. 269 As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, 270 with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}~=~67.83 corresponds 271 to $\alpha_{CB} = 100$. further setting \np{ln\_lsurf} to true applies \eqref{ZDF_Lsbc} 272 as surface boundary condition on length scale, with $\beta$ hard coded to the Stacet's value. 273 Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) 274 is applied on surface $\bar{e}$ value. 275 276 277 \subsubsection{Langmuir cells} 278 %--------------------------------------% 279 Langmuir circulations (LC) can be described as ordered large-scale vertical motions 280 in the surface layer of the oceans. Although LC have nothing to do with convection, 281 the circulation pattern is rather similar to so-called convective rolls in the atmospheric 282 boundary layer. The detailed physics behind LC is described in, for example, 283 \citet{Craik_Leibovich_JFM76}. The prevailing explanation is that LC arise from 284 a nonlinear interaction between the Stokes drift and wind drift currents. 285 286 Here we introduced in the TKE turbulent closure the simple parameterization of 287 Langmuir circulations proposed by \citep{Axell_JGR02} for a $k-\epsilon$ turbulent closure. 288 The parameterization, tuned against large-eddy simulation, includes the whole effect 289 of LC in an extra source terms of TKE, $P_{LC}$. 290 The presence of $P_{LC}$ in \eqref{Eq_zdftke_e}, the TKE equation, is controlled 291 by setting \np{ln\_lc} to \textit{true} in the namtke namelist. 292 293 By making an analogy with the characteristic convective velocity scale 294 ($e.g.$, \citet{D'Alessio_al_JPO98}), $P_{LC}$ is assumed to be : 295 \begin{equation} 296 P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}} 297 \end{equation} 298 where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth. 299 With no information about the wave field, $w_{LC}$ is assumed to be proportional to 300 the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module 301 \footnote{Following \citet{Li_Garrett_JMR93}, the surface Stoke drift velocity 302 may be expressed as $u_s = 0.016 \,|U_{10m}|$. Assuming an air density of 303 $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of $1.5~10^{-3}$ give the expression 304 used of $u_s$ as a function of the module of surface stress}. 305 For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as 306 at a finite depth $H_{LC}$ (which is often close to the mixed layer depth), and simply 307 varies as a sine function in between (a first-order profile for the Langmuir cell structures). 308 The resulting expression for $w_{LC}$ is : 309 \begin{equation} 310 w_{LC} = \begin{cases} 311 c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} ) & \text{if $-z \leq H_{LC}$} \\ 312 0 & \text{otherwise} 313 \end{cases} 314 \end{equation} 315 where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise 316 to fit LES data. The chosen value yields maximum vertical velocities $w_{LC}$ of the order 317 of a few centimeters per second. The value of $c_{LC}$ is set through the \np{rn\_lc} 318 namelist parameter, having in mind that it should stay between 0.15 and 0.54 \citep{Axell_JGR02}. 319 320 The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations: 321 $H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift 322 can reach on its own by converting its kinetic energy to potential energy, according to 323 \begin{equation} 324 - \int_{-H_{LC}}^0 { N^2\;z \;dz} = \frac{1}{2} u_s^2 325 \end{equation} 326 327 328 %\subsubsection{Mixing just below the mixed layer} 329 %---------------------------------------------------------------% 330 331 % add here a description of "penetration of TKE" and the associated namelist parameters 332 333 % ------------------------------------------------------------------------------------------------------------- 334 % TKE discretization considerations 236 335 % ------------------------------------------------------------------------------------------------------------- 237 336 \subsection{TKE discretization considerations (\key{zdftke})} … … 239 338 240 339 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 241 \begin{figure}[!t] \label{Fig_TKE_time_scheme}\begin{center}340 \begin{figure}[!t] \begin{center} 242 341 \includegraphics[width=1.00\textwidth]{./TexFiles/Figures/Fig_ZDF_TKE_time_scheme.pdf} 243 \caption {Illustration of the TKE time integration and its links to the momentum and tracer time integration. } 342 \caption{ \label{Fig_TKE_time_scheme} 343 Illustration of the TKE time integration and its links to the momentum and tracer time integration. } 244 344 \end{center} 245 345 \end{figure} … … 389 489 390 490 %--------------------------------------------------TABLE-------------------------------------------------- 391 \begin{table}[htbp] \label{Tab_GLS} 392 \begin{center} 491 \begin{table}[htbp] \begin{center} 393 492 %\begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c} 394 493 \begin{tabular}{ccccc} … … 408 507 \hline 409 508 \end{tabular} 410 \caption {Set of predefined GLS parameters, or equivalently predefined turbulence models available with \key{zdfgls} and controlled by the \np{nn\_clos} namelist parameter.} 411 \end{center} 412 \end{table} 509 \caption{ \label{Tab_GLS} 510 Set of predefined GLS parameters, or equivalently predefined turbulence models available 511 with \key{zdfgls} and controlled by the \np{nn\_clos} namelist parameter.} 512 \end{center} \end{table} 413 513 %-------------------------------------------------------------------------------------------------------------- 414 514 … … 417 517 value near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$ 418 518 are calculated from stability function proposed by \citet{Galperin_al_JAS88}, or by \citet{Kantha_Clayson_1994} 419 or 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. 519 or one of the two functions suggested by \citet{Canuto_2001} (\np{nn\_stab\_func} = 0, 1, 2 or 3, resp.}). 520 The value of $C_{0\mu}$ depends of the choice of the stability function. 420 521 421 522 The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated 422 523 thanks to Dirichlet or Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp. 423 The wave effect on the mixing could be also being considered \citep{Craig_Banner_1994}. 524 As for TKE closure , the wave effect on the mixing is considered when \np{ln\_crban}~=~true 525 \citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}. The \np{rn\_crban} namelist parameter 526 is $\alpha_{CB}$ in \eqref{ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \eqref{ZDF_Lsbc}. 424 527 425 528 The $\psi$ equation is known to fail in stably stratified flows, and for this reason … … 433 536 if \np{ln\_length\_lim}=true, and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value. 434 537 538 The time and space discretization of the GLS equations follows the same energetic 539 consideration as for the TKE case described in \S\ref{ZDF_tke_ene} \citep{Burchard_OM02}. 540 Examples of performance of the 4 turbulent closure scheme can be found in \citet{Warner_al_OM05}. 541 435 542 % ------------------------------------------------------------------------------------------------------------- 436 543 % K Profile Parametrisation (KPP) … … 479 586 480 587 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 481 \begin{figure}[!htb] \label{Fig_npc}\begin{center}588 \begin{figure}[!htb] \begin{center} 482 589 \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_npc.pdf} 483 \caption {Example of an unstable density profile treated by the non penetrative 590 \caption{ \label{Fig_npc} 591 Example of an unstable density profile treated by the non penetrative 484 592 convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from 485 593 the surface to the bottom. It is found to be unstable between levels 3 and 4. … … 641 749 642 750 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 643 \begin{figure}[!t] \label{Fig_zdfddm}\begin{center}751 \begin{figure}[!t] \begin{center} 644 752 \includegraphics[width=0.99\textwidth]{./TexFiles/Figures/Fig_zdfddm.pdf} 645 \caption {From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ 753 \caption{ \label{Fig_zdfddm} 754 From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ 646 755 and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy 647 756 curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves … … 986 1095 987 1096 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 988 \begin{figure}[!t] \label{Fig_ZDF_M2_K1_tmx}\begin{center}1097 \begin{figure}[!t] \begin{center} 989 1098 \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_ZDF_M2_K1_tmx.pdf} 990 \caption {(a) M2 and (b) K2 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). }991 \end{center} 992 \end{ figure}1099 \caption{ \label{Fig_ZDF_M2_K1_tmx} 1100 (a) M2 and (b) K2 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). } 1101 \end{center} \end{figure} 993 1102 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 994 1103
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