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branches/2016/dev_INGV_UKMO_2016/DOC/TexFiles/Chapters/Chap_ZDF.tex
r5120 r7351 1 \documentclass[NEMO_book]{subfiles} 2 \begin{document} 1 3 % ================================================================ 2 4 % Chapter Vertical Ocean Physics (ZDF) … … 34 36 coefficients can be assumed to be either constant, or a function of the local 35 37 Richardson number, or computed from a turbulent closure model (either 36 TKE or KPPformulation). The computation of these coefficients is initialized38 TKE or GLS formulation). The computation of these coefficients is initialized 37 39 in the \mdl{zdfini} module and performed in the \mdl{zdfric}, \mdl{zdftke} or 38 \mdl{zdf kpp} modules. The trends due to the vertical momentum and tracer40 \mdl{zdfgls} modules. The trends due to the vertical momentum and tracer 39 41 diffusion, including the surface forcing, are computed and added to the 40 42 general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively. … … 234 236 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 235 237 \begin{figure}[!t] \begin{center} 236 \includegraphics[width=1.00\textwidth]{ ./TexFiles/Figures/Fig_mixing_length.pdf}238 \includegraphics[width=1.00\textwidth]{Fig_mixing_length} 237 239 \caption{ \label{Fig_mixing_length} 238 240 Illustration of the mixing length computation. } … … 262 264 \end{equation} 263 265 264 At the ocean surface, a non zero length scale is set through the \np{rn\_ lmin0} namelist266 At the ocean surface, a non zero length scale is set through the \np{rn\_mxl0} namelist 265 267 parameter. Usually the surface scale is given by $l_o = \kappa \,z_o$ 266 268 where $\kappa = 0.4$ is von Karman's constant and $z_o$ the roughness 267 269 parameter of the surface. Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94} 268 leads to a 0.04~m, the default value of \np{rn\_ lsurf}. In the ocean interior270 leads to a 0.04~m, the default value of \np{rn\_mxl0}. In the ocean interior 269 271 a minimum length scale is set to recover the molecular viscosity when $\bar{e}$ 270 272 reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ). … … 295 297 As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, 296 298 with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}~=~67.83 corresponds 297 to $\alpha_{CB} = 100$. further setting \np{ln\_lsurf} to true applies \eqref{ZDF_Lsbc}298 as surface boundary condition on length scale, with $\beta$ hard coded to the Stace t's value.299 to $\alpha_{CB} = 100$. Further setting \np{ln\_mxl0} to true applies \eqref{ZDF_Lsbc} 300 as surface boundary condition on length scale, with $\beta$ hard coded to the Stacey's value. 299 301 Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) 300 302 is applied on surface $\bar{e}$ value. … … 355 357 %--------------------------------------------------------------% 356 358 357 To be add here a description of "penetration of TKE" and the associated namelist parameters 358 \np{nn\_etau}, \np{rn\_efr} and \np{nn\_htau}. 359 Vertical mixing parameterizations commonly used in ocean general circulation models 360 tend to produce mixed-layer depths that are too shallow during summer months and windy conditions. 361 This bias is particularly acute over the Southern Ocean. 362 To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{Rodgers_2014}. 363 The parameterization is an empirical one, $i.e.$ not derived from theoretical considerations, 364 but rather is meant to account for observed processes that affect the density structure of 365 the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme 366 ($i.e.$ near-inertial oscillations and ocean swells and waves). 367 368 When using this parameterization ($i.e.$ when \np{nn\_etau}~=~1), the TKE input to the ocean ($S$) 369 imposed by the winds in the form of near-inertial oscillations, swell and waves is parameterized 370 by \eqref{ZDF_Esbc} the standard TKE surface boundary condition, plus a depth depend one given by: 371 \begin{equation} \label{ZDF_Ehtau} 372 S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau} 373 \end{equation} 374 where 375 $z$ is the depth, 376 $e_s$ is TKE surface boundary condition, 377 $f_r$ is the fraction of the surface TKE that penetrate in the ocean, 378 $h_\tau$ is a vertical mixing length scale that controls exponential shape of the penetration, 379 and $f_i$ is the ice concentration (no penetration if $f_i=1$, that is if the ocean is entirely 380 covered by sea-ice). 381 The value of $f_r$, usually a few percents, is specified through \np{rn\_efr} namelist parameter. 382 The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np{nn\_etau}~=~0) 383 or a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m 384 at high latitudes (\np{nn\_etau}~=~1). 385 386 Note that two other option existe, \np{nn\_etau}~=~2, or 3. They correspond to applying 387 \eqref{ZDF_Ehtau} only at the base of the mixed layer, or to using the high frequency part 388 of the stress to evaluate the fraction of TKE that penetrate the ocean. 389 Those two options are obsolescent features introduced for test purposes. 390 They will be removed in the next release. 391 392 359 393 360 394 % from Burchard et al OM 2008 : 361 % the most critical process not reproduced by statistical turbulence models is the activity of internal waves and their interaction with turbulence. After the Reynolds decomposition, internal waves are in principle included in the RANS equations, but later partially excluded by the hydrostatic assumption and the model resolution. Thus far, the representation of internal wave mixing in ocean models has been relatively crude (e.g. Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002). 395 % the most critical process not reproduced by statistical turbulence models is the activity of 396 % internal waves and their interaction with turbulence. After the Reynolds decomposition, 397 % internal waves are in principle included in the RANS equations, but later partially 398 % excluded by the hydrostatic assumption and the model resolution. 399 % Thus far, the representation of internal wave mixing in ocean models has been relatively crude 400 % (e.g. Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002). 362 401 363 402 … … 371 410 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 372 411 \begin{figure}[!t] \begin{center} 373 \includegraphics[width=1.00\textwidth]{ ./TexFiles/Figures/Fig_ZDF_TKE_time_scheme.pdf}412 \includegraphics[width=1.00\textwidth]{Fig_ZDF_TKE_time_scheme} 374 413 \caption{ \label{Fig_TKE_time_scheme} 375 414 Illustration of the TKE time integration and its links to the momentum and tracer time integration. } … … 550 589 value near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$ 551 590 are calculated from stability function proposed by \citet{Galperin_al_JAS88}, or by \citet{Kantha_Clayson_1994} 552 or one of the two functions suggested by \citet{Canuto_2001} (\np{nn\_stab\_func} = 0, 1, 2 or 3, resp. }).591 or one of the two functions suggested by \citet{Canuto_2001} (\np{nn\_stab\_func} = 0, 1, 2 or 3, resp.). 553 592 The value of $C_{0\mu}$ depends of the choice of the stability function. 554 593 … … 573 612 Examples of performance of the 4 turbulent closure scheme can be found in \citet{Warner_al_OM05}. 574 613 575 % -------------------------------------------------------------------------------------------------------------576 % K Profile Parametrisation (KPP)577 % -------------------------------------------------------------------------------------------------------------578 \subsection{K Profile Parametrisation (KPP) (\key{zdfkpp}) }579 \label{ZDF_kpp}580 581 %--------------------------------------------namkpp--------------------------------------------------------582 \namdisplay{namzdf_kpp}583 %--------------------------------------------------------------------------------------------------------------584 585 The KKP scheme has been implemented by J. Chanut ...586 Options are defined through the \ngn{namzdf\_kpp} namelist variables.587 588 \colorbox{yellow}{Add a description of KPP here.}589 590 614 591 615 % ================================================================ … … 621 645 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 622 646 \begin{figure}[!htb] \begin{center} 623 \includegraphics[width=0.90\textwidth]{ ./TexFiles/Figures/Fig_npc.pdf}647 \includegraphics[width=0.90\textwidth]{Fig_npc} 624 648 \caption{ \label{Fig_npc} 625 649 Example of an unstable density profile treated by the non penetrative … … 636 660 637 661 Options are defined through the \ngn{namzdf} namelist variables. 638 The non-penetrative convective adjustment is used when \np{ln\_zdfnpc} =true.662 The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}~=~\textit{true}. 639 663 It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously 640 664 the statically unstable portion of the water column, but only until the density … … 644 668 (Fig. \ref{Fig_npc}): starting from the top of the ocean, the first instability is 645 669 found. Assume in the following that the instability is located between levels 646 $k$ and $k+1$. The potentialtemperature and salinity in the two levels are670 $k$ and $k+1$. The temperature and salinity in the two levels are 647 671 vertically mixed, conserving the heat and salt contents of the water column. 648 672 The new density is then computed by a linear approximation. If the new … … 664 688 \citep{Madec_al_JPO91, Madec_al_DAO91, Madec_Crepon_Bk91}. 665 689 666 Note that in the current implementation of this algorithm presents several 667 limitations. First, potential density referenced to the sea surface is used to 668 check whether the density profile is stable or not. This is a strong 669 simplification which leads to large errors for realistic ocean simulations. 670 Indeed, many water masses of the world ocean, especially Antarctic Bottom 671 Water, are unstable when represented in surface-referenced potential density. 672 The scheme will erroneously mix them up. Second, the mixing of potential 673 density is assumed to be linear. This assures the convergence of the algorithm 674 even when the equation of state is non-linear. Small static instabilities can thus 675 persist due to cabbeling: they will be treated at the next time step. 676 Third, temperature and salinity, and thus density, are mixed, but the 677 corresponding velocity fields remain unchanged. When using a Richardson 678 Number dependent eddy viscosity, the mixing of momentum is done through 679 the vertical diffusion: after a static adjustment, the Richardson Number is zero 680 and thus the eddy viscosity coefficient is at a maximum. When this convective 681 adjustment algorithm is used with constant vertical eddy viscosity, spurious 682 solutions can occur since the vertical momentum diffusion remains small even 683 after a static adjustment. In that case, we recommend the addition of momentum 684 mixing in a manner that mimics the mixing in temperature and salinity 685 \citep{Speich_PhD92, Speich_al_JPO96}. 690 The current implementation has been modified in order to deal with any non linear 691 equation of seawater (L. Brodeau, personnal communication). 692 Two main differences have been introduced compared to the original algorithm: 693 $(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency 694 (not the the difference in potential density) ; 695 $(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients 696 are vertically mixed in the same way their temperature and salinity has been mixed. 697 These two modifications allow the algorithm to perform properly and accurately 698 with TEOS10 or EOS-80 without having to recompute the expansion coefficients at each 699 mixing iteration. 686 700 687 701 % ------------------------------------------------------------------------------------------------------------- … … 689 703 % ------------------------------------------------------------------------------------------------------------- 690 704 \subsection [Enhanced Vertical Diffusion (\np{ln\_zdfevd})] 691 705 {Enhanced Vertical Diffusion (\np{ln\_zdfevd}=true)} 692 706 \label{ZDF_evd} 693 707 … … 787 801 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 788 802 \begin{figure}[!t] \begin{center} 789 \includegraphics[width=0.99\textwidth]{ ./TexFiles/Figures/Fig_zdfddm.pdf}803 \includegraphics[width=0.99\textwidth]{Fig_zdfddm} 790 804 \caption{ \label{Fig_zdfddm} 791 805 From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ … … 830 844 % Bottom Friction 831 845 % ================================================================ 832 \section [Bottom and top Friction (\textit{zdfbfr})] {BottomFriction (\mdl{zdfbfr} module)}846 \section [Bottom and Top Friction (\textit{zdfbfr})] {Bottom and Top Friction (\mdl{zdfbfr} module)} 833 847 \label{ZDF_bfr} 834 848 … … 838 852 839 853 Options to define the top and bottom friction are defined through the \ngn{nambfr} namelist variables. 840 The top friction is activated only if the ice shelf cavities are opened (\np{ln\_isfcav}~=~true). 841 As the friction processes at the top and bottom are the represented similarly, only the bottom friction is described in detail. 854 The bottom friction represents the friction generated by the bathymetry. 855 The top friction represents the friction generated by the ice shelf/ocean interface. 856 As the friction processes at the top and bottom are treated in similar way, 857 only the bottom friction is described in detail below. 858 842 859 843 860 Both the surface momentum flux (wind stress) and the bottom momentum … … 912 929 $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$. 913 930 This is the default value used in \NEMO. It corresponds to a decay time scale 914 of 115~days. It can be changed by specifying \np{rn\_bfri c1} (namelist parameter).931 of 115~days. It can be changed by specifying \np{rn\_bfri1} (namelist parameter). 915 932 916 933 For the linear friction case the coefficients defined in the general … … 922 939 \end{split} 923 940 \end{equation} 924 When \np{nn\_botfr}=1, the value of $r$ used is \np{rn\_bfri c1}.941 When \np{nn\_botfr}=1, the value of $r$ used is \np{rn\_bfri1}. 925 942 Setting \np{nn\_botfr}=0 is equivalent to setting $r=0$ and leads to a free-slip 926 943 bottom boundary condition. These values are assigned in \mdl{zdfbfr}. … … 929 946 in the \ifile{bfr\_coef} input NetCDF file. The mask values should vary from 0 to 1. 930 947 Locations with a non-zero mask value will have the friction coefficient increased 931 by $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri c1}.948 by $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri1}. 932 949 933 950 % ------------------------------------------------------------------------------------------------------------- … … 949 966 $e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992} 950 967 uses $C_D = 1.4\;10^{-3}$ and $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$. 951 The CME choices have been set as default values (\np{rn\_bfri c2} and \np{rn\_bfeb2}968 The CME choices have been set as default values (\np{rn\_bfri2} and \np{rn\_bfeb2} 952 969 namelist parameters). 953 970 … … 964 981 \end{equation} 965 982 966 The coefficients that control the strength of the non-linear bottom friction are 967 initialised as namelist parameters: $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}. 968 Note for applications which treat tides explicitly a low or even zero value of 969 \np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$ 970 is possible via an externally defined 2D mask array (\np{ln\_bfr2d}=true). 971 See previous section for details. 983 The coefficients that control the strength of the non-linear bottom friction are 984 initialised as namelist parameters: $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}. 985 Note for applications which treat tides explicitly a low or even zero value of 986 \np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$ is possible 987 via an externally defined 2D mask array (\np{ln\_bfr2d}=true). This works in the same way 988 as for the linear bottom friction case with non-zero masked locations increased by 989 $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri2}. 990 991 % ------------------------------------------------------------------------------------------------------------- 992 % Bottom Friction Log-layer 993 % ------------------------------------------------------------------------------------------------------------- 994 \subsection{Log-layer Bottom Friction enhancement (\np{nn\_botfr} = 2, \np{ln\_loglayer} = .true.)} 995 \label{ZDF_bfr_loglayer} 996 997 In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally 998 enhanced using a "law of the wall" scaling. If \np{ln\_loglayer} = .true., $C_D$ is no 999 longer constant but is related to the thickness of the last wet layer in each column by: 1000 1001 \begin{equation} 1002 C_D = \left ( {\kappa \over {\rm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2 1003 \end{equation} 1004 1005 \noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness 1006 length provided via the namelist. 1007 1008 For stability, the drag coefficient is bounded such that it is kept greater or equal to 1009 the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional 1010 namelist parameter: \np{rn\_bfri2\_max}, i.e.: 1011 1012 \begin{equation} 1013 rn\_bfri2 \leq C_D \leq rn\_bfri2\_max 1014 \end{equation} 1015 1016 \noindent Note also that a log-layer enhancement can also be applied to the top boundary 1017 friction if under ice-shelf cavities are in use (\np{ln\_isfcav}=.true.). In this case, the 1018 relevant namelist parameters are \np{rn\_tfrz0}, \np{rn\_tfri2} 1019 and \np{rn\_tfri2\_max}. 972 1020 973 1021 % ------------------------------------------------------------------------------------------------------------- … … 1083 1131 baroclinic and barotropic components which is appropriate when using either the 1084 1132 explicit or filtered surface pressure gradient algorithms (\key{dynspg\_exp} or 1085 {\key{dynspg\_flt}). Extra attention is required, however, when using1133 \key{dynspg\_flt}). Extra attention is required, however, when using 1086 1134 split-explicit time stepping (\key{dynspg\_ts}). In this case the free surface 1087 1135 equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, while the three … … 1198 1246 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1199 1247 \begin{figure}[!t] \begin{center} 1200 \includegraphics[width=0.90\textwidth]{ ./TexFiles/Figures/Fig_ZDF_M2_K1_tmx.pdf}1248 \includegraphics[width=0.90\textwidth]{Fig_ZDF_M2_K1_tmx} 1201 1249 \caption{ \label{Fig_ZDF_M2_K1_tmx} 1202 1250 (a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). } … … 1253 1301 1254 1302 % ================================================================ 1303 % Internal wave-driven mixing 1304 % ================================================================ 1305 \section{Internal wave-driven mixing (\key{zdftmx\_new})} 1306 \label{ZDF_tmx_new} 1307 1308 %--------------------------------------------namzdf_tmx_new------------------------------------------ 1309 \namdisplay{namzdf_tmx_new} 1310 %-------------------------------------------------------------------------------------------------------------- 1311 1312 The parameterization of mixing induced by breaking internal waves is a generalization 1313 of the approach originally proposed by \citet{St_Laurent_al_GRL02}. 1314 A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed, 1315 and the resulting diffusivity is obtained as 1316 \begin{equation} \label{Eq_Kwave} 1317 A^{vT}_{wave} = R_f \,\frac{ \epsilon }{ \rho \, N^2 } 1318 \end{equation} 1319 where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution 1320 of the energy available for mixing. If the \np{ln\_mevar} namelist parameter is set to false, 1321 the mixing efficiency is taken as constant and equal to 1/6 \citep{Osborn_JPO80}. 1322 In the opposite (recommended) case, $R_f$ is instead a function of the turbulence intensity parameter 1323 $Re_b = \frac{ \epsilon}{\nu \, N^2}$, with $\nu$ the molecular viscosity of seawater, 1324 following the model of \cite{Bouffard_Boegman_DAO2013} 1325 and the implementation of \cite{de_lavergne_JPO2016_efficiency}. 1326 Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when the mixing efficiency is constant. 1327 1328 In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary 1329 as a function of $Re_b$ by setting the \np{ln\_tsdiff} parameter to true, a recommended choice). 1330 This parameterization of differential mixing, due to \cite{Jackson_Rehmann_JPO2014}, 1331 is implemented as in \cite{de_lavergne_JPO2016_efficiency}. 1332 1333 The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$, is constructed 1334 from three static maps of column-integrated internal wave energy dissipation, $E_{cri}(i,j)$, 1335 $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures 1336 (de Lavergne et al., in prep): 1337 \begin{align*} 1338 F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\ 1339 F_{pyc}(i,j,k) &\propto N^{n\_p}\\ 1340 F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} } 1341 \end{align*} 1342 In the above formula, $h_{ab}$ denotes the height above bottom, 1343 $h_{wkb}$ denotes the WKB-stretched height above bottom, defined by 1344 \begin{equation*} 1345 h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz' } \; , 1346 \end{equation*} 1347 The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist) controls the stratification-dependence of the pycnocline-intensified dissipation. 1348 It can take values of 1 (recommended) or 2. 1349 Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of 1350 the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps. 1351 $h_{cri}$ is related to the large-scale topography of the ocean (etopo2) 1352 and $h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of 1353 the abyssal hill topography \citep{Goff_JGR2010} and the latitude. 1354 1355 % ================================================================ 1356 1357 1358 1359 \end{document}
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