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NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex
r11584 r11596 4 4 \begin{document} 5 5 6 % ================================================================7 % Chapter 1 Model Basics8 % ================================================================9 6 \chapter{Model Basics} 10 7 \label{chap:MB} … … 12 9 \chaptertoc 13 10 14 \newpage 15 16 % ================================================================ 17 % Primitive Equations 18 % ================================================================ 11 %% ================================================================================================= 19 12 \section{Primitive equations} 20 13 \label{sec:MB_PE} 21 14 22 % ------------------------------------------------------------------------------------------------------------- 23 % Vector Invariant Formulation 24 % ------------------------------------------------------------------------------------------------------------- 25 15 %% ================================================================================================= 26 16 \subsection{Vector invariant formulation} 27 17 \label{subsec:MB_PE_vector} … … 33 23 34 24 \begin{enumerate} 35 \item 36 \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods 25 \item \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods 37 26 that follow the Earth's bulge; these spheroids are approximated by spheres with 38 27 gravity locally vertical (parallel to the Earth's radius) and independent of latitude 39 28 \citep[][section 2]{white.hoskins.ea_QJRMS05}. 40 \item 41 \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius 42 \item 43 \textit{turbulent closure hypothesis}: the turbulent fluxes 29 \item \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius 30 \item \textit{turbulent closure hypothesis}: the turbulent fluxes 44 31 (which represent the effect of small scale processes on the large-scale) 45 32 are expressed in terms of large-scale features 46 \item 47 \textit{Boussinesq hypothesis}: density variations are neglected except in their contribution to 33 \item \textit{Boussinesq hypothesis}: density variations are neglected except in their contribution to 48 34 the buoyancy force 49 35 \begin{equation} … … 51 37 \rho = \rho \ (T,S,p) 52 38 \end{equation} 53 \item 54 \textit{Hydrostatic hypothesis}: the vertical momentum equation is reduced to a balance between 39 \item \textit{Hydrostatic hypothesis}: the vertical momentum equation is reduced to a balance between 55 40 the vertical pressure gradient and the buoyancy force 56 41 (this removes convective processes from the initial Navier-Stokes equations and so … … 60 45 \pd[p]{z} = - \rho \ g 61 46 \end{equation} 62 \item 63 \textit{Incompressibility hypothesis}: the three dimensional divergence of the velocity vector $\vect U$ 47 \item \textit{Incompressibility hypothesis}: the three dimensional divergence of the velocity vector $\vect U$ 64 48 is assumed to be zero. 65 49 \begin{equation} … … 67 51 \nabla \cdot \vect U = 0 68 52 \end{equation} 69 \item 70 \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected. 53 \item \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected. 71 54 These terms may be non-negligible where the Brunt-Vaisala frequency $N$ is small, either in the deep ocean or 72 55 in the sub-mesoscale motions of the mixed layer, or near the equator \citep[][section 1]{white.hoskins.ea_QJRMS05}. … … 108 91 Their nature and formulation are discussed in \autoref{sec:MB_zdf_ldf} and \autoref{subsec:MB_boundary_condition}. 109 92 110 % ------------------------------------------------------------------------------------------------------------- 111 % Boundary condition 112 % ------------------------------------------------------------------------------------------------------------- 93 %% ================================================================================================= 113 94 \subsection{Boundary conditions} 114 95 \label{subsec:MB_boundary_condition} … … 128 109 the other components of the earth system. 129 110 130 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>131 111 \begin{figure}[!ht] 132 112 \centering … … 138 118 \label{fig:MB_ocean_bc} 139 119 \end{figure} 140 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>141 120 142 121 \begin{description} 143 \item[Land - ocean interface:] 144 the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. 122 \item [Land - ocean interface:] the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. 145 123 Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths. 146 124 It can be neglected for short range integrations but has to be taken into account for long term integrations as … … 148 126 It is required in order to close the water cycle of the climate system. 149 127 It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths. 150 \item[Solid earth - ocean interface:] 151 heat and salt fluxes through the sea floor are small, except in special areas of little extent. 128 \item [Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small, except in special areas of little extent. 152 129 They are usually neglected in the model 153 130 \footnote{ … … 171 148 $\vect D^{\vect U}$ in \autoref{eq:MB_PE_dyn}. 172 149 It is discussed in \autoref{eq:MB_zdf}.% and Chap. III.6 to 9. 173 \item[Atmosphere - ocean interface:] 174 the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) 150 \item [Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) 175 151 leads to: 176 152 \[ … … 181 157 leads to the continuity of pressure across the interface $z = \eta$. 182 158 The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat. 183 \item[Sea ice - ocean interface:] 184 the ocean and sea ice exchange heat, salt, fresh water and momentum. 159 \item [Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water and momentum. 185 160 The sea surface temperature is constrained to be at the freezing point at the interface. 186 161 Sea ice salinity is very low ($\sim4-6 \, psu$) compared to those of the ocean ($\sim34 \, psu$). … … 188 163 \end{description} 189 164 190 % ================================================================ 191 % The Horizontal Pressure Gradient 192 % ================================================================ 165 %% ================================================================================================= 193 166 \section{Horizontal pressure gradient} 194 167 \label{sec:MB_hor_pg} 195 168 196 % ------------------------------------------------------------------------------------------------------------- 197 % Pressure Formulation 198 % ------------------------------------------------------------------------------------------------------------- 169 %% ================================================================================================= 199 170 \subsection{Pressure formulation} 200 171 \label{subsec:MB_p_formulation} … … 228 199 Only the free surface formulation is now described in this document (see the next sub-section). 229 200 230 % ------------------------------------------------------------------------------------------------------------- 231 % Free Surface Formulation 232 % ------------------------------------------------------------------------------------------------------------- 201 %% ================================================================================================= 233 202 \subsection{Free surface formulation} 234 203 \label{subsec:MB_free_surface} … … 280 249 (see \autoref{subsec:DYN_spg_ts}). 281 250 282 % ================================================================ 283 % Curvilinear z-coordinate System 284 % ================================================================ 251 %% ================================================================================================= 285 252 \section{Curvilinear \textit{z-}coordinate system} 286 253 \label{sec:MB_zco} 287 254 288 % ------------------------------------------------------------------------------------------------------------- 289 % Tensorial Formalism 290 % ------------------------------------------------------------------------------------------------------------- 255 %% ================================================================================================= 291 256 \subsection{Tensorial formalism} 292 257 \label{subsec:MB_tensorial} … … 338 303 \label{fig:MB_referential} 339 304 \end{figure} 340 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>341 305 342 306 Since the ocean depth is far smaller than the earth's radius, $a + z$, can be replaced by $a$ in … … 373 337 where $q$ is a scalar quantity and $\vect A = (a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinates system. 374 338 375 % ------------------------------------------------------------------------------------------------------------- 376 % Continuous Model Equations 377 % ------------------------------------------------------------------------------------------------------------- 339 %% ================================================================================================= 378 340 \subsection{Continuous model equations} 379 341 \label{subsec:MB_zco_Eq} … … 496 458 497 459 \begin{itemize} 498 \item 499 \textbf{Vector invariant form of the momentum equations}: 460 \item \textbf{Vector invariant form of the momentum equations}: 500 461 \begin{equation} 501 462 \label{eq:MB_dyn_vect} … … 510 471 \end{split} 511 472 \end{equation} 512 \item 513 \textbf{flux form of the momentum equations}: 473 \item \textbf{flux form of the momentum equations}: 514 474 % \label{eq:MB_dyn_flux} 515 475 \begin{multline*} … … 544 504 where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:MB_div_Uh}. 545 505 546 \item 547 \textbf{tracer equations}: 506 \item \textbf{tracer equations}: 548 507 \begin{equation} 549 508 \begin{split} … … 562 521 are discussed in \autoref{chap:SBC}. 563 522 564 \newpage 565 566 % ================================================================ 567 % Curvilinear generalised vertical coordinate System 568 % ================================================================ 523 %% ================================================================================================= 569 524 \section{Curvilinear generalised vertical coordinate system} 570 525 \label{sec:MB_gco} … … 647 602 %} 648 603 649 % ------------------------------------------------------------------------------------------------------------- 650 % The s-coordinate Formulation 651 % ------------------------------------------------------------------------------------------------------------- 604 %% ================================================================================================= 652 605 \subsection{\textit{S}-coordinate formulation} 653 606 … … 737 690 } 738 691 739 % ------------------------------------------------------------------------------------------------------------- 740 % Curvilinear \zstar-coordinate System 741 % ------------------------------------------------------------------------------------------------------------- 692 %% ================================================================================================= 742 693 \subsection{Curvilinear \zstar-coordinate system} 743 694 \label{subsec:MB_zco_star} 744 695 745 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>746 696 \begin{figure}[!b] 747 697 \centering … … 754 704 \label{fig:MB_z_zstar} 755 705 \end{figure} 756 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>757 706 758 707 In this case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. … … 827 776 %end MOM doc %%% 828 777 829 \newpage 830 831 % ------------------------------------------------------------------------------------------------------------- 832 % Terrain following coordinate System 833 % ------------------------------------------------------------------------------------------------------------- 778 %% ================================================================================================= 834 779 \subsection{Curvilinear terrain-following \textit{s}--coordinate} 835 780 \label{subsec:MB_sco} 836 781 837 % ------------------------------------------------------------------------------------------------------------- 838 % Introduction 839 % ------------------------------------------------------------------------------------------------------------- 782 %% ================================================================================================= 840 783 \subsubsection{Introduction} 841 784 … … 918 861 It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate. 919 862 920 % ------------------------------------------------------------------------------------------------------------- 921 % Curvilinear z-tilde coordinate System 922 % ------------------------------------------------------------------------------------------------------------- 863 %% ================================================================================================= 923 864 \subsection{\texorpdfstring{Curvilinear \ztilde-coordinate}{}} 924 865 \label{subsec:MB_zco_tilde} … … 929 870 Its use is therefore not recommended. 930 871 931 \newpage 932 933 % ================================================================ 934 % Subgrid Scale Physics 935 % ================================================================ 872 %% ================================================================================================= 936 873 \section{Subgrid scale physics} 937 874 \label{sec:MB_zdf_ldf} … … 957 894 The formulation of these terms and their underlying physics are briefly discussed in the next two subsections. 958 895 959 % ------------------------------------------------------------------------------------------------------------- 960 % Vertical Subgrid Scale Physics 961 % ------------------------------------------------------------------------------------------------------------- 896 %% ================================================================================================= 962 897 \subsection{Vertical subgrid scale physics} 963 898 \label{subsec:MB_zdf} … … 991 926 The choices available in \NEMO\ are discussed in \autoref{chap:ZDF}). 992 927 993 % ------------------------------------------------------------------------------------------------------------- 994 % Lateral Diffusive and Viscous Operators Formulation 995 % ------------------------------------------------------------------------------------------------------------- 928 %% ================================================================================================= 996 929 \subsection{Formulation of the lateral diffusive and viscous operators} 997 930 \label{subsec:MB_ldf} … … 1047 980 and UBS advection schemes when flux form is chosen for the momentum advection. 1048 981 982 %% ================================================================================================= 1049 983 \subsubsection{Lateral laplacian tracer diffusive operator} 1050 984 … … 1088 1022 while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$. 1089 1023 1024 %% ================================================================================================= 1090 1025 \subsubsection{Eddy induced velocity} 1091 1026 … … 1124 1059 The latter strategy is used in \NEMO\ (cf. \autoref{chap:LDF}). 1125 1060 1061 %% ================================================================================================= 1126 1062 \subsubsection{Lateral bilaplacian tracer diffusive operator} 1127 1063 … … 1135 1071 the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. 1136 1072 1073 %% ================================================================================================= 1137 1074 \subsubsection{Lateral Laplacian momentum diffusive operator} 1138 1075 … … 1167 1104 a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}. 1168 1105 1106 %% ================================================================================================= 1169 1107 \subsubsection{Lateral bilaplacian momentum diffusive operator} 1170 1108
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