Changeset 11512 for NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/chap_model_basics.tex
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NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/chap_model_basics.tex
r11353 r11512 1 1 2 \documentclass[../main/NEMO_manual]{subfiles} 2 3 … … 8 9 \chapter{Model Basics} 9 10 \label{chap:PE} 10 \minitoc 11 12 \chaptertoc 11 13 12 14 \newpage … … 19 21 20 22 % ------------------------------------------------------------------------------------------------------------- 21 % Vector Invariant Formulation 23 % Vector Invariant Formulation 22 24 % ------------------------------------------------------------------------------------------------------------- 23 25 … … 26 28 27 29 The ocean is a fluid that can be described to a good approximation by the primitive equations, 28 \ie the Navier-Stokes equations along with a nonlinear equation of state which30 \ie\ the Navier-Stokes equations along with a nonlinear equation of state which 29 31 couples the two active tracers (temperature and salinity) to the fluid velocity, 30 32 plus the following additional assumptions made from scale considerations: … … 34 36 \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods 35 37 that follow the Earth's bulge; these spheroids are approximated by spheres with 36 gravity locally vertical (parallel to the Earth's radius) and independent of latitude 37 \citep[][section 2]{white.hoskins.ea_QJRMS05}. 38 gravity locally vertical (parallel to the Earth's radius) and independent of latitude 39 \citep[][section 2]{white.hoskins.ea_QJRMS05}. 38 40 \item 39 41 \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius … … 65 67 \nabla \cdot \vect U = 0 66 68 \end{equation} 67 \item 68 \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected. 69 \item 70 \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected. 69 71 These terms may be non-negligible where the Brunt-Vaisala frequency $N$ is small, either in the deep ocean or 70 in the sub-mesoscale motions of the mixed layer, or near the equator \citep[][section 1]{white.hoskins.ea_QJRMS05}. 71 They can be consistently included as part of the ocean dynamics \citep[][section 3(d)]{white.hoskins.ea_QJRMS05} and are 72 retained in the MIT ocean model. 72 in the sub-mesoscale motions of the mixed layer, or near the equator \citep[][section 1]{white.hoskins.ea_QJRMS05}. 73 They can be consistently included as part of the ocean dynamics \citep[][section 3(d)]{white.hoskins.ea_QJRMS05} and are 74 retained in the MIT ocean model. 73 75 \end{enumerate} 74 76 … … 76 78 it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the Earth such that 77 79 $k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, 78 \ie tangent to the geopotential surfaces.80 \ie\ tangent to the geopotential surfaces. 79 81 Let us define the following variables: $\vect U$ the vector velocity, $\vect U = \vect U_h + w \, \vect k$ 80 (the subscript $h$ denotes the local horizontal vector, \ie over the $(i,j)$ plane),82 (the subscript $h$ denotes the local horizontal vector, \ie\ over the $(i,j)$ plane), 81 83 $T$ the potential temperature, $S$ the salinity, $\rho$ the \textit{in situ} density. 82 84 The vector invariant form of the primitive equations in the $(i,j,k)$ vector system provides … … 115 117 an air-sea or ice-sea interface at its top. 116 118 These boundaries can be defined by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,k,t)$, 117 where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface 118 (discretisation can introduce additional artificial ``side-wall'' boundaries). 119 Both $H$ and $\eta$ are referenced to a surface of constant geopotential (\ie a mean sea surface height) on which $z = 0$.119 where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface 120 (discretisation can introduce additional artificial ``side-wall'' boundaries). 121 Both $H$ and $\eta$ are referenced to a surface of constant geopotential (\ie\ a mean sea surface height) on which $z = 0$. 120 122 (\autoref{fig:ocean_bc}). 121 123 Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with … … 153 155 \footnote{ 154 156 In fact, it has been shown that the heat flux associated with the solid Earth cooling 155 (\ie the geothermal heating) is not negligible for the thermohaline circulation of the world ocean157 (\ie\ the geothermal heating) is not negligible for the thermohaline circulation of the world ocean 156 158 (see \autoref{subsec:TRA_bbc}). 157 159 }. 158 160 The boundary condition is thus set to no flux of heat and salt across solid boundaries. 159 161 For momentum, the situation is different. There is no flow across solid boundaries, 160 \ie the velocity normal to the ocean bottom and coastlines is zero (in other words,162 \ie\ the velocity normal to the ocean bottom and coastlines is zero (in other words, 161 163 the bottom velocity is parallel to solid boundaries). This kinematic boundary condition 162 164 can be expressed as: … … 221 223 the time step has to be very short when they are present in the model. 222 224 The latter strategy filters out these waves since the rigid lid approximation implies $\eta = 0$, 223 \ie the sea surface is the surface $z = 0$.225 \ie\ the sea surface is the surface $z = 0$. 224 226 This well known approximation increases the surface wave speed to infinity and 225 modifies certain other longwave dynamics (\eg barotropic Rossby or planetary waves).227 modifies certain other longwave dynamics (\eg\ barotropic Rossby or planetary waves). 226 228 The rigid-lid hypothesis is an obsolescent feature in modern OGCMs. 227 229 It has been available until the release 3.1 of \NEMO, and it has been removed in release 3.2 and followings. … … 246 248 Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as 247 249 a class of solution of the primitive equations. 248 These waves are barotropic (\ie nearly independent of depth) and their phase speed is quite high.250 These waves are barotropic (\ie\ nearly independent of depth) and their phase speed is quite high. 249 251 Their time scale is short with respect to the other processes described by the primitive equations. 250 252 … … 275 277 the implicit scheme \citep{dukowicz.smith_JGR94} or 276 278 the addition of a filtering force in the momentum equation \citep{roullet.madec_JGR00}. 277 With the present release, \NEMO offers the choice between279 With the present release, \NEMO\ offers the choice between 278 280 an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or 279 281 a split-explicit scheme strongly inspired the one proposed by \citet{shchepetkin.mcwilliams_OM05} … … 293 295 294 296 In many ocean circulation problems, the flow field has regions of enhanced dynamics 295 (\ie surface layers, western boundary currents, equatorial currents, or ocean fronts).297 (\ie\ surface layers, western boundary currents, equatorial currents, or ocean fronts). 296 298 The representation of such dynamical processes can be improved by 297 299 specifically increasing the model resolution in these regions. … … 313 315 $(i,j,k)$ linked to the earth such that 314 316 $k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, 315 \ie along geopotential surfaces (\autoref{fig:referential}).317 \ie\ along geopotential surfaces (\autoref{fig:referential}). 316 318 Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by 317 319 the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and … … 445 447 This is the so-called \textit{vector invariant form} of the momentum advection term. 446 448 For some purposes, it can be advantageous to write this term in the so-called flux form, 447 \ie to write it as the divergence of fluxes.449 \ie\ to write it as the divergence of fluxes. 448 450 For example, the first component of \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows: 449 451 \begin{alignat*}{2} … … 484 486 the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation) 485 487 and the second one is due to the curvilinear nature of the coordinate system used. 486 The latter is called the \textit{metric} term and can be viewed as a modification of the Coriolis parameter: 488 The latter is called the \textit{metric} term and can be viewed as a modification of the Coriolis parameter: 487 489 \[ 488 490 % \label{eq:PE_cor+metric} … … 491 493 492 494 Note that in the case of geographical coordinate, 493 \ie when $(i,j) \to (\lambda,\varphi)$ and $(e_1,e_2) \to (a \, \cos \varphi,a)$,495 \ie\ when $(i,j) \to (\lambda,\varphi)$ and $(e_1,e_2) \to (a \, \cos \varphi,a)$, 494 496 we recover the commonly used modification of the Coriolis parameter $f \to f + (u / a) \tan \varphi$. 495 497 … … 508 510 &+ D_u^{\vect U} + F_u^{\vect U} \\ 509 511 \pd[v]{t} = &- (\zeta + f) \, u - \frac{1}{2 e_2} \pd[]{j} (u^2 + v^2) 510 - \frac{1}{e_3} w \pd[v]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) \\ 512 - \frac{1}{e_3} w \pd[v]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) \\ 511 513 &+ D_v^{\vect U} + F_v^{\vect U} 512 514 \end{split} … … 540 542 \[ 541 543 % \label{eq:w_diag} 542 \pd[w]{k} = - \chi \; e_3 \qquad 544 \pd[w]{k} = - \chi \; e_3 \qquad 543 545 % \label{eq:hp_diag} 544 546 \pd[p_h]{k} = - \rho \; g \; e_3 … … 546 548 where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. 547 549 548 \item 550 \item 549 551 \textbf{tracer equations}: 550 552 \begin{equation} … … 579 581 Therefore, in order to represent the ocean with respect to 580 582 the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height 581 \eg an \zstar-coordinate;583 \eg\ an \zstar-coordinate; 582 584 for the second point, a space variation to fit the change of bottom topography 583 \eg a terrain-following or $\sigma$-coordinate;585 \eg\ a terrain-following or $\sigma$-coordinate; 584 586 and for the third point, one will be tempted to use a space and time dependent coordinate that 585 follows the isopycnal surfaces, \eg an isopycnic coordinate.587 follows the isopycnal surfaces, \eg\ an isopycnic coordinate. 586 588 587 589 In order to satisfy two or more constraints one can even be tempted to mixed these coordinate systems, as in … … 666 668 \sigma_2 = \frac{1}{e_2} \; \lt. \pd[z]{j} \rt|_s 667 669 \end{equation} 668 We also introduce $\omega$, a dia-surface velocity component, defined as the velocity 670 We also introduce $\omega$, a dia-surface velocity component, defined as the velocity 669 671 relative to the moving $s$-surfaces and normal to them: 670 672 \[ … … 761 763 762 764 In this case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 763 These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO web site.765 These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 764 766 765 767 The \zstar coordinate approach is an unapproximated, non-linear free surface implementation which allows one to … … 784 786 \[ 785 787 % \label{eq:PE_zstar_2} 786 \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) . 788 \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) . 787 789 \] 788 790 Since the vertical displacement of the free surface is incorporated in the vertical coordinate \zstar, … … 831 833 %end MOM doc %%% 832 834 833 \newpage 835 \newpage 834 836 835 837 % ------------------------------------------------------------------------------------------------------------- … … 911 913 In contrast, the ocean will stay at rest in a $z$-model. 912 914 As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below 913 the strongly stratified portion of the water column (\ie the main thermocline) \citep{madec.delecluse.ea_JPO96}.915 the strongly stratified portion of the water column (\ie\ the main thermocline) \citep{madec.delecluse.ea_JPO96}. 914 916 An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces 915 917 (see \autoref{subsec:PE_ldf}). … … 929 931 930 932 The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}. 931 It is available in \NEMO since the version 3.4 and is more robust in version 4.0 than previously.933 It is available in \NEMO\ since the version 3.4 and is more robust in version 4.0 than previously. 932 934 Nevertheless, it is currently not robust enough to be used in all possible configurations. 933 935 Its use is therefore not recommended. 934 936 935 \newpage 937 \newpage 936 938 937 939 % ================================================================ … … 947 949 must be represented entirely in terms of large-scale patterns to close the equations. 948 950 These effects appear in the equations as the divergence of turbulent fluxes 949 (\ie fluxes associated with the mean correlation of small scale perturbations).951 (\ie\ fluxes associated with the mean correlation of small scale perturbations). 950 952 Assuming a turbulent closure hypothesis is equivalent to choose a formulation for these fluxes. 951 953 It is usually called the subgrid scale physics. … … 991 993 All the vertical physics is embedded in the specification of the eddy coefficients. 992 994 They can be assumed to be either constant, or function of the local fluid properties 993 (\eg Richardson number, Brunt-Vais\"{a}l\"{a} frequency, distance from the boundary ...),995 (\eg\ Richardson number, Brunt-Vais\"{a}l\"{a} frequency, distance from the boundary ...), 994 996 or computed from a turbulent closure model. 995 The choices available in \NEMO are discussed in \autoref{chap:ZDF}).997 The choices available in \NEMO\ are discussed in \autoref{chap:ZDF}). 996 998 997 999 % ------------------------------------------------------------------------------------------------------------- … … 1006 1008 and a sub mesoscale turbulence which is never explicitly solved even partially, but always parameterized. 1007 1009 The formulation of lateral eddy fluxes depends on whether the mesoscale is below or above the grid-spacing 1008 (\ie the model is eddy-resolving or not).1010 (\ie\ the model is eddy-resolving or not). 1009 1011 1010 1012 In non-eddy-resolving configurations, the closure is similar to that used for the vertical physics. … … 1014 1016 (or more precisely neutral surfaces \cite{mcdougall_JPO87}) rather than across them. 1015 1017 As the slope of neutral surfaces is small in the ocean, a common approximation is to assume that 1016 the `lateral' direction is the horizontal, \ie the lateral mixing is performed along geopotential surfaces.1018 the `lateral' direction is the horizontal, \ie\ the lateral mixing is performed along geopotential surfaces. 1017 1019 This leads to a geopotential second order operator for lateral subgrid scale physics. 1018 1020 This assumption can be relaxed: the eddy-induced turbulent fluxes can be better approached by assuming that … … 1021 1023 it has components in the three space directions. 1022 1024 However, 1023 both horizontal and isoneutral operators have no effect on mean (\ie large scale) potential energy whereas1025 both horizontal and isoneutral operators have no effect on mean (\ie\ large scale) potential energy whereas 1024 1026 potential energy is a main source of turbulence (through baroclinic instabilities). 1025 1027 \citet{gent.mcwilliams_JPO90} proposed a parameterisation of mesoscale eddy-induced turbulence which … … 1065 1067 \end{equation} 1066 1068 where $r_1$ and $r_2$ are the slopes between the surface along which the diffusive operator acts and 1067 the model level (\eg $z$- or $s$-surfaces).1069 the model level (\eg\ $z$- or $s$-surfaces). 1068 1070 Note that the formulation \autoref{eq:PE_iso_tensor} is exact for 1069 1071 the rotation between geopotential and $s$-surfaces, … … 1112 1114 (or equivalently the isoneutral thickness diffusivity coefficient), 1113 1115 and $\tilde r_1$ and $\tilde r_2$ are the slopes between isoneutral and \textit{geopotential} surfaces. 1114 Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate: 1116 Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate: 1115 1117 \begin{align} 1116 1118 \label{eq:PE_slopes_eiv} … … 1126 1128 This can be achieved in a model by tapering either the eddy coefficient or the slopes to zero in the vicinity of 1127 1129 the boundaries. 1128 The latter strategy is used in \NEMO (cf. \autoref{chap:LDF}).1130 The latter strategy is used in \NEMO\ (cf. \autoref{chap:LDF}). 1129 1131 1130 1132 \subsubsection{Lateral bilaplacian tracer diffusive operator} … … 1148 1150 - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\ 1149 1151 &= \lt( \frac{1}{e_1} \pd[ \lt( A^{lm} \chi \rt) ]{i} \rt. 1150 - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} , 1152 - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} , 1151 1153 \frac{1}{e_2} \pd[ \lt( A^{lm} \chi \rt) ]{j} 1152 1154 \lt. + \frac{1}{e_1 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{i} \rt) … … 1157 1159 Unfortunately, it is only available in \textit{iso-level} direction. 1158 1160 When a rotation is required 1159 (\ie geopotential diffusion in $s$-coordinates or isoneutral diffusion in both $z$- and $s$-coordinates),1161 (\ie\ geopotential diffusion in $s$-coordinates or isoneutral diffusion in both $z$- and $s$-coordinates), 1160 1162 the $u$ and $v$-fields are considered as independent scalar fields, so that the diffusive operator is given by: 1161 1163 \begin{gather*} … … 1167 1169 It is the same expression as those used for diffusive operator on tracers. 1168 1170 It must be emphasised that such a formulation is only exact in a Cartesian coordinate system, 1169 \ie on a $f$- or $\beta$-plane, not on the sphere.1171 \ie\ on a $f$- or $\beta$-plane, not on the sphere. 1170 1172 It is also a very good approximation in vicinity of the Equator in 1171 1173 a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}.
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