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NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex
r10544 r11123 258 258 If further, an approximative conservation of heat and salt contents is sufficient for the problem solved, 259 259 then it is sufficient to solve a linearized version of \autoref{eq:PE_ssh}, 260 which still allows to take into account freshwater fluxes applied at the ocean surface \citep{ Roullet_Madec_JGR00}.260 which still allows to take into account freshwater fluxes applied at the ocean surface \citep{roullet.madec_JGR00}. 261 261 Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost. 262 262 263 263 The filtering of EGWs in models with a free surface is usually a matter of discretisation of 264 264 the temporal derivatives, 265 using a split-explicit method \citep{ Killworth_al_JPO91, Zhang_Endoh_JGR92} or266 the implicit scheme \citep{ Dukowicz1994} or267 the addition of a filtering force in the momentum equation \citep{ Roullet_Madec_JGR00}.265 using a split-explicit method \citep{killworth.webb.ea_JPO91, zhang.endoh_JGR92} or 266 the implicit scheme \citep{dukowicz.smith_JGR94} or 267 the addition of a filtering force in the momentum equation \citep{roullet.madec_JGR00}. 268 268 With the present release, \NEMO offers the choice between 269 269 an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or 270 a split-explicit scheme strongly inspired the one proposed by \citet{ Shchepetkin_McWilliams_OM05}270 a split-explicit scheme strongly inspired the one proposed by \citet{shchepetkin.mcwilliams_OM05} 271 271 (see \autoref{subsec:DYN_spg_ts}). 272 272 … … 292 292 cannot be easily treated in a global model without filtering. 293 293 A solution consists of introducing an appropriate coordinate transformation that 294 shifts the singular point onto land \citep{ Madec_Imbard_CD96, Murray_JCP96}.294 shifts the singular point onto land \citep{madec.imbard_CD96, murray_JCP96}. 295 295 As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems. 296 296 An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism. … … 298 298 Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation), 299 299 with preservation of the local vertical. Here we give the simplified equations for this particular case. 300 The general case is detailed by \citet{ Eiseman1980} in their survey of the conservation laws of fluid dynamics.300 The general case is detailed by \citet{eiseman.stone_SR80} in their survey of the conservation laws of fluid dynamics. 301 301 302 302 Let $(i,j,k)$ be a set of orthogonal curvilinear coordinates on … … 577 577 In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in 578 578 HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at 579 the ocean bottom) \citep{ Chassignet_al_JPO03} or579 the ocean bottom) \citep{chassignet.smith.ea_JPO03} or 580 580 OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere) 581 \citep{ Madec_al_JPO96} among others.581 \citep{madec.delecluse.ea_JPO96} among others. 582 582 583 583 In fact one is totally free to choose any space and time vertical coordinate by … … 592 592 the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through 593 593 \autoref{eq:PE_s}. 594 This so-called \textit{generalised vertical coordinate} \citep{ Kasahara_MWR74} is in fact594 This so-called \textit{generalised vertical coordinate} \citep{kasahara_MWR74} is in fact 595 595 an Arbitrary Lagrangian--Eulerian (ALE) coordinate. 596 596 Indeed, choosing an expression for $s$ is an arbitrary choice that determines 597 597 which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and 598 598 which part will be used to move them (Lagrangian part). 599 The coordinate is also sometime referenced as an adaptive coordinate \citep{ Hofmeister_al_OM09},599 The coordinate is also sometime referenced as an adaptive coordinate \citep{hofmeister.burchard.ea_OM10}, 600 600 since the coordinate system is adapted in the course of the simulation. 601 601 Its most often used implementation is via an ALE algorithm, 602 602 in which a pure lagrangian step is followed by regridding and remapping steps, 603 603 the later step implicitly embedding the vertical advection 604 \citep{ Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}.605 Here we follow the \citep{ Kasahara_MWR74} strategy:604 \citep{hirt.amsden.ea_JCP74, chassignet.smith.ea_JPO03, white.adcroft.ea_JCP09}. 605 Here we follow the \citep{kasahara_MWR74} strategy: 606 606 a regridding step (an update of the vertical coordinate) followed by an eulerian step with 607 607 an explicit computation of vertical advection relative to the moving s-surfaces. … … 744 744 (b) $z$-coordinate in non-linear free surface case ; 745 745 (c) re-scaled height coordinate 746 (become popular as the \zstar-coordinate \citep{ Adcroft_Campin_OM04}).746 (become popular as the \zstar-coordinate \citep{adcroft.campin_OM04}). 747 747 } 748 748 \end{center} … … 751 751 752 752 In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 753 These coordinates systems is presented in a report \citep{ Levier2007} available on the \NEMO web site.753 These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO web site. 754 754 755 755 The \zstar coordinate approach is an unapproximated, non-linear free surface implementation which allows one to 756 deal with large amplitude free-surface variations relative to the vertical resolution \citep{ Adcroft_Campin_OM04}.756 deal with large amplitude free-surface variations relative to the vertical resolution \citep{adcroft.campin_OM04}. 757 757 In the \zstar formulation, 758 758 the variation of the column thickness due to sea-surface undulations is not concentrated in the surface level, … … 805 805 The quasi -horizontal nature of the coordinate surfaces also facilitates the implementation of 806 806 neutral physics parameterizations in \zstar models using the same techniques as in $z$-models 807 (see Chapters 13-16 of \cite{ Griffies_Bk04}) for a discussion of neutral physics in $z$-models,807 (see Chapters 13-16 of \cite{griffies_bk04}) for a discussion of neutral physics in $z$-models, 808 808 as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 809 809 … … 849 849 The response to such a velocity field often leads to numerical dispersion effects. 850 850 One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of 851 a full step one \cite{ Pacanowski_Gnanadesikan_MWR98}.851 a full step one \cite{pacanowski.gnanadesikan_MWR98}. 852 852 Another solution is to introduce a terrain-following coordinate system (hereafter $s$-coordinate). 853 853 … … 876 876 introduces a truncation error that is not present in a $z$-model. 877 877 In the special case of a $\sigma$-coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), 878 \citet{ Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude of this truncation error.878 \citet{haney_JPO91} and \citet{beckmann.haidvogel_JPO93} have given estimates of the magnitude of this truncation error. 879 879 It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state, 880 880 and the finite difference scheme. … … 884 884 The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive. 885 885 This problem can be at least partially overcome by mixing $s$-coordinate and 886 step-like representation of bottom topography \citep{ Gerdes1993a,Gerdes1993b,Madec_al_JPO96}.886 step-like representation of bottom topography \citep{gerdes_JGR93*a,gerdes_JGR93*b,madec.delecluse.ea_JPO96}. 887 887 However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for 888 888 a realistic bottom topography: … … 904 904 In contrast, the ocean will stay at rest in a $z$-model. 905 905 As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below 906 the strongly stratified portion of the water column (\ie the main thermocline) \citep{ Madec_al_JPO96}.906 the strongly stratified portion of the water column (\ie the main thermocline) \citep{madec.delecluse.ea_JPO96}. 907 907 An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces 908 908 (see \autoref{subsec:PE_ldf}). … … 910 910 strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}). 911 911 912 The $s$-coordinates introduced here \citep{ Lott_al_OM90,Madec_al_JPO96} differ mainly in two aspects from912 The $s$-coordinates introduced here \citep{lott.madec.ea_OM90,madec.delecluse.ea_JPO96} differ mainly in two aspects from 913 913 similar models: 914 914 it allows a representation of bottom topography with mixed full or partial step-like/terrain following topography; … … 921 921 \label{subsec:PE_zco_tilde} 922 922 923 The \ztilde -coordinate has been developed by \citet{ Leclair_Madec_OM11}.923 The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}. 924 924 It is available in \NEMO since the version 3.4. 925 925 Nevertheless, it is currently not robust enough to be used in all possible configurations. … … 1005 1005 The resulting lateral diffusive and dissipative operators are of second order. 1006 1006 Observations show that lateral mixing induced by mesoscale turbulence tends to be along isopycnal surfaces 1007 (or more precisely neutral surfaces \cite{ McDougall1987}) rather than across them.1007 (or more precisely neutral surfaces \cite{mcdougall_JPO87}) rather than across them. 1008 1008 As the slope of neutral surfaces is small in the ocean, a common approximation is to assume that 1009 1009 the `lateral' direction is the horizontal, \ie the lateral mixing is performed along geopotential surfaces. … … 1016 1016 both horizontal and isoneutral operators have no effect on mean (\ie large scale) potential energy whereas 1017 1017 potential energy is a main source of turbulence (through baroclinic instabilities). 1018 \citet{ Gent1990} have proposed a parameterisation of mesoscale eddy-induced turbulence which1018 \citet{gent.mcwilliams_JPO90} have proposed a parameterisation of mesoscale eddy-induced turbulence which 1019 1019 associates an eddy-induced velocity to the isoneutral diffusion. 1020 1020 Its mean effect is to reduce the mean potential energy of the ocean. … … 1040 1040 There are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are: 1041 1041 Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces, 1042 \citet{ Gent1990} parameterisation, and various slightly diffusive advection schemes.1042 \citet{gent.mcwilliams_JPO90} parameterisation, and various slightly diffusive advection schemes. 1043 1043 For momentum, the main ones are: Laplacian and bilaplacian operators acting along geopotential surfaces, 1044 1044 and UBS advection schemes when flux form is chosen for the momentum advection. … … 1062 1062 the rotation between geopotential and $s$-surfaces, 1063 1063 while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces. 1064 Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{ Cox1987}.1064 Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{cox_OM87}. 1065 1065 First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and 1066 1066 dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. … … 1087 1087 \subsubsection{Eddy induced velocity} 1088 1088 1089 When the \textit{eddy induced velocity} parametrisation (eiv) \citep{ Gent1990} is used,1089 When the \textit{eddy induced velocity} parametrisation (eiv) \citep{gent.mcwilliams_JPO90} is used, 1090 1090 an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 1091 1091 \[ … … 1162 1162 \ie on a $f$- or $\beta$-plane, not on the sphere. 1163 1163 It is also a very good approximation in vicinity of the Equator in 1164 a geographical coordinate system \citep{ Lengaigne_al_JGR03}.1164 a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}. 1165 1165 1166 1166 \subsubsection{lateral bilaplacian momentum diffusive operator}
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