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Changeset 9414 – NEMO

# Changeset 9414

Ignore:
Timestamp:
2018-03-21T15:39:48+01:00 (6 years ago)
Message:

Fix multiple defined references

Location:
branches/2017/dev_merge_2017/DOC
Files:
9 edited

Unmodified
Removed
• ## branches/2017/dev_merge_2017/DOC/tex_main/NEMO_manual.sty

 r9394 \usepackage{subfiles}   %% subdocs %% Extensions in bundle package \usepackage{amssymb, graphicx, longtable, makeidx, xspace} \usepackage{amssymb, graphicx, makeidx, tabularx, xspace}
• ## branches/2017/dev_merge_2017/DOC/tex_sub/annex_A.tex

 r9408 % ================================================================ \section{Chain rule for $s-$coordinates} \label{sec:A_continuity} \label{sec:A_chain} In order to establish the set of Primitive Equation in curvilinear $s$-coordinates Applying similar manipulation to the second component and replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: \label{apdx:A_grad_p} \label{apdx:A_grad_p_1} \begin{split} -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z An additional term appears in (\autoref{apdx:A_grad_p}) which accounts for the An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. \end{equation*} Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p} and using the definition of Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of the density anomaly it comes the expression in two parts: \label{apdx:A_grad_p} \label{apdx:A_grad_p_2} \begin{split} -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z This formulation of the pressure gradient is characterised by the appearance of a term depending on the the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p}). the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). This term will be loosely termed \textit{surface pressure gradient} whereas the first term will be termed the
• ## branches/2017/dev_merge_2017/DOC/tex_sub/annex_C.tex

 r9408 advection term (vector invariant form): \label{eq:E_tot_vect_vor} \label{eq:E_tot_vect_vor_1} \int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\ % \label{eq:E_tot_vect_adv} \label{eq:E_tot_vect_adv_1} \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv + \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv pressure gradient: \label{eq:E_tot_pg} \label{eq:E_tot_pg_1} - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv Vector invariant form: \begin{subequations} \label{eq:E_tot_vect} \label{eq:E_tot_vect_vor} \label{eq:E_tot_vect_vor_2} \int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0   \\ \label{eq:E_tot_vect_adv} \label{eq:E_tot_vect_adv_2} \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv + \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0   \\ \label{eq:E_tot_pg} \label{eq:E_tot_pg_2} - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv Flux form: \begin{subequations} \label{eq:E_tot_flux} \label{eq:E_tot_flux_metric} \label{eq:E_tot_flux_metric_2} \int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0   \\ \label{eq:E_tot_flux_adv} \label{eq:E_tot_flux_adv_2} \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0  \\ \label{eq:E_tot_pg} \label{eq:E_tot_pg_3} - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv \autoref{eq:E_tot_pg} is the balance between the conversion KE to PE and PE to KE. Indeed the left hand side of \autoref{eq:E_tot_pg} can be transformed as follows: \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: \begin{flalign*} \partial_t  \left( \int\limits_D { \rho \, g \, z  \;dv} \right) the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). The left hand side of \autoref{eq:E_tot_pg} can be transformed as follows: The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: \begin{flalign*} - \int\limits_D  \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv % ================================================================ \section{Discrete total energy conservation: vector invariant form} \label{sec:C.1} \label{sec:C.2} % ------------------------------------------------------------------------------------------------------------- % ------------------------------------------------------------------------------------------------------------- \subsection{Total energy conservation} \label{subsec:C_KE+PE} \label{subsec:C_KE+PE_vect} The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: % ------------------------------------------------------------------------------------------------------------- \subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} \label{subsec:C_vorEEN} \label{subsec:C_vorEEN_vect} With the EEN scheme, the vorticity terms are represented as: \label{eq:dynvor_een} \tag{\ref{eq:dynvor_een}} \left\{ {    \begin{aligned} +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: \label{eq:Q_triads} \tag{\ref{eq:Q_triads}} _i^j \mathbb{Q}^{i_p}_{j_p} = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) % ------------------------------------------------------------------------------------------------------------- \subsection{Pressure gradient term} \label{subsec:C.1.4} \label{subsec:C.2.6} \gmcomment{ % ================================================================ \section{Discrete total energy conservation: flux form} \label{sec:C.1} \label{sec:C.3} % ------------------------------------------------------------------------------------------------------------- % ------------------------------------------------------------------------------------------------------------- \subsection{Total energy conservation} \label{subsec:C_KE+PE} \label{subsec:C_KE+PE_flux} The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: % ------------------------------------------------------------------------------------------------------------- \subsection{Coriolis and advection terms: flux form} \label{subsec:C.1.3} \label{subsec:C.3.2} % ------------------------------------------------------------------------------------------------------------- % ------------------------------------------------------------------------------------------------------------- \subsubsection{Coriolis plus metric'' term} \label{subsec:C.1.3.1} \label{subsec:C.3.3} In flux from the vorticity term reduces to a Coriolis term in which the Coriolis % ------------------------------------------------------------------------------------------------------------- \subsubsection{Flux form advection} \label{subsec:C.1.3.2} \label{subsec:C.3.4} The flux form operator of the momentum advection is evaluated using a % ================================================================ \section{Discrete enstrophy conservation} \label{sec:C.1} \label{sec:C.4} In the ENS scheme, the vorticity term is descretized as follows: \label{eq:dynvor_ens} \tag{\ref{eq:dynvor_ens}} \left\{   \begin{aligned} +\frac{1}{e_{1u}} & \overline{q}^{\,i}  & {\overline{ \overline{\left( e_{1v}\,e_{3v}\;  v \right) } } }^{\,i, j+1/2}    \\ With the EEN scheme, the vorticity terms are represented as: \label{eq:dynvor_een} \tag{\ref{eq:dynvor_een}} \left\{ {    \begin{aligned} +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: \label{eq:Q_triads} \tag{\ref{eq:Q_triads}} _i^j \mathbb{Q}^{i_p}_{j_p} = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) % ================================================================ \section{Conservation properties on tracers} \label{sec:C.2} \label{sec:C.5} % ------------------------------------------------------------------------------------------------------------- \subsection{Advection term} \label{subsec:C.2.1} \label{subsec:C.5.1} conservation of a tracer, $T$: % ------------------------------------------------------------------------------------------------------------- \subsection{Conservation of potential vorticity} \label{subsec:C.3.1} \label{subsec:C.6.1} The lateral momentum diffusion term conserves the potential vorticity : % ------------------------------------------------------------------------------------------------------------- \subsection{Dissipation of horizontal kinetic energy} \label{subsec:C.3.2} \label{subsec:C.6.2} The lateral momentum diffusion term dissipates the horizontal kinetic energy: % ------------------------------------------------------------------------------------------------------------- \subsection{Dissipation of enstrophy} \label{subsec:C.3.3} \label{subsec:C.6.3} The lateral momentum diffusion term dissipates the enstrophy when the eddy % ------------------------------------------------------------------------------------------------------------- \subsection{Conservation of horizontal divergence} \label{subsec:C.3.4} \label{subsec:C.6.4} When the horizontal divergence of the horizontal diffusion of momentum % ------------------------------------------------------------------------------------------------------------- \subsection{Dissipation of horizontal divergence variance} \label{subsec:C.3.5} \label{subsec:C.6.5} \begin{flalign*} % ================================================================ \section{Conservation properties on vertical momentum physics} \label{sec:C_4} \label{sec:C.7} As for the lateral momentum physics, the continuous form of the vertical diffusion % ================================================================ \section{Conservation properties on tracer physics} \label{sec:C.5} \label{sec:C.8} The numerical schemes used for tracer subgridscale physics are written such % ------------------------------------------------------------------------------------------------------------- \subsection{Conservation of tracers} \label{subsec:C.5.1} \label{subsec:C.8.1} constraint of conservation of tracers: % ------------------------------------------------------------------------------------------------------------- \subsection{Dissipation of tracer variance} \label{subsec:C.5.2} \label{subsec:C.8.2} constraint on the dissipation of tracer variance:
• ## branches/2017/dev_merge_2017/DOC/tex_sub/annex_iso.tex

 r9407 The iso-neutral second order tracer diffusive operator for small angles between iso-neutral surfaces and geopotentials is given by \autoref{eq:PE_iso_tensor}: \begin{subequations} \label{eq:PE_iso_tensor} \autoref{eq:iso_tensor_1}: \begin{subequations} \label{eq:iso_tensor_1} D^{lT}=-\Div\vect{f}^{lT}\equiv \label{eq:PE_iso_tensor:c} \label{eq:iso_tensor_2} \mbox{with}\quad \;\;\Re = \begin{pmatrix} The off-diagonal terms of the small angle diffusion tensor \autoref{eq:PE_iso_tensor}, \autoref{eq:PE_iso_tensor:c} produce skew-fluxes along the \autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along the $i$- and $j$-directions resulting from the vertical tracer gradient: \begin{align}
• ## branches/2017/dev_merge_2017/DOC/tex_sub/chap_ASM.tex

 r9407 % Divergence damping description %%% \section{Divergence damping initialisation} \label{sec:ASM_details} \label{sec:ASM_div_dmp} The velocity increments may be initialized by the iterative application of
• ## branches/2017/dev_merge_2017/DOC/tex_sub/chap_DIA.tex

 r9413 A non-Boussinesq fluid conserves mass. It satisfies the following relations: \label{eq:MV_nBq} \begin{split} $\begin{split} \mathcal{M} &= \mathcal{V} \;\bar{\rho} \\ \mathcal{V} &= \mathcal{A} \;\bar{\eta} \end{split} \end{equation} \end{split} \label{eq:MV_nBq}$ Temporal changes in total mass is obtained from the density conservation equation :
• ## branches/2017/dev_merge_2017/DOC/tex_sub/chap_DOM.tex

 r9407 Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from \autoref{eq:DOM_zgr_ana} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} from \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} \end{center}   \end{figure} %>>>>>>>>>>>>>>>>>>>>>>>>>>>> near the ocean surface. The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps): \label{eq:DOM_zgr_ana} \label{eq:DOM_zgr_ana_1} \begin{split} z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same. However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: \label{eq:DOM_zgr_ana} \label{eq:DOM_zgr_ana_2} \begin{split} e_3^T(k) &= z_W (k+1) - z_W (k)   \\ With the choice of the stretching $h_{cr} =3$ and the number of levels \jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in \autoref{eq:DOM_zgr_ana} have been determined such that \autoref{eq:DOM_zgr_coef} is \autoref{eq:DOM_zgr_ana_2} have been determined such that \autoref{eq:DOM_zgr_coef} is satisfied, through an optimisation procedure using a bisection method. For the first standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$, \caption{ \protect\label{tab:orca_zgr} Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from \autoref{eq:DOM_zgr_ana} using the coefficients given in \autoref{eq:DOM_zgr_coef}} from \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}} \end{table} %>>>>>>>>>>>>>>>>>>>>>>>>>>>> s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1 \label{eq:s} \label{eq:DOM_s}
• ## branches/2017/dev_merge_2017/DOC/tex_sub/chap_DYN.tex

 r9407 Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to compute the product of the Coriolis parameter and the vorticity. However, the energy-conserving scheme  (\autoref{eq:dynvor_een}) has vorticity. However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has exclusively been used to date. This term is evaluated using a leapfrog scheme, $i.e.$ the velocity is centred in time (\textit{now} velocity).
• ## branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics.tex

 r9407 In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate : \label{eq:s} \label{eq:PE_s} s=s(i,j,k,t) with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:s}. with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:PE_s}. This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part). The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces. \vspace{0.5cm} $\bullet$ Vector invariant form of the momentum equation : \begin{multline} \label{eq:PE_sco_u} \begin{multline} \label{eq:PE_sco_u_vector} \frac{\partial  u   }{\partial t}= +   \left( {\zeta +f} \right)\,v +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad \end{multline} \begin{multline} \label{eq:PE_sco_v} \begin{multline} \label{eq:PE_sco_v_vector} \frac{\partial v }{\partial t}= -   \left( {\zeta +f} \right)\,u \vspace{0.5cm} $\bullet$ Vector invariant form of the momentum equation : \begin{multline} \label{eq:PE_sco_u} $\bullet$ Flux form of the momentum equation : \begin{multline} \label{eq:PE_sco_u_flux} \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}= +   \left( { f + \frac{1}{e_1 \; e_2 } +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad \end{multline} \begin{multline} \label{eq:PE_sco_v} \begin{multline} \label{eq:PE_sco_v_flux} \frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}= -   \left( { f + \frac{1}{e_1 \; e_2} rotation between geopotential and $s$-surfaces, while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter case, two assumptions are made to simplify  \autoref{eq:PE_iso_tensor} \citep{Cox1987}. case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}. First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and dia-neutral diffusive coefficients is known to be several orders of
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