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Changeset 11543 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex – NEMO

Ignore:
Timestamp:
2019-09-13T15:57:52+02:00 (5 years ago)
Author:
nicolasmartin
Message:

Implementation of convention for labelling references + files renaming
Now each reference is supposed to have the information of the chapter in its name
to identify quickly which file contains the reference (\label{$prefix:$chap_...)

Rename the appendices from 'annex_' to 'apdx_' to conform with the prefix used in labels (apdx:...)
Suppress the letter numbering

File:
1 edited

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  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex

    r11537 r11543  
    1313\newpage 
    1414 
    15 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and 
     15The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and 
    1616their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). 
    1717In this section we further discuss each lateral physics option. 
    … …  
    2525Note that this chapter describes the standard implementation of iso-neutral tracer mixing.  
    2626Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{=.true.}, 
    27 is described in \autoref{apdx:triad} 
     27is described in \autoref{apdx:TRIADS} 
    2828 
    2929%-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- 
    … …  
    8282the cell of the quantity to be diffused. 
    8383For a tracer, this leads to the following four slopes: 
    84 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}), 
     84$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), 
    8585while for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and 
    8686$r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.  
    … …  
    9292In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between 
    9393the geopotential and computational surfaces. 
    94 Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso} when 
     94Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when 
    9595the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, 
    9696\ie\ a linear function of $z_T$, the depth of a $T$-point.  
    … …  
    9898 
    9999\begin{equation} 
    100   \label{eq:ldfslp_geo} 
     100  \label{eq:LDF_slp_geo} 
    101101  \begin{aligned} 
    102102    r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} 
    … …  
    125125Their discrete formulation is found using the fact that the diffusive fluxes of 
    126126locally referenced potential density (\ie\ $in situ$ density) vanish. 
    127 So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the diffusive fluxes in 
     127So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in 
    128128the three directions to zero leads to the following definition for the neutral slopes: 
    129129 
    130130\begin{equation} 
    131   \label{eq:ldfslp_iso} 
     131  \label{eq:LDF_slp_iso} 
    132132  \begin{split} 
    133133    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} 
    … …  
    145145 
    146146%gm% rewrite this as the explanation is not very clear !!! 
    147 %In practice, \autoref{eq:ldfslp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:ldfslp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth.  
    148  
    149 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:ldfslp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). 
    150  
    151 %In the $z$-coordinate, the derivative of the  \autoref{eq:ldfslp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so  the $in situ$ density can be used for its evaluation.  
    152  
    153 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:ldfslp_iso} has to 
     147%In practice, \autoref{eq:LDF_slp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:LDF_slp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth.  
     148 
     149%By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:LDF_slp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). 
     150 
     151%In the $z$-coordinate, the derivative of the  \autoref{eq:LDF_slp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so  the $in situ$ density can be used for its evaluation.  
     152 
     153As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to 
    154154be evaluated at the same local pressure 
    155155(which, in decibars, is approximated by the depth in meters in the model). 
    156 Therefore \autoref{eq:ldfslp_iso} cannot be used as such, 
     156Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, 
    157157but further transformation is needed depending on the vertical coordinate used: 
    158158 
    … …  
    160160 
    161161\item[$z$-coordinate with full step: ] 
    162   in \autoref{eq:ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth, 
     162  in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth, 
    163163  thus the $in situ$ density can be used. 
    164164  This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, 
    … …  
    173173  in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 
    174174  the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{=.true.}; 
    175   see \autoref{apdx:triad}). 
     175  see \autoref{apdx:TRIADS}). 
    176176  In other words, iso-neutral mixing will only be accurately represented with a linear equation of state 
    177177  (\np{ln\_seos}\forcode{=.true.}). 
    178   In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso} 
     178  In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} 
    179179  will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 
    180180 
    … …  
    193193 
    194194\[ 
    195   % \label{eq:ldfslp_iso2} 
     195  % \label{eq:LDF_slp_iso2} 
    196196  \begin{split} 
    197197    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac 
    … …  
    230230To overcome this problem, several techniques have been proposed in which the numerical schemes of 
    231231the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. 
    232 Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{=.true.}; see \autoref{apdx:triad}. 
     232Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:TRIADS}. 
    233233Here, another strategy is presented \citep{lazar_phd97}: 
    234234a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of 
    … …  
    280280    \includegraphics[width=\textwidth]{Fig_eiv_slp} 
    281281    \caption{ 
    282       \protect\label{fig:eiv_slp} 
     282      \protect\label{fig:LDF_eiv_slp} 
    283283      Vertical profile of the slope used for lateral mixing in the mixed layer: 
    284284      \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 
    … …  
    304304The iso-neutral diffusion operator on momentum is the same as the one used on tracers but 
    305305applied to each component of the velocity separately 
    306 (see \autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). 
     306(see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). 
    307307The slopes between the surface along which the diffusion operator acts and the surface of computation 
    308308($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and 
    309309\textit{vw}- points for the $v$-component. 
    310310They are computed from the slopes used for tracer diffusion, 
    311 \ie\ \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso}: 
     311\ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: 
    312312 
    313313\[ 
    314   % \label{eq:ldfslp_dyn} 
     314  % \label{eq:LDF_slp_dyn} 
    315315  \begin{aligned} 
    316316    &r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ 
    … …  
    371371 
    372372\begin{equation} 
    373   \label{eq:constantah} 
     373  \label{eq:LDF_constantah} 
    374374  A_o^l = \left\{ 
    375375    \begin{aligned} 
    … …  
    386386 
    387387In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which 
    388 the surface value is given by \autoref{eq:constantah}, the bottom value is 1/4 of the surface value, 
     388the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, 
    389389and the transition takes place around z=500~m with a width of 200~m. 
    390390This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. 
    … …  
    396396the type of operator used: 
    397397\begin{equation} 
    398   \label{eq:title} 
     398  \label{eq:LDF_title} 
    399399  A_l = \left\{ 
    400400    \begin{aligned} 
    … …  
    411411model configurations presenting large changes in grid spacing such as global ocean models. 
    412412Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to 
    413 large coefficient compare to the smallest grid size (see \autoref{sec:STP_forward_imp}), 
     413large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}), 
    414414especially when using a bilaplacian operator. 
    415415 
    … …  
    429429 
    430430\begin{equation} 
    431   \label{eq:flowah} 
     431  \label{eq:LDF_flowah} 
    432432  A_l = \left\{ 
    433433    \begin{aligned} 
    … …  
    445445 
    446446\begin{equation} 
    447   \label{eq:smag1} 
     447  \label{eq:LDF_smag1} 
    448448  \begin{split} 
    449449    T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2  } \\ 
    … …  
    455455 
    456456\begin{equation} 
    457   \label{eq:smag2} 
     457  \label{eq:LDF_smag2} 
    458458  A_{smag} = \left\{ 
    459459    \begin{aligned} 
    … …  
    464464\end{equation} 
    465465 
    466 For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:STP_forward_imp}) so that: 
    467 \begin{equation} 
    468   \label{eq:smag3} 
     466For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that: 
     467\begin{equation} 
     468  \label{eq:LDF_smag3} 
    469469    \begin{aligned} 
    470470      & C_{min} \frac{1}{2}   \lvert U \rvert  e    < A_{smag} < C_{max} \frac{e^2}{   8\rdt}                 & \text{for laplacian operator } \\ 
    … …  
    480480 
    481481(1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and 
    482 divergent components of the horizontal current (see \autoref{subsec:PE_ldf}). 
     482divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). 
    483483Although the eddy coefficient could be set to different values in these two terms, 
    484484this option is not currently available.  
    … …  
    486486(2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of 
    487487the horizontal divergence for operators acting along model-surfaces are no longer satisfied 
    488 (\autoref{sec:dynldf_properties}). 
     488(\autoref{sec:INVARIANTS_dynldf_properties}). 
    489489 
    490490% ================================================================ 
    … …  
    527527the formulation of which depends on the slopes of iso-neutral surfaces. 
    528528Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 
    529 \ie\ \autoref{eq:ldfslp_geo} is used in $z$-coordinates, 
    530 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates. 
     529\ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates, 
     530and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. 
    531531 
    532532If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by:  
    533533\begin{equation} 
    534   \label{eq:ldfeiv} 
     534  \label{eq:LDF_eiv} 
    535535  \begin{split} 
    536536    u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ 
    … …  
    554554\colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} 
    555555 
    556 In case of setting \np{ln\_traldf\_triad}\forcode{=.true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:triad}. 
     556In case of setting \np{ln\_traldf\_triad}\forcode{ = .true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}. 
    557557 
    558558% ================================================================ 
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