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Changeset 11543 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.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_ZDF.tex

    r11537 r11543  
    1818% ================================================================ 
    1919\section{Vertical mixing} 
    20 \label{sec:ZDF_zdf} 
     20\label{sec:ZDF} 
    2121 
    2222The discrete form of the ocean subgrid scale physics has been presented in 
    … …  
    4141%(namelist parameter \np{ln\_zdfexp}\forcode{=.true.}) or a backward time stepping scheme 
    4242%(\np{ln\_zdfexp}\forcode{=.false.}) depending on the magnitude of the mixing coefficients, 
    43 %and thus of the formulation used (see \autoref{chap:STP}). 
     43%and thus of the formulation used (see \autoref{chap:TD}). 
    4444 
    4545%--------------------------------------------namzdf-------------------------------------------------------- 
    … …  
    9292Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented: 
    9393\[ 
    94   % \label{eq:zdfric} 
     94  % \label{eq:ZDF_ric} 
    9595  \left\{ 
    9696    \begin{aligned} 
    … …  
    151151its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type: 
    152152\begin{equation} 
    153   \label{eq:zdftke_e} 
     153  \label{eq:ZDF_tke_e} 
    154154  \frac{\partial \bar{e}}{\partial t} = 
    155155  \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 
    … …  
    161161\end{equation} 
    162162\[ 
    163   % \label{eq:zdftke_kz} 
     163  % \label{eq:ZDF_tke_kz} 
    164164  \begin{split} 
    165165    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }    \\ 
    … …  
    175175$P_{rt}$ can be set to unity or, following \citet{blanke.delecluse_JPO93}, be a function of the local Richardson number, $R_i$: 
    176176\begin{align*} 
    177   % \label{eq:prt} 
     177  % \label{eq:ZDF_prt} 
    178178  P_{rt} = 
    179179  \begin{cases} 
    … …  
    208208The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}: 
    209209\begin{equation} 
    210   \label{eq:tke_mxl0_1} 
     210  \label{eq:ZDF_tke_mxl0_1} 
    211211  l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N 
    212212\end{equation} 
    … …  
    219219To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np{nn\_mxl}\forcode{=2, 3} cases, 
    220220which add an extra assumption concerning the vertical gradient of the computed length scale. 
    221 So, the length scales are first evaluated as in \autoref{eq:tke_mxl0_1} and then bounded such that: 
     221So, the length scales are first evaluated as in \autoref{eq:ZDF_tke_mxl0_1} and then bounded such that: 
    222222\begin{equation} 
    223   \label{eq:tke_mxl_constraint} 
     223  \label{eq:ZDF_tke_mxl_constraint} 
    224224  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 
    225225  \qquad \text{with }\  l =  l_k = l_\epsilon 
    226226\end{equation} 
    227 \autoref{eq:tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than 
     227\autoref{eq:ZDF_tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than 
    228228the variations of depth. 
    229229It provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less 
    … …  
    231231In particular, it allows the length scale to be limited not only by the distance to the surface or 
    232232to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as 
    233 the thermocline (\autoref{fig:mixing_length}). 
    234 In order to impose the \autoref{eq:tke_mxl_constraint} constraint, we introduce two additional length scales: 
     233the thermocline (\autoref{fig:ZDF_mixing_length}). 
     234In order to impose the \autoref{eq:ZDF_tke_mxl_constraint} constraint, we introduce two additional length scales: 
    235235$l_{up}$ and $l_{dwn}$, the upward and downward length scales, and 
    236236evaluate the dissipation and mixing length scales as 
    … …  
    241241    \includegraphics[width=\textwidth]{Fig_mixing_length} 
    242242    \caption{ 
    243       \protect\label{fig:mixing_length} 
     243      \protect\label{fig:ZDF_mixing_length} 
    244244      Illustration of the mixing length computation. 
    245245    } 
    … …  
    248248%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    249249\[ 
    250   % \label{eq:tke_mxl2} 
     250  % \label{eq:ZDF_tke_mxl2} 
    251251  \begin{aligned} 
    252252    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right) 
    … …  
    256256  \end{aligned} 
    257257\] 
    258 where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 
     258where $l^{(k)}$ is computed using \autoref{eq:ZDF_tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 
    259259 
    260260In the \np{nn\_mxl}\forcode{=2} case, the dissipation and mixing length scales take the same value: 
    … …  
    262262the dissipation and mixing turbulent length scales are give as in \citet{gaspar.gregoris.ea_JGR90}: 
    263263\[ 
    264   % \label{eq:tke_mxl_gaspar} 
     264  % \label{eq:ZDF_tke_mxl_gaspar} 
    265265  \begin{aligned} 
    266266    & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }   \\ 
    … …  
    325325The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in 
    326326an extra source term of TKE, $P_{LC}$. 
    327 The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to 
     327The presence of $P_{LC}$ in \autoref{eq:ZDF_tke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to 
    328328\forcode{.true.} in the \nam{zdf\_tke} namelist. 
    329329 
    … …  
    428428$\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}. 
    429429This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, 
    430 where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover a number of 
     430where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:ZDF_GLS} allows to recover a number of 
    431431well-known turbulent closures ($k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87}, 
    432432$k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}). 
    433433The GLS scheme is given by the following set of equations: 
    434434\begin{equation} 
    435   \label{eq:zdfgls_e} 
     435  \label{eq:ZDF_gls_e} 
    436436  \frac{\partial \bar{e}}{\partial t} = 
    437437  \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 
    … …  
    443443 
    444444\[ 
    445   % \label{eq:zdfgls_psi} 
     445  % \label{eq:ZDF_gls_psi} 
    446446  \begin{split} 
    447447    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ 
    … …  
    455455 
    456456\[ 
    457   % \label{eq:zdfgls_kz} 
     457  % \label{eq:ZDF_gls_kz} 
    458458  \begin{split} 
    459459    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\ 
    … …  
    463463 
    464464\[ 
    465   % \label{eq:zdfgls_eps} 
     465  % \label{eq:ZDF_gls_eps} 
    466466  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 
    467467\] 
    … …  
    470470The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of 
    471471the choice of the turbulence model. 
    472 Four different turbulent models are pre-defined (\autoref{tab:GLS}). 
     472Four different turbulent models are pre-defined (\autoref{tab:ZDF_GLS}). 
    473473They are made available through the \np{nn\_clo} namelist parameter. 
    474474 
    … …  
    495495    \end{tabular} 
    496496    \caption{ 
    497       \protect\label{tab:GLS} 
     497      \protect\label{tab:ZDF_GLS} 
    498498      Set of predefined GLS parameters, or equivalently predefined turbulence models available with 
    499499      \protect\np{ln\_zdfgls}\forcode{=.true.} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\nam{zdf\_gls}. 
    … …  
    559559    \includegraphics[width=\textwidth]{Fig_ZDF_TKE_time_scheme} 
    560560    \caption{ 
    561       \protect\label{fig:TKE_time_scheme} 
     561      \protect\label{fig:ZDF_TKE_time_scheme} 
    562562      Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and its links to the momentum and tracer time integration. 
    563563    } 
    … …  
    567567 
    568568The production of turbulence by vertical shear (the first term of the right hand side of 
    569 \autoref{eq:zdftke_e}) and  \autoref{eq:zdfgls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion 
    570 (first line in \autoref{eq:PE_zdf}). 
     569\autoref{eq:ZDF_tke_e}) and  \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion 
     570(first line in \autoref{eq:MB_zdf}). 
    571571To do so a special care has to be taken for both the time and space discretization of 
    572572the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}. 
    573573 
    574 Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows how 
     574Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how 
    575575the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with 
    576576the one-level forward time stepping of the equation for $\bar{e}$. 
    … …  
    579579summing the result vertically: 
    580580\begin{equation} 
    581   \label{eq:energ1} 
     581  \label{eq:ZDF_energ1} 
    582582  \begin{split} 
    583583    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\ 
    … …  
    587587\end{equation} 
    588588Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$, 
    589 known at time $t$ (\autoref{fig:TKE_time_scheme}), as it is required when using the TKE scheme 
    590 (see \autoref{sec:STP_forward_imp}). 
    591 The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy transfer at 
     589known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme 
     590(see \autoref{sec:TD_forward_imp}). 
     591The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at 
    592592the surface (atmospheric forcing) and at the bottom (friction effect). 
    593593The second term is always negative. 
    594594It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$. 
    595 \autoref{eq:energ1} implies that, to be energetically consistent, 
     595\autoref{eq:ZDF_energ1} implies that, to be energetically consistent, 
    596596the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as 
    597597${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ 
    … …  
    599599 
    600600A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification 
    601 (second term of the right hand side of \autoref{eq:zdftke_e} and \autoref{eq:zdfgls_e}). 
     601(second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}). 
    602602This term must balance the input of potential energy resulting from vertical mixing. 
    603603The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by 
    604604multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically: 
    605605\begin{equation} 
    606   \label{eq:energ2} 
     606  \label{eq:ZDF_energ2} 
    607607  \begin{split} 
    608608    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\ 
    … …  
    614614\end{equation} 
    615615where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. 
    616 The first term of the right hand side of \autoref{eq:energ2} is always zero because 
     616The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because 
    617617there is no diffusive flux through the ocean surface and bottom). 
    618618The second term is minus the destruction rate of  $\bar{e}$ due to stratification. 
    619 Therefore \autoref{eq:energ1} implies that, to be energetically consistent, 
    620 the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e} and  \autoref{eq:zdfgls_e}. 
     619Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent, 
     620the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and  \autoref{eq:ZDF_gls_e}. 
    621621 
    622622Let us now address the space discretization issue. 
    623623The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in 
    624 the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:cell}). 
     624the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}). 
    625625A space averaging is thus required to obtain the shear TKE production term. 
    626 By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of eddy coefficient by 
     626By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by 
    627627the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. 
    628628Furthermore, the time variation of $e_3$ has be taken into account. 
    … …  
    630630The above energetic considerations leads to the following final discrete form for the TKE equation: 
    631631\begin{equation} 
    632   \label{eq:zdftke_ene} 
     632  \label{eq:ZDF_tke_ene} 
    633633  \begin{split} 
    634634    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv 
    … …  
    647647  \end{split} 
    648648\end{equation} 
    649 where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation) 
    650 are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}). 
     649where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation) 
     650are time stepped using a backward scheme (see\autoref{sec:TD_forward_imp}). 
    651651Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible. 
    652652%The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as 
    653 %they all appear in the right hand side of \autoref{eq:zdftke_ene}. 
     653%they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}. 
    654654%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 
    655655 
    … …  
    679679    \includegraphics[width=\textwidth]{Fig_npc} 
    680680    \caption{ 
    681       \protect\label{fig:npc} 
     681      \protect\label{fig:ZDF_npc} 
    682682      Example of an unstable density profile treated by the non penetrative convective adjustment algorithm. 
    683683      $1^{st}$ step: the initial profile is checked from the surface to the bottom. 
    … …  
    702702(\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below) 
    703703\citep{madec.delecluse.ea_JPO91}. 
    704 The associated algorithm is an iterative process used in the following way (\autoref{fig:npc}): 
     704The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}): 
    705705starting from the top of the ocean, the first instability is found. 
    706706Assume in the following that the instability is located between levels $k$ and $k+1$. 
    … …  
    759759Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$. 
    760760This removes a potential source of divergence of odd and even time step in 
    761 a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:STP_mLF}). 
     761a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}). 
    762762 
    763763% ------------------------------------------------------------------------------------------------------------- 
    … …  
    772772with statically unstable density profiles. 
    773773In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in 
    774 \autoref{eq:zdftke_e} or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative. 
     774\autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative. 
    775775It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also of the four neighboring values at 
    776776velocity points $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1}$). 
    … …  
    814814Diapycnal mixing of S and T are described by diapycnal diffusion coefficients 
    815815\begin{align*} 
    816   % \label{eq:zdfddm_Kz} 
     816  % \label{eq:ZDF_ddm_Kz} 
    817817  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ 
    818818  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} 
    … …  
    826826(1981): 
    827827\begin{align} 
    828   \label{eq:zdfddm_f} 
     828  \label{eq:ZDF_ddm_f} 
    829829  A_f^{vS} &= 
    830830             \begin{cases} 
    … …  
    832832               0                              &\text{otherwise} 
    833833             \end{cases} 
    834   \\         \label{eq:zdfddm_f_T} 
     834  \\         \label{eq:ZDF_ddm_f_T} 
    835835  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 
    836836\end{align} 
    … …  
    841841    \includegraphics[width=\textwidth]{Fig_zdfddm} 
    842842    \caption{ 
    843       \protect\label{fig:zdfddm} 
     843      \protect\label{fig:ZDF_ddm} 
    844844      From \citet{merryfield.holloway.ea_JPO99} : 
    845845      (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering. 
    … …  
    854854%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    855855 
    856 The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of 
     856The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of 
    857857buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}). 
    858858Following  \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$. 
    … …  
    861861Federov (1988) is used: 
    862862\begin{align} 
    863   % \label{eq:zdfddm_d} 
     863  % \label{eq:ZDF_ddm_d} 
    864864  A_d^{vT} &= 
    865865             \begin{cases} 
    … …  
    869869             \end{cases} 
    870870                                       \nonumber \\ 
    871   \label{eq:zdfddm_d_S} 
     871  \label{eq:ZDF_ddm_d_S} 
    872872  A_d^{vS} &= 
    873873             \begin{cases} 
    … …  
    878878\end{align} 
    879879 
    880 The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ are illustrated in 
    881 \autoref{fig:zdfddm}. 
     880The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in 
     881\autoref{fig:ZDF_ddm}. 
    882882Implementing this requires computing $R_\rho$ at each grid point on every time step. 
    883883This is done in \mdl{eosbn2} at the same time as $N^2$ is computed. 
    … …  
    891891 \label{sec:ZDF_drg} 
    892892 
    893 %--------------------------------------------nambfr-------------------------------------------------------- 
     893%--------------------------------------------namdrg-------------------------------------------------------- 
    894894% 
    895895\nlst{namdrg} 
    … …  
    910910For the bottom boundary layer, one has: 
    911911 \[ 
    912    % \label{eq:zdfbfr_flux} 
     912   % \label{eq:ZDF_bfr_flux} 
    913913   A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 
    914914 \] 
    … …  
    926926To illustrate this, consider the equation for $u$ at $k$, the last ocean level: 
    927927\begin{equation} 
    928   \label{eq:zdfdrg_flux2} 
     928  \label{eq:ZDF_drg_flux2} 
    929929  \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}} 
    930930\end{equation} 
    … …  
    946946 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where: 
    947947\begin{equation} 
    948   \label{eq:zdfbfr_bdef} 
     948  \label{eq:ZDF_bfr_bdef} 
    949949  \frac{\partial {\textbf U_h}}{\partial t} = 
    950950  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b 
    … …  
    963963the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level): 
    964964\[ 
    965   % \label{eq:zdfbfr_linear} 
     965  % \label{eq:ZDF_bfr_linear} 
    966966  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 
    967967\] 
    … …  
    978978It can be changed by specifying \np{rn\_Uc0} (namelist parameter). 
    979979 
    980  For the linear friction case the drag coefficient used in the general expression \autoref{eq:zdfbfr_bdef} is: 
    981 \[ 
    982   % \label{eq:zdfbfr_linbfr_b} 
     980 For the linear friction case the drag coefficient used in the general expression \autoref{eq:ZDF_bfr_bdef} is: 
     981\[ 
     982  % \label{eq:ZDF_bfr_linbfr_b} 
    983983    c_b^T = - r 
    984984\] 
    … …  
    10021002The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic: 
    10031003\[ 
    1004   % \label{eq:zdfdrg_nonlinear} 
     1004  % \label{eq:ZDF_drg_nonlinear} 
    10051005  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 
    10061006  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b 
    … …  
    10181018For the non-linear friction case the term computed in \mdl{zdfdrg} is: 
    10191019\[ 
    1020   % \label{eq:zdfdrg_nonlinbfr} 
     1020  % \label{eq:ZDF_drg_nonlinbfr} 
    10211021    c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2} 
    10221022\] 
    … …  
    10771077 
    10781078Since this is conditionally stable, some care needs to exercised over the choice of parameters to ensure that the implementation of explicit top/bottom friction does not induce numerical instability. 
    1079 For the purposes of stability analysis, an approximation to \autoref{eq:zdfdrg_flux2} is: 
     1079For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is: 
    10801080\begin{equation} 
    1081   \label{eq:Eqn_drgstab} 
     1081  \label{eq:ZDF_Eqn_drgstab} 
    10821082  \begin{split} 
    10831083    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\ 
    … …  
    10901090  |\Delta u| < \;|u| 
    10911091\] 
    1092 \noindent which, using \autoref{eq:Eqn_drgstab}, gives: 
     1092\noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives: 
    10931093\[ 
    10941094  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ 
    … …  
    11331133At the top (below an ice shelf cavity): 
    11341134\[ 
    1135   % \label{eq:dynzdf_drg_top} 
     1135  % \label{eq:ZDF_dynZDF__drg_top} 
    11361136  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t} 
    11371137  = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t} 
    … …  
    11401140At the bottom (above the sea floor): 
    11411141\[ 
    1142   % \label{eq:dynzdf_drg_bot} 
     1142  % \label{eq:ZDF_dynZDF__drg_bot} 
    11431143  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b} 
    11441144  = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b} 
    … …  
    11831183and the resulting diffusivity is obtained as 
    11841184\[ 
    1185   % \label{eq:Kwave} 
     1185  % \label{eq:ZDF_Kwave} 
    11861186  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 } 
    11871187\] 
    … …  
    12421242 
    12431243\begin{equation} 
    1244   \label{eq:Bv} 
     1244  \label{eq:ZDF_Bv} 
    12451245  B_{v} = \alpha {A} {U}_{st} {exp(3kz)} 
    12461246\end{equation} 
    … …  
    12761276criteria for a range of advection schemes. The values for the Leap-Frog with Robert 
    12771277asselin filter time-stepping (as used in NEMO) are reproduced in 
    1278 \autoref{tab:zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these 
     1278\autoref{tab:ZDF_zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these 
    12791279restrictions but at the cost of large dispersive errors and, possibly, large numerical 
    12801280viscosity. The adaptive-implicit vertical advection option provides a targetted use of the 
    … …  
    12961296    \end{tabular} 
    12971297    \caption{ 
    1298       \protect\label{tab:zad_Aimp_CFLcrit} 
     1298      \protect\label{tab:ZDF_zad_Aimp_CFLcrit} 
    12991299      The advective CFL criteria for a range of spatial discretizations for the Leap-Frog with Robert Asselin filter time-stepping 
    13001300      ($\nu=0.1$) as given in \citep{lemarie.debreu.ea_OM15}. 
    … …  
    13131313 
    13141314\begin{equation} 
    1315   \label{eq:Eqn_zad_Aimp_Courant} 
     1315  \label{eq:ZDF_Eqn_zad_Aimp_Courant} 
    13161316  \begin{split} 
    13171317    Cu &= {2 \rdt \over e^n_{3t_{ijk}}} \bigg (\big [ \texttt{Max}(w^n_{ijk},0.0) - \texttt{Min}(w^n_{ijk+1},0.0) \big ]    \\ 
    … …  
    13261326 
    13271327\begin{align} 
    1328   \label{eq:Eqn_zad_Aimp_partition} 
     1328  \label{eq:ZDF_Eqn_zad_Aimp_partition} 
    13291329Cu_{min} &= 0.15 \nonumber \\ 
    13301330Cu_{max} &= 0.3  \nonumber \\ 
    … …  
    13431343    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_coeff} 
    13441344    \caption{ 
    1345       \protect\label{fig:zad_Aimp_coeff} 
     1345      \protect\label{fig:ZDF_zad_Aimp_coeff} 
    13461346      The value of the partitioning coefficient ($C\kern-0.14em f$) used to partition vertical velocities into parts to 
    13471347      be treated implicitly and explicitly for a range of typical Courant numbers (\forcode{ln_zad_Aimp=.true.}) 
    … …  
    13551355 
    13561356\begin{align} 
    1357   \label{eq:Eqn_zad_Aimp_partition2} 
     1357  \label{eq:ZDF_Eqn_zad_Aimp_partition2} 
    13581358    w_{i_{ijk}} &= C\kern-0.14em f_{ijk} w_{n_{ijk}}     \nonumber \\ 
    1359     w_{n_{ijk}} &= (1-C\kern-0.14em f_{ijk}) w_{n_{ijk}}            
     1359    w_{n_{ijk}} &= (1-C\kern-0.14em f_{ijk}) w_{n_{ijk}} 
    13601360\end{align} 
    13611361 
    13621362\noindent Note that the coefficient is such that the treatment is never fully implicit; 
    1363 the three cases from \autoref{eq:Eqn_zad_Aimp_partition} can be considered as: 
     1363the three cases from \autoref{eq:ZDF_Eqn_zad_Aimp_partition} can be considered as: 
    13641364fully-explicit; mixed explicit/implicit and mostly-implicit.  With the settings shown the 
    1365 coefficient ($C\kern-0.14em f$) varies as shown in \autoref{fig:zad_Aimp_coeff}. Note with these values 
     1365coefficient ($C\kern-0.14em f$) varies as shown in \autoref{fig:ZDF_zad_Aimp_coeff}. Note with these values 
    13661366the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly 
    13671367implicit' is 0.45 which is just below the stability limited given in 
    1368 \autoref{tab:zad_Aimp_CFLcrit}  for a 3rd order scheme. 
     1368\autoref{tab:ZDF_zad_Aimp_CFLcrit}  for a 3rd order scheme. 
    13691369 
    13701370The $w_i$ component is added to the implicit solvers for the vertical mixing in 
    … …  
    13761376vertical fluxes are then removed since they are added by the implicit solver later on. 
    13771377 
    1378 The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be  
     1378The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be 
    13791379used in a wide range of simulations. The following test simulation, however, does illustrate 
    13801380the potential benefits and will hopefully encourage further testing and feedback from users: 
    … …  
    13841384    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames} 
    13851385    \caption{ 
    1386       \protect\label{fig:zad_Aimp_overflow_frames} 
     1386      \protect\label{fig:ZDF_zad_Aimp_overflow_frames} 
    13871387      A time-series of temperature vertical cross-sections for the OVERFLOW test case. These results are for the default 
    13881388      settings with \forcode{nn_rdt=10.0} and without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}). 
    … …  
    14081408\noindent which were chosen to provide a slightly more stable and less noisy solution. The 
    14091409result when using the default value of \forcode{nn_rdt=10.} without adaptive-implicit 
    1410 vertical velocity is illustrated in \autoref{fig:zad_Aimp_overflow_frames}. The mass of 
     1410vertical velocity is illustrated in \autoref{fig:ZDF_zad_Aimp_overflow_frames}. The mass of 
    14111411cold water, initially sitting on the shelf, moves down the slope and forms a 
    14121412bottom-trapped, dense plume. Even with these extra physics choices the model is close to 
    … …  
    14181418 
    14191419The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps 
    1420 are shown in \autoref{fig:zad_Aimp_overflow_all_rdt} (together with the equivalent 
     1420are shown in \autoref{fig:ZDF_zad_Aimp_overflow_all_rdt} (together with the equivalent 
    14211421frames from the base run).  In this simple example the use of the adaptive-implicit 
    14221422vertcal advection scheme has enabled a 12x increase in the model timestep without 
    … …  
    14341434\autoref{sec:MISC_opt} for activation details). 
    14351435 
    1436 \autoref{fig:zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for 
     1436\autoref{fig:ZDF_zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for 
    14371437the various overflow tests.  Note that the adaptive-implicit vertical advection scheme is 
    14381438active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the 
    … …  
    14411441oscillatory nature of this measure appears to be linked to the progress of the plume front 
    14421442as each cusp is associated with the location of the maximum shifting to the adjacent cell. 
    1443 This is illustrated in \autoref{fig:zad_Aimp_maxCf_loc} where the i- and k- locations of the 
     1443This is illustrated in \autoref{fig:ZDF_zad_Aimp_maxCf_loc} where the i- and k- locations of the 
    14441444maximum have been overlaid for the base run case. 
    14451445 
    … …  
    14631463    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_overflow_all_rdt} 
    14641464    \caption{ 
    1465       \protect\label{fig:zad_Aimp_overflow_all_rdt} 
    1466       Sample temperature vertical cross-sections from mid- and end-run using different values for \forcode{nn_rdt}  
     1465      \protect\label{fig:ZDF_zad_Aimp_overflow_all_rdt} 
     1466      Sample temperature vertical cross-sections from mid- and end-run using different values for \forcode{nn_rdt} 
    14671467      and with or without adaptive implicit vertical advection. Without the adaptive implicit vertical advection only 
    14681468      the run with the shortest timestep is able to run to completion. Note also that the colour-scale has been 
    … …  
    14761476    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_maxCf} 
    14771477    \caption{ 
    1478       \protect\label{fig:zad_Aimp_maxCf} 
     1478      \protect\label{fig:ZDF_zad_Aimp_maxCf} 
    14791479      The maximum partitioning coefficient during a series of test runs with increasing model timestep length. 
    1480       At the larger timesteps, the vertical velocity is treated mostly implicitly at some location throughout  
     1480      At the larger timesteps, the vertical velocity is treated mostly implicitly at some location throughout 
    14811481      the run. 
    14821482    } 
    … …  
    14881488    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_maxCf_loc} 
    14891489    \caption{ 
    1490       \protect\label{fig:zad_Aimp_maxCf_loc} 
     1490      \protect\label{fig:ZDF_zad_Aimp_maxCf_loc} 
    14911491      The maximum partitioning coefficient for the  \forcode{nn_rdt=10.0s} case overlaid with  information on the gridcell i- and k- 
    1492       locations of the maximum value.  
     1492      locations of the maximum value. 
    14931493    } 
    14941494  \end{center} 
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