Changeset 11544


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2019-09-13T16:37:38+02:00 (12 months ago)
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nicolasmartin
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Missing modifications from previous commit

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

    r11543 r11544  
    1818% ------------------------------------------------------------------------------------------------------------- 
    1919\section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} 
    20 \label{sec:TRA_adv_ubs} 
     20\label{sec:ALGOS_tra_adv_ubs} 
    2121 
    2222The UBS advection scheme is an upstream biased third order scheme based on 
     
    2525For example, in the $i$-direction: 
    2626\begin{equation} 
    27   \label{eq:tra_adv_ubs2} 
     27  \label{eq:ALGOS_tra_adv_ubs2} 
    2828  \tau_u^{ubs} = \left\{ 
    2929    \begin{aligned} 
     
    3535or equivalently, the advective flux is 
    3636\begin{equation} 
    37   \label{eq:tra_adv_ubs2} 
     37  \label{eq:ALGOS_tra_adv_ubs2} 
    3838  U_{i+1/2} \ \tau_u^{ubs} 
    3939  =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} 
     
    8585NB 3: It is straight forward to rewrite \autoref{eq:TRA_adv_ubs} as follows: 
    8686\begin{equation} 
    87   \label{eq:tra_adv_ubs2} 
     87  \label{eq:ALGOS_tra_adv_ubs2} 
    8888  \tau_u^{ubs} = \left\{ 
    8989    \begin{aligned} 
     
    9595or equivalently 
    9696\begin{equation} 
    97   \label{eq:tra_adv_ubs2} 
     97  \label{eq:ALGOS_tra_adv_ubs2} 
    9898  \begin{split} 
    9999    e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs} 
     
    112112laplacian diffusion: 
    113113\begin{equation} 
    114   \label{eq:tra_ldf_lap} 
     114  \label{eq:ALGOS_tra_ldf_lap} 
    115115  \begin{split} 
    116116    D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta_i 
     
    125125bilaplacian: 
    126126\begin{equation} 
    127   \label{eq:tra_ldf_lap} 
     127  \label{eq:ALGOS_tra_ldf_lap} 
    128128  \begin{split} 
    129129    D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\ 
     
    138138it comes: 
    139139\begin{equation} 
    140   \label{eq:tra_ldf_lap} 
     140  \label{eq:ALGOS_tra_ldf_lap} 
    141141  \begin{split} 
    142142    D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\ 
     
    149149if the velocity is uniform (\ie\ $|u|=cst$) then the diffusive flux is 
    150150\begin{equation} 
    151   \label{eq:tra_ldf_lap} 
     151  \label{eq:ALGOS_tra_ldf_lap} 
    152152  \begin{split} 
    153153    F_u^{lT} = - \frac{1}{12} 
     
    161161 
    162162\begin{equation} 
    163   \label{eq:tra_adv_ubs2} 
     163  \label{eq:ALGOS_tra_adv_ubs2} 
    164164  \begin{split} 
    165165    F_u^{lT} &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
     
    171171sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): 
    172172\begin{equation} 
    173   \label{eq:tra_adv_ubs2} 
     173  \label{eq:ALGOS_tra_adv_ubs2} 
    174174  \begin{split} 
    175175    F_u^{lT} &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] 
     
    180180sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ 
    181181\begin{equation} 
    182   \label{eq:tra_adv_ubs2} 
     182  \label{eq:ALGOS_tra_adv_ubs2} 
    183183  \begin{split} 
    184184    F_u^{lT} &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta_i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\ 
     
    192192% ------------------------------------------------------------------------------------------------------------- 
    193193\section{Leapfrog energetic} 
    194 \label{sec:LF} 
     194\label{sec:ALGOS_LF} 
    195195 
    196196We adopt the following semi-discrete notation for time derivative. 
     
    198198the time derivation and averaging operators at the mid time step are: 
    199199\[ 
    200   % \label{eq:dt_mt} 
     200  % \label{eq:ALGOS_dt_mt} 
    201201  \begin{split} 
    202202    \delta_{t+\rdt/2} [q]     &=  \  \ \,   q^{t+\rdt}  - q^{t}      \\ 
     
    210210The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: 
    211211\[ 
    212   % \label{eq:LF} 
     212  % \label{eq:ALGOS_LF} 
    213213  \frac{\partial q}{\partial t} 
    214214  \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} 
     
    220220As such it respects the quadratic invariant in integral forms, \ie\ the following continuous property, 
    221221\[ 
    222   % \label{eq:Energy} 
     222  % \label{eq:ALGOS_Energy} 
    223223  \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
    224224  =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} 
     
    278278For example in the (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: 
    279279\begin{equation} 
    280   \label{eq:Gf_triads} 
     280  \label{eq:ALGOS_Gf_triads} 
    281281  _i^k \mathbb{T}_{i_p}^{k_p} (T) 
    282282  = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k     \left( 
     
    291291and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad: 
    292292\begin{equation} 
    293   \label{eq:Gf_slopes} 
     293  \label{eq:ALGOS_Gf_slopes} 
    294294  _i^k \mathbb{R}_{i_p}^{k_p} 
    295295  =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac 
     
    297297  {\left(\alpha / \beta \right)_i^k  \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } 
    298298\end{equation} 
    299 Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of 
     299Note that in \autoref{eq:ALGOS_Gf_slopes} we use the ratio $\alpha / \beta$ instead of 
    300300multiplying the temperature derivative by $\alpha$ and the salinity derivative by $\beta$. 
    301301This is more efficient as the ratio $\alpha / \beta$ can to be evaluated directly. 
    302302 
    303 Note that in \autoref{eq:Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. 
     303Note that in \autoref{eq:ALGOS_Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. 
    304304This choice has been motivated by the decrease of tracer variance and 
    305 the presence of partial cell at the ocean bottom (see \autoref{apdx:Gf_operator}). 
     305the presence of partial cell at the ocean bottom (see \autoref{subsec:ALGOS_Gf_operator}). 
    306306 
    307307%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    310310    \includegraphics[width=\textwidth]{Fig_ISO_triad} 
    311311    \caption{ 
    312       \protect\label{fig:ISO_triad} 
     312      \protect\label{fig:ALGOS_ISO_triad} 
    313313      Triads used in the Griffies's like iso-neutral diffision scheme for 
    314314      $u$-component (upper panel) and $w$-component (lower panel). 
     
    321321They take the following expression: 
    322322\begin{flalign*} 
    323   % \label{eq:Gf_fluxes} 
     323  % \label{eq:ALGOS_Gf_fluxes} 
    324324  \begin{split} 
    325325    {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 
     
    334334the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:TRIADS_ISO_triad}): 
    335335\begin{flalign} 
    336   \label{eq:iso_flux} 
     336  \label{eq:ALGOS_iso_flux} 
    337337  \textbf{F}_{iso}(T) 
    338338  &\equiv  \sum_{\substack{i_p,\,k_p}} 
     
    364364the divergence of the sum of all the four triad fluxes: 
    365365\begin{equation} 
    366   \label{eq:Gf_operator} 
     366  \label{eq:ALGOS_Gf_operator} 
    367367  D_l^T = \frac{1}{b_T}  \sum_{\substack{i_p,\,k_p}} \left\{ 
    368368    \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 
     
    377377  the limit of flat iso-neutral direction: 
    378378  \[ 
    379     % \label{eq:Gf_property1a} 
     379    % \label{eq:ALGOS_Gf_property1a} 
    380380    D_l^T = \frac{1}{b_T}  \ \delta_{i} 
    381381    \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] 
     
    401401  The iso-neutral flux of locally referenced potential density is zero, \ie 
    402402  \begin{align*} 
    403     % \label{eq:Gf_property2} 
     403    % \label{eq:ALGOS_Gf_property2} 
    404404    \begin{matrix} 
    405405      &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} 
     
    411411    \end{matrix} 
    412412  \end{align*} 
    413   This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ and 
    414   the definition of the triads' slopes \autoref{eq:Gf_slopes}. 
     413  This result is trivially obtained using the \autoref{eq:ALGOS_Gf_triads} applied to $T$ and $S$ and 
     414  the definition of the triads' slopes \autoref{eq:ALGOS_Gf_slopes}. 
    415415 
    416416\item[$\bullet$ conservation of tracer] 
    417417  The iso-neutral diffusion term conserve the total tracer content, \ie 
    418418  \[ 
    419     % \label{eq:Gf_property1} 
     419    % \label{eq:ALGOS_Gf_property1} 
    420420    \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 
    421421  \] 
     
    425425  The iso-neutral diffusion term does not increase the total tracer variance, \ie 
    426426  \[ 
    427     % \label{eq:Gf_property1} 
     427    % \label{eq:ALGOS_Gf_property1} 
    428428    \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 
    429429  \] 
    430 The property is demonstrated in the \autoref{apdx:Gf_operator}. 
     430The property is demonstrated in the \autoref{subsec:ALGOS_Gf_operator}. 
    431431It is a key property for a diffusion term. 
    432432It means that the operator is also a dissipation term, 
     
    438438  The iso-neutral diffusion operator is self-adjoint, \ie 
    439439  \[ 
    440     % \label{eq:Gf_property1} 
     440    % \label{eq:ALGOS_Gf_property1} 
    441441    \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 
    442442  \] 
     
    444444We just have to apply the same routine. 
    445445This properties can be demonstrated quite easily in a similar way the "non increase of tracer variance" property 
    446 has been proved (see \autoref{apdx:Gf_operator}). 
     446has been proved (see \autoref{apdx:ALGOS_Gf_operator}). 
    447447\end{description} 
    448448 
     
    462462The eddy induced velocity is given by: 
    463463\begin{equation} 
    464   \label{eq:eiv_v} 
     464  \label{eq:ALGOS_eiv_v} 
    465465  \begin{split} 
    466466    u^* & = - \frac{1}{e_2\,e_{3}}          \;\partial_k \left( e_2 \, A_e \; r_i  \right) 
     
    487487% give here the expression using the triads. It is different from the one given in \autoref{eq:LDF_eiv} 
    488488% see just below a copy of this equation: 
    489 %\begin{equation} \label{eq:ldfeiv} 
     489%\begin{equation} \label{eq:ALGOS_ldfeiv} 
    490490%\begin{split} 
    491491% u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ 
     
    495495%\end{equation} 
    496496\[ 
    497   % \label{eq:eiv_vd} 
     497  % \label{eq:ALGOS_eiv_vd} 
    498498  \textbf{F}_{eiv}^T   \equiv   \left( 
    499499    \begin{aligned} 
     
    540540we end up with the skew form of the eddy induced advective fluxes: 
    541541\begin{equation} 
    542   \label{eq:eiv_skew_continuous} 
     542  \label{eq:ALGOS_eiv_skew_continuous} 
    543543  \textbf{F}_{eiv}^T = 
    544544  \begin{pmatrix} 
     
    548548\end{equation} 
    549549The tendency associated with eddy induced velocity is then simply the divergence of 
    550 the \autoref{eq:eiv_skew_continuous} fluxes. 
     550the \autoref{eq:ALGOS_eiv_skew_continuous} fluxes. 
    551551It naturally conserves the tracer content, as it is expressed in flux form and, 
    552552as the advective form, it preserves the tracer variance. 
    553 Another interesting property of \autoref{eq:eiv_skew_continuous} form is that when $A=A_e$, 
     553Another interesting property of \autoref{eq:ALGOS_eiv_skew_continuous} form is that when $A=A_e$, 
    554554a simplification occurs in the sum of the iso-neutral diffusion and eddy induced velocity terms: 
    555555\begin{flalign*} 
    556   % \label{eq:eiv_skew+eiv_continuous} 
     556  % \label{eq:ALGOS_eiv_skew+eiv_continuous} 
    557557  \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= 
    558558  \begin{pmatrix} 
     
    576576This property has been used to reduce the computational time \citep{griffies_JPO98}, 
    577577but it is not of practical use as usually $A \neq A_e$. 
    578 Nevertheless this property can be used to choose a discret form of \autoref{eq:eiv_skew_continuous} which 
    579 is consistent with the iso-neutral operator \autoref{eq:Gf_operator}. 
    580 Using the slopes \autoref{eq:Gf_slopes} and defining $A_e$ at $T$-point(\ie\ as $A$, 
     578Nevertheless this property can be used to choose a discret form of \autoref{eq:ALGOS_eiv_skew_continuous} which 
     579is consistent with the iso-neutral operator \autoref{eq:ALGOS_Gf_operator}. 
     580Using the slopes \autoref{eq:ALGOS_Gf_slopes} and defining $A_e$ at $T$-point(\ie\ as $A$, 
    581581the eddy diffusivity coefficient), the resulting discret form is given by: 
    582582\begin{equation} 
    583   \label{eq:eiv_skew} 
     583  \label{eq:ALGOS_eiv_skew} 
    584584  \textbf{F}_{eiv}^T   \equiv   \frac{1}{4} \left( 
    585585    \begin{aligned} 
     
    593593  \right) 
    594594\end{equation} 
    595 Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. 
     595Note that \autoref{eq:ALGOS_eiv_skew} is valid in $z$-coordinate with or without partial cells. 
    596596In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces must be added to 
    597597$\mathbb{R}$ for the discret form to be exact. 
     
    599599Such a choice of discretisation is consistent with the iso-neutral operator as 
    600600it uses the same definition for the slopes. 
    601 It also ensures the conservation of the tracer variance (see Appendix \autoref{apdx:eiv_skew}), 
     601It also ensures the conservation of the tracer variance (see \autoref{subsec:ALGOS_eiv_skew}), 
    602602\ie\ it does not include a diffusive component but is a "pure" advection term. 
    603603 
     
    607607% ================================================================ 
    608608\subsection{Discrete invariants of the iso-neutral diffrusion} 
    609 \label{subsec:Gf_operator} 
     609\label{subsec:ALGOS_Gf_operator} 
    610610 
    611611Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. 
     
    702702  \intertext{ 
    703703  Then outing in factor the triad in each of the four terms of the summation and 
    704   substituting the triads by their expression given in \autoref{eq:Gf_triads}. 
     704  substituting the triads by their expression given in \autoref{eq:ALGOS_Gf_triads}. 
    705705  It becomes: 
    706706  } 
     
    774774% ================================================================ 
    775775\subsection{Discrete invariants of the skew flux formulation} 
    776 \label{subsec:eiv_skew} 
     776\label{subsec:ALGOS_eiv_skew} 
    777777 
    778778Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 
     
    784784  \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv  \equiv 0 
    785785\end{align*} 
    786 The discrete form of its left hand side is obtained using \autoref{eq:eiv_skew} 
     786The discrete form of its left hand side is obtained using \autoref{eq:ALGOS_eiv_skew} 
    787787\begin{align*} 
    788788  \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\; 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIU.tex

    r11435 r11544  
    5757%=============================================================== 
    5858\section{Warm layer model} 
    59 \label{sec:warm_layer_sec} 
     59\label{sec:DIU_warm_layer_sec} 
    6060%=============================================================== 
    6161 
     
    6565\frac{\partial{\Delta T_{\mathrm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p 
    6666\nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,} 
    67 \label{eq:ecmwf1} \\ 
    68 L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2} 
     67\label{eq:DIU_ecmwf1} \\ 
     68L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:DIU_ecmwf2} 
    6969\end{align} 
    7070where $\Delta T_{\mathrm{wl}}$ is the temperature difference between the top of the warm layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal. 
    71 In equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water, 
     71In equation (\autoref{eq:DIU_ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water, 
    7272$\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat capacity at constant pressure of sea water, 
    7373$\rho_w$ is the water density, and $L$ is the Monin-Obukhov length. 
     
    7979the relationship $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is the drag coefficient, 
    8080and $\rho_a$ is the density of air. 
    81 The symbol $Q$ in equation (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy flux into 
     81The symbol $Q$ in equation (\autoref{eq:DIU_ecmwf1}) is the instantaneous total thermal energy flux into 
    8282the diurnal layer, \ie 
    8383\[ 
    8484  Q = Q_{\mathrm{sol}} + Q_{\mathrm{lw}} + Q_{\mathrm{h}}\mbox{,} 
    85   % \label{eq:e_flux_eqn} 
     85  % \label{eq:DIU_e_flux_eqn} 
    8686\] 
    8787where $Q_{\mathrm{h}}$ is the sensible and latent heat flux, $Q_{\mathrm{lw}}$ is the long wave flux, 
    8888and $Q_{\mathrm{sol}}$ is the solar flux absorbed within the diurnal warm layer. 
    8989For $Q_{\mathrm{sol}}$ the 9 term representation of \citet{gentemann.minnett.ea_JGR09} is used. 
    90 In equation \autoref{eq:ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, 
     90In equation \autoref{eq:DIU_ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, 
    9191where $L_a=0.3$\footnote{ 
    9292  This is a global average value, more accurately $L_a$ could be computed as $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$, 
     
    99994\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\ 
    100100                                    (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,} 
    101                                     \end{array} \right. \label{eq:stab_func_eqn} 
     101                                    \end{array} \right. \label{eq:DIU_stab_func_eqn} 
    102102\end{equation} 
    103 where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of (\autoref{eq:stab_func_eqn}), 
    104 and thus of (\autoref{eq:ecmwf1}), is discontinuous at $\zeta=0$ (\ie\ $Q\rightarrow0$ in 
    105 equation (\autoref{eq:ecmwf2})). 
     103where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of (\autoref{eq:DIU_stab_func_eqn}), 
     104and thus of (\autoref{eq:DIU_ecmwf1}), is discontinuous at $\zeta=0$ (\ie\ $Q\rightarrow0$ in 
     105equation (\autoref{eq:DIU_ecmwf2})). 
    106106 
    107 The two terms on the right hand side of (\autoref{eq:ecmwf1}) represent different processes. 
     107The two terms on the right hand side of (\autoref{eq:DIU_ecmwf1}) represent different processes. 
    108108The first term is simply the diabatic heating or cooling of the diurnal warm layer due to 
    109109thermal energy fluxes into and out of the layer. 
     
    114114 
    115115\section{Cool skin model} 
    116 \label{sec:cool_skin_sec} 
     116\label{sec:DIU_cool_skin_sec} 
    117117 
    118118%=============================================================== 
     
    121121As the cool skin is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the Saunders equation for the cool skin temperature difference $\Delta T_{\mathrm{cs}}$ becomes 
    122122\[ 
    123   % \label{eq:sunders_eqn} 
     123  % \label{eq:DIU_sunders_eqn} 
    124124  \Delta T_{\mathrm{cs}}=\frac{Q_{\mathrm{ns}}\delta}{k_t} \mbox{,} 
    125125\] 
     
    128128$\delta$ is the thickness of the skin layer and is given by 
    129129\begin{equation} 
    130 \label{eq:sunders_thick_eqn} 
     130\label{eq:DIU_sunders_thick_eqn} 
    131131\delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,} 
    132132\end{equation} 
     
    134134\citet{saunders_JAS67} suggested varied between 5 and 10. 
    135135 
    136 The value of $\lambda$ used in equation (\autoref{eq:sunders_thick_eqn}) is that of \citet{artale.iudicone.ea_JGR02}, 
     136The value of $\lambda$ used in equation (\autoref{eq:DIU_sunders_thick_eqn}) is that of \citet{artale.iudicone.ea_JGR02}, 
    137137which is shown in \citet{tu.tsuang_GRL05} to outperform a number of other parametrisations at 
    138138both low and high wind speeds. 
    139139Specifically, 
    140140\[ 
    141   % \label{eq:artale_lambda_eqn} 
     141  % \label{eq:DIU_artale_lambda_eqn} 
    142142  \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} 
    143143\] 
     
    145145$\gamma$ is a dimensionless function of wind speed $u$: 
    146146\[ 
    147   % \label{eq:artale_gamma_eqn} 
     147  % \label{eq:DIU_artale_gamma_eqn} 
    148148  \gamma = 
    149149  \begin{cases} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_STO.tex

    r11435 r11544  
    3030As detailed in \cite{brankart_OM13}, the stochastic formulation of the equation of state can be written as: 
    3131\begin{equation} 
    32   \label{eq:eos_sto} 
     32  \label{eq:STO_eos_sto} 
    3333  \rho = \frac{1}{2} \sum_{i=1}^m\{ \rho[T+\Delta T_i,S+\Delta S_i,p_o(z)] + \rho[T-\Delta T_i,S-\Delta S_i,p_o(z)] \} 
    3434\end{equation} 
     
    3737the scalar product of the respective local T/S gradients with random walks $\mathbf{\xi}$: 
    3838\begin{equation} 
    39   \label{eq:sto_pert} 
     39  \label{eq:STO_sto_pert} 
    4040  \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S 
    4141\end{equation} 
     
    5959 
    6060\begin{equation} 
    61   \label{eq:autoreg} 
     61  \label{eq:STO_autoreg} 
    6262  \xi^{(i)}_{k+1} = a^{(i)} \xi^{(i)}_k + b^{(i)} w^{(i)} + c^{(i)} 
    6363\end{equation} 
     
    7474 
    7575  \[ 
    76     % \label{eq:ord1} 
     76    % \label{eq:STO_ord1} 
    7777    \left\{ 
    7878      \begin{array}{l} 
     
    9191 
    9292  \begin{equation} 
    93     \label{eq:ord2} 
     93    \label{eq:STO_ord2} 
    9494    \left\{ 
    9595      \begin{array}{l} 
     
    107107\noindent 
    108108In this way, higher order processes can be easily generated recursively using the same piece of code implementing 
    109 \autoref{eq:autoreg}, and using successive processes from order $0$ to~$n-1$ as~$w^{(i)}$. 
    110 The parameters in \autoref{eq:ord2} are computed so that this recursive application of 
    111 \autoref{eq:autoreg} leads to processes with the required standard deviation and correlation timescale, 
     109\autoref{eq:STO_autoreg}, and using successive processes from order $0$ to~$n-1$ as~$w^{(i)}$. 
     110The parameters in \autoref{eq:STO_ord2} are computed so that this recursive application of 
     111\autoref{eq:STO_autoreg} leads to processes with the required standard deviation and correlation timescale, 
    112112with the additional condition that the $n-1$ first derivatives of the autocorrelation function are equal to 
    113113zero at~$t=0$, so that the resulting processes become smoother and smoother as $n$ increases. 
     
    135135 
    136136\mdl{stopts} : stochastic parametrisation associated with the non-linearity of the equation of 
    137  seawater, implementing \autoref{eq:sto_pert} so as specifics in the equation of state 
    138  implementing \autoref{eq:eos_sto}. 
     137 seawater, implementing \autoref{eq:STO_sto_pert} so as specifics in the equation of state 
     138 implementing \autoref{eq:STO_eos_sto}. 
    139139% \end{description} 
    140140 
    141141The \mdl{stopar} module includes three public routines called in the model: 
    142142 
    143 (\rou{sto\_par}) is a direct implementation of \autoref{eq:autoreg}, 
     143(\rou{sto\_par}) is a direct implementation of \autoref{eq:STO_autoreg}, 
    144144applied at each model grid point (in 2D or 3D), and called at each model time step ($k$) to 
    145145update every autoregressive process ($i=1,\ldots,m$). 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_conservation.tex

    r11435 r11544  
    77% ================================================================ 
    88\chapter{Invariants of the Primitive Equations} 
    9 \label{chap:Invariant} 
     9\label{chap:CONS} 
     10 
    1011\chaptertoc 
    1112 
     
    4546% ------------------------------------------------------------------------------------------------------------- 
    4647\section{Conservation properties on ocean dynamics} 
    47 \label{sec:Invariant_dyn} 
     48\label{sec:CONS_Invariant_dyn} 
    4849 
    4950The non linear term of the momentum equations has been split into a vorticity term, 
     
    6364The continuous formulation of the vorticity term satisfies following integral constraints: 
    6465\[ 
    65   % \label{eq:vor_vorticity} 
     66  % \label{eq:CONS_vor_vorticity} 
    6667  \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma 
    6768        \;{\mathrm {\mathbf k}}\times {\textbf {U}}_h } \right)\;dv} =0 
     
    6970 
    7071\[ 
    71   % \label{eq:vor_enstrophy} 
     72  % \label{eq:CONS_vor_enstrophy} 
    7273  if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot 
    7374    \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} 
     
    7677 
    7778\[ 
    78   % \label{eq:vor_energy} 
     79  % \label{eq:CONS_vor_energy} 
    7980  \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 
    8081\] 
     
    8889Using the symmetry or anti-symmetry properties of the operators (Eqs II.1.10 and 11), 
    8990it can be shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), 
    90 while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C).  
     91while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). 
    9192Note that the enstrophy conserving scheme on total vorticity has been chosen as the standard discrete form of 
    9293the vorticity term. 
     
    102103the horizontal gradient of horizontal kinetic energy: 
    103104 
    104 \begin{equation} \label{eq:keg_zad} 
    105 \int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial  
     105\begin{equation} \label{eq:CONS_keg_zad} 
     106\int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial 
    106107{\textbf{U}}_h }{\partial k}\;dv} 
    107108\end{equation} 
    108109 
    109110Using the discrete form given in {\S}II.2-a and the symmetry or anti-symmetry properties of 
    110 the mean and difference operators, \autoref{eq:keg_zad} is demonstrated in the Appendix C. 
    111 The main point here is that satisfying \autoref{eq:keg_zad} links the choice of the discrete forms of 
     111the mean and difference operators, \autoref{eq:CONS_keg_zad} is demonstrated in the Appendix C. 
     112The main point here is that satisfying \autoref{eq:CONS_keg_zad} links the choice of the discrete forms of 
    112113the vertical advection and of the horizontal gradient of horizontal kinetic energy. 
    113114Choosing one imposes the other. 
     
    127128 
    128129\[ 
    129   % \label{eq:hpg_pe} 
     130  % \label{eq:CONS_hpg_pe} 
    130131  \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} 
    131132\] 
     
    133134Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of 
    134135the mean and difference operators, (II.4.3) is demonstrated in the Appendix C. 
    135 The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of  
     136The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of 
    136137$T$-points and of the term added to the pressure gradient in $s-$coordinates: the depth of a $T$-point, $z_T$, 
    137138is defined as the sum the vertical scale factors at $w$-points starting from the surface. 
     
    145146Nevertheless, the $\psi$-equation is solved numerically by an iterative solver (see {\S}~III.5), 
    146147thus the property is only satisfied with the accuracy required on the solver. 
    147 In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of  
     148In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of 
    148149surface pressure forces is exactly zero: 
    149150\[ 
    150   % \label{eq:spg} 
     151  % \label{eq:CONS_spg} 
    151152  \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 
    152153\] 
     
    161162% ------------------------------------------------------------------------------------------------------------- 
    162163\section{Conservation properties on ocean thermodynamics} 
    163 \label{sec:Invariant_tra} 
     164\label{sec:CONS_Invariant_tra} 
    164165 
    165166In continuous formulation, the advective terms of the tracer equations conserve the tracer content and 
    166167the quadratic form of the tracer, \ie 
    167168\[ 
    168   % \label{eq:tra_tra2} 
     169  % \label{eq:CONS_tra_tra2} 
    169170  \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 
    170171  \;\text{and} 
     
    176177Note that in both continuous and discrete formulations, there is generally no strict conservation of mass, 
    177178since the equation of state is non linear with respect to $T$ and $S$. 
    178 In practice, the mass is conserved with a very good accuracy.  
     179In practice, the mass is conserved with a very good accuracy. 
    179180 
    180181% ------------------------------------------------------------------------------------------------------------- 
     
    182183% ------------------------------------------------------------------------------------------------------------- 
    183184\subsection{Conservation properties on momentum physics} 
    184 \label{subsec:Invariant_dyn_physics} 
     185\label{subsec:CONS_Invariant_dyn_physics} 
    185186 
    186187\textbf{* lateral momentum diffusion term} 
     
    188189The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~: 
    189190\[ 
    190   % \label{eq:dynldf_dyn} 
     191  % \label{eq:CONS_dynldf_dyn} 
    191192  \int\limits_D {\frac{1}{e_3 }{\mathrm {\mathbf k}}\cdot \nabla \times \left[ {\nabla 
    192193        _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta 
     
    195196 
    196197\[ 
    197   % \label{eq:dynldf_div} 
     198  % \label{eq:CONS_dynldf_div} 
    198199  \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
    199200        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} 
     
    202203 
    203204\[ 
    204   % \label{eq:dynldf_curl} 
     205  % \label{eq:CONS_dynldf_curl} 
    205206  \int_D {{\mathrm {\mathbf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
    206207        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} 
     
    209210 
    210211\[ 
    211   % \label{eq:dynldf_curl2} 
     212  % \label{eq:CONS_dynldf_curl2} 
    212213  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\mathrm {\mathbf k}}\cdot 
    213214    \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h 
     
    217218 
    218219\[ 
    219   % \label{eq:dynldf_div2} 
     220  % \label{eq:CONS_dynldf_div2} 
    220221  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ 
    221222      {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( 
     
    250251 
    251252\[ 
    252   % \label{eq:dynzdf_dyn} 
     253  % \label{eq:CONS_dynzdf_dyn} 
    253254  \begin{aligned} 
    254255    & \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ 
     
    258259conservation of vorticity, dissipation of enstrophy 
    259260\[ 
    260   % \label{eq:dynzdf_vor} 
     261  % \label{eq:CONS_dynzdf_vor} 
    261262  \begin{aligned} 
    262263    & \int_D {\frac{1}{e_3 }{\mathrm {\mathbf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
     
    270271conservation of horizontal divergence, dissipation of square of the horizontal divergence 
    271272\[ 
    272   % \label{eq:dynzdf_div} 
     273  % \label{eq:CONS_dynzdf_div} 
    273274  \begin{aligned} 
    274275    &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
     
    289290% ------------------------------------------------------------------------------------------------------------- 
    290291\subsection{Conservation properties on tracer physics} 
    291 \label{subsec:Invariant_tra_physics} 
     292\label{subsec:CONS_Invariant_tra_physics} 
    292293 
    293294The numerical schemes used for tracer subgridscale physics are written in such a way that 
     
    296297the quadratic form of these quantities (\ie\ their variance) globally tends to diminish. 
    297298As for the advective term, there is generally no strict conservation of mass even if, 
    298 in practice, the mass is conserved with a very good accuracy.  
    299  
    300 \textbf{* lateral physics: }conservation of tracer, dissipation of tracer  
     299in practice, the mass is conserved with a very good accuracy. 
     300 
     301\textbf{* lateral physics: }conservation of tracer, dissipation of tracer 
    301302variance, i.e. 
    302303 
    303304\[ 
    304   % \label{eq:traldf_t_t2} 
     305  % \label{eq:CONS_traldf_t_t2} 
    305306  \begin{aligned} 
    306307    &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ 
     
    312313 
    313314\[ 
    314   % \label{eq:trazdf_t_t2} 
     315  % \label{eq:CONS_trazdf_t_t2} 
    315316  \begin{aligned} 
    316317    & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\ 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

    r11543 r11544  
    2626To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 
    2727\[ 
    28   % \label{eq:PE_} 
     28  % \label{eq:MBZ_PE_} 
    2929  z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 
    3030\] 
     
    7575\section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})] 
    7676{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} 
    77 \label{sec:DYN_hpg_spg} 
     77\label{sec:MBZ_dyn_hpg_spg} 
    7878%-----------------------------------------nam_dynspg---------------------------------------------------- 
    7979 
     
    100100\subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})] 
    101101{Explicit (\protect\key{dynspg\_exp})} 
    102 \label{subsec:DYN_spg_exp} 
     102\label{subsec:MBZ_dyn_spg_exp} 
    103103 
    104104In the explicit free surface formulation, the model time step is chosen small enough to 
     
    106106The sea surface height is given by: 
    107107\begin{equation} 
    108   \label{eq:dynspg_ssh} 
     108  \label{eq:MBZ_dynspg_ssh} 
    109109  \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T} 
    110110    e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u} 
     
    120120The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: 
    121121\begin{equation} 
    122   \label{eq:dynspg_exp} 
     122  \label{eq:MBZ_dynspg_exp} 
    123123  \left\{ 
    124124    \begin{aligned} 
     
    137137\subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})] 
    138138{Split-explicit time-stepping (\protect\key{dynspg\_ts})} 
    139 \label{subsec:DYN_spg_ts} 
     139\label{subsec:MBZ_dyn_spg_ts} 
    140140%--------------------------------------------namdom---------------------------------------------------- 
    141141 
     
    152152    \includegraphics[width=\textwidth]{Fig_DYN_dynspg_ts} 
    153153    \caption{ 
    154       \protect\label{fig:DYN_dynspg_ts} 
     154      \protect\label{fig:MBZ_dyn_dynspg_ts} 
    155155      Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 
    156156      after \citet{Griffies2004?}. 
     
    186186We have 
    187187\[ 
    188   % \label{eq:DYN_spg_ts_eta} 
     188  % \label{eq:MBZ_dyn_spg_ts_eta} 
    189189  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 
    190190  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
    191191\] 
    192192\begin{multline*} 
    193   % \label{eq:DYN_spg_ts_u} 
     193  % \label{eq:MBZ_dyn_spg_ts_u} 
    194194  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\ 
    195195  = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) 
     
    207207This is also the time that sets the barotropic time steps via 
    208208\[ 
    209   % \label{eq:DYN_spg_ts_t} 
     209  % \label{eq:MBZ_dyn_spg_ts_t} 
    210210  t_n=\tau+n\Delta t 
    211211\] 
     
    213213The density scaled surface pressure is evaluated via 
    214214\[ 
    215   % \label{eq:DYN_spg_ts_ps} 
     215  % \label{eq:MBZ_dyn_spg_ts_ps} 
    216216  p_s^{(b)}(\tau,t_{n}) = 
    217217  \begin{cases} 
     
    222222To get started, we assume the following initial conditions 
    223223\[ 
    224   % \label{eq:DYN_spg_ts_eta} 
     224  % \label{eq:MBZ_dyn_spg_ts_eta} 
    225225  \begin{split} 
    226226    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ 
     
    230230with 
    231231\[ 
    232   % \label{eq:DYN_spg_ts_etaF} 
     232  % \label{eq:MBZ_dyn_spg_ts_etaF} 
    233233  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 
    234234\] 
     
    236236Likewise, 
    237237\[ 
    238   % \label{eq:DYN_spg_ts_u} 
     238  % \label{eq:MBZ_dyn_spg_ts_u} 
    239239  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ 
    240240  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 
     
    242242with 
    243243\[ 
    244   % \label{eq:DYN_spg_ts_u} 
     244  % \label{eq:MBZ_dyn_spg_ts_u} 
    245245  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) 
    246246\] 
     
    251251produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 
    252252\[ 
    253   % \label{eq:DYN_spg_ts_u} 
     253  % \label{eq:MBZ_dyn_spg_ts_u} 
    254254  \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 
    255255  = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 
     
    258258a baroclinic leap-frog using the following form 
    259259\begin{equation} 
    260   \label{eq:DYN_spg_ts_ssh} 
     260  \label{eq:MBZ_dyn_spg_ts_ssh} 
    261261  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
    262262\end{equation} 
     
    267267 
    268268In general, some form of time filter is needed to maintain integrity of the surface height field due to 
    269 the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. 
     269the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}. 
    270270We have tried various forms of such filtering, 
    271271with the following method discussed in Griffies et al. (2001) chosen due to its stability and 
     
    273273 
    274274\begin{equation} 
    275   \label{eq:DYN_spg_ts_sshf} 
     275  \label{eq:MBZ_dyn_spg_ts_sshf} 
    276276  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
    277277\end{equation} 
     
    279279 
    280280\[ 
    281   % \label{eq:DYN_spg_ts_sshf2} 
     281  % \label{eq:MBZ_dyn_spg_ts_sshf2} 
    282282  \eta^{F}(\tau-\Delta) = \eta(\tau) 
    283283  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) 
     
    288288This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and 
    289289surface height time filtering (see ?? for more complete discussion). 
    290 However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to 
     290However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to 
    291291be more conservative, and so is recommended. 
    292292 
     
    296296\subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})] 
    297297{Filtered formulation (\protect\key{dynspg\_flt})} 
    298 \label{subsec:DYN_spg_flt} 
     298\label{subsec:MBZ_dyn_spg_flt} 
    299299 
    300300The filtered formulation follows the \citet{Roullet2000?} implementation. 
     
    311311\subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})] 
    312312{Non-linear free surface formulation (\protect\key{vvl})} 
    313 \label{subsec:DYN_spg_vvl} 
     313\label{subsec:MBZ_dyn_spg_vvl} 
    314314 
    315315In the non-linear free surface formulation, the variations of volume are fully taken into account. 
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