# Changeset 6055

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Timestamp:
2015-12-15T17:19:09+01:00 (5 years ago)
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#1613: vvl by default : update DOC on TRA advection + change in namelist comment

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branches/2015/dev_r5836_NOC3_vvl_by_default
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• ## branches/2015/dev_r5836_NOC3_vvl_by_default/DOC/TexFiles/Biblio/Biblio.bib

 r6040 } @PHDTHESIS{Demange_PhD2014, author = {J. Farge}, title = {Sch\'{e}́mas num\'{e}́riques d’advection et de propagation d’ondes de gravit\'{e}́ dans les mod\`{e}les de circulation oc\'{e}́anique.}, school = {Doctorat es Applied Mathematiques, Grenoble University, France}, year = {2014}, pages = {138pp} } @ARTICLE{Dobricic_al_OS07, author = {M. Farge}, title = {Dynamique non lineaire des ondes et des tourbillons dans les equations de Saint Venant}, school = {Doctorat es Mathematiques, Paris VI University}, school = {Doctorat es Mathematiques, Paris VI University, France}, year = {1987}, pages = {401pp} } @ARTICLE{Lemarie_OM2015, author = {F. Lemari\'{e} and L. Debreu and J. Demange and  G. Madec and J.M. Molines and M. Honnorat}, title = {Stability Constraints for Oceanic Numerical Models: Implications for the Formulation of time and space Discretizations}, journal = OM, year = {2015}, volume = {92}, pages = {124--148}, doi = {10.1016/j.ocemod.2015.06.006}, url = {http://dx.doi.org/10.1016/j.ocemod.2015.06.006} } @ARTICLE{Lermusiaux2001, author = {P. F. J. Lermusiaux}, author = {M. L\'{e}vy}, title = {Mod\'{e}lisation des processus biog\'{e}ochimiques en M\'{e}diterran\'{e}e nord-occidentale. Cycle saisonnier et variabilit\'{e} m\'{e}so\'{e}chelle}, nord-occidentale. Cycle saisonnier et variabilit\'{e} m\'{e}so\'{e}chelle}, school = {Universit\'{e} Pierre et Marie Curie, Paris, France, 207pp}, year = {1996} year = {2010}, pages = {submitted}, } @ARTICLE{Lele_JCP1992, author = {S.K. Lele}, title = {Compact finite difference schemes with spectral-like resolution}, journal = JCP, year = {1992}, volume = {103} pages = {16--42} }
• ## branches/2015/dev_r5836_NOC3_vvl_by_default/DOC/TexFiles/Chapters/Chap_TRA.tex

 r6040 when those parameterisations are used (see Chap.~\ref{LDF}). The choice of an advection scheme is made in the \textit{\ngn{namtra\_adv}} namelist, by setting to \textit{true} one of the logicals \textit{ln\_traadv\_xxx}. The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST). The choice is made in the \textit{\ngn{namtra\_adv}} namelist, by setting to \textit{true} one of the logicals \textit{ln\_traadv\_xxx}. The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. By default ($i.e.$ in the reference namelist, \ngn{namelist\_ref}), all the logicals are set to \textit{false}. If the user does not select an advection scheme in the configuration namelist (\ngn{namelist\_cfg}), the tracers will not be advected ! Details of the advection schemes are given below. The choice of an advection scheme in the configuration namelist (\ngn{namelist\_cfg}), the tracers will \textit{not} be advected ! Details of the advection schemes are given below. The choosing an advection scheme is a complex matter which depends on the model physics, model resolution, type of tracer, as well as the issue of numerical cost. Note that type of tracer, as well as the issue of numerical cost. In particular, we note that (1) CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that they do not necessarily require additional diffusion ; while the other schemes are diffusive enough so that they do not necessarily need additional diffusion ; (2) CEN and UBS are not \textit{positive} schemes \footnote{negative values can appear in an initially strictly positive tracer field %        2nd and 4th order centred schemes % ------------------------------------------------------------------------------------------------------------- \subsection   [$2^{nd}$ and $4^{th}$ order centred schemes (CEN) (\np{ln\_traadv\_cen})] {$2^{nd}$ and $4^{th}$ order centred schemes (CEN) (\np{ln\_traadv\_cen}=true)} \subsection [centred schemes (CEN) (\np{ln\_traadv\_cen})] {centred schemes (CEN) (\np{ln\_traadv\_cen}=true)} \label{TRA_adv_cen} %        2nd order centred scheme In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of the two neighbouring $T$-point values. The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}~=~\textit{true}. Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$. CEN implementation can be found in the \mdl{traadv\_cen} module. In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of the two neighbouring $T$-point values. For example, in the $i$-direction : \label{Eq_tra_adv_cen2} a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. CEN2 is computed in the \mdl{traadv\_cen} module. Note that using the CEN2, the overall tracer advection is of second order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) have this order of accuracy. \gmcomment{Note also that ... blah, blah} have this order of accuracy. %        4nd order centred scheme In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at velocity points as In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. For example, in the $i$-direction: =\overline{   T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2} In the vertical direction (\np{nn\_cen\_v}=$4$), a $4^{th}$ COMPACT interpolation has been prefered \citep{Demange_PhD2014}. In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion, spectral characteristics similar to schemes of higher order \citep{Lele_JCP1992}. Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive, $i.e.$ the global variance of a tracer is not preserved using CEN4. Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer. At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This hypothesis usually reduces the order of the scheme. Here we choose to set the gradient of $T$ across the boundary to zero. Alternative conditions can be specified, such as a reduction to a second order scheme for these near boundary grid points. it is not non-diffusive, $i.e.$ the global variance of a tracer is not preserved using CEN4. Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer. At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This hypothesis usually reduces the order of the scheme. Here we choose to set the gradient of $T$ across the boundary to zero. Alternative conditions can be specified, such as a reduction to a second order scheme for these near boundary grid points. % ------------------------------------------------------------------------------------------------------------- %        FCT scheme % ------------------------------------------------------------------------------------------------------------- \subsection   [$2^{nd}$ and $4^{th}$ Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct})] {$2^{nd}$ and $4^{th}$ Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct}=true)} \subsection   [Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct})] {Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct}=true)} \label{TRA_adv_tvd} In the Flux Corrected Transport formulation, the tracer at velocity points is evaluated using a combination of an upstream and a centred scheme. For example, in the $i$-direction : The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}~=~\textit{true}. Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$. FCT implementation can be found in the \mdl{traadv\_fct} module. In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and a centred scheme. For example, in the $i$-direction : \label{Eq_tra_adv_fct} \begin{split} where $c_u$ is a flux limiter function taking values between 0 and 1. The FCT order is the one of the centred scheme used ($i.e.$ it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}. There exist many ways to define $c_u$, each corresponding to a different total variance decreasing scheme. The one chosen in \NEMO is described in \citet{Zalesak_JCP79}. $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. The resulting scheme is quite expensive but \emph{positive}. It can be used on both active and passive tracers. This scheme is tested and compared with MUSCL and a MPDATA scheme in \citet{Levy_al_GRL01}. The FCT scheme is implemented in the \mdl{traadv\_fct} module. For stability reasons (see \S\ref{STP}), $\tau _u^{cen}$ is evaluated  in (\ref{Eq_tra_adv_fct}) using the \textit{now} tracer while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, FCT scheme. The one chosen in \NEMO is described in \citet{Zalesak_JCP79}. $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. The resulting scheme is quite expensive but \emph{positive}. It can be used on both active and passive tracers. A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}. An additional option has been added controlled by \np{nn\_fct\_zts}. By setting this integer to a value larger than zero, a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, but on the latter, a split-explicit time stepping is used, with a number of sub-timestep equals to \np{nn\_fct\_zts}. This option can be useful when the size of the timestep is limited by vertical advection \citep{Lemarie_OM2015)}. Note that in this case, a similar split-explicit time stepping should be used on vertical advection of momentum to insure a better stability (see \S\ref{DYN_zad}). For stability reasons (see \S\ref{STP}), $\tau _u^{cen}$ is evaluated in (\ref{Eq_tra_adv_fct}) using the \textit{now} tracer while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, the advective part of the scheme is time stepped with a leap-frog scheme while a forward scheme is used for the diffusive part. \label{TRA_adv_mus} The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been implemented by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}~=~\textit{true}. MUSCL implementation can be found in the \mdl{traadv\_mus} module. MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between two $T$-points (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction : directed toward land, an upstream flux is used. This choice ensure the \textit{positive} character of the scheme. In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes (\np{ln\_mus\_ups}~=~\textit{true}). % ------------------------------------------------------------------------------------------------------------- \label{TRA_adv_ubs} The UBS advection scheme (also often called UP3) is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation. It is also known as the Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics). For example, in the $i$-direction : The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}~=~\textit{true}. UBS implementation can be found in the \mdl{traadv\_mus} module. The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics). It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation. For example, in the $i$-direction : \label{Eq_tra_adv_ubs} \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ the advection scheme is similar to that reported in \cite{Farrow1995}. It is a relatively good compromise between accuracy and smoothness. It is not a \emph{positive} scheme, meaning that false extrema are permitted, Nevertheless the scheme is not \emph{positive}, meaning that false extrema are permitted, but the amplitude of such are significantly reduced over the centred second or fourth order method. Nevertheless it is not recommended that it should be or fourth order method. therefore it is not recommended that it should be applied to a passive tracer that requires positivity. The intrinsic diffusion of UBS makes its use risky in the vertical direction where the control of artificial diapycnal fluxes is of paramount importance. Therefore the vertical flux is evaluated using either a 2nd order FCT scheme or a 4th order COMPACT scheme (\np{nn\_cen\_v}=2 or 4). where the control of artificial diapycnal fluxes is of paramount importance \citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}. Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme (\np{nn\_cen\_v}=2 or 4). For stability reasons  (see \S\ref{STP}), substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. ??? Four different options are possible for the vertical component used in the UBS scheme. $\tau _w^{ubs}$ can be evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme, or  \textit{(b)} a FCT scheme, or  \textit{(c)} an interpolation based on conservative parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS, or  \textit{(d)} a UBS. The $3^{rd}$ case has dispersion properties similar to an eighth-order accurate conventional scheme. The current reference version uses method (b). ??? Note that : (1) When a high vertical resolution $O(1m)$ is used, the model stability can be controlled by vertical advection (not vertical diffusion which is usually solved using an implicit scheme). Computer time can be saved by using a time-splitting technique on vertical advection. Such a technique has been implemented and validated in ORCA05 with 301 levels. It is not available in the current reference version. (2) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: Note that it is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: \label{Eq_traadv_ubs2} \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which is simply proportional to the velocity: $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO still uses \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. %%% \gmcomment{the change in UBS scheme has to be done} %%% $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO uses the computationally more efficient formulation \eqref{Eq_tra_adv_ubs}. % ------------------------------------------------------------------------------------------------------------- The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST is used when \np{ln\_traadv\_qck}~=~\textit{true}. QUICKEST implementation can be found in the \mdl{traadv\_mus} module. QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. This no longer guarantees the positivity of the scheme. The use of TVD in the vertical direction (as for the UBS case) should be implemented to restore this property. %%%gmcomment   :  Cross term are missing in the current implementation.... \hline coeff.   & computer name   & S-EOS     &  description                      \\ \hline $a_0$       & \np{nn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff.    \\ \hline $b_0$       & \np{nn\_b0}     & 7.6554 $10^{-1}$ &  linear haline  expansion coeff.    \\ \hline $\lambda_1$ & \np{nn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$          \\ \hline $\lambda_2$ & \np{nn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$       \\ \hline $\nu$       & \np{nn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$       \\ \hline $\mu_1$     & \np{nn\_mu1}    & 1.4970 $10^{-4}$ &  thermobaric coeff. in T         \\ \hline $\mu_2$     & \np{nn\_mu2}    & 1.1090 $10^{-5}$ &  thermobaric coeff. in S            \\ \hline $a_0$       & \np{rn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff.    \\ \hline $b_0$       & \np{rn\_b0}     & 7.6554 $10^{-1}$ &  linear haline  expansion coeff.    \\ \hline $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$          \\ \hline $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$       \\ \hline $\nu$       & \np{rn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$       \\ \hline $\mu_1$     & \np{rn\_mu1}    & 1.4970 $10^{-4}$ &  thermobaric coeff. in T         \\ \hline $\mu_2$     & \np{rn\_mu2}    & 1.1090 $10^{-5}$ &  thermobaric coeff. in S            \\ \hline \end{tabular} \caption{ \label{Tab_SEOS}