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branches/2015/dev_r5836_NOC3_vvl_by_default/DOC/TexFiles/Biblio/Biblio.bib
r6040 r6055 763 763 } 764 764 765 @PHDTHESIS{Demange_PhD2014, 766 author = {J. Farge}, 767 title = {Sch\'{e}́mas num\'{e}́riques d’advection et de propagation d’ondes de gravit\'{e}́ 768 dans les mod\`{e}les de circulation oc\'{e}́anique.}, 769 school = {Doctorat es Applied Mathematiques, Grenoble University, France}, 770 year = {2014}, 771 pages = {138pp} 772 } 765 773 766 774 @ARTICLE{Dobricic_al_OS07, … … 983 991 author = {M. Farge}, 984 992 title = {Dynamique non lineaire des ondes et des tourbillons dans les equations de Saint Venant}, 985 school = {Doctorat es Mathematiques, Paris VI University },993 school = {Doctorat es Mathematiques, Paris VI University, France}, 986 994 year = {1987}, 987 995 pages = {401pp} … … 1663 1671 } 1664 1672 1673 1674 @ARTICLE{Lemarie_OM2015, 1675 author = {F. Lemari\'{e} and L. Debreu and J. Demange and G. Madec and J.M. Molines and M. Honnorat}, 1676 title = {Stability Constraints for Oceanic Numerical Models: 1677 Implications for the Formulation of time and space Discretizations}, 1678 journal = OM, 1679 year = {2015}, 1680 volume = {92}, 1681 pages = {124--148}, 1682 doi = {10.1016/j.ocemod.2015.06.006}, 1683 url = {http://dx.doi.org/10.1016/j.ocemod.2015.06.006} 1684 } 1685 1665 1686 @ARTICLE{Lermusiaux2001, 1666 1687 author = {P. F. J. Lermusiaux}, … … 1675 1696 author = {M. L\'{e}vy}, 1676 1697 title = {Mod\'{e}lisation des processus biog\'{e}ochimiques en M\'{e}diterran\'{e}e 1677 nord-occidentale. Cycle saisonnier et variabilit\'{e} m\'{e}so\'{e}chelle},1698 nord-occidentale. Cycle saisonnier et variabilit\'{e} m\'{e}so\'{e}chelle}, 1678 1699 school = {Universit\'{e} Pierre et Marie Curie, Paris, France, 207pp}, 1679 1700 year = {1996} … … 1814 1835 year = {2010}, 1815 1836 pages = {submitted}, 1837 } 1838 1839 @ARTICLE{Lele_JCP1992, 1840 author = {S.K. Lele}, 1841 title = {Compact finite difference schemes with spectral-like resolution}, 1842 journal = JCP, 1843 year = {1992}, 1844 volume = {103} 1845 pages = {16--42} 1816 1846 } 1817 1847 -
branches/2015/dev_r5836_NOC3_vvl_by_default/DOC/TexFiles/Chapters/Chap_TRA.tex
r6040 r6055 135 135 when those parameterisations are used (see Chap.~\ref{LDF}). 136 136 137 The choice of an advection scheme is made in the \textit{\ngn{namtra\_adv}} namelist, by 138 setting to \textit{true} one of the logicals \textit{ln\_traadv\_xxx}. The 139 corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 140 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 137 Several tracer advection scheme are proposed, namely 138 a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), 139 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), 140 a Monotone Upstream Scheme for Conservative Laws scheme (MUSCL), 141 a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), and 142 a Quadratic Upstream Interpolation for Convective Kinematics with 143 Estimated Streaming Terms scheme (QUICKEST). 144 The choice is made in the \textit{\ngn{namtra\_adv}} namelist, by 145 setting to \textit{true} one of the logicals \textit{ln\_traadv\_xxx}. 146 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, 147 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 141 148 By default ($i.e.$ in the reference namelist, \ngn{namelist\_ref}), all the logicals 142 149 are set to \textit{false}. If the user does not select an advection scheme 143 in the configuration namelist (\ngn{namelist\_cfg}), the tracers will notbe advected !144 145 Details of the advection schemes are given below. The cho ice ofan advection scheme150 in the configuration namelist (\ngn{namelist\_cfg}), the tracers will \textit{not} be advected ! 151 152 Details of the advection schemes are given below. The choosing an advection scheme 146 153 is a complex matter which depends on the model physics, model resolution, 147 type of tracer, as well as the issue of numerical cost. 148 149 Note that 154 type of tracer, as well as the issue of numerical cost. In particular, we note that 150 155 (1) CEN and FCT schemes require an explicit diffusion operator 151 while the other schemes are diffusive enough so that they do not necessarily requireadditional diffusion ;156 while the other schemes are diffusive enough so that they do not necessarily need additional diffusion ; 152 157 (2) CEN and UBS are not \textit{positive} schemes 153 158 \footnote{negative values can appear in an initially strictly positive tracer field … … 166 171 % 2nd and 4th order centred schemes 167 172 % ------------------------------------------------------------------------------------------------------------- 168 \subsection [$2^{nd}$ and $4^{th}$ ordercentred schemes (CEN) (\np{ln\_traadv\_cen})]169 {$2^{nd}$ and $4^{th}$ ordercentred schemes (CEN) (\np{ln\_traadv\_cen}=true)}173 \subsection [centred schemes (CEN) (\np{ln\_traadv\_cen})] 174 {centred schemes (CEN) (\np{ln\_traadv\_cen}=true)} 170 175 \label{TRA_adv_cen} 171 176 172 177 % 2nd order centred scheme 173 178 174 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is 175 evaluated as the mean of the two neighbouring $T$-point values. 179 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}~=~\textit{true}. 180 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) 181 and vertical direction by setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$. 182 CEN implementation can be found in the \mdl{traadv\_cen} module. 183 184 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points 185 is evaluated as the mean of the two neighbouring $T$-point values. 176 186 For example, in the $i$-direction : 177 187 \begin{equation} \label{Eq_tra_adv_cen2} … … 185 195 a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in 186 196 (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. 187 CEN2 is computed in the \mdl{traadv\_cen} module.188 197 189 198 Note that using the CEN2, the overall tracer advection is of second 190 199 order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) 191 have this order of accuracy. \gmcomment{Note also that ... blah, blah}200 have this order of accuracy. 192 201 193 202 % 4nd order centred scheme 194 203 195 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at velocitypoints as204 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as 196 205 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 197 206 For example, in the $i$-direction: … … 200 209 =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 201 210 \end{equation} 211 In the vertical direction (\np{nn\_cen\_v}=$4$), a $4^{th}$ COMPACT interpolation 212 has been prefered \citep{Demange_PhD2014}. 213 In the COMPACT scheme, both the field and its derivative are interpolated, 214 which leads, after a matrix inversion, spectral characteristics 215 similar to schemes of higher order \citep{Lele_JCP1992}. 216 202 217 203 218 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme … … 212 227 213 228 A direct consequence of the pseudo-fourth order nature of the scheme is that 214 it is not non-diffusive, $i.e.$ the global variance of a tracer is not preserved using 215 CEN4. Furthermore, it must be used in conjunction with an explicit216 diffusion operator to produce a sensible solution. As in CEN2 case, the time-stepping is217 performed using a leapfrog scheme in conjunction with an Asselin time-filter,218 so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.219 220 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an221 a dditional hypothesis must be made to evaluate $\tau _u^{cen4}$. This222 hypothesis usually reduces the order of the scheme. Here we choose to set223 the gradient of $T$ across the boundary to zero. Alternative conditions can be224 specified, such as a reduction to a second order scheme for these near boundary225 grid points.229 it is not non-diffusive, $i.e.$ the global variance of a tracer is not preserved using CEN4. 230 Furthermore, it must be used in conjunction with an explicit diffusion operator 231 to produce a sensible solution. 232 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction 233 with an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer. 234 235 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), 236 an additional hypothesis must be made to evaluate $\tau _u^{cen4}$. 237 This hypothesis usually reduces the order of the scheme. 238 Here we choose to set the gradient of $T$ across the boundary to zero. 239 Alternative conditions can be specified, such as a reduction to a second order scheme 240 for these near boundary grid points. 226 241 227 242 % ------------------------------------------------------------------------------------------------------------- 228 243 % FCT scheme 229 244 % ------------------------------------------------------------------------------------------------------------- 230 \subsection [ $2^{nd}$ and $4^{th}$Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct})]231 { $2^{nd}$ and $4^{th}$Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct}=true)}245 \subsection [Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct})] 246 {Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct}=true)} 232 247 \label{TRA_adv_tvd} 233 248 234 In the Flux Corrected Transport formulation, the tracer at velocity 235 points is evaluated using a combination of an upstream and a centred scheme. 236 For example, in the $i$-direction : 249 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}~=~\textit{true}. 250 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) 251 and vertical direction by setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$. 252 FCT implementation can be found in the \mdl{traadv\_fct} module. 253 254 In FCT formulation, the tracer at velocity points is evaluated using a combination of 255 an upstream and a centred scheme. For example, in the $i$-direction : 237 256 \begin{equation} \label{Eq_tra_adv_fct} 238 257 \begin{split} … … 246 265 \end{equation} 247 266 where $c_u$ is a flux limiter function taking values between 0 and 1. 267 The FCT order is the one of the centred scheme used ($i.e.$ it depends on the setting of 268 \np{nn\_fct\_h} and \np{nn\_fct\_v}. 248 269 There exist many ways to define $c_u$, each corresponding to a different 249 total variance decreasing scheme. The one chosen in \NEMO is described in 250 \citet{Zalesak_JCP79}. $c_u$ only departs from $1$ when the advective term 251 produces a local extremum in the tracer field. The resulting scheme is quite 252 expensive but \emph{positive}. It can be used on both active and passive tracers. 253 This scheme is tested and compared with MUSCL and a MPDATA scheme in \citet{Levy_al_GRL01}. 254 The FCT scheme is implemented in the \mdl{traadv\_fct} module. 255 256 For stability reasons (see \S\ref{STP}), 257 $\tau _u^{cen}$ is evaluated in (\ref{Eq_tra_adv_fct}) using the \textit{now} tracer 258 while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, 270 FCT scheme. The one chosen in \NEMO is described in \citet{Zalesak_JCP79}. 271 $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. 272 The resulting scheme is quite expensive but \emph{positive}. 273 It can be used on both active and passive tracers. 274 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}. 275 276 An additional option has been added controlled by \np{nn\_fct\_zts}. By setting this integer to 277 a value larger than zero, a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, 278 but on the latter, a split-explicit time stepping is used, with a number of sub-timestep equals 279 to \np{nn\_fct\_zts}. This option can be useful when the size of the timestep is limited 280 by vertical advection \citep{Lemarie_OM2015)}. Note that in this case, a similar split-explicit 281 time stepping should be used on vertical advection of momentum to insure a better stability 282 (see \S\ref{DYN_zad}). 283 284 For stability reasons (see \S\ref{STP}), $\tau _u^{cen}$ is evaluated in (\ref{Eq_tra_adv_fct}) 285 using the \textit{now} tracer while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, 259 286 the advective part of the scheme is time stepped with a leap-frog scheme 260 287 while a forward scheme is used for the diffusive part. … … 267 294 \label{TRA_adv_mus} 268 295 269 The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been 270 implemented by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points 296 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}~=~\textit{true}. 297 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 298 299 MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points 271 300 is evaluated assuming a linear tracer variation between two $T$-points 272 301 (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction : … … 288 317 directed toward land, an upstream flux is used. This choice ensure 289 318 the \textit{positive} character of the scheme. 319 In addition, fluxes round a grid-point where a runoff is applied can optionally be 320 computed using upstream fluxes (\np{ln\_mus\_ups}~=~\textit{true}). 290 321 291 322 % ------------------------------------------------------------------------------------------------------------- … … 296 327 \label{TRA_adv_ubs} 297 328 298 The UBS advection scheme (also often called UP3) is an upstream-biased third order 299 scheme based on an upstream-biased parabolic interpolation. It is also known as 300 the Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective 301 Kinematics). For example, in the $i$-direction : 329 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}~=~\textit{true}. 330 UBS implementation can be found in the \mdl{traadv\_mus} module. 331 332 The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme 333 (Quadratic Upstream Interpolation for Convective Kinematics). It is an upstream-biased 334 third order scheme based on an upstream-biased parabolic interpolation. 335 For example, in the $i$-direction : 302 336 \begin{equation} \label{Eq_tra_adv_ubs} 303 337 \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ … … 313 347 the advection scheme is similar to that reported in \cite{Farrow1995}. 314 348 It is a relatively good compromise between accuracy and smoothness. 315 It is not a \emph{positive} scheme, meaning that false extrema are permitted,349 Nevertheless the scheme is not \emph{positive}, meaning that false extrema are permitted, 316 350 but the amplitude of such are significantly reduced over the centred second 317 or fourth order method. Neverthelessit is not recommended that it should be351 or fourth order method. therefore it is not recommended that it should be 318 352 applied to a passive tracer that requires positivity. 319 353 320 354 The intrinsic diffusion of UBS makes its use risky in the vertical direction 321 where the control of artificial diapycnal fluxes is of paramount importance .322 Therefore the vertical flux is evaluated using either a 2ndorder FCT scheme323 or a 4thorder COMPACT scheme (\np{nn\_cen\_v}=2 or 4).355 where the control of artificial diapycnal fluxes is of paramount importance \citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}. 356 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme 357 or a $4^th$ order COMPACT scheme (\np{nn\_cen\_v}=2 or 4). 324 358 325 359 For stability reasons (see \S\ref{STP}), … … 336 370 substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 337 371 338 ??? 339 340 Four different options are possible for the vertical 341 component used in the UBS scheme. $\tau _w^{ubs}$ can be evaluated 342 using either \textit{(a)} a centred $2^{nd}$ order scheme, or \textit{(b)} 343 a FCT scheme, or \textit{(c)} an interpolation based on conservative 344 parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} 345 implementation of UBS in ROMS, or \textit{(d)} a UBS. The $3^{rd}$ case 346 has dispersion properties similar to an eighth-order accurate conventional scheme. 347 The current reference version uses method (b). 348 349 ??? 350 351 Note that : 352 353 (1) When a high vertical resolution $O(1m)$ is used, the model stability can 354 be controlled by vertical advection (not vertical diffusion which is usually 355 solved using an implicit scheme). Computer time can be saved by using a 356 time-splitting technique on vertical advection. Such a technique has been 357 implemented and validated in ORCA05 with 301 levels. It is not available 358 in the current reference version. 359 360 (2) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: 372 Note that it is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: 361 373 \begin{equation} \label{Eq_traadv_ubs2} 362 374 \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ … … 380 392 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 381 393 coefficient which is simply proportional to the velocity: 382 $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO still uses 383 \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. 384 %%% 385 \gmcomment{the change in UBS scheme has to be done} 386 %%% 394 $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO uses 395 the computationally more efficient formulation \eqref{Eq_tra_adv_ubs}. 387 396 388 397 % ------------------------------------------------------------------------------------------------------------- … … 395 404 The Quadratic Upstream Interpolation for Convective Kinematics with 396 405 Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} 397 is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST 406 is used when \np{ln\_traadv\_qck}~=~\textit{true}. 407 QUICKEST implementation can be found in the \mdl{traadv\_mus} module. 408 409 QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST 398 410 limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray 399 411 (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. … … 405 417 This no longer guarantees the positivity of the scheme. The use of TVD in the vertical 406 418 direction (as for the UBS case) should be implemented to restore this property. 419 420 %%%gmcomment : Cross term are missing in the current implementation.... 407 421 408 422 … … 1241 1255 \hline 1242 1256 coeff. & computer name & S-EOS & description \\ \hline 1243 $a_0$ & \np{ nn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline1244 $b_0$ & \np{ nn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline1245 $\lambda_1$ & \np{ nn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline1246 $\lambda_2$ & \np{ nn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline1247 $\nu$ & \np{ nn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline1248 $\mu_1$ & \np{ nn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline1249 $\mu_2$ & \np{ nn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline1257 $a_0$ & \np{rn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline 1258 $b_0$ & \np{rn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline 1259 $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline 1260 $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline 1261 $\nu$ & \np{rn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline 1262 $\mu_1$ & \np{rn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline 1263 $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1250 1264 \end{tabular} 1251 1265 \caption{ \label{Tab_SEOS} -
branches/2015/dev_r5836_NOC3_vvl_by_default/DOC/TexFiles/Namelist/namtra_adv
r6040 r6055 13 13 ln_mus_ups = .false. ! use upstream scheme near river mouths 14 14 ln_traadv_ubs = .false. ! UBS scheme 15 nn_ubs_v = 2 ! =2 , vertical 2nd order FCT 15 nn_ubs_v = 2 ! =2 , vertical 2nd order FCT / COMPACT 4th order 16 16 ln_traadv_qck = .false. ! QUICKEST scheme 17 17 / -
branches/2015/dev_r5836_NOC3_vvl_by_default/NEMOGCM/CONFIG/SHARED/namelist_ref
r6004 r6055 765 765 ln_mus_ups = .false. ! use upstream scheme near river mouths 766 766 ln_traadv_ubs = .false. ! UBS scheme 767 nn_ubs_v = 2 ! =2 , vertical 2nd order FCT 767 nn_ubs_v = 2 ! =2 , vertical 2nd order FCT / COMPACT 4th order 768 768 ln_traadv_qck = .false. ! QUICKEST scheme 769 769 /
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