Changeset 11123 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex
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r10544 r11123 136 136 Nevertheless, in the latter case, it is achieved to a good approximation since 137 137 the non-conservative term is the product of the time derivative of the tracer and the free surface height, 138 two quantities that are not correlated \citep{ Roullet_Madec_JGR00, Griffies_al_MWR01, Campin2004}.139 140 The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco }) is138 two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 139 140 The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco?}) is 141 141 the centred (\textit{now}) \textit{effective} ocean velocity, \ie the \textit{eulerian} velocity 142 142 (see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or … … 221 221 \end{equation} 222 222 In the vertical direction (\np{nn\_cen\_v}~\forcode{= 4}), 223 a $4^{th}$ COMPACT interpolation has been prefered \citep{ Demange_PhD2014}.223 a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}. 224 224 In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion, 225 spectral characteristics similar to schemes of higher order \citep{ Lele_JCP1992}.225 spectral characteristics similar to schemes of higher order \citep{lele_JCP92}. 226 226 227 227 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but … … 277 277 (\ie it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}). 278 278 There exist many ways to define $c_u$, each corresponding to a different FCT scheme. 279 The one chosen in \NEMO is described in \citet{ Zalesak_JCP79}.279 The one chosen in \NEMO is described in \citet{zalesak_JCP79}. 280 280 $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. 281 281 The resulting scheme is quite expensive but \textit{positive}. 282 282 It can be used on both active and passive tracers. 283 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{ Levy_al_GRL01}.283 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{levy.estublier.ea_GRL01}. 284 284 285 285 An additional option has been added controlled by \np{nn\_fct\_zts}. … … 287 287 a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, but on the latter, 288 288 a split-explicit time stepping is used, with a number of sub-timestep equals to \np{nn\_fct\_zts}. 289 This option can be useful when the size of the timestep is limited by vertical advection \citep{ Lemarie_OM2015}.289 This option can be useful when the size of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}. 290 290 Note that in this case, a similar split-explicit time stepping should be used on vertical advection of momentum to 291 291 insure a better stability (see \autoref{subsec:DYN_zad}). … … 306 306 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 307 307 308 MUSCL has been first implemented in \NEMO by \citet{ Levy_al_GRL01}.308 MUSCL has been first implemented in \NEMO by \citet{levy.estublier.ea_GRL01}. 309 309 In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between 310 310 two $T$-points (\autoref{fig:adv_scheme}). … … 358 358 359 359 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error 360 \citep{ Shchepetkin_McWilliams_OM05}.361 The overall performance of the advection scheme is similar to that reported in \cite{ Farrow1995}.360 \citep{shchepetkin.mcwilliams_OM05}. 361 The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}. 362 362 It is a relatively good compromise between accuracy and smoothness. 363 363 Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted, … … 367 367 The intrinsic diffusion of UBS makes its use risky in the vertical direction where 368 368 the control of artificial diapycnal fluxes is of paramount importance 369 \citep{ Shchepetkin_McWilliams_OM05, Demange_PhD2014}.369 \citep{shchepetkin.mcwilliams_OM05, demange_phd14}. 370 370 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme 371 371 (\np{nn\_cen\_v}~\forcode{= 2 or 4}). … … 376 376 (which is the diffusive part of the scheme), 377 377 is evaluated using the \textit{before} tracer (forward in time). 378 This choice is discussed by \citet{ Webb_al_JAOT98} in the context of the QUICK advection scheme.378 This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the QUICK advection scheme. 379 379 UBS and QUICK schemes only differ by one coefficient. 380 Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{ Webb_al_JAOT98}.380 Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 381 381 This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 382 382 Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. … … 412 412 413 413 The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme 414 proposed by \citet{ Leonard1979} is used when \np{ln\_traadv\_qck}~\forcode{= .true.}.414 proposed by \citet{leonard_CMAME79} is used when \np{ln\_traadv\_qck}~\forcode{= .true.}. 415 415 QUICKEST implementation can be found in the \mdl{traadv\_qck} module. 416 416 417 417 QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter 418 \citep{ Leonard1991}.418 \citep{leonard_CMAME91}. 419 419 It has been implemented in NEMO by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. 420 420 The resulting scheme is quite expensive but \textit{positive}. … … 454 454 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). 455 455 When \np{ln\_traldf\_msc}~\forcode{= .true.}, a Method of Stabilizing Correction is used in which 456 the pure vertical component is split into an explicit and an implicit part \citep{ Lemarie_OM2012}.456 the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}. 457 457 458 458 % ------------------------------------------------------------------------------------------------------------- … … 590 590 This formulation conserves the tracer but does not ensure the decrease of the tracer variance. 591 591 Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without 592 any additional background horizontal diffusion \citep{ Guilyardi_al_CD01}.592 any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}. 593 593 594 594 Note that in the partial step $z$-coordinate (\np{ln\_zps}~\forcode{= .true.}), … … 603 603 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}~\forcode{= .true.}; see \autoref{apdx:triad}) 604 604 605 An alternative scheme developed by \cite{ Griffies_al_JPO98} which ensures tracer variance decreases605 An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases 606 606 is also available in \NEMO (\np{ln\_traldf\_grif}~\forcode{= .true.}). 607 607 A complete description of the algorithm is given in \autoref{apdx:triad}. … … 747 747 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 748 748 In the linear free surface case, there is a small imbalance. 749 The imbalance is larger than the imbalance associated with the Asselin time filter \citep{ Leclair_Madec_OM09}.749 The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}. 750 750 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}). 751 751 … … 794 794 In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}~\forcode{= .true.}) 795 795 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 796 leading to the following expression \citep{ Paulson1977}:796 leading to the following expression \citep{paulson.simpson_JPO77}: 797 797 \[ 798 798 % \label{eq:traqsr_iradiance} … … 805 805 806 806 Such assumptions have been shown to provide a very crude and simplistic representation of 807 observed light penetration profiles (\cite{ Morel_JGR88}, see also \autoref{fig:traqsr_irradiance}).807 observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig:traqsr_irradiance}). 808 808 Light absorption in the ocean depends on particle concentration and is spectrally selective. 809 \cite{ Morel_JGR88} has shown that an accurate representation of light penetration can be provided by809 \cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by 810 810 a 61 waveband formulation. 811 811 Unfortunately, such a model is very computationally expensive. 812 Thus, \cite{ Lengaigne_al_CD07} have constructed a simplified version of this formulation in which812 Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of this formulation in which 813 813 visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm). 814 814 For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from 815 the full spectral model of \cite{ Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}),815 the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}), 816 816 assuming the same power-law relationship. 817 817 As shown in \autoref{fig:traqsr_irradiance}, this formulation, called RGB (Red-Green-Blue), … … 834 834 \item[\np{nn\_chdta}~\forcode{= 2}] 835 835 same as previous case except that a vertical profile of chlorophyl is used. 836 Following \cite{ Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value;836 Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value; 837 837 \item[\np{ln\_qsr\_bio}~\forcode{= .true.}] 838 838 simulated time varying chlorophyll by TOP biogeochemical model. … … 865 865 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 866 866 (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. 867 From \citet{ Lengaigne_al_CD07}.867 From \citet{lengaigne.menkes.ea_CD07}. 868 868 } 869 869 \end{center} … … 886 886 \caption{ 887 887 \protect\label{fig:geothermal} 888 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{ Emile-Geay_Madec_OS09}.889 It is inferred from the age of the sea floor and the formulae of \citet{ Stein_Stein_Nat92}.888 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}. 889 It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}. 890 890 } 891 891 \end{center} … … 897 897 This is the default option in \NEMO, and it is implemented using the masking technique. 898 898 However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling. 899 This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{ Stein_Stein_Nat92}),899 This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}), 900 900 but it warms systematically the ocean and acts on the densest water masses. 901 901 Taking this flux into account in a global ocean model increases the deepest overturning cell 902 (\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{ Emile-Geay_Madec_OS09}.902 (\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}. 903 903 904 904 Options are defined through the \ngn{namtra\_bbc} namelist variables. … … 907 907 the \np{nn\_geoflx\_cst}, which is also a namelist parameter. 908 908 When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in 909 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{ Emile-Geay_Madec_OS09}.909 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{emile-geay.madec_OS09}. 910 910 911 911 % ================================================================ … … 931 931 sometimes over a thickness much larger than the thickness of the observed gravity plume. 932 932 A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of 933 a sill \citep{ Willebrand_al_PO01}, and the thickness of the plume is not resolved.934 935 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{ Beckmann_Doscher1997},933 a sill \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved. 934 935 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{beckmann.doscher_JPO97}, 936 936 is to allow a direct communication between two adjacent bottom cells at different levels, 937 937 whenever the densest water is located above the less dense water. … … 939 939 In the current implementation of the BBL, only the tracers are modified, not the velocities. 940 940 Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by 941 \citet{ Campin_Goosse_Tel99}.941 \citet{campin.goosse_T99}. 942 942 943 943 % ------------------------------------------------------------------------------------------------------------- … … 955 955 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and 956 956 $A_l^\sigma$ the lateral diffusivity in the BBL. 957 Following \citet{ Beckmann_Doscher1997}, the latter is prescribed with a spatial dependence,957 Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence, 958 958 \ie in the conditional form 959 959 \begin{equation} … … 1020 1020 \np{nn\_bbl\_adv}~\forcode{= 1}: 1021 1021 the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step 1022 (see black arrow in \autoref{fig:bbl}) \citep{ Beckmann_Doscher1997}.1022 (see black arrow in \autoref{fig:bbl}) \citep{beckmann.doscher_JPO97}. 1023 1023 It is a \textit{conditional advection}, that is, advection is allowed only 1024 1024 if dense water overlies less dense water on the slope (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$) and … … 1027 1027 \np{nn\_bbl\_adv}~\forcode{= 2}: 1028 1028 the downslope velocity is chosen to be proportional to $\Delta \rho$, 1029 the density difference between the higher cell and lower cell densities \citep{ Campin_Goosse_Tel99}.1029 the density difference between the higher cell and lower cell densities \citep{campin.goosse_T99}. 1030 1030 The advection is allowed only if dense water overlies less dense water on the slope 1031 1031 (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$). … … 1041 1041 The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity, 1042 1042 and because no direct estimation of this parameter is available, a uniform value has been assumed. 1043 The possible values for $\gamma$ range between 1 and $10~s$ \citep{ Campin_Goosse_Tel99}.1043 The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}. 1044 1044 1045 1045 Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme. … … 1109 1109 In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas 1110 1110 it is zero in the interior of the model domain. 1111 The second case corresponds to the use of the robust diagnostic method \citep{ Sarmiento1982}.1111 The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}. 1112 1112 It allows us to find the velocity field consistent with the model dynamics whilst 1113 1113 having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$). … … 1121 1121 only below the mixed layer (defined either on a density or $S_o$ criterion). 1122 1122 It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here 1123 \citep{ Madec_al_JPO96}.1123 \citep{madec.delecluse.ea_JPO96}. 1124 1124 1125 1125 For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under … … 1137 1137 1138 1138 Options are defined through the \ngn{namdom} namelist variables. 1139 The general framework for tracer time stepping is a modified leap-frog scheme \citep{ Leclair_Madec_OM09},1139 The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09}, 1140 1140 \ie a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 1141 1141 \begin{equation} … … 1186 1186 Nonlinearities of the EOS are of major importance, in particular influencing the circulation through 1187 1187 determination of the static stability below the mixed layer, 1188 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{ Roquet_JPO2015}.1189 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{ UNESCO1983}) or1190 TEOS-10 \citep{ TEOS10} standards should be used anytime a simulation of the real ocean circulation is attempted1191 \citep{ Roquet_JPO2015}.1188 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{roquet.madec.ea_JPO15}. 1189 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{fofonoff.millard_bk83}) or 1190 TEOS-10 \citep{ioc.iapso_bk10} standards should be used anytime a simulation of the real ocean circulation is attempted 1191 \citep{roquet.madec.ea_JPO15}. 1192 1192 The use of TEOS-10 is highly recommended because 1193 1193 \textit{(i)} it is the new official EOS, … … 1195 1195 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and 1196 1196 practical salinity for EOS-980, both variables being more suitable for use as model variables 1197 \citep{ TEOS10, Graham_McDougall_JPO13}.1197 \citep{ioc.iapso_bk10, graham.mcdougall_JPO13}. 1198 1198 EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. 1199 1199 For process studies, it is often convenient to use an approximation of the EOS. 1200 To that purposed, a simplified EOS (S-EOS) inspired by \citet{ Vallis06} is also available.1200 To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available. 1201 1201 1202 1202 In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density. … … 1204 1204 This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as, 1205 1205 with the exception of only a small percentage of the ocean, 1206 density in the World Ocean varies by no more than 2$\%$ from that value \citep{ Gill1982}.1206 density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}. 1207 1207 1208 1208 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} which … … 1211 1211 \begin{description} 1212 1212 \item[\np{nn\_eos}~\forcode{= -1}] 1213 the polyTEOS10-bsq equation of seawater \citep{ Roquet_OM2015} is used.1213 the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used. 1214 1214 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 1215 1215 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and … … 1217 1217 use in ocean models. 1218 1218 Note that a slightly higher precision polynomial form is now used replacement of 1219 the TEOS-10 rational function approximation for hydrographic data analysis \citep{ TEOS10}.1219 the TEOS-10 rational function approximation for hydrographic data analysis \citep{ioc.iapso_bk10}. 1220 1220 A key point is that conservative state variables are used: 1221 1221 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg{C}, notation: $\Theta$). 1222 1222 The pressure in decibars is approximated by the depth in meters. 1223 1223 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. 1224 It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ TEOS10}.1224 It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ioc.iapso_bk10}. 1225 1225 Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$. 1226 1226 In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and … … 1238 1238 The pressure in decibars is approximated by the depth in meters. 1239 1239 With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and 1240 pressure \citep{ UNESCO1983}.1240 pressure \citep{fofonoff.millard_bk83}. 1241 1241 Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which 1242 1242 is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 1243 1243 \item[\np{nn\_eos}~\forcode{= 1}] 1244 a simplified EOS (S-EOS) inspired by \citet{ Vallis06} is chosen,1244 a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen, 1245 1245 the coefficients of which has been optimized to fit the behavior of TEOS10 1246 (Roquet, personal comm.) (see also \citet{ Roquet_JPO2015}).1246 (Roquet, personal comm.) (see also \citet{roquet.madec.ea_JPO15}). 1247 1247 It provides a simplistic linear representation of both cabbeling and thermobaricity effects which 1248 is enough for a proper treatment of the EOS in theoretical studies \citep{ Roquet_JPO2015}.1248 is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}. 1249 1249 With such an equation of state there is no longer a distinction between 1250 1250 \textit{conservative} and \textit{potential} temperature, … … 1329 1329 \label{subsec:TRA_fzp} 1330 1330 1331 The freezing point of seawater is a function of salinity and pressure \citep{ UNESCO1983}:1331 The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}: 1332 1332 \begin{equation} 1333 1333 \label{eq:tra_eos_fzp}
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