Changeset 10442 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex
r10414 r10442 39 39 The terms QSR, BBC, BBL and DMP are optional. 40 40 The external forcings and parameterisations require complex inputs and complex calculations 41 ( $e.g.$bulk formulae, estimation of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and41 (\eg bulk formulae, estimation of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 42 42 described in \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. 43 43 Note that \mdl{tranpc}, the non-penetrative convection module, although located in the NEMO/OPA/TRA directory as … … 69 69 %------------------------------------------------------------------------------------------------------------- 70 70 71 When considered ( $i.e.$when \np{ln\_traadv\_NONE} is not set to \forcode{.true.}),71 When considered (\ie when \np{ln\_traadv\_NONE} is not set to \forcode{.true.}), 72 72 the advection tendency of a tracer is expressed in flux form, 73 $i.e.$as the divergence of the advective fluxes.73 \ie as the divergence of the advective fluxes. 74 74 Its discrete expression is given by : 75 75 \begin{equation} … … 85 85 $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ which 86 86 results from the use of the continuity equation, $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ 87 (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, $i.e.$\np{ln\_linssh}\forcode{ = .true.}).87 (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, \ie \np{ln\_linssh}\forcode{ = .true.}). 88 88 Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that 89 89 it is consistent with the continuity equation in order to enforce the conservation properties of … … 127 127 There is a non-zero advective flux which is set for all advection schemes as 128 128 $\left. {\tau_w } \right|_{k=1/2} =T_{k=1} $, 129 $i.e.$the product of surface velocity (at $z=0$) by the first level tracer value.129 \ie the product of surface velocity (at $z=0$) by the first level tracer value. 130 130 \item[non-linear free surface:] 131 131 (\np{ln\_linssh}\forcode{ = .false.}) … … 141 141 The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco}) 142 142 is the centred (\textit{now}) \textit{effective} ocean velocity, 143 $i.e.$the \textit{eulerian} velocity (see \autoref{chap:DYN}) plus143 \ie the \textit{eulerian} velocity (see \autoref{chap:DYN}) plus 144 144 the eddy induced velocity (\textit{eiv}) and/or 145 145 the mixed layer eddy induced velocity (\textit{eiv}) when … … 156 156 The corresponding code can be found in the \mdl{traadv\_xxx} module, 157 157 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 158 By default ( $i.e.$in the reference namelist, \ngn{namelist\_ref}), all the logicals are set to \forcode{.false.}.158 By default (\ie in the reference namelist, \ngn{namelist\_ref}), all the logicals are set to \forcode{.false.}. 159 159 If the user does not select an advection scheme in the configuration namelist (\ngn{namelist\_cfg}), 160 160 the tracers will \textit{not} be advected! … … 199 199 \end{equation} 200 200 201 CEN2 is non diffusive ( $i.e.$it conserves the tracer variance, $\tau^2)$ but dispersive202 ( $i.e.$it may create false extrema).201 CEN2 is non diffusive (\ie it conserves the tracer variance, $\tau^2)$ but dispersive 202 (\ie it may create false extrema). 203 203 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 204 204 produce a sensible solution. … … 234 234 235 235 A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive, 236 $i.e.$the global variance of a tracer is not preserved using CEN4.236 \ie the global variance of a tracer is not preserved using CEN4. 237 237 Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. 238 238 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, … … 274 274 where $c_u$ is a flux limiter function taking values between 0 and 1. 275 275 The FCT order is the one of the centred scheme used 276 ( $i.e.$it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}).276 (\ie it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}). 277 277 There exist many ways to define $c_u$, each corresponding to a different FCT scheme. 278 278 The one chosen in \NEMO is described in \citet{Zalesak_JCP79}. … … 356 356 where $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. 357 357 358 This results in a dissipatively dominant ( i.e.hyper-diffusive) truncation error358 This results in a dissipatively dominant (\ie hyper-diffusive) truncation error 359 359 \citep{Shchepetkin_McWilliams_OM05}. 360 360 The overall performance of the advection scheme is similar to that reported in \cite{Farrow1995}. … … 447 447 $(i)$ the type of operator used (none, laplacian, bilaplacian), 448 448 $(ii)$ the direction along which the operator acts (iso-level, horizontal, iso-neutral), 449 $(iii)$ some specific options related to the rotated operators ( $i.e.$non-iso-level operator), and449 $(iii)$ some specific options related to the rotated operators (\ie non-iso-level operator), and 450 450 $(iv)$ the specification of eddy diffusivity coefficient (either constant or variable in space and time). 451 451 Item $(iv)$ will be described in \autoref{chap:LDF}. … … 455 455 456 456 The lateral diffusion of tracers is evaluated using a forward scheme, 457 $i.e.$the tracers appearing in its expression are the \textit{before} tracers in time,457 \ie the tracers appearing in its expression are the \textit{before} tracers in time, 458 458 except for the pure vertical component that appears when a rotation tensor is used. 459 459 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). … … 491 491 minimizing the impact on the larger scale features. 492 492 The main difference between the two operators is the scale selectiveness. 493 The bilaplacian damping time ( $i.e.$its spin down time) scales like $\lambda^{-4}$ for493 The bilaplacian damping time (\ie its spin down time) scales like $\lambda^{-4}$ for 494 494 disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones), 495 495 whereas the laplacian damping time scales only like $\lambda^{-2}$. … … 506 506 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when 507 507 iso-level option is used (\np{ln\_traldf\_lev}\forcode{ = .true.}) or 508 when a horizontal ( $i.e.$ geopotential) operator is demanded in \textit{z}-coordinate508 when a horizontal (\ie geopotential) operator is demanded in \zstar-coordinate 509 509 (\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}). 510 510 The associated code can be found in the \mdl{traldf\_lap\_blp} module. … … 514 514 (\np{ln\_traldf\_iso} or \np{ln\_traldf\_triad} equals \forcode{.true.}, 515 515 see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or 516 when a horizontal ( $i.e.$geopotential) operator is demanded in \textit{s}-coordinate516 when a horizontal (\ie geopotential) operator is demanded in \textit{s}-coordinate 517 517 (\np{ln\_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.}) 518 518 \footnote{In this case, the standard iso-neutral operator will be automatically selected}. … … 544 544 compute the iso-level bilaplacian operator. 545 545 546 It is a \emph{horizontal} operator ( $i.e.$acting along geopotential surfaces) in546 It is a \emph{horizontal} operator (\ie acting along geopotential surfaces) in 547 547 the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 548 548 It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}\forcode{ = .true.}, … … 593 593 where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells, 594 594 $r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and 595 the surface along which the diffusion operator acts ( $i.e.$horizontal or iso-neutral surfaces).595 the surface along which the diffusion operator acts (\ie horizontal or iso-neutral surfaces). 596 596 It is thus used when, in addition to \np{ln\_traldf\_lap}\forcode{ = .true.}, 597 597 we have \np{ln\_traldf\_iso}\forcode{ = .true.}, … … 676 676 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity, 677 677 respectively. 678 Generally, $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is parameterised ( $i.e.$\key{zdfddm} is defined).678 Generally, $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is parameterised (\ie \key{zdfddm} is defined). 679 679 The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF). 680 680 Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by … … 715 715 716 716 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 717 ( $i.e.$atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due717 (\ie atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due 718 718 both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and 719 719 to the heat and salt content of the mass exchange. … … 725 725 726 726 $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 727 ( i.e.the difference between the total surface heat flux and the fraction of the short wave flux that727 (\ie the difference between the total surface heat flux and the fraction of the short wave flux that 728 728 penetrates into the water column, see \autoref{subsec:TRA_qsr}) 729 729 plus the heat content associated with of the mass exchange with the atmosphere and lands. … … 796 796 \end{split} 797 797 \end{equation} 798 where $Q_{sr}$ is the penetrative part of the surface heat flux ( $i.e.$the shortwave radiation) and798 where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie the shortwave radiation) and 799 799 $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). 800 800 The additional term in \autoref{eq:PE_qsr} is discretized as follows: … … 843 843 844 844 The RGB formulation is used when \np{ln\_qsr\_rgb}\forcode{ = .true.}. 845 The RGB attenuation coefficients ( $i.e.$the inverses of the extinction length scales) are tabulated over845 The RGB attenuation coefficients (\ie the inverses of the extinction length scales) are tabulated over 846 846 61 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L 847 847 (see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module). … … 867 867 the depth of $w-$levels does not significantly vary with location. 868 868 The level at which the light has been totally absorbed 869 ( $i.e.$it is less than the computer precision) is computed once,869 (\ie it is less than the computer precision) is computed once, 870 870 and the trend associated with the penetration of the solar radiation is only added down to that level. 871 871 Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor. 872 872 In this case, we have chosen that all remaining radiation is absorbed in the last ocean level 873 ( $i.e.$$I$ is masked).873 (\ie $I$ is masked). 874 874 875 875 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 914 914 915 915 Usually it is assumed that there is no exchange of heat or salt through the ocean bottom, 916 $i.e.$a no flux boundary condition is applied on active tracers at the bottom.916 \ie a no flux boundary condition is applied on active tracers at the bottom. 917 917 This is the default option in \NEMO, and it is implemented using the masking technique. 918 918 However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling. … … 920 920 but it warms systematically the ocean and acts on the densest water masses. 921 921 Taking this flux into account in a global ocean model increases the deepest overturning cell 922 ( $i.e.$the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}.922 (\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}. 923 923 924 924 Options are defined through the \ngn{namtra\_bbc} namelist variables. … … 976 976 and $A_l^\sigma$ the lateral diffusivity in the BBL. 977 977 Following \citet{Beckmann_Doscher1997}, the latter is prescribed with a spatial dependence, 978 $i.e.$in the conditional form978 \ie in the conditional form 979 979 \begin{equation} 980 980 \label{eq:tra_bbl_coef} … … 1006 1006 \label{subsec:TRA_bbl_adv} 1007 1007 1008 \sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following1009 if this is not what is meant then "downwards sloping flow" is also a possibility"}1008 %\sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following 1009 %if this is not what is meant then "downwards sloping flow" is also a possibility"} 1010 1010 1011 1011 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 1029 1029 %!! nn_bbl_adv = 1 use of the ocean velocity as bbl velocity 1030 1030 %!! nn_bbl_adv = 2 follow Campin and Goosse (1999) implentation 1031 %!! i.e.transport proportional to the along-slope density gradient1031 %!! \ie transport proportional to the along-slope density gradient 1032 1032 1033 1033 %%%gmcomment : this section has to be really written … … 1041 1041 (see black arrow in \autoref{fig:bbl}) \citep{Beckmann_Doscher1997}. 1042 1042 It is a \textit{conditional advection}, that is, advection is allowed only 1043 if dense water overlies less dense water on the slope ( $i.e.$$\nabla_\sigma \rho \cdot \nabla H<0$) and1044 if the velocity is directed towards greater depth ( $i.e.$$\vect{U} \cdot \nabla H>0$).1043 if dense water overlies less dense water on the slope (\ie $\nabla_\sigma \rho \cdot \nabla H<0$) and 1044 if the velocity is directed towards greater depth (\ie $\vect{U} \cdot \nabla H>0$). 1045 1045 1046 1046 \np{nn\_bbl\_adv}\forcode{ = 2}: … … 1048 1048 the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}. 1049 1049 The advection is allowed only if dense water overlies less dense water on the slope 1050 ( $i.e.$$\nabla_\sigma \rho \cdot \nabla H<0$).1050 (\ie $\nabla_\sigma \rho \cdot \nabla H<0$). 1051 1051 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}), 1052 1052 is simply given by the following expression: … … 1112 1112 It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true in 1113 1113 \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set 1114 ( $i.e.$that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread},1114 (\ie that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread}, 1115 1115 see \autoref{subsec:SBC_fldread}). 1116 1116 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. … … 1148 1148 This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. 1149 1149 The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. 1150 The \n l{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for1150 The \ngn{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for 1151 1151 the restoration coefficient. 1152 1152 … … 1156 1156 1157 1157 \np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and 1158 should be the same as specified in \n l{namcfg}.1159 The variable \n l{lzoom} is used to specify that the damping is being used as in case \textit{a} above to1158 should be the same as specified in \ngn{namcfg}. 1159 The variable \np{lzoom} is used to specify that the damping is being used as in case \textit{a} above to 1160 1160 provide boundary conditions to a zoom configuration. 1161 1161 In the case of the arctic or antarctic zoom configurations this includes some specific treatment. 1162 1162 Otherwise damping is applied to the 6 grid points along the ocean boundaries. 1163 1163 The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in 1164 the \n l{nam\_zoom\_dmp} name list.1164 the \ngn{nam\_zoom\_dmp} name list. 1165 1165 1166 1166 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in … … 1201 1201 Options are defined through the \ngn{namdom} namelist variables. 1202 1202 The general framework for tracer time stepping is a modified leap-frog scheme \citep{Leclair_Madec_OM09}, 1203 $i.e.$a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}):1203 \ie a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 1204 1204 \begin{equation} 1205 1205 \label{eq:tra_nxt} … … 1213 1213 where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, 1214 1214 $\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$ 1215 ( $i.e.$fluxes plus content in mass exchanges).1215 (\ie fluxes plus content in mass exchanges). 1216 1216 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1217 1217 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}. … … 1282 1282 the TEOS-10 rational function approximation for hydrographic data analysis \citep{TEOS10}. 1283 1283 A key point is that conservative state variables are used: 1284 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg C, notation: $\Theta$).1284 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg{C}, notation: $\Theta$). 1285 1285 The pressure in decibars is approximated by the depth in meters. 1286 1286 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. … … 1363 1363 \label{subsec:TRA_bn2} 1364 1364 1365 An accurate computation of the ocean stability ( i.e.of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of1365 An accurate computation of the ocean stability (\ie of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of 1366 1366 paramount importance as determine the ocean stratification and is used in several ocean parameterisations 1367 1367 (namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion, … … 1378 1378 The coefficients are a polynomial function of temperature, salinity and depth which 1379 1379 expression depends on the chosen EOS. 1380 They are computed through \textit{eos\_rab}, a \ textsc{Fortran}function that can be found in \mdl{eosbn2}.1380 They are computed through \textit{eos\_rab}, a \fortran function that can be found in \mdl{eosbn2}. 1381 1381 1382 1382 % ------------------------------------------------------------------------------------------------------------- … … 1396 1396 1397 1397 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water 1398 ( $i.e.$referenced to the surface $p=0$),1398 (\ie referenced to the surface $p=0$), 1399 1399 thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped. 1400 1400 The freezing point is computed through \textit{eos\_fzp}, 1401 a \ textsc{Fortran}function that can be found in \mdl{eosbn2}.1401 a \fortran function that can be found in \mdl{eosbn2}. 1402 1402 1403 1403 … … 1511 1511 \biblio 1512 1512 1513 \pindex 1514 1513 1515 \end{document}
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