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branches/2015/nemo_v3_6_STABLE/DOC/TexFiles/Chapters/Chap_TRA.tex
r6039 r6275 36 36 (BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, 37 37 BBC, BBL and DMP are optional. The external forcings and parameterisations 38 require complex inputs and complex calculations ( e.g.bulk formulae, estimation38 require complex inputs and complex calculations ($e.g.$ bulk formulae, estimation 39 39 of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 40 40 described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively. 41 Note that \mdl{tranpc}, the non-penetrative convection module, although 42 (temporarily) located in the NEMO/OPA/TRA directory, is described with the 43 model vertical physics (ZDF). 44 %%% 45 \gmcomment{change the position of eosbn2 in the reference code} 46 %%% 41 Note that \mdl{tranpc}, the non-penetrative convection module, although 42 located in the NEMO/OPA/TRA directory as it directly modifies the tracer fields, 43 is described with the model vertical physics (ZDF) together with other available 44 parameterization of convection. 47 45 48 46 In the present chapter we also describe the diagnostic equations used to compute 49 the sea-water properties (density, Brunt-V ais\"{a}l\"{a} frequency, specific heat and47 the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and 50 48 freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). 51 49 … … 56 54 found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory. 57 55 58 The user has the option of extracting each tendency term on the rhsof the tracer59 equation for output (\ key{trdtra} is defined), as described in Chap.~\ref{MISC}.56 The user has the option of extracting each tendency term on the RHS of the tracer 57 equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}~=~true), as described in Chap.~\ref{DIA}. 60 58 61 59 $\ $\newline % force a new ligne … … 125 123 \end{description} 126 124 In all cases, this boundary condition retains local conservation of tracer. 127 Global conservation is obtained in both rigid-lid and non-linear free surface128 cases, but not in the linear free surface case. Nevertheless, in the latter 129 case,it is achieved to a good approximation since the non-conservative125 Global conservation is obtained in non-linear free surface case, 126 but \textit{not} in the linear free surface case. Nevertheless, in the latter case, 127 it is achieved to a good approximation since the non-conservative 130 128 term is the product of the time derivative of the tracer and the free surface 131 129 height, two quantities that are not correlated (see \S\ref{PE_free_surface}, … … 133 131 134 132 The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) 135 is the centred (\textit{now}) \textit{eulerian} ocean velocity (see Chap.~\ref{DYN}). 136 When eddy induced velocity (\textit{eiv}) parameterisation is used it is the \textit{now} 137 \textit{effective} velocity ($i.e.$ the sum of the eulerian and eiv velocities) which is used. 133 is the centred (\textit{now}) \textit{effective} ocean velocity, $i.e.$ the \textit{eulerian} velocity 134 (see Chap.~\ref{DYN}) plus the eddy induced velocity (\textit{eiv}) 135 and/or the mixed layer eddy induced velocity (\textit{eiv}) 136 when those parameterisations are used (see Chap.~\ref{LDF}). 138 137 139 138 The choice of an advection scheme is made in the \textit{\ngn{nam\_traadv}} namelist, by … … 146 145 147 146 Note that 148 (1) cen2 , cen4and TVD schemes require an explicit diffusion147 (1) cen2 and TVD schemes require an explicit diffusion 149 148 operator while the other schemes are diffusive enough so that they do not 150 149 require additional diffusion ; 151 (2) cen2, cen4,MUSCL2, and UBS are not \textit{positive} schemes150 (2) cen2, MUSCL2, and UBS are not \textit{positive} schemes 152 151 \footnote{negative values can appear in an initially strictly positive tracer field 153 152 which is advected} … … 189 188 temperature is close to the freezing point). 190 189 This combined scheme has been included for specific grid points in the ORCA2 191 and ORCA4 configurationsonly. This is an obsolescent feature as the recommended190 configuration only. This is an obsolescent feature as the recommended 192 191 advection scheme for the ORCA configuration is TVD (see \S\ref{TRA_adv_tvd}). 193 192 … … 196 195 have this order of accuracy. \gmcomment{Note also that ... blah, blah} 197 196 198 % -------------------------------------------------------------------------------------------------------------199 % 4nd order centred scheme200 % -------------------------------------------------------------------------------------------------------------201 \subsection [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})]202 {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=true)}203 \label{TRA_adv_cen4}204 205 In the $4^{th}$ order formulation (to be implemented), tracer values are206 evaluated at velocity points as a $4^{th}$ order interpolation, and thus depend on207 the four neighbouring $T$-points. For example, in the $i$-direction:208 \begin{equation} \label{Eq_tra_adv_cen4}209 \tau _u^{cen4}210 =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2}211 \end{equation}212 213 Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme214 but a $4^{th}$ order evaluation of advective fluxes, since the divergence of215 advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. The phrase ``$4^{th}$216 order scheme'' used in oceanographic literature is usually associated217 with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection218 scheme is feasible but, for consistency reasons, it requires changes in the219 discretisation of the tracer advection together with changes in both the220 continuity equation and the momentum advection terms.221 222 A direct consequence of the pseudo-fourth order nature of the scheme is that223 it is not non-diffusive, i.e. the global variance of a tracer is not preserved using224 \textit{cen4}. Furthermore, it must be used in conjunction with an explicit225 diffusion operator to produce a sensible solution. The time-stepping is also226 performed using a leapfrog scheme in conjunction with an Asselin time-filter,227 so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.228 229 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an230 additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This231 hypothesis usually reduces the order of the scheme. Here we choose to set232 the gradient of $T$ across the boundary to zero. Alternative conditions can be233 specified, such as a reduction to a second order scheme for these near boundary234 grid points.235 197 236 198 % ------------------------------------------------------------------------------------------------------------- … … 270 232 used for the diffusive part. 271 233 234 An additional option has been added controlled by \np{ln\_traadv\_tvd\_zts}. 235 By setting this logical to true, a TVD scheme is used on both horizontal and vertical direction, 236 but on the latter, a split-explicit time stepping is used, with 5 sub-timesteps. 237 This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 238 Note that in this case, a similar split-explicit time stepping should be used on 239 vertical advection of momentum to ensure a better stability (see \np{ln\_dynzad\_zts} in \S\ref{DYN_zad}). 240 241 272 242 % ------------------------------------------------------------------------------------------------------------- 273 243 % MUSCL scheme … … 296 266 297 267 For an ocean grid point adjacent to land and where the ocean velocity is 298 directed toward land, two choices are available: an upstream flux 299 (\np{ln\_traadv\_muscl}=true) or a second order flux 300 (\np{ln\_traadv\_muscl2}=true). Note that the latter choice does not ensure 301 the \textit{positive} character of the scheme. Only the former can be used 302 on both active and passive tracers. The two MUSCL schemes are implemented 303 in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules. 268 directed toward land, two choices are available: an upstream flux (\np{ln\_traadv\_muscl}=true) 269 or a second order flux (\np{ln\_traadv\_muscl2}=true). 270 Note that the latter choice does not ensure the \textit{positive} character of the scheme. 271 Only the former can be used on both active and passive tracers. 272 The two MUSCL schemes are implemented in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules. 273 274 Note that when using np{ln\_traadv\_msc\_ups}~=~true in addition to \np{ln\_traadv\_muscl}=true, 275 the MUSCL fluxes are replaced by upstream fluxes in vicinity of river mouths. 304 276 305 277 % ------------------------------------------------------------------------------------------------------------- … … 416 388 direction (as for the UBS case) should be implemented to restore this property. 417 389 418 419 % -------------------------------------------------------------------------------------------------------------420 % PPM scheme421 % -------------------------------------------------------------------------------------------------------------422 \subsection [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})]423 {Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=true)}424 \label{TRA_adv_ppm}425 426 The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984)427 \sgacomment{reference?}428 is based on a quadradic piecewise construction. Like the QCK scheme, it is associated429 with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented430 in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference431 version 3.3.432 390 433 391 % ================================================================ … … 464 422 surfaces is given by: 465 423 \begin{equation} \label{Eq_tra_ldf_lap} 466 D_T^{lT} =\frac{1}{b_t T} \left( \;424 D_T^{lT} =\frac{1}{b_t} \left( \; 467 425 \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] 468 426 + \delta _{j}\left[ A_v^{lT} \; \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right] \;\right) … … 661 619 the thickness of the top model layer. 662 620 663 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components ($i.e.$ atmosphere, sea-ice, land), 664 the change in the heat and salt content of the surface layer of the ocean is due both 665 to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 666 and to the heat and salt content of the mass exchange. 667 \sgacomment{ the following does not apply to the release to which this documentation is 668 attached and so should not be included .... 669 In a forthcoming release, these two parts, computed in the surface module (SBC), will be included directly 670 in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux. 671 The specification of these fluxes is further detailed in the SBC chapter (see \S\ref{SBC}). 672 This change will provide a forcing formulation which is the same for any tracer (including temperature and salinity). 673 674 In the current version, the situation is a little bit more complicated. } 621 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 622 ($i.e.$ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer 623 of the ocean is due both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 624 and to the heat and salt content of the mass exchange. They are both included directly in $Q_{ns}$, 625 the surface heat flux, and $F_{salt}$, the surface salt flux (see \S\ref{SBC} for further details). 626 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 675 627 676 628 The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following … … 679 631 $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 680 632 (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that 681 penetrates into the water column, see \S\ref{TRA_qsr}) 682 683 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) 684 685 $\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange 686 687 $\bullet$ \textit{rnf}, the mass flux associated with runoff (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 688 689 The $\textit{emp}_S$ field is not simply the budget of evaporation-precipitation+freezing-melting because 690 the sea-ice is not currently embedded in the ocean but levitates above it. There is no mass 691 exchanged between the sea-ice and the ocean. Instead we only take into account the salt 692 flux associated with the non-zero salinity of sea-ice, and the concentration/dilution effect 693 due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into 694 an equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, 695 the surface boundary condition on temperature and salinity is applied as follows: 696 697 In the nonlinear free surface case (\key{vvl} is defined): 633 penetrates into the water column, see \S\ref{TRA_qsr}) plus the heat content associated with 634 of the mass exchange with the atmosphere and lands. 635 636 $\bullet$ $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 637 638 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) 639 and possibly with the sea-ice and ice-shelves. 640 641 $\bullet$ \textit{rnf}, the mass flux associated with runoff 642 (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 643 644 In the non-linear free surface case (\key{vvl} is defined), the surface boundary condition 645 on temperature and salinity is applied as follows: 698 646 \begin{equation} \label{Eq_tra_sbc} 647 \begin{aligned} 648 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ 649 & F^S =\frac{ 1 }{\rho _o \, \left. e_{3t} \right|_{k=1} } &\overline{ \textit{sfx} }^t & \\ 650 \end{aligned} 651 \end{equation} 652 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 653 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 654 divergence of odd and even time step (see \S\ref{STP}). 655 656 In the linear free surface case (\key{vvl} is \textit{not} defined), 657 an additional term has to be added on both temperature and salinity. 658 On temperature, this term remove the heat content associated with mass exchange 659 that has been added to $Q_{ns}$. On salinity, this term mimics the concentration/dilution effect that 660 would have resulted from a change in the volume of the first level. 661 The resulting surface boundary condition is applied as follows: 662 \begin{equation} \label{Eq_tra_sbc_lin} 699 663 \begin{aligned} 700 664 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } … … 702 666 % 703 667 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 704 &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1} \right) }^t & \\668 &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1} \right) }^t & \\ 705 669 \end{aligned} 706 670 \end{equation} 707 708 In the linear free surface case (\key{vvl} not defined): 709 \begin{equation} \label{Eq_tra_sbc_lin} 710 \begin{aligned} 711 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ 712 % 713 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 714 &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1} \right) }^t & \\ 715 \end{aligned} 716 \end{equation} 717 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 718 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 719 divergence of odd and even time step (see \S\ref{STP}). 720 721 The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained 722 by assuming that the temperature of precipitation and evaporation are equal to 723 the ocean surface temperature and that their salinity is zero. Therefore, the heat content 724 of the \textit{emp} budget must be added to the temperature equation in the variable volume case, 725 while it does not appear in the constant volume case. Similarly, the \textit{emp} budget affects 726 the ocean surface salinity in the constant volume case (through the concentration dilution effect) 727 while it does not appears explicitly in the variable volume case since salinity change will be 728 induced by volume change. In both constant and variable volume cases, surface salinity 729 will change with ice-ocean salt flux and F/M flux (both contained in $\textit{emp}_S - \textit{emp}$) without mass exchanges. 730 731 Note that the concentration/dilution effect due to F/M is computed using 732 a constant ice salinity as well as a constant ocean salinity. 733 This approximation suppresses the correlation between \textit{SSS} 734 and F/M flux, allowing the ice-ocean salt exchanges to be conservative. 735 Indeed, if this approximation is not made, even if the F/M budget is zero 736 on average over the whole ocean domain and over the seasonal cycle, 737 the associated salt flux is not zero, since sea-surface salinity and F/M flux are 738 intrinsically correlated (high \textit{SSS} are found where freezing is 739 strong whilst low \textit{SSS} is usually associated with high melting areas). 740 741 Even using this approximation, an exact conservation of heat and salt content 742 is only achieved in the variable volume case. In the constant volume case, 743 there is a small imbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$. 744 Nevertheless, the salt content variation is quite small and will not induce 745 a long term drift as there is no physical reason for $(\partial_t\eta - \textit{emp})$ 746 and \textit{SSS} to be correlated \citep{Roullet_Madec_JGR00}. 747 Note that, while quite small, the imbalance in the constant volume case is larger 671 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 672 In the linear free surface case, there is a small imbalance. The imbalance is larger 748 673 than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 749 This is the reason why the modified filter is not applied in the constant volume case.674 This is the reason why the modified filter is not applied in the linear free surface case (see \S\ref{STP}). 750 675 751 676 % ------------------------------------------------------------------------------------------------------------- … … 1103 1028 \subsection[DMP\_TOOLS]{Generating resto.nc using DMP\_TOOLS} 1104 1029 1105 DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled and run on the same machine as the NEMO model. A mesh\_mask.nc file for the model configuration is required as an input. This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient. 1030 DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. 1031 Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled 1032 and run on the same machine as the NEMO model. A mesh\_mask.nc file for the model configuration is required as an input. 1033 This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. 1034 The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. 1035 The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient. 1106 1036 1107 1037 %--------------------------------------------nam_dmp_create------------------------------------------------- … … 1111 1041 \np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and should be the same as specified in \nl{namcfg}. The variable \nl{lzoom} is used to specify that the damping is being used as in case \textit{a} above to provide boundary conditions to a zoom configuration. In the case of the arctic or antarctic zoom configurations this includes some specific treatment. Otherwise damping is applied to the 6 grid points along the ocean boundaries. The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in the \nl{nam\_zoom\_dmp} name list. 1112 1042 1113 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations. \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea for the ORCA4, ORCA2 and ORCA05 configurations. If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference configurations with previous model versions. \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. This option only has an effect if \np{ln\_full\_field} is true. \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. Finally \np{ln\_custom} specifies that the custom module will be called. This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. 1114 1115 The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}. Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to the full values of a 10$^{\circ}$ latitud band. This is often used because of the short adjustment time scale in the equatorial region \citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. 1043 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations. 1044 \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. 1045 \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea 1046 for the ORCA4, ORCA2 and ORCA05 configurations. 1047 If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as 1048 a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference 1049 configurations with previous model versions. 1050 \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. 1051 This option only has an effect if \np{ln\_full\_field} is true. 1052 \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. 1053 Finally \np{ln\_custom} specifies that the custom module will be called. 1054 This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. 1055 1056 The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}. 1057 Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to 1058 the full values of a 10$^{\circ}$ latitud band. 1059 This is often used because of the short adjustment time scale in the equatorial region 1060 \citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a 1061 hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. 1116 1062 1117 1063 % ================================================================ … … 1265 1211 \hline 1266 1212 coeff. & computer name & S-EOS & description \\ \hline 1267 $a_0$ & \np{ nn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline1268 $b_0$ & \np{ nn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline1269 $\lambda_1$ & \np{ nn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline1270 $\lambda_2$ & \np{ nn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline1271 $\nu$ & \np{ nn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline1272 $\mu_1$ & \np{ nn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline1273 $\mu_2$ & \np{ nn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline1213 $a_0$ & \np{rn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline 1214 $b_0$ & \np{rn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline 1215 $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline 1216 $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline 1217 $\nu$ & \np{rn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline 1218 $\mu_1$ & \np{rn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline 1219 $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1274 1220 \end{tabular} 1275 1221 \caption{ \label{Tab_SEOS} … … 1281 1227 1282 1228 % ------------------------------------------------------------------------------------------------------------- 1283 % Brunt-V ais\"{a}l\"{a} Frequency1284 % ------------------------------------------------------------------------------------------------------------- 1285 \subsection{Brunt-V ais\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)}1229 % Brunt-V\"{a}is\"{a}l\"{a} Frequency 1230 % ------------------------------------------------------------------------------------------------------------- 1231 \subsection{Brunt-V\"{a}is\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)} 1286 1232 \label{TRA_bn2} 1287 1233 1288 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V ais\"{a}l\"{a}1234 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} 1289 1235 frequency) is of paramount importance as determine the ocean stratification and 1290 1236 is used in several ocean parameterisations (namely TKE, GLS, Richardson number dependent … … 1302 1248 function that can be found in \mdl{eosbn2}. 1303 1249 1304 1305 % -------------------------------------------------------------------------------------------------------------1306 % Potential Energy1307 % -------------------------------------------------------------------------------------------------------------1308 %\subsection{Potential Energy anomalies}1309 %\label{TRA_bn2}1310 1311 % =====>>>>> TO BE written1312 %1313 1314 1250 % ------------------------------------------------------------------------------------------------------------- 1315 1251 % Freezing Point of Seawater … … 1341 1277 \label{TRA_zpshde} 1342 1278 1343 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"} 1279 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, 1280 I've changed "derivative" to "difference" and "mean" to "average"} 1344 1281 1345 1282 With partial bottom cells (\np{ln\_zps}=true), in general, tracers in horizontally
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