2015-12-12T15:13:32+01:00 (5 years ago)

#1629: DOC of v3.6_stable : update EOS presentation

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  • branches/2015/nemo_v3_6_STABLE/DOC/TexFiles/Chapters/Chap_TRA.tex

    r5890 r6039  
    11% ================================================================ 
    2 % Chapter 1 Ocean Tracers (TRA) 
     2% Chapter 1 ——— Ocean Tracers (TRA) 
    33% ================================================================ 
    44\chapter{Ocean Tracers (TRA)} 
    11671167%        Equation of State 
    11681168% ------------------------------------------------------------------------------------------------------------- 
    1169 \subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)} 
     1169\subsection{Equation Of Seawater (\np{nn\_eos} = -1, 0, or 1)} 
    1172 It is necessary to know the equation of state for the ocean very accurately  
    1173 to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency),  
    1174 particularly in the deep ocean. The ocean seawater volumic mass, $\rho$,  
    1175 abusively called density, is a non linear empirical function of \textit{in situ}  
    1176 temperature, salinity and pressure. The reference equation of state is that  
    1177 defined by the Joint Panel on Oceanographic Tables and Standards  
    1178 \citep{UNESCO1983}. It was the standard equation of state used in early  
    1179 releases of OPA. However, even though this computation is fully vectorised,  
    1180 it is quite time consuming ($15$ to $20${\%} of the total CPU time) since  
    1181 it requires the prior computation of the \textit{in situ} temperature from the  
    1182 model \textit{potential} temperature using the \citep{Bryden1973} polynomial  
    1183 for adiabatic lapse rate and a $4^th$ order Runge-Kutta integration scheme.  
    1184 Since OPA6, we have used the \citet{JackMcD1995} equation of state for  
    1185 seawater instead. It allows the computation of the \textit{in situ} ocean density  
    1186 directly as a function of \textit{potential} temperature relative to the surface  
    1187 (an \NEMO variable), the practical salinity (another \NEMO variable) and the  
    1188 pressure (assuming no pressure variation along geopotential surfaces, $i.e.$  
    1189 the pressure in decibars is approximated by the depth in meters).  
    1190 Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state  
    1191 have exactly the same except that the values of the various coefficients have  
    1192 been adjusted by \citet{JackMcD1995} in order to directly use the \textit{potential}  
    1193 temperature instead of the \textit{in situ} one. This reduces the CPU time of the  
    1194 \textit{in situ} density computation to about $3${\%} of the total CPU time,  
    1195 while maintaining a quite accurate equation of state. 
    1197 In the computer code, a \textit{true} density anomaly, $d_a= \rho / \rho_o - 1$,  
    1198 is computed, with $\rho_o$ a reference volumic mass. Called \textit{rau0}  
    1199 in the code, $\rho_o$ is defined in \mdl{phycst}, and a value of $1,035~Kg/m^3$.  
     1172The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship  
     1173linking seawater density, $\rho$, to a number of state variables,  
     1174most typically temperature, salinity and pressure.  
     1175Because density gradients control the pressure gradient force through the hydrostatic balance,  
     1176the equation of state provides a fundamental bridge between the distribution of active tracers  
     1177and the fluid dynamics. Nonlinearities of the EOS are of major importance, in particular  
     1178influencing the circulation through determination of the static stability below the mixed layer,  
     1179thus controlling rates of exchange between the atmosphere  and the ocean interior \citep{Roquet_JPO2015}.  
     1180Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983})  
     1181or TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real  
     1182ocean circulation is attempted \citep{Roquet_JPO2015}.  
     1183The use of TEOS-10 is highly recommended because  
     1184\textit{(i)} it is the new official EOS,  
     1185\textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and  
     1186\textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature  
     1187and practical salinity for EOS-980, both variables being more suitable for use as model variables  
     1188\citep{TEOS10, Graham_McDougall_JPO13}.  
     1189EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. 
     1190For process studies, it is often convenient to use an approximation of the EOS. To that purposed,  
     1191a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available. 
     1193In the computer code, a density anomaly, $d_a= \rho / \rho_o - 1$,  
     1194is computed, with $\rho_o$ a reference density. Called \textit{rau0}  
     1195in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$.  
    12001196This is a sensible choice for the reference density used in a Boussinesq ocean  
    12011197climate model, as, with the exception of only a small percentage of the ocean,  
    1202 density in the World Ocean varies by no more than 2$\%$ from $1,035~kg/m^3$  
    1203 \citep{Gill1982}. 
    1205 Options are defined through the  \ngn{nameos} namelist variables. 
    1206 The default option (namelist parameter \np{nn\_eos}=0) is the \citet{JackMcD1995}  
    1207 equation of state. Its use is highly recommended. However, for process studies,  
    1208 it is often convenient to use a linear approximation of the density. 
     1198density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. 
     1200Options are defined through the  \ngn{nameos} namelist variables, and in particular \np{nn\_eos}  
     1201which controls the EOS used (=-1 for TEOS10 ; =0 for EOS-80 ; =1 for S-EOS). 
     1204\item[\np{nn\_eos}$=-1$] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used.   
     1205The accuracy of this approximation is comparable to the TEOS-10 rational function approximation,  
     1206but it is optimized for a boussinesq fluid and the polynomial expressions have simpler  
     1207and more computationally efficient expressions for their derived quantities  
     1208which make them more adapted for use in ocean models.  
     1209Note that a slightly higher precision polynomial form is now used replacement of the TEOS-10  
     1210rational function approximation for hydrographic data analysis  \citep{TEOS10}.  
     1211A key point is that conservative state variables are used:  
     1212Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: $\degres C$, notation: $\Theta$). 
     1213The pressure in decibars is approximated by the depth in meters.  
     1214With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. It is set to  
     1215$C_p=3991.86795711963~J\,Kg^{-1}\,\degres K^{-1}$, according to \citet{TEOS10}. 
     1217Choosing polyTEOS10-bsq implies that the state variables used by the model are  
     1218$\Theta$ and $S_A$. In particular, the initial state deined by the user have to be given as  
     1219\textit{Conservative} Temperature and \textit{Absolute} Salinity.  
     1220In addition, setting \np{ln\_useCT} to \textit{true} convert the Conservative SST to potential SST  
     1221prior to either computing the air-sea and ice-sea fluxes (forced mode)  
     1222or sending the SST field to the atmosphere (coupled mode). 
     1224\item[\np{nn\_eos}$=0$] the polyEOS80-bsq equation of seawater is used. 
     1225It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized  
     1226to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80  
     1227and the ocean model are:  
     1228the Practical Salinity ((unit: psu, notation: $S_p$)) and Potential Temperature (unit: $\degres C$, notation: $\theta$). 
     1229The pressure in decibars is approximated by the depth in meters.   
     1230With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature,  
     1231salinity and pressure \citep{UNESCO1983}. Nevertheless, a severe assumption is made in order to  
     1232have a heat content ($C_p T_p$) which is conserved by the model: $C_p$ is set to a constant  
     1233value, the TEOS10 value.  
     1235\item[\np{nn\_eos}$=1$] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen,  
     1236the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.)  
     1237(see also \citet{Roquet_JPO2015}). It provides a simplistic linear representation of both  
     1238cabbeling and thermobaricity effects which is enough for a proper treatment of the EOS  
     1239in theoretical studies \citep{Roquet_JPO2015}. 
    12091240With such an equation of state there is no longer a distinction between  
    1210 \textit{in situ} and \textit{potential} density and both cabbeling and thermobaric 
    1211 effects are removed. 
    1212 Two linear formulations are available: a function of $T$ only (\np{nn\_eos}=1)  
    1213 and a function of both $T$ and $S$ (\np{nn\_eos}=2): 
    1214 \begin{equation} \label{Eq_tra_eos_linear} 
     1241\textit{conservative} and \textit{potential} temperature, as well as between \textit{absolute}  
     1242and \textit{practical} salinity. 
     1243S-EOS takes the following expression: 
     1244\begin{equation} \label{Eq_tra_S-EOS} 
    1216   d_a(T)       &=  \rho (T)      /  \rho_o   - 1     =  \  0.0285         -  \alpha   \;T     \\  
    1217   d_a(T,S)    &=  \rho (T,S)   /  \rho_o   - 1     =  \  \beta \; S       -  \alpha   \;T     
     1246  d_a(T,S,z)  =  ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a  \\ 
     1247                                & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a  \\ 
     1248                                & - \nu \; T_a \; S_a \;  ) \; / \; \rho_o                     \\ 
     1249  with \ \  T_a = T-10  \; ;  & \;  S_a = S-35  \; ;\;  \rho_o = 1026~Kg/m^3 
    1220 where $\alpha$ and $\beta$ are the thermal and haline expansion  
    1221 coefficients, and $\rho_o$, the reference volumic mass, $rau0$.  
    1222 ($\alpha$ and $\beta$ can be modified through the \np{rn\_alpha} and  
    1223 \np{rn\_beta} namelist variables). Note that when $d_a$ is a function  
    1224 of $T$ only (\np{nn\_eos}=1), the salinity is a passive tracer and can be  
    1225 used as such. 
     1252where the computer name of the coefficients as well as their standard value are given in \ref{Tab_SEOS}. 
     1253In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing  
     1254the associated coefficients.  
     1255Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS. 
     1256setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS. 
     1257Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 
     1264\begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} 
     1266coeff.   & computer name   & S-EOS     &  description                      \\ \hline 
     1267$a_0$       & \np{nn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff.    \\ \hline 
     1268$b_0$       & \np{nn\_b0}     & 7.6554 $10^{-1}$ &  linear haline  expansion coeff.    \\ \hline 
     1269$\lambda_1$ & \np{nn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$          \\ \hline 
     1270$\lambda_2$ & \np{nn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$       \\ \hline 
     1271$\nu$       & \np{nn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$       \\ \hline 
     1272$\mu_1$     & \np{nn\_mu1}    & 1.4970 $10^{-4}$ &  thermobaric coeff. in T         \\ \hline 
     1273$\mu_2$     & \np{nn\_mu2}    & 1.1090 $10^{-5}$ &  thermobaric coeff. in S            \\ \hline 
     1275\caption{ \label{Tab_SEOS} 
     1276Standard value of S-EOS coefficients. } 
    12271282% ------------------------------------------------------------------------------------------------------------- 
    12331288An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a} 
    1234  frequency) is of paramount importance as it is used in several ocean  
    1235  parameterisations (namely TKE, KPP, Richardson number dependent  
    1236  vertical diffusion, enhanced vertical diffusion, non-penetrative convection,  
    1237  iso-neutral diffusion). In particular, one must be aware that $N^2$ has to  
    1238  be computed with an \textit{in situ} reference. The expression for $N^2$  
    1239  depends on the type of equation of state used (\np{nn\_eos} namelist parameter). 
    1241 For \np{nn\_eos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987}  
    1242 polynomial expression is used (with the pressure in decibar approximated by  
    1243 the depth in meters):  
     1289 frequency) is of paramount importance as determine the ocean stratification and  
     1290 is used in several ocean parameterisations (namely TKE, GLS, Richardson number dependent  
     1291 vertical diffusion, enhanced vertical diffusion, non-penetrative convection, tidal mixing  
     1292 parameterisation, iso-neutral diffusion). In particular, $N^2$ has to be computed at the local pressure  
     1293 (pressure in decibar being approximated by the depth in meters). The expression for $N^2$  
     1294 is given by:  
    12441295\begin{equation} \label{Eq_tra_bn2} 
    1245 N^2 = \frac{g}{e_{3w}} \; \beta   \  
    1246       \left(  \alpha / \beta \ \delta_{k+1/2}[T]     - \delta_{k+1/2}[S]   \right)  
    1247 \end{equation}  
    1248 where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients.  
    1249 They are a function of  $\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$,  
    1250 and  $z_w$, with $T$ the \textit{potential} temperature and $\widetilde{S}$ a salinity anomaly.  
    1251 Note that both $\alpha$ and $\beta$ depend on \textit{potential}  
    1252 temperature and salinity which are averaged at $w$-points prior  
    1253 to the computation instead of being computed at $T$-points and  
    1254 then averaged to $w$-points. 
    1256 When a linear equation of state is used (\np{nn\_eos}=1 or 2,  
    1257 \eqref{Eq_tra_bn2} reduces to: 
    1258 \begin{equation} \label{Eq_tra_bn2_linear} 
    12591296N^2 = \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right) 
    1261 where $\alpha$ and $\beta $ are the constant coefficients used to  
    1262 defined the linear equation of state \eqref{Eq_tra_eos_linear}. 
    1264 % ------------------------------------------------------------------------------------------------------------- 
    1265 %        Specific Heat 
    1266 % ------------------------------------------------------------------------------------------------------------- 
    1267 \subsection    [Specific Heat (\textit{phycst})] 
    1268          {Specific Heat (\mdl{phycst})} 
    1269 \label{TRA_adv_ldf} 
    1271 The specific heat of sea water, $C_p$, is a function of temperature, salinity  
    1272 and pressure \citep{UNESCO1983}. It is only used in the model to convert  
    1273 surface heat fluxes into surface temperature increase and so the pressure  
    1274 dependence is neglected. The dependence on $T$ and $S$ is weak.  
    1275 For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$  
    1276 when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has  
    1277 been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$.  
    1278 Its value is set in \mdl{phycst} module.  
     1298where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS,  
     1299and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients.  
     1300The coefficients are a polynomial function of temperature, salinity and depth which expression  
     1301depends on the chosen EOS. They are computed through \textit{eos\_rab}, a \textsc{Fortran}  
     1302function that can be found in \mdl{eosbn2}. 
     1305% ------------------------------------------------------------------------------------------------------------- 
     1306%        Potential Energy      
     1307% ------------------------------------------------------------------------------------------------------------- 
     1308%\subsection{Potential Energy anomalies} 
     1311%    =====>>>>> TO BE written 
    12811314% ------------------------------------------------------------------------------------------------------------- 
    12981331sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent  
    12991332terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing 
    1300 point is computed through \textit{tfreez}, a \textsc{Fortran} function that can be found  
     1333point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found  
    13011334in \mdl{eosbn2}.   
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