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branches/2015/nemo_v3_6_STABLE/DOC/TexFiles/Chapters/Chap_TRA.tex
r5890 r6039 1 1 % ================================================================ 2 % Chapter 1 �Ocean Tracers (TRA)2 % Chapter 1 ——— Ocean Tracers (TRA) 3 3 % ================================================================ 4 4 \chapter{Ocean Tracers (TRA)} … … 1167 1167 % Equation of State 1168 1168 %  1169 \subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)}1169 \subsection{Equation Of Seawater (\np{nn\_eos} = 1, 0, or 1)} 1170 1170 \label{TRA_eos} 1171 1171 1172 It is necessary to know the equation of state for the ocean very accurately 1173 to determine stability properties (especially the BruntVais\"{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 RungeKutta 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. 1196 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 $1,035~Kg/m^3$. 1172 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship 1173 linking seawater density, $\rho$, to a number of state variables, 1174 most typically temperature, salinity and pressure. 1175 Because density gradients control the pressure gradient force through the hydrostatic balance, 1176 the equation of state provides a fundamental bridge between the distribution of active tracers 1177 and the fluid dynamics. Nonlinearities of the EOS are of major importance, in particular 1178 influencing the circulation through determination of the static stability below the mixed layer, 1179 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{Roquet_JPO2015}. 1180 Therefore an accurate EOS based on either the 1980 equation of state (EOS80, \cite{UNESCO1983}) 1181 or TEOS10 \citep{TEOS10} standards should be used anytime a simulation of the real 1182 ocean circulation is attempted \citep{Roquet_JPO2015}. 1183 The use of TEOS10 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 1187 and practical salinity for EOS980, both variables being more suitable for use as model variables 1188 \citep{TEOS10, Graham_McDougall_JPO13}. 1189 EOS80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. 1190 For process studies, it is often convenient to use an approximation of the EOS. To that purposed, 1191 a simplified EOS (SEOS) inspired by \citet{Vallis06} is also available. 1192 1193 In the computer code, a density anomaly, $d_a= \rho / \rho_o  1$, 1194 is computed, with $\rho_o$ a reference density. Called \textit{rau0} 1195 in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 1200 1196 This is a sensible choice for the reference density used in a Boussinesq ocean 1201 1197 climate 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}. 1204 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. 1198 density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. 1199 1200 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} 1201 which controls the EOS used (=1 for TEOS10 ; =0 for EOS80 ; =1 for SEOS). 1202 \begin{description} 1203 1204 \item[\np{nn\_eos}$=1$] the polyTEOS10bsq equation of seawater \citep{Roquet_OM2015} is used. 1205 The accuracy of this approximation is comparable to the TEOS10 rational function approximation, 1206 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler 1207 and more computationally efficient expressions for their derived quantities 1208 which make them more adapted for use in ocean models. 1209 Note that a slightly higher precision polynomial form is now used replacement of the TEOS10 1210 rational function approximation for hydrographic data analysis \citep{TEOS10}. 1211 A key point is that conservative state variables are used: 1212 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: $\degres C$, notation: $\Theta$). 1213 The pressure in decibars is approximated by the depth in meters. 1214 With 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}. 1216 1217 Choosing polyTEOS10bsq 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. 1220 In addition, setting \np{ln\_useCT} to \textit{true} convert the Conservative SST to potential SST 1221 prior to either computing the airsea and icesea fluxes (forced mode) 1222 or sending the SST field to the atmosphere (coupled mode). 1223 1224 \item[\np{nn\_eos}$=0$] the polyEOS80bsq equation of seawater is used. 1225 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized 1226 to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80 1227 and the ocean model are: 1228 the Practical Salinity ((unit: psu, notation: $S_p$)) and Potential Temperature (unit: $\degres C$, notation: $\theta$). 1229 The pressure in decibars is approximated by the depth in meters. 1230 With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, 1231 salinity and pressure \citep{UNESCO1983}. Nevertheless, a severe assumption is made in order to 1232 have a heat content ($C_p T_p$) which is conserved by the model: $C_p$ is set to a constant 1233 value, the TEOS10 value. 1234 1235 \item[\np{nn\_eos}$=1$] a simplified EOS (SEOS) inspired by \citet{Vallis06} is chosen, 1236 the 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 1238 cabbeling and thermobaricity effects which is enough for a proper treatment of the EOS 1239 in theoretical studies \citep{Roquet_JPO2015}. 1209 1240 With 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} 1242 and \textit{practical} salinity. 1243 SEOS takes the following expression: 1244 \begin{equation} \label{Eq_tra_SEOS} 1215 1245 \begin{split} 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 = T10 \; ; & \; S_a = S35 \; ;\; \rho_o = 1026~Kg/m^3 1218 1250 \end{split} 1219 1251 \end{equation} 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. 1252 where the computer name of the coefficients as well as their standard value are given in \ref{Tab_SEOS}. 1253 In fact, when choosing SEOS, various approximation of EOS can be specified simply by changing 1254 the associated coefficients. 1255 Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from SEOS. 1256 setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from SEOS. 1257 Keeping nonzero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 1258 1259 \end{description} 1260 1261 1262 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1263 \begin{table}[!tb] 1264 \begin{center} \begin{tabular}{p{26pt}p{72pt}p{56pt}p{136pt}} 1265 \hline 1266 coeff. & computer name & SEOS & 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 1274 \end{tabular} 1275 \caption{ \label{Tab_SEOS} 1276 Standard value of SEOS coefficients. } 1277 \end{center} 1278 \end{table} 1279 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1280 1226 1281 1227 1282 %  … … 1232 1287 1233 1288 An accurate computation of the ocean stability (i.e. of $N$, the bruntVais\"{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, nonpenetrative convection, 1237 isoneutral 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). 1240 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, nonpenetrative convection, tidal mixing 1292 parameterisation, isoneutral 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: 1244 1295 \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 prior1253 to the computation instead of being computed at $T$points and1254 then averaged to $w$points.1255 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}1259 1296 N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S]  \alpha \;\delta_{k+1/2}[T] \right) 1260 1297 \end{equation} 1261 where $\alpha$ and $\beta $ are the constant coefficients used to 1262 defined the linear equation of state \eqref{Eq_tra_eos_linear}. 1263 1264 %  1265 % Specific Heat 1266 %  1267 \subsection [Specific Heat (\textit{phycst})] 1268 {Specific Heat (\mdl{phycst})} 1269 \label{TRA_adv_ldf} 1270 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. 1279 1298 where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS80, or $=(T,S)$ for SEOS, 1299 and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 1300 The coefficients are a polynomial function of temperature, salinity and depth which expression 1301 depends on the chosen EOS. They are computed through \textit{eos\_rab}, a \textsc{Fortran} 1302 function that can be found in \mdl{eosbn2}. 1303 1304 1305 %  1306 % Potential Energy 1307 %  1308 %\subsection{Potential Energy anomalies} 1309 %\label{TRA_bn2} 1310 1311 % =====>>>>> TO BE written 1312 % 1280 1313 1281 1314 %  … … 1298 1331 sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent 1299 1332 terms 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 found1333 point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found 1301 1334 in \mdl{eosbn2}. 1302 1335
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