# Changeset 6064 for branches/2015/dev_r5918_nemo_v3_6_STABLE_XIOS2/DOC/TexFiles/Chapters/Chap_TRA.tex

Ignore:
Timestamp:
2015-12-16T12:14:31+01:00 (5 years ago)
Message:

dev_xios2 : update to the head of v3.6 stable branch

File:
1 edited

### Legend:

Unmodified
 r5890 % ================================================================ % Chapter 1 � Ocean Tracers (TRA) % Chapter 1 ——— Ocean Tracers (TRA) % ================================================================ \chapter{Ocean Tracers (TRA)} %        Equation of State % ------------------------------------------------------------------------------------------------------------- \subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)} \subsection{Equation Of Seawater (\np{nn\_eos} = -1, 0, or 1)} \label{TRA_eos} It is necessary to know the equation of state for the ocean very accurately to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), particularly in the deep ocean. The ocean seawater volumic mass, $\rho$, abusively called density, is a non linear empirical function of \textit{in situ} temperature, salinity and pressure. The reference equation of state is that defined by the Joint Panel on Oceanographic Tables and Standards \citep{UNESCO1983}. It was the standard equation of state used in early releases of OPA. However, even though this computation is fully vectorised, it is quite time consuming ($15$ to $20${\%} of the total CPU time) since it requires the prior computation of the \textit{in situ} temperature from the model \textit{potential} temperature using the \citep{Bryden1973} polynomial for adiabatic lapse rate and a $4^th$ order Runge-Kutta integration scheme. Since OPA6, we have used the \citet{JackMcD1995} equation of state for seawater instead. It allows the computation of the \textit{in situ} ocean density directly as a function of \textit{potential} temperature relative to the surface (an \NEMO variable), the practical salinity (another \NEMO variable) and the pressure (assuming no pressure variation along geopotential surfaces, $i.e.$ the pressure in decibars is approximated by the depth in meters). Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state have exactly the same except that the values of the various coefficients have been adjusted by \citet{JackMcD1995} in order to directly use the \textit{potential} temperature instead of the \textit{in situ} one. This reduces the CPU time of the \textit{in situ} density computation to about $3${\%} of the total CPU time, while maintaining a quite accurate equation of state. In the computer code, a \textit{true} density anomaly, $d_a= \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference volumic mass. Called \textit{rau0} in the code, $\rho_o$ is defined in \mdl{phycst}, and a value of $1,035~Kg/m^3$. The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density, $\rho$, to a number of state variables, most typically temperature, salinity and pressure. Because density gradients control the pressure gradient force through the hydrostatic balance, the equation of state provides a fundamental bridge between the distribution of active tracers and the fluid dynamics. Nonlinearities of the EOS are of major importance, in particular influencing the circulation through determination of the static stability below the mixed layer, thus controlling rates of exchange between the atmosphere  and the ocean interior \citep{Roquet_JPO2015}. Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983}) or TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real ocean circulation is attempted \citep{Roquet_JPO2015}. The use of TEOS-10 is highly recommended because \textit{(i)} it is the new official EOS, \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and practical salinity for EOS-980, both variables being more suitable for use as model variables \citep{TEOS10, Graham_McDougall_JPO13}. EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. For process studies, it is often convenient to use an approximation of the EOS. To that purposed, a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available. In the computer code, a density anomaly, $d_a= \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density. Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as, with the exception of only a small percentage of the ocean, density in the World Ocean varies by no more than 2$\%$ from $1,035~kg/m^3$ \citep{Gill1982}. Options are defined through the  \ngn{nameos} namelist variables. The default option (namelist parameter \np{nn\_eos}=0) is the \citet{JackMcD1995} equation of state. Its use is highly recommended. However, for process studies, it is often convenient to use a linear approximation of the density. density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. Options are defined through the  \ngn{nameos} namelist variables, and in particular \np{nn\_eos} which controls the EOS used (=-1 for TEOS10 ; =0 for EOS-80 ; =1 for S-EOS). \begin{description} \item[\np{nn\_eos}$=-1$] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used. The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and more computationally efficient expressions for their derived quantities which make them more adapted for use in ocean models. Note that a slightly higher precision polynomial form is now used replacement of the TEOS-10 rational function approximation for hydrographic data analysis  \citep{TEOS10}. A key point is that conservative state variables are used: Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: $\degres C$, notation: $\Theta$). The pressure in decibars is approximated by the depth in meters. With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. It is set to $C_p=3991.86795711963~J\,Kg^{-1}\,\degres K^{-1}$, according to \citet{TEOS10}. Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$. In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and \textit{Absolute} Salinity. In addition, setting \np{ln\_useCT} to \textit{true} convert the Conservative SST to potential SST prior to either computing the air-sea and ice-sea fluxes (forced mode) or sending the SST field to the atmosphere (coupled mode). \item[\np{nn\_eos}$=0$] the polyEOS80-bsq equation of seawater is used. It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80 and the ocean model are: the Practical Salinity ((unit: psu, notation: $S_p$)) and Potential Temperature (unit: $\degres C$, notation: $\theta$). The pressure in decibars is approximated by the depth in meters. With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and pressure \citep{UNESCO1983}. Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. \item[\np{nn\_eos}$=1$] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen, the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) (see also \citet{Roquet_JPO2015}). It provides a simplistic linear representation of both cabbeling and thermobaricity effects which is enough for a proper treatment of the EOS in theoretical studies \citep{Roquet_JPO2015}. With such an equation of state there is no longer a distinction between \textit{in situ} and \textit{potential} density and both cabbeling and thermobaric effects are removed. Two linear formulations are available: a function of $T$ only (\np{nn\_eos}=1) and a function of both $T$ and $S$ (\np{nn\_eos}=2): \label{Eq_tra_eos_linear} \textit{conservative} and \textit{potential} temperature, as well as between \textit{absolute} and \textit{practical} salinity. S-EOS takes the following expression: \label{Eq_tra_S-EOS} \begin{split} d_a(T)       &=  \rho (T)      /  \rho_o   - 1     =  \  0.0285         -  \alpha   \;T     \\ d_a(T,S)    &=  \rho (T,S)   /  \rho_o   - 1     =  \  \beta \; S       -  \alpha   \;T d_a(T,S,z)  =  ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a  \\ & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a  \\ & - \nu \; T_a \; S_a \;  ) \; / \; \rho_o                     \\ with \ \  T_a = T-10  \; ;  & \;  S_a = S-35  \; ;\;  \rho_o = 1026~Kg/m^3 \end{split} where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients, and $\rho_o$, the reference volumic mass, $rau0$. ($\alpha$ and $\beta$ can be modified through the \np{rn\_alpha} and \np{rn\_beta} namelist variables). Note that when $d_a$ is a function of $T$ only (\np{nn\_eos}=1), the salinity is a passive tracer and can be used as such. where the computer name of the coefficients as well as their standard value are given in \ref{Tab_SEOS}. In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing the associated coefficients. Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS. setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS. Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. \end{description} %>>>>>>>>>>>>>>>>>>>>>>>>>>>> \begin{table}[!tb] \begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} \hline coeff.   & computer name   & S-EOS     &  description                      \\ \hline $a_0$       & \np{nn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff.    \\ \hline $b_0$       & \np{nn\_b0}     & 7.6554 $10^{-1}$ &  linear haline  expansion coeff.    \\ \hline $\lambda_1$ & \np{nn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$          \\ \hline $\lambda_2$ & \np{nn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$       \\ \hline $\nu$       & \np{nn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$       \\ \hline $\mu_1$     & \np{nn\_mu1}    & 1.4970 $10^{-4}$ &  thermobaric coeff. in T         \\ \hline $\mu_2$     & \np{nn\_mu2}    & 1.1090 $10^{-5}$ &  thermobaric coeff. in S            \\ \hline \end{tabular} \caption{ \label{Tab_SEOS} Standard value of S-EOS coefficients. } \end{center} \end{table} %>>>>>>>>>>>>>>>>>>>>>>>>>>>> % ------------------------------------------------------------------------------------------------------------- An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a} frequency) is of paramount importance as it is used in several ocean parameterisations (namely TKE, KPP, Richardson number dependent vertical diffusion, enhanced vertical diffusion, non-penetrative convection, iso-neutral diffusion). In particular, one must be aware that $N^2$ has to be computed with an \textit{in situ} reference. The expression for $N^2$ depends on the type of equation of state used (\np{nn\_eos} namelist parameter). For \np{nn\_eos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} polynomial expression is used (with the pressure in decibar approximated by the depth in meters): frequency) is of paramount importance as determine the ocean stratification and is used in several ocean parameterisations (namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion, non-penetrative convection, tidal mixing parameterisation, iso-neutral diffusion). In particular, $N^2$ has to be computed at the local pressure (pressure in decibar being approximated by the depth in meters). The expression for $N^2$ is given by: \label{Eq_tra_bn2} N^2 = \frac{g}{e_{3w}} \; \beta   \ \left(  \alpha / \beta \ \delta_{k+1/2}[T]     - \delta_{k+1/2}[S]   \right) where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. They are a function of  $\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$, and  $z_w$, with $T$ the \textit{potential} temperature and $\widetilde{S}$ a salinity anomaly. Note that both $\alpha$ and $\beta$ depend on \textit{potential} temperature and salinity which are averaged at $w$-points prior to the computation instead of being computed at $T$-points and then averaged to $w$-points. When a linear equation of state is used (\np{nn\_eos}=1 or 2, \eqref{Eq_tra_bn2} reduces to: \label{Eq_tra_bn2_linear} N^2 = \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right) where $\alpha$ and $\beta$ are the constant coefficients used to defined the linear equation of state \eqref{Eq_tra_eos_linear}. % ------------------------------------------------------------------------------------------------------------- %        Specific Heat % ------------------------------------------------------------------------------------------------------------- \subsection    [Specific Heat (\textit{phycst})] {Specific Heat (\mdl{phycst})} \label{TRA_adv_ldf} The specific heat of sea water, $C_p$, is a function of temperature, salinity and pressure \citep{UNESCO1983}. It is only used in the model to convert surface heat fluxes into surface temperature increase and so the pressure dependence is neglected. The dependence on $T$ and $S$ is weak. For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. Its value is set in \mdl{phycst} module. where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS, and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. The coefficients are a polynomial function of temperature, salinity and depth which expression depends on the chosen EOS. They are computed through \textit{eos\_rab}, a \textsc{Fortran} function that can be found in \mdl{eosbn2}. % ------------------------------------------------------------------------------------------------------------- %        Potential Energy % ------------------------------------------------------------------------------------------------------------- %\subsection{Potential Energy anomalies} %\label{TRA_bn2} %    =====>>>>> TO BE written % % ------------------------------------------------------------------------------------------------------------- sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing point is computed through \textit{tfreez}, a \textsc{Fortran} function that can be found point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found in \mdl{eosbn2}.