Changeset 10251 for branches/UKMO/dev_r5518_AMM15_package/DOC/TexFiles/Chapters/Chap_TRA.tex

Timestamp:

2018-10-29T15:20:26+01:00 (5 years ago)

Author:

kingr

Message:

Rolled back to r10247 - i.e., undid merge of pkg br and 3.6_stable br

File:

• branches/UKMO/dev_r5518_AMM15_package/DOC/TexFiles/Chapters/Chap_TRA.tex (modified) (5 diffs)

branches/UKMO/dev_r5518_AMM15_package/DOC/TexFiles/Chapters/Chap_TRA.tex

@@ -1 +1 @@
 % ================================================================
-% Chapter 1 ——— Ocean Tracers (TRA)
+% Chapter 1 � Ocean Tracers (TRA)
 % ================================================================
 \chapter{Ocean Tracers (TRA)}
@@ -859 +859 @@
 \label{TRA_bbc}
 %--------------------------------------------nambbc--------------------------------------------------------
-\namdisplay{nambbc}
+\namdisplay{namtra_bbc}
 %--------------------------------------------------------------------------------------------------------------
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -1167 +1167 @@
 %        Equation of State
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Equation Of Seawater (\np{nn\_eos} = -1, 0, or 1)}
+\subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)}
 \label{TRA_eos}
 
-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 $1,026~Kg/m^3$. 
+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 $1,035~Kg/m^3$. 
 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 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}.
+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.
 With such an equation of state there is no longer a distinction between 
-\textit{conservative} and \textit{potential} temperature, as well as between \textit{absolute} 
-and \textit{practical} salinity.
-S-EOS takes the following expression:
-\begin{equation} \label{Eq_tra_S-EOS}
+\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):
+\begin{equation} \label{Eq_tra_eos_linear}
 \begin{split}
-  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
+  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    
 \end{split}
 \end{equation} 
-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}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
+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.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -1287 +1232 @@
 
 An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a}
- 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: 
+ 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): 
 \begin{equation} \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) 
+\end{equation} 
+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:
+\begin{equation} \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)
 \end{equation} 
-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
-%
+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. 
+
 
 % -------------------------------------------------------------------------------------------------------------
@@ -1331 +1298 @@
 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{eos\_fzp}, a \textsc{Fortran} function that can be found 
+point is computed through \textit{tfreez}, a \textsc{Fortran} function that can be found 
 in \mdl{eosbn2}.