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branches/2015/dev_r5836_NOC3_vvl_by_default/DOC/TexFiles/Chapters/Chap_TRA.tex
r5102 r6040 1 1 % ================================================================ 2 % Chapter 1 �Ocean Tracers (TRA)2 % Chapter 1 ——— Ocean Tracers (TRA) 3 3 % ================================================================ 4 4 \chapter{Ocean Tracers (TRA)} … … 40 40 described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively. 41 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 %%% 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 … … 50 48 freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). 51 49 52 The different options available to the user are managed by namelist logicals or 53 CPP keys. For each equation term \textit{ttt}, the namelist logicals are \textit{ln\_trattt\_xxx},50 The different options available to the user are managed by namelist logicals or CPP keys. 51 For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx}, 54 52 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 55 The CPP key (when it exists) is \textbf{key\_tra ttt}. The equivalent code can be56 found in the \textit{tra ttt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory.53 The CPP key (when it exists) is \textbf{key\_traTTT}. The equivalent code can be 54 found in the \textit{traTTT} or \textit{traTTT\_xxx} module, in the NEMO/OPA/TRA directory. 57 55 58 56 The user has the option of extracting each tendency term on the rhs of the tracer 59 equation for output (\ key{trdtra} is defined), as described in Chap.~\ref{MISC}.57 equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}~=~true), as described in Chap.~\ref{MISC}. 60 58 61 59 $\ $\newline % force a new ligne … … 82 80 implicitly requires the use of the continuity equation. Indeed, it is obtained 83 81 by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 84 which results from the use of the continuity equation, $\nabla \cdot \vect{U}=0$ or85 $ \partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant volume or variable volume case, respectively.82 which results from the use of the continuity equation, $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ 83 (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, $i.e.$ \np{ln\_linssh}=true). 86 84 Therefore it is of paramount importance to design the discrete analogue of the 87 85 advection tendency so that it is consistent with the continuity equation in order to 88 86 enforce the conservation properties of the continuous equations. In other words, 89 by replacing $\tau$ by the number 1in (\ref{Eq_tra_adv}) we recover the discrete form of87 by setting $\tau = 1$ in (\ref{Eq_tra_adv}) we recover the discrete form of 90 88 the continuity equation which is used to calculate the vertical velocity. 91 89 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 113 111 boundary condition depends on the type of sea surface chosen: 114 112 \begin{description} 115 \item [linear free surface:] the first level thickness is constant in time:113 \item [linear free surface:] (\np{ln\_linssh}=true) the first level thickness is constant in time: 116 114 the vertical boundary condition is applied at the fixed surface $z=0$ 117 115 rather than on the moving surface $z=\eta$. There is a non-zero advective … … 119 117 $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$ 120 118 the product of surface velocity (at $z=0$) by the first level tracer value. 121 \item [non-linear free surface:] (\ key{vvl} is defined)119 \item [non-linear free surface:] (\np{ln\_linssh}=false) 122 120 convergence/divergence in the first ocean level moves the free surface 123 121 up/down. There is no tracer advection through it so that the advective … … 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 height, two quantities that are not correlated (see \S\ref{PE_free_surface}, 132 and also \citet{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}). 129 height, two quantities that are not correlated \citep{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}. 133 130 134 131 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. 138 139 The choice of an advection scheme is made in the \textit{\ngn{nam\_traadv}} namelist, by 140 setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The 132 is the centred (\textit{now}) \textit{effective} ocean velocity, $i.e.$ the \textit{eulerian} velocity 133 (see Chap.~\ref{DYN}) plus the eddy induced velocity (\textit{eiv}) 134 and/or the mixed layer eddy induced velocity (\textit{eiv}) 135 when those parameterisations are used (see Chap.~\ref{LDF}). 136 137 The choice of an advection scheme is made in the \textit{\ngn{namtra\_adv}} namelist, by 138 setting to \textit{true} one of the logicals \textit{ln\_traadv\_xxx}. The 141 139 corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 142 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. Details 143 of the advection schemes are given below. The choice of an advection scheme 140 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 141 By default ($i.e.$ in the reference namelist, \ngn{namelist\_ref}), all the logicals 142 are set to \textit{false}. If the user does not select an advection scheme 143 in the configuration namelist (\ngn{namelist\_cfg}), the tracers will not be advected ! 144 145 Details of the advection schemes are given below. The choice of an advection scheme 144 146 is a complex matter which depends on the model physics, model resolution, 145 147 type of tracer, as well as the issue of numerical cost. 146 148 147 149 Note that 148 (1) cen2, cen4 and TVD schemes require an explicit diffusion 149 operator while the other schemes are diffusive enough so that they do not 150 require additional diffusion ; 151 (2) cen2, cen4, MUSCL2, and UBS are not \textit{positive} schemes 150 (1) CEN and FCT schemes require an explicit diffusion operator 151 while the other schemes are diffusive enough so that they do not necessarily require additional diffusion ; 152 (2) CEN and UBS are not \textit{positive} schemes 152 153 \footnote{negative values can appear in an initially strictly positive tracer field 153 154 which is advected} … … 163 164 164 165 % ------------------------------------------------------------------------------------------------------------- 166 % 2nd and 4th order centred schemes 167 % ------------------------------------------------------------------------------------------------------------- 168 \subsection [$2^{nd}$ and $4^{th}$ order centred schemes (CEN) (\np{ln\_traadv\_cen})] 169 {$2^{nd}$ and $4^{th}$ order centred schemes (CEN) (\np{ln\_traadv\_cen}=true)} 170 \label{TRA_adv_cen} 171 165 172 % 2nd order centred scheme 166 % ------------------------------------------------------------------------------------------------------------- 167 \subsection [$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2})] 168 {$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=true)} 169 \label{TRA_adv_cen2} 170 171 In the centred second order formulation, the tracer at velocity points is 173 174 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is 172 175 evaluated as the mean of the two neighbouring $T$-point values. 173 176 For example, in the $i$-direction : … … 176 179 \end{equation} 177 180 178 The schemeis non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$181 CEN2 is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ 179 182 but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously 180 183 noisy and must be used in conjunction with an explicit diffusion operator to 181 184 produce a sensible solution. The associated time-stepping is performed using 182 185 a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in 183 (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. The centered second 184 order advection is computed in the \mdl{traadv\_cen2} module. In this module, 185 it is advantageous to combine the \textit{cen2} scheme with an upstream scheme 186 in specific areas which require a strong diffusion in order to avoid the generation 187 of false extrema. These areas are the vicinity of large river mouths, some straits 188 with coarse resolution, and the vicinity of ice cover area ($i.e.$ when the ocean 189 temperature is close to the freezing point). 190 This combined scheme has been included for specific grid points in the ORCA2 191 and ORCA4 configurations only. This is an obsolescent feature as the recommended 192 advection scheme for the ORCA configuration is TVD (see \S\ref{TRA_adv_tvd}). 193 194 Note that using the cen2 scheme, the overall tracer advection is of second 186 (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. 187 CEN2 is computed in the \mdl{traadv\_cen} module. 188 189 Note that using the CEN2, the overall tracer advection is of second 195 190 order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) 196 191 have this order of accuracy. \gmcomment{Note also that ... blah, blah} 197 192 198 % -------------------------------------------------------------------------------------------------------------199 193 % 4nd order centred scheme 200 % ------------------------------------------------------------------------------------------------------------- 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 are 206 evaluated at velocity points as a $4^{th}$ order interpolation, and thus depend on 207 the four neighbouring $T$-points. For example, in the $i$-direction: 194 195 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at velocity points as 196 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 197 For example, in the $i$-direction: 208 198 \begin{equation} \label{Eq_tra_adv_cen4} 209 199 \tau _u^{cen4} … … 211 201 \end{equation} 212 202 213 Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme203 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme 214 204 but a $4^{th}$ order evaluation of advective fluxes, since the divergence of 215 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 associated 217 with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection 218 scheme is feasible but, for consistency reasons, it requires changes in the 219 discretisation of the tracer advection together with changes in both the 220 continuity equation and the momentum advection terms. 205 advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. 206 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature 207 is usually associated with the scheme presented here. 208 Introducing a \textit{true} $4^{th}$ order advection scheme is feasible but, 209 for consistency reasons, it requires changes in the discretisation of the tracer 210 advection together with changes in the continuity equation, 211 and the momentum advection and pressure terms. 221 212 222 213 A direct consequence of the pseudo-fourth order nature of the scheme is that 223 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 also214 it is not non-diffusive, $i.e.$ the global variance of a tracer is not preserved using 215 CEN4. Furthermore, it must be used in conjunction with an explicit 216 diffusion operator to produce a sensible solution. As in CEN2 case, the time-stepping is 226 217 performed using a leapfrog scheme in conjunction with an Asselin time-filter, 227 218 so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer. … … 235 226 236 227 % ------------------------------------------------------------------------------------------------------------- 237 % TVDscheme238 % ------------------------------------------------------------------------------------------------------------- 239 \subsection [ Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})]240 { Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=true)}228 % FCT scheme 229 % ------------------------------------------------------------------------------------------------------------- 230 \subsection [$2^{nd}$ and $4^{th}$ Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct})] 231 {$2^{nd}$ and $4^{th}$ Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct}=true)} 241 232 \label{TRA_adv_tvd} 242 233 243 In the Total Variance Dissipation (TVD)formulation, the tracer at velocity234 In the Flux Corrected Transport formulation, the tracer at velocity 244 235 points is evaluated using a combination of an upstream and a centred scheme. 245 236 For example, in the $i$-direction : 246 \begin{equation} \label{Eq_tra_adv_ tvd}237 \begin{equation} \label{Eq_tra_adv_fct} 247 238 \begin{split} 248 239 \tau _u^{ups}&= \begin{cases} … … 251 242 \end{cases} \\ 252 243 \\ 253 \tau _u^{ tvd}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen2} -\tau _u^{ups} } \right)244 \tau _u^{fct}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen} -\tau _u^{ups} } \right) 254 245 \end{split} 255 246 \end{equation} … … 260 251 produces a local extremum in the tracer field. The resulting scheme is quite 261 252 expensive but \emph{positive}. It can be used on both active and passive tracers. 262 This scheme is tested and compared with MUSCL and the MPDATA scheme in 263 \citet{Levy_al_GRL01}; note that in this paper it is referred to as "FCT" (Flux corrected 264 transport) rather than TVD. The TVD scheme is implemented in the \mdl{traadv\_tvd} module. 253 This scheme is tested and compared with MUSCL and a MPDATA scheme in \citet{Levy_al_GRL01}. 254 The FCT scheme is implemented in the \mdl{traadv\_fct} module. 265 255 266 256 For stability reasons (see \S\ref{STP}), 267 $\tau _u^{cen 2}$ is evaluated in (\ref{Eq_tra_adv_tvd}) using the \textit{now} tracer while $\tau _u^{ups}$268 is evaluated using the \textit{before} tracer. In other words, the advective part of269 the scheme is time stepped with a leap-frog scheme while a forward scheme is270 used for the diffusive part.257 $\tau _u^{cen}$ is evaluated in (\ref{Eq_tra_adv_fct}) using the \textit{now} tracer 258 while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, 259 the advective part of the scheme is time stepped with a leap-frog scheme 260 while a forward scheme is used for the diffusive part. 271 261 272 262 % ------------------------------------------------------------------------------------------------------------- 273 263 % MUSCL scheme 274 264 % ------------------------------------------------------------------------------------------------------------- 275 \subsection[MUSCL scheme (\np{ln\_traadv\_mus cl})]276 {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_mus cl}=T)}277 \label{TRA_adv_mus cl}265 \subsection[MUSCL scheme (\np{ln\_traadv\_mus})] 266 {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_mus}=T)} 267 \label{TRA_adv_mus} 278 268 279 269 The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been … … 281 271 is evaluated assuming a linear tracer variation between two $T$-points 282 272 (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction : 283 \begin{equation} \label{Eq_tra_adv_mus cl}273 \begin{equation} \label{Eq_tra_adv_mus} 284 274 \tau _u^{mus} = \left\{ \begin{aligned} 285 275 &\tau _i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) … … 296 286 297 287 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. 288 directed toward land, an upstream flux is used. This choice ensure 289 the \textit{positive} character of the scheme. 304 290 305 291 % ------------------------------------------------------------------------------------------------------------- … … 310 296 \label{TRA_adv_ubs} 311 297 312 The UBS advection scheme is an upstream-biased third order scheme based on313 an upstream-biased parabolic interpolation. It is also known as the Cell314 Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective298 The UBS advection scheme (also often called UP3) is an upstream-biased third order 299 scheme based on an upstream-biased parabolic interpolation. It is also known as 300 the Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective 315 301 Kinematics). For example, in the $i$-direction : 316 302 \begin{equation} \label{Eq_tra_adv_ubs} … … 324 310 325 311 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation 326 error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the advection327 scheme is similar to that reported in \cite{Farrow1995}.312 error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of 313 the advection scheme is similar to that reported in \cite{Farrow1995}. 328 314 It is a relatively good compromise between accuracy and smoothness. 329 315 It is not a \emph{positive} scheme, meaning that false extrema are permitted, 330 316 but the amplitude of such are significantly reduced over the centred second 331 or der method. Nevertheless it is not recommended that it should be applied332 to a passive tracer that requires positivity.317 or fourth order method. Nevertheless it is not recommended that it should be 318 applied to a passive tracer that requires positivity. 333 319 334 320 The intrinsic diffusion of UBS makes its use risky in the vertical direction 335 321 where the control of artificial diapycnal fluxes is of paramount importance. 336 Therefore the vertical flux is evaluated using the TVD scheme when337 \np{ln\_traadv\_ubs}=true.322 Therefore the vertical flux is evaluated using either a 2nd order FCT scheme 323 or a 4th order COMPACT scheme (\np{nn\_cen\_v}=2 or 4). 338 324 339 325 For stability reasons (see \S\ref{STP}), 340 the first term in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order centred scheme)341 is evaluated using the \textit{now} tracer (centred in time) while the342 second term (which is the diffusive part of the scheme), is326 the first term in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order 327 centred scheme) is evaluated using the \textit{now} tracer (centred in time) 328 while the second term (which is the diffusive part of the scheme), is 343 329 evaluated using the \textit{before} tracer (forward in time). 344 330 This choice is discussed by \citet{Webb_al_JAOT98} in the context of the … … 350 336 substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 351 337 338 ??? 339 352 340 Four different options are possible for the vertical 353 341 component used in the UBS scheme. $\tau _w^{ubs}$ can be evaluated 354 342 using either \textit{(a)} a centred $2^{nd}$ order scheme, or \textit{(b)} 355 a TVDscheme, or \textit{(c)} an interpolation based on conservative343 a FCT scheme, or \textit{(c)} an interpolation based on conservative 356 344 parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} 357 345 implementation of UBS in ROMS, or \textit{(d)} a UBS. The $3^{rd}$ case 358 346 has dispersion properties similar to an eighth-order accurate conventional scheme. 359 The current reference version uses method b) 347 The current reference version uses method (b). 348 349 ??? 360 350 361 351 Note that : … … 390 380 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 391 381 coefficient which is simply proportional to the velocity: 392 $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v3.4still uses382 $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO still uses 393 383 \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. 394 384 %%% … … 416 406 direction (as for the UBS case) should be implemented to restore this property. 417 407 418 419 % -------------------------------------------------------------------------------------------------------------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 3.3.432 408 433 409 % ================================================================ … … 1167 1143 % Equation of State 1168 1144 % ------------------------------------------------------------------------------------------------------------- 1169 \subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)}1145 \subsection{Equation Of Seawater (\np{nn\_eos} = -1, 0, or 1)} 1170 1146 \label{TRA_eos} 1171 1147 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. 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 $1,035~Kg/m^3$. 1148 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship 1149 linking seawater density, $\rho$, to a number of state variables, 1150 most typically temperature, salinity and pressure. 1151 Because density gradients control the pressure gradient force through the hydrostatic balance, 1152 the equation of state provides a fundamental bridge between the distribution of active tracers 1153 and the fluid dynamics. Nonlinearities of the EOS are of major importance, in particular 1154 influencing the circulation through determination of the static stability below the mixed layer, 1155 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{Roquet_JPO2015}. 1156 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983}) 1157 or TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real 1158 ocean circulation is attempted \citep{Roquet_JPO2015}. 1159 The use of TEOS-10 is highly recommended because 1160 \textit{(i)} it is the new official EOS, 1161 \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and 1162 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature 1163 and practical salinity for EOS-980, both variables being more suitable for use as model variables 1164 \citep{TEOS10, Graham_McDougall_JPO13}. 1165 EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. 1166 For process studies, it is often convenient to use an approximation of the EOS. To that purposed, 1167 a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available. 1168 1169 In the computer code, a density anomaly, $d_a= \rho / \rho_o - 1$, 1170 is computed, with $\rho_o$ a reference density. Called \textit{rau0} 1171 in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 1200 1172 This is a sensible choice for the reference density used in a Boussinesq ocean 1201 1173 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. 1174 density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. 1175 1176 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} 1177 which controls the EOS used (=-1 for TEOS10 ; =0 for EOS-80 ; =1 for S-EOS). 1178 \begin{description} 1179 1180 \item[\np{nn\_eos}$=-1$] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used. 1181 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 1182 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler 1183 and more computationally efficient expressions for their derived quantities 1184 which make them more adapted for use in ocean models. 1185 Note that a slightly higher precision polynomial form is now used replacement of the TEOS-10 1186 rational function approximation for hydrographic data analysis \citep{TEOS10}. 1187 A key point is that conservative state variables are used: 1188 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: $\degres C$, notation: $\Theta$). 1189 The pressure in decibars is approximated by the depth in meters. 1190 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. It is set to 1191 $C_p=3991.86795711963~J\,Kg^{-1}\,\degres K^{-1}$, according to \citet{TEOS10}. 1192 1193 Choosing polyTEOS10-bsq implies that the state variables used by the model are 1194 $\Theta$ and $S_A$. In particular, the initial state deined by the user have to be given as 1195 \textit{Conservative} Temperature and \textit{Absolute} Salinity. 1196 In addition, setting \np{ln\_useCT} to \textit{true} convert the Conservative SST to potential SST 1197 prior to either computing the air-sea and ice-sea fluxes (forced mode) 1198 or sending the SST field to the atmosphere (coupled mode). 1199 1200 \item[\np{nn\_eos}$=0$] the polyEOS80-bsq equation of seawater is used. 1201 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized 1202 to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80 1203 and the ocean model are: 1204 the Practical Salinity ((unit: psu, notation: $S_p$)) and Potential Temperature (unit: $\degres C$, notation: $\theta$). 1205 The pressure in decibars is approximated by the depth in meters. 1206 With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, 1207 salinity and pressure \citep{UNESCO1983}. Nevertheless, a severe assumption is made in order to 1208 have a heat content ($C_p T_p$) which is conserved by the model: $C_p$ is set to a constant 1209 value, the TEOS10 value. 1210 1211 \item[\np{nn\_eos}$=1$] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen, 1212 the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) 1213 (see also \citet{Roquet_JPO2015}). It provides a simplistic linear representation of both 1214 cabbeling and thermobaricity effects which is enough for a proper treatment of the EOS 1215 in theoretical studies \citep{Roquet_JPO2015}. 1209 1216 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} 1217 \textit{conservative} and \textit{potential} temperature, as well as between \textit{absolute} 1218 and \textit{practical} salinity. 1219 S-EOS takes the following expression: 1220 \begin{equation} \label{Eq_tra_S-EOS} 1215 1221 \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 1222 d_a(T,S,z) = ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a \\ 1223 & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a \\ 1224 & - \nu \; T_a \; S_a \; ) \; / \; \rho_o \\ 1225 with \ \ T_a = T-10 \; ; & \; S_a = S-35 \; ;\; \rho_o = 1026~Kg/m^3 1218 1226 \end{split} 1219 1227 \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. 1228 where the computer name of the coefficients as well as their standard value are given in \ref{Tab_SEOS}. 1229 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing 1230 the associated coefficients. 1231 Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS. 1232 setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS. 1233 Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 1234 1235 \end{description} 1236 1237 1238 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1239 \begin{table}[!tb] 1240 \begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} 1241 \hline 1242 coeff. & computer name & S-EOS & description \\ \hline 1243 $a_0$ & \np{nn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline 1244 $b_0$ & \np{nn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline 1245 $\lambda_1$ & \np{nn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline 1246 $\lambda_2$ & \np{nn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline 1247 $\nu$ & \np{nn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline 1248 $\mu_1$ & \np{nn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline 1249 $\mu_2$ & \np{nn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1250 \end{tabular} 1251 \caption{ \label{Tab_SEOS} 1252 Standard value of S-EOS coefficients. } 1253 \end{center} 1254 \end{table} 1255 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1256 1226 1257 1227 1258 % ------------------------------------------------------------------------------------------------------------- … … 1232 1263 1233 1264 An 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). 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): 1265 frequency) is of paramount importance as determine the ocean stratification and 1266 is used in several ocean parameterisations (namely TKE, GLS, Richardson number dependent 1267 vertical diffusion, enhanced vertical diffusion, non-penetrative convection, tidal mixing 1268 parameterisation, iso-neutral diffusion). In particular, $N^2$ has to be computed at the local pressure 1269 (pressure in decibar being approximated by the depth in meters). The expression for $N^2$ 1270 is given by: 1244 1271 \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 1272 N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) 1260 1273 \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 1274 where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS, 1275 and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 1276 The coefficients are a polynomial function of temperature, salinity and depth which expression 1277 depends on the chosen EOS. They are computed through \textit{eos\_rab}, a \textsc{Fortran} 1278 function that can be found in \mdl{eosbn2}. 1280 1279 1281 1280 % ------------------------------------------------------------------------------------------------------------- … … 1298 1297 sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent 1299 1298 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 found1299 point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found 1301 1300 in \mdl{eosbn2}. 1301 1302 1303 % ------------------------------------------------------------------------------------------------------------- 1304 % Potential Energy 1305 % ------------------------------------------------------------------------------------------------------------- 1306 %\subsection{Potential Energy anomalies} 1307 %\label{TRA_bn2} 1308 1309 % =====>>>>> TO BE written 1310 % 1311 1302 1312 1303 1313 % ================================================================
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