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branches/2016/dev_INGV_UKMO_2016/DOC/TexFiles/Chapters/Chap_TRA.tex
r5102 r7351 1 % ================================================================ 2 % Chapter 1 � Ocean Tracers (TRA) 1 \documentclass[NEMO_book]{subfiles} 2 \begin{document} 3 % ================================================================ 4 % Chapter 1 ——— Ocean Tracers (TRA) 3 5 % ================================================================ 4 6 \chapter{Ocean Tracers (TRA)} … … 36 38 (BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, 37 39 BBC, BBL and DMP are optional. The external forcings and parameterisations 38 require complex inputs and complex calculations ( e.g.bulk formulae, estimation40 require complex inputs and complex calculations ($e.g.$ bulk formulae, estimation 39 41 of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 40 42 described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively. 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 %%% 43 Note that \mdl{tranpc}, the non-penetrative convection module, although 44 located in the NEMO/OPA/TRA directory as it directly modifies the tracer fields, 45 is described with the model vertical physics (ZDF) together with other available 46 parameterization of convection. 47 47 48 48 In the present chapter we also describe the diagnostic equations used to compute 49 the sea-water properties (density, Brunt-V ais\"{a}l\"{a} frequency, specific heat and49 the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and 50 50 freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). 51 51 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},52 The different options available to the user are managed by namelist logicals or CPP keys. 53 For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx}, 54 54 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.57 58 The user has the option of extracting each tendency term on the rhsof the tracer59 equation for output (\ key{trdtra} is defined), as described in Chap.~\ref{MISC}.55 The CPP key (when it exists) is \textbf{key\_traTTT}. The equivalent code can be 56 found in the \textit{traTTT} or \textit{traTTT\_xxx} module, in the NEMO/OPA/TRA directory. 57 58 The user has the option of extracting each tendency term on the RHS of the tracer 59 equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}~=~true), as described in Chap.~\ref{DIA}. 60 60 61 61 $\ $\newline % force a new ligne … … 70 70 %------------------------------------------------------------------------------------------------------------- 71 71 72 The advection tendency of a tracer in flux form is the divergence of the advective 73 fluxes. Its discrete expression is given by : 72 When considered ($i.e.$ when \np{ln\_traadv\_NONE} is not set to \textit{true}), 73 the advection tendency of a tracer is expressed in flux form, 74 $i.e.$ as the divergence of the advective fluxes. Its discrete expression is given by : 74 75 \begin{equation} \label{Eq_tra_adv} 75 76 ADV_\tau =-\frac{1}{b_t} \left( … … 82 83 implicitly requires the use of the continuity equation. Indeed, it is obtained 83 84 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.85 which results from the use of the continuity equation, $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ 86 (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, $i.e.$ \np{ln\_linssh}=true). 86 87 Therefore it is of paramount importance to design the discrete analogue of the 87 88 advection tendency so that it is consistent with the continuity equation in order to 88 89 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 of90 by setting $\tau = 1$ in (\ref{Eq_tra_adv}) we recover the discrete form of 90 91 the continuity equation which is used to calculate the vertical velocity. 91 92 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 92 93 \begin{figure}[!t] \begin{center} 93 \includegraphics[width=0.9\textwidth]{ ./TexFiles/Figures/Fig_adv_scheme.pdf}94 \includegraphics[width=0.9\textwidth]{Fig_adv_scheme} 94 95 \caption{ \label{Fig_adv_scheme} 95 96 Schematic representation of some ways used to evaluate the tracer value … … 113 114 boundary condition depends on the type of sea surface chosen: 114 115 \begin{description} 115 \item [linear free surface:] the first level thickness is constant in time:116 \item [linear free surface:] (\np{ln\_linssh}=true) the first level thickness is constant in time: 116 117 the vertical boundary condition is applied at the fixed surface $z=0$ 117 118 rather than on the moving surface $z=\eta$. There is a non-zero advective … … 119 120 $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$ 120 121 the product of surface velocity (at $z=0$) by the first level tracer value. 121 \item [non-linear free surface:] (\ key{vvl} is defined)122 \item [non-linear free surface:] (\np{ln\_linssh}=false) 122 123 convergence/divergence in the first ocean level moves the free surface 123 124 up/down. There is no tracer advection through it so that the advective … … 125 126 \end{description} 126 127 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-conservative128 Global conservation is obtained in non-linear free surface case, 129 but \textit{not} in the linear free surface case. Nevertheless, in the latter case, 130 it is achieved to a good approximation since the non-conservative 130 131 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}). 132 height, two quantities that are not correlated \citep{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}. 133 133 134 134 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 141 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 135 is the centred (\textit{now}) \textit{effective} ocean velocity, $i.e.$ the \textit{eulerian} velocity 136 (see Chap.~\ref{DYN}) plus the eddy induced velocity (\textit{eiv}) 137 and/or the mixed layer eddy induced velocity (\textit{eiv}) 138 when those parameterisations are used (see Chap.~\ref{LDF}). 139 140 Several tracer advection scheme are proposed, namely 141 a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), 142 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), 143 a Monotone Upstream Scheme for Conservative Laws scheme (MUSCL), 144 a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), and 145 a Quadratic Upstream Interpolation for Convective Kinematics with 146 Estimated Streaming Terms scheme (QUICKEST). 147 The choice is made in the \textit{\ngn{namtra\_adv}} namelist, by 148 setting to \textit{true} one of the logicals \textit{ln\_traadv\_xxx}. 149 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, 150 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 151 By default ($i.e.$ in the reference namelist, \ngn{namelist\_ref}), all the logicals 152 are set to \textit{false}. If the user does not select an advection scheme 153 in the configuration namelist (\ngn{namelist\_cfg}), the tracers will \textit{not} be advected ! 154 155 Details of the advection schemes are given below. The choosing an advection scheme 144 156 is a complex matter which depends on the model physics, model resolution, 145 type of tracer, as well as the issue of numerical cost. 146 147 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 157 type of tracer, as well as the issue of numerical cost. In particular, we note that 158 (1) CEN and FCT schemes require an explicit diffusion operator 159 while the other schemes are diffusive enough so that they do not necessarily need additional diffusion ; 160 (2) CEN and UBS are not \textit{positive} schemes 152 161 \footnote{negative values can appear in an initially strictly positive tracer field 153 162 which is advected} … … 163 172 164 173 % ------------------------------------------------------------------------------------------------------------- 174 % 2nd and 4th order centred schemes 175 % ------------------------------------------------------------------------------------------------------------- 176 \subsection [Centred schemes (CEN) (\np{ln\_traadv\_cen})] 177 {Centred schemes (CEN) (\np{ln\_traadv\_cen}=true)} 178 \label{TRA_adv_cen} 179 165 180 % 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 172 evaluated as the mean of the two neighbouring $T$-point values. 181 182 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}~=~\textit{true}. 183 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) 184 and vertical direction by setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$. 185 CEN implementation can be found in the \mdl{traadv\_cen} module. 186 187 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points 188 is evaluated as the mean of the two neighbouring $T$-point values. 173 189 For example, in the $i$-direction : 174 190 \begin{equation} \label{Eq_tra_adv_cen2} … … 176 192 \end{equation} 177 193 178 The schemeis non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$194 CEN2 is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ 179 195 but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously 180 196 noisy and must be used in conjunction with an explicit diffusion operator to 181 197 produce a sensible solution. The associated time-stepping is performed using 182 198 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 199 (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. 200 201 Note that using the CEN2, the overall tracer advection is of second 195 202 order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) 196 have this order of accuracy. \gmcomment{Note also that ... blah, blah} 197 198 % ------------------------------------------------------------------------------------------------------------- 203 have this order of accuracy. 204 199 205 % 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: 206 207 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as 208 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 209 For example, in the $i$-direction: 208 210 \begin{equation} \label{Eq_tra_adv_cen4} 209 211 \tau _u^{cen4} 210 212 =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 211 213 \end{equation} 212 213 Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme 214 In the vertical direction (\np{nn\_cen\_v}=$4$), a $4^{th}$ COMPACT interpolation 215 has been prefered \citep{Demange_PhD2014}. 216 In the COMPACT scheme, both the field and its derivative are interpolated, 217 which leads, after a matrix inversion, spectral characteristics 218 similar to schemes of higher order \citep{Lele_JCP1992}. 219 220 221 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme 214 222 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. 223 advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. 224 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature 225 is usually associated with the scheme presented here. 226 Introducing a \textit{true} $4^{th}$ order advection scheme is feasible but, 227 for consistency reasons, it requires changes in the discretisation of the tracer 228 advection together with changes in the continuity equation, 229 and the momentum advection and pressure terms. 221 230 222 231 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 also226 performed using a leapfrog scheme in conjunction with an Asselin time-filter,227 so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.228 229 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an230 a dditional hypothesis must be made to evaluate $\tau _u^{cen4}$. This231 hypothesis usually reduces the order of the scheme. Here we choose to set232 the gradient of $T$ across the boundary to zero. Alternative conditions can be233 specified, such as a reduction to a second order scheme for these near boundary234 grid points.235 236 % ------------------------------------------------------------------------------------------------------------- 237 % TVDscheme238 % ------------------------------------------------------------------------------------------------------------- 239 \subsection [ Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})]240 { Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=true)}232 it is not non-diffusive, $i.e.$ the global variance of a tracer is not preserved using CEN4. 233 Furthermore, it must be used in conjunction with an explicit diffusion operator 234 to produce a sensible solution. 235 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction 236 with an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer. 237 238 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), 239 an additional hypothesis must be made to evaluate $\tau _u^{cen4}$. 240 This hypothesis usually reduces the order of the scheme. 241 Here we choose to set the gradient of $T$ across the boundary to zero. 242 Alternative conditions can be specified, such as a reduction to a second order scheme 243 for these near boundary grid points. 244 245 % ------------------------------------------------------------------------------------------------------------- 246 % FCT scheme 247 % ------------------------------------------------------------------------------------------------------------- 248 \subsection [Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct})] 249 {Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct}=true)} 241 250 \label{TRA_adv_tvd} 242 251 243 In the Total Variance Dissipation (TVD) formulation, the tracer at velocity 244 points is evaluated using a combination of an upstream and a centred scheme. 245 For example, in the $i$-direction : 246 \begin{equation} \label{Eq_tra_adv_tvd} 252 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}~=~\textit{true}. 253 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) 254 and vertical direction by setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$. 255 FCT implementation can be found in the \mdl{traadv\_fct} module. 256 257 In FCT formulation, the tracer at velocity points is evaluated using a combination of 258 an upstream and a centred scheme. For example, in the $i$-direction : 259 \begin{equation} \label{Eq_tra_adv_fct} 247 260 \begin{split} 248 261 \tau _u^{ups}&= \begin{cases} … … 251 264 \end{cases} \\ 252 265 \\ 253 \tau _u^{ tvd}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen2} -\tau _u^{ups} } \right)266 \tau _u^{fct}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen} -\tau _u^{ups} } \right) 254 267 \end{split} 255 268 \end{equation} 256 269 where $c_u$ is a flux limiter function taking values between 0 and 1. 270 The FCT order is the one of the centred scheme used ($i.e.$ it depends on the setting of 271 \np{nn\_fct\_h} and \np{nn\_fct\_v}. 257 272 There exist many ways to define $c_u$, each corresponding to a different 258 total variance decreasing scheme. The one chosen in \NEMO is described in 259 \citet{Zalesak_JCP79}. $c_u$ only departs from $1$ when the advective term 260 produces a local extremum in the tracer field. The resulting scheme is quite 261 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. 265 266 For stability reasons (see \S\ref{STP}), 267 $\tau _u^{cen2}$ 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 of 269 the scheme is time stepped with a leap-frog scheme while a forward scheme is 270 used for the diffusive part. 273 FCT scheme. The one chosen in \NEMO is described in \citet{Zalesak_JCP79}. 274 $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. 275 The resulting scheme is quite expensive but \emph{positive}. 276 It can be used on both active and passive tracers. 277 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}. 278 279 An additional option has been added controlled by \np{nn\_fct\_zts}. By setting this integer to 280 a value larger than zero, a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, 281 but on the latter, a split-explicit time stepping is used, with a number of sub-timestep equals 282 to \np{nn\_fct\_zts}. This option can be useful when the size of the timestep is limited 283 by vertical advection \citep{Lemarie_OM2015}. Note that in this case, a similar split-explicit 284 time stepping should be used on vertical advection of momentum to insure a better stability 285 (see \S\ref{DYN_zad}). 286 287 For stability reasons (see \S\ref{STP}), $\tau _u^{cen}$ is evaluated in (\ref{Eq_tra_adv_fct}) 288 using the \textit{now} tracer while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, 289 the advective part of the scheme is time stepped with a leap-frog scheme 290 while a forward scheme is used for the diffusive part. 271 291 272 292 % ------------------------------------------------------------------------------------------------------------- 273 293 % MUSCL scheme 274 294 % ------------------------------------------------------------------------------------------------------------- 275 \subsection[MUSCL scheme (\np{ln\_traadv\_muscl})] 276 {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_muscl}=T)} 277 \label{TRA_adv_muscl} 278 279 The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been 280 implemented by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points 295 \subsection[MUSCL scheme (\np{ln\_traadv\_mus})] 296 {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_mus}=T)} 297 \label{TRA_adv_mus} 298 299 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}~=~\textit{true}. 300 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 301 302 MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points 281 303 is evaluated assuming a linear tracer variation between two $T$-points 282 304 (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction : 283 \begin{equation} \label{Eq_tra_adv_mus cl}305 \begin{equation} \label{Eq_tra_adv_mus} 284 306 \tau _u^{mus} = \left\{ \begin{aligned} 285 307 &\tau _i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) … … 296 318 297 319 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. 320 directed toward land, an upstream flux is used. This choice ensure 321 the \textit{positive} character of the scheme. 322 In addition, fluxes round a grid-point where a runoff is applied can optionally be 323 computed using upstream fluxes (\np{ln\_mus\_ups}~=~\textit{true}). 304 324 305 325 % ------------------------------------------------------------------------------------------------------------- … … 310 330 \label{TRA_adv_ubs} 311 331 312 The UBS advection scheme is an upstream-biased third order scheme based on 313 an upstream-biased parabolic interpolation. It is also known as the Cell 314 Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective 315 Kinematics). For example, in the $i$-direction : 332 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}~=~\textit{true}. 333 UBS implementation can be found in the \mdl{traadv\_mus} module. 334 335 The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme 336 (Quadratic Upstream Interpolation for Convective Kinematics). It is an upstream-biased 337 third order scheme based on an upstream-biased parabolic interpolation. 338 For example, in the $i$-direction : 316 339 \begin{equation} \label{Eq_tra_adv_ubs} 317 340 \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ … … 324 347 325 348 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}.349 error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of 350 the advection scheme is similar to that reported in \cite{Farrow1995}. 328 351 It is a relatively good compromise between accuracy and smoothness. 329 It is not a \emph{positive} scheme, meaning that false extrema are permitted,352 Nevertheless the scheme is not \emph{positive}, meaning that false extrema are permitted, 330 353 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.354 or fourth order method. therefore it is not recommended that it should be 355 applied to a passive tracer that requires positivity. 333 356 334 357 The intrinsic diffusion of UBS makes its use risky in the vertical direction 335 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.358 where the control of artificial diapycnal fluxes is of paramount importance \citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}. 359 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme 360 or a $4^th$ order COMPACT scheme (\np{nn\_cen\_v}=2 or 4). 338 361 339 362 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), is363 the first term in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order 364 centred scheme) is evaluated using the \textit{now} tracer (centred in time) 365 while the second term (which is the diffusive part of the scheme), is 343 366 evaluated using the \textit{before} tracer (forward in time). 344 367 This choice is discussed by \citet{Webb_al_JAOT98} in the context of the … … 350 373 substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 351 374 352 Four different options are possible for the vertical 353 component used in the UBS scheme. $\tau _w^{ubs}$ can be evaluated 354 using either \textit{(a)} a centred $2^{nd}$ order scheme, or \textit{(b)} 355 a TVD scheme, or \textit{(c)} an interpolation based on conservative 356 parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} 357 implementation of UBS in ROMS, or \textit{(d)} a UBS. The $3^{rd}$ case 358 has dispersion properties similar to an eighth-order accurate conventional scheme. 359 The current reference version uses method b) 360 361 Note that : 362 363 (1) When a high vertical resolution $O(1m)$ is used, the model stability can 364 be controlled by vertical advection (not vertical diffusion which is usually 365 solved using an implicit scheme). Computer time can be saved by using a 366 time-splitting technique on vertical advection. Such a technique has been 367 implemented and validated in ORCA05 with 301 levels. It is not available 368 in the current reference version. 369 370 (2) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: 375 Note that it is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: 371 376 \begin{equation} \label{Eq_traadv_ubs2} 372 377 \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ … … 390 395 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 391 396 coefficient which is simply proportional to the velocity: 392 $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v3.4 still uses 393 \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. 394 %%% 395 \gmcomment{the change in UBS scheme has to be done} 396 %%% 397 $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO uses 398 the computationally more efficient formulation \eqref{Eq_tra_adv_ubs}. 397 399 398 400 % ------------------------------------------------------------------------------------------------------------- … … 405 407 The Quadratic Upstream Interpolation for Convective Kinematics with 406 408 Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} 407 is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST 409 is used when \np{ln\_traadv\_qck}~=~\textit{true}. 410 QUICKEST implementation can be found in the \mdl{traadv\_qck} module. 411 412 QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST 408 413 limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray 409 414 (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. … … 413 418 direction where the control of artificial diapycnal fluxes is of paramount importance. 414 419 Therefore the vertical flux is evaluated using the CEN2 scheme. 415 This no longer guarantees the positivity of the scheme. The use of TVD in the vertical 416 direction (as for the UBS case) should be implemented to restore this property. 417 418 419 % ------------------------------------------------------------------------------------------------------------- 420 % PPM scheme 421 % ------------------------------------------------------------------------------------------------------------- 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 associated 429 with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented 430 in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference 431 version 3.3. 420 This no longer guarantees the positivity of the scheme. 421 The use of FCT in the vertical direction (as for the UBS case) should be implemented 422 to restore this property. 423 424 %%%gmcomment : Cross term are missing in the current implementation.... 425 432 426 433 427 % ================================================================ … … 441 435 %------------------------------------------------------------------------------------------------------------- 442 436 443 Options are defined through the \ngn{namtra\_ldf} namelist variables. 444 The options available for lateral diffusion are a laplacian (rotated or not) 445 or a biharmonic operator, the latter being more scale-selective (more 446 diffusive at small scales). The specification of eddy diffusivity 447 coefficients (either constant or variable in space and time) as well as the 448 computation of the slope along which the operators act, are performed in the 449 \mdl{ldftra} and \mdl{ldfslp} modules, respectively. This is described in Chap.~\ref{LDF}. 437 Options are defined through the \ngn{namtra\_ldf} namelist variables. 438 They are regrouped in four items, allowing to specify 439 $(i)$ the type of operator used (none, laplacian, bilaplacian), 440 $(ii)$ the direction along which the operator acts (iso-level, horizontal, iso-neutral), 441 $(iii)$ some specific options related to the rotated operators ($i.e.$ non-iso-level operator), and 442 $(iv)$ the specification of eddy diffusivity coefficient (either constant or variable in space and time). 443 Item $(iv)$ will be described in Chap.\ref{LDF} . 444 The direction along which the operators act is defined through the slope between this direction and the iso-level surfaces. 445 The slope is computed in the \mdl{ldfslp} module and will also be described in Chap.~\ref{LDF}. 446 450 447 The lateral diffusion of tracers is evaluated using a forward scheme, 451 448 $i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, 452 except for the pure vertical component that appears when a rotation tensor 453 is used. This latter term is solved implicitly together with the 454 vertical diffusion term (see \S\ref{STP}). 455 456 % ------------------------------------------------------------------------------------------------------------- 457 % Iso-level laplacian operator 458 % ------------------------------------------------------------------------------------------------------------- 459 \subsection [Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap})] 460 {Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap}=true) } 461 \label{TRA_ldf_lap} 462 463 A laplacian diffusion operator ($i.e.$ a harmonic operator) acting along the model 464 surfaces is given by: 449 except for the pure vertical component that appears when a rotation tensor is used. 450 This latter component is solved implicitly together with the vertical diffusion term (see \S\ref{STP}). 451 When \np{ln\_traldf\_msc}~=~\textit{true}, a Method of Stabilizing Correction is used in which 452 the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}. 453 454 % ------------------------------------------------------------------------------------------------------------- 455 % Type of operator 456 % ------------------------------------------------------------------------------------------------------------- 457 \subsection [Type of operator (\np{ln\_traldf\_NONE}, \np{ln\_traldf\_lap}, \np{ln\_traldf\_blp})] 458 {Type of operator (\np{ln\_traldf\_NONE}, \np{ln\_traldf\_lap}, or \np{ln\_traldf\_blp} = true) } 459 \label{TRA_ldf_op} 460 461 Three operator options are proposed and, one and only one of them must be selected: 462 \begin{description} 463 \item [\np{ln\_traldf\_NONE}] = true : no operator selected, the lateral diffusive tendency will not be 464 applied to the tracer equation. This option can be used when the selected advection scheme 465 is diffusive enough (MUSCL scheme for example). 466 \item [ \np{ln\_traldf\_lap}] = true : a laplacian operator is selected. This harmonic operator 467 takes the following expression: $\mathpzc{L}(T)=\nabla \cdot A_{ht}\;\nabla T $, 468 where the gradient operates along the selected direction (see \S\ref{TRA_ldf_dir}), 469 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see Chap.~\ref{LDF}). 470 \item [\np{ln\_traldf\_blp}] = true : a bilaplacian operator is selected. This biharmonic operator 471 takes the following expression: 472 $\mathpzc{B}=- \mathpzc{L}\left(\mathpzc{L}(T) \right) = -\nabla \cdot b\nabla \left( {\nabla \cdot b\nabla T} \right)$ 473 where the gradient operats along the selected direction, 474 and $b^2=B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see Chap.~\ref{LDF}). 475 In the code, the bilaplacian operator is obtained by calling the laplacian twice. 476 \end{description} 477 478 Both laplacian and bilaplacian operators ensure the total tracer variance decrease. 479 Their primary role is to provide strong dissipation at the smallest scale supported 480 by the grid while minimizing the impact on the larger scale features. 481 The main difference between the two operators is the scale selectiveness. 482 The bilaplacian damping time ($i.e.$ its spin down time) scales like $\lambda^{-4}$ 483 for disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones), 484 whereas the laplacian damping time scales only like $\lambda^{-2}$. 485 486 487 % ------------------------------------------------------------------------------------------------------------- 488 % Direction of action 489 % ------------------------------------------------------------------------------------------------------------- 490 \subsection [Direction of action (\np{ln\_traldf\_lev}, \np{ln\_traldf\_hor}, \np{ln\_traldf\_iso}, \np{ln\_traldf\_triad})] 491 {Direction of action (\np{ln\_traldf\_lev}, \textit{...\_hor}, \textit{...\_iso}, or \textit{...\_triad} = true) } 492 \label{TRA_ldf_dir} 493 494 The choice of a direction of action determines the form of operator used. 495 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane 496 when iso-level option is used (\np{ln\_traldf\_lev}~=~\textit{true}) 497 or when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{z}-coordinate 498 (\np{ln\_traldf\_hor} and \np{ln\_zco} equal \textit{true}). 499 The associated code can be found in the \mdl{traldf\_lap\_blp} module. 500 The operator is a rotated (re-entrant) laplacian when the direction along which it acts 501 does not coincide with the iso-level surfaces, 502 that is when standard or triad iso-neutral option is used (\np{ln\_traldf\_iso} or 503 \np{ln\_traldf\_triad} equals \textit{true}, see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), 504 or when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{s}-coordinate 505 (\np{ln\_traldf\_hor} and \np{ln\_sco} equal \textit{true}) 506 \footnote{In this case, the standard iso-neutral operator will be automatically selected}. 507 In that case, a rotation is applied to the gradient(s) that appears in the operator 508 so that diffusive fluxes acts on the three spatial direction. 509 510 The resulting discret form of the three operators (one iso-level and two rotated one) 511 is given in the next two sub-sections. 512 513 514 % ------------------------------------------------------------------------------------------------------------- 515 % iso-level operator 516 % ------------------------------------------------------------------------------------------------------------- 517 \subsection [Iso-level (bi-)laplacian operator ( \np{ln\_traldf\_iso})] 518 {Iso-level (bi-)laplacian operator ( \np{ln\_traldf\_iso}) } 519 \label{TRA_ldf_lev} 520 521 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 465 522 \begin{equation} \label{Eq_tra_ldf_lap} 466 D_ T^{lT} =\frac{1}{b_tT} \left( \;523 D_t^{lT} =\frac{1}{b_t} \left( \; 467 524 \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] 468 525 + \delta _{j}\left[ A_v^{lT} \; \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right] \;\right) 469 526 \end{equation} 470 where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 471 It is implemented in the \mdl{traadv\_lap} module. 472 473 This lateral operator is computed in \mdl{traldf\_lap}. It is a \emph{horizontal} 474 operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate with 475 or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 476 It is thus used when, in addition to \np{ln\_traldf\_lap}=true, we have 477 \np{ln\_traldf\_level}=true or \np{ln\_traldf\_hor}=\np{ln\_zco}=true. 527 where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells 528 and where zero diffusive fluxes is assumed across solid boundaries, 529 first (and third in bilaplacian case) horizontal tracer derivative are masked. 530 It is implemented in the \rou{traldf\_lap} subroutine found in the \mdl{traldf\_lap} module. 531 The module also contains \rou{traldf\_blp}, the subroutine calling twice \rou{traldf\_lap} 532 in order to compute the iso-level bilaplacian operator. 533 534 It is a \emph{horizontal} operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate 535 with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 536 It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}~=~\textit{true}, 537 we have \np{ln\_traldf\_lev}~=~\textit{true} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}~=~\textit{true}. 478 538 In both cases, it significantly contributes to diapycnal mixing. 479 It is therefore n ot recommended.539 It is therefore never recommended, even when using it in the bilaplacian case. 480 540 481 541 Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally … … 485 545 described in \S\ref{TRA_zpshde}. 486 546 487 % ------------------------------------------------------------------------------------------------------------- 488 % Rotated laplacian operator 489 % ------------------------------------------------------------------------------------------------------------- 490 \subsection [Rotated laplacian operator (iso) (\np{ln\_traldf\_lap})] 491 {Rotated laplacian operator (iso) (\np{ln\_traldf\_lap}=true)} 547 548 % ------------------------------------------------------------------------------------------------------------- 549 % Rotated laplacian operator 550 % ------------------------------------------------------------------------------------------------------------- 551 \subsection [Standard and triad rotated (bi-)laplacian operator (\mdl{traldf\_iso}, \mdl{traldf\_triad})] 552 {Standard and triad (bi-)laplacian operator (\mdl{traldf\_iso}, \mdl{traldf\_triad}))} 553 \label{TRA_ldf_iso_triad} 554 555 %&& Standard rotated (bi-)laplacian operator 556 %&& ---------------------------------------------- 557 \subsubsection [Standard rotated (bi-)laplacian operator (\mdl{traldf\_iso})] 558 {Standard rotated (bi-)laplacian operator (\mdl{traldf\_iso})} 492 559 \label{TRA_ldf_iso} 493 494 If the Griffies trad scheme is not employed 495 (\np{ln\_traldf\_grif}=true; see App.\ref{sec:triad}) the general form of the second order lateral tracer subgrid scale physics 496 (\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and 497 $s$-coordinates: 560 The general form of the second order lateral tracer subgrid scale physics 561 (\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates: 498 562 \begin{equation} \label{Eq_tra_ldf_iso} 499 563 \begin{split} … … 537 601 of the tracer variance. Nevertheless the treatment performed on the slopes 538 602 (see \S\ref{LDF}) allows the model to run safely without any additional 539 background horizontal diffusion \citep{Guilyardi_al_CD01}. An alternative scheme 540 developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases 603 background horizontal diffusion \citep{Guilyardi_al_CD01}. 604 605 Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal derivatives 606 at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific treatment. 607 They are calculated in module zpshde, described in \S\ref{TRA_zpshde}. 608 609 %&& Triad rotated (bi-)laplacian operator 610 %&& ------------------------------------------- 611 \subsubsection [Triad rotated (bi-)laplacian operator (\np{ln\_traldf\_triad})] 612 {Triad rotated (bi-)laplacian operator (\np{ln\_traldf\_triad})} 613 \label{TRA_ldf_triad} 614 615 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}=true ; see App.\ref{sec:triad}) 616 617 An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases 541 618 is also available in \NEMO (\np{ln\_traldf\_grif}=true). A complete description of 542 619 the algorithm is given in App.\ref{sec:triad}. 543 544 Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal545 derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific546 treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.547 548 % -------------------------------------------------------------------------------------------------------------549 % Iso-level bilaplacian operator550 % -------------------------------------------------------------------------------------------------------------551 \subsection [Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap})]552 {Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap}=true)}553 \label{TRA_ldf_bilap}554 620 555 621 The lateral fourth order bilaplacian operator on tracers is obtained by 556 622 applying (\ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption 557 623 on boundary conditions: both first and third derivative terms normal to the 558 coast are set to zero. It is used when, in addition to \np{ln\_traldf\_bilap}=true, 559 we have \np{ln\_traldf\_level}=true, or both \np{ln\_traldf\_hor}=true and 560 \np{ln\_zco}=false. In both cases, it can contribute diapycnal mixing, 561 although less than in the laplacian case. It is therefore not recommended. 562 563 Note that in the code, the bilaplacian routine does not call the laplacian 564 routine twice but is rather a separate routine that can be found in the 565 \mdl{traldf\_bilap} module. This is due to the fact that we introduce the 566 eddy diffusivity coefficient, A, in the operator as: 567 $\nabla \cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$, 568 instead of 569 $-\nabla \cdot a\nabla \left( {\nabla \cdot a\nabla T} \right)$ 570 where $a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations 571 ensure the total variance decrease, but the former requires a larger 572 number of code-lines. 573 574 % ------------------------------------------------------------------------------------------------------------- 575 % Rotated bilaplacian operator 576 % ------------------------------------------------------------------------------------------------------------- 577 \subsection [Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap})] 578 {Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap}=true)} 579 \label{TRA_ldf_bilapg} 624 coast are set to zero. 580 625 581 626 The lateral fourth order operator formulation on tracers is obtained by 582 627 applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption 583 628 on boundary conditions: first and third derivative terms normal to the 584 coast, normal to the bottom and normal to the surface are set to zero. It can be found in the 585 \mdl{traldf\_bilapg}. 586 587 It is used when, in addition to \np{ln\_traldf\_bilap}=true, we have 588 \np{ln\_traldf\_iso}= .true, or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. 589 This rotated bilaplacian operator has never been seriously 590 tested. There are no guarantees that it is either free of bugs or correctly formulated. 591 Moreover, the stability range of such an operator will be probably quite 592 narrow, requiring a significantly smaller time-step than the one used with an 593 unrotated operator. 629 coast, normal to the bottom and normal to the surface are set to zero. 630 631 %&& Option for the rotated operators 632 %&& ---------------------------------------------- 633 \subsubsection [Option for the rotated operators] 634 {Option for the rotated operators} 635 \label{TRA_ldf_options} 636 637 \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) 638 639 \np{rn\_slpmax} = slope limit (both operators) 640 641 \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 642 643 \np{rn\_sw\_triad} =1 switching triad ; =0 all 4 triads used (triad only) 644 645 \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 594 646 595 647 % ================================================================ … … 603 655 %-------------------------------------------------------------------------------------------------------------- 604 656 605 Options are defined through the 657 Options are defined through the \ngn{namzdf} namelist variables. 606 658 The formulation of the vertical subgrid scale tracer physics is the same 607 659 for all the vertical coordinates, and is based on a laplacian operator. … … 661 713 the thickness of the top model layer. 662 714 663 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components ($i.e.$ atmosphere, sea-ice, land), 664 the change in the heat and salt content of the surface layer of the ocean is due both 665 to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 666 and to the heat and salt content of the mass exchange. 667 \sgacomment{ the following does not apply to the release to which this documentation is 668 attached and so should not be included .... 669 In a forthcoming release, these two parts, computed in the surface module (SBC), will be included directly 670 in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux. 671 The specification of these fluxes is further detailed in the SBC chapter (see \S\ref{SBC}). 672 This change will provide a forcing formulation which is the same for any tracer (including temperature and salinity). 673 674 In the current version, the situation is a little bit more complicated. } 715 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 716 ($i.e.$ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer 717 of the ocean is due both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) 718 and to the heat and salt content of the mass exchange. They are both included directly in $Q_{ns}$, 719 the surface heat flux, and $F_{salt}$, the surface salt flux (see \S\ref{SBC} for further details). 720 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 675 721 676 722 The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following … … 679 725 $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 680 726 (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that 681 penetrates into the water column, see \S\ref{TRA_qsr}) 682 683 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) 684 685 $\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange 686 687 $\bullet$ \textit{rnf}, the mass flux associated with runoff (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 688 689 The $\textit{emp}_S$ field is not simply the budget of evaporation-precipitation+freezing-melting because 690 the sea-ice is not currently embedded in the ocean but levitates above it. There is no mass 691 exchanged between the sea-ice and the ocean. Instead we only take into account the salt 692 flux associated with the non-zero salinity of sea-ice, and the concentration/dilution effect 693 due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into 694 an equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, 695 the surface boundary condition on temperature and salinity is applied as follows: 696 697 In the nonlinear free surface case (\key{vvl} is defined): 727 penetrates into the water column, see \S\ref{TRA_qsr}) plus the heat content associated with 728 of the mass exchange with the atmosphere and lands. 729 730 $\bullet$ $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 731 732 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) 733 and possibly with the sea-ice and ice-shelves. 734 735 $\bullet$ \textit{rnf}, the mass flux associated with runoff 736 (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 737 738 $\bullet$ \textit{fwfisf}, the mass flux associated with ice shelf melt, 739 (see \S\ref{SBC_isf} for further details on how the ice shelf melt is computed and applied). 740 741 The surface boundary condition on temperature and salinity is applied as follows: 698 742 \begin{equation} \label{Eq_tra_sbc} 743 \begin{aligned} 744 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ 745 & F^S =\frac{ 1 }{\rho _o \, \left. e_{3t} \right|_{k=1} } &\overline{ \textit{sfx} }^t & \\ 746 \end{aligned} 747 \end{equation} 748 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 749 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 750 divergence of odd and even time step (see \S\ref{STP}). 751 752 In the linear free surface case (\np{ln\_linssh}~=~\textit{true}), 753 an additional term has to be added on both temperature and salinity. 754 On temperature, this term remove the heat content associated with mass exchange 755 that has been added to $Q_{ns}$. On salinity, this term mimics the concentration/dilution effect that 756 would have resulted from a change in the volume of the first level. 757 The resulting surface boundary condition is applied as follows: 758 \begin{equation} \label{Eq_tra_sbc_lin} 699 759 \begin{aligned} 700 760 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } … … 702 762 % 703 763 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 704 &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1} \right) }^t & \\764 &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1} \right) }^t & \\ 705 765 \end{aligned} 706 766 \end{equation} 707 708 In the linear free surface case (\key{vvl} not defined): 709 \begin{equation} \label{Eq_tra_sbc_lin} 710 \begin{aligned} 711 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ 712 % 713 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 714 &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1} \right) }^t & \\ 715 \end{aligned} 716 \end{equation} 717 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 718 ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 719 divergence of odd and even time step (see \S\ref{STP}). 720 721 The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained 722 by assuming that the temperature of precipitation and evaporation are equal to 723 the ocean surface temperature and that their salinity is zero. Therefore, the heat content 724 of the \textit{emp} budget must be added to the temperature equation in the variable volume case, 725 while it does not appear in the constant volume case. Similarly, the \textit{emp} budget affects 726 the ocean surface salinity in the constant volume case (through the concentration dilution effect) 727 while it does not appears explicitly in the variable volume case since salinity change will be 728 induced by volume change. In both constant and variable volume cases, surface salinity 729 will change with ice-ocean salt flux and F/M flux (both contained in $\textit{emp}_S - \textit{emp}$) without mass exchanges. 730 731 Note that the concentration/dilution effect due to F/M is computed using 732 a constant ice salinity as well as a constant ocean salinity. 733 This approximation suppresses the correlation between \textit{SSS} 734 and F/M flux, allowing the ice-ocean salt exchanges to be conservative. 735 Indeed, if this approximation is not made, even if the F/M budget is zero 736 on average over the whole ocean domain and over the seasonal cycle, 737 the associated salt flux is not zero, since sea-surface salinity and F/M flux are 738 intrinsically correlated (high \textit{SSS} are found where freezing is 739 strong whilst low \textit{SSS} is usually associated with high melting areas). 740 741 Even using this approximation, an exact conservation of heat and salt content 742 is only achieved in the variable volume case. In the constant volume case, 743 there is a small imbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$. 744 Nevertheless, the salt content variation is quite small and will not induce 745 a long term drift as there is no physical reason for $(\partial_t\eta - \textit{emp})$ 746 and \textit{SSS} to be correlated \citep{Roullet_Madec_JGR00}. 747 Note that, while quite small, the imbalance in the constant volume case is larger 767 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 768 In the linear free surface case, there is a small imbalance. The imbalance is larger 748 769 than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 749 This is the reason why the modified filter is not applied in the constant volume case.770 This is the reason why the modified filter is not applied in the linear free surface case (see \S\ref{STP}). 750 771 751 772 % ------------------------------------------------------------------------------------------------------------- … … 821 842 ($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform 822 843 chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} 823 in \mdl{trc\_oce} module). Three types of chlorophyll can be chosen in the RGB formulation: 824 (1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) an observed 825 time varying chlorophyll (\np{nn\_chdta}=1) ; (3) simulated time varying chlorophyll 826 by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the latter case, the RGB 827 formulation is used to calculate both the phytoplankton light limitation in PISCES 828 or LOBSTER and the oceanic heating rate. 829 844 in \mdl{trc\_oce} module). Four types of chlorophyll can be chosen in the RGB formulation: 845 \begin{description} 846 \item[\np{nn\_chdta}=0] 847 a constant 0.05 g.Chl/L value everywhere ; 848 \item[\np{nn\_chdta}=1] 849 an observed time varying chlorophyll deduced from satellite surface ocean color measurement 850 spread uniformly in the vertical direction ; 851 \item[\np{nn\_chdta}=2] 852 same as previous case except that a vertical profile of chlorophyl is used. 853 Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value ; 854 \item[\np{ln\_qsr\_bio}=true] 855 simulated time varying chlorophyll by TOP biogeochemical model. 856 In this case, the RGB formulation is used to calculate both the phytoplankton 857 light limitation in PISCES or LOBSTER and the oceanic heating rate. 858 \end{description} 830 859 The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation 831 860 is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. … … 842 871 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 843 872 \begin{figure}[!t] \begin{center} 844 \includegraphics[width=1.0\textwidth]{ ./TexFiles/Figures/Fig_TRA_Irradiance.pdf}873 \includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance} 845 874 \caption{ \label{Fig_traqsr_irradiance} 846 875 Penetration profile of the downward solar irradiance calculated by four models. … … 859 888 \label{TRA_bbc} 860 889 %--------------------------------------------nambbc-------------------------------------------------------- 861 \namdisplay{nam tra_bbc}890 \namdisplay{nambbc} 862 891 %-------------------------------------------------------------------------------------------------------------- 863 892 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 864 893 \begin{figure}[!t] \begin{center} 865 \includegraphics[width=1.0\textwidth]{ ./TexFiles/Figures/Fig_TRA_geoth.pdf}894 \includegraphics[width=1.0\textwidth]{Fig_TRA_geoth} 866 895 \caption{ \label{Fig_geothermal} 867 896 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}. … … 973 1002 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 974 1003 \begin{figure}[!t] \begin{center} 975 \includegraphics[width=0.7\textwidth]{ ./TexFiles/Figures/Fig_BBL_adv.pdf}1004 \includegraphics[width=0.7\textwidth]{Fig_BBL_adv} 976 1005 \caption{ \label{Fig_bbl} 977 1006 Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is … … 1103 1132 \subsection[DMP\_TOOLS]{Generating resto.nc using DMP\_TOOLS} 1104 1133 1105 DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled and run on the same machine as the NEMO model. A mesh\_mask.nc file for the model configuration is required as an input. This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient. 1134 DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. 1135 Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled 1136 and run on the same machine as the NEMO model. A mesh\_mask.nc file for the model configuration is required as an input. 1137 This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. 1138 The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. 1139 The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient. 1106 1140 1107 1141 %--------------------------------------------nam_dmp_create------------------------------------------------- 1108 \nam display{nam_dmp_create}1142 \namtools{namelist_dmp} 1109 1143 %------------------------------------------------------------------------------------------------------- 1110 1144 1111 1145 \np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and should be the same as specified in \nl{namcfg}. The variable \nl{lzoom} is used to specify that the damping is being used as in case \textit{a} above to provide boundary conditions to a zoom configuration. In the case of the arctic or antarctic zoom configurations this includes some specific treatment. Otherwise damping is applied to the 6 grid points along the ocean boundaries. The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in the \nl{nam\_zoom\_dmp} name list. 1112 1146 1113 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations. \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea for the ORCA4, ORCA2 and ORCA05 configurations. If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference configurations with previous model versions. \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. This option only has an effect if \np{ln\_full\_field} is true. \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. Finally \np{ln\_custom} specifies that the custom module will be called. This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. 1114 1115 The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}. Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to the full values of a 10$^{\circ}$ latitud band. This is often used because of the short adjustment time scale in the equatorial region \citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. 1147 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations. 1148 \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. 1149 \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea 1150 for the ORCA4, ORCA2 and ORCA05 configurations. 1151 If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as 1152 a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference 1153 configurations with previous model versions. 1154 \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. 1155 This option only has an effect if \np{ln\_full\_field} is true. 1156 \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. 1157 Finally \np{ln\_custom} specifies that the custom module will be called. 1158 This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. 1159 1160 The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}. 1161 Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to 1162 the full values of a 10\deg latitud band. 1163 This is often used because of the short adjustment time scale in the equatorial region 1164 \citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a 1165 hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. 1116 1166 1117 1167 % ================================================================ … … 1167 1217 % Equation of State 1168 1218 % ------------------------------------------------------------------------------------------------------------- 1169 \subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)}1219 \subsection{Equation Of Seawater (\np{nn\_eos} = -1, 0, or 1)} 1170 1220 \label{TRA_eos} 1171 1221 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 $1,035~Kg/m^3$. 1222 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship 1223 linking seawater density, $\rho$, to a number of state variables, 1224 most typically temperature, salinity and pressure. 1225 Because density gradients control the pressure gradient force through the hydrostatic balance, 1226 the equation of state provides a fundamental bridge between the distribution of active tracers 1227 and the fluid dynamics. Nonlinearities of the EOS are of major importance, in particular 1228 influencing the circulation through determination of the static stability below the mixed layer, 1229 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{Roquet_JPO2015}. 1230 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983}) 1231 or TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real 1232 ocean circulation is attempted \citep{Roquet_JPO2015}. 1233 The use of TEOS-10 is highly recommended because 1234 \textit{(i)} it is the new official EOS, 1235 \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and 1236 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature 1237 and practical salinity for EOS-980, both variables being more suitable for use as model variables 1238 \citep{TEOS10, Graham_McDougall_JPO13}. 1239 EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. 1240 For process studies, it is often convenient to use an approximation of the EOS. To that purposed, 1241 a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available. 1242 1243 In the computer code, a density anomaly, $d_a= \rho / \rho_o - 1$, 1244 is computed, with $\rho_o$ a reference density. Called \textit{rau0} 1245 in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 1200 1246 This is a sensible choice for the reference density used in a Boussinesq ocean 1201 1247 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. 1248 density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. 1249 1250 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} 1251 which controls the EOS used (=-1 for TEOS10 ; =0 for EOS-80 ; =1 for S-EOS). 1252 \begin{description} 1253 1254 \item[\np{nn\_eos}$=-1$] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used. 1255 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 1256 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler 1257 and more computationally efficient expressions for their derived quantities 1258 which make them more adapted for use in ocean models. 1259 Note that a slightly higher precision polynomial form is now used replacement of the TEOS-10 1260 rational function approximation for hydrographic data analysis \citep{TEOS10}. 1261 A key point is that conservative state variables are used: 1262 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \degC, notation: $\Theta$). 1263 The pressure in decibars is approximated by the depth in meters. 1264 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. It is set to 1265 $C_p=3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{TEOS10}. 1266 1267 Choosing polyTEOS10-bsq implies that the state variables used by the model are 1268 $\Theta$ and $S_A$. In particular, the initial state deined by the user have to be given as 1269 \textit{Conservative} Temperature and \textit{Absolute} Salinity. 1270 In addition, setting \np{ln\_useCT} to \textit{true} convert the Conservative SST to potential SST 1271 prior to either computing the air-sea and ice-sea fluxes (forced mode) 1272 or sending the SST field to the atmosphere (coupled mode). 1273 1274 \item[\np{nn\_eos}$=0$] the polyEOS80-bsq equation of seawater is used. 1275 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized 1276 to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80 1277 and the ocean model are: 1278 the Practical Salinity ((unit: psu, notation: $S_p$)) and Potential Temperature (unit: $^{\circ}C$, notation: $\theta$). 1279 The pressure in decibars is approximated by the depth in meters. 1280 With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, 1281 salinity and pressure \citep{UNESCO1983}. Nevertheless, a severe assumption is made in order to 1282 have a heat content ($C_p T_p$) which is conserved by the model: $C_p$ is set to a constant 1283 value, the TEOS10 value. 1284 1285 \item[\np{nn\_eos}$=1$] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen, 1286 the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) 1287 (see also \citet{Roquet_JPO2015}). It provides a simplistic linear representation of both 1288 cabbeling and thermobaricity effects which is enough for a proper treatment of the EOS 1289 in theoretical studies \citep{Roquet_JPO2015}. 1209 1290 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} 1291 \textit{conservative} and \textit{potential} temperature, as well as between \textit{absolute} 1292 and \textit{practical} salinity. 1293 S-EOS takes the following expression: 1294 \begin{equation} \label{Eq_tra_S-EOS} 1215 1295 \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 1296 d_a(T,S,z) = ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a \\ 1297 & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a \\ 1298 & - \nu \; T_a \; S_a \; ) \; / \; \rho_o \\ 1299 with \ \ T_a = T-10 \; ; & \; S_a = S-35 \; ;\; \rho_o = 1026~Kg/m^3 1218 1300 \end{split} 1219 1301 \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. 1226 1227 % ------------------------------------------------------------------------------------------------------------- 1228 % Brunt-Vais\"{a}l\"{a} Frequency 1229 % ------------------------------------------------------------------------------------------------------------- 1230 \subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)} 1302 where the computer name of the coefficients as well as their standard value are given in \ref{Tab_SEOS}. 1303 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing 1304 the associated coefficients. 1305 Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS. 1306 setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS. 1307 Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 1308 1309 \end{description} 1310 1311 1312 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1313 \begin{table}[!tb] 1314 \begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} 1315 \hline 1316 coeff. & computer name & S-EOS & description \\ \hline 1317 $a_0$ & \np{rn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline 1318 $b_0$ & \np{rn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline 1319 $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline 1320 $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline 1321 $\nu$ & \np{rn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline 1322 $\mu_1$ & \np{rn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline 1323 $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1324 \end{tabular} 1325 \caption{ \label{Tab_SEOS} 1326 Standard value of S-EOS coefficients. } 1327 \end{center} 1328 \end{table} 1329 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1330 1331 1332 % ------------------------------------------------------------------------------------------------------------- 1333 % Brunt-V\"{a}is\"{a}l\"{a} Frequency 1334 % ------------------------------------------------------------------------------------------------------------- 1335 \subsection{Brunt-V\"{a}is\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)} 1231 1336 \label{TRA_bn2} 1232 1337 1233 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): 1338 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} 1339 frequency) is of paramount importance as determine the ocean stratification and 1340 is used in several ocean parameterisations (namely TKE, GLS, Richardson number dependent 1341 vertical diffusion, enhanced vertical diffusion, non-penetrative convection, tidal mixing 1342 parameterisation, iso-neutral diffusion). In particular, $N^2$ has to be computed at the local pressure 1343 (pressure in decibar being approximated by the depth in meters). The expression for $N^2$ 1344 is given by: 1244 1345 \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 1346 N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) 1260 1347 \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 1348 where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS, 1349 and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 1350 The coefficients are a polynomial function of temperature, salinity and depth which expression 1351 depends on the chosen EOS. They are computed through \textit{eos\_rab}, a \textsc{Fortran} 1352 function that can be found in \mdl{eosbn2}. 1280 1353 1281 1354 % ------------------------------------------------------------------------------------------------------------- … … 1298 1371 sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent 1299 1372 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 found1373 point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found 1301 1374 in \mdl{eosbn2}. 1375 1376 1377 % ------------------------------------------------------------------------------------------------------------- 1378 % Potential Energy 1379 % ------------------------------------------------------------------------------------------------------------- 1380 %\subsection{Potential Energy anomalies} 1381 %\label{TRA_bn2} 1382 1383 % =====>>>>> TO BE written 1384 % 1385 1302 1386 1303 1387 % ================================================================ … … 1308 1392 \label{TRA_zpshde} 1309 1393 1310 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"} 1311 1312 With partial bottom cells (\np{ln\_zps}=true), in general, tracers in horizontally 1313 adjacent cells live at different depths. Horizontal gradients of tracers are needed 1314 for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure 1315 gradient (\mdl{dynhpg} module) to be active. 1316 \gmcomment{STEVEN from gm : question: not sure of what -to be active- means} 1394 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, 1395 I've changed "derivative" to "difference" and "mean" to "average"} 1396 1397 With partial cells (\np{ln\_zps}=true) at bottom and top (\np{ln\_isfcav}=true), in general, 1398 tracers in horizontally adjacent cells live at different depths. 1399 Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) 1400 and the hydrostatic pressure gradient calculations (\mdl{dynhpg} module). 1401 The partial cell properties at the top (\np{ln\_isfcav}=true) are computed in the same way as for the bottom. 1402 So, only the bottom interpolation is explained below. 1403 1317 1404 Before taking horizontal gradients between the tracers next to the bottom, a linear 1318 1405 interpolation in the vertical is used to approximate the deeper tracer as if it actually … … 1323 1410 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1324 1411 \begin{figure}[!p] \begin{center} 1325 \includegraphics[width=0.9\textwidth]{ ./TexFiles/Figures/Partial_step_scheme.pdf}1412 \includegraphics[width=0.9\textwidth]{Partial_step_scheme} 1326 1413 \caption{ \label{Fig_Partial_step_scheme} 1327 1414 Discretisation of the horizontal difference and average of tracers in the $z$-partial … … 1390 1477 \gmcomment{gm : this last remark has to be done} 1391 1478 %%% 1479 \end{document}
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