Changeset 10502 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex
r10442 r10502 15 15 %STEVEN : is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below 16 16 17 %\newpage 18 19 Using the representation described in \autoref{chap:DOM}, 20 several semi-discrete space forms of the tracer equations are available depending on 21 the vertical coordinate used and on the physics used. 17 Using the representation described in \autoref{chap:DOM}, several semi -discrete space forms of 18 the tracer equations are available depending on the vertical coordinate used and on the physics used. 22 19 In all the equations presented here, the masking has been omitted for simplicity. 23 One must be aware that all the quantities are masked fields and 24 that each time a mean or difference operator is used, 25 the resulting field is multiplied by a mask. 20 One must be aware that all the quantities are masked fields and that each time a mean or 21 difference operator is used, the resulting field is multiplied by a mask. 26 22 27 23 The two active tracers are potential temperature and salinity. 28 24 Their prognostic equations can be summarized as follows: 29 25 \[ 30 \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}31 \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP})26 \text{NXT} = \text{ADV} + \text{LDF} + \text{ZDF} + \text{SBC} 27 + \{\text{QSR}, \text{BBC}, \text{BBL}, \text{DMP}\} 32 28 \] 33 29 … … 39 35 The terms QSR, BBC, BBL and DMP are optional. 40 36 The external forcings and parameterisations require complex inputs and complex calculations 41 (\eg bulk formulae, estimation of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 42 described in \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. 43 Note that \mdl{tranpc}, the non-penetrative convection module, although located in the NEMO/OPA/TRA directory as 44 it directly modifies the tracer fields, is described with the model vertical physics (ZDF) together with 37 (\eg bulk formulae, estimation of mixing coefficients) that are carried out in the SBC, 38 LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and 39 \autoref{chap:ZDF}, respectively. 40 Note that \mdl{tranpc}, the non-penetrative convection module, although located in 41 the \path{./src/OCE/TRA} directory as it directly modifies the tracer fields, 42 is described with the model vertical physics (ZDF) together with 45 43 other available parameterization of convection. 46 44 … … 50 48 51 49 The different options available to the user are managed by namelist logicals or CPP keys. 52 For each equation term 50 For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx}, 53 51 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 54 52 The CPP key (when it exists) is \key{traTTT}. 55 53 The equivalent code can be found in the \textit{traTTT} or \textit{traTTT\_xxx} module, 56 in the NEMO/OPA/TRAdirectory.54 in the \path{./src/OCE/TRA} directory. 57 55 58 56 The user has the option of extracting each tendency term on the RHS of the tracer equation for output 59 (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl} \forcode{= .true.}), as described in \autoref{chap:DIA}.57 (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}~\forcode{= .true.}), as described in \autoref{chap:DIA}. 60 58 61 59 % ================================================================ … … 75 73 \begin{equation} 76 74 \label{eq:tra_adv} 77 ADV_\tau =-\frac{1}{b_t} \left( 78 \;\delta_i \left[ e_{2u}\,e_{3u} \; u\; \tau_u \right] 79 +\delta_j \left[ e_{1v}\,e_{3v} \; v\; \tau_v \right] \; \right) 80 -\frac{1}{e_{3t}} \;\delta_k \left[ w\; \tau_w \right] 75 ADV_\tau = - \frac{1}{b_t} \Big( \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u] 76 + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big) 77 - \frac{1}{e_{3t}} \delta_k [w \; \tau_w] 81 78 \end{equation} 82 where $\tau$ is either T or S, and $b_t = e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells.79 where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells. 83 80 The flux form in \autoref{eq:tra_adv} implicitly requires the use of the continuity equation. 84 Indeed, it is obtained by using the following equality: 85 $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ which 86 results from the use of the continuity equation, $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ 87 (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, \ie \np{ln\_linssh}\forcode{ = .true.}). 81 Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 82 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, \ie \np{ln\_linssh}~\forcode{= .true.}). 88 84 Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that 89 85 it is consistent with the continuity equation in order to enforce the conservation properties of … … 94 90 \begin{figure}[!t] 95 91 \begin{center} 96 \includegraphics[ width=0.9\textwidth]{Fig_adv_scheme}92 \includegraphics[]{Fig_adv_scheme} 97 93 \caption{ 98 94 \protect\label{fig:adv_scheme} … … 120 116 since the normal velocity is zero there. 121 117 At the sea surface the boundary condition depends on the type of sea surface chosen: 118 122 119 \begin{description} 123 120 \item[linear free surface:] 124 (\np{ln\_linssh} \forcode{= .true.})121 (\np{ln\_linssh}~\forcode{= .true.}) 125 122 the first level thickness is constant in time: 126 the vertical boundary condition is applied at the fixed surface $z=0$ rather than on the moving surface $z=\eta$. 123 the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on 124 the moving surface $z = \eta$. 127 125 There is a non-zero advective flux which is set for all advection schemes as 128 $\ left. {\tau_w } \right|_{k=1/2} =T_{k=1} $,129 \ie the product of surface velocity (at $z=0$) bythe first level tracer value.126 $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie the product of surface velocity (at $z = 0$) by 127 the first level tracer value. 130 128 \item[non-linear free surface:] 131 (\np{ln\_linssh} \forcode{= .false.})129 (\np{ln\_linssh}~\forcode{= .false.}) 132 130 convergence/divergence in the first ocean level moves the free surface up/down. 133 131 There is no tracer advection through it so that the advective fluxes through the surface are also zero. 134 132 \end{description} 133 135 134 In all cases, this boundary condition retains local conservation of tracer. 136 135 Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case. 137 136 Nevertheless, in the latter case, it is achieved to a good approximation since 138 137 the non-conservative term is the product of the time derivative of the tracer and the free surface height, 139 two quantities that are not correlated \citep{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}. 140 141 The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco}) 142 is the centred (\textit{now}) \textit{effective} ocean velocity, 143 \ie the \textit{eulerian} velocity (see \autoref{chap:DYN}) plus 144 the eddy induced velocity (\textit{eiv}) and/or 145 the mixed layer eddy induced velocity (\textit{eiv}) when 146 those parameterisations are used (see \autoref{chap:LDF}). 138 two quantities that are not correlated \citep{Roullet_Madec_JGR00, Griffies_al_MWR01, Campin2004}. 139 140 The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco}) is 141 the centred (\textit{now}) \textit{effective} ocean velocity, \ie the \textit{eulerian} velocity 142 (see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or 143 the mixed layer eddy induced velocity (\textit{eiv}) when those parameterisations are used 144 (see \autoref{chap:LDF}). 147 145 148 146 Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), 149 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), 150 a Monotone Upstream Scheme for Conservative Laws scheme (MUSCL), 151 a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), 152 and a Quadratic Upstream Interpolation for Convective Kinematics with 153 Estimated Streaming Terms scheme (QUICKEST). 154 The choice is made in the \textit{\ngn{namtra\_adv}} namelist, 155 by setting to \forcode{.true.} one of the logicals \textit{ln\_traadv\_xxx}. 156 The corresponding code can be found in the \mdl{traadv\_xxx} module, 157 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 158 By default (\ie in the reference namelist, \ngn{namelist\_ref}), all the logicals are set to \forcode{.false.}. 159 If the user does not select an advection scheme in the configuration namelist (\ngn{namelist\_cfg}), 147 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for 148 Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), 149 and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST). 150 The choice is made in the \ngn{namtra\_adv} namelist, by setting to \forcode{.true.} one of 151 the logicals \textit{ln\_traadv\_xxx}. 152 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 153 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 154 By default (\ie in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}. 155 If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}), 160 156 the tracers will \textit{not} be advected! 161 157 … … 163 159 The choosing an advection scheme is a complex matter which depends on the model physics, model resolution, 164 160 type of tracer, as well as the issue of numerical cost. In particular, we note that 165 (1) CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that 166 they do not necessarily need additional diffusion; 167 (2) CEN and UBS are not \textit{positive} schemes 168 \footnote{negative values can appear in an initially strictly positive tracer field which is advected}, 169 implying that false extrema are permitted. 170 Their use is not recommended on passive tracers; 171 (3) It is recommended that the same advection-diffusion scheme is used on both active and passive tracers. 172 Indeed, if a source or sink of a passive tracer depends on an active one, 173 the difference of treatment of active and passive tracers can create very nice-looking frontal structures that 174 are pure numerical artefacts. 161 162 \begin{enumerate} 163 \item 164 CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that 165 they do not necessarily need additional diffusion; 166 \item 167 CEN and UBS are not \textit{positive} schemes 168 \footnote{negative values can appear in an initially strictly positive tracer field which is advected}, 169 implying that false extrema are permitted. 170 Their use is not recommended on passive tracers; 171 \item 172 It is recommended that the same advection-diffusion scheme is used on both active and passive tracers. 173 \end{enumerate} 174 175 Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and 176 passive tracers can create very nice-looking frontal structures that are pure numerical artefacts. 175 177 Nevertheless, most of our users set a different treatment on passive and active tracers, 176 178 that's the reason why this possibility is offered. 177 We strongly suggest them to perform a sensitivity experiment using a same treatment to 178 assess the robustness oftheir results.179 We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of 180 their results. 179 181 180 182 % ------------------------------------------------------------------------------------------------------------- 181 183 % 2nd and 4th order centred schemes 182 184 % ------------------------------------------------------------------------------------------------------------- 183 \subsection{CEN: Centred scheme (\protect\np{ln\_traadv\_cen} \forcode{= .true.})}185 \subsection{CEN: Centred scheme (\protect\np{ln\_traadv\_cen}~\forcode{= .true.})} 184 186 \label{subsec:TRA_adv_cen} 185 187 186 188 % 2nd order centred scheme 187 189 188 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen} \forcode{= .true.}.190 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}~\forcode{= .true.}. 189 191 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 190 192 setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$. … … 196 198 \begin{equation} 197 199 \label{eq:tra_adv_cen2} 198 \tau_u^{cen2} = \overline T ^{i+1/2}200 \tau_u^{cen2} = \overline T ^{i + 1/2} 199 201 \end{equation} 200 202 201 CEN2 is non diffusive (\ie it conserves the tracer variance, $\tau^2 )$but dispersive203 CEN2 is non diffusive (\ie it conserves the tracer variance, $\tau^2$) but dispersive 202 204 (\ie it may create false extrema). 203 205 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 204 206 produce a sensible solution. 205 207 The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 206 so $T$ in (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value. 208 so $T$ in (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value. 207 209 208 210 Note that using the CEN2, the overall tracer advection is of second order accuracy since … … 216 218 \begin{equation} 217 219 \label{eq:tra_adv_cen4} 218 \tau_u^{cen4} = \overline{ T - \frac{1}{6}\,\delta_i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2}220 \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2} 219 221 \end{equation} 220 In the vertical direction (\np{nn\_cen\_v} \forcode{= 4}),222 In the vertical direction (\np{nn\_cen\_v}~\forcode{= 4}), 221 223 a $4^{th}$ COMPACT interpolation has been prefered \citep{Demange_PhD2014}. 222 224 In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion, 223 spectral characteristics similar to schemes of higher order \citep{Lele_JCP1992}. 224 225 spectral characteristics similar to schemes of higher order \citep{Lele_JCP1992}. 225 226 226 227 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but … … 249 250 % FCT scheme 250 251 % ------------------------------------------------------------------------------------------------------------- 251 \subsection{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct} \forcode{= .true.})}252 \subsection{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct}~\forcode{= .true.})} 252 253 \label{subsec:TRA_adv_tvd} 253 254 254 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct} \forcode{= .true.}.255 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}~\forcode{= .true.}. 255 256 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 256 257 setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$. … … 263 264 \label{eq:tra_adv_fct} 264 265 \begin{split} 265 \tau_u^{ups} &=266 \tau_u^{ups} &= 266 267 \begin{cases} 267 T_{i+1} & \text{if $\ u_{i+1/2} < 0$} \hfill\\268 T_i & \text{if $\ u_{i+1/2} \geq 0$} \hfill\\268 T_{i + 1} & \text{if~} u_{i + 1/2} < 0 \\ 269 T_i & \text{if~} u_{i + 1/2} \geq 0 \\ 269 270 \end{cases} 270 \\ \\271 \tau_u^{fct} &=\tau_u^{ups} +c_u \;\left( {\tau_u^{cen} -\tau_u^{ups} } \right)271 \\ 272 \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big) 272 273 \end{split} 273 274 \end{equation} … … 278 279 The one chosen in \NEMO is described in \citet{Zalesak_JCP79}. 279 280 $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. 280 The resulting scheme is quite expensive but \ emph{positive}.281 The resulting scheme is quite expensive but \textit{positive}. 281 282 It can be used on both active and passive tracers. 282 283 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}. … … 294 295 $\tau_u^{ups}$ is evaluated using the \textit{before} tracer. 295 296 In other words, the advective part of the scheme is time stepped with a leap-frog scheme 296 while a forward scheme is used for the diffusive part. 297 while a forward scheme is used for the diffusive part. 297 298 298 299 % ------------------------------------------------------------------------------------------------------------- 299 300 % MUSCL scheme 300 301 % ------------------------------------------------------------------------------------------------------------- 301 \subsection{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus} \forcode{= .true.})}302 \subsection{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}~\forcode{= .true.})} 302 303 \label{subsec:TRA_adv_mus} 303 304 304 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus} \forcode{= .true.}.305 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}~\forcode{= .true.}. 305 306 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 306 307 … … 309 310 two $T$-points (\autoref{fig:adv_scheme}). 310 311 For example, in the $i$-direction : 311 \ [312 \begin{equation} 312 313 % \label{eq:tra_adv_mus} 313 \tau_u^{mus} = \l eft\{314 \begin{aligned}315 &\tau_i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right)316 &\ \widetilde{\partial _i \tau} & \quad \text{if }\;u_{i+1/2} \geqslant 0\\317 &\tau_{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right)318 &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0319 \end{aligned}320 \right.321 \ ]322 where $\widetilde{\partial 314 \tau_u^{mus} = \lt\{ 315 \begin{split} 316 \tau_i &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 317 \widetilde{\partial_i \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\ 318 \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 319 \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} < 0 320 \end{split} 321 \rt. 322 \end{equation} 323 where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to 323 324 ensure the \textit{positive} character of the scheme. 324 325 325 The time stepping is performed using a forward scheme, 326 that is the \textit{before} tracer field is used toevaluate $\tau_u^{mus}$.326 The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to 327 evaluate $\tau_u^{mus}$. 327 328 328 329 For an ocean grid point adjacent to land and where the ocean velocity is directed toward land, … … 330 331 This choice ensure the \textit{positive} character of the scheme. 331 332 In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes 332 (\np{ln\_mus\_ups} \forcode{= .true.}).333 (\np{ln\_mus\_ups}~\forcode{= .true.}). 333 334 334 335 % ------------------------------------------------------------------------------------------------------------- 335 336 % UBS scheme 336 337 % ------------------------------------------------------------------------------------------------------------- 337 \subsection{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs} \forcode{= .true.})}338 \subsection{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}~\forcode{= .true.})} 338 339 \label{subsec:TRA_adv_ubs} 339 340 340 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs} \forcode{= .true.}.341 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}~\forcode{= .true.}. 341 342 UBS implementation can be found in the \mdl{traadv\_mus} module. 342 343 … … 347 348 \begin{equation} 348 349 \label{eq:tra_adv_ubs} 349 \tau_u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ 350 \begin{aligned} 351 &\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 352 &\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 353 \end{aligned} 354 \right. 350 \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6} 351 \begin{cases} 352 \tau"_i & \text{if~} u_{i + 1/2} \geqslant 0 \\ 353 \tau"_{i + 1} & \text{if~} u_{i + 1/2} < 0 354 \end{cases} 355 \quad 356 \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt] 355 357 \end{equation} 356 where $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. 357 358 This results in a dissipatively dominant (\ie hyper-diffusive) truncation error 358 359 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error 359 360 \citep{Shchepetkin_McWilliams_OM05}. 360 361 The overall performance of the advection scheme is similar to that reported in \cite{Farrow1995}. 361 362 It is a relatively good compromise between accuracy and smoothness. 362 Nevertheless the scheme is not \ emph{positive}, meaning that false extrema are permitted,363 Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted, 363 364 but the amplitude of such are significantly reduced over the centred second or fourth order method. 364 365 Therefore it is not recommended that it should be applied to a passive tracer that requires positivity. … … 368 369 \citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}. 369 370 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme 370 (\np{nn\_cen\_v} \forcode{= 2 or 4}).371 (\np{nn\_cen\_v}~\forcode{= 2 or 4}). 371 372 372 373 For stability reasons (see \autoref{chap:STP}), the first term in \autoref{eq:tra_adv_ubs} … … 382 383 383 384 Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows: 384 \ [385 \begin{gather} 385 386 \label{eq:traadv_ubs2} 386 \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} \left\{ 387 \begin{aligned} 388 & + \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 389 & - \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 390 \end{aligned} 391 \right. 392 \] 393 or equivalently 394 \[ 387 \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} 388 \begin{cases} 389 + \tau"_i & \text{if} \ u_{i + 1/2} \geqslant 0 \\ 390 - \tau"_{i + 1} & \text{if} \ u_{i + 1/2} < 0 391 \end{cases} 392 \intertext{or equivalently} 395 393 % \label{eq:traadv_ubs2b} 396 u_{i+1/2} \ \tau_u^{ubs} 397 =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta_i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 398 - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 399 \] 394 u_{i + 1/2} \ \tau_u^{ubs} = u_{i + 1/2} \, \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \Big]}^{\,i + 1/2} 395 - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber 396 \end{gather} 400 397 401 398 \autoref{eq:traadv_ubs2} has several advantages. … … 403 400 an upstream-biased diffusion term is added. 404 401 Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to 405 be evaluated at the \ emph{now} time step using \autoref{eq:tra_adv_ubs}.402 be evaluated at the \textit{now} time step using \autoref{eq:tra_adv_ubs}. 406 403 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which 407 is simply proportional to the velocity: 408 $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. 404 is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 409 405 Note the current version of NEMO uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}. 410 406 … … 412 408 % QCK scheme 413 409 % ------------------------------------------------------------------------------------------------------------- 414 \subsection{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck} \forcode{= .true.})}410 \subsection{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}~\forcode{= .true.})} 415 411 \label{subsec:TRA_adv_qck} 416 412 417 413 The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme 418 proposed by \citet{Leonard1979} is used when \np{ln\_traadv\_qck} \forcode{= .true.}.414 proposed by \citet{Leonard1979} is used when \np{ln\_traadv\_qck}~\forcode{= .true.}. 419 415 QUICKEST implementation can be found in the \mdl{traadv\_qck} module. 420 416 … … 422 418 \citep{Leonard1991}. 423 419 It has been implemented in NEMO by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. 424 The resulting scheme is quite expensive but \ emph{positive}.420 The resulting scheme is quite expensive but \textit{positive}. 425 421 It can be used on both active and passive tracers. 426 422 However, the intrinsic diffusion of QCK makes its use risky in the vertical direction where … … 431 427 432 428 %%%gmcomment : Cross term are missing in the current implementation.... 433 434 429 435 430 % ================================================================ … … 458 453 except for the pure vertical component that appears when a rotation tensor is used. 459 454 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). 460 When \np{ln\_traldf\_msc} \forcode{= .true.}, a Method of Stabilizing Correction is used in which455 When \np{ln\_traldf\_msc}~\forcode{= .true.}, a Method of Stabilizing Correction is used in which 461 456 the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}. 462 457 … … 464 459 % Type of operator 465 460 % ------------------------------------------------------------------------------------------------------------- 466 \subsection[Type of operator (\protect\np{ln\_traldf}\{\_NONE,\_lap,\_blp\}\})] 467 {Type of operator (\protect\np{ln\_traldf\_NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) } 461 \subsection[Type of operator (\protect\np{ln\_traldf}\{\_NONE,\_lap,\_blp\}\})]{Type of operator (\protect\np{ln\_traldf\_NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) } 468 462 \label{subsec:TRA_ldf_op} 469 463 470 464 Three operator options are proposed and, one and only one of them must be selected: 465 471 466 \begin{description} 472 \item[\np{ln\_traldf\_NONE} \forcode{= .true.}:]467 \item[\np{ln\_traldf\_NONE}~\forcode{= .true.}:] 473 468 no operator selected, the lateral diffusive tendency will not be applied to the tracer equation. 474 469 This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example). 475 \item[\np{ln\_traldf\_lap} \forcode{= .true.}:]470 \item[\np{ln\_traldf\_lap}~\forcode{= .true.}:] 476 471 a laplacian operator is selected. 477 This harmonic operator takes the following expression: $\mathpzc{L}(T) =\nabla \cdot A_{ht}\;\nabla T $,472 This harmonic operator takes the following expression: $\mathpzc{L}(T) = \nabla \cdot A_{ht} \; \nabla T $, 478 473 where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}), 479 474 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}). 480 \item[\np{ln\_traldf\_blp} \forcode{= .true.}]:475 \item[\np{ln\_traldf\_blp}~\forcode{= .true.}]: 481 476 a bilaplacian operator is selected. 482 477 This biharmonic operator takes the following expression: 483 $\mathpzc{B} =- \mathpzc{L}\left(\mathpzc{L}(T) \right) = -\nabla \cdot b\nabla \left( {\nabla \cdot b\nabla T} \right)$478 $\mathpzc{B} = - \mathpzc{L}(\mathpzc{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$ 484 479 where the gradient operats along the selected direction, 485 and $b^2 =B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$(see \autoref{chap:LDF}).480 and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}). 486 481 In the code, the bilaplacian operator is obtained by calling the laplacian twice. 487 482 \end{description} … … 495 490 whereas the laplacian damping time scales only like $\lambda^{-2}$. 496 491 497 498 492 % ------------------------------------------------------------------------------------------------------------- 499 493 % Direction of action 500 494 % ------------------------------------------------------------------------------------------------------------- 501 \subsection[Action direction (\protect\np{ln\_traldf}\{\_lev,\_hor,\_iso,\_triad\})] 502 {Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 495 \subsection[Action direction (\protect\np{ln\_traldf}\{\_lev,\_hor,\_iso,\_triad\})]{Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 503 496 \label{subsec:TRA_ldf_dir} 504 497 505 498 The choice of a direction of action determines the form of operator used. 506 499 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when 507 iso-level option is used (\np{ln\_traldf\_lev} \forcode{= .true.}) or508 when a horizontal (\ie geopotential) operator is demanded in \ zstar-coordinate500 iso-level option is used (\np{ln\_traldf\_lev}~\forcode{= .true.}) or 501 when a horizontal (\ie geopotential) operator is demanded in \textit{z}-coordinate 509 502 (\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}). 510 503 The associated code can be found in the \mdl{traldf\_lap\_blp} module. … … 521 514 522 515 The resulting discret form of the three operators (one iso-level and two rotated one) is given in 523 the next two sub-sections. 524 516 the next two sub-sections. 525 517 526 518 % ------------------------------------------------------------------------------------------------------------- 527 519 % iso-level operator 528 520 % ------------------------------------------------------------------------------------------------------------- 529 \subsection{Iso-level (bi -)laplacian operator ( \protect\np{ln\_traldf\_iso}) }521 \subsection{Iso-level (bi -)laplacian operator ( \protect\np{ln\_traldf\_iso}) } 530 522 \label{subsec:TRA_ldf_lev} 531 523 … … 533 525 \begin{equation} 534 526 \label{eq:tra_ldf_lap} 535 D_t^{lT} =\frac{1}{b_t} \left( \; 536 \delta_{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta_{i+1/2} [T] \right] 537 + \delta_{j}\left[ A_v^{lT} \; \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta_{j+1/2} [T] \right] \;\right) 527 D_t^{lT} = \frac{1}{b_t} \Bigg( \delta_{i} \lt[ A_u^{lT} \; \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [T] \rt] 528 + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg) 538 529 \end{equation} 539 where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells and530 where $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells and 540 531 where zero diffusive fluxes is assumed across solid boundaries, 541 532 first (and third in bilaplacian case) horizontal tracer derivative are masked. 542 533 It is implemented in the \rou{traldf\_lap} subroutine found in the \mdl{traldf\_lap} module. 543 534 The module also contains \rou{traldf\_blp}, the subroutine calling twice \rou{traldf\_lap} in order to 544 compute the iso-level bilaplacian operator. 545 546 It is a \ emph{horizontal} operator (\ie acting along geopotential surfaces) in535 compute the iso-level bilaplacian operator. 536 537 It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in 547 538 the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 548 It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp} \forcode{= .true.},549 we have \np{ln\_traldf\_lev} \forcode{ = .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}\forcode{= .true.}.539 It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}~\forcode{= .true.}, 540 we have \np{ln\_traldf\_lev}~\forcode{= .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}~\forcode{= .true.}. 550 541 In both cases, it significantly contributes to diapycnal mixing. 551 542 It is therefore never recommended, even when using it in the bilaplacian case. 552 543 553 Note that in the partial step $z$-coordinate (\np{ln\_zps} \forcode{= .true.}),544 Note that in the partial step $z$-coordinate (\np{ln\_zps}~\forcode{= .true.}), 554 545 tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom. 555 546 In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level require a specific treatment. 556 547 They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}. 557 548 558 559 549 % ------------------------------------------------------------------------------------------------------------- 560 550 % Rotated laplacian operator 561 551 % ------------------------------------------------------------------------------------------------------------- 562 \subsection{Standard and triad (bi -)laplacian operator}552 \subsection{Standard and triad (bi -)laplacian operator} 563 553 \label{subsec:TRA_ldf_iso_triad} 564 554 565 %&& Standard rotated (bi -)laplacian operator555 %&& Standard rotated (bi -)laplacian operator 566 556 %&& ---------------------------------------------- 567 \subsubsection{Standard rotated (bi -)laplacian operator (\protect\mdl{traldf\_iso})}557 \subsubsection{Standard rotated (bi -)laplacian operator (\protect\mdl{traldf\_iso})} 568 558 \label{subsec:TRA_ldf_iso} 569 570 559 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf}) 571 takes the following semi -discrete space form in $z$- and $s$-coordinates:560 takes the following semi -discrete space form in $z$- and $s$-coordinates: 572 561 \begin{equation} 573 562 \label{eq:tra_ldf_iso} 574 563 \begin{split} 575 D_T^{lT} = \frac{1}{b_t} & \left\{ \,\;\delta_i \left[ A_u^{lT} \left( 576 \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T] 577 - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k} 578 \right) \right] \right. \\ 579 & +\delta_j \left[ A_v^{lT} \left( 580 \frac{e_{1v}\,e_{3v}}{e_{2v}} \,\delta_{j+1/2} [T] 581 - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} 582 \right) \right] \\ 583 & +\delta_k \left[ A_w^{lT} \left( 584 -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2} 585 \right. \right. \\ 586 & \qquad \qquad \quad 587 - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2} \\ 588 & \left. {\left. { \qquad \qquad \ \ \ \left. { 589 +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right) 590 \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} 564 D_T^{lT} = \frac{1}{b_t} \Bigg[ \quad &\delta_i A_u^{lT} \lt( \frac{e_{2u} e_{3u}}{e_{1u}} \, \delta_{i + 1/2} [T] 565 - e_{2u} r_{1u} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,i + 1/2,k} \rt) \Bigg. \\ 566 + &\delta_j A_v^{lT} \lt( \frac{e_{1v} e_{3v}}{e_{2v}} \, \delta_{j + 1/2} [T] 567 - e_{1v} r_{2v} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,j + 1/2,k} \rt) \\ 568 + &\delta_k A_w^{lT} \lt( \frac{e_{1w} e_{2w}}{e_{3w}} (r_{1w}^2 + r_{2w}^2) \, \delta_{k + 1/2} [T] \rt. \\ 569 & \qquad \quad \Bigg. \lt. - e_{2w} r_{1w} \, \overline{\overline{\delta_{i + 1/2} [T]}}^{\,i,k + 1/2} 570 - e_{1w} r_{2w} \, \overline{\overline{\delta_{j + 1/2} [T]}}^{\,j,k + 1/2} \rt) \Bigg] 591 571 \end{split} 592 572 \end{equation} 593 where $b_t $=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells,573 where $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells, 594 574 $r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and 595 575 the surface along which the diffusion operator acts (\ie horizontal or iso-neutral surfaces). 596 It is thus used when, in addition to \np{ln\_traldf\_lap} \forcode{= .true.},597 we have \np{ln\_traldf\_iso} \forcode{= .true.},598 or both \np{ln\_traldf\_hor} \forcode{ = .true.} and \np{ln\_zco}\forcode{= .true.}.576 It is thus used when, in addition to \np{ln\_traldf\_lap}~\forcode{= .true.}, 577 we have \np{ln\_traldf\_iso}~\forcode{= .true.}, 578 or both \np{ln\_traldf\_hor}~\forcode{= .true.} and \np{ln\_zco}~\forcode{= .true.}. 599 579 The way these slopes are evaluated is given in \autoref{sec:LDF_slp}. 600 580 At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using 601 the mask technique (see \autoref{sec:LBC_coast}). 581 the mask technique (see \autoref{sec:LBC_coast}). 602 582 603 583 The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical derivatives. … … 606 586 For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module, 607 587 but in the \mdl{trazdf} module where, if iso-neutral mixing is used, 608 the vertical mixing coefficient is simply increased by 609 $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. 588 the vertical mixing coefficient is simply increased by $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$. 610 589 611 590 This formulation conserves the tracer but does not ensure the decrease of the tracer variance. 612 591 Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without 613 any additional background horizontal diffusion \citep{Guilyardi_al_CD01}. 614 615 Note that in the partial step $z$-coordinate (\np{ln\_zps} \forcode{= .true.}),592 any additional background horizontal diffusion \citep{Guilyardi_al_CD01}. 593 594 Note that in the partial step $z$-coordinate (\np{ln\_zps}~\forcode{= .true.}), 616 595 the horizontal derivatives at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment. 617 596 They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 618 597 619 %&& Triad rotated (bi -)laplacian operator598 %&& Triad rotated (bi -)laplacian operator 620 599 %&& ------------------------------------------- 621 \subsubsection{Triad rotated (bi -)laplacian operator (\protect\np{ln\_traldf\_triad})}600 \subsubsection{Triad rotated (bi -)laplacian operator (\protect\np{ln\_traldf\_triad})} 622 601 \label{subsec:TRA_ldf_triad} 623 602 624 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad} \forcode{= .true.}; see \autoref{apdx:triad})603 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}~\forcode{= .true.}; see \autoref{apdx:triad}) 625 604 626 605 An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases 627 is also available in \NEMO (\np{ln\_traldf\_grif} \forcode{= .true.}).606 is also available in \NEMO (\np{ln\_traldf\_grif}~\forcode{= .true.}). 628 607 A complete description of the algorithm is given in \autoref{apdx:triad}. 629 608 … … 635 614 It requires an additional assumption on boundary conditions: 636 615 first and third derivative terms normal to the coast, 637 normal to the bottom and normal to the surface are set to zero. 616 normal to the bottom and normal to the surface are set to zero. 638 617 639 618 %&& Option for the rotated operators … … 642 621 \label{subsec:TRA_ldf_options} 643 622 644 \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) 645 646 \np{rn\_slpmax} = slope limit (both operators) 647 648 \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 649 650 \np{rn\_sw\_triad} =1 switching triad; 651 =0 all 4 triads used (triad only) 652 653 \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 623 \begin{itemize} 624 \item \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) 625 \item \np{rn\_slpmax} = slope limit (both operators) 626 \item \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 627 \item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only) 628 \item \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 629 \end{itemize} 654 630 655 631 % ================================================================ … … 666 642 The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, 667 643 and is based on a laplacian operator. 668 The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi -discrete space form:669 \ [644 The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi -discrete space form: 645 \begin{gather*} 670 646 % \label{eq:tra_zdf} 671 \begin{split} 672 D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}} \delta_{k+1/2}[T] \;\right] \\ 673 D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}} \delta_{k+1/2}[S] \;\right] 674 \end{split} 675 \] 647 D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\ 648 D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] 649 \end{gather*} 676 650 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity, 677 651 respectively. 678 Generally, $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is parameterised (\ie \key{zdfddm} is defined). 652 Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised 653 (\ie \key{zdfddm} is defined). 679 654 The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF). 680 655 Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by 681 $\frac{e_{1w} \,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ to account for682 the vertical second derivative of\autoref{eq:tra_ldf_iso}.656 $\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of 657 \autoref{eq:tra_ldf_iso}. 683 658 684 659 At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified. 685 660 At the surface they are prescribed from the surface forcing and added in a dedicated routine 686 661 (see \autoref{subsec:TRA_sbc}), whilst at the bottom they are set to zero for heat and salt unless 687 a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}). 662 a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}). 688 663 689 664 The large eddy coefficient found in the mixed layer together with high vertical resolution implies that 690 in the case of explicit time stepping (\np{ln\_zdfexp} \forcode{= .true.})665 in the case of explicit time stepping (\np{ln\_zdfexp}~\forcode{= .true.}) 691 666 there would be too restrictive a constraint on the time step. 692 667 Therefore, the default implicit time stepping is preferred for the vertical diffusion since 693 668 it overcomes the stability constraint. 694 A forward time differencing scheme (\np{ln\_zdfexp} \forcode{= .true.}) using669 A forward time differencing scheme (\np{ln\_zdfexp}~\forcode{= .true.}) using 695 670 a time splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative. 696 Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 671 Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 697 672 698 673 % ================================================================ … … 712 687 This has been found to enhance readability of the code. 713 688 The two formulations are completely equivalent; 714 the forcing terms in trasbc are the surface fluxes divided by the thickness of the top model layer. 689 the forcing terms in trasbc are the surface fluxes divided by the thickness of the top model layer. 715 690 716 691 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components … … 724 699 The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers): 725 700 726 $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 727 (\ie the difference between the total surface heat flux and the fraction of the short wave flux that 728 penetrates into the water column, see \autoref{subsec:TRA_qsr}) 729 plus the heat content associated with of the mass exchange with the atmosphere and lands. 730 731 $\bullet$ $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 732 733 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and 734 possibly with the sea-ice and ice-shelves. 735 736 $\bullet$ \textit{rnf}, the mass flux associated with runoff 737 (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 738 739 $\bullet$ \textit{fwfisf}, the mass flux associated with ice shelf melt, 740 (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 701 \begin{itemize} 702 \item 703 $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 704 (\ie the difference between the total surface heat flux and the fraction of the short wave flux that 705 penetrates into the water column, see \autoref{subsec:TRA_qsr}) 706 plus the heat content associated with of the mass exchange with the atmosphere and lands. 707 \item 708 $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 709 \item 710 \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and 711 possibly with the sea-ice and ice-shelves. 712 \item 713 \textit{rnf}, the mass flux associated with runoff 714 (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 715 \item 716 \textit{fwfisf}, the mass flux associated with ice shelf melt, 717 (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 718 \end{itemize} 741 719 742 720 The surface boundary condition on temperature and salinity is applied as follows: 743 721 \begin{equation} 744 722 \label{eq:tra_sbc} 745 \begin{aligned} 746 &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ 747 & F^S =\frac{ 1 }{\rho_o \, \left. e_{3t} \right|_{k=1} } &\overline{ \textit{sfx} }^t & \\ 748 \end{aligned} 749 \end{equation} 750 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps ($t-\rdt/2$ and $t+\rdt/2$). 723 \begin{alignedat}{2} 724 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns} }^t \\ 725 F^S &= &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t 726 \end{alignedat} 727 \end{equation} 728 where $\overline x^t$ means that $x$ is averaged over two consecutive time steps 729 ($t - \rdt / 2$ and $t + \rdt / 2$). 751 730 Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:STP}). 752 731 753 In the linear free surface case (\np{ln\_linssh} \forcode{= .true.}), an additional term has to be added on732 In the linear free surface case (\np{ln\_linssh}~\forcode{= .true.}), an additional term has to be added on 754 733 both temperature and salinity. 755 734 On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$. … … 759 738 \begin{equation} 760 739 \label{eq:tra_sbc_lin} 761 \begin{aligned} 762 &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} } 763 &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t & \\ 764 % 765 & F^S =\frac{ 1 }{\rho_o \,\left. e_{3t} \right|_{k=1} } 766 &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1} \right) }^t & \\ 767 \end{aligned} 740 \begin{alignedat}{2} 741 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 742 &\overline{(Q_{ns} - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\ 743 F^S &= &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 744 &\overline{(\textit{sfx} - \textit{emp} \lt. S \rt|_{k = 1})}^t 745 \end{alignedat} 768 746 \end{equation} 769 747 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. … … 783 761 784 762 Options are defined through the \ngn{namtra\_qsr} namelist variables. 785 When the penetrative solar radiation option is used (\np{ln\_flxqsr} \forcode{= .true.}),763 When the penetrative solar radiation option is used (\np{ln\_flxqsr}~\forcode{= .true.}), 786 764 the solar radiation penetrates the top few tens of meters of the ocean. 787 If it is not used (\np{ln\_flxqsr} \forcode{= .false.}) all the heat flux is absorbed in the first ocean level.765 If it is not used (\np{ln\_flxqsr}~\forcode{= .false.}) all the heat flux is absorbed in the first ocean level. 788 766 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and 789 767 the surface boundary condition is modified to take into account only the non-penetrative part of the surface … … 791 769 \begin{equation} 792 770 \label{eq:PE_qsr} 793 \begin{ split}794 \ frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}\\795 Q_{ns} &= Q_\text{Total} - Q_{sr}796 \end{ split}771 \begin{gathered} 772 \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\ 773 Q_{ns} = Q_\text{Total} - Q_{sr} 774 \end{gathered} 797 775 \end{equation} 798 776 where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie the shortwave radiation) and 799 $I$ is the downward irradiance ($\l eft. I \right|_{z=\eta}=Q_{sr}$).777 $I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$). 800 778 The additional term in \autoref{eq:PE_qsr} is discretized as follows: 801 779 \begin{equation} 802 780 \label{eq:tra_qsr} 803 \frac{1}{\rho_o \, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right]781 \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w] 804 782 \end{equation} 805 783 … … 810 788 (specified through namelist parameter \np{rn\_abs}). 811 789 It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 812 of a few tens of centimetres (typically $\xi_0 =0.35~m$ set as \np{rn\_si0} in the \ngn{namtra\_qsr} namelist).790 of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \ngn{namtra\_qsr} namelist). 813 791 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to 814 792 larger depths where it contributes to local heating. 815 793 The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen. 816 In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd} \forcode{= .true.})794 In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}~\forcode{= .true.}) 817 795 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 818 796 leading to the following expression \citep{Paulson1977}: 819 797 \[ 820 798 % \label{eq:traqsr_iradiance} 821 I(z) = Q_{sr} \l eft[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right]799 I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt] 822 800 \] 823 801 where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 824 802 It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. 825 The set of default values ($\xi_0 $, $\xi_1$, $R$) corresponds to a Type I water in Jerlov's (1968) classification803 The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification 826 804 (oligotrophic waters). 827 805 … … 840 818 reproduces quite closely the light penetration profiles predicted by the full spectal model, 841 819 but with much greater computational efficiency. 842 The 2-bands formulation does not reproduce the full model very well. 843 844 The RGB formulation is used when \np{ln\_qsr\_rgb} \forcode{= .true.}.820 The 2-bands formulation does not reproduce the full model very well. 821 822 The RGB formulation is used when \np{ln\_qsr\_rgb}~\forcode{= .true.}. 845 823 The RGB attenuation coefficients (\ie the inverses of the extinction length scales) are tabulated over 846 824 61 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L 847 825 (see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module). 848 826 Four types of chlorophyll can be chosen in the RGB formulation: 849 \begin{description} 850 \item[\np{nn\_chdta}\forcode{ = 0}] 827 828 \begin{description} 829 \item[\np{nn\_chdta}~\forcode{= 0}] 851 830 a constant 0.05 g.Chl/L value everywhere ; 852 \item[\np{nn\_chdta} \forcode{= 1}]831 \item[\np{nn\_chdta}~\forcode{= 1}] 853 832 an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in 854 833 the vertical direction; 855 \item[\np{nn\_chdta} \forcode{= 2}]834 \item[\np{nn\_chdta}~\forcode{= 2}] 856 835 same as previous case except that a vertical profile of chlorophyl is used. 857 836 Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value; 858 \item[\np{ln\_qsr\_bio} \forcode{= .true.}]837 \item[\np{ln\_qsr\_bio}~\forcode{= .true.}] 859 838 simulated time varying chlorophyll by TOP biogeochemical model. 860 839 In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in 861 PISCES or LOBSTER and the oceanic heating rate. 840 PISCES or LOBSTER and the oceanic heating rate. 862 841 \end{description} 842 863 843 The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation is added to 864 844 the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. … … 871 851 Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor. 872 852 In this case, we have chosen that all remaining radiation is absorbed in the last ocean level 873 (\ie $I$ is masked). 853 (\ie $I$ is masked). 874 854 875 855 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 876 856 \begin{figure}[!t] 877 857 \begin{center} 878 \includegraphics[ width=1.0\textwidth]{Fig_TRA_Irradiance}858 \includegraphics[]{Fig_TRA_Irradiance} 879 859 \caption{ 880 860 \protect\label{fig:traqsr_irradiance} … … 903 883 \begin{figure}[!t] 904 884 \begin{center} 905 \includegraphics[ width=1.0\textwidth]{Fig_TRA_geoth}885 \includegraphics[]{Fig_TRA_geoth} 906 886 \caption{ 907 887 \protect\label{fig:geothermal} … … 917 897 This is the default option in \NEMO, and it is implemented using the masking technique. 918 898 However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling. 919 This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1\;W/m^2$ \citep{Stein_Stein_Nat92}),899 This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{Stein_Stein_Nat92}), 920 900 but it warms systematically the ocean and acts on the densest water masses. 921 901 Taking this flux into account in a global ocean model increases the deepest overturning cell 922 (\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}.902 (\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}. 923 903 924 904 Options are defined through the \ngn{namtra\_bbc} namelist variables. … … 939 919 %-------------------------------------------------------------------------------------------------------------- 940 920 941 Options are defined through the 921 Options are defined through the \ngn{nambbl} namelist variables. 942 922 In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps. 943 923 This is not adequate to represent gravity driven downslope flows. … … 951 931 sometimes over a thickness much larger than the thickness of the observed gravity plume. 952 932 A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of 953 a sill \citep{Willebrand_al_PO01}, and the thickness of the plume is not resolved. 933 a sill \citep{Willebrand_al_PO01}, and the thickness of the plume is not resolved. 954 934 955 935 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{Beckmann_Doscher1997}, … … 964 944 % Diffusive BBL 965 945 % ------------------------------------------------------------------------------------------------------------- 966 \subsection{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf} \forcode{= 1})}946 \subsection{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf}~\forcode{= 1})} 967 947 \label{subsec:TRA_bbl_diff} 968 948 … … 971 951 \[ 972 952 % \label{eq:tra_bbl_diff} 973 {\rm {\bf F}}_\sigma=A_l^\sigma \;\nabla_\sigma T953 \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T 974 954 \] 975 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 976 and$A_l^\sigma$ the lateral diffusivity in the BBL.955 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and 956 $A_l^\sigma$ the lateral diffusivity in the BBL. 977 957 Following \citet{Beckmann_Doscher1997}, the latter is prescribed with a spatial dependence, 978 958 \ie in the conditional form 979 959 \begin{equation} 980 960 \label{eq:tra_bbl_coef} 981 A_l^\sigma (i,j,t) =\left\{ {982 \begin{ array}{l}983 A_{bbl} \quad \quad \mbox{if} \quad \nabla_\sigma \rho \cdot \nabla H<0 \\\\984 0\quad \quad \;\,\mbox{otherwise}\\985 \end{array}}986 \right.987 \end{equation} 961 A_l^\sigma (i,j,t) = 962 \begin{cases} 963 A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\ 964 \\ 965 0 & \text{otherwise} \\ 966 \end{cases} 967 \end{equation} 988 968 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and 989 969 usually set to a value much larger than the one used for lateral mixing in the open ocean. … … 995 975 \[ 996 976 % \label{eq:tra_bbl_Drho} 997 \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \,\nabla_\sigma S977 \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S 998 978 \] 999 where $\rho$, $\alpha$ and $\beta$ are functions of $\overline {T}^\sigma$,1000 $\overline {S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, salinity and depth, respectively.979 where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and 980 $\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively. 1001 981 1002 982 % ------------------------------------------------------------------------------------------------------------- 1003 983 % Advective BBL 1004 984 % ------------------------------------------------------------------------------------------------------------- 1005 \subsection{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv} \forcode{= 1..2})}985 \subsection{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}~\forcode{= 1..2})} 1006 986 \label{subsec:TRA_bbl_adv} 1007 987 1008 %\sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following 1009 %if this is not what is meant then "downwards sloping flow" is also a possibility"} 988 %\sgacomment{ 989 % "downsloping flow" has been replaced by "downslope flow" in the following 990 % if this is not what is meant then "downwards sloping flow" is also a possibility" 991 %} 1010 992 1011 993 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1012 994 \begin{figure}[!t] 1013 995 \begin{center} 1014 \includegraphics[ width=0.7\textwidth]{Fig_BBL_adv}996 \includegraphics[]{Fig_BBL_adv} 1015 997 \caption{ 1016 998 \protect\label{fig:bbl} 1017 999 Advective/diffusive Bottom Boundary Layer. 1018 The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i +1}_{kdnw}$.1000 The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$. 1019 1001 Red arrows indicate the additional overturning circulation due to the advective BBL. 1020 1002 The transport of the downslope flow is defined either as the transport of the bottom ocean cell (black arrow), … … 1026 1008 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1027 1009 1028 1029 1010 %!! nn_bbl_adv = 1 use of the ocean velocity as bbl velocity 1030 1011 %!! nn_bbl_adv = 2 follow Campin and Goosse (1999) implentation 1031 %!! \ietransport proportional to the along-slope density gradient1012 %!! i.e. transport proportional to the along-slope density gradient 1032 1013 1033 1014 %%%gmcomment : this section has to be really written 1034 1015 1035 When applying an advective BBL (\np{nn\_bbl\_adv} \forcode{= 1..2}), an overturning circulation is added which1016 When applying an advective BBL (\np{nn\_bbl\_adv}~\forcode{= 1..2}), an overturning circulation is added which 1036 1017 connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope. 1037 The density difference causes dense water to move down the slope. 1038 1039 \np{nn\_bbl\_adv} \forcode{= 1}:1018 The density difference causes dense water to move down the slope. 1019 1020 \np{nn\_bbl\_adv}~\forcode{= 1}: 1040 1021 the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step 1041 1022 (see black arrow in \autoref{fig:bbl}) \citep{Beckmann_Doscher1997}. 1042 1023 It is a \textit{conditional advection}, that is, advection is allowed only 1043 if dense water overlies less dense water on the slope (\ie $\nabla_\sigma \rho \cdot \nabla H<0$) and1044 if the velocity is directed towards greater depth (\ie $\vect {U} \cdot \nabla H>0$).1045 1046 \np{nn\_bbl\_adv} \forcode{= 2}:1024 if dense water overlies less dense water on the slope (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$) and 1025 if the velocity is directed towards greater depth (\ie $\vect U \cdot \nabla H > 0$). 1026 1027 \np{nn\_bbl\_adv}~\forcode{= 2}: 1047 1028 the downslope velocity is chosen to be proportional to $\Delta \rho$, 1048 1029 the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}. 1049 1030 The advection is allowed only if dense water overlies less dense water on the slope 1050 (\ie $\nabla_\sigma \rho \cdot \nabla H<0$).1031 (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$). 1051 1032 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}), 1052 1033 is simply given by the following expression: 1053 1034 \[ 1054 1035 % \label{eq:bbl_Utr} 1055 u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o} e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right)1036 u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 1056 1037 \] 1057 1038 where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl}, … … 1062 1043 The possible values for $\gamma$ range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}. 1063 1044 1064 Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl})$ using the upwind scheme.1045 Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme. 1065 1046 Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and 1066 1047 the surrounding water at intermediate depths. … … 1071 1052 the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and 1072 1053 the upward \autoref{eq:bbl_up} return flows as follows: 1073 \begin{align} 1054 \begin{alignat}{3} 1055 \label{eq:bbl_dw} 1074 1056 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 1075 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} \left( T^{sh}_{kup} - T^{do}_{kdw} \right) \label{eq:bbl_dw}\\1076 %1057 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 1058 \label{eq:bbl_hor} 1077 1059 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 1078 + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} \left( T^{do}_{kup} - T^{sh}_{kup} \right) \label{eq:bbl_hor}\\1079 1060 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ 1061 % 1080 1062 \intertext{and for $k =kdw-1,\;..., \; kup$ :} 1081 1063 % 1064 \label{eq:bbl_up} 1082 1065 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 1083 + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} \left( T^{do}_{k+1} - T^{sh}_{k} \right) \label{eq:bbl_up}1084 \end{align }1085 where $b_t$ is the $T$-cell volume. 1086 1087 Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl})$, is available in the model outputs.1066 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} &&\lt( T^{do}_{k +1} - T^{sh}_{k} \rt) 1067 \end{alignat} 1068 where $b_t$ is the $T$-cell volume. 1069 1070 Note that the BBL transport, $(u^{tr}_{bbl},v^{tr}_{bbl})$, is available in the model outputs. 1088 1071 It has to be used to compute the effective velocity as well as the effective overturning circulation. 1089 1072 … … 1101 1084 \begin{equation} 1102 1085 \label{eq:tra_dmp} 1103 \begin{ split}1104 \ frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)\\1105 \ frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right)1106 \end{ split}1086 \begin{gathered} 1087 \pd[T]{t} = \cdots - \gamma (T - T_o) \\ 1088 \pd[S]{t} = \cdots - \gamma (S - S_o) 1089 \end{gathered} 1107 1090 \end{equation} 1108 1091 where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields … … 1111 1094 The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true. 1112 1095 It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true in 1113 \ textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set1096 \ngn{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set 1114 1097 (\ie that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread}, 1115 1098 see \autoref{subsec:SBC_fldread}). … … 1128 1111 The second case corresponds to the use of the robust diagnostic method \citep{Sarmiento1982}. 1129 1112 It allows us to find the velocity field consistent with the model dynamics whilst 1130 having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$). 1113 having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$). 1131 1114 1132 1115 The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but … … 1140 1123 \citep{Madec_al_JPO96}. 1141 1124 1142 \subsection{Generating \ifile{resto} using DMP\_TOOLS} 1143 1144 DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. 1145 Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled and 1146 run on the same machine as the NEMO model. 1147 A \ifile{mesh\_mask} file for the model configuration is required as an input. 1148 This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. 1149 The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. 1150 The \ngn{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for 1151 the restoration coefficient. 1152 1153 %--------------------------------------------nam_dmp_create------------------------------------------------- 1154 %\namtools{namelist_dmp} 1155 %------------------------------------------------------------------------------------------------------- 1156 1157 \np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and 1158 should be the same as specified in \ngn{namcfg}. 1159 The variable \np{lzoom} is used to specify that the damping is being used as in case \textit{a} above to 1160 provide boundary conditions to a zoom configuration. 1161 In the case of the arctic or antarctic zoom configurations this includes some specific treatment. 1162 Otherwise damping is applied to the 6 grid points along the ocean boundaries. 1163 The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in 1164 the \ngn{nam\_zoom\_dmp} name list. 1165 1166 The remaining switch namelist variables determine the spatial variation of the restoration coefficient in 1167 non-zoom configurations. 1168 \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. 1169 \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea for 1170 the ORCA4, ORCA2 and ORCA05 configurations. 1171 If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as 1172 a function of the model number. 1173 This option is included to allow backwards compatability of the ORCA2 reference configurations with 1174 previous model versions. 1175 \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. 1176 This option only has an effect if \np{ln\_full\_field} is true. 1177 \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. 1178 Finally \np{ln\_custom} specifies that the custom module will be called. 1179 This module is contained in the file \mdl{custom} and can be edited by users. 1180 For example damping could be applied in a specific region. 1181 1182 The restoration coefficient can be set to zero in equatorial regions by 1183 specifying a positive value of \np{nn\_hdmp}. 1184 Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to 1185 the full values of a 10\deg latitud band. 1186 This is often used because of the short adjustment time scale in the equatorial region 1187 \citep{Reverdin1991, Fujio1991, Marti_PhD92}. 1188 The time scale associated with the damping depends on the depth as a hyperbolic tangent, 1189 with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. 1125 For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under 1126 \path{./tools/DMP_TOOLS}. 1190 1127 1191 1128 % ================================================================ … … 1199 1136 %-------------------------------------------------------------------------------------------------------------- 1200 1137 1201 Options are defined through the 1138 Options are defined through the \ngn{namdom} namelist variables. 1202 1139 The general framework for tracer time stepping is a modified leap-frog scheme \citep{Leclair_Madec_OM09}, 1203 1140 \ie a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 1204 1141 \begin{equation} 1205 1142 \label{eq:tra_nxt} 1206 \begin{aligned} 1207 (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt \,e_{3t}^t\ \text{RHS}^t & \\ \\ 1208 (e_{3t}T)_f^t \;\ \quad &= (e_{3t}T)^t \;\quad 1209 &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] & \\ 1210 & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] & 1211 \end{aligned} 1143 \begin{alignedat}{3} 1144 &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\ 1145 &(e_{3t}T)_f^t &&= (e_{3t}T)^t &&+ \, \gamma \, \lt[ (e_{3t}T)_f^{t - \rdt} - 2(e_{3t}T)^t + (e_{3t}T)^{t + \rdt} \rt] \\ 1146 & && &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt] 1147 \end{alignedat} 1212 1148 \end{equation} 1213 1149 where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, … … 1215 1151 (\ie fluxes plus content in mass exchanges). 1216 1152 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1217 Its default value is \np{rn\_atfp} \forcode{= 10.e-3}.1153 Its default value is \np{rn\_atfp}~\forcode{= 10.e-3}. 1218 1154 Note that the forcing correction term in the filter is not applied in linear free surface 1219 (\jp{lk\_vvl} \forcode{ = .false.}) (see \autoref{subsec:TRA_sbc}.1155 (\jp{lk\_vvl}~\forcode{= .false.}) (see \autoref{subsec:TRA_sbc}). 1220 1156 Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$. 1221 1157 1222 When the vertical mixing is solved implicitly, 1223 the update of the \textit{next} tracer fields is done in module \mdl{trazdf}.1158 When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in 1159 \mdl{trazdf} module. 1224 1160 In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module. 1225 1161 1226 1162 In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed: 1227 $T^{t -\rdt} = T^t$ and $T^t = T_f$.1163 $T^{t - \rdt} = T^t$ and $T^t = T_f$. 1228 1164 1229 1165 % ================================================================ … … 1240 1176 % Equation of State 1241 1177 % ------------------------------------------------------------------------------------------------------------- 1242 \subsection{Equation of seawater (\protect\np{nn\_eos} \forcode{= -1..1})}1178 \subsection{Equation of seawater (\protect\np{nn\_eos}~\forcode{= -1..1})} 1243 1179 \label{subsec:TRA_eos} 1244 1180 … … 1264 1200 To that purposed, a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available. 1265 1201 1266 In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density.1202 In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density. 1267 1203 Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 1268 1204 This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as, … … 1270 1206 density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. 1271 1207 1272 Options are defined through the 1208 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} which 1273 1209 controls the EOS used (\forcode{= -1} for TEOS10 ; \forcode{= 0} for EOS-80 ; \forcode{= 1} for S-EOS). 1210 1274 1211 \begin{description} 1275 \item[\np{nn\_eos} \forcode{= -1}]1212 \item[\np{nn\_eos}~\forcode{= -1}] 1276 1213 the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used. 1277 1214 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, … … 1282 1219 the TEOS-10 rational function approximation for hydrographic data analysis \citep{TEOS10}. 1283 1220 A key point is that conservative state variables are used: 1284 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg {C}, notation: $\Theta$).1221 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \degC, notation: $\Theta$). 1285 1222 The pressure in decibars is approximated by the depth in meters. 1286 1223 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. 1287 It is set to $C_p=3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{TEOS10}. 1288 1224 It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{TEOS10}. 1289 1225 Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$. 1290 1226 In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and … … 1293 1229 either computing the air-sea and ice-sea fluxes (forced mode) or 1294 1230 sending the SST field to the atmosphere (coupled mode). 1295 1296 \item[\np{nn\_eos}\forcode{ = 0}] 1231 \item[\np{nn\_eos}~\forcode{= 0}] 1297 1232 the polyEOS80-bsq equation of seawater is used. 1298 1233 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to … … 1305 1240 pressure \citep{UNESCO1983}. 1306 1241 Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which 1307 is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 1308 1309 \item[\np{nn\_eos}\forcode{ = 1}] 1242 is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 1243 \item[\np{nn\_eos}~\forcode{= 1}] 1310 1244 a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen, 1311 1245 the coefficients of which has been optimized to fit the behavior of TEOS10 … … 1317 1251 as well as between \textit{absolute} and \textit{practical} salinity. 1318 1252 S-EOS takes the following expression: 1319 \ [1253 \begin{gather*} 1320 1254 % \label{eq:tra_S-EOS} 1321 \begin{split} 1322 d_a(T,S,z) = ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a \\ 1323 & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a \\ 1324 & - \nu \; T_a \; S_a \; ) \; / \; \rho_o \\ 1325 with \ \ T_a = T-10 \; ; & \; S_a = S-35 \; ;\; \rho_o = 1026~Kg/m^3 1326 \end{split} 1327 \] 1255 \begin{alignedat}{2} 1256 &d_a(T,S,z) = \frac{1}{\rho_o} \big[ &- a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * &T_a \big. \\ 1257 & &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a \\ 1258 & \big. &- \nu \; T_a &S_a \big] \\ 1259 \end{alignedat} 1260 \\ 1261 \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3 1262 \end{gather*} 1328 1263 where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}. 1329 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing the associated coefficients. 1330 Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS. 1331 setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS. 1264 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by 1265 changing the associated coefficients. 1266 Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS. 1267 setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from 1268 S-EOS. 1332 1269 Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 1333 1270 \end{description} 1334 1335 1271 1336 1272 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1337 1273 \begin{table}[!tb] 1338 1274 \begin{center} 1339 \begin{tabular}{| p{26pt}|p{72pt}|p{56pt}|p{136pt}|}1275 \begin{tabular}{|l|l|l|l|} 1340 1276 \hline 1341 coeff. & computer name & S-EOS & description \\ \hline 1342 $a_0$ & \np{rn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline 1343 $b_0$ & \np{rn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline 1344 $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline 1345 $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline 1346 $\nu$ & \np{rn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline 1347 $\mu_1$ & \np{rn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline 1348 $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline 1277 coeff. & computer name & S-EOS & description \\ 1278 \hline 1279 $a_0$ & \np{rn\_a0} & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\ 1280 \hline 1281 $b_0$ & \np{rn\_b0} & $7.6554~10^{-1}$ & linear haline expansion coeff. \\ 1282 \hline 1283 $\lambda_1$ & \np{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$ \\ 1284 \hline 1285 $\lambda_2$ & \np{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$ \\ 1286 \hline 1287 $\nu$ & \np{rn\_nu} & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$ \\ 1288 \hline 1289 $\mu_1$ & \np{rn\_mu1} & $1.4970~10^{-4}$ & thermobaric coeff. in T \\ 1290 \hline 1291 $\mu_2$ & \np{rn\_mu2} & $1.1090~10^{-5}$ & thermobaric coeff. in S \\ 1292 \hline 1349 1293 \end{tabular} 1350 1294 \caption{ … … 1352 1296 Standard value of S-EOS coefficients. 1353 1297 } 1354 1298 \end{center} 1355 1299 \end{table} 1356 1300 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1357 1301 1358 1359 1302 % ------------------------------------------------------------------------------------------------------------- 1360 1303 % Brunt-V\"{a}is\"{a}l\"{a} Frequency 1361 1304 % ------------------------------------------------------------------------------------------------------------- 1362 \subsection{Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos} \forcode{= 0..2})}1305 \subsection{Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos}~\forcode{= 0..2})} 1363 1306 \label{subsec:TRA_bn2} 1364 1307 1365 An accurate computation of the ocean stability ( \ieof $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of1308 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of 1366 1309 paramount importance as determine the ocean stratification and is used in several ocean parameterisations 1367 1310 (namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion, … … 1372 1315 \[ 1373 1316 % \label{eq:tra_bn2} 1374 N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right)1317 N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt) 1375 1318 \] 1376 where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS,1377 and,$\alpha$ and $\beta$ are the thermal and haline expansion coefficients.1378 The coefficients are a polynomial function of temperature, salinity and depth which 1379 expression depends onthe chosen EOS.1319 where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and, 1320 $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 1321 The coefficients are a polynomial function of temperature, salinity and depth which expression depends on 1322 the chosen EOS. 1380 1323 They are computed through \textit{eos\_rab}, a \fortran function that can be found in \mdl{eosbn2}. 1381 1324 … … 1390 1333 \label{eq:tra_eos_fzp} 1391 1334 \begin{split} 1392 T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} - 2.154996 \;10^{-4} \,S \right) \ S \\ 1393 - 7.53\,10^{-3} \ \ p 1394 \end{split} 1335 &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 1336 &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\ 1337 &\text{and~} d = -7.53~10^{-3} 1338 \end{split} 1395 1339 \end{equation} 1396 1340 1397 1341 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water 1398 (\ie referenced to the surface $p =0$),1342 (\ie referenced to the surface $p = 0$), 1399 1343 thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped. 1400 1344 The freezing point is computed through \textit{eos\_fzp}, 1401 a \fortran function that can be found in \mdl{eosbn2}. 1402 1345 a \fortran function that can be found in \mdl{eosbn2}. 1403 1346 1404 1347 % ------------------------------------------------------------------------------------------------------------- … … 1411 1354 % 1412 1355 1413 1414 1356 % ================================================================ 1415 1357 % Horizontal Derivative in zps-coordinate … … 1421 1363 I've changed "derivative" to "difference" and "mean" to "average"} 1422 1364 1423 With partial cells (\np{ln\_zps} \forcode{ = .true.}) at bottom and top (\np{ln\_isfcav}\forcode{= .true.}),1365 With partial cells (\np{ln\_zps}~\forcode{= .true.}) at bottom and top (\np{ln\_isfcav}~\forcode{= .true.}), 1424 1366 in general, tracers in horizontally adjacent cells live at different depths. 1425 1367 Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and 1426 1368 the hydrostatic pressure gradient calculations (\mdl{dynhpg} module). 1427 The partial cell properties at the top (\np{ln\_isfcav} \forcode{= .true.}) are computed in the same way as1369 The partial cell properties at the top (\np{ln\_isfcav}~\forcode{= .true.}) are computed in the same way as 1428 1370 for the bottom. 1429 1371 So, only the bottom interpolation is explained below. … … 1432 1374 a linear interpolation in the vertical is used to approximate the deeper tracer as if 1433 1375 it actually lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}). 1434 For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde {T}$, is:1376 For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is: 1435 1377 1436 1378 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1437 1379 \begin{figure}[!p] 1438 1380 \begin{center} 1439 \includegraphics[ width=0.9\textwidth]{Fig_partial_step_scheme}1381 \includegraphics[]{Fig_partial_step_scheme} 1440 1382 \caption{ 1441 1383 \protect\label{fig:Partial_step_scheme} 1442 1384 Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate 1443 (\protect\np{ln\_zps} \forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i )>0$.1444 A linear interpolation is used to estimate $\widetilde {T}_k^{i+1}$,1385 (\protect\np{ln\_zps}~\forcode{= .true.}) in the case $(e3w_k^{i + 1} - e3w_k^i) > 0$. 1386 A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$, 1445 1387 the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. 1446 The horizontal difference is then given by: $\delta_{i +1/2} T_k= \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and1447 the average by: $\overline {T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i}) / 2$.1388 The horizontal difference is then given by: $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and 1389 the average by: $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$. 1448 1390 } 1449 1391 \end{center} … … 1451 1393 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1452 1394 \[ 1453 \widetilde {T}= \left\{1454 \begin{aligned }1455 &T^{\, i+1} -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta_k T^{i+1}1456 & & \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ }\\ \\1457 &T^{\, i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i }\;\delta_k T^{i+1}1458 & & \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$}1459 \end{aligned }1460 \r ight.1395 \widetilde T = \lt\{ 1396 \begin{alignedat}{2} 1397 &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1} 1398 & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\ 1399 &T^{\, i} &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i } \; \delta_k T^{i + 1} 1400 & \quad \text{if $e_{3w}^{i + 1} < e_{3w}^i$} 1401 \end{alignedat} 1402 \rt. 1461 1403 \] 1462 1404 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 1463 1405 \begin{equation} 1464 1406 \label{eq:zps_hde} 1465 \begin{ aligned}1466 \delta_{i +1/2} T=1407 \begin{split} 1408 \delta_{i + 1/2} T &= 1467 1409 \begin{cases} 1468 \ \ \ \widetilde {T}\quad\ -T^i & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ \\ 1469 \ \ \ T^{\,i+1}-\widetilde{T} & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } 1410 \widetilde T - T^i & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1411 \\ 1412 T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 1470 1413 \end{cases} 1471 \\ \\1472 \overline {T}^{\,i+1/2} \=1414 \\ 1415 \overline T^{\, i + 1/2} &= 1473 1416 \begin{cases} 1474 ( \widetilde {T}\ \ \;\,-T^{\,i}) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ \\ 1475 ( T^{\,i+1}-\widetilde{T} ) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } 1417 (\widetilde T - T^{\, i} ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1418 \\ 1419 (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 1476 1420 \end{cases} 1477 \end{ aligned}1421 \end{split} 1478 1422 \end{equation} 1479 1423 1480 1424 The computation of horizontal derivative of tracers as well as of density is performed once for all at 1481 1425 each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed. 1482 It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde {\rho}$,1426 It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$, 1483 1427 is not the same as that used for $T$ and $S$. 1484 Instead of forming a linear approximation of density, we compute $\widetilde {\rho }$ from the interpolated values of1428 Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of 1485 1429 $T$ and $S$, and the pressure at a $u$-point 1486 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos} 1430 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 1487 1431 \[ 1488 1432 % \label{eq:zps_hde_rho} 1489 \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 1490 \quad \text{where }\ z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) 1433 \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt) 1491 1434 \] 1492 1435
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