Changeset 11630
- Timestamp:
- 2019-10-01T19:37:49+02:00 (5 years ago)
- Location:
- NEMO/trunk/doc/latex/NEMO/subfiles
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-
- 4 edited
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex
r11622 r11630 22 22 23 23 {\footnotesize 24 \begin{tabularx}{ \textwidth}{l||X|X}24 \begin{tabularx}{0.8\textwidth}{l||X|X} 25 25 Release & 26 26 Author(s) & … … 182 182 its Laplacian is defined at the $t$-point. 183 183 These operators have the following discrete forms in the curvilinear $s$-coordinates system: 184 \ [184 \begin{gather*} 185 185 % \label{eq:DOM_grad} 186 186 \nabla q \equiv \frac{1}{e_{1u}} \delta_{i + 1/2} [q] \; \, \vect i 187 187 + \frac{1}{e_{2v}} \delta_{j + 1/2} [q] \; \, \vect j 188 + \frac{1}{e_{3w}} \delta_{k + 1/2} [q] \; \, \vect k 189 \] 190 \[ 188 + \frac{1}{e_{3w}} \delta_{k + 1/2} [q] \; \, \vect k \\ 191 189 % \label{eq:DOM_lap} 192 190 \Delta q \equiv \frac{1}{e_{1t} \, e_{2t} \, e_{3t}} 193 191 \; \lt[ \delta_i \lt( \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [q] \rt) 194 + \delta_j \lt( \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [q] \rt) \; \rt] \\192 + \delta_j \lt( \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [q] \rt) \; \rt] 195 193 + \frac{1}{e_{3t}} 196 194 \delta_k \lt[ \frac{1 }{e_{3w}} \; \delta_{k + 1/2} [q] \rt] 197 \ ]195 \end{gather*} 198 196 199 197 Following \autoref{eq:MB_curl} and \autoref{eq:MB_div}, … … 248 246 and further that the averaging operators ($\overline{\cdots}^{\, i}$, $\overline{\cdots}^{\, j}$ and 249 247 $\overline{\cdots}^{\, k}$) are symmetric linear operators, \ie 250 \begin{alignat}{ 4}248 \begin{alignat}{5} 251 249 \label{eq:DOM_di_adj} 252 250 &\sum \limits_i a_i \; \delta_i [b] &\equiv &- &&\sum \limits_i \delta _{ i + 1/2} [a] &b_{i + 1/2} \\ … … 479 477 \caption[Ocean bottom regarding coordinate systems ($z$, $s$ and hybrid $s-z$)]{ 480 478 The ocean bottom as seen by the model: 481 \begin{enumerate*}[label= {(\alph*)}]479 \begin{enumerate*}[label=(\textit{\alph*})] 482 480 \item $z$-coordinate with full step, 483 481 \item $z$-coordinate with partial step, -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex
r11599 r11630 14 14 {\footnotesize 15 15 \begin{tabularx}{\textwidth}{l||X|X} 16 Release & Author(s) & Modifications\\16 Release & Author(s) & Modifications \\ 17 17 \hline 18 {\em 4.0} & {\em ...} & {\em ...} \\ 19 {\em 3.6} & {\em ...} & {\em ...} \\ 20 {\em 3.4} & {\em ...} & {\em ...} \\ 21 {\em <=3.4} & {\em ...} & {\em ...} 18 {\em 4.0} & {\em Christian \'{E}th\'{e} } & {\em Review } \\ 19 {\em 3.6} & {\em Gurvan Madec } & {\em Update } \\ 20 {\em $\leq$ 3.4} & {\em Gurvan Madec and S\'{e}bastien Masson} & {\em First version} \\ 22 21 \end{tabularx} 23 22 } … … 34 33 the tracer equations are available depending on the vertical coordinate used and on the physics used. 35 34 In all the equations presented here, the masking has been omitted for simplicity. 36 One must be aware that all the quantities are masked fields and that each time a mean or37 difference operator is used, the resulting field is multiplied by a mask.35 One must be aware that all the quantities are masked fields and that 36 each time a mean or difference operator is used, the resulting field is multiplied by a mask. 38 37 39 38 The two active tracers are potential temperature and salinity. … … 46 45 NXT stands for next, referring to the time-stepping. 47 46 From left to right, the terms on the rhs of the tracer equations are the advection (ADV), 48 the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings 49 (SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition), 50 the contribution from the bottom boundary Layer (BBL) parametrisation, and an internal damping (DMP) term. 47 the lateral diffusion (LDF), the vertical diffusion (ZDF), 48 the contributions from the external forcings (SBC: Surface Boundary Condition, 49 QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition), 50 the contribution from the bottom boundary Layer (BBL) parametrisation, 51 and an internal damping (DMP) term. 51 52 The terms QSR, BBC, BBL and DMP are optional. 52 53 The external forcings and parameterisations require complex inputs and complex calculations … … 54 55 LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and 55 56 \autoref{chap:ZDF}, respectively. 56 Note that \mdl{tranpc}, the non-penetrative convection module, although located in57 the \path{./src/OCE/TRA} directory as it directly modifies the tracer fields,57 Note that \mdl{tranpc}, the non-penetrative convection module, 58 although located in the \path{./src/OCE/TRA} directory as it directly modifies the tracer fields, 58 59 is described with the model vertical physics (ZDF) together with 59 60 other available parameterization of convection. 60 61 61 In the present chapter we also describe the diagnostic equations used to compute the sea-water properties62 (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and freezing point with 63 associated modules \mdl{eosbn2} and \mdl{phycst}).62 In the present chapter we also describe the diagnostic equations used to 63 compute the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and 64 freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). 64 65 65 66 The different options available to the user are managed by namelist logicals. … … 70 71 71 72 The user has the option of extracting each tendency term on the RHS of the tracer equation for output 72 (\np{ln_tra_trd}{ln\_tra\_trd} or \np[=.true.]{ln_tra_mxl}{ln\_tra\_mxl}), as described in \autoref{chap:DIA}. 73 (\np{ln_tra_trd}{ln\_tra\_trd} or \np[=.true.]{ln_tra_mxl}{ln\_tra\_mxl}), 74 as described in \autoref{chap:DIA}. 73 75 74 76 %% ================================================================================================= … … 85 87 the advection tendency of a tracer is expressed in flux form, 86 88 \ie\ as the divergence of the advective fluxes. 87 Its discrete expression is given by 89 Its discrete expression is given by: 88 90 \begin{equation} 89 91 \label{eq:TRA_adv} … … 94 96 where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells. 95 97 The flux form in \autoref{eq:TRA_adv} implicitly requires the use of the continuity equation. 96 Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 97 results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ 98 (which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie\ \np[=.true.]{ln_linssh}{ln\_linssh}). 99 Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that 100 it is consistent with the continuity equation in order to enforce the conservation properties of 101 the continuous equations. 102 In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover the discrete form of 103 the continuity equation which is used to calculate the vertical velocity. 104 \begin{figure}[!t] 98 Indeed, it is obtained by using the following equality: 99 $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 100 results from the use of the continuity equation, 101 $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ 102 (which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, 103 \ie\ \np[=.true.]{ln_linssh}{ln\_linssh}). 104 Therefore it is of paramount importance to 105 design the discrete analogue of the advection tendency so that 106 it is consistent with the continuity equation in order to 107 enforce the conservation properties of the continuous equations. 108 In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover 109 the discrete form of the continuity equation which is used to calculate the vertical velocity. 110 \begin{figure} 105 111 \centering 106 112 \includegraphics[width=0.66\textwidth]{Fig_adv_scheme} … … 120 126 \end{figure} 121 127 122 The key difference between the advection schemes available in \NEMO\ is the choice made in space and123 time interpolation to define the value of the tracer at the velocity points128 The key difference between the advection schemes available in \NEMO\ is the choice made in 129 space and time interpolation to define the value of the tracer at the velocity points 124 130 (\autoref{fig:TRA_adv_scheme}). 125 131 … … 129 135 130 136 \begin{description} 131 \item [linear free surface :] (\np[=.true.]{ln_linssh}{ln\_linssh})137 \item [linear free surface] (\np[=.true.]{ln_linssh}{ln\_linssh}) 132 138 the first level thickness is constant in time: 133 the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on134 the moving surface $z = \eta$.135 There is a non-zero advective flux which is set for all advection schemes as136 $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie\ the product of surface velocity (at $z = 0$) by137 the first level tracer value.138 \item [non-linear free surface :] (\np[=.false.]{ln_linssh}{ln\_linssh})139 the vertical boundary condition is applied at the fixed surface $z = 0$ rather than 140 on the moving surface $z = \eta$. 141 There is a non-zero advective flux which is set for 142 all advection schemes as $\tau_w|_{k = 1/2} = T_{k = 1}$, 143 \ie\ the product of surface velocity (at $z = 0$) by the first level tracer value. 144 \item [non-linear free surface] (\np[=.false.]{ln_linssh}{ln\_linssh}) 139 145 convergence/divergence in the first ocean level moves the free surface up/down. 140 There is no tracer advection through it so that the advective fluxes through the surface are also zero. 146 There is no tracer advection through it so that 147 the advective fluxes through the surface are also zero. 141 148 \end{description} 142 149 143 150 In all cases, this boundary condition retains local conservation of tracer. 144 Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case. 145 Nevertheless, in the latter case, it is achieved to a good approximation since 146 the non-conservative term is the product of the time derivative of the tracer and the free surface height, 147 two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 151 Global conservation is obtained in non-linear free surface case, 152 but \textit{not} in the linear free surface case. 153 Nevertheless, in the latter case, 154 it is achieved to a good approximation since the non-conservative term is 155 the product of the time derivative of the tracer and the free surface height, 156 two quantities that are not correlated 157 \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 148 158 149 159 The velocity field that appears in (\autoref{eq:TRA_adv} is … … 153 163 (see \autoref{chap:LDF}). 154 164 155 Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), 156 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for 157 Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), 158 and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST). 159 The choice is made in the \nam{tra_adv}{tra\_adv} namelist, by setting to \forcode{.true.} one of 160 the logicals \textit{ln\_traadv\_xxx}. 161 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 162 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 163 By default (\ie\ in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}. 164 If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}), 165 the tracers will \textit{not} be advected! 165 Several tracer advection scheme are proposed, 166 namely a $2^{nd}$ or $4^{th}$ order \textbf{CEN}tred schemes (CEN), 167 a $2^{nd}$ or $4^{th}$ order \textbf{F}lux \textbf{C}orrected \textbf{T}ransport scheme (FCT), 168 a \textbf{M}onotone \textbf{U}pstream \textbf{S}cheme for 169 \textbf{C}onservative \textbf{L}aws scheme (MUSCL), 170 a $3^{rd}$ \textbf{U}pstream \textbf{B}iased \textbf{S}cheme (UBS, also often called UP3), 171 and a \textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for 172 \textbf{C}onvective \textbf{K}inematics with 173 \textbf{E}stimated \textbf{S}treaming \textbf{T}erms scheme (QUICKEST). 174 The choice is made in the \nam{tra_adv}{tra\_adv} namelist, 175 by setting to \forcode{.true.} one of the logicals \textit{ln\_traadv\_xxx}. 176 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, 177 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 178 By default (\ie\ in the reference namelist, \textit{namelist\_ref}), 179 all the logicals are set to \forcode{.false.}. 180 If the user does not select an advection scheme in the configuration namelist 181 (\textit{namelist\_cfg}), the tracers will \textit{not} be advected! 166 182 167 183 Details of the advection schemes are given below. 168 The choosing an advection scheme is a complex matter which depends on the model physics, model resolution, 169 type of tracer, as well as the issue of numerical cost. In particular, we note that 184 The choosing an advection scheme is a complex matter which depends on the 185 model physics, model resolution, type of tracer, as well as the issue of numerical cost. 186 In particular, we note that 170 187 171 188 \begin{enumerate} 172 \item CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that173 the y do not necessarily need additional diffusion;174 \item CEN and UBS are not \textit{positive} schemes 175 \footnote{negative values can appear inan initially strictly positive tracer field which is advected},189 \item CEN and FCT schemes require an explicit diffusion operator while 190 the other schemes are diffusive enough so that they do not necessarily need additional diffusion; 191 \item CEN and UBS are not \textit{positive} schemes \footnote{negative values can appear in 192 an initially strictly positive tracer field which is advected}, 176 193 implying that false extrema are permitted. 177 194 Their use is not recommended on passive tracers; 178 \item It is recommended that the same advection-diffusion scheme is used on both active and passive tracers. 195 \item It is recommended that the same advection-diffusion scheme is used on 196 both active and passive tracers. 179 197 \end{enumerate} 180 198 181 Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and 182 passive tracers can create very nice-looking frontal structures that are pure numerical artefacts. 199 Indeed, if a source or sink of a passive tracer depends on an active one, 200 the difference of treatment of active and passive tracers can create 201 very nice-looking frontal structures that are pure numerical artefacts. 183 202 Nevertheless, most of our users set a different treatment on passive and active tracers, 184 203 that's the reason why this possibility is offered. 185 We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of186 their results.204 We strongly suggest them to perform a sensitivity experiment using a same treatment to 205 assess the robustness of their results. 187 206 188 207 %% ================================================================================================= … … 192 211 % 2nd order centred scheme 193 212 194 The centred advection scheme (CEN) is used when \np[=.true.]{ln_traadv_cen}{ln\_traadv\_cen}. 195 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 213 The \textbf{CEN}tred advection scheme (CEN) is used when \np[=.true.]{ln_traadv_cen}{ln\_traadv\_cen}. 214 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on 215 horizontal (iso-level) and vertical direction by 196 216 setting \np{nn_cen_h}{nn\_cen\_h} and \np{nn_cen_v}{nn\_cen\_v} to $2$ or $4$. 197 217 CEN implementation can be found in the \mdl{traadv\_cen} module. 198 218 199 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of200 the two neighbouring $T$-point values.219 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as 220 the mean of the two neighbouring $T$-point values. 201 221 For example, in the $i$-direction : 202 222 \begin{equation} … … 205 225 \end{equation} 206 226 207 CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but dispersive 208 (\ie\ it may create false extrema). 209 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 210 produce a sensible solution. 211 The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 227 CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but 228 dispersive (\ie\ it may create false extrema). 229 It is therefore notoriously noisy and must be used in conjunction with 230 an explicit diffusion operator to produce a sensible solution. 231 The associated time-stepping is performed using 232 a leapfrog scheme in conjunction with an Asselin time-filter, 212 233 so $T$ in (\autoref{eq:TRA_adv_cen2}) is the \textit{now} tracer value. 213 234 … … 217 238 % 4nd order centred scheme 218 239 219 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as 220 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 240 In the $4^{th}$ order formulation (CEN4), 241 tracer values are evaluated at u- and v-points as a $4^{th}$ order interpolation, 242 and thus depend on the four neighbouring $T$-points. 221 243 For example, in the $i$-direction: 222 244 \begin{equation} … … 226 248 In the vertical direction (\np[=4]{nn_cen_v}{nn\_cen\_v}), 227 249 a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}. 228 In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion, 229 spectral characteristics similar to schemes of higher order \citep{lele_JCP92}. 250 In the COMPACT scheme, both the field and its derivative are interpolated, 251 which leads, after a matrix inversion, spectral characteristics similar to schemes of higher order 252 \citep{lele_JCP92}. 230 253 231 254 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but 232 255 a $4^{th}$ order evaluation of advective fluxes, 233 256 since the divergence of advective fluxes \autoref{eq:TRA_adv} is kept at $2^{nd}$ order. 234 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with235 the scheme presented here.236 Introducing a \forcode{.true.}$4^{th}$ order advection scheme is feasible but, for consistency reasons,237 it requires changes in the discretisation of the tracer advection together with changes in the continuity equation,238 and the momentum advection and pressure terms.257 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is 258 usually associated with the scheme presented here. 259 Introducing a ``true'' $4^{th}$ order advection scheme is feasible but, for consistency reasons, 260 it requires changes in the discretisation of the tracer advection together with 261 changes in the continuity equation, and the momentum advection and pressure terms. 239 262 240 263 A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive, 241 264 \ie\ the global variance of a tracer is not preserved using CEN4. 242 Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. 243 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 244 so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer. 265 Furthermore, it must be used in conjunction with an explicit diffusion operator to 266 produce a sensible solution. 267 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with 268 an Asselin time-filter, so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer. 245 269 246 270 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), … … 248 272 This hypothesis usually reduces the order of the scheme. 249 273 Here we choose to set the gradient of $T$ across the boundary to zero. 250 Alternative conditions can be specified, such as a reduction to a second order scheme for251 these near boundary grid points.274 Alternative conditions can be specified, 275 such as a reduction to a second order scheme for these near boundary grid points. 252 276 253 277 %% ================================================================================================= … … 255 279 \label{subsec:TRA_adv_tvd} 256 280 257 The Flux Corrected Transport schemes (FCT) is used when \np[=.true.]{ln_traadv_fct}{ln\_traadv\_fct}. 258 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 281 The \textbf{F}lux \textbf{C}orrected \textbf{T}ransport schemes (FCT) is used when 282 \np[=.true.]{ln_traadv_fct}{ln\_traadv\_fct}. 283 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on 284 horizontal (iso-level) and vertical direction by 259 285 setting \np{nn_fct_h}{nn\_fct\_h} and \np{nn_fct_v}{nn\_fct\_v} to $2$ or $4$. 260 286 FCT implementation can be found in the \mdl{traadv\_fct} module. 261 287 262 In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and263 a c entred scheme.288 In FCT formulation, the tracer at velocity points is evaluated using 289 a combination of an upstream and a centred scheme. 264 290 For example, in the $i$-direction : 265 291 \begin{equation} … … 270 296 T_{i + 1} & \text{if~} u_{i + 1/2} < 0 \\ 271 297 T_i & \text{if~} u_{i + 1/2} \geq 0 \\ 272 \end{cases} 273 \\ 298 \end{cases} \\ 274 299 \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big) 275 300 \end{split} … … 288 313 $\tau_u^{cen}$ is evaluated in (\autoref{eq:TRA_adv_fct}) using the \textit{now} tracer while 289 314 $\tau_u^{ups}$ is evaluated using the \textit{before} tracer. 290 In other words, the advective part of the scheme is time stepped with a leap-frog scheme 291 whilea forward scheme is used for the diffusive part.315 In other words, the advective part of the scheme is time stepped with a leap-frog scheme while 316 a forward scheme is used for the diffusive part. 292 317 293 318 %% ================================================================================================= … … 295 320 \label{subsec:TRA_adv_mus} 296 321 297 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np[=.true.]{ln_traadv_mus}{ln\_traadv\_mus}. 322 The \textbf{M}onotone \textbf{U}pstream \textbf{S}cheme for \textbf{C}onservative \textbf{L}aws 323 (MUSCL) is used when \np[=.true.]{ln_traadv_mus}{ln\_traadv\_mus}. 298 324 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 299 325 300 326 MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}. 301 In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between302 two $T$-points (\autoref{fig:TRA_adv_scheme}).327 In its formulation, the tracer at velocity points is evaluated assuming 328 a linear tracer variation between two $T$-points (\autoref{fig:TRA_adv_scheme}). 303 329 For example, in the $i$-direction : 304 \ begin{equation}330 \[ 305 331 % \label{eq:TRA_adv_mus} 306 332 \tau_u^{mus} = \lt\{ 307 333 \begin{split} 308 \tau_i&+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)309 \widetilde{\partial_i\tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\310 311 334 \tau_i &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 335 \widetilde{\partial_i \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\ 336 \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 337 \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} < 0 312 338 \end{split} 313 339 \rt. 314 \ end{equation}315 where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to316 ensure the \textit{positive} character of the scheme.317 318 The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to319 evaluate $\tau_u^{mus}$.340 \] 341 where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which 342 a limitation is imposed to ensure the \textit{positive} character of the scheme. 343 344 The time stepping is performed using a forward scheme, 345 that is the \textit{before} tracer field is used to evaluate $\tau_u^{mus}$. 320 346 321 347 For an ocean grid point adjacent to land and where the ocean velocity is directed toward land, 322 348 an upstream flux is used. 323 349 This choice ensure the \textit{positive} character of the scheme. 324 In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes325 (\np[=.true.]{ln_mus_ups}{ln\_mus\_ups}).350 In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using 351 upstream fluxes (\np[=.true.]{ln_mus_ups}{ln\_mus\_ups}). 326 352 327 353 %% ================================================================================================= … … 329 355 \label{subsec:TRA_adv_ubs} 330 356 331 The Upstream-Biased Scheme (UBS) is used when \np[=.true.]{ln_traadv_ubs}{ln\_traadv\_ubs}. 357 The \textbf{U}pstream-\textbf{B}iased \textbf{S}cheme (UBS) is used when 358 \np[=.true.]{ln_traadv_ubs}{ln\_traadv\_ubs}. 332 359 UBS implementation can be found in the \mdl{traadv\_mus} module. 333 360 334 361 The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme 335 (Quadratic Upstream Interpolation for Convective Kinematics). 362 (\textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for 363 \textbf{C}onvective \textbf{K}inematics). 336 364 It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation. 337 365 For example, in the $i$-direction: … … 340 368 \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6} 341 369 \begin{cases} 342 343 370 \tau"_i & \text{if~} u_{i + 1/2} \geqslant 0 \\ 371 \tau"_{i + 1} & \text{if~} u_{i + 1/2} < 0 344 372 \end{cases} 345 \quad 346 \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt] 373 \quad \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt] 347 374 \end{equation} 348 375 349 376 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error 350 377 \citep{shchepetkin.mcwilliams_OM05}. 351 The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}. 378 The overall performance of the advection scheme is similar to that reported in 379 \cite{farrow.stevens_JPO95}. 352 380 It is a relatively good compromise between accuracy and smoothness. 353 381 Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted, 354 382 but the amplitude of such are significantly reduced over the centred second or fourth order method. 355 Therefore it is not recommended that it should be applied to a passive tracer that requires positivity. 383 Therefore it is not recommended that it should be applied to 384 a passive tracer that requires positivity. 356 385 357 386 The intrinsic diffusion of UBS makes its use risky in the vertical direction where 358 387 the control of artificial diapycnal fluxes is of paramount importance 359 388 \citep{shchepetkin.mcwilliams_OM05, demange_phd14}. 360 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme361 (\np[=2 or 4]{nn_ubs_v}{nn\_ubs\_v}).362 363 For stability reasons (see \autoref{chap:TD}), the first term in \autoref{eq:TRA_adv_ubs}364 (which corresponds to a second order centred scheme)365 is evaluated using the \textit{now} tracer (centred in time) while the second term366 (which is the diffusive part of the scheme),389 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or 390 a $4^th$ order COMPACT scheme (\np[=2 or 4]{nn_ubs_v}{nn\_ubs\_v}). 391 392 For stability reasons (see \autoref{chap:TD}), 393 the first term in \autoref{eq:TRA_adv_ubs} (which corresponds to a second order centred scheme) 394 is evaluated using the \textit{now} tracer (centred in time) while 395 the second term (which is the diffusive part of the scheme), 367 396 is evaluated using the \textit{before} tracer (forward in time). 368 This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the QUICK advection scheme. 397 This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in 398 the context of the QUICK advection scheme. 369 399 UBS and QUICK schemes only differ by one coefficient. 370 Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 400 Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme 401 \citep{webb.de-cuevas.ea_JAOT98}. 371 402 This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 372 Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 403 Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and 404 obtain a QUICK scheme. 373 405 374 406 Note that it is straightforward to rewrite \autoref{eq:TRA_adv_ubs} as follows: … … 389 421 Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which 390 422 an upstream-biased diffusion term is added. 391 Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to 392 be evaluated at the \textit{now} time step using \autoref{eq:TRA_adv_ubs}. 393 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which 394 is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 395 Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:TRA_adv_ubs}. 423 Secondly, 424 this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to be evaluated at the \textit{now} time step using \autoref{eq:TRA_adv_ubs}. 425 Thirdly, the diffusion term is in fact a biharmonic operator with 426 an eddy coefficient which is simply proportional to the velocity: 427 $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 428 Note the current version of \NEMO\ uses the computationally more efficient formulation 429 \autoref{eq:TRA_adv_ubs}. 396 430 397 431 %% ================================================================================================= … … 399 433 \label{subsec:TRA_adv_qck} 400 434 401 The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme 402 proposed by \citet{leonard_CMAME79} is used when \np[=.true.]{ln_traadv_qck}{ln\_traadv\_qck}. 435 The \textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for 436 \textbf{C}onvective \textbf{K}inematics with \textbf{E}stimated \textbf{S}treaming \textbf{T}erms 437 (QUICKEST) scheme proposed by \citet{leonard_CMAME79} is used when 438 \np[=.true.]{ln_traadv_qck}{ln\_traadv\_qck}. 403 439 QUICKEST implementation can be found in the \mdl{traadv\_qck} module. 404 440 405 441 QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter 406 442 \citep{leonard_CMAME91}. 407 It has been implemented in \NEMO\ by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. 443 It has been implemented in \NEMO\ by G. Reffray (Mercator Ocean) and 444 can be found in the \mdl{traadv\_qck} module. 408 445 The resulting scheme is quite expensive but \textit{positive}. 409 446 It can be used on both active and passive tracers. … … 412 449 Therefore the vertical flux is evaluated using the CEN2 scheme. 413 450 This no longer guarantees the positivity of the scheme. 414 The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property. 451 The use of FCT in the vertical direction (as for the UBS case) should be implemented to 452 restore this property. 415 453 416 454 %%%gmcomment : Cross term are missing in the current implementation.... … … 428 466 Options are defined through the \nam{tra_ldf}{tra\_ldf} namelist variables. 429 467 They are regrouped in four items, allowing to specify 430 $(i)$ the type of operator used (none, laplacian, bilaplacian), 431 $(ii)$ the direction along which the operator acts (iso-level, horizontal, iso-neutral), 432 $(iii)$ some specific options related to the rotated operators (\ie\ non-iso-level operator), and 433 $(iv)$ the specification of eddy diffusivity coefficient (either constant or variable in space and time). 434 Item $(iv)$ will be described in \autoref{chap:LDF}. 468 \begin{enumerate*}[label=(\textit{\roman*})] 469 \item the type of operator used (none, laplacian, bilaplacian), 470 \item the direction along which the operator acts (iso-level, horizontal, iso-neutral), 471 \item some specific options related to the rotated operators (\ie\ non-iso-level operator), and 472 \item the specification of eddy diffusivity coefficient 473 (either constant or variable in space and time). 474 \end{enumerate*} 475 Item (iv) will be described in \autoref{chap:LDF}. 435 476 The direction along which the operators act is defined through the slope between 436 477 this direction and the iso-level surfaces. … … 440 481 \ie\ the tracers appearing in its expression are the \textit{before} tracers in time, 441 482 except for the pure vertical component that appears when a rotation tensor is used. 442 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}). 443 When \np[=.true.]{ln_traldf_msc}{ln\_traldf\_msc}, a Method of Stabilizing Correction is used in which 444 the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}. 483 This latter component is solved implicitly together with the vertical diffusion term 484 (see \autoref{chap:TD}). 485 When \np[=.true.]{ln_traldf_msc}{ln\_traldf\_msc}, 486 a Method of Stabilizing Correction is used in which the pure vertical component is split into 487 an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}. 445 488 446 489 %% ================================================================================================= … … 451 494 452 495 \begin{description} 453 \item [{\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}}] no operator selected, the lateral diffusive tendency will not be applied to the tracer equation. 454 This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example). 496 \item [{\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}}] no operator selected, 497 the lateral diffusive tendency will not be applied to the tracer equation. 498 This option can be used when the selected advection scheme is diffusive enough 499 (MUSCL scheme for example). 455 500 \item [{\np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap}}] a laplacian operator is selected. 456 This harmonic operator takes the following expression: $\mathcal{L}(T) = \nabla \cdot A_{ht} \; \nabla T $, 501 This harmonic operator takes the following expression: 502 $\mathcal{L}(T) = \nabla \cdot A_{ht} \; \nabla T $, 457 503 where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}), 458 504 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}). … … 461 507 $\mathcal{B} = - \mathcal{L}(\mathcal{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$ 462 508 where the gradient operats along the selected direction, 463 and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}). 509 and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ 510 (see \autoref{chap:LDF}). 464 511 In the code, the bilaplacian operator is obtained by calling the laplacian twice. 465 512 \end{description} … … 469 516 minimizing the impact on the larger scale features. 470 517 The main difference between the two operators is the scale selectiveness. 471 The bilaplacian damping time (\ie\ its spin down time) scales like $\lambda^{-4}$ for 472 disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones), 518 The bilaplacian damping time (\ie\ its spin down time) scales like 519 $\lambda^{-4}$ for disturbances of wavelength $\lambda$ 520 (so that short waves damped more rapidelly than long ones), 473 521 whereas the laplacian damping time scales only like $\lambda^{-2}$. 474 522 … … 479 527 The choice of a direction of action determines the form of operator used. 480 528 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when 481 iso-level option is used (\np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev}) or 482 whena horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate529 iso-level option is used (\np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev}) or when 530 a horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate 483 531 (\np{ln_traldf_hor}{ln\_traldf\_hor} and \np[=.true.]{ln_zco}{ln\_zco}). 484 532 The associated code can be found in the \mdl{traldf\_lap\_blp} module. … … 489 537 see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or 490 538 when a horizontal (\ie\ geopotential) operator is demanded in \textit{s}-coordinate 491 (\np{ln_traldf_hor}{ln\_traldf\_hor} and \np{ln_sco}{ln\_sco} = \forcode{.true.}) 492 \footnote{In this case, the standard iso-neutral operator will be automatically selected}.539 (\np{ln_traldf_hor}{ln\_traldf\_hor} and \np{ln_sco}{ln\_sco} = \forcode{.true.}) \footnote{ 540 In this case, the standard iso-neutral operator will be automatically selected}. 493 541 In that case, a rotation is applied to the gradient(s) that appears in the operator so that 494 542 diffusive fluxes acts on the three spatial direction. … … 511 559 first (and third in bilaplacian case) horizontal tracer derivative are masked. 512 560 It is implemented in the \rou{tra\_ldf\_lap} subroutine found in the \mdl{traldf\_lap\_blp} module. 513 The module also contains \rou{tra\_ldf\_blp}, the subroutine calling twice \rou{tra\_ldf\_lap} in order to 561 The module also contains \rou{tra\_ldf\_blp}, 562 the subroutine calling twice \rou{tra\_ldf\_lap} in order to 514 563 compute the iso-level bilaplacian operator. 515 564 516 565 It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in 517 the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 518 It is thus used when, in addition to \np{ln_traldf_lap}{ln\_traldf\_lap} or \np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}, 519 we have \np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev} or \np{ln_traldf_hor}{ln\_traldf\_hor}~=~\np[=.true.]{ln_zco}{ln\_zco}. 566 the $z$-coordinate with or without partial steps, 567 but is simply an iso-level operator in the $s$-coordinate. 568 It is thus used when, 569 in addition to \np{ln_traldf_lap}{ln\_traldf\_lap} or \np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}, 570 we have \np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev} or 571 \np[=]{ln_traldf_hor}{ln\_traldf\_hor}\np[=.true.]{ln_zco}{ln\_zco}. 520 572 In both cases, it significantly contributes to diapycnal mixing. 521 573 It is therefore never recommended, even when using it in the bilaplacian case. … … 523 575 Note that in the partial step $z$-coordinate (\np[=.true.]{ln_zps}{ln\_zps}), 524 576 tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom. 525 In this case, horizontal derivatives in (\autoref{eq:TRA_ldf_lap}) at the bottom level require a specific treatment. 577 In this case, 578 horizontal derivatives in (\autoref{eq:TRA_ldf_lap}) at the bottom level require a specific treatment. 526 579 They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}. 527 580 … … 533 586 \subsubsection[Standard rotated (bi-)laplacian operator (\textit{traldf\_iso.F90})]{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 534 587 \label{subsec:TRA_ldf_iso} 588 535 589 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:MB_zdf}) 536 takes the following semi 590 takes the following semi-discrete space form in $z$- and $s$-coordinates: 537 591 \begin{equation} 538 592 \label{eq:TRA_ldf_iso} … … 554 608 or both \np[=.true.]{ln_traldf_hor}{ln\_traldf\_hor} and \np[=.true.]{ln_zco}{ln\_zco}. 555 609 The way these slopes are evaluated is given in \autoref{sec:LDF_slp}. 556 At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using 557 the mask technique (see \autoref{sec:LBC_coast}). 610 At the surface, bottom and lateral boundaries, 611 the turbulent fluxes of heat and salt are set to zero using the mask technique 612 (see \autoref{sec:LBC_coast}). 558 613 559 614 The operator in \autoref{eq:TRA_ldf_iso} involves both lateral and vertical derivatives. 560 For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that561 used in the vertical physics (see \autoref{sec:TRA_zdf}).615 For numerical stability, the vertical second derivative must be solved using 616 the same implicit time scheme as that used in the vertical physics (see \autoref{sec:TRA_zdf}). 562 617 For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module, 563 618 but in the \mdl{trazdf} module where, if iso-neutral mixing is used, 564 the vertical mixing coefficient is simply increased by $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$. 619 the vertical mixing coefficient is simply increased by 620 $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$. 565 621 566 622 This formulation conserves the tracer but does not ensure the decrease of the tracer variance. 567 Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without568 any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}.623 Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to 624 run safely without any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}. 569 625 570 626 Note that in the partial step $z$-coordinate (\np[=.true.]{ln_zps}{ln\_zps}), 571 the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require a specific treatment. 627 the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require 628 a specific treatment. 572 629 They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 573 630 … … 576 633 \label{subsec:TRA_ldf_triad} 577 634 578 An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases 579 is also available in \NEMO\ (\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}). 635 An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which 636 ensures tracer variance decreases is also available in \NEMO\ 637 (\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}). 580 638 A complete description of the algorithm is given in \autoref{apdx:TRIADS}. 581 639 582 The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:TRA_ldf_lap}) twice. 640 The lateral fourth order bilaplacian operator on tracers is obtained by 641 applying (\autoref{eq:TRA_ldf_lap}) twice. 583 642 The operator requires an additional assumption on boundary conditions: 584 643 both first and third derivative terms normal to the coast are set to zero. 585 644 586 The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:TRA_ldf_iso}) twice. 645 The lateral fourth order operator formulation on tracers is obtained by 646 applying (\autoref{eq:TRA_ldf_iso}) twice. 587 647 It requires an additional assumption on boundary conditions: 588 648 first and third derivative terms normal to the coast, … … 593 653 \label{subsec:TRA_ldf_options} 594 654 595 \begin{itemize} 596 \item \np{ln_traldf_msc}{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) 597 \item \np{rn_slpmax}{rn\_slpmax} = slope limit (both operators) 598 \item \np{ln_triad_iso}{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 599 \item \np{rn_sw_triad}{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only) 600 \item \np{ln_botmix_triad}{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 601 \end{itemize} 655 \begin{labeling}{{\np{ln_botmix_triad}{ln\_botmix\_triad}}} 656 \item [{\np{ln_traldf_msc}{ln\_traldf\_msc} }] Method of Stabilizing Correction (both operators) 657 \item [{\np{rn_slpmax}{rn\_slpmax} }] Slope limit (both operators) 658 \item [{\np{ln_triad_iso}{ln\_triad\_iso} }] Pure horizontal mixing in ML (triad only) 659 \item [{\np{rn_sw_triad}{rn\_sw\_triad} }] \forcode{=1} switching triad; 660 \forcode{= 0} all 4 triads used (triad only) 661 \item [{\np{ln_botmix_triad}{ln\_botmix\_triad}}] Lateral mixing on bottom (triad only) 662 \end{labeling} 602 663 603 664 %% ================================================================================================= … … 606 667 607 668 Options are defined through the \nam{zdf}{zdf} namelist variables. 608 The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, 609 and is based on a laplacian operator. 610 The vertical diffusion operator given by (\autoref{eq:MB_zdf}) takes the following semi -discrete space form: 611 \begin{gather*} 669 The formulation of the vertical subgrid scale tracer physics is the same for 670 all the vertical coordinates, and is based on a laplacian operator. 671 The vertical diffusion operator given by (\autoref{eq:MB_zdf}) takes 672 the following semi-discrete space form: 673 \[ 612 674 % \label{eq:TRA_zdf} 613 D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\614 615 \ end{gather*}616 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity,617 respectively.675 D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \quad 676 D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] 677 \] 678 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on 679 temperature and salinity, respectively. 618 680 Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised 619 681 (\ie\ \np[=.true.]{ln_zdfddm}{ln\_zdfddm},). 620 682 The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF). 621 Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by622 $\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of 623 \autoref{eq:TRA_ldf_iso}.683 Furthermore, when iso-neutral mixing is used, 684 both mixing coefficients are increased by $\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to 685 account for the vertical second derivative of \autoref{eq:TRA_ldf_iso}. 624 686 625 687 At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified. … … 628 690 a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}). 629 691 630 The large eddy coefficient found in the mixed layer together with high vertical resolution implies that631 th ere would be too restrictive constraint on the time step if we use explicit time stepping.692 The large eddy coefficient found in the mixed layer together with high vertical resolution implies 693 that there would be too restrictive constraint on the time step if we use explicit time stepping. 632 694 Therefore an implicit time stepping is preferred for the vertical diffusion since 633 695 it overcomes the stability constraint. … … 648 710 649 711 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 650 (\ie\ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due 651 both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and 712 (\ie\ atmosphere, sea-ice, land), 713 the change in the heat and salt content of the surface layer of the ocean is due both to 714 the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and 652 715 to the heat and salt content of the mass exchange. 653 716 They are both included directly in $Q_{ns}$, the surface heat flux, 654 717 and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details). 655 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 656 657 The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers): 658 659 \begin{itemize} 660 \item $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 661 (\ie\ the difference between the total surface heat flux and the fraction of the short wave flux that 662 penetrates into the water column, see \autoref{subsec:TRA_qsr}) 718 By doing this, the forcing formulation is the same for any tracer 719 (including temperature and salinity). 720 721 The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields 722 (used on tracers): 723 724 \begin{labeling}{\textit{fwfisf}} 725 \item [$Q_{ns}$] The non-solar part of the net surface heat flux that crosses the sea surface 726 (\ie\ the difference between the total surface heat flux and 727 the fraction of the short wave flux that penetrates into the water column, 728 see \autoref{subsec:TRA_qsr}) 663 729 plus the heat content associated with of the mass exchange with the atmosphere and lands. 664 \item $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 665 \item \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and 730 \item [\textit{sfx}] The salt flux resulting from ice-ocean mass exchange 731 (freezing, melting, ridging...) 732 \item [\textit{emp}] The mass flux exchanged with the atmosphere (evaporation minus precipitation) and 666 733 possibly with the sea-ice and ice-shelves. 667 \item \textit{rnf}, the mass flux associated with runoff734 \item [\textit{rnf}] The mass flux associated with runoff 668 735 (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 669 \item \textit{fwfisf}, the mass flux associated with ice shelf melt,736 \item [\textit{fwfisf}] The mass flux associated with ice shelf melt, 670 737 (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 671 \end{ itemize}738 \end{labeling} 672 739 673 740 The surface boundary condition on temperature and salinity is applied as follows: 674 741 \begin{equation} 675 742 \label{eq:TRA_sbc} 676 \begin{alignedat}{2} 677 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns} }^t \\ 678 F^S &= &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t 679 \end{alignedat} 743 F^T = \frac{1}{C_p} \frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} \overline{Q_{ns} }^t \qquad 744 F^S = \frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} \overline{\textit{sfx}}^t 680 745 \end{equation} 681 746 where $\overline x^t$ means that $x$ is averaged over two consecutive time steps … … 683 748 Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:TD}). 684 749 685 In the linear free surface case (\np[=.true.]{ln_linssh}{ln\_linssh}), an additional term has to be added on 686 both temperature and salinity. 687 On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$. 688 On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in 689 the volume of the first level. 750 In the linear free surface case (\np[=.true.]{ln_linssh}{ln\_linssh}), 751 an additional term has to be added on both temperature and salinity. 752 On temperature, this term remove the heat content associated with 753 mass exchange that has been added to $Q_{ns}$. 754 On salinity, this term mimics the concentration/dilution effect that would have resulted from 755 a change in the volume of the first level. 690 756 The resulting surface boundary condition is applied as follows: 691 757 \begin{equation} 692 758 \label{eq:TRA_sbc_lin} 693 \begin{alignedat}{2} 694 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 695 &\overline{(Q_{ns} - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\ 696 F^S &= &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 697 &\overline{(\textit{sfx} - \textit{emp} \lt. S \rt|_{k = 1})}^t 698 \end{alignedat} 699 \end{equation} 700 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 759 F^T = \frac{1}{C_p} \frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 760 \overline{(Q_{ns} - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \qquad 761 F^S = \frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 762 \overline{(\textit{sfx} - \textit{emp} \lt. S \rt|_{k = 1})}^t 763 \end{equation} 764 Note that an exact conservation of heat and salt content is only achieved with 765 non-linear free surface. 701 766 In the linear free surface case, there is a small imbalance. 702 The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}. 703 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:TD}). 767 The imbalance is larger than the imbalance associated with the Asselin time filter 768 \citep{leclair.madec_OM09}. 769 This is the reason why the modified filter is not applied in the linear free surface case 770 (see \autoref{chap:TD}). 704 771 705 772 %% ================================================================================================= … … 716 783 When the penetrative solar radiation option is used (\np[=.true.]{ln_traqsr}{ln\_traqsr}), 717 784 the solar radiation penetrates the top few tens of meters of the ocean. 718 If it is not used (\np[=.false.]{ln_traqsr}{ln\_traqsr}) all the heat flux is absorbed in the first ocean level. 719 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:MB_PE_tra_T} and 720 the surface boundary condition is modified to take into account only the non-penetrative part of the surface 721 heat flux: 785 If it is not used (\np[=.false.]{ln_traqsr}{ln\_traqsr}) all the heat flux is absorbed in 786 the first ocean level. 787 Thus, in the former case a term is added to the time evolution equation of temperature 788 \autoref{eq:MB_PE_tra_T} and the surface boundary condition is modified to 789 take into account only the non-penetrative part of the surface heat flux: 722 790 \begin{equation} 723 791 \label{eq:TRA_PE_qsr} … … 736 804 737 805 The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. 738 The ocean is strongly absorbing for wavelengths longer than 700 ~nm and these wavelengths contribute to739 heatingthe upper few tens of centimetres.740 The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$806 The ocean is strongly absorbing for wavelengths longer than 700 $nm$ and 807 these wavelengths contribute to heat the upper few tens of centimetres. 808 The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim$ 58\% 741 809 (specified through namelist parameter \np{rn_abs}{rn\_abs}). 742 It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 743 of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn_si0}{rn\_si0} in the \nam{tra_qsr}{tra\_qsr} namelist). 744 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to 745 larger depths where it contributes to local heating. 746 The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen. 810 It is assumed to penetrate the ocean with a decreasing exponential profile, 811 with an e-folding depth scale, $\xi_0$, of a few tens of centimetres 812 (typically $\xi_0 = 0.35~m$ set as \np{rn_si0}{rn\_si0} in the \nam{tra_qsr}{tra\_qsr} namelist). 813 For shorter wavelengths (400-700 $nm$), the ocean is more transparent, 814 and solar energy propagates to larger depths where it contributes to local heating. 815 The way this second part of the solar energy penetrates into 816 the ocean depends on which formulation is chosen. 747 817 In the simple 2-waveband light penetration scheme (\np[=.true.]{ln_qsr_2bd}{ln\_qsr\_2bd}) 748 818 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, … … 754 824 where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 755 825 It is usually chosen to be 23~m by setting the \np{rn_si0}{rn\_si0} namelist parameter. 756 The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification757 (oligotrophic waters).826 The set of default values ($\xi_0, \xi_1, R$) corresponds to 827 a Type I water in Jerlov's (1968) classification (oligotrophic waters). 758 828 759 829 Such assumptions have been shown to provide a very crude and simplistic representation of … … 763 833 a 61 waveband formulation. 764 834 Unfortunately, such a model is very computationally expensive. 765 Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of this formulation in which 766 visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm). 767 For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from 768 the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}), 769 assuming the same power-law relationship. 770 As shown in \autoref{fig:TRA_qsr_irradiance}, this formulation, called RGB (Red-Green-Blue), 835 Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of 836 this formulation in which visible light is split into three wavebands: 837 blue (400-500 $nm$), green (500-600 $nm$) and red (600-700 $nm$). 838 For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to 839 the coefficients computed from the full spectral model of \cite{morel_JGR88} 840 (as modified by \cite{morel.maritorena_JGR01}), assuming the same power-law relationship. 841 As shown in \autoref{fig:TRA_qsr_irradiance}, this formulation, 842 called RGB (\textbf{R}ed-\textbf{G}reen-\textbf{B}lue), 771 843 reproduces quite closely the light penetration profiles predicted by the full spectal model, 772 844 but with much greater computational efficiency. … … 774 846 775 847 The RGB formulation is used when \np[=.true.]{ln_qsr_rgb}{ln\_qsr\_rgb}. 776 The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are tabulated over777 61 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L 848 The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are 849 tabulated over 61 nonuniform chlorophyll classes ranging from 0.01 to 10 $g.Chl/L$ 778 850 (see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module). 779 851 Four types of chlorophyll can be chosen in the RGB formulation: 780 852 781 853 \begin{description} 782 \item [{\np[=0]{nn_chldta}{nn\_chldta}}] a constant 0.05 g.Chl/L value everywhere ; 783 \item [{\np[=1]{nn_chldta}{nn\_chldta}}] an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in the vertical direction; 784 \item [{\np[=2]{nn_chldta}{nn\_chldta}}] same as previous case except that a vertical profile of chlorophyl is used. 785 Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value; 786 \item [{\np[=.true.]{ln_qsr_bio}{ln\_qsr\_bio}}] simulated time varying chlorophyll by TOP biogeochemical model. 787 In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in 788 PISCES and the oceanic heating rate. 854 \item [{\np[=0]{nn_chldta}{nn\_chldta}}] a constant 0.05 $g.Chl/L$ value everywhere; 855 \item [{\np[=1]{nn_chldta}{nn\_chldta}}] an observed time varying chlorophyll deduced from 856 satellite surface ocean color measurement spread uniformly in the vertical direction; 857 \item [{\np[=2]{nn_chldta}{nn\_chldta}}] same as previous case except that 858 a vertical profile of chlorophyl is used. 859 Following \cite{morel.berthon_LO89}, 860 the profile is computed from the local surface chlorophyll value; 861 \item [{\np[=.true.]{ln_qsr_bio}{ln\_qsr\_bio}}] simulated time varying chlorophyll by 862 \TOP\ biogeochemical model. 863 In this case, the RGB formulation is used to calculate both 864 the phytoplankton light limitation in \PISCES\ and the oceanic heating rate. 789 865 \end{description} 790 866 … … 797 873 (\ie\ it is less than the computer precision) is computed once, 798 874 and the trend associated with the penetration of the solar radiation is only added down to that level. 799 Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor. 875 Finally, note that when the ocean is shallow ($<$ 200~m), 876 part of the solar radiation can reach the ocean floor. 800 877 In this case, we have chosen that all remaining radiation is absorbed in the last ocean level 801 878 (\ie\ $I$ is masked). 802 879 803 \begin{figure} [!t]880 \begin{figure} 804 881 \centering 805 882 \includegraphics[width=0.66\textwidth]{Fig_TRA_Irradiance} … … 810 887 4 waveband RGB formulation (red), 811 888 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 812 (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$.889 (a) Chl=0.05 $mg/m^3$ and (b) Chl=0.5 $mg/m^3$. 813 890 From \citet{lengaigne.menkes.ea_CD07}.} 814 891 \label{fig:TRA_qsr_irradiance} … … 824 901 \label{lst:nambbc} 825 902 \end{listing} 826 \begin{figure}[!t] 903 904 \begin{figure} 827 905 \centering 828 906 \includegraphics[width=0.66\textwidth]{Fig_TRA_geoth} … … 836 914 \ie\ a no flux boundary condition is applied on active tracers at the bottom. 837 915 This is the default option in \NEMO, and it is implemented using the masking technique. 838 However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling. 839 This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}), 916 However, there is a non-zero heat flux across the seafloor that 917 is associated with solid earth cooling. 918 This flux is weak compared to surface fluxes 919 (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}), 840 920 but it warms systematically the ocean and acts on the densest water masses. 841 921 Taking this flux into account in a global ocean model increases the deepest overturning cell 842 (\ie\ the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}. 922 (\ie\ the one associated with the Antarctic Bottom Water) by 923 a few Sverdrups \citep{emile-geay.madec_OS09}. 843 924 844 925 Options are defined through the \nam{bbc}{bbc} namelist variables. 845 The presence of geothermal heating is controlled by setting the namelist parameter \np{ln_trabbc}{ln\_trabbc} to true. 846 Then, when \np{nn_geoflx}{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by 847 the \np{rn_geoflx_cst}{rn\_geoflx\_cst}, which is also a namelist parameter. 848 When \np{nn_geoflx}{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in 849 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}. 926 The presence of geothermal heating is controlled by 927 setting the namelist parameter \np{ln_trabbc}{ln\_trabbc} to true. 928 Then, when \np{nn_geoflx}{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose 929 value is given by the \np{rn_geoflx_cst}{rn\_geoflx\_cst}, which is also a namelist parameter. 930 When \np{nn_geoflx}{nn\_geoflx} is set to 2, 931 a spatially varying geothermal heat flux is introduced which is provided in 932 the \ifile{geothermal\_heating} NetCDF file 933 (\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}. 850 934 851 935 %% ================================================================================================= … … 865 949 where dense water formed in marginal seas flows into a basin filled with less dense water, 866 950 or along the continental slope when dense water masses are formed on a continental shelf. 867 The amount of entrainment that occurs in these gravity plumes is critical in determining the density and 868 volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, or North Atlantic Deep Water. 951 The amount of entrainment that occurs in these gravity plumes is critical in 952 determining the density and volume flux of the densest waters of the ocean, 953 such as Antarctic Bottom Water, or North Atlantic Deep Water. 869 954 $z$-coordinate models tend to overestimate the entrainment, 870 because the gravity flow is mixed vertically by convection as it goes ''downstairs'' following the step topography, 955 because the gravity flow is mixed vertically by convection as 956 it goes ''downstairs'' following the step topography, 871 957 sometimes over a thickness much larger than the thickness of the observed gravity plume. 872 A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of 873 a sill \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved. 874 875 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{beckmann.doscher_JPO97}, 958 A similar problem occurs in the $s$-coordinate when 959 the thickness of the bottom level varies rapidly downstream of a sill 960 \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved. 961 962 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by 963 \citet{beckmann.doscher_JPO97}, 876 964 is to allow a direct communication between two adjacent bottom cells at different levels, 877 965 whenever the densest water is located above the less dense water. 878 The communication can be by a diffusive flux (diffusive BBL), an advective flux (advective BBL), or both. 966 The communication can be by a diffusive flux (diffusive BBL), 967 an advective flux (advective BBL), or both. 879 968 In the current implementation of the BBL, only the tracers are modified, not the velocities. 880 Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by881 \citet{campin.goosse_T99}.969 Furthermore, it only connects ocean bottom cells, 970 and therefore does not include all the improvements introduced by \citet{campin.goosse_T99}. 882 971 883 972 %% ================================================================================================= … … 885 974 \label{subsec:TRA_bbl_diff} 886 975 887 When applying sigma-diffusion (\np[=.true.]{ln_trabbl}{ln\_trabbl} and \np{nn_bbl_ldf}{nn\_bbl\_ldf} set to 1), 976 When applying sigma-diffusion 977 (\np[=.true.]{ln_trabbl}{ln\_trabbl} and \np{nn_bbl_ldf}{nn\_bbl\_ldf} set to 1), 888 978 the diffusive flux between two adjacent cells at the ocean floor is given by 889 979 \[ … … 891 981 \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T 892 982 \] 893 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and894 $A_l^\sigma$ the lateral diffusivity in the BBL.983 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 984 and $A_l^\sigma$ the lateral diffusivity in the BBL. 895 985 Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence, 896 986 \ie\ in the conditional form … … 900 990 \begin{cases} 901 991 A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\ 902 \\ 903 0 & \text{otherwise} \\ 992 0 & \text{otherwise} 904 993 \end{cases} 905 994 \end{equation} 906 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn_ahtbbl}{rn\_ahtbbl} and 995 where $A_{bbl}$ is the BBL diffusivity coefficient, 996 given by the namelist parameter \np{rn_ahtbbl}{rn\_ahtbbl} and 907 997 usually set to a value much larger than the one used for lateral mixing in the open ocean. 908 998 The constraint in \autoref{eq:TRA_bbl_coef} implies that sigma-like diffusion only occurs when … … 915 1005 \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S 916 1006 \] 917 where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and 918 $\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively. 1007 where $\rho$, $\alpha$ and $\beta$ are functions of 1008 $\overline T^\sigma$, $\overline S^\sigma$ and $\overline H^\sigma$, 1009 the along bottom mean temperature, salinity and depth, respectively. 919 1010 920 1011 %% ================================================================================================= … … 927 1018 %} 928 1019 929 \begin{figure} [!t]1020 \begin{figure} 930 1021 \centering 931 \includegraphics[width=0. 66\textwidth]{Fig_BBL_adv}1022 \includegraphics[width=0.33\textwidth]{Fig_BBL_adv} 932 1023 \caption[Advective/diffusive bottom boundary layer]{ 933 1024 Advective/diffusive Bottom Boundary Layer. … … 948 1039 %%%gmcomment : this section has to be really written 949 1040 950 When applying an advective BBL (\np[=1..2]{nn_bbl_adv}{nn\_bbl\_adv}), an overturning circulation is added which 951 connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope. 1041 When applying an advective BBL (\np[=1..2]{nn_bbl_adv}{nn\_bbl\_adv}), 1042 an overturning circulation is added which connects two adjacent bottom grid-points only if 1043 dense water overlies less dense water on the slope. 952 1044 The density difference causes dense water to move down the slope. 953 1045 954 \np[=1]{nn_bbl_adv}{nn\_bbl\_adv}: 955 the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step 956 (see black arrow in \autoref{fig:TRA_bbl}) \citep{beckmann.doscher_JPO97}. 957 It is a \textit{conditional advection}, that is, advection is allowed only 958 if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and 959 if the velocity is directed towards greater depth (\ie\ $\vect U \cdot \nabla H > 0$). 960 961 \np[=2]{nn_bbl_adv}{nn\_bbl\_adv}: 962 the downslope velocity is chosen to be proportional to $\Delta \rho$, 963 the density difference between the higher cell and lower cell densities \citep{campin.goosse_T99}. 964 The advection is allowed only if dense water overlies less dense water on the slope 965 (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$). 966 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:TRA_bbl}), 967 is simply given by the following expression: 968 \[ 969 % \label{eq:TRA_bbl_Utr} 970 u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 971 \] 972 where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn_gambbl}{rn\_gambbl}, 973 a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells, 974 respectively. 975 The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity, 976 and because no direct estimation of this parameter is available, a uniform value has been assumed. 977 The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}. 978 979 Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme. 980 Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and 981 the surrounding water at intermediate depths. 1046 \begin{description} 1047 \item [{\np[=1]{nn_bbl_adv}{nn\_bbl\_adv}}] the downslope velocity is chosen to 1048 be the Eulerian ocean velocity just above the topographic step 1049 (see black arrow in \autoref{fig:TRA_bbl}) \citep{beckmann.doscher_JPO97}. 1050 It is a \textit{conditional advection}, that is, 1051 advection is allowed only if dense water overlies less dense water on the slope 1052 (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and if the velocity is directed towards greater depth 1053 (\ie\ $\vect U \cdot \nabla H > 0$). 1054 \item [{\np[=2]{nn_bbl_adv}{nn\_bbl\_adv}}] the downslope velocity is chosen to be proportional to 1055 $\Delta \rho$, the density difference between the higher cell and lower cell densities 1056 \citep{campin.goosse_T99}. 1057 The advection is allowed only if dense water overlies less dense water on the slope 1058 (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$). 1059 For example, the resulting transport of the downslope flow, here in the $i$-direction 1060 (\autoref{fig:TRA_bbl}), is simply given by the following expression: 1061 \[ 1062 % \label{eq:TRA_bbl_Utr} 1063 u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 1064 \] 1065 where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as 1066 \np{rn_gambbl}{rn\_gambbl}, a namelist parameter, and 1067 \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells, respectively. 1068 The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity, 1069 and because no direct estimation of this parameter is available, a uniform value has been assumed. 1070 The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}. 1071 \end{description} 1072 1073 Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using 1074 the upwind scheme. 1075 Such a diffusive advective scheme has been chosen to mimic the entrainment between 1076 the downslope plume and the surrounding water at intermediate depths. 982 1077 The entrainment is replaced by the vertical mixing implicit in the advection scheme. 983 1078 Let us consider as an example the case displayed in \autoref{fig:TRA_bbl} where 984 1079 the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$. 985 The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by986 the downslope flow \autoref{eq:TRA_bbl_dw}, the horizontal \autoref{eq:TRA_bbl_hor} and987 the upward \autoref{eq:TRA_bbl_up} return flows as follows:988 \begin{alignat}{ 3}1080 The advective BBL scheme modifies the tracer time tendency of 1081 the ocean cells near the topographic step by the downslope flow \autoref{eq:TRA_bbl_dw}, 1082 the horizontal \autoref{eq:TRA_bbl_hor} and the upward \autoref{eq:TRA_bbl_up} return flows as follows: 1083 \begin{alignat}{5} 989 1084 \label{eq:TRA_bbl_dw} 990 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 991 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 1085 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 992 1086 \label{eq:TRA_bbl_hor} 993 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 994 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ 995 % 996 \intertext{and for $k =kdw-1,\;..., \; kup$ :} 997 % 1087 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ 1088 \shortintertext{and for $k =kdw-1,\;..., \; kup$ :} 998 1089 \label{eq:TRA_bbl_up} 999 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 1000 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} &&\lt( T^{do}_{k +1} - T^{sh}_{k} \rt) 1090 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} &&\lt( T^{do}_{k +1} - T^{sh}_{k} \rt) 1001 1091 \end{alignat} 1002 1092 where $b_t$ is the $T$-cell volume. … … 1015 1105 \end{listing} 1016 1106 1017 In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations: 1107 In some applications it can be useful to add a Newtonian damping term into 1108 the temperature and salinity equations: 1018 1109 \begin{equation} 1019 1110 \label{eq:TRA_dmp} 1020 \begin{gathered} 1021 \pd[T]{t} = \cdots - \gamma (T - T_o) \\ 1022 \pd[S]{t} = \cdots - \gamma (S - S_o) 1023 \end{gathered} 1024 \end{equation} 1025 where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields 1026 (usually a climatology). 1027 Options are defined through the \nam{tra_dmp}{tra\_dmp} namelist variables. 1111 \pd[T]{t} = \cdots - \gamma (T - T_o) \qquad \pd[S]{t} = \cdots - \gamma (S - S_o) 1112 \end{equation} 1113 where $\gamma$ is the inverse of a time scale, 1114 and $T_o$ and $S_o$ are given temperature and salinity fields (usually a climatology). 1115 Options are defined through the \nam{tra_dmp}{tra\_dmp} namelist variables. 1028 1116 The restoring term is added when the namelist parameter \np{ln_tradmp}{ln\_tradmp} is set to true. 1029 It also requires that both \np{ln_tsd_init}{ln\_tsd\_init} and \np{ln_tsd_dmp}{ln\_tsd\_dmp} are set to true in 1030 \nam{tsd}{tsd} namelist as well as \np{sn_tem}{sn\_tem} and \np{sn_sal}{sn\_sal} structures are correctly set 1117 It also requires that both \np{ln_tsd_init}{ln\_tsd\_init} and 1118 \np{ln_tsd_dmp}{ln\_tsd\_dmp} are set to true in \nam{tsd}{tsd} namelist as well as 1119 \np{sn_tem}{sn\_tem} and \np{sn_sal}{sn\_sal} structures are correctly set 1031 1120 (\ie\ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread}, 1032 1121 see \autoref{subsec:SBC_fldread}). 1033 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. 1122 The restoring coefficient $\gamma$ is a three-dimensional array read in during 1123 the \rou{tra\_dmp\_init} routine. 1034 1124 The file name is specified by the namelist variable \np{cn_resto}{cn\_resto}. 1035 The DMP\_TOOLS tool isprovided to allow users to generate the netcdf file.1125 The \texttt{DMP\_TOOLS} are provided to allow users to generate the netcdf file. 1036 1126 1037 1127 The two main cases in which \autoref{eq:TRA_dmp} is used are 1038 \textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and 1039 \textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field 1040 (for example to build the initial state of a prognostic simulation, 1041 or to use the resulting velocity field for a passive tracer study). 1128 \begin{enumerate*}[label=(\textit{\alph*})] 1129 \item the specification of the boundary conditions along 1130 artificial walls of a limited domain basin and 1131 \item the computation of the velocity field associated with a given $T$-$S$ field 1132 (for example to build the initial state of a prognostic simulation, 1133 or to use the resulting velocity field for a passive tracer study). 1134 \end{enumerate*} 1042 1135 The first case applies to regional models that have artificial walls instead of open boundaries. 1043 In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas1044 it is zero in the interior of the model domain.1136 In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) 1137 whereas it is zero in the interior of the model domain. 1045 1138 The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}. 1046 1139 It allows us to find the velocity field consistent with the model dynamics whilst 1047 1140 having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$). 1048 1141 1049 The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but1050 i t produces artificial sources of heat and salt within the ocean.1142 The robust diagnostic method is very efficient in preventing temperature drift in 1143 intermediate waters but it produces artificial sources of heat and salt within the ocean. 1051 1144 It also has undesirable effects on the ocean convection. 1052 It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much. 1053 1054 The namelist parameter \np{nn_zdmp}{nn\_zdmp} sets whether the damping should be applied in the whole water column or 1055 only below the mixed layer (defined either on a density or $S_o$ criterion). 1145 It tends to prevent deep convection and subsequent deep-water formation, 1146 by stabilising the water column too much. 1147 1148 The namelist parameter \np{nn_zdmp}{nn\_zdmp} sets whether the damping should be applied in 1149 the whole water column or only below the mixed layer (defined either on a density or $S_o$ criterion). 1056 1150 It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here 1057 1151 \citep{madec.delecluse.ea_JPO96}. 1058 1152 1059 For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under1060 \path{./tools/DMP_TOOLS}.1153 For generating \ifile{resto}, 1154 see the documentation for the DMP tools provided with the source code under \path{./tools/DMP_TOOLS}. 1061 1155 1062 1156 %% ================================================================================================= … … 1065 1159 1066 1160 Options are defined through the \nam{dom}{dom} namelist variables. 1067 The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09}, 1068 \ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:TD_mLF}): 1161 The general framework for tracer time stepping is a modified leap-frog scheme 1162 \citep{leclair.madec_OM09}, \ie\ a three level centred time scheme associated with 1163 a Asselin time filter (cf. \autoref{sec:TD_mLF}): 1069 1164 \begin{equation} 1070 1165 \label{eq:TRA_nxt} 1071 \begin{alignedat}{ 3}1166 \begin{alignedat}{5} 1072 1167 &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\ 1073 1168 &(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] \\ … … 1075 1170 \end{alignedat} 1076 1171 \end{equation} 1077 where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, 1078 $\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$ 1079 (\ie\ fluxes plus content in mass exchanges). 1080 $\gamma$ is initialized as \np{rn_atfp}{rn\_atfp} (\textbf{namelist} parameter). 1081 Its default value is \np[=10.e-3]{rn_atfp}{rn\_atfp}. 1172 where RHS is the right hand side of the temperature equation, 1173 the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient, 1174 and $S$ is the total forcing applied on $T$ (\ie\ fluxes plus content in mass exchanges). 1175 $\gamma$ is initialized as \np{rn_atfp}{rn\_atfp}, its default value is \forcode{10.e-3}. 1082 1176 Note that the forcing correction term in the filter is not applied in linear free surface 1083 1177 (\jp{ln\_linssh}\forcode{=.true.}) (see \autoref{subsec:TRA_sbc}). 1084 Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$. 1085 1086 When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in 1087 \mdl{trazdf} module. 1178 Not also that in constant volume case, the time stepping is performed on $T$, 1179 not on its content, $e_{3t}T$. 1180 1181 When the vertical mixing is solved implicitly, 1182 the update of the \textit{next} tracer fields is done in \mdl{trazdf} module. 1088 1183 In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module. 1089 1184 1090 In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed:1091 $T^{t - \rdt} = T^t$ and $T^t = T_f$.1185 In order to prepare for the computation of the \textit{next} time step, 1186 a swap of tracer arrays is performed: $T^{t - \rdt} = T^t$ and $T^t = T_f$. 1092 1187 1093 1188 %% ================================================================================================= … … 1105 1200 \label{subsec:TRA_eos} 1106 1201 1107 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density, 1108 $\rho$, to a number of state variables, most typically temperature, salinity and pressure. 1202 The \textbf{E}quation \textbf{O}f \textbf{S}eawater (EOS) is 1203 an empirical nonlinear thermodynamic relationship linking 1204 seawater density, $\rho$, to a number of state variables, 1205 most typically temperature, salinity and pressure. 1109 1206 Because density gradients control the pressure gradient force through the hydrostatic balance, 1110 the equation of state provides a fundamental bridge between the distribution of active tracers and1111 the fluid dynamics.1207 the equation of state provides a fundamental bridge between 1208 the distribution of active tracers and the fluid dynamics. 1112 1209 Nonlinearities of the EOS are of major importance, in particular influencing the circulation through 1113 1210 determination of the static stability below the mixed layer, 1114 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{roquet.madec.ea_JPO15}. 1115 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{fofonoff.millard_bk83}) or 1116 TEOS-10 \citep{ioc.iapso_bk10} standards should be used anytime a simulation of the real ocean circulation is attempted 1211 thus controlling rates of exchange between the atmosphere and the ocean interior 1117 1212 \citep{roquet.madec.ea_JPO15}. 1213 Therefore an accurate EOS based on either the 1980 equation of state 1214 (EOS-80, \cite{fofonoff.millard_bk83}) or TEOS-10 \citep{ioc.iapso_bk10} standards should 1215 be used anytime a simulation of the real ocean circulation is attempted \citep{roquet.madec.ea_JPO15}. 1118 1216 The use of TEOS-10 is highly recommended because 1119 \textit{(i)} it is the new official EOS, 1120 \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and 1121 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and 1122 practical salinity for EOS-80, both variables being more suitable for use as model variables 1123 \citep{ioc.iapso_bk10, graham.mcdougall_JPO13}. 1217 \begin{enumerate*}[label=(\textit{\roman*})] 1218 \item it is the new official EOS, 1219 \item it is more accurate, being based on an updated database of laboratory measurements, and 1220 \item it uses Conservative Temperature and Absolute Salinity 1221 (instead of potential temperature and practical salinity for EOS-80), 1222 both variables being more suitable for use as model variables 1223 \citep{ioc.iapso_bk10, graham.mcdougall_JPO13}. 1224 \end{enumerate*} 1124 1225 EOS-80 is an obsolescent feature of the \NEMO\ system, kept only for backward compatibility. 1125 1226 For process studies, it is often convenient to use an approximation of the EOS. 1126 1227 To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available. 1127 1228 1128 In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density. 1129 Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 1130 This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as, 1131 with the exception of only a small percentage of the ocean, 1132 density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}. 1229 In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, 1230 with $\rho_o$ a reference density. 1231 Called \textit{rau0} in the code, 1232 $\rho_o$ is set in \mdl{phycst} to a value of \texttt{1,026} $Kg/m^3$. 1233 This is a sensible choice for the reference density used in a Boussinesq ocean climate model, 1234 as, with the exception of only a small percentage of the ocean, 1235 density in the World Ocean varies by no more than 2\% from that value \citep{gill_bk82}. 1133 1236 1134 1237 Options which control the EOS used are defined through the \nam{eos}{eos} namelist variables. 1135 1238 1136 1239 \begin{description} 1137 \item [{\np[=.true.]{ln_teos10}{ln\_teos10}}] the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used. 1240 \item [{\np[=.true.]{ln_teos10}{ln\_teos10}}] the polyTEOS10-bsq equation of seawater 1241 \citep{roquet.madec.ea_OM15} is used. 1138 1242 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 1139 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and 1140 more computationally efficient expressions for their derived quantities which make them more adapted for 1141 use in ocean models. 1142 Note that a slightly higher precision polynomial form is now used replacement of 1143 the TEOS-10 rational function approximation for hydrographic data analysis \citep{ioc.iapso_bk10}. 1243 but it is optimized for a Boussinesq fluid and 1244 the polynomial expressions have simpler and more computationally efficient expressions for 1245 their derived quantities which make them more adapted for use in ocean models. 1246 Note that a slightly higher precision polynomial form is now used 1247 replacement of the TEOS-10 rational function approximation for hydrographic data analysis 1248 \citep{ioc.iapso_bk10}. 1144 1249 A key point is that conservative state variables are used: 1145 Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg{C}, notation: $\Theta$). 1250 Absolute Salinity (unit: $g/kg$, notation: $S_A$) and 1251 Conservative Temperature (unit: $\deg{C}$, notation: $\Theta$). 1146 1252 The pressure in decibars is approximated by the depth in meters. 1147 1253 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. 1148 It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ioc.iapso_bk10}. 1254 It is set to $C_p$ = 3991.86795711963 $J.Kg^{-1}.\deg{K}^{-1}$, 1255 according to \citet{ioc.iapso_bk10}. 1149 1256 Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$. 1150 In particular, the initial state de ined by the user have to be given as \textit{Conservative} Temperature and1151 \textit{ Absolute} Salinity.1257 In particular, the initial state defined by the user have to be given as 1258 \textit{Conservative} Temperature and \textit{Absolute} Salinity. 1152 1259 In addition, when using TEOS10, the Conservative SST is converted to potential SST prior to 1153 1260 either computing the air-sea and ice-sea fluxes (forced mode) or 1154 1261 sending the SST field to the atmosphere (coupled mode). 1155 1262 \item [{\np[=.true.]{ln_eos80}{ln\_eos80}}] the polyEOS80-bsq equation of seawater is used. 1156 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to1157 accurately fit EOS80 (Roquet, personal comm.).1263 It takes the same polynomial form as the polyTEOS10, 1264 but the coefficients have been optimized to accurately fit EOS80 (Roquet, personal comm.). 1158 1265 The state variables used in both the EOS80 and the ocean model are: 1159 the Practical Salinity ( (unit: psu, notation: $S_p$)) and1160 Potential Temperature (unit: $ ^{\circ}C$, notation: $\theta$).1266 the Practical Salinity (unit: $psu$, notation: $S_p$) and 1267 Potential Temperature (unit: $\deg{C}$, notation: $\theta$). 1161 1268 The pressure in decibars is approximated by the depth in meters. 1162 With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and1163 pressure \citep{fofonoff.millard_bk83}.1269 With EOS, the specific heat capacity of sea water, $C_p$, is a function of 1270 temperature, salinity and pressure \citep{fofonoff.millard_bk83}. 1164 1271 Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which 1165 1272 is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 1166 \item [{\np[=.true.]{ln_seos}{ln\_seos}}] a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen, 1167 the coefficients of which has been optimized to fit the behavior of TEOS10 1168 (Roquet, personal comm.) (see also \citet{roquet.madec.ea_JPO15}). 1273 \item [{\np[=.true.]{ln_seos}{ln\_seos}}] a simplified EOS (S-EOS) inspired by 1274 \citet{vallis_bk06} is chosen, 1275 the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) 1276 (see also \citet{roquet.madec.ea_JPO15}). 1169 1277 It provides a simplistic linear representation of both cabbeling and thermobaricity effects which 1170 1278 is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}. 1171 With such an equation of state there is no longer a distinction between 1172 \textit{ conservative} and \textit{potential} temperature,1173 as well as between \textit{absolute} and\textit{practical} salinity.1279 With such an equation of state there is no longer a distinction between \textit{conservative} and 1280 \textit{potential} temperature, as well as between \textit{absolute} and 1281 \textit{practical} salinity. 1174 1282 S-EOS takes the following expression: 1175 1176 1283 \begin{gather*} 1177 1284 % \label{eq:TRA_S-EOS} 1178 \begin{alignedat}{2} 1179 &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. \\ 1180 & &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a \\ 1181 & \big. &- \nu \; T_a &S_a \big] \\ 1182 \end{alignedat} 1183 \\ 1285 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. 1286 + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a 1287 \big. - \nu \; T_a S_a \big] \\ 1184 1288 \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3 1185 1289 \end{gather*} 1186 where the computer name of the coefficients as well as their standard value are given in \autoref{tab:TRA_SEOS}. 1290 where the computer name of the coefficients as well as their standard value are given in 1291 \autoref{tab:TRA_SEOS}. 1187 1292 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by 1188 1293 changing the associated coefficients. 1189 Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS. 1190 setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from 1191 S-EOS. 1294 Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ 1295 remove thermobaric effect from S-EOS. 1296 Setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ 1297 remove cabbeling effect from S-EOS. 1192 1298 Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 1193 1299 \end{description} 1194 1300 1195 \begin{table} [!tb]1301 \begin{table} 1196 1302 \centering 1197 1303 \begin{tabular}{|l|l|l|l|} 1198 1304 \hline 1199 coeff. & computer name & S-EOS & description\\1305 coeff. & computer name & S-EOS & description \\ 1200 1306 \hline 1201 $a_0 $ & \np{rn_a0}{rn\_a0}& $1.6550~10^{-1}$ & linear thermal expansion coeff. \\1307 $a_0 $ & \np{rn_a0}{rn\_a0} & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\ 1202 1308 \hline 1203 $b_0 $ & \np{rn_b0}{rn\_b0}& $7.6554~10^{-1}$ & linear haline expansion coeff. \\1309 $b_0 $ & \np{rn_b0}{rn\_b0} & $7.6554~10^{-1}$ & linear haline expansion coeff. \\ 1204 1310 \hline 1205 $\lambda_1$ & \np{rn_lambda1}{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$\\1311 $\lambda_1$ & \np{rn_lambda1}{rn\_lambda1} & $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$ \\ 1206 1312 \hline 1207 $\lambda_2$ & \np{rn_lambda2}{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$\\1313 $\lambda_2$ & \np{rn_lambda2}{rn\_lambda2} & $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$ \\ 1208 1314 \hline 1209 $\nu $ & \np{rn_nu}{rn\_nu} & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$\\1315 $\nu $ & \np{rn_nu}{rn\_nu} & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$ \\ 1210 1316 \hline 1211 $\mu_1 $ & \np{rn_mu1}{rn\_mu1} & $1.4970~10^{-4}$ & thermobaric coeff. in T\\1317 $\mu_1 $ & \np{rn_mu1}{rn\_mu1} & $1.4970~10^{-4}$ & thermobaric coeff. in T \\ 1212 1318 \hline 1213 $\mu_2 $ & \np{rn_mu2}{rn\_mu2} & $1.1090~10^{-5}$ & thermobaric coeff. in S\\1319 $\mu_2 $ & \np{rn_mu2}{rn\_mu2} & $1.1090~10^{-5}$ & thermobaric coeff. in S \\ 1214 1320 \hline 1215 1321 \end{tabular} … … 1222 1328 \label{subsec:TRA_bn2} 1223 1329 1224 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of1225 paramount importance as determine the ocean stratification andis used in several ocean parameterisations1330 An accurate computation of the ocean stability (i.e. of $N$, the Brunt-V\"{a}is\"{a}l\"{a} frequency) is of paramount importance as determine the ocean stratification and 1331 is used in several ocean parameterisations 1226 1332 (namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion, 1227 1333 non-penetrative convection, tidal mixing parameterisation, iso-neutral diffusion). … … 1235 1341 where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and, 1236 1342 $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 1237 The coefficients are a polynomial function of temperature, salinity and depth which expression depends on1238 the chosen EOS.1343 The coefficients are a polynomial function of temperature, salinity and depth which 1344 expression depends on the chosen EOS. 1239 1345 They are computed through \textit{eos\_rab}, a \fortran\ function that can be found in \mdl{eosbn2}. 1240 1346 … … 1246 1352 \begin{equation} 1247 1353 \label{eq:TRA_eos_fzp} 1248 \begin{split} 1249 &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 1250 &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\ 1251 &\text{and~} d = -7.53~10^{-3} 1252 \end{split} 1354 \begin{gathered} 1355 T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 1356 \text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \text{and~} d = -7.53~10^{-3} 1357 \end{gathered} 1253 1358 \end{equation} 1254 1359 … … 1272 1377 I've changed "derivative" to "difference" and "mean" to "average"} 1273 1378 1274 With partial cells (\np[=.true.]{ln_zps}{ln\_zps}) at bottom and top (\np[=.true.]{ln_isfcav}{ln\_isfcav}), 1379 With partial cells (\np[=.true.]{ln_zps}{ln\_zps}) at bottom and top 1380 (\np[=.true.]{ln_isfcav}{ln\_isfcav}), 1275 1381 in general, tracers in horizontally adjacent cells live at different depths. 1276 Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and1277 the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).1278 The partial cell properties at the top (\np[=.true.]{ln_isfcav}{ln\_isfcav}) are computed in the same way as1279 for the bottom.1382 Horizontal gradients of tracers are needed for horizontal diffusion 1383 (\mdl{traldf} module) and the hydrostatic pressure gradient calculations (\mdl{dynhpg} module). 1384 The partial cell properties at the top (\np[=.true.]{ln_isfcav}{ln\_isfcav}) are computed in 1385 the same way as for the bottom. 1280 1386 So, only the bottom interpolation is explained below. 1281 1387 … … 1283 1389 a linear interpolation in the vertical is used to approximate the deeper tracer as if 1284 1390 it actually lived at the depth of the shallower tracer point (\autoref{fig:TRA_Partial_step_scheme}). 1285 For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is: 1286 1287 \begin{figure}[!p] 1391 For example, for temperature in the $i$-direction the needed interpolated temperature, 1392 $\widetilde T$, is: 1393 1394 \begin{figure} 1288 1395 \centering 1289 \includegraphics[width=0. 66\textwidth]{Fig_partial_step_scheme}1396 \includegraphics[width=0.33\textwidth]{Fig_partial_step_scheme} 1290 1397 \caption[Discretisation of the horizontal difference and average of tracers in 1291 1398 the $z$-partial step coordinate]{ … … 1294 1401 the case $(e3w_k^{i + 1} - e3w_k^i) > 0$. 1295 1402 A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$, 1296 the tracer value at the depth of the shallower tracer point of 1297 the two adjacent bottom $T$-points. 1403 the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. 1298 1404 The horizontal difference is then given by: 1299 $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and 1300 the average by: 1405 $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and the average by: 1301 1406 $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$.} 1302 1407 \label{fig:TRA_Partial_step_scheme} 1303 1408 \end{figure} 1409 1304 1410 \[ 1305 1411 \widetilde T = \lt\{ 1306 1412 \begin{alignedat}{2} 1307 1413 &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1} 1308 & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\1414 & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ 1309 1415 &T^{\, i} &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i } \; \delta_k T^{i + 1} 1310 1416 & \quad \text{if $e_{3w}^{i + 1} < e_{3w}^i$} … … 1312 1418 \rt. 1313 1419 \] 1314 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 1420 and the resulting forms for the horizontal difference and the horizontal average value of 1421 $T$ at a $U$-point are: 1315 1422 \begin{equation} 1316 1423 \label{eq:TRA_zps_hde} … … 1318 1425 \delta_{i + 1/2} T &= 1319 1426 \begin{cases} 1320 \widetilde T - T^i & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1321 \\ 1322 T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 1323 \end{cases} 1324 \\ 1427 \widetilde T - T^i & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1428 T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 1429 \end{cases} \\ 1325 1430 \overline T^{\, i + 1/2} &= 1326 1431 \begin{cases} 1327 (\widetilde T - T^{\, i} ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1328 \\ 1329 (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 1432 (\widetilde T - T^{\, i} ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1433 (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 1330 1434 \end{cases} 1331 1435 \end{split} … … 1334 1438 The computation of horizontal derivative of tracers as well as of density is performed once for all at 1335 1439 each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed. 1336 It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$, 1337 is not the same as that used for $T$ and $S$. 1338 Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of 1339 $T$ and $S$, and the pressure at a $u$-point 1440 It has to be emphasized that the procedure used to compute the interpolated density, 1441 $\widetilde \rho$, is not the same as that used for $T$ and $S$. 1442 Instead of forming a linear approximation of density, 1443 we compute $\widetilde \rho$ from the interpolated values of $T$ and $S$, 1444 and the pressure at a $u$-point 1340 1445 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 1341 1446 \[ … … 1345 1450 1346 1451 This is a much better approximation as the variation of $\rho$ with depth (and thus pressure) 1347 is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation. 1348 This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and 1349 the slopes of neutral surfaces (\autoref{sec:LDF_slp}). 1350 1351 Note that in almost all the advection schemes presented in this Chapter, 1452 is highly non-linear with a true equation of state and thus is badly approximated with 1453 a linear interpolation. 1454 This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) 1455 and the slopes of neutral surfaces (\autoref{sec:LDF_slp}). 1456 1457 Note that in almost all the advection schemes presented in this chapter, 1352 1458 both averaging and differencing operators appear. 1353 1459 Yet \autoref{eq:TRA_zps_hde} has not been used in these schemes: … … 1356 1462 The main motivation is to preserve the domain averaged mean variance of the advected field when 1357 1463 using the $2^{nd}$ order centred scheme. 1358 Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of1359 partial cells should be further investigated in the near future.1464 Sensitivity of the advection schemes to the way horizontal averages are performed in 1465 the vicinity of partial cells should be further investigated in the near future. 1360 1466 \gmcomment{gm : this last remark has to be done} 1361 1467 -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex
r11622 r11630 208 208 \] 209 209 Two strategies can be considered for the surface pressure term: 210 \begin{enumerate*}[label= {(\alph*)}]210 \begin{enumerate*}[label=(\textit{\alph*})] 211 211 \item introduce of a new variable $\eta$, the free-surface elevation, 212 212 for which a prognostic equation can be established and solved; … … 486 486 \item [Flux form of the momentum equations] 487 487 % \label{eq:MB_dyn_flux} 488 \begin{ multline*}488 \begin{alignat*}{2} 489 489 % \label{eq:MB_dyn_flux_u} 490 \pd[u]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt)\\491 - \frac{1}{e_3} \pd[(w \, u)]{k}- \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_u^{\vect U} + F_u^{\vect U}492 \end{ multline*}493 \begin{ multline*}490 \pd[u]{t} = &+ \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) - \frac{1}{e_3} \pd[(w \, u)]{k} \\ 491 &- \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_u^{\vect U} + F_u^{\vect U} 492 \end{alignat*} 493 \begin{alignat*}{2} 494 494 % \label{eq:MB_dyn_flux_v} 495 \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt)\\496 - \frac{1}{e_3} \pd[(w \, v)]{k}- \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_v^{\vect U} + F_v^{\vect U}497 \end{ multline*}495 \pd[v]{t} = &- \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) - \frac{1}{e_3} \pd[(w \, v)]{k} \\ 496 &- \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_v^{\vect U} + F_v^{\vect U} 497 \end{alignat*} 498 498 where $\zeta$, the relative vorticity, is given by \autoref{eq:MB_curl_Uh} and 499 499 $p_s$, the surface pressure, is given by: … … 650 650 \end{gather*} 651 651 \item [Flux form of the momentum equation] 652 \begin{ multline*}652 \begin{alignat*}{2} 653 653 % \label{eq:MB_sco_u_flux} 654 \frac{1}{e_3} \pd[(e_3 \, u)]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt)\\655 - \frac{1}{e_3} \pd[(\omega \, u)]{k}- \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U}656 \end{ multline*}657 \begin{ multline*}654 \frac{1}{e_3} \pd[(e_3 \, u)]{t} = &+ \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt) - \frac{1}{e_3} \pd[(\omega \, u)]{k} \\ 655 &- \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} 656 \end{alignat*} 657 \begin{alignat*}{2} 658 658 % \label{eq:MB_sco_v_flux} 659 \frac{1}{e_3} \pd[(e_3 \, v)]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt)\\660 - \frac{1}{e_3} \pd[(\omega \, v)]{k}- \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U}661 \end{ multline*}659 \frac{1}{e_3} \pd[(e_3 \, v)]{t} = &- \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt) - \frac{1}{e_3} \pd[(\omega \, v)]{k} \\ 660 &- \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} 661 \end{alignat*} 662 662 where the relative vorticity, $\zeta$, the surface pressure gradient, 663 663 and the hydrostatic pressure have the same expressions as in $z$-coordinates although … … 694 694 \includegraphics[width=0.66\textwidth]{Fig_z_zstar} 695 695 \caption[Curvilinear z-coordinate systems (\{non-\}linear free-surface cases and re-scaled \zstar)]{ 696 \begin{enumerate*}[label= {(\alph*)}]696 \begin{enumerate*}[label=(\textit{\alph*})] 697 697 \item $z$-coordinate in linear free-surface case; 698 698 \item $z$-coordinate in non-linear free surface case; … … 1067 1067 where $ \vect U^\ast = \lt( u^\ast,v^\ast,w^\ast \rt)$ is a non-divergent, 1068 1068 eddy-induced transport velocity. This velocity field is defined by: 1069 \ begin{gather*}1069 \[ 1070 1070 % \label{eq:MB_eiv} 1071 1071 u^\ast = \frac{1}{e_3} \pd[]{k} \lt( A^{eiv} \; \tilde{r}_1 \rt) \quad 1072 v^\ast = \frac{1}{e_3} \pd[]{k} \lt( A^{eiv} \; \tilde{r}_2 \rt) \ \1072 v^\ast = \frac{1}{e_3} \pd[]{k} \lt( A^{eiv} \; \tilde{r}_2 \rt) \quad 1073 1073 w^\ast = - \frac{1}{e_1 e_2} \lt[ \pd[]{i} \lt( A^{eiv} \; e_2 \, \tilde{r}_1 \rt) 1074 1074 + \pd[]{j} \lt( A^{eiv} \; e_1 \, \tilde{r}_2 \rt) \rt] 1075 \ end{gather*}1075 \] 1076 1076 where $A^{eiv}$ is the eddy induced velocity coefficient 1077 1077 (or equivalently the isoneutral thickness diffusivity coefficient), … … 1130 1130 the $u$ and $v$-fields are considered as independent scalar fields, 1131 1131 so that the diffusive operator is given by: 1132 \ begin{gather*}1132 \[ 1133 1133 % \label{eq:MB_lapU_iso} 1134 D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \ \1134 D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \quad 1135 1135 D_v^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla v \rt) 1136 \ end{gather*}1136 \] 1137 1137 where $\Re$ is given by \autoref{eq:MB_iso_tensor}. 1138 1138 It is the same expression as those used for diffusive operator on tracers. -
NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex
r11622 r11630 13 13 14 14 {\footnotesize 15 \begin{tabular}{l||l|l} 16 Release & Author(s) & Modifications \\ 15 \begin{tabularx}{0.5\textwidth}{l||X|X} 16 Release & Author(s) & 17 Modifications \\ 17 18 \hline 18 {\em 4.0} & {\em J\'{e}r\^{o}me Chanut and Tim Graham} & {\em Review } \\ 19 {\em 3.6} & {\em Christian \'{E}th\'{e} } & {\em Update } \\ 20 {\em $\leq$ 3.4} & {\em Gurvan Madec } & {\em First version} \\ 21 \end{tabular} 19 {\em 4.0} & {\em J\'{e}r\^{o}me Chanut \newline Tim Graham} & 20 {\em Review \newline Update } \\ 21 {\em 3.6} & {\em Christian \'{E}th\'{e} } & 22 {\em Update } \\ 23 {\em $\leq$ 3.4} & {\em Gurvan Madec } & 24 {\em First version } \\ 25 \end{tabularx} 22 26 } 23 27 … … 173 177 \end{equation} 174 178 where 175 \ begin{align*}176 c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k) \\177 d(k) &= e_{3t} (k) \, / \, (2 \rdt) + c_k + c_{k + 1} \\178 b(k) &= e_{3t} (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt)179 \ end{align*}179 \[ 180 c(k) = A_w^{vT} (k) \, / \, e_{3w} (k) \text{,} \quad 181 d(k) = e_{3t} (k) \, / \, (2 \rdt) + c_k + c_{k + 1} \quad \text{and} \quad 182 b(k) = e_{3t} (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt) 183 \] 180 184 181 185 \autoref{eq:TD_imp_mat} is a linear system of equations with
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