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r10544 r12065 2 2 3 3 \begin{document} 4 % ================================================================ 5 % Chapter 1 ——— Ocean Tracers (TRA) 6 % ================================================================ 4 7 5 \chapter{Ocean Tracers (TRA)} 8 6 \label{chap:TRA} 9 7 10 \minitoc 11 12 % missing/update 8 \thispagestyle{plain} 9 10 \chaptertoc 11 12 \paragraph{Changes record} ~\\ 13 14 {\footnotesize 15 \begin{tabularx}{\textwidth}{l||X|X} 16 Release & Author(s) & Modifications \\ 17 \hline 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} \\ 21 \end{tabularx} 22 } 23 24 \clearpage 25 26 % missing/update 13 27 % traqsr: need to coordinate with SBC module 14 28 15 %STEVEN : is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below 29 %STEVEN : is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? 30 %I added a comment to this effect on some instances of this below 16 31 17 32 Using the representation described in \autoref{chap:DOM}, several semi -discrete space forms of 18 33 the tracer equations are available depending on the vertical coordinate used and on the physics used. 19 34 In all the equations presented here, the masking has been omitted for simplicity. 20 One must be aware that all the quantities are masked fields and that each time a mean or21 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. 22 37 23 38 The two active tracers are potential temperature and salinity. … … 30 45 NXT stands for next, referring to the time-stepping. 31 46 From left to right, the terms on the rhs of the tracer equations are the advection (ADV), 32 the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings 33 (SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition), 34 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. 35 52 The terms QSR, BBC, BBL and DMP are optional. 36 53 The external forcings and parameterisations require complex inputs and complex calculations 37 (\eg bulk formulae, estimation of mixing coefficients) that are carried out in the SBC,54 (\eg\ bulk formulae, estimation of mixing coefficients) that are carried out in the SBC, 38 55 LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and 39 56 \autoref{chap:ZDF}, respectively. 40 Note that \mdl{tranpc}, the non-penetrative convection module, although located in41 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, 42 59 is described with the model vertical physics (ZDF) together with 43 60 other available parameterization of convection. 44 61 45 In the present chapter we also describe the diagnostic equations used to compute the sea-water properties46 (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and freezing point with 47 associated modules \mdl{eosbn2} and \mdl{phycst}).48 49 The different options available to the user are managed by namelist logicals or CPP keys.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}). 65 66 The different options available to the user are managed by namelist logicals. 50 67 For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx}, 51 68 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 52 The CPP key (when it exists) is \key{traTTT}.53 69 The equivalent code can be found in the \textit{traTTT} or \textit{traTTT\_xxx} module, 54 70 in the \path{./src/OCE/TRA} directory. 55 71 56 72 The user has the option of extracting each tendency term on the RHS of the tracer equation for output 57 (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}~\forcode{= .true.}), as described in \autoref{chap:DIA}. 58 59 % ================================================================ 60 % Tracer Advection 61 % ================================================================ 62 \section{Tracer advection (\protect\mdl{traadv})} 73 (\np{ln_tra_trd}{ln\_tra\_trd} or \np[=.true.]{ln_tra_mxl}{ln\_tra\_mxl}), 74 as described in \autoref{chap:DIA}. 75 76 %% ================================================================================================= 77 \section[Tracer advection (\textit{traadv.F90})]{Tracer advection (\protect\mdl{traadv})} 63 78 \label{sec:TRA_adv} 64 %------------------------------------------namtra_adv----------------------------------------------------- 65 66 \nlst{namtra_adv} 67 %------------------------------------------------------------------------------------------------------------- 68 69 When considered (\ie when \np{ln\_traadv\_NONE} is not set to \forcode{.true.}), 79 80 \begin{listing} 81 \nlst{namtra_adv} 82 \caption{\forcode{&namtra_adv}} 83 \label{lst:namtra_adv} 84 \end{listing} 85 86 When considered (\ie\ when \np{ln_traadv_OFF}{ln\_traadv\_OFF} is not set to \forcode{.true.}), 70 87 the advection tendency of a tracer is expressed in flux form, 71 \ie as the divergence of the advective fluxes.72 Its discrete expression is given by 73 \begin{equation} 74 \label{eq: tra_adv}88 \ie\ as the divergence of the advective fluxes. 89 Its discrete expression is given by: 90 \begin{equation} 91 \label{eq:TRA_adv} 75 92 ADV_\tau = - \frac{1}{b_t} \Big( \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u] 76 93 + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big) … … 78 95 \end{equation} 79 96 where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells. 80 The flux form in \autoref{eq:tra_adv} implicitly requires the use of the continuity equation. 81 Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 82 results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ 83 (which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie \np{ln\_linssh}~\forcode{= .true.}). 84 Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that 85 it is consistent with the continuity equation in order to enforce the conservation properties of 86 the continuous equations. 87 In other words, by setting $\tau = 1$ in (\autoref{eq:tra_adv}) we recover the discrete form of 88 the continuity equation which is used to calculate the vertical velocity. 89 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 90 \begin{figure}[!t] 91 \begin{center} 92 \includegraphics[]{Fig_adv_scheme} 93 \caption{ 94 \protect\label{fig:adv_scheme} 95 Schematic representation of some ways used to evaluate the tracer value at $u$-point and 96 the amount of tracer exchanged between two neighbouring grid points. 97 Upsteam biased scheme (ups): 98 the upstream value is used and the black area is exchanged. 99 Piecewise parabolic method (ppm): 100 a parabolic interpolation is used and the black and dark grey areas are exchanged. 101 Monotonic upstream scheme for conservative laws (muscl): 102 a parabolic interpolation is used and black, dark grey and grey areas are exchanged. 103 Second order scheme (cen2): 104 the mean value is used and black, dark grey, grey and light grey areas are exchanged. 105 Note that this illustration does not include the flux limiter used in ppm and muscl schemes. 106 } 107 \end{center} 97 The flux form in \autoref{eq:TRA_adv} implicitly requires the use of the continuity equation. 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} 111 \centering 112 \includegraphics[width=0.66\textwidth]{TRA_adv_scheme} 113 \caption[Ways to evaluate the tracer value and the amount of tracer exchanged]{ 114 Schematic representation of some ways used to evaluate the tracer value at $u$-point and 115 the amount of tracer exchanged between two neighbouring grid points. 116 Upsteam biased scheme (ups): 117 the upstream value is used and the black area is exchanged. 118 Piecewise parabolic method (ppm): 119 a parabolic interpolation is used and the black and dark grey areas are exchanged. 120 Monotonic upstream scheme for conservative laws (muscl): 121 a parabolic interpolation is used and black, dark grey and grey areas are exchanged. 122 Second order scheme (cen2): 123 the mean value is used and black, dark grey, grey and light grey areas are exchanged. 124 Note that this illustration does not include the flux limiter used in ppm and muscl schemes.} 125 \label{fig:TRA_adv_scheme} 108 126 \end{figure} 109 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 110 111 The key difference between the advection schemes available in \NEMO is the choice made in space and 112 time interpolation to define the value of the tracer at the velocity points 113 (\autoref{fig:adv_scheme}). 127 128 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 130 (\autoref{fig:TRA_adv_scheme}). 114 131 115 132 Along solid lateral and bottom boundaries a zero tracer flux is automatically specified, … … 118 135 119 136 \begin{description} 120 \item[linear free surface:] 121 (\np{ln\_linssh}~\forcode{= .true.}) 137 \item [linear free surface] (\np[=.true.]{ln_linssh}{ln\_linssh}) 122 138 the first level thickness is constant in time: 123 the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on 124 the moving surface $z = \eta$. 125 There is a non-zero advective flux which is set for all advection schemes as 126 $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie the product of surface velocity (at $z = 0$) by 127 the first level tracer value. 128 \item[non-linear free surface:] 129 (\np{ln\_linssh}~\forcode{= .false.}) 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}) 130 145 convergence/divergence in the first ocean level moves the free surface up/down. 131 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. 132 148 \end{description} 133 149 134 150 In all cases, this boundary condition retains local conservation of tracer. 135 Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case. 136 Nevertheless, in the latter case, it is achieved to a good approximation since 137 the non-conservative term is the product of the time derivative of the tracer and the free surface height, 138 two quantities that are not correlated \citep{Roullet_Madec_JGR00, Griffies_al_MWR01, Campin2004}. 139 140 The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco}) is 141 the centred (\textit{now}) \textit{effective} ocean velocity, \ie the \textit{eulerian} velocity 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}. 158 159 The velocity field that appears in (\autoref{eq:TRA_adv} is 160 the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity 142 161 (see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or 143 162 the mixed layer eddy induced velocity (\textit{eiv}) when those parameterisations are used 144 163 (see \autoref{chap:LDF}). 145 164 146 Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), 147 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for 148 Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), 149 and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST). 150 The choice is made in the \ngn{namtra\_adv} namelist, by setting to \forcode{.true.} one of 151 the logicals \textit{ln\_traadv\_xxx}. 152 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 153 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 154 By default (\ie in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}. 155 If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}), 156 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! 157 182 158 183 Details of the advection schemes are given below. 159 The choosing an advection scheme is a complex matter which depends on the model physics, model resolution, 160 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 161 187 162 188 \begin{enumerate} 163 \item 164 CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that 165 they do not necessarily need additional diffusion; 166 \item 167 CEN and UBS are not \textit{positive} schemes 168 \footnote{negative values can appear in an initially strictly positive tracer field which is advected}, 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}, 169 193 implying that false extrema are permitted. 170 194 Their use is not recommended on passive tracers; 171 \item 172 It is recommended that the same advection-diffusion scheme is used onboth active and passive tracers.195 \item It is recommended that the same advection-diffusion scheme is used on 196 both active and passive tracers. 173 197 \end{enumerate} 174 198 175 Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and 176 passive tracers can create very nice-looking frontal structures that are pure numerical artefacts. 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. 177 202 Nevertheless, most of our users set a different treatment on passive and active tracers, 178 203 that's the reason why this possibility is offered. 179 We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of 180 their results. 181 182 % ------------------------------------------------------------------------------------------------------------- 183 % 2nd and 4th order centred schemes 184 % ------------------------------------------------------------------------------------------------------------- 185 \subsection{CEN: Centred scheme (\protect\np{ln\_traadv\_cen}~\forcode{= .true.})} 204 We strongly suggest them to perform a sensitivity experiment using a same treatment to 205 assess the robustness of their results. 206 207 %% ================================================================================================= 208 \subsection[CEN: Centred scheme (\forcode{ln_traadv_cen})]{CEN: Centred scheme (\protect\np{ln_traadv_cen}{ln\_traadv\_cen})} 186 209 \label{subsec:TRA_adv_cen} 187 210 188 % 2nd order centred scheme 189 190 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}~\forcode{= .true.}. 191 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 192 setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$. 211 % 2nd order centred scheme 212 213 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 216 setting \np{nn_cen_h}{nn\_cen\_h} and \np{nn_cen_v}{nn\_cen\_v} to $2$ or $4$. 193 217 CEN implementation can be found in the \mdl{traadv\_cen} module. 194 218 195 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of196 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. 197 221 For example, in the $i$-direction : 198 222 \begin{equation} 199 \label{eq: tra_adv_cen2}223 \label{eq:TRA_adv_cen2} 200 224 \tau_u^{cen2} = \overline T ^{i + 1/2} 201 225 \end{equation} 202 226 203 CEN2 is non diffusive (\ie it conserves the tracer variance, $\tau^2$) but dispersive 204 (\ie it may create false extrema). 205 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 206 produce a sensible solution. 207 The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 208 so $T$ in (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value. 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, 233 so $T$ in (\autoref{eq:TRA_adv_cen2}) is the \textit{now} tracer value. 209 234 210 235 Note that using the CEN2, the overall tracer advection is of second order accuracy since 211 both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2}) have this order of accuracy. 212 213 % 4nd order centred scheme 214 215 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as 216 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 236 both (\autoref{eq:TRA_adv}) and (\autoref{eq:TRA_adv_cen2}) have this order of accuracy. 237 238 % 4nd order centred scheme 239 240 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. 217 243 For example, in the $i$-direction: 218 244 \begin{equation} 219 \label{eq: tra_adv_cen4}245 \label{eq:TRA_adv_cen4} 220 246 \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2} 221 247 \end{equation} 222 In the vertical direction (\np{nn\_cen\_v}~\forcode{= 4}), 223 a $4^{th}$ COMPACT interpolation has been prefered \citep{Demange_PhD2014}. 224 In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion, 225 spectral characteristics similar to schemes of higher order \citep{Lele_JCP1992}. 248 In the vertical direction (\np[=4]{nn_cen_v}{nn\_cen\_v}), 249 a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}. 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}. 226 253 227 254 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but 228 255 a $4^{th}$ order evaluation of advective fluxes, 229 since the divergence of advective fluxes \autoref{eq: tra_adv} is kept at $2^{nd}$ order.230 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with231 the scheme presented here.232 Introducing a \forcode{.true.}$4^{th}$ order advection scheme is feasible but, for consistency reasons,233 it requires changes in the discretisation of the tracer advection together with changes in the continuity equation,234 and the momentum advection and pressure terms.256 since the divergence of advective fluxes \autoref{eq:TRA_adv} is kept at $2^{nd}$ order. 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. 235 262 236 263 A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive, 237 \ie the global variance of a tracer is not preserved using CEN4. 238 Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. 239 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 240 so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer. 264 \ie\ the global variance of a tracer is not preserved using CEN4. 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. 241 269 242 270 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), … … 244 272 This hypothesis usually reduces the order of the scheme. 245 273 Here we choose to set the gradient of $T$ across the boundary to zero. 246 Alternative conditions can be specified, such as a reduction to a second order scheme for 247 these near boundary grid points. 248 249 % ------------------------------------------------------------------------------------------------------------- 250 % FCT scheme 251 % ------------------------------------------------------------------------------------------------------------- 252 \subsection{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct}~\forcode{= .true.})} 274 Alternative conditions can be specified, 275 such as a reduction to a second order scheme for these near boundary grid points. 276 277 %% ================================================================================================= 278 \subsection[FCT: Flux Corrected Transport scheme (\forcode{ln_traadv_fct})]{FCT: Flux Corrected Transport scheme (\protect\np{ln_traadv_fct}{ln\_traadv\_fct})} 253 279 \label{subsec:TRA_adv_tvd} 254 280 255 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}~\forcode{= .true.}. 256 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 257 setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$. 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 285 setting \np{nn_fct_h}{nn\_fct\_h} and \np{nn_fct_v}{nn\_fct\_v} to $2$ or $4$. 258 286 FCT implementation can be found in the \mdl{traadv\_fct} module. 259 287 260 In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and261 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. 262 290 For example, in the $i$-direction : 263 291 \begin{equation} 264 \label{eq: tra_adv_fct}292 \label{eq:TRA_adv_fct} 265 293 \begin{split} 266 294 \tau_u^{ups} &= … … 268 296 T_{i + 1} & \text{if~} u_{i + 1/2} < 0 \\ 269 297 T_i & \text{if~} u_{i + 1/2} \geq 0 \\ 270 \end{cases} 271 \\ 298 \end{cases} \\ 272 299 \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big) 273 300 \end{split} … … 275 302 where $c_u$ is a flux limiter function taking values between 0 and 1. 276 303 The FCT order is the one of the centred scheme used 277 (\ie it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}).304 (\ie\ it depends on the setting of \np{nn_fct_h}{nn\_fct\_h} and \np{nn_fct_v}{nn\_fct\_v}). 278 305 There exist many ways to define $c_u$, each corresponding to a different FCT scheme. 279 The one chosen in \NEMO is described in \citet{Zalesak_JCP79}.306 The one chosen in \NEMO\ is described in \citet{zalesak_JCP79}. 280 307 $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. 281 308 The resulting scheme is quite expensive but \textit{positive}. 282 309 It can be used on both active and passive tracers. 283 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}. 284 285 An additional option has been added controlled by \np{nn\_fct\_zts}. 286 By setting this integer to a value larger than zero, 287 a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, but on the latter, 288 a split-explicit time stepping is used, with a number of sub-timestep equals to \np{nn\_fct\_zts}. 289 This option can be useful when the size of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 290 Note that in this case, a similar split-explicit time stepping should be used on vertical advection of momentum to 291 insure a better stability (see \autoref{subsec:DYN_zad}). 292 293 For stability reasons (see \autoref{chap:STP}), 294 $\tau_u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while 310 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{levy.estublier.ea_GRL01}. 311 312 For stability reasons (see \autoref{chap:TD}), 313 $\tau_u^{cen}$ is evaluated in (\autoref{eq:TRA_adv_fct}) using the \textit{now} tracer while 295 314 $\tau_u^{ups}$ is evaluated using the \textit{before} tracer. 296 In other words, the advective part of the scheme is time stepped with a leap-frog scheme 297 while a forward scheme is used for the diffusive part. 298 299 % ------------------------------------------------------------------------------------------------------------- 300 % MUSCL scheme 301 % ------------------------------------------------------------------------------------------------------------- 302 \subsection{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}~\forcode{= .true.})} 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. 317 318 %% ================================================================================================= 319 \subsection[MUSCL: Monotone Upstream Scheme for Conservative Laws (\forcode{ln_traadv_mus})]{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln_traadv_mus}{ln\_traadv\_mus})} 303 320 \label{subsec:TRA_adv_mus} 304 321 305 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}~\forcode{= .true.}. 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}. 306 324 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 307 325 308 MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}.309 In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between310 two $T$-points (\autoref{fig:adv_scheme}).326 MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}. 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}). 311 329 For example, in the $i$-direction : 312 \ begin{equation}313 % \label{eq: tra_adv_mus}330 \[ 331 % \label{eq:TRA_adv_mus} 314 332 \tau_u^{mus} = \lt\{ 315 333 \begin{split} 316 \tau_i&+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)317 \widetilde{\partial_i\tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\318 319 334 \tau_i &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 335 \widetilde{\partial_i \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\ 336 \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 337 \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} < 0 320 338 \end{split} 321 339 \rt. 322 \ end{equation}323 where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to324 ensure the \textit{positive} character of the scheme.325 326 The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to327 evaluate $\tau_u^{mus}$.340 \] 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 $\tau_u^{mus}$. 328 346 329 347 For an ocean grid point adjacent to land and where the ocean velocity is directed toward land, 330 348 an upstream flux is used. 331 349 This choice ensure the \textit{positive} character of the scheme. 332 In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes 333 (\np{ln\_mus\_ups}~\forcode{= .true.}). 334 335 % ------------------------------------------------------------------------------------------------------------- 336 % UBS scheme 337 % ------------------------------------------------------------------------------------------------------------- 338 \subsection{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}~\forcode{= .true.})} 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}). 352 353 %% ================================================================================================= 354 \subsection[UBS a.k.a. UP3: Upstream-Biased Scheme (\forcode{ln_traadv_ubs})]{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln_traadv_ubs}{ln\_traadv\_ubs})} 339 355 \label{subsec:TRA_adv_ubs} 340 356 341 The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}~\forcode{= .true.}. 357 The \textbf{U}pstream-\textbf{B}iased \textbf{S}cheme (UBS) is used when 358 \np[=.true.]{ln_traadv_ubs}{ln\_traadv\_ubs}. 342 359 UBS implementation can be found in the \mdl{traadv\_mus} module. 343 360 344 361 The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme 345 (Quadratic Upstream Interpolation for Convective Kinematics). 362 (\textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for 363 \textbf{C}onvective \textbf{K}inematics). 346 364 It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation. 347 365 For example, in the $i$-direction: 348 366 \begin{equation} 349 \label{eq: tra_adv_ubs}367 \label{eq:TRA_adv_ubs} 350 368 \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6} 351 369 \begin{cases} 352 353 370 \tau"_i & \text{if~} u_{i + 1/2} \geqslant 0 \\ 371 \tau"_{i + 1} & \text{if~} u_{i + 1/2} < 0 354 372 \end{cases} 355 \quad 356 \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] 357 374 \end{equation} 358 375 359 376 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error 360 \citep{Shchepetkin_McWilliams_OM05}. 361 The overall performance of the advection scheme is similar to that reported in \cite{Farrow1995}. 377 \citep{shchepetkin.mcwilliams_OM05}. 378 The overall performance of the advection scheme is similar to that reported in 379 \cite{farrow.stevens_JPO95}. 362 380 It is a relatively good compromise between accuracy and smoothness. 363 381 Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted, 364 382 but the amplitude of such are significantly reduced over the centred second or fourth order method. 365 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. 366 385 367 386 The intrinsic diffusion of UBS makes its use risky in the vertical direction where 368 387 the control of artificial diapycnal fluxes is of paramount importance 369 \citep{ Shchepetkin_McWilliams_OM05, Demange_PhD2014}.370 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme371 (\np{nn\_cen\_v}~\forcode{= 2 or 4}).372 373 For stability reasons (see \autoref{chap: STP}), the first term in \autoref{eq:tra_adv_ubs}374 (which corresponds to a second order centred scheme)375 is evaluated using the \textit{now} tracer (centred in time) while the second term376 (which is the diffusive part of the scheme),388 \citep{shchepetkin.mcwilliams_OM05, demange_phd14}. 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), 377 396 is evaluated using the \textit{before} tracer (forward in time). 378 This choice is discussed by \citet{Webb_al_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. 379 399 UBS and QUICK schemes only differ by one coefficient. 380 Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{Webb_al_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}. 381 402 This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 382 Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 383 384 Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows: 403 Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and 404 obtain a QUICK scheme. 405 406 Note that it is straightforward to rewrite \autoref{eq:TRA_adv_ubs} as follows: 385 407 \begin{gather} 386 \label{eq: traadv_ubs2}408 \label{eq:TRA_adv_ubs2} 387 409 \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} 388 410 \begin{cases} … … 391 413 \end{cases} 392 414 \intertext{or equivalently} 393 % \label{eq: traadv_ubs2b}415 % \label{eq:TRA_adv_ubs2b} 394 416 u_{i + 1/2} \ \tau_u^{ubs} = u_{i + 1/2} \, \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \Big]}^{\,i + 1/2} 395 417 - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber 396 418 \end{gather} 397 419 398 \autoref{eq: traadv_ubs2} has several advantages.420 \autoref{eq:TRA_adv_ubs2} has several advantages. 399 421 Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which 400 422 an upstream-biased diffusion term is added. 401 Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to402 be evaluated at the \textit{now} time step using \autoref{eq:tra_adv_ubs}.403 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which404 is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 405 Note the current version of NEMO uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}.406 407 % ------------------------------------------------------------------------------------------------------------- 408 % QCK scheme 409 % -------------------------------------------------------------------------------------------------------------410 \subsection {QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}~\forcode{= .true.})}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}. 430 431 %% ================================================================================================= 432 \subsection[QCK: QuiCKest scheme (\forcode{ln_traadv_qck})]{QCK: QuiCKest scheme (\protect\np{ln_traadv_qck}{ln\_traadv\_qck})} 411 433 \label{subsec:TRA_adv_qck} 412 434 413 The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme 414 proposed by \citet{Leonard1979} is used when \np{ln\_traadv\_qck}~\forcode{= .true.}. 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}. 415 439 QUICKEST implementation can be found in the \mdl{traadv\_qck} module. 416 440 417 441 QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter 418 \citep{Leonard1991}. 419 It has been implemented in NEMO by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. 442 \citep{leonard_CMAME91}. 443 It has been implemented in \NEMO\ by G. Reffray (Mercator Ocean) and 444 can be found in the \mdl{traadv\_qck} module. 420 445 The resulting scheme is quite expensive but \textit{positive}. 421 446 It can be used on both active and passive tracers. … … 424 449 Therefore the vertical flux is evaluated using the CEN2 scheme. 425 450 This no longer guarantees the positivity of the scheme. 426 The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property. 427 428 %%%gmcomment : Cross term are missing in the current implementation.... 429 430 % ================================================================ 431 % Tracer Lateral Diffusion 432 % ================================================================ 433 \section{Tracer lateral diffusion (\protect\mdl{traldf})} 451 The use of FCT in the vertical direction (as for the UBS case) should be implemented to 452 restore this property. 453 454 \cmtgm{Cross term are missing in the current implementation....} 455 456 %% ================================================================================================= 457 \section[Tracer lateral diffusion (\textit{traldf.F90})]{Tracer lateral diffusion (\protect\mdl{traldf})} 434 458 \label{sec:TRA_ldf} 435 %-----------------------------------------nam_traldf------------------------------------------------------ 436 437 \nlst{namtra_ldf} 438 %------------------------------------------------------------------------------------------------------------- 439 440 Options are defined through the \ngn{namtra\_ldf} namelist variables. 441 They are regrouped in four items, allowing to specify 442 $(i)$ the type of operator used (none, laplacian, bilaplacian), 443 $(ii)$ the direction along which the operator acts (iso-level, horizontal, iso-neutral), 444 $(iii)$ some specific options related to the rotated operators (\ie non-iso-level operator), and 445 $(iv)$ the specification of eddy diffusivity coefficient (either constant or variable in space and time). 446 Item $(iv)$ will be described in \autoref{chap:LDF}. 459 460 \begin{listing} 461 \nlst{namtra_ldf} 462 \caption{\forcode{&namtra_ldf}} 463 \label{lst:namtra_ldf} 464 \end{listing} 465 466 Options are defined through the \nam{tra_ldf}{tra\_ldf} namelist variables. 467 They are regrouped in four items, allowing to specify 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}. 447 476 The direction along which the operators act is defined through the slope between 448 477 this direction and the iso-level surfaces. … … 450 479 451 480 The lateral diffusion of tracers is evaluated using a forward scheme, 452 \ie the tracers appearing in its expression are the \textit{before} tracers in time,481 \ie\ the tracers appearing in its expression are the \textit{before} tracers in time, 453 482 except for the pure vertical component that appears when a rotation tensor is used. 454 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}).455 When \np{ln\_traldf\_msc}~\forcode{= .true.}, a Method of Stabilizing Correction is used in which 456 the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}. 457 458 % ------------------------------------------------------------------------------------------------------------- 459 % Type of operator 460 % -------------------------------------------------------------------------------------------------------------461 \subsection[Type of operator (\ protect\np{ln\_traldf}\{\_NONE,\_lap,\_blp\}\})]{Type of operator (\protect\np{ln\_traldf\_NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) }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}. 488 489 %% ================================================================================================= 490 \subsection[Type of operator (\forcode{ln_traldf_}\{\forcode{OFF,lap,blp}\})]{Type of operator (\protect\np{ln_traldf_OFF}{ln\_traldf\_OFF}, \protect\np{ln_traldf_lap}{ln\_traldf\_lap}, or \protect\np{ln_traldf_blp}{ln\_traldf\_blp})} 462 491 \label{subsec:TRA_ldf_op} 463 492 … … 465 494 466 495 \begin{description} 467 \item[\np{ln\_traldf\_NONE}~\forcode{= .true.}:] 468 no operator selected, the lateral diffusive tendency will not be applied to the tracer equation. 469 This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example). 470 \item[\np{ln\_traldf\_lap}~\forcode{= .true.}:] 471 a laplacian operator is selected. 472 This harmonic operator takes the following expression: $\mathpzc{L}(T) = \nabla \cdot A_{ht} \; \nabla T $, 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). 500 \item [{\np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap}}] a laplacian operator is selected. 501 This harmonic operator takes the following expression: 502 $\mathcal{L}(T) = \nabla \cdot A_{ht} \; \nabla T $, 473 503 where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}), 474 504 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}). 475 \item[\np{ln\_traldf\_blp}~\forcode{= .true.}]: 476 a bilaplacian operator is selected. 505 \item [{\np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}}] a bilaplacian operator is selected. 477 506 This biharmonic operator takes the following expression: 478 $\math pzc{B} = - \mathpzc{L}(\mathpzc{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$507 $\mathcal{B} = - \mathcal{L}(\mathcal{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$ 479 508 where the gradient operats along the selected direction, 480 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}). 481 511 In the code, the bilaplacian operator is obtained by calling the laplacian twice. 482 512 \end{description} … … 486 516 minimizing the impact on the larger scale features. 487 517 The main difference between the two operators is the scale selectiveness. 488 The bilaplacian damping time (\ie its spin down time) scales like $\lambda^{-4}$ for 489 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), 490 521 whereas the laplacian damping time scales only like $\lambda^{-2}$. 491 522 492 % ------------------------------------------------------------------------------------------------------------- 493 % Direction of action 494 % ------------------------------------------------------------------------------------------------------------- 495 \subsection[Action direction (\protect\np{ln\_traldf}\{\_lev,\_hor,\_iso,\_triad\})]{Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 523 %% ================================================================================================= 524 \subsection[Action direction (\forcode{ln_traldf_}\{\forcode{lev,hor,iso,triad}\})]{Direction of action (\protect\np{ln_traldf_lev}{ln\_traldf\_lev}, \protect\np{ln_traldf_hor}{ln\_traldf\_hor}, \protect\np{ln_traldf_iso}{ln\_traldf\_iso}, or \protect\np{ln_traldf_triad}{ln\_traldf\_triad})} 496 525 \label{subsec:TRA_ldf_dir} 497 526 498 527 The choice of a direction of action determines the form of operator used. 499 528 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when 500 iso-level option is used (\np {ln\_traldf\_lev}~\forcode{= .true.}) or501 when a horizontal (\iegeopotential) operator is demanded in \textit{z}-coordinate502 (\np{ln \_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}).529 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 531 (\np{ln_traldf_hor}{ln\_traldf\_hor} and \np[=.true.]{ln_zco}{ln\_zco}). 503 532 The associated code can be found in the \mdl{traldf\_lap\_blp} module. 504 533 The operator is a rotated (re-entrant) laplacian when 505 534 the direction along which it acts does not coincide with the iso-level surfaces, 506 535 that is when standard or triad iso-neutral option is used 507 (\np{ln \_traldf\_iso} or \np{ln\_traldf\_triad} equals\forcode{.true.},536 (\np{ln_traldf_iso}{ln\_traldf\_iso} or \np{ln_traldf_triad}{ln\_traldf\_triad} = \forcode{.true.}, 508 537 see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or 509 when a horizontal (\ie geopotential) operator is demanded in \textit{s}-coordinate510 (\np{ln \_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.})511 \footnote{In this case, the standard iso-neutral operator will be automatically selected}.538 when a horizontal (\ie\ geopotential) operator is demanded in \textit{s}-coordinate 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}. 512 541 In that case, a rotation is applied to the gradient(s) that appears in the operator so that 513 542 diffusive fluxes acts on the three spatial direction. … … 516 545 the next two sub-sections. 517 546 518 % ------------------------------------------------------------------------------------------------------------- 519 % iso-level operator 520 % ------------------------------------------------------------------------------------------------------------- 521 \subsection{Iso-level (bi -)laplacian operator ( \protect\np{ln\_traldf\_iso}) } 547 %% ================================================================================================= 548 \subsection[Iso-level (bi-)laplacian operator (\forcode{ln_traldf_iso})]{Iso-level (bi-)laplacian operator ( \protect\np{ln_traldf_iso}{ln\_traldf\_iso})} 522 549 \label{subsec:TRA_ldf_lev} 523 550 524 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 525 \begin{equation} 526 \label{eq: tra_ldf_lap}551 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 552 \begin{equation} 553 \label{eq:TRA_ldf_lap} 527 554 D_t^{lT} = \frac{1}{b_t} \Bigg( \delta_{i} \lt[ A_u^{lT} \; \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [T] \rt] 528 555 + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg) … … 531 558 where zero diffusive fluxes is assumed across solid boundaries, 532 559 first (and third in bilaplacian case) horizontal tracer derivative are masked. 533 It is implemented in the \rou{traldf\_lap} subroutine found in the \mdl{traldf\_lap} module. 534 The module also contains \rou{traldf\_blp}, the subroutine calling twice \rou{traldf\_lap} in order to 560 It is implemented in the \rou{tra\_ldf\_lap} subroutine found in the \mdl{traldf\_lap\_blp} module. 561 The module also contains \rou{tra\_ldf\_blp}, 562 the subroutine calling twice \rou{tra\_ldf\_lap} in order to 535 563 compute the iso-level bilaplacian operator. 536 564 537 565 It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in 538 the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 539 It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}~\forcode{= .true.}, 540 we have \np{ln\_traldf\_lev}~\forcode{= .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}~\forcode{= .true.}. 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}. 541 572 In both cases, it significantly contributes to diapycnal mixing. 542 573 It is therefore never recommended, even when using it in the bilaplacian case. 543 574 544 Note that in the partial step $z$-coordinate (\np {ln\_zps}~\forcode{= .true.}),575 Note that in the partial step $z$-coordinate (\np[=.true.]{ln_zps}{ln\_zps}), 545 576 tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom. 546 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. 547 579 They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}. 548 580 549 % ------------------------------------------------------------------------------------------------------------- 550 % Rotated laplacian operator 551 % ------------------------------------------------------------------------------------------------------------- 552 \subsection{Standard and triad (bi -)laplacian operator} 581 %% ================================================================================================= 582 \subsection{Standard and triad (bi-)laplacian operator} 553 583 \label{subsec:TRA_ldf_iso_triad} 554 584 555 %&& Standard rotated (bi -)laplacian operator 556 %&& ---------------------------------------------- 557 \subsubsection{Standard rotated (bi -)laplacian operator (\protect\mdl{traldf\_iso})} 585 %% ================================================================================================= 586 \subsubsection[Standard rotated (bi-)laplacian operator (\textit{traldf\_iso.F90})]{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 558 587 \label{subsec:TRA_ldf_iso} 559 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf}) 560 takes the following semi -discrete space form in $z$- and $s$-coordinates: 561 \begin{equation} 562 \label{eq:tra_ldf_iso} 588 589 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:MB_zdf}) 590 takes the following semi-discrete space form in $z$- and $s$-coordinates: 591 \begin{equation} 592 \label{eq:TRA_ldf_iso} 563 593 \begin{split} 564 594 D_T^{lT} = \frac{1}{b_t} \Bigg[ \quad &\delta_i A_u^{lT} \lt( \frac{e_{2u} e_{3u}}{e_{1u}} \, \delta_{i + 1/2} [T] … … 573 603 where $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells, 574 604 $r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and 575 the surface along which the diffusion operator acts (\ie horizontal or iso-neutral surfaces).576 It is thus used when, in addition to \np {ln\_traldf\_lap}~\forcode{= .true.},577 we have \np {ln\_traldf\_iso}~\forcode{= .true.},578 or both \np {ln\_traldf\_hor}~\forcode{= .true.} and \np{ln\_zco}~\forcode{= .true.}.605 the surface along which the diffusion operator acts (\ie\ horizontal or iso-neutral surfaces). 606 It is thus used when, in addition to \np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap}, 607 we have \np[=.true.]{ln_traldf_iso}{ln\_traldf\_iso}, 608 or both \np[=.true.]{ln_traldf_hor}{ln\_traldf\_hor} and \np[=.true.]{ln_zco}{ln\_zco}. 579 609 The way these slopes are evaluated is given in \autoref{sec:LDF_slp}. 580 At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using 581 the mask technique (see \autoref{sec:LBC_coast}). 582 583 The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical derivatives. 584 For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that 585 used in the vertical physics (see \autoref{sec:TRA_zdf}). 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}). 613 614 The operator in \autoref{eq:TRA_ldf_iso} involves both lateral and vertical derivatives. 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}). 586 617 For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module, 587 618 but in the \mdl{trazdf} module where, if iso-neutral mixing is used, 588 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 $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$. 589 621 590 622 This formulation conserves the tracer but does not ensure the decrease of the tracer variance. 591 Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without 592 any additional background horizontal diffusion \citep{Guilyardi_al_CD01}. 593 594 Note that in the partial step $z$-coordinate (\np{ln\_zps}~\forcode{= .true.}), 595 the horizontal derivatives at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment. 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}. 625 626 Note that in the partial step $z$-coordinate (\np[=.true.]{ln_zps}{ln\_zps}), 627 the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require 628 a specific treatment. 596 629 They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 597 630 598 %&& Triad rotated (bi -)laplacian operator 599 %&& ------------------------------------------- 600 \subsubsection{Triad rotated (bi -)laplacian operator (\protect\np{ln\_traldf\_triad})} 631 %% ================================================================================================= 632 \subsubsection[Triad rotated (bi-)laplacian operator (\forcode{ln_traldf_triad})]{Triad rotated (bi-)laplacian operator (\protect\np{ln_traldf_triad}{ln\_traldf\_triad})} 601 633 \label{subsec:TRA_ldf_triad} 602 634 603 If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}~\forcode{= .true.}; see \autoref{apdx:triad}) 604 605 An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases 606 is also available in \NEMO (\np{ln\_traldf\_grif}~\forcode{= .true.}).607 A complete description of the algorithm is given in \autoref{apdx:triad}. 608 609 The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:tra_ldf_lap}) twice.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}). 638 A complete description of the algorithm is given in \autoref{apdx:TRIADS}. 639 640 The lateral fourth order bilaplacian operator on tracers is obtained by 641 applying (\autoref{eq:TRA_ldf_lap}) twice. 610 642 The operator requires an additional assumption on boundary conditions: 611 643 both first and third derivative terms normal to the coast are set to zero. 612 644 613 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. 614 647 It requires an additional assumption on boundary conditions: 615 648 first and third derivative terms normal to the coast, 616 649 normal to the bottom and normal to the surface are set to zero. 617 650 618 %&& Option for the rotated operators 619 %&& ---------------------------------------------- 651 %% ================================================================================================= 620 652 \subsubsection{Option for the rotated operators} 621 653 \label{subsec:TRA_ldf_options} 622 654 623 \begin{itemize} 624 \item \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) 625 \item \np{rn\_slpmax} = slope limit (both operators) 626 \item \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 627 \item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only) 628 \item \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 629 \end{itemize} 630 631 % ================================================================ 632 % Tracer Vertical Diffusion 633 % ================================================================ 634 \section{Tracer vertical diffusion (\protect\mdl{trazdf})} 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} 663 664 %% ================================================================================================= 665 \section[Tracer vertical diffusion (\textit{trazdf.F90})]{Tracer vertical diffusion (\protect\mdl{trazdf})} 635 666 \label{sec:TRA_zdf} 636 %--------------------------------------------namzdf--------------------------------------------------------- 637 638 \nlst{namzdf} 639 %-------------------------------------------------------------------------------------------------------------- 640 641 Options are defined through the \ngn{namzdf} namelist variables. 642 The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, 643 and is based on a laplacian operator. 644 The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi -discrete space form: 645 \begin{gather*} 646 % \label{eq:tra_zdf} 647 D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\ 648 D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] 649 \end{gather*} 650 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity, 651 respectively. 667 668 Options are defined through the \nam{zdf}{zdf} namelist variables. 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 \[ 674 % \label{eq:TRA_zdf} 675 D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \quad 676 D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] 677 \] 678 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on 679 temperature and salinity, respectively. 652 680 Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised 653 (\ie \key{zdfddm} is defined).681 (\ie\ \np[=.true.]{ln_zdfddm}{ln\_zdfddm},). 654 682 The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF). 655 Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by656 $\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of 657 \autoref{eq:tra_ldf_iso}.683 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}. 658 686 659 687 At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified. … … 662 690 a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}). 663 691 664 The large eddy coefficient found in the mixed layer together with high vertical resolution implies that 665 in the case of explicit time stepping (\np{ln\_zdfexp}~\forcode{= .true.}) 666 there would be too restrictive a constraint on the time step. 667 Therefore, the default implicit time stepping is preferred for the vertical diffusion since 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. 694 Therefore an implicit time stepping is preferred for the vertical diffusion since 668 695 it overcomes the stability constraint. 669 A forward time differencing scheme (\np{ln\_zdfexp}~\forcode{= .true.}) using 670 a time splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative. 671 Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 672 673 % ================================================================ 674 % External Forcing 675 % ================================================================ 696 697 %% ================================================================================================= 676 698 \section{External forcing} 677 699 \label{sec:TRA_sbc_qsr_bbc} 678 700 679 % ------------------------------------------------------------------------------------------------------------- 680 % surface boundary condition 681 % ------------------------------------------------------------------------------------------------------------- 682 \subsection{Surface boundary condition (\protect\mdl{trasbc})} 701 %% ================================================================================================= 702 \subsection[Surface boundary condition (\textit{trasbc.F90})]{Surface boundary condition (\protect\mdl{trasbc})} 683 703 \label{subsec:TRA_sbc} 684 704 … … 690 710 691 711 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 692 (\ie atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due 693 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 694 715 to the heat and salt content of the mass exchange. 695 716 They are both included directly in $Q_{ns}$, the surface heat flux, 696 717 and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details). 697 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 698 699 The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers): 700 701 \begin{itemize} 702 \item 703 $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 704 (\ie the difference between the total surface heat flux and the fraction of the short wave flux that 705 penetrates into the water column, see \autoref{subsec:TRA_qsr}) 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}) 706 729 plus the heat content associated with of the mass exchange with the atmosphere and lands. 707 \item 708 $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 709 \item 710 \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and 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 711 733 possibly with the sea-ice and ice-shelves. 712 \item 713 \textit{rnf}, the mass flux associated with runoff 734 \item [\textit{rnf}] The mass flux associated with runoff 714 735 (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 715 \item 716 \textit{fwfisf}, the mass flux associated with ice shelf melt, 736 \item [\textit{fwfisf}] The mass flux associated with ice shelf melt, 717 737 (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 718 \end{ itemize}738 \end{labeling} 719 739 720 740 The surface boundary condition on temperature and salinity is applied as follows: 721 741 \begin{equation} 722 \label{eq:tra_sbc} 723 \begin{alignedat}{2} 724 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns} }^t \\ 725 F^S &= &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t 726 \end{alignedat} 742 \label{eq:TRA_sbc} 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 727 745 \end{equation} 728 746 where $\overline x^t$ means that $x$ is averaged over two consecutive time steps 729 747 ($t - \rdt / 2$ and $t + \rdt / 2$). 730 Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:STP}). 731 732 In the linear free surface case (\np{ln\_linssh}~\forcode{= .true.}), an additional term has to be added on 733 both temperature and salinity. 734 On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$. 735 On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in 736 the volume of the first level. 748 Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:TD}). 749 750 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. 737 756 The resulting surface boundary condition is applied as follows: 738 757 \begin{equation} 739 \label{eq:tra_sbc_lin} 740 \begin{alignedat}{2} 741 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 742 &\overline{(Q_{ns} - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\ 743 F^S &= &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 744 &\overline{(\textit{sfx} - \textit{emp} \lt. S \rt|_{k = 1})}^t 745 \end{alignedat} 746 \end{equation} 747 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 758 \label{eq:TRA_sbc_lin} 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. 748 766 In the linear free surface case, there is a small imbalance. 749 The imbalance is larger than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}.750 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}).751 752 % ------------------------------------------------------------------------------------------------------------- 753 % Solar Radiation Penetration 754 % -------------------------------------------------------------------------------------------------------------755 \subsection {Solar radiation penetration (\protect\mdl{traqsr})}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}). 771 772 %% ================================================================================================= 773 \subsection[Solar radiation penetration (\textit{traqsr.F90})]{Solar radiation penetration (\protect\mdl{traqsr})} 756 774 \label{subsec:TRA_qsr} 757 %--------------------------------------------namqsr-------------------------------------------------------- 758 759 \nlst{namtra_qsr} 760 %-------------------------------------------------------------------------------------------------------------- 761 762 Options are defined through the \ngn{namtra\_qsr} namelist variables. 763 When the penetrative solar radiation option is used (\np{ln\_flxqsr}~\forcode{= .true.}), 775 776 \begin{listing} 777 \nlst{namtra_qsr} 778 \caption{\forcode{&namtra_qsr}} 779 \label{lst:namtra_qsr} 780 \end{listing} 781 782 Options are defined through the \nam{tra_qsr}{tra\_qsr} namelist variables. 783 When the penetrative solar radiation option is used (\np[=.true.]{ln_traqsr}{ln\_traqsr}), 764 784 the solar radiation penetrates the top few tens of meters of the ocean. 765 If it is not used (\np{ln\_flxqsr}~\forcode{= .false.}) all the heat flux is absorbed in the first ocean level. 766 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and 767 the surface boundary condition is modified to take into account only the non-penetrative part of the surface 768 heat flux: 769 \begin{equation} 770 \label{eq:PE_qsr} 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: 790 \begin{equation} 791 \label{eq:TRA_PE_qsr} 771 792 \begin{gathered} 772 793 \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\ … … 774 795 \end{gathered} 775 796 \end{equation} 776 where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie the shortwave radiation) and797 where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie\ the shortwave radiation) and 777 798 $I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$). 778 The additional term in \autoref{eq: PE_qsr} is discretized as follows:779 \begin{equation} 780 \label{eq: tra_qsr}799 The additional term in \autoref{eq:TRA_PE_qsr} is discretized as follows: 800 \begin{equation} 801 \label{eq:TRA_qsr} 781 802 \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w] 782 803 \end{equation} 783 804 784 805 The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. 785 The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to 786 heating the upper few tens of centimetres. 787 The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ 788 (specified through namelist parameter \np{rn\_abs}). 789 It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 790 of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \ngn{namtra\_qsr} namelist). 791 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to 792 larger depths where it contributes to local heating. 793 The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen. 794 In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}~\forcode{= .true.}) 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\% 809 (specified through namelist parameter \np{rn_abs}{rn\_abs}). 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. 817 In the simple 2-waveband light penetration scheme (\np[=.true.]{ln_qsr_2bd}{ln\_qsr\_2bd}) 795 818 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 796 leading to the following expression \citep{ Paulson1977}:819 leading to the following expression \citep{paulson.simpson_JPO77}: 797 820 \[ 798 % \label{eq: traqsr_iradiance}821 % \label{eq:TRA_qsr_iradiance} 799 822 I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt] 800 823 \] 801 824 where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 802 It is usually chosen to be 23~m by setting the \np{rn \_si0} namelist parameter.803 The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification804 (oligotrophic waters).825 It is usually chosen to be 23~m by setting the \np{rn_si0}{rn\_si0} namelist parameter. 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). 805 828 806 829 Such assumptions have been shown to provide a very crude and simplistic representation of 807 observed light penetration profiles (\cite{ Morel_JGR88}, see also \autoref{fig:traqsr_irradiance}).830 observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig:TRA_qsr_irradiance}). 808 831 Light absorption in the ocean depends on particle concentration and is spectrally selective. 809 \cite{ Morel_JGR88} has shown that an accurate representation of light penetration can be provided by832 \cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by 810 833 a 61 waveband formulation. 811 834 Unfortunately, such a model is very computationally expensive. 812 Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this formulation in which 813 visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm). 814 For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from 815 the full spectral model of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), 816 assuming the same power-law relationship. 817 As shown in \autoref{fig:traqsr_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), 818 843 reproduces quite closely the light penetration profiles predicted by the full spectal model, 819 844 but with much greater computational efficiency. 820 845 The 2-bands formulation does not reproduce the full model very well. 821 846 822 The RGB formulation is used when \np {ln\_qsr\_rgb}~\forcode{= .true.}.823 The RGB attenuation coefficients (\ie the inverses of the extinction length scales) are tabulated over824 61 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L 847 The RGB formulation is used when \np[=.true.]{ln_qsr_rgb}{ln\_qsr\_rgb}. 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 $g.Chl/L$ 825 850 (see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module). 826 851 Four types of chlorophyll can be chosen in the RGB formulation: 827 852 828 853 \begin{description} 829 \item[\np{nn\_chdta}~\forcode{= 0}] 830 a constant 0.05 g.Chl/L value everywhere ; 831 \item[\np{nn\_chdta}~\forcode{= 1}] 832 an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in 833 the vertical direction; 834 \item[\np{nn\_chdta}~\forcode{= 2}] 835 same as previous case except that a vertical profile of chlorophyl is used. 836 Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value; 837 \item[\np{ln\_qsr\_bio}~\forcode{= .true.}] 838 simulated time varying chlorophyll by TOP biogeochemical model. 839 In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in 840 PISCES or LOBSTER and the oceanic heating rate. 841 \end{description} 842 843 The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation is added to 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. 865 \end{description} 866 867 The trend in \autoref{eq:TRA_qsr} associated with the penetration of the solar radiation is added to 844 868 the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 845 869 … … 847 871 the depth of $w-$levels does not significantly vary with location. 848 872 The level at which the light has been totally absorbed 849 (\ie it is less than the computer precision) is computed once,873 (\ie\ it is less than the computer precision) is computed once, 850 874 and the trend associated with the penetration of the solar radiation is only added down to that level. 851 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. 852 877 In this case, we have chosen that all remaining radiation is absorbed in the last ocean level 853 (\ie $I$ is masked). 854 855 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 856 \begin{figure}[!t] 857 \begin{center} 858 \includegraphics[]{Fig_TRA_Irradiance} 859 \caption{ 860 \protect\label{fig:traqsr_irradiance} 861 Penetration profile of the downward solar irradiance calculated by four models. 862 Two waveband chlorophyll-independent formulation (blue), 863 a chlorophyll-dependent monochromatic formulation (green), 864 4 waveband RGB formulation (red), 865 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 866 (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. 867 From \citet{Lengaigne_al_CD07}. 868 } 869 \end{center} 878 (\ie\ $I$ is masked). 879 880 \begin{figure} 881 \centering 882 \includegraphics[width=0.66\textwidth]{TRA_Irradiance} 883 \caption[Penetration profile of the downward solar irradiance calculated by four models]{ 884 Penetration profile of the downward solar irradiance calculated by four models. 885 Two waveband chlorophyll-independent formulation (blue), 886 a chlorophyll-dependent monochromatic formulation (green), 887 4 waveband RGB formulation (red), 888 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 889 (a) Chl=0.05 $mg/m^3$ and (b) Chl=0.5 $mg/m^3$. 890 From \citet{lengaigne.menkes.ea_CD07}.} 891 \label{fig:TRA_qsr_irradiance} 870 892 \end{figure} 871 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 872 873 % ------------------------------------------------------------------------------------------------------------- 874 % Bottom Boundary Condition 875 % ------------------------------------------------------------------------------------------------------------- 876 \subsection{Bottom boundary condition (\protect\mdl{trabbc})} 893 894 %% ================================================================================================= 895 \subsection[Bottom boundary condition (\textit{trabbc.F90}) - \forcode{ln_trabbc})]{Bottom boundary condition (\protect\mdl{trabbc} - \protect\np{ln_trabbc}{ln\_trabbc})} 877 896 \label{subsec:TRA_bbc} 878 %--------------------------------------------nambbc-------------------------------------------------------- 879 880 \nlst{nambbc}881 %-------------------------------------------------------------------------------------------------------------- 882 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 883 \ begin{figure}[!t]884 \begin{center} 885 \includegraphics[]{Fig_TRA_geoth}886 \caption{887 \protect\label{fig:geothermal}888 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}.889 It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.890 }891 \ end{center}897 898 \begin{listing} 899 \nlst{nambbc} 900 \caption{\forcode{&nambbc}} 901 \label{lst:nambbc} 902 \end{listing} 903 904 \begin{figure} 905 \centering 906 \includegraphics[width=0.66\textwidth]{TRA_geoth} 907 \caption[Geothermal heat flux]{ 908 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}. 909 It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}.} 910 \label{fig:TRA_geothermal} 892 911 \end{figure} 893 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>894 912 895 913 Usually it is assumed that there is no exchange of heat or salt through the ocean bottom, 896 \ie a no flux boundary condition is applied on active tracers at the bottom.914 \ie\ a no flux boundary condition is applied on active tracers at the bottom. 897 915 This is the default option in \NEMO, and it is implemented using the masking technique. 898 However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling. 899 This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{Stein_Stein_Nat92}), 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}), 900 920 but it warms systematically the ocean and acts on the densest water masses. 901 921 Taking this flux into account in a global ocean model increases the deepest overturning cell 902 (\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}. 903 904 Options are defined through the \ngn{namtra\_bbc} namelist variables. 905 The presence of geothermal heating is controlled by setting the namelist parameter \np{ln\_trabbc} to true. 906 Then, when \np{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by 907 the \np{nn\_geoflx\_cst}, which is also a namelist parameter. 908 When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in 909 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{Emile-Geay_Madec_OS09}. 910 911 % ================================================================ 912 % Bottom Boundary Layer 913 % ================================================================ 914 \section{Bottom boundary layer (\protect\mdl{trabbl} - \protect\key{trabbl})} 922 (\ie\ the one associated with the Antarctic Bottom Water) by 923 a few Sverdrups \citep{emile-geay.madec_OS09}. 924 925 Options are defined through the \nam{bbc}{bbc} namelist variables. 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}. 934 935 %% ================================================================================================= 936 \section[Bottom boundary layer (\textit{trabbl.F90} - \forcode{ln_trabbl})]{Bottom boundary layer (\protect\mdl{trabbl} - \protect\np{ln_trabbl}{ln\_trabbl})} 915 937 \label{sec:TRA_bbl} 916 %--------------------------------------------nambbl--------------------------------------------------------- 917 918 \nlst{nambbl} 919 %-------------------------------------------------------------------------------------------------------------- 920 921 Options are defined through the \ngn{nambbl} namelist variables. 938 939 \begin{listing} 940 \nlst{nambbl} 941 \caption{\forcode{&nambbl}} 942 \label{lst:nambbl} 943 \end{listing} 944 945 Options are defined through the \nam{bbl}{bbl} namelist variables. 922 946 In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps. 923 947 This is not adequate to represent gravity driven downslope flows. … … 925 949 where dense water formed in marginal seas flows into a basin filled with less dense water, 926 950 or along the continental slope when dense water masses are formed on a continental shelf. 927 The amount of entrainment that occurs in these gravity plumes is critical in determining the density and 928 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. 929 954 $z$-coordinate models tend to overestimate the entrainment, 930 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, 931 957 sometimes over a thickness much larger than the thickness of the observed gravity plume. 932 A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of 933 a sill \citep{Willebrand_al_PO01}, and the thickness of the plume is not resolved. 934 935 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{Beckmann_Doscher1997}, 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}, 936 964 is to allow a direct communication between two adjacent bottom cells at different levels, 937 965 whenever the densest water is located above the less dense water. 938 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. 939 968 In the current implementation of the BBL, only the tracers are modified, not the velocities. 940 Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by 941 \citet{Campin_Goosse_Tel99}. 942 943 % ------------------------------------------------------------------------------------------------------------- 944 % Diffusive BBL 945 % ------------------------------------------------------------------------------------------------------------- 946 \subsection{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf}~\forcode{= 1})} 969 Furthermore, it only connects ocean bottom cells, 970 and therefore does not include all the improvements introduced by \citet{campin.goosse_T99}. 971 972 %% ================================================================================================= 973 \subsection[Diffusive bottom boundary layer (\forcode{nn_bbl_ldf=1})]{Diffusive bottom boundary layer (\protect\np[=1]{nn_bbl_ldf}{nn\_bbl\_ldf})} 947 974 \label{subsec:TRA_bbl_diff} 948 975 949 When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 950 the diffusive flux between two adjacent cells at the ocean floor is given by 976 When applying sigma-diffusion 977 (\np[=.true.]{ln_trabbl}{ln\_trabbl} and \np{nn_bbl_ldf}{nn\_bbl\_ldf} set to 1), 978 the diffusive flux between two adjacent cells at the ocean floor is given by 951 979 \[ 952 % \label{eq: tra_bbl_diff}980 % \label{eq:TRA_bbl_diff} 953 981 \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T 954 982 \] 955 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and956 $A_l^\sigma$ the lateral diffusivity in the BBL.957 Following \citet{ Beckmann_Doscher1997}, the latter is prescribed with a spatial dependence,958 \ie in the conditional form959 \begin{equation} 960 \label{eq: tra_bbl_coef}983 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 984 and $A_l^\sigma$ the lateral diffusivity in the BBL. 985 Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence, 986 \ie\ in the conditional form 987 \begin{equation} 988 \label{eq:TRA_bbl_coef} 961 989 A_l^\sigma (i,j,t) = 962 990 \begin{cases} 963 991 A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\ 964 \\ 965 0 & \text{otherwise} \\ 992 0 & \text{otherwise} 966 993 \end{cases} 967 994 \end{equation} 968 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and 995 where $A_{bbl}$ is the BBL diffusivity coefficient, 996 given by the namelist parameter \np{rn_ahtbbl}{rn\_ahtbbl} and 969 997 usually set to a value much larger than the one used for lateral mixing in the open ocean. 970 The constraint in \autoref{eq: tra_bbl_coef} implies that sigma-like diffusion only occurs when998 The constraint in \autoref{eq:TRA_bbl_coef} implies that sigma-like diffusion only occurs when 971 999 the density above the sea floor, at the top of the slope, is larger than in the deeper ocean 972 (see green arrow in \autoref{fig: bbl}).1000 (see green arrow in \autoref{fig:TRA_bbl}). 973 1001 In practice, this constraint is applied separately in the two horizontal directions, 974 and the density gradient in \autoref{eq: tra_bbl_coef} is evaluated with the log gradient formulation:1002 and the density gradient in \autoref{eq:TRA_bbl_coef} is evaluated with the log gradient formulation: 975 1003 \[ 976 % \label{eq: tra_bbl_Drho}1004 % \label{eq:TRA_bbl_Drho} 977 1005 \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S 978 1006 \] 979 where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and 980 $\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively. 981 982 % ------------------------------------------------------------------------------------------------------------- 983 % Advective BBL 984 % ------------------------------------------------------------------------------------------------------------- 985 \subsection{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}~\forcode{= 1..2})} 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. 1010 1011 %% ================================================================================================= 1012 \subsection[Advective bottom boundary layer (\forcode{nn_bbl_adv=1,2})]{Advective bottom boundary layer (\protect\np[=1,2]{nn_bbl_adv}{nn\_bbl\_adv})} 986 1013 \label{subsec:TRA_bbl_adv} 987 1014 … … 991 1018 %} 992 1019 993 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 994 \begin{figure}[!t] 995 \begin{center} 996 \includegraphics[]{Fig_BBL_adv} 997 \caption{ 998 \protect\label{fig:bbl} 999 Advective/diffusive Bottom Boundary Layer. 1000 The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$. 1001 Red arrows indicate the additional overturning circulation due to the advective BBL. 1002 The transport of the downslope flow is defined either as the transport of the bottom ocean cell (black arrow), 1003 or as a function of the along slope density gradient. 1004 The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ ocean bottom cells. 1005 } 1006 \end{center} 1020 \begin{figure} 1021 \centering 1022 \includegraphics[width=0.33\textwidth]{TRA_BBL_adv} 1023 \caption[Advective/diffusive bottom boundary layer]{ 1024 Advective/diffusive Bottom Boundary Layer. 1025 The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$. 1026 Red arrows indicate the additional overturning circulation due to the advective BBL. 1027 The transport of the downslope flow is defined either 1028 as the transport of the bottom ocean cell (black arrow), 1029 or as a function of the along slope density gradient. 1030 The green arrow indicates the diffusive BBL flux directly connecting 1031 $kup$ and $kdwn$ ocean bottom cells.} 1032 \label{fig:TRA_bbl} 1007 1033 \end{figure} 1008 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>1009 1034 1010 1035 %!! nn_bbl_adv = 1 use of the ocean velocity as bbl velocity … … 1012 1037 %!! i.e. transport proportional to the along-slope density gradient 1013 1038 1014 %%%gmcomment : this section has to be really written 1015 1016 When applying an advective BBL (\np{nn\_bbl\_adv}~\forcode{= 1..2}), an overturning circulation is added which 1017 connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope. 1039 \cmtgm{This section has to be really written} 1040 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. 1018 1044 The density difference causes dense water to move down the slope. 1019 1045 1020 \np{nn\_bbl\_adv}~\forcode{= 1}: 1021 the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step 1022 (see black arrow in \autoref{fig:bbl}) \citep{Beckmann_Doscher1997}. 1023 It is a \textit{conditional advection}, that is, advection is allowed only 1024 if dense water overlies less dense water on the slope (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$) and 1025 if the velocity is directed towards greater depth (\ie $\vect U \cdot \nabla H > 0$). 1026 1027 \np{nn\_bbl\_adv}~\forcode{= 2}: 1028 the downslope velocity is chosen to be proportional to $\Delta \rho$, 1029 the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}. 1030 The advection is allowed only if dense water overlies less dense water on the slope 1031 (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$). 1032 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}), 1033 is simply given by the following expression: 1034 \[ 1035 % \label{eq:bbl_Utr} 1036 u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 1037 \] 1038 where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl}, 1039 a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells, 1040 respectively. 1041 The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity, 1042 and because no direct estimation of this parameter is available, a uniform value has been assumed. 1043 The possible values for $\gamma$ range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}. 1044 1045 Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme. 1046 Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and 1047 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. 1048 1077 The entrainment is replaced by the vertical mixing implicit in the advection scheme. 1049 Let us consider as an example the case displayed in \autoref{fig: bbl} where1078 Let us consider as an example the case displayed in \autoref{fig:TRA_bbl} where 1050 1079 the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$. 1051 The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by 1052 the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and 1053 the upward \autoref{eq:bbl_up} return flows as follows: 1054 \begin{alignat}{3} 1055 \label{eq:bbl_dw} 1056 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 1057 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 1058 \label{eq:bbl_hor} 1059 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 1060 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ 1061 % 1062 \intertext{and for $k =kdw-1,\;..., \; kup$ :} 1063 % 1064 \label{eq:bbl_up} 1065 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 1066 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} &&\lt( T^{do}_{k +1} - T^{sh}_{k} \rt) 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} 1084 \label{eq:TRA_bbl_dw} 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) \\ 1086 \label{eq:TRA_bbl_hor} 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 $k =kdw-1,\;..., \; kup$ :} 1089 \label{eq:TRA_bbl_up} 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) 1067 1091 \end{alignat} 1068 1092 where $b_t$ is the $T$-cell volume. … … 1071 1095 It has to be used to compute the effective velocity as well as the effective overturning circulation. 1072 1096 1073 % ================================================================ 1074 % Tracer damping 1075 % ================================================================ 1076 \section{Tracer damping (\protect\mdl{tradmp})} 1097 %% ================================================================================================= 1098 \section[Tracer damping (\textit{tradmp.F90})]{Tracer damping (\protect\mdl{tradmp})} 1077 1099 \label{sec:TRA_dmp} 1078 %--------------------------------------------namtra_dmp------------------------------------------------- 1079 1080 \nlst{namtra_dmp} 1081 %-------------------------------------------------------------------------------------------------------------- 1082 1083 In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations: 1084 \begin{equation} 1085 \label{eq:tra_dmp} 1086 \begin{gathered} 1087 \pd[T]{t} = \cdots - \gamma (T - T_o) \\ 1088 \pd[S]{t} = \cdots - \gamma (S - S_o) 1089 \end{gathered} 1090 \end{equation} 1091 where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields 1092 (usually a climatology). 1093 Options are defined through the \ngn{namtra\_dmp} namelist variables. 1094 The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true. 1095 It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true in 1096 \ngn{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set 1097 (\ie that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread}, 1100 1101 \begin{listing} 1102 \nlst{namtra_dmp} 1103 \caption{\forcode{&namtra_dmp}} 1104 \label{lst:namtra_dmp} 1105 \end{listing} 1106 1107 In some applications it can be useful to add a Newtonian damping term into 1108 the temperature and salinity equations: 1109 \begin{equation} 1110 \label{eq:TRA_dmp} 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. 1116 The restoring term is added when the namelist parameter \np{ln_tradmp}{ln\_tradmp} is set to true. 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 1120 (\ie\ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread}, 1098 1121 see \autoref{subsec:SBC_fldread}). 1099 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. 1100 The file name is specified by the namelist variable \np{cn\_resto}. 1101 The DMP\_TOOLS tool is provided to allow users to generate the netcdf file. 1102 1103 The two main cases in which \autoref{eq:tra_dmp} is used are 1104 \textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and 1105 \textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field 1106 (for example to build the initial state of a prognostic simulation, 1107 or to use the resulting velocity field for a passive tracer study). 1122 The restoring coefficient $\gamma$ is a three-dimensional array read in during 1123 the \rou{tra\_dmp\_init} routine. 1124 The file name is specified by the namelist variable \np{cn_resto}{cn\_resto}. 1125 The \texttt{DMP\_TOOLS} are provided to allow users to generate the netcdf file. 1126 1127 The two main cases in which \autoref{eq:TRA_dmp} is used are 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*} 1108 1135 The first case applies to regional models that have artificial walls instead of open boundaries. 1109 In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas1110 it is zero in the interior of the model domain.1111 The second case corresponds to the use of the robust diagnostic method \citep{ Sarmiento1982}.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. 1138 The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}. 1112 1139 It allows us to find the velocity field consistent with the model dynamics whilst 1113 1140 having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$). 1114 1141 1115 The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but1116 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. 1117 1144 It also has undesirable effects on the ocean convection. 1118 It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much. 1119 1120 The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or 1121 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). 1122 1150 It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here 1123 \citep{Madec_al_JPO96}. 1124 1125 For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under 1126 \path{./tools/DMP_TOOLS}. 1127 1128 % ================================================================ 1129 % Tracer time evolution 1130 % ================================================================ 1131 \section{Tracer time evolution (\protect\mdl{tranxt})} 1151 \citep{madec.delecluse.ea_JPO96}. 1152 1153 For generating \ifile{resto}, 1154 see the documentation for the DMP tools provided with the source code under \path{./tools/DMP_TOOLS}. 1155 1156 %% ================================================================================================= 1157 \section[Tracer time evolution (\textit{tranxt.F90})]{Tracer time evolution (\protect\mdl{tranxt})} 1132 1158 \label{sec:TRA_nxt} 1133 %--------------------------------------------namdom----------------------------------------------------- 1134 1135 \nlst{namdom} 1136 %-------------------------------------------------------------------------------------------------------------- 1137 1138 Options are defined through the \ngn{namdom} namelist variables. 1139 The general framework for tracer time stepping is a modified leap-frog scheme \citep{Leclair_Madec_OM09}, 1140 \ie a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 1141 \begin{equation} 1142 \label{eq:tra_nxt} 1143 \begin{alignedat}{3} 1159 1160 Options are defined through the \nam{dom}{dom} namelist variables. 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}): 1164 \begin{equation} 1165 \label{eq:TRA_nxt} 1166 \begin{alignedat}{5} 1144 1167 &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\ 1145 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] \\ 1146 & && &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt] 1169 & && &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt] 1147 1170 \end{alignedat} 1148 \end{equation} 1149 where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, 1150 $\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$ 1151 (\ie fluxes plus content in mass exchanges). 1152 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1153 Its default value is \np{rn\_atfp}~\forcode{= 10.e-3}. 1171 \end{equation} 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}. 1154 1176 Note that the forcing correction term in the filter is not applied in linear free surface 1155 (\jp{lk\_vvl}~\forcode{= .false.}) (see \autoref{subsec:TRA_sbc}). 1156 Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$. 1157 1158 When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in 1159 \mdl{trazdf} module. 1177 (\jp{ln\_linssh}\forcode{=.true.}) (see \autoref{subsec:TRA_sbc}). 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. 1160 1183 In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module. 1161 1184 1162 In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed: 1163 $T^{t - \rdt} = T^t$ and $T^t = T_f$. 1164 1165 % ================================================================ 1166 % Equation of State (eosbn2) 1167 % ================================================================ 1168 \section{Equation of state (\protect\mdl{eosbn2}) } 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 $T^t = T_f$. 1187 1188 %% ================================================================================================= 1189 \section[Equation of state (\textit{eosbn2.F90})]{Equation of state (\protect\mdl{eosbn2})} 1169 1190 \label{sec:TRA_eosbn2} 1170 %--------------------------------------------nameos----------------------------------------------------- 1171 1172 \nlst{nameos}1173 %-------------------------------------------------------------------------------------------------------------- 1174 1175 % ------------------------------------------------------------------------------------------------------------- 1176 % Equation of State 1177 % -------------------------------------------------------------------------------------------------------------1178 \subsection {Equation of seawater (\protect\np{nn\_eos}~\forcode{= -1..1})}1191 1192 \begin{listing} 1193 \nlst{nameos} 1194 \caption{\forcode{&nameos}} 1195 \label{lst:nameos} 1196 \end{listing} 1197 1198 %% ================================================================================================= 1199 \subsection[Equation of seawater (\forcode{ln_}\{\forcode{teos10,eos80,seos}\})]{Equation of seawater (\protect\np{ln_teos10}{ln\_teos10}, \protect\np{ln_teos80}{ln\_teos80}, or \protect\np{ln_seos}{ln\_seos})} 1179 1200 \label{subsec:TRA_eos} 1180 1201 1181 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density, 1182 $\rho$, 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. 1183 1206 Because density gradients control the pressure gradient force through the hydrostatic balance, 1184 the equation of state provides a fundamental bridge between the distribution of active tracers and1185 the fluid dynamics.1207 the equation of state provides a fundamental bridge between 1208 the distribution of active tracers and the fluid dynamics. 1186 1209 Nonlinearities of the EOS are of major importance, in particular influencing the circulation through 1187 1210 determination of the static stability below the mixed layer, 1188 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{Roquet_JPO2015}. 1189 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983}) or 1190 TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real ocean circulation is attempted 1191 \citep{Roquet_JPO2015}. 1211 thus controlling rates of exchange between the atmosphere and the ocean interior 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}. 1192 1216 The use of TEOS-10 is highly recommended because 1193 \textit{(i)} it is the new official EOS, 1194 \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and 1195 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and 1196 practical salinity for EOS-980, both variables being more suitable for use as model variables 1197 \citep{TEOS10, Graham_McDougall_JPO13}. 1198 EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. 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*} 1225 EOS-80 is an obsolescent feature of the \NEMO\ system, kept only for backward compatibility. 1199 1226 For process studies, it is often convenient to use an approximation of the EOS. 1200 To that purposed, a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available. 1201 1202 In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density. 1203 Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 1204 This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as, 1205 with the exception of only a small percentage of the ocean, 1206 density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. 1207 1208 Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} which 1209 controls the EOS used (\forcode{= -1} for TEOS10 ; \forcode{= 0} for EOS-80 ; \forcode{= 1} for S-EOS). 1227 To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available. 1228 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}. 1236 1237 Options which control the EOS used are defined through the \nam{eos}{eos} namelist variables. 1210 1238 1211 1239 \begin{description} 1212 \item [\np{nn\_eos}~\forcode{= -1}]1213 the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used.1240 \item [{\np[=.true.]{ln_teos10}{ln\_teos10}}] the polyTEOS10-bsq equation of seawater 1241 \citep{roquet.madec.ea_OM15} is used. 1214 1242 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 1215 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and 1216 more computationally efficient expressions for their derived quantities which make them more adapted for 1217 use in ocean models. 1218 Note that a slightly higher precision polynomial form is now used replacement of 1219 the TEOS-10 rational function approximation for hydrographic data analysis \citep{TEOS10}. 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}. 1220 1249 A key point is that conservative state variables are used: 1221 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: $\Theta$). 1222 1252 The pressure in decibars is approximated by the depth in meters. 1223 1253 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. 1224 It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{TEOS10}. 1254 It is set to $C_p$ = 3991.86795711963 $J.Kg^{-1}.\deg{K}^{-1}$, 1255 according to \citet{ioc.iapso_bk10}. 1225 1256 Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$. 1226 In particular, the initial state de ined by the user have to be given as \textit{Conservative} Temperature and1227 \textit{ Absolute} Salinity.1228 In addition, setting \np{ln\_useCT} to \forcode{.true.} convert the Conservative SSTto potential SST prior to1257 In particular, the initial state defined by the user have to be given as 1258 \textit{Conservative} Temperature and \textit{Absolute} Salinity. 1259 In addition, when using TEOS10, the Conservative SST is converted to potential SST prior to 1229 1260 either computing the air-sea and ice-sea fluxes (forced mode) or 1230 1261 sending the SST field to the atmosphere (coupled mode). 1231 \item[\np{nn\_eos}~\forcode{= 0}] 1232 the polyEOS80-bsq equation of seawater is used. 1233 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to 1234 accurately fit EOS80 (Roquet, personal comm.). 1262 \item [{\np[=.true.]{ln_eos80}{ln\_eos80}}] the polyEOS80-bsq equation of seawater is used. 1263 It takes the same polynomial form as the polyTEOS10, 1264 but the coefficients have been optimized to accurately fit EOS80 (Roquet, personal comm.). 1235 1265 The state variables used in both the EOS80 and the ocean model are: 1236 the Practical Salinity ( (unit: psu, notation: $S_p$)) and1237 Potential Temperature (unit: $ ^{\circ}C$, notation: $\theta$).1266 the Practical Salinity (unit: $psu$, notation: $S_p$) and 1267 Potential Temperature (unit: $\deg{C}$, notation: $\theta$). 1238 1268 The pressure in decibars is approximated by the depth in meters. 1239 With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and1240 pressure \citep{UNESCO1983}.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}. 1241 1271 Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which 1242 1272 is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 1243 \item [\np{nn\_eos}~\forcode{= 1}]1244 a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen,1245 the coefficients of which has been optimized to fit the behavior of TEOS10 1246 ( Roquet, personal comm.) (see also \citet{Roquet_JPO2015}).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}). 1247 1277 It provides a simplistic linear representation of both cabbeling and thermobaricity effects which 1248 is enough for a proper treatment of the EOS in theoretical studies \citep{ Roquet_JPO2015}.1249 With such an equation of state there is no longer a distinction between 1250 \textit{ conservative} and \textit{potential} temperature,1251 as well as between \textit{absolute} and\textit{practical} salinity.1278 is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}. 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. 1252 1282 S-EOS takes the following expression: 1253 1283 \begin{gather*} 1254 % \label{eq:tra_S-EOS} 1255 \begin{alignedat}{2} 1256 &d_a(T,S,z) = \frac{1}{\rho_o} \big[ &- a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * &T_a \big. \\ 1257 & &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a \\ 1258 & \big. &- \nu \; T_a &S_a \big] \\ 1259 \end{alignedat} 1260 \\ 1284 % \label{eq:TRA_S-EOS} 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] \\ 1261 1288 \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3 1262 1289 \end{gather*} 1263 where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}. 1290 where the computer name of the coefficients as well as their standard value are given in 1291 \autoref{tab:TRA_SEOS}. 1264 1292 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by 1265 1293 changing the associated coefficients. 1266 Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS. 1267 setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from 1268 S-EOS. 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. 1269 1298 Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 1270 1299 \end{description} 1271 1300 1272 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1273 \begin{table}[!tb] 1274 \begin{center} 1275 \begin{tabular}{|l|l|l|l|} 1276 \hline 1277 coeff. & computer name & S-EOS & description \\ 1278 \hline 1279 $a_0$ & \np{rn\_a0} & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\ 1280 \hline 1281 $b_0$ & \np{rn\_b0} & $7.6554~10^{-1}$ & linear haline expansion coeff. \\ 1282 \hline 1283 $\lambda_1$ & \np{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$ \\ 1284 \hline 1285 $\lambda_2$ & \np{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$ \\ 1286 \hline 1287 $\nu$ & \np{rn\_nu} & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$ \\ 1288 \hline 1289 $\mu_1$ & \np{rn\_mu1} & $1.4970~10^{-4}$ & thermobaric coeff. in T \\ 1290 \hline 1291 $\mu_2$ & \np{rn\_mu2} & $1.1090~10^{-5}$ & thermobaric coeff. in S \\ 1292 \hline 1293 \end{tabular} 1294 \caption{ 1295 \protect\label{tab:SEOS} 1296 Standard value of S-EOS coefficients. 1297 } 1298 \end{center} 1301 \begin{table} 1302 \centering 1303 \begin{tabular}{|l|l|l|l|} 1304 \hline 1305 coeff. & computer name & S-EOS & description \\ 1306 \hline 1307 $a_0 $ & \np{rn_a0}{rn\_a0} & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\ 1308 \hline 1309 $b_0 $ & \np{rn_b0}{rn\_b0} & $7.6554~10^{-1}$ & linear haline expansion coeff. \\ 1310 \hline 1311 $\lambda_1$ & \np{rn_lambda1}{rn\_lambda1} & $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$ \\ 1312 \hline 1313 $\lambda_2$ & \np{rn_lambda2}{rn\_lambda2} & $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$ \\ 1314 \hline 1315 $\nu $ & \np{rn_nu}{rn\_nu} & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$ \\ 1316 \hline 1317 $\mu_1 $ & \np{rn_mu1}{rn\_mu1} & $1.4970~10^{-4}$ & thermobaric coeff. in T \\ 1318 \hline 1319 $\mu_2 $ & \np{rn_mu2}{rn\_mu2} & $1.1090~10^{-5}$ & thermobaric coeff. in S \\ 1320 \hline 1321 \end{tabular} 1322 \caption{Standard value of S-EOS coefficients} 1323 \label{tab:TRA_SEOS} 1299 1324 \end{table} 1300 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1301 1302 % ------------------------------------------------------------------------------------------------------------- 1303 % Brunt-V\"{a}is\"{a}l\"{a} Frequency 1304 % ------------------------------------------------------------------------------------------------------------- 1305 \subsection{Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos}~\forcode{= 0..2})} 1325 1326 %% ================================================================================================= 1327 \subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency]{Brunt-V\"{a}is\"{a}l\"{a} frequency} 1306 1328 \label{subsec:TRA_bn2} 1307 1329 1308 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of1309 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 1310 1332 (namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion, 1311 1333 non-penetrative convection, tidal mixing parameterisation, iso-neutral diffusion). 1312 1334 In particular, $N^2$ has to be computed at the local pressure 1313 1335 (pressure in decibar being approximated by the depth in meters). 1314 The expression for $N^2$ is given by: 1336 The expression for $N^2$ is given by: 1315 1337 \[ 1316 % \label{eq: tra_bn2}1338 % \label{eq:TRA_bn2} 1317 1339 N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt) 1318 1340 \] 1319 1341 where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and, 1320 1342 $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 1321 The coefficients are a polynomial function of temperature, salinity and depth which expression depends on 1322 the chosen EOS. 1323 They are computed through \textit{eos\_rab}, a \fortran function that can be found in \mdl{eosbn2}. 1324 1325 % ------------------------------------------------------------------------------------------------------------- 1326 % Freezing Point of Seawater 1327 % ------------------------------------------------------------------------------------------------------------- 1343 The coefficients are a polynomial function of temperature, salinity and depth which 1344 expression depends on the chosen EOS. 1345 They are computed through \textit{eos\_rab}, a \fortran\ function that can be found in \mdl{eosbn2}. 1346 1347 %% ================================================================================================= 1328 1348 \subsection{Freezing point of seawater} 1329 1349 \label{subsec:TRA_fzp} 1330 1350 1331 The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: 1332 \begin{equation} 1333 \label{eq:tra_eos_fzp} 1334 \begin{split} 1335 &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 1336 &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\ 1337 &\text{and~} d = -7.53~10^{-3} 1338 \end{split} 1339 \end{equation} 1340 1341 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water 1342 (\ie referenced to the surface $p = 0$), 1343 thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped. 1351 The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}: 1352 \begin{equation} 1353 \label{eq:TRA_eos_fzp} 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} 1358 \end{equation} 1359 1360 \autoref{eq:TRA_eos_fzp} is only used to compute the potential freezing point of sea water 1361 (\ie\ referenced to the surface $p = 0$), 1362 thus the pressure dependent terms in \autoref{eq:TRA_eos_fzp} (last term) have been dropped. 1344 1363 The freezing point is computed through \textit{eos\_fzp}, 1345 a \fortran function that can be found in \mdl{eosbn2}. 1346 1347 % ------------------------------------------------------------------------------------------------------------- 1348 % Potential Energy 1349 % ------------------------------------------------------------------------------------------------------------- 1364 a \fortran\ function that can be found in \mdl{eosbn2}. 1365 1366 %% ================================================================================================= 1350 1367 %\subsection{Potential Energy anomalies} 1351 1368 %\label{subsec:TRA_bn2} 1352 1369 1353 1370 % =====>>>>> TO BE written 1354 % 1355 1356 % ================================================================ 1357 % Horizontal Derivative in zps-coordinate 1358 % ================================================================ 1359 \section{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})} 1371 1372 %% ================================================================================================= 1373 \section[Horizontal derivative in \textit{zps}-coordinate (\textit{zpshde.F90})]{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})} 1360 1374 \label{sec:TRA_zpshde} 1361 1375 1362 \ gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators,1376 \cmtgm{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, 1363 1377 I've changed "derivative" to "difference" and "mean" to "average"} 1364 1378 1365 With partial cells (\np{ln\_zps}~\forcode{= .true.}) at bottom and top (\np{ln\_isfcav}~\forcode{= .true.}), 1379 With partial cells (\np[=.true.]{ln_zps}{ln\_zps}) at bottom and top 1380 (\np[=.true.]{ln_isfcav}{ln\_isfcav}), 1366 1381 in general, tracers in horizontally adjacent cells live at different depths. 1367 Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and1368 the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).1369 The partial cell properties at the top (\np {ln\_isfcav}~\forcode{= .true.}) are computed in the same way as1370 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. 1371 1386 So, only the bottom interpolation is explained below. 1372 1387 1373 1388 Before taking horizontal gradients between the tracers next to the bottom, 1374 1389 a linear interpolation in the vertical is used to approximate the deeper tracer as if 1375 it actually lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}). 1376 For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is: 1377 1378 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1379 \begin{figure}[!p] 1380 \begin{center} 1381 \includegraphics[]{Fig_partial_step_scheme} 1382 \caption{ 1383 \protect\label{fig:Partial_step_scheme} 1384 Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate 1385 (\protect\np{ln\_zps}~\forcode{= .true.}) in the case $(e3w_k^{i + 1} - e3w_k^i) > 0$. 1386 A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$, 1387 the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. 1388 The horizontal difference is then given by: $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and 1389 the average by: $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$. 1390 } 1391 \end{center} 1390 it actually lived at the depth of the shallower tracer point (\autoref{fig:TRA_Partial_step_scheme}). 1391 For example, for temperature in the $i$-direction the needed interpolated temperature, 1392 $\widetilde T$, is: 1393 1394 \begin{figure} 1395 \centering 1396 \includegraphics[width=0.33\textwidth]{TRA_partial_step_scheme} 1397 \caption[Discretisation of the horizontal difference and average of tracers in 1398 the $z$-partial step coordinate]{ 1399 Discretisation of the horizontal difference and average of tracers in 1400 the $z$-partial step coordinate (\protect\np[=.true.]{ln_zps}{ln\_zps}) in 1401 the case $(e3w_k^{i + 1} - e3w_k^i) > 0$. 1402 A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$, 1403 the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. 1404 The horizontal difference is then given by: 1405 $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and the average by: 1406 $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$.} 1407 \label{fig:TRA_Partial_step_scheme} 1392 1408 \end{figure} 1393 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1409 1394 1410 \[ 1395 1411 \widetilde T = \lt\{ 1396 1412 \begin{alignedat}{2} 1397 1413 &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1} 1398 & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\1414 & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ 1399 1415 &T^{\, i} &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i } \; \delta_k T^{i + 1} 1400 1416 & \quad \text{if $e_{3w}^{i + 1} < e_{3w}^i$} … … 1402 1418 \rt. 1403 1419 \] 1404 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 1405 \begin{equation} 1406 \label{eq:zps_hde} 1420 and the resulting forms for the horizontal difference and the horizontal average value of 1421 $T$ at a $U$-point are: 1422 \begin{equation} 1423 \label{eq:TRA_zps_hde} 1407 1424 \begin{split} 1408 1425 \delta_{i + 1/2} T &= 1409 1426 \begin{cases} 1410 \widetilde T - T^i & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1411 \\ 1412 T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 1413 \end{cases} 1414 \\ 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} \\ 1415 1430 \overline T^{\, i + 1/2} &= 1416 1431 \begin{cases} 1417 (\widetilde T - T^{\, i} ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1418 \\ 1419 (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 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 1420 1434 \end{cases} 1421 1435 \end{split} … … 1424 1438 The computation of horizontal derivative of tracers as well as of density is performed once for all at 1425 1439 each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed. 1426 It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$, 1427 is not the same as that used for $T$ and $S$. 1428 Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of 1429 $T$ and $S$, and the pressure at a $u$-point 1430 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 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 1445 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 1431 1446 \[ 1432 % \label{eq: zps_hde_rho}1447 % \label{eq:TRA_zps_hde_rho} 1433 1448 \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt) 1434 1449 \] 1435 1450 1436 1451 This is a much better approximation as the variation of $\rho$ with depth (and thus pressure) 1437 is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation. 1438 This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and 1439 the slopes of neutral surfaces (\autoref{sec:LDF_slp}). 1440 1441 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, 1442 1458 both averaging and differencing operators appear. 1443 Yet \autoref{eq: zps_hde} has not been used in these schemes:1459 Yet \autoref{eq:TRA_zps_hde} has not been used in these schemes: 1444 1460 in contrast to diffusion and pressure gradient computations, 1445 1461 no correction for partial steps is applied for advection. 1446 1462 The main motivation is to preserve the domain averaged mean variance of the advected field when 1447 1463 using the $2^{nd}$ order centred scheme. 1448 Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of 1449 partial cells should be further investigated in the near future. 1450 %%% 1451 \gmcomment{gm : this last remark has to be done} 1452 %%% 1453 1454 \biblio 1455 1456 \pindex 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. 1466 \cmtgm{gm : this last remark has to be done} 1467 1468 \subinc{\input{../../global/epilogue}} 1457 1469 1458 1470 \end{document}
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