- Timestamp:
- 2019-10-25T16:27:34+02:00 (5 years ago)
- Location:
- NEMO/branches/2019/dev_r11470_HPC_12_mpi3/doc
- Files:
-
- 5 edited
Legend:
- Unmodified
- Added
- Removed
-
NEMO/branches/2019/dev_r11470_HPC_12_mpi3/doc
-
Property
svn:externals
set to
^/utils/badges badges
^/utils/logos logos
-
Property
svn:externals
set to
-
NEMO/branches/2019/dev_r11470_HPC_12_mpi3/doc/latex
- Property svn:ignore deleted
-
NEMO/branches/2019/dev_r11470_HPC_12_mpi3/doc/latex/NEMO
-
Property
svn:externals
set to
^/utils/figures/NEMO figures
-
Property
svn:externals
set to
-
NEMO/branches/2019/dev_r11470_HPC_12_mpi3/doc/latex/NEMO/subfiles
- Property svn:ignore
-
old new 1 *.aux 2 *.bbl 3 *.blg 4 *.dvi 5 *.fdb* 6 *.fls 7 *.idx 1 *.ind 8 2 *.ilg 9 *.ind10 *.log11 *.maf12 *.mtc*13 *.out14 *.pdf15 *.toc16 _minted-*
-
- Property svn:ignore
-
NEMO/branches/2019/dev_r11470_HPC_12_mpi3/doc/latex/NEMO/subfiles/chap_TRA.tex
r11459 r11799 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 8 \thispagestyle{plain} 9 10 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 11 25 12 26 % missing/update … … 19 33 the tracer equations are available depending on the vertical coordinate used and on the physics used. 20 34 In all the equations presented here, the masking has been omitted for simplicity. 21 One must be aware that all the quantities are masked fields and that each time a mean or22 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. 23 37 24 38 The two active tracers are potential temperature and salinity. … … 31 45 NXT stands for next, referring to the time-stepping. 32 46 From left to right, the terms on the rhs of the tracer equations are the advection (ADV), 33 the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings 34 (SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition), 35 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. 36 52 The terms QSR, BBC, BBL and DMP are optional. 37 53 The external forcings and parameterisations require complex inputs and complex calculations … … 39 55 LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and 40 56 \autoref{chap:ZDF}, respectively. 41 Note that \mdl{tranpc}, the non-penetrative convection module, although located in42 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, 43 59 is described with the model vertical physics (ZDF) together with 44 60 other available parameterization of convection. 45 61 46 In the present chapter we also describe the diagnostic equations used to compute the sea-water properties47 (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and freezing point with 48 associated modules \mdl{eosbn2} and \mdl{phycst}).62 In the present chapter we also describe the diagnostic equations used to 63 compute the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and 64 freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). 49 65 50 66 The different options available to the user are managed by namelist logicals. … … 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 (\textit{traadv.F90})] 63 {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})} 64 78 \label{sec:TRA_adv} 65 %------------------------------------------namtra_adv----------------------------------------------------- 66 67 \nlst{namtra_adv} 68 %------------------------------------------------------------------------------------------------------------- 69 70 When considered (\ie\ when \np{ln\_traadv\_OFF} 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.}), 71 87 the advection tendency of a tracer is expressed in flux form, 72 88 \ie\ as the divergence of the advective fluxes. 73 Its discrete expression is given by 74 \begin{equation} 75 \label{eq: tra_adv}89 Its discrete expression is given by: 90 \begin{equation} 91 \label{eq:TRA_adv} 76 92 ADV_\tau = - \frac{1}{b_t} \Big( \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u] 77 93 + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big) … … 79 95 \end{equation} 80 96 where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells. 81 The flux form in \autoref{eq:tra_adv} implicitly requires the use of the continuity equation. 82 Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 83 results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ 84 (which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie\ \np{ln\_linssh}\forcode{ = .true.}). 85 Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that 86 it is consistent with the continuity equation in order to enforce the conservation properties of 87 the continuous equations. 88 In other words, by setting $\tau = 1$ in (\autoref{eq:tra_adv}) we recover the discrete form of 89 the continuity equation which is used to calculate the vertical velocity. 90 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 91 \begin{figure}[!t] 92 \begin{center} 93 \includegraphics[width=\textwidth]{Fig_adv_scheme} 94 \caption{ 95 \protect\label{fig:adv_scheme} 96 Schematic representation of some ways used to evaluate the tracer value at $u$-point and 97 the amount of tracer exchanged between two neighbouring grid points. 98 Upsteam biased scheme (ups): 99 the upstream value is used and the black area is exchanged. 100 Piecewise parabolic method (ppm): 101 a parabolic interpolation is used and the black and dark grey areas are exchanged. 102 Monotonic upstream scheme for conservative laws (muscl): 103 a parabolic interpolation is used and black, dark grey and grey areas are exchanged. 104 Second order scheme (cen2): 105 the mean value is used and black, dark grey, grey and light grey areas are exchanged. 106 Note that this illustration does not include the flux limiter used in ppm and muscl schemes. 107 } 108 \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]{Fig_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} 109 126 \end{figure} 110 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 111 112 The key difference between the advection schemes available in \NEMO\ is the choice made in space and 113 time interpolation to define the value of the tracer at the velocity points 114 (\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}). 115 131 116 132 Along solid lateral and bottom boundaries a zero tracer flux is automatically specified, … … 119 135 120 136 \begin{description} 121 \item[linear free surface:] 122 (\np{ln\_linssh}\forcode{ = .true.}) 137 \item [linear free surface] (\np[=.true.]{ln_linssh}{ln\_linssh}) 123 138 the first level thickness is constant in time: 124 the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on 125 the moving surface $z = \eta$. 126 There is a non-zero advective flux which is set for all advection schemes as 127 $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie\ the product of surface velocity (at $z = 0$) by 128 the first level tracer value. 129 \item[non-linear free surface:] 130 (\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}) 131 145 convergence/divergence in the first ocean level moves the free surface up/down. 132 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. 133 148 \end{description} 134 149 135 150 In all cases, this boundary condition retains local conservation of tracer. 136 Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case. 137 Nevertheless, in the latter case, it is achieved to a good approximation since 138 the non-conservative term is the product of the time derivative of the tracer and the free surface height, 139 two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 140 141 The velocity field that appears in (\autoref{eq:tra_adv} is 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 142 160 the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity 143 161 (see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or … … 145 163 (see \autoref{chap:LDF}). 146 164 147 Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), 148 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for 149 Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), 150 and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST). 151 The choice is made in the \nam{tra\_adv} namelist, by setting to \forcode{.true.} one of 152 the logicals \textit{ln\_traadv\_xxx}. 153 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 154 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 155 By default (\ie\ in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}. 156 If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}), 157 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! 158 182 159 183 Details of the advection schemes are given below. 160 The choosing an advection scheme is a complex matter which depends on the model physics, model resolution, 161 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 162 187 163 188 \begin{enumerate} 164 \item 165 CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that 166 they do not necessarily need additional diffusion; 167 \item 168 CEN and UBS are not \textit{positive} schemes 169 \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}, 170 193 implying that false extrema are permitted. 171 194 Their use is not recommended on passive tracers; 172 \item 173 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. 174 197 \end{enumerate} 175 198 176 Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and 177 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. 178 202 Nevertheless, most of our users set a different treatment on passive and active tracers, 179 203 that's the reason why this possibility is offered. 180 We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of 181 their results. 182 183 % ------------------------------------------------------------------------------------------------------------- 184 % 2nd and 4th order centred schemes 185 % ------------------------------------------------------------------------------------------------------------- 186 \subsection[CEN: Centred scheme (\forcode{ln_traadv_cen = .true.})] 187 {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})} 188 209 \label{subsec:TRA_adv_cen} 189 210 190 211 % 2nd order centred scheme 191 212 192 The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}\forcode{ = .true.}. 193 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 194 setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$. 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 $4$. 195 217 CEN implementation can be found in the \mdl{traadv\_cen} module. 196 218 197 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of198 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. 199 221 For example, in the $i$-direction : 200 222 \begin{equation} 201 \label{eq: tra_adv_cen2}223 \label{eq:TRA_adv_cen2} 202 224 \tau_u^{cen2} = \overline T ^{i + 1/2} 203 225 \end{equation} 204 226 205 CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but dispersive 206 (\ie\ it may create false extrema). 207 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 208 produce a sensible solution. 209 The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 210 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. 211 234 212 235 Note that using the CEN2, the overall tracer advection is of second order accuracy since 213 both (\autoref{eq: tra_adv}) and (\autoref{eq:tra_adv_cen2}) have this order of accuracy.236 both (\autoref{eq:TRA_adv}) and (\autoref{eq:TRA_adv_cen2}) have this order of accuracy. 214 237 215 238 % 4nd order centred scheme 216 239 217 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as 218 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 240 In the $4^{th}$ order formulation (CEN4), 241 tracer values are evaluated at u- and v-points as a $4^{th}$ order interpolation, 242 and thus depend on the four neighbouring $T$-points. 219 243 For example, in the $i$-direction: 220 244 \begin{equation} 221 \label{eq: tra_adv_cen4}245 \label{eq:TRA_adv_cen4} 222 246 \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2} 223 247 \end{equation} 224 In the vertical direction (\np {nn\_cen\_v}\forcode{ = 4}),248 In the vertical direction (\np[=4]{nn_cen_v}{nn\_cen\_v}), 225 249 a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}. 226 In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion, 227 spectral characteristics similar to schemes of higher order \citep{lele_JCP92}. 250 In the COMPACT scheme, both the field and its derivative are interpolated, 251 which leads, after a matrix inversion, spectral characteristics similar to schemes of higher order 252 \citep{lele_JCP92}. 228 253 229 254 Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but 230 255 a $4^{th}$ order evaluation of advective fluxes, 231 since the divergence of advective fluxes \autoref{eq: tra_adv} is kept at $2^{nd}$ order.232 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with233 the scheme presented here.234 Introducing a \forcode{.true.}$4^{th}$ order advection scheme is feasible but, for consistency reasons,235 it requires changes in the discretisation of the tracer advection together with changes in the continuity equation,236 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. 237 262 238 263 A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive, 239 264 \ie\ the global variance of a tracer is not preserved using CEN4. 240 Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. 241 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 242 so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer. 265 Furthermore, it must be used in conjunction with an explicit diffusion operator to 266 produce a sensible solution. 267 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with 268 an Asselin time-filter, so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer. 243 269 244 270 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), … … 246 272 This hypothesis usually reduces the order of the scheme. 247 273 Here we choose to set the gradient of $T$ across the boundary to zero. 248 Alternative conditions can be specified, such as a reduction to a second order scheme for 249 these near boundary grid points. 250 251 % ------------------------------------------------------------------------------------------------------------- 252 % FCT scheme 253 % ------------------------------------------------------------------------------------------------------------- 254 \subsection[FCT: Flux Corrected Transport scheme (\forcode{ln_traadv_fct = .true.})] 255 {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})} 256 279 \label{subsec:TRA_adv_tvd} 257 280 258 The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}\forcode{ = .true.}. 259 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 260 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 $4$. 261 286 FCT implementation can be found in the \mdl{traadv\_fct} module. 262 287 263 In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and264 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. 265 290 For example, in the $i$-direction : 266 291 \begin{equation} 267 \label{eq: tra_adv_fct}292 \label{eq:TRA_adv_fct} 268 293 \begin{split} 269 294 \tau_u^{ups} &= … … 271 296 T_{i + 1} & \text{if~} u_{i + 1/2} < 0 \\ 272 297 T_i & \text{if~} u_{i + 1/2} \geq 0 \\ 273 \end{cases} 274 \\ 298 \end{cases} \\ 275 299 \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big) 276 300 \end{split} … … 278 302 where $c_u$ is a flux limiter function taking values between 0 and 1. 279 303 The FCT order is the one of the centred scheme used 280 (\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}). 281 305 There exist many ways to define $c_u$, each corresponding to a different FCT scheme. 282 306 The one chosen in \NEMO\ is described in \citet{zalesak_JCP79}. … … 286 310 A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{levy.estublier.ea_GRL01}. 287 311 288 289 For stability reasons (see \autoref{chap:STP}), 290 $\tau_u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while 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 291 314 $\tau_u^{ups}$ is evaluated using the \textit{before} tracer. 292 In other words, the advective part of the scheme is time stepped with a leap-frog scheme 293 while a forward scheme is used for the diffusive part. 294 295 % ------------------------------------------------------------------------------------------------------------- 296 % MUSCL scheme 297 % ------------------------------------------------------------------------------------------------------------- 298 \subsection[MUSCL: Monotone Upstream Scheme for Conservative Laws (\forcode{ln_traadv_mus = .true.})] 299 {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})} 300 320 \label{subsec:TRA_adv_mus} 301 321 302 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}. 303 324 MUSCL implementation can be found in the \mdl{traadv\_mus} module. 304 325 305 326 MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}. 306 In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between307 two $T$-points (\autoref{fig:adv_scheme}).327 In its formulation, the tracer at velocity points is evaluated assuming 328 a linear tracer variation between two $T$-points (\autoref{fig:TRA_adv_scheme}). 308 329 For example, in the $i$-direction : 309 \ begin{equation}310 % \label{eq: tra_adv_mus}330 \[ 331 % \label{eq:TRA_adv_mus} 311 332 \tau_u^{mus} = \lt\{ 312 333 \begin{split} 313 \tau_i&+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)314 \widetilde{\partial_i\tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\315 316 334 \tau_i &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 335 \widetilde{\partial_i \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\ 336 \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 337 \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} < 0 317 338 \end{split} 318 339 \rt. 319 \ end{equation}320 where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to321 ensure the \textit{positive} character of the scheme.322 323 The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to324 evaluate $\tau_u^{mus}$.340 \] 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 $\tau_u^{mus}$. 325 346 326 347 For an ocean grid point adjacent to land and where the ocean velocity is directed toward land, 327 348 an upstream flux is used. 328 349 This choice ensure the \textit{positive} character of the scheme. 329 In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes 330 (\np{ln\_mus\_ups}\forcode{ = .true.}). 331 332 % ------------------------------------------------------------------------------------------------------------- 333 % UBS scheme 334 % ------------------------------------------------------------------------------------------------------------- 335 \subsection[UBS a.k.a. UP3: Upstream-Biased Scheme (\forcode{ln_traadv_ubs = .true.})] 336 {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})} 337 355 \label{subsec:TRA_adv_ubs} 338 356 339 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}. 340 359 UBS implementation can be found in the \mdl{traadv\_mus} module. 341 360 342 361 The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme 343 (Quadratic Upstream Interpolation for Convective Kinematics). 362 (\textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for 363 \textbf{C}onvective \textbf{K}inematics). 344 364 It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation. 345 365 For example, in the $i$-direction: 346 366 \begin{equation} 347 \label{eq: tra_adv_ubs}367 \label{eq:TRA_adv_ubs} 348 368 \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6} 349 369 \begin{cases} 350 351 370 \tau"_i & \text{if~} u_{i + 1/2} \geqslant 0 \\ 371 \tau"_{i + 1} & \text{if~} u_{i + 1/2} < 0 352 372 \end{cases} 353 \quad 354 \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] 355 374 \end{equation} 356 375 357 376 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error 358 377 \citep{shchepetkin.mcwilliams_OM05}. 359 The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}. 378 The overall performance of the advection scheme is similar to that reported in 379 \cite{farrow.stevens_JPO95}. 360 380 It is a relatively good compromise between accuracy and smoothness. 361 381 Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted, 362 382 but the amplitude of such are significantly reduced over the centred second or fourth order method. 363 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. 364 385 365 386 The intrinsic diffusion of UBS makes its use risky in the vertical direction where 366 387 the control of artificial diapycnal fluxes is of paramount importance 367 388 \citep{shchepetkin.mcwilliams_OM05, demange_phd14}. 368 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme369 (\np{nn\_ubs\_v}\forcode{ = 2 or 4}).370 371 For stability reasons (see \autoref{chap: STP}), the first term in \autoref{eq:tra_adv_ubs}372 (which corresponds to a second order centred scheme)373 is evaluated using the \textit{now} tracer (centred in time) while the second term374 (which is the diffusive part of the scheme),389 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or 390 a $4^th$ order COMPACT scheme (\np[=2 or 4]{nn_ubs_v}{nn\_ubs\_v}). 391 392 For stability reasons (see \autoref{chap:TD}), 393 the first term in \autoref{eq:TRA_adv_ubs} (which corresponds to a second order centred scheme) 394 is evaluated using the \textit{now} tracer (centred in time) while 395 the second term (which is the diffusive part of the scheme), 375 396 is evaluated using the \textit{before} tracer (forward in time). 376 This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the QUICK advection scheme. 397 This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in 398 the context of the QUICK advection scheme. 377 399 UBS and QUICK schemes only differ by one coefficient. 378 Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 400 Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme 401 \citep{webb.de-cuevas.ea_JAOT98}. 379 402 This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 380 Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 381 382 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: 383 407 \begin{gather} 384 \label{eq: traadv_ubs2}408 \label{eq:TRA_adv_ubs2} 385 409 \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} 386 410 \begin{cases} … … 389 413 \end{cases} 390 414 \intertext{or equivalently} 391 % \label{eq: traadv_ubs2b}415 % \label{eq:TRA_adv_ubs2b} 392 416 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} 393 417 - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber 394 418 \end{gather} 395 419 396 \autoref{eq: traadv_ubs2} has several advantages.420 \autoref{eq:TRA_adv_ubs2} has several advantages. 397 421 Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which 398 422 an upstream-biased diffusion term is added. 399 Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to 400 be evaluated at the \textit{now} time step using \autoref{eq:tra_adv_ubs}. 401 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which 402 is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 403 Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}. 404 405 % ------------------------------------------------------------------------------------------------------------- 406 % QCK scheme 407 % ------------------------------------------------------------------------------------------------------------- 408 \subsection[QCK: QuiCKest scheme (\forcode{ln_traadv_qck = .true.})] 409 {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})} 410 433 \label{subsec:TRA_adv_qck} 411 434 412 The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme 413 proposed by \citet{leonard_CMAME79} 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}. 414 439 QUICKEST implementation can be found in the \mdl{traadv\_qck} module. 415 440 416 441 QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter 417 442 \citep{leonard_CMAME91}. 418 It has been implemented in \NEMO\ by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. 443 It has been implemented in \NEMO\ by G. Reffray (Mercator Ocean) and 444 can be found in the \mdl{traadv\_qck} module. 419 445 The resulting scheme is quite expensive but \textit{positive}. 420 446 It can be used on both active and passive tracers. … … 423 449 Therefore the vertical flux is evaluated using the CEN2 scheme. 424 450 This no longer guarantees the positivity of the scheme. 425 The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property. 451 The use of FCT in the vertical direction (as for the UBS case) should be implemented to 452 restore this property. 426 453 427 454 %%%gmcomment : Cross term are missing in the current implementation.... 428 455 429 % ================================================================ 430 % Tracer Lateral Diffusion 431 % ================================================================ 432 \section[Tracer lateral diffusion (\textit{traldf.F90})] 433 {Tracer lateral diffusion (\protect\mdl{traldf})} 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 \nam{tra\_ldf} namelist variables. 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. 441 467 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}. 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. … … 452 481 \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.debreu.ea_OM12}. 457 458 % ------------------------------------------------------------------------------------------------------------- 459 % Type of operator 460 % ------------------------------------------------------------------------------------------------------------- 461 \subsection[Type of operator (\texttt{ln\_traldf}\{\texttt{\_OFF,\_lap,\_blp}\})] 462 {Type of operator (\protect\np{ln\_traldf\_OFF}, \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})} 463 491 \label{subsec:TRA_ldf_op} 464 492 … … 466 494 467 495 \begin{description} 468 \item[\np{ln\_traldf\_OFF}\forcode{ = .true.}:] 469 no operator selected, the lateral diffusive tendency will not be applied to the tracer equation. 470 This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example). 471 \item[\np{ln\_traldf\_lap}\forcode{ = .true.}:] 472 a laplacian operator is selected. 473 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 $, 474 503 where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}), 475 504 and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}). 476 \item[\np{ln\_traldf\_blp}\forcode{ = .true.}]: 477 a bilaplacian operator is selected. 505 \item [{\np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}}] a bilaplacian operator is selected. 478 506 This biharmonic operator takes the following expression: 479 $\math pzc{B} = - \mathpzc{L}(\mathpzc{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$507 $\mathcal{B} = - \mathcal{L}(\mathcal{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$ 480 508 where the gradient operats along the selected direction, 481 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}). 482 511 In the code, the bilaplacian operator is obtained by calling the laplacian twice. 483 512 \end{description} … … 487 516 minimizing the impact on the larger scale features. 488 517 The main difference between the two operators is the scale selectiveness. 489 The bilaplacian damping time (\ie\ its spin down time) scales like $\lambda^{-4}$ for 490 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), 491 521 whereas the laplacian damping time scales only like $\lambda^{-2}$. 492 522 493 % ------------------------------------------------------------------------------------------------------------- 494 % Direction of action 495 % ------------------------------------------------------------------------------------------------------------- 496 \subsection[Action direction (\texttt{ln\_traldf}\{\texttt{\_lev,\_hor,\_iso,\_triad}\})] 497 {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})} 498 525 \label{subsec:TRA_ldf_dir} 499 526 500 527 The choice of a direction of action determines the form of operator used. 501 528 The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when 502 iso-level option is used (\np {ln\_traldf\_lev}\forcode{ = .true.}) or503 whena horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate504 (\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}). 505 532 The associated code can be found in the \mdl{traldf\_lap\_blp} module. 506 533 The operator is a rotated (re-entrant) laplacian when 507 534 the direction along which it acts does not coincide with the iso-level surfaces, 508 535 that is when standard or triad iso-neutral option is used 509 (\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.}, 510 537 see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or 511 538 when a horizontal (\ie\ geopotential) operator is demanded in \textit{s}-coordinate 512 (\np{ln \_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.})513 \footnote{In this case, the standard iso-neutral operator will be automatically selected}.539 (\np{ln_traldf_hor}{ln\_traldf\_hor} and \np{ln_sco}{ln\_sco} = \forcode{.true.}) \footnote{ 540 In this case, the standard iso-neutral operator will be automatically selected}. 514 541 In that case, a rotation is applied to the gradient(s) that appears in the operator so that 515 542 diffusive fluxes acts on the three spatial direction. … … 518 545 the next two sub-sections. 519 546 520 % ------------------------------------------------------------------------------------------------------------- 521 % iso-level operator 522 % ------------------------------------------------------------------------------------------------------------- 523 \subsection[Iso-level (bi-)laplacian operator (\texttt{ln\_traldf\_iso})] 524 {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})} 525 549 \label{subsec:TRA_ldf_lev} 526 550 527 551 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 528 552 \begin{equation} 529 \label{eq: tra_ldf_lap}553 \label{eq:TRA_ldf_lap} 530 554 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] 531 555 + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg) … … 534 558 where zero diffusive fluxes is assumed across solid boundaries, 535 559 first (and third in bilaplacian case) horizontal tracer derivative are masked. 536 It is implemented in the \rou{tra\_ldf\_lap} subroutine found in the \mdl{traldf\_lap\_blp}} module. 537 The module also contains \rou{tra\_ldf\_blp}, the subroutine calling twice \rou{tra\_ldf\_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 538 563 compute the iso-level bilaplacian operator. 539 564 540 565 It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in 541 the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 542 It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}\forcode{ = .true.}, 543 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}. 544 572 In both cases, it significantly contributes to diapycnal mixing. 545 573 It is therefore never recommended, even when using it in the bilaplacian case. 546 574 547 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}), 548 576 tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom. 549 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. 550 579 They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}. 551 580 552 % ------------------------------------------------------------------------------------------------------------- 553 % Rotated laplacian operator 554 % ------------------------------------------------------------------------------------------------------------- 581 %% ================================================================================================= 555 582 \subsection{Standard and triad (bi-)laplacian operator} 556 583 \label{subsec:TRA_ldf_iso_triad} 557 584 558 %&& Standard rotated (bi-)laplacian operator 559 %&& ---------------------------------------------- 560 \subsubsection[Standard rotated (bi-)laplacian operator (\textit{traldf\_iso.F90})] 561 {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})} 562 587 \label{subsec:TRA_ldf_iso} 563 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf}) 564 takes the following semi -discrete space form in $z$- and $s$-coordinates: 565 \begin{equation} 566 \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} 567 593 \begin{split} 568 594 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] … … 578 604 $r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and 579 605 the surface along which the diffusion operator acts (\ie\ horizontal or iso-neutral surfaces). 580 It is thus used when, in addition to \np {ln\_traldf\_lap}\forcode{ = .true.},581 we have \np {ln\_traldf\_iso}\forcode{ = .true.},582 or both \np {ln\_traldf\_hor}\forcode{ = .true.} and \np{ln\_zco}\forcode{ = .true.}.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}. 583 609 The way these slopes are evaluated is given in \autoref{sec:LDF_slp}. 584 At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using 585 the mask technique (see \autoref{sec:LBC_coast}). 586 587 The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical derivatives. 588 For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that 589 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}). 590 617 For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module, 591 618 but in the \mdl{trazdf} module where, if iso-neutral mixing is used, 592 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 $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$. 593 621 594 622 This formulation conserves the tracer but does not ensure the decrease of the tracer variance. 595 Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without 596 any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}. 597 598 Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}), 599 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. 600 629 They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 601 630 602 %&& Triad rotated (bi-)laplacian operator 603 %&& ------------------------------------------- 604 \subsubsection[Triad rotated (bi-)laplacian operator (\textit{ln\_traldf\_triad})] 605 {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})} 606 633 \label{subsec:TRA_ldf_triad} 607 634 608 An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases 609 is also available in \NEMO\ (\np{ln\_traldf\_triad}\forcode{ = .true.}). 610 A complete description of the algorithm is given in \autoref{apdx:triad}. 611 612 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. 613 642 The operator requires an additional assumption on boundary conditions: 614 643 both first and third derivative terms normal to the coast are set to zero. 615 644 616 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. 617 647 It requires an additional assumption on boundary conditions: 618 648 first and third derivative terms normal to the coast, 619 649 normal to the bottom and normal to the surface are set to zero. 620 650 621 %&& Option for the rotated operators 622 %&& ---------------------------------------------- 651 %% ================================================================================================= 623 652 \subsubsection{Option for the rotated operators} 624 653 \label{subsec:TRA_ldf_options} 625 654 626 \begin{itemize} 627 \item \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) 628 \item \np{rn\_slpmax} = slope limit (both operators) 629 \item \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 630 \item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only) 631 \item \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 632 \end{itemize} 633 634 % ================================================================ 635 % Tracer Vertical Diffusion 636 % ================================================================ 637 \section[Tracer vertical diffusion (\textit{trazdf.F90})] 638 {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})} 639 666 \label{sec:TRA_zdf} 640 %--------------------------------------------namzdf--------------------------------------------------------- 641 642 \nlst{namzdf} 643 %-------------------------------------------------------------------------------------------------------------- 644 645 Options are defined through the \nam{zdf} namelist variables. 646 The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, 647 and is based on a laplacian operator. 648 The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi -discrete space form: 649 \begin{gather*} 650 % \label{eq:tra_zdf} 651 D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\ 652 D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] 653 \end{gather*} 654 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity, 655 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. 656 680 Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised 657 (\ie\ \np {ln\_zdfddm} equals \forcode{.true.},).681 (\ie\ \np[=.true.]{ln_zdfddm}{ln\_zdfddm},). 658 682 The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF). 659 Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by660 $\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of 661 \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}. 662 686 663 687 At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified. … … 666 690 a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}). 667 691 668 The large eddy coefficient found in the mixed layer together with high vertical resolution implies that669 th ere would be too restrictive constraint on the time step if we use explicit time stepping.692 The large eddy coefficient found in the mixed layer together with high vertical resolution implies 693 that there would be too restrictive constraint on the time step if we use explicit time stepping. 670 694 Therefore an implicit time stepping is preferred for the vertical diffusion since 671 695 it overcomes the stability constraint. 672 696 673 % ================================================================ 674 % External Forcing 675 % ================================================================ 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 (\textit{trasbc.F90})] 683 {Surface boundary condition (\protect\mdl{trasbc})} 701 %% ================================================================================================= 702 \subsection[Surface boundary condition (\textit{trasbc.F90})]{Surface boundary condition (\protect\mdl{trasbc})} 684 703 \label{subsec:TRA_sbc} 685 704 … … 691 710 692 711 Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 693 (\ie\ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due 694 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 695 715 to the heat and salt content of the mass exchange. 696 716 They are both included directly in $Q_{ns}$, the surface heat flux, 697 717 and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details). 698 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 699 700 The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers): 701 702 \begin{itemize} 703 \item 704 $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 705 (\ie\ the difference between the total surface heat flux and the fraction of the short wave flux that 706 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}) 707 729 plus the heat content associated with of the mass exchange with the atmosphere and lands. 708 \item 709 $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 710 \item 711 \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 712 733 possibly with the sea-ice and ice-shelves. 713 \item 714 \textit{rnf}, the mass flux associated with runoff 734 \item [\textit{rnf}] The mass flux associated with runoff 715 735 (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 716 \item 717 \textit{fwfisf}, the mass flux associated with ice shelf melt, 736 \item [\textit{fwfisf}] The mass flux associated with ice shelf melt, 718 737 (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 719 \end{ itemize}738 \end{labeling} 720 739 721 740 The surface boundary condition on temperature and salinity is applied as follows: 722 741 \begin{equation} 723 \label{eq:tra_sbc} 724 \begin{alignedat}{2} 725 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns} }^t \\ 726 F^S &= &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t 727 \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 728 745 \end{equation} 729 746 where $\overline x^t$ means that $x$ is averaged over two consecutive time steps 730 747 ($t - \rdt / 2$ and $t + \rdt / 2$). 731 Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:STP}). 732 733 In the linear free surface case (\np{ln\_linssh}\forcode{ = .true.}), an additional term has to be added on 734 both temperature and salinity. 735 On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$. 736 On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in 737 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. 738 756 The resulting surface boundary condition is applied as follows: 739 757 \begin{equation} 740 \label{eq:tra_sbc_lin} 741 \begin{alignedat}{2} 742 F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 743 &\overline{(Q_{ns} - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\ 744 F^S &= &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 745 &\overline{(\textit{sfx} - \textit{emp} \lt. S \rt|_{k = 1})}^t 746 \end{alignedat} 747 \end{equation} 748 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. 749 766 In the linear free surface case, there is a small imbalance. 750 The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}. 751 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}). 752 753 % ------------------------------------------------------------------------------------------------------------- 754 % Solar Radiation Penetration 755 % ------------------------------------------------------------------------------------------------------------- 756 \subsection[Solar radiation penetration (\textit{traqsr.F90})] 757 {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})} 758 774 \label{subsec:TRA_qsr} 759 %--------------------------------------------namqsr-------------------------------------------------------- 760 761 \nlst{namtra_qsr} 762 %-------------------------------------------------------------------------------------------------------------- 763 764 Options are defined through the \nam{tra\_qsr} namelist variables. 765 When the penetrative solar radiation option is used (\np{ln\_traqsr}\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}), 766 784 the solar radiation penetrates the top few tens of meters of the ocean. 767 If it is not used (\np{ln\_traqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level. 768 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and 769 the surface boundary condition is modified to take into account only the non-penetrative part of the surface 770 heat flux: 771 \begin{equation} 772 \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} 773 792 \begin{gathered} 774 793 \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\ … … 778 797 where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie\ the shortwave radiation) and 779 798 $I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$). 780 The additional term in \autoref{eq: PE_qsr} is discretized as follows:781 \begin{equation} 782 \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} 783 802 \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w] 784 803 \end{equation} 785 804 786 805 The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. 787 The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to 788 heating the upper few tens of centimetres. 789 The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ 790 (specified through namelist parameter \np{rn\_abs}). 791 It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 792 of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \nam{tra\_qsr} namelist). 793 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to 794 larger depths where it contributes to local heating. 795 The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen. 796 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}) 797 818 a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 798 819 leading to the following expression \citep{paulson.simpson_JPO77}: 799 820 \[ 800 % \label{eq: traqsr_iradiance}821 % \label{eq:TRA_qsr_iradiance} 801 822 I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt] 802 823 \] 803 824 where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 804 It is usually chosen to be 23~m by setting the \np{rn \_si0} namelist parameter.805 The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification806 (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). 807 828 808 829 Such assumptions have been shown to provide a very crude and simplistic representation of 809 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}). 810 831 Light absorption in the ocean depends on particle concentration and is spectrally selective. 811 832 \cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by 812 833 a 61 waveband formulation. 813 834 Unfortunately, such a model is very computationally expensive. 814 Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of this formulation in which 815 visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm). 816 For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from 817 the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}), 818 assuming the same power-law relationship. 819 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), 820 843 reproduces quite closely the light penetration profiles predicted by the full spectal model, 821 844 but with much greater computational efficiency. 822 845 The 2-bands formulation does not reproduce the full model very well. 823 846 824 The RGB formulation is used when \np {ln\_qsr\_rgb}\forcode{ = .true.}.825 The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are tabulated over826 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 $g.Chl/L$ 827 850 (see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module). 828 851 Four types of chlorophyll can be chosen in the RGB formulation: 829 852 830 853 \begin{description} 831 \item[\np{nn\_chldta}\forcode{ = 0}] 832 a constant 0.05 g.Chl/L value everywhere ; 833 \item[\np{nn\_chldta}\forcode{ = 1}] 834 an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in 835 the vertical direction; 836 \item[\np{nn\_chldta}\forcode{ = 2}] 837 same as previous case except that a vertical profile of chlorophyl is used. 838 Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value; 839 \item[\np{ln\_qsr\_bio}\forcode{ = .true.}] 840 simulated time varying chlorophyll by TOP biogeochemical model. 841 In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in 842 PISCES and the oceanic heating rate. 854 \item [{\np[=0]{nn_chldta}{nn\_chldta}}] a constant 0.05 $g.Chl/L$ value everywhere; 855 \item [{\np[=1]{nn_chldta}{nn\_chldta}}] an observed time varying chlorophyll deduced from 856 satellite surface ocean color measurement spread uniformly in the vertical direction; 857 \item [{\np[=2]{nn_chldta}{nn\_chldta}}] same as previous case except that 858 a vertical profile of chlorophyl is used. 859 Following \cite{morel.berthon_LO89}, 860 the profile is computed from the local surface chlorophyll value; 861 \item [{\np[=.true.]{ln_qsr_bio}{ln\_qsr\_bio}}] simulated time varying chlorophyll by 862 \TOP\ biogeochemical model. 863 In this case, the RGB formulation is used to calculate both 864 the phytoplankton light limitation in \PISCES\ and the oceanic heating rate. 843 865 \end{description} 844 866 845 The trend in \autoref{eq: tra_qsr} associated with the penetration of the solar radiation is added to867 The trend in \autoref{eq:TRA_qsr} associated with the penetration of the solar radiation is added to 846 868 the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 847 869 … … 851 873 (\ie\ it is less than the computer precision) is computed once, 852 874 and the trend associated with the penetration of the solar radiation is only added down to that level. 853 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. 854 877 In this case, we have chosen that all remaining radiation is absorbed in the last ocean level 855 878 (\ie\ $I$ is masked). 856 879 857 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 858 \begin{figure}[!t] 859 \begin{center} 860 \includegraphics[width=\textwidth]{Fig_TRA_Irradiance} 861 \caption{ 862 \protect\label{fig:traqsr_irradiance} 863 Penetration profile of the downward solar irradiance calculated by four models. 864 Two waveband chlorophyll-independent formulation (blue), 865 a chlorophyll-dependent monochromatic formulation (green), 866 4 waveband RGB formulation (red), 867 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 868 (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. 869 From \citet{lengaigne.menkes.ea_CD07}. 870 } 871 \end{center} 880 \begin{figure} 881 \centering 882 \includegraphics[width=0.66\textwidth]{Fig_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} 872 892 \end{figure} 873 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 874 875 % ------------------------------------------------------------------------------------------------------------- 876 % Bottom Boundary Condition 877 % ------------------------------------------------------------------------------------------------------------- 878 \subsection[Bottom boundary condition (\textit{trabbc.F90}) - \forcode{ln_trabbc = .true.})] 879 {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})} 880 896 \label{subsec:TRA_bbc} 881 %--------------------------------------------nambbc-------------------------------------------------------- 882 883 \nlst{nambbc}884 %-------------------------------------------------------------------------------------------------------------- 885 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 886 \ begin{figure}[!t]887 \begin{center} 888 \includegraphics[width=\textwidth]{Fig_TRA_geoth}889 \caption{890 \protect\label{fig:geothermal}891 Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}.892 It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}.893 }894 \ 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]{Fig_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} 895 911 \end{figure} 896 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>897 912 898 913 Usually it is assumed that there is no exchange of heat or salt through the ocean bottom, 899 914 \ie\ a no flux boundary condition is applied on active tracers at the bottom. 900 915 This is the default option in \NEMO, and it is implemented using the masking technique. 901 However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling. 902 This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}), 916 However, there is a non-zero heat flux across the seafloor that 917 is associated with solid earth cooling. 918 This flux is weak compared to surface fluxes 919 (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}), 903 920 but it warms systematically the ocean and acts on the densest water masses. 904 921 Taking this flux into account in a global ocean model increases the deepest overturning cell 905 (\ie\ the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}. 906 907 Options are defined through the \nam{bbc} namelist variables. 908 The presence of geothermal heating is controlled by setting the namelist parameter \np{ln\_trabbc} to true. 909 Then, when \np{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by 910 the \np{rn\_geoflx\_cst}, which is also a namelist parameter. 911 When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in 912 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{emile-geay.madec_OS09}. 913 914 % ================================================================ 915 % Bottom Boundary Layer 916 % ================================================================ 917 \section[Bottom boundary layer (\textit{trabbl.F90} - \forcode{ln_trabbl = .true.})] 918 {Bottom boundary layer (\protect\mdl{trabbl} - \protect\np{ln\_trabbl}\forcode{ = .true.})} 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})} 919 937 \label{sec:TRA_bbl} 920 %--------------------------------------------nambbl--------------------------------------------------------- 921 922 \nlst{nambbl} 923 %-------------------------------------------------------------------------------------------------------------- 924 925 Options are defined through the \nam{bbl} 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. 926 946 In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps. 927 947 This is not adequate to represent gravity driven downslope flows. … … 929 949 where dense water formed in marginal seas flows into a basin filled with less dense water, 930 950 or along the continental slope when dense water masses are formed on a continental shelf. 931 The amount of entrainment that occurs in these gravity plumes is critical in determining the density and 932 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. 933 954 $z$-coordinate models tend to overestimate the entrainment, 934 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, 935 957 sometimes over a thickness much larger than the thickness of the observed gravity plume. 936 A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of 937 a sill \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved. 938 939 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{beckmann.doscher_JPO97}, 958 A similar problem occurs in the $s$-coordinate when 959 the thickness of the bottom level varies rapidly downstream of a sill 960 \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved. 961 962 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by 963 \citet{beckmann.doscher_JPO97}, 940 964 is to allow a direct communication between two adjacent bottom cells at different levels, 941 965 whenever the densest water is located above the less dense water. 942 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. 943 968 In the current implementation of the BBL, only the tracers are modified, not the velocities. 944 Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by 945 \citet{campin.goosse_T99}. 946 947 % ------------------------------------------------------------------------------------------------------------- 948 % Diffusive BBL 949 % ------------------------------------------------------------------------------------------------------------- 950 \subsection[Diffusive bottom boundary layer (\forcode{nn_bbl_ldf = 1})] 951 {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})} 952 974 \label{subsec:TRA_bbl_diff} 953 975 954 When applying sigma-diffusion (\np{ln\_trabbl}\forcode{ = .true.} and \np{nn\_bbl\_ldf} set to 1), 976 When applying sigma-diffusion 977 (\np[=.true.]{ln_trabbl}{ln\_trabbl} and \np{nn_bbl_ldf}{nn\_bbl\_ldf} set to 1), 955 978 the diffusive flux between two adjacent cells at the ocean floor is given by 956 979 \[ 957 % \label{eq: tra_bbl_diff}980 % \label{eq:TRA_bbl_diff} 958 981 \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T 959 982 \] 960 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and961 $A_l^\sigma$ the lateral diffusivity in the BBL.983 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 984 and $A_l^\sigma$ the lateral diffusivity in the BBL. 962 985 Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence, 963 986 \ie\ in the conditional form 964 987 \begin{equation} 965 \label{eq: tra_bbl_coef}988 \label{eq:TRA_bbl_coef} 966 989 A_l^\sigma (i,j,t) = 967 990 \begin{cases} 968 991 A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\ 969 \\ 970 0 & \text{otherwise} \\ 992 0 & \text{otherwise} 971 993 \end{cases} 972 994 \end{equation} 973 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 974 997 usually set to a value much larger than the one used for lateral mixing in the open ocean. 975 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 976 999 the density above the sea floor, at the top of the slope, is larger than in the deeper ocean 977 (see green arrow in \autoref{fig: bbl}).1000 (see green arrow in \autoref{fig:TRA_bbl}). 978 1001 In practice, this constraint is applied separately in the two horizontal directions, 979 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: 980 1003 \[ 981 % \label{eq: tra_bbl_Drho}1004 % \label{eq:TRA_bbl_Drho} 982 1005 \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S 983 1006 \] 984 where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and 985 $\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively. 986 987 % ------------------------------------------------------------------------------------------------------------- 988 % Advective BBL 989 % ------------------------------------------------------------------------------------------------------------- 990 \subsection[Advective bottom boundary layer (\forcode{nn_bbl_adv = [12]})] 991 {Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}\forcode{ = [12]})} 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})} 992 1013 \label{subsec:TRA_bbl_adv} 993 1014 … … 997 1018 %} 998 1019 999 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1000 \begin{figure}[!t] 1001 \begin{center} 1002 \includegraphics[width=\textwidth]{Fig_BBL_adv} 1003 \caption{ 1004 \protect\label{fig:bbl} 1005 Advective/diffusive Bottom Boundary Layer. 1006 The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$. 1007 Red arrows indicate the additional overturning circulation due to the advective BBL. 1008 The transport of the downslope flow is defined either as the transport of the bottom ocean cell (black arrow), 1009 or as a function of the along slope density gradient. 1010 The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ ocean bottom cells. 1011 } 1012 \end{center} 1020 \begin{figure} 1021 \centering 1022 \includegraphics[width=0.33\textwidth]{Fig_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} 1013 1033 \end{figure} 1014 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>1015 1034 1016 1035 %!! nn_bbl_adv = 1 use of the ocean velocity as bbl velocity … … 1020 1039 %%%gmcomment : this section has to be really written 1021 1040 1022 When applying an advective BBL (\np{nn\_bbl\_adv}\forcode{ = 1..2}), an overturning circulation is added which 1023 connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope. 1041 When applying an advective BBL (\np[=1..2]{nn_bbl_adv}{nn\_bbl\_adv}), 1042 an overturning circulation is added which connects two adjacent bottom grid-points only if 1043 dense water overlies less dense water on the slope. 1024 1044 The density difference causes dense water to move down the slope. 1025 1045 1026 \np{nn\_bbl\_adv}\forcode{ = 1}: 1027 the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step 1028 (see black arrow in \autoref{fig:bbl}) \citep{beckmann.doscher_JPO97}. 1029 It is a \textit{conditional advection}, that is, advection is allowed only 1030 if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and 1031 if the velocity is directed towards greater depth (\ie\ $\vect U \cdot \nabla H > 0$). 1032 1033 \np{nn\_bbl\_adv}\forcode{ = 2}: 1034 the downslope velocity is chosen to be proportional to $\Delta \rho$, 1035 the density difference between the higher cell and lower cell densities \citep{campin.goosse_T99}. 1036 The advection is allowed only if dense water overlies less dense water on the slope 1037 (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$). 1038 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}), 1039 is simply given by the following expression: 1040 \[ 1041 % \label{eq:bbl_Utr} 1042 u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 1043 \] 1044 where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl}, 1045 a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells, 1046 respectively. 1047 The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity, 1048 and because no direct estimation of this parameter is available, a uniform value has been assumed. 1049 The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}. 1050 1051 Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme. 1052 Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and 1053 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. 1054 1077 The entrainment is replaced by the vertical mixing implicit in the advection scheme. 1055 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 1056 1079 the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$. 1057 The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by 1058 the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and 1059 the upward \autoref{eq:bbl_up} return flows as follows: 1060 \begin{alignat}{3} 1061 \label{eq:bbl_dw} 1062 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 1063 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 1064 \label{eq:bbl_hor} 1065 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 1066 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ 1067 % 1068 \intertext{and for $k =kdw-1,\;..., \; kup$ :} 1069 % 1070 \label{eq:bbl_up} 1071 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 1072 &&+ \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) 1073 1091 \end{alignat} 1074 1092 where $b_t$ is the $T$-cell volume. … … 1077 1095 It has to be used to compute the effective velocity as well as the effective overturning circulation. 1078 1096 1079 % ================================================================ 1080 % Tracer damping 1081 % ================================================================ 1082 \section[Tracer damping (\textit{tradmp.F90})] 1083 {Tracer damping (\protect\mdl{tradmp})} 1097 %% ================================================================================================= 1098 \section[Tracer damping (\textit{tradmp.F90})]{Tracer damping (\protect\mdl{tradmp})} 1084 1099 \label{sec:TRA_dmp} 1085 %--------------------------------------------namtra_dmp------------------------------------------------- 1086 1087 \nlst{namtra_dmp} 1088 %-------------------------------------------------------------------------------------------------------------- 1089 1090 In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations: 1091 \begin{equation} 1092 \label{eq:tra_dmp} 1093 \begin{gathered} 1094 \pd[T]{t} = \cdots - \gamma (T - T_o) \\ 1095 \pd[S]{t} = \cdots - \gamma (S - S_o) 1096 \end{gathered} 1097 \end{equation} 1098 where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields 1099 (usually a climatology). 1100 Options are defined through the \nam{tra\_dmp} namelist variables. 1101 The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true. 1102 It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_dmp} are set to true in 1103 \nam{tsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set 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 1104 1120 (\ie\ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread}, 1105 1121 see \autoref{subsec:SBC_fldread}). 1106 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. 1107 The file name is specified by the namelist variable \np{cn\_resto}. 1108 The DMP\_TOOLS tool is provided to allow users to generate the netcdf file. 1109 1110 The two main cases in which \autoref{eq:tra_dmp} is used are 1111 \textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and 1112 \textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field 1113 (for example to build the initial state of a prognostic simulation, 1114 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*} 1115 1135 The first case applies to regional models that have artificial walls instead of open boundaries. 1116 In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas1117 it is zero in the interior of the model domain.1136 In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) 1137 whereas it is zero in the interior of the model domain. 1118 1138 The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}. 1119 1139 It allows us to find the velocity field consistent with the model dynamics whilst 1120 1140 having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$). 1121 1141 1122 The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but1123 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. 1124 1144 It also has undesirable effects on the ocean convection. 1125 It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much. 1126 1127 The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or 1128 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). 1129 1150 It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here 1130 1151 \citep{madec.delecluse.ea_JPO96}. 1131 1152 1132 For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under 1133 \path{./tools/DMP_TOOLS}. 1134 1135 % ================================================================ 1136 % Tracer time evolution 1137 % ================================================================ 1138 \section[Tracer time evolution (\textit{tranxt.F90})] 1139 {Tracer time evolution (\protect\mdl{tranxt})} 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})} 1140 1158 \label{sec:TRA_nxt} 1141 %--------------------------------------------namdom----------------------------------------------------- 1142 1143 \nlst{namdom} 1144 %-------------------------------------------------------------------------------------------------------------- 1145 1146 Options are defined through the \nam{dom} namelist variables. 1147 The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09}, 1148 \ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 1149 \begin{equation} 1150 \label{eq:tra_nxt} 1151 \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} 1152 1167 &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\ 1153 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] \\ … … 1155 1170 \end{alignedat} 1156 1171 \end{equation} 1157 where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, 1158 $\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$ 1159 (\ie\ fluxes plus content in mass exchanges). 1160 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1161 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}. 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}. 1162 1176 Note that the forcing correction term in the filter is not applied in linear free surface 1163 (\jp{ln\_linssh}\forcode{ = .true.}) (see \autoref{subsec:TRA_sbc}). 1164 Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$. 1165 1166 When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in 1167 \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. 1168 1183 In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module. 1169 1184 1170 In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed: 1171 $T^{t - \rdt} = T^t$ and $T^t = T_f$. 1172 1173 % ================================================================ 1174 % Equation of State (eosbn2) 1175 % ================================================================ 1176 \section[Equation of state (\textit{eosbn2.F90})] 1177 {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})} 1178 1190 \label{sec:TRA_eosbn2} 1179 %--------------------------------------------nameos----------------------------------------------------- 1180 1181 \nlst{nameos} 1182 %-------------------------------------------------------------------------------------------------------------- 1183 1184 % ------------------------------------------------------------------------------------------------------------- 1185 % Equation of State 1186 % ------------------------------------------------------------------------------------------------------------- 1187 \subsection[Equation of seawater (\texttt{ln}\{\texttt{\_teso10,\_eos80,\_seos}\})] 1188 {Equation of seawater (\protect\np{ln\_teos10}, \protect\np{ln\_teos80}, or \protect\np{ln\_seos}) } 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})} 1189 1200 \label{subsec:TRA_eos} 1190 1201 1191 1192 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density, 1193 $\rho$, 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. 1194 1206 Because density gradients control the pressure gradient force through the hydrostatic balance, 1195 the equation of state provides a fundamental bridge between the distribution of active tracers and1196 the fluid dynamics.1207 the equation of state provides a fundamental bridge between 1208 the distribution of active tracers and the fluid dynamics. 1197 1209 Nonlinearities of the EOS are of major importance, in particular influencing the circulation through 1198 1210 determination of the static stability below the mixed layer, 1199 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{roquet.madec.ea_JPO15}. 1200 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{fofonoff.millard_bk83}) or 1201 TEOS-10 \citep{ioc.iapso_bk10} standards should be used anytime a simulation of the real ocean circulation is attempted 1211 thus controlling rates of exchange between the atmosphere and the ocean interior 1202 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}. 1203 1216 The use of TEOS-10 is highly recommended because 1204 \textit{(i)} it is the new official EOS, 1205 \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and 1206 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and 1207 practical salinity for EOS-80, both variables being more suitable for use as model variables 1208 \citep{ioc.iapso_bk10, graham.mcdougall_JPO13}. 1217 \begin{enumerate*}[label=(\textit{\roman*})] 1218 \item it is the new official EOS, 1219 \item it is more accurate, being based on an updated database of laboratory measurements, and 1220 \item it uses Conservative Temperature and Absolute Salinity 1221 (instead of potential temperature and practical salinity for EOS-80), 1222 both variables being more suitable for use as model variables 1223 \citep{ioc.iapso_bk10, graham.mcdougall_JPO13}. 1224 \end{enumerate*} 1209 1225 EOS-80 is an obsolescent feature of the \NEMO\ system, kept only for backward compatibility. 1210 1226 For process studies, it is often convenient to use an approximation of the EOS. 1211 1227 To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available. 1212 1228 1213 In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density. 1214 Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 1215 This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as, 1216 with the exception of only a small percentage of the ocean, 1217 density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}. 1218 1219 Options which control the EOS used are defined through the \ngn{nameos} namelist variables. 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. 1220 1238 1221 1239 \begin{description} 1222 \item [\np{ln\_teos10}\forcode{ = .true.}]1223 the polyTEOS10-bsq equation of seawater\citep{roquet.madec.ea_OM15} is used.1240 \item [{\np[=.true.]{ln_teos10}{ln\_teos10}}] the polyTEOS10-bsq equation of seawater 1241 \citep{roquet.madec.ea_OM15} is used. 1224 1242 The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 1225 but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and 1226 more computationally efficient expressions for their derived quantities which make them more adapted for 1227 use in ocean models. 1228 Note that a slightly higher precision polynomial form is now used replacement of 1229 the TEOS-10 rational function approximation for hydrographic data analysis \citep{ioc.iapso_bk10}. 1243 but it is optimized for a Boussinesq fluid and 1244 the polynomial expressions have simpler and more computationally efficient expressions for 1245 their derived quantities which make them more adapted for use in ocean models. 1246 Note that a slightly higher precision polynomial form is now used 1247 replacement of the TEOS-10 rational function approximation for hydrographic data analysis 1248 \citep{ioc.iapso_bk10}. 1230 1249 A key point is that conservative state variables are used: 1231 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: $\Theta$). 1232 1252 The pressure in decibars is approximated by the depth in meters. 1233 1253 With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. 1234 It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ioc.iapso_bk10}. 1254 It is set to $C_p$ = 3991.86795711963 $J.Kg^{-1}.\deg{K}^{-1}$, 1255 according to \citet{ioc.iapso_bk10}. 1235 1256 Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$. 1236 In particular, the initial state de ined by the user have to be given as \textit{Conservative} Temperature and1237 \textit{ Absolute} Salinity.1257 In particular, the initial state defined by the user have to be given as 1258 \textit{Conservative} Temperature and \textit{Absolute} Salinity. 1238 1259 In addition, when using TEOS10, the Conservative SST is converted to potential SST prior to 1239 1260 either computing the air-sea and ice-sea fluxes (forced mode) or 1240 1261 sending the SST field to the atmosphere (coupled mode). 1241 \item[\np{ln\_eos80}\forcode{ = .true.}] 1242 the polyEOS80-bsq equation of seawater is used. 1243 It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to 1244 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.). 1245 1265 The state variables used in both the EOS80 and the ocean model are: 1246 the Practical Salinity ( (unit: psu, notation: $S_p$)) and1247 Potential Temperature (unit: $ ^{\circ}C$, notation: $\theta$).1266 the Practical Salinity (unit: $psu$, notation: $S_p$) and 1267 Potential Temperature (unit: $\deg{C}$, notation: $\theta$). 1248 1268 The pressure in decibars is approximated by the depth in meters. 1249 With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and1250 pressure \citep{fofonoff.millard_bk83}.1269 With EOS, the specific heat capacity of sea water, $C_p$, is a function of 1270 temperature, salinity and pressure \citep{fofonoff.millard_bk83}. 1251 1271 Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which 1252 1272 is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 1253 \item [\np{ln\_seos}\forcode{ = .true.}]1254 a simplified EOS (S-EOS) inspired by\citet{vallis_bk06} is chosen,1255 the coefficients of which has been optimized to fit the behavior of TEOS10 1256 ( Roquet, personal comm.) (see also \citet{roquet.madec.ea_JPO15}).1273 \item [{\np[=.true.]{ln_seos}{ln\_seos}}] a simplified EOS (S-EOS) inspired by 1274 \citet{vallis_bk06} is chosen, 1275 the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) 1276 (see also \citet{roquet.madec.ea_JPO15}). 1257 1277 It provides a simplistic linear representation of both cabbeling and thermobaricity effects which 1258 1278 is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}. 1259 With such an equation of state there is no longer a distinction between 1260 \textit{ conservative} and \textit{potential} temperature,1261 as well as between \textit{absolute} and\textit{practical} salinity.1279 With such an equation of state there is no longer a distinction between \textit{conservative} and 1280 \textit{potential} temperature, as well as between \textit{absolute} and 1281 \textit{practical} salinity. 1262 1282 S-EOS takes the following expression: 1263 1264 1283 \begin{gather*} 1265 % \label{eq:tra_S-EOS} 1266 \begin{alignedat}{2} 1267 &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. \\ 1268 & &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a \\ 1269 & \big. &- \nu \; T_a &S_a \big] \\ 1270 \end{alignedat} 1271 \\ 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] \\ 1272 1288 \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3 1273 1289 \end{gather*} 1274 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}. 1275 1292 In fact, when choosing S-EOS, various approximation of EOS can be specified simply by 1276 1293 changing the associated coefficients. 1277 Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS. 1278 setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from 1279 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. 1280 1298 Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 1281 1299 \end{description} 1282 1300 1283 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1284 \begin{table}[!tb] 1285 \begin{center} 1286 \begin{tabular}{|l|l|l|l|} 1287 \hline 1288 coeff. & computer name & S-EOS & description \\ 1289 \hline 1290 $a_0$ & \np{rn\_a0} & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\ 1291 \hline 1292 $b_0$ & \np{rn\_b0} & $7.6554~10^{-1}$ & linear haline expansion coeff. \\ 1293 \hline 1294 $\lambda_1$ & \np{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$ \\ 1295 \hline 1296 $\lambda_2$ & \np{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$ \\ 1297 \hline 1298 $\nu$ & \np{rn\_nu} & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$ \\ 1299 \hline 1300 $\mu_1$ & \np{rn\_mu1} & $1.4970~10^{-4}$ & thermobaric coeff. in T \\ 1301 \hline 1302 $\mu_2$ & \np{rn\_mu2} & $1.1090~10^{-5}$ & thermobaric coeff. in S \\ 1303 \hline 1304 \end{tabular} 1305 \caption{ 1306 \protect\label{tab:SEOS} 1307 Standard value of S-EOS coefficients. 1308 } 1309 \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} 1310 1324 \end{table} 1311 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1312 1313 % ------------------------------------------------------------------------------------------------------------- 1314 % Brunt-V\"{a}is\"{a}l\"{a} Frequency 1315 % ------------------------------------------------------------------------------------------------------------- 1316 \subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency] 1317 {Brunt-V\"{a}is\"{a}l\"{a} frequency} 1325 1326 %% ================================================================================================= 1327 \subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency]{Brunt-V\"{a}is\"{a}l\"{a} frequency} 1318 1328 \label{subsec:TRA_bn2} 1319 1329 1320 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of1321 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 1322 1332 (namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion, 1323 1333 non-penetrative convection, tidal mixing parameterisation, iso-neutral diffusion). … … 1326 1336 The expression for $N^2$ is given by: 1327 1337 \[ 1328 % \label{eq: tra_bn2}1338 % \label{eq:TRA_bn2} 1329 1339 N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt) 1330 1340 \] 1331 1341 where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and, 1332 1342 $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 1333 The coefficients are a polynomial function of temperature, salinity and depth which expression depends on 1334 the chosen EOS. 1335 They are computed through \textit{eos\_rab}, a \fortran function that can be found in \mdl{eosbn2}. 1336 1337 % ------------------------------------------------------------------------------------------------------------- 1338 % Freezing Point of Seawater 1339 % ------------------------------------------------------------------------------------------------------------- 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 %% ================================================================================================= 1340 1348 \subsection{Freezing point of seawater} 1341 1349 \label{subsec:TRA_fzp} … … 1343 1351 The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}: 1344 1352 \begin{equation} 1345 \label{eq:tra_eos_fzp} 1346 \begin{split} 1347 &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 1348 &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\ 1349 &\text{and~} d = -7.53~10^{-3} 1350 \end{split} 1351 \end{equation} 1352 1353 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water 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 1354 1361 (\ie\ referenced to the surface $p = 0$), 1355 thus the pressure dependent terms in \autoref{eq: tra_eos_fzp} (last term) have been dropped.1362 thus the pressure dependent terms in \autoref{eq:TRA_eos_fzp} (last term) have been dropped. 1356 1363 The freezing point is computed through \textit{eos\_fzp}, 1357 a \fortran function that can be found in \mdl{eosbn2}. 1358 1359 % ------------------------------------------------------------------------------------------------------------- 1360 % Potential Energy 1361 % ------------------------------------------------------------------------------------------------------------- 1364 a \fortran\ function that can be found in \mdl{eosbn2}. 1365 1366 %% ================================================================================================= 1362 1367 %\subsection{Potential Energy anomalies} 1363 1368 %\label{subsec:TRA_bn2} 1364 1369 1365 1370 % =====>>>>> TO BE written 1366 % 1367 1368 % ================================================================ 1369 % Horizontal Derivative in zps-coordinate 1370 % ================================================================ 1371 \section[Horizontal derivative in \textit{zps}-coordinate (\textit{zpshde.F90})] 1372 {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})} 1373 1374 \label{sec:TRA_zpshde} 1374 1375 … … 1376 1377 I've changed "derivative" to "difference" and "mean" to "average"} 1377 1378 1378 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}), 1379 1381 in general, tracers in horizontally adjacent cells live at different depths. 1380 Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and1381 the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).1382 The partial cell properties at the top (\np {ln\_isfcav}\forcode{ = .true.}) are computed in the same way as1383 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. 1384 1386 So, only the bottom interpolation is explained below. 1385 1387 1386 1388 Before taking horizontal gradients between the tracers next to the bottom, 1387 1389 a linear interpolation in the vertical is used to approximate the deeper tracer as if 1388 it actually lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}). 1389 For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is: 1390 1391 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1392 \begin{figure}[!p] 1393 \begin{center} 1394 \includegraphics[width=\textwidth]{Fig_partial_step_scheme} 1395 \caption{ 1396 \protect\label{fig:Partial_step_scheme} 1397 Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate 1398 (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $(e3w_k^{i + 1} - e3w_k^i) > 0$. 1399 A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$, 1400 the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. 1401 The horizontal difference is then given by: $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and 1402 the average by: $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$. 1403 } 1404 \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]{Fig_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} 1405 1408 \end{figure} 1406 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1409 1407 1410 \[ 1408 1411 \widetilde T = \lt\{ 1409 1412 \begin{alignedat}{2} 1410 1413 &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1} 1411 & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\1414 & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ 1412 1415 &T^{\, i} &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i } \; \delta_k T^{i + 1} 1413 1416 & \quad \text{if $e_{3w}^{i + 1} < e_{3w}^i$} … … 1415 1418 \rt. 1416 1419 \] 1417 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 1418 \begin{equation} 1419 \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} 1420 1424 \begin{split} 1421 1425 \delta_{i + 1/2} T &= 1422 1426 \begin{cases} 1423 \widetilde T - T^i & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1424 \\ 1425 T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} < e_{3w}^i 1426 \end{cases} 1427 \\ 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} \\ 1428 1430 \overline T^{\, i + 1/2} &= 1429 1431 \begin{cases} 1430 (\widetilde T - T^{\, i} ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 1431 \\ 1432 (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 1433 1434 \end{cases} 1434 1435 \end{split} … … 1437 1438 The computation of horizontal derivative of tracers as well as of density is performed once for all at 1438 1439 each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed. 1439 It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$, 1440 is not the same as that used for $T$ and $S$. 1441 Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of 1442 $T$ and $S$, and the pressure at a $u$-point 1440 It has to be emphasized that the procedure used to compute the interpolated density, 1441 $\widetilde \rho$, is not the same as that used for $T$ and $S$. 1442 Instead of forming a linear approximation of density, 1443 we compute $\widetilde \rho$ from the interpolated values of $T$ and $S$, 1444 and the pressure at a $u$-point 1443 1445 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 1444 1446 \[ 1445 % \label{eq: zps_hde_rho}1447 % \label{eq:TRA_zps_hde_rho} 1446 1448 \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt) 1447 1449 \] 1448 1450 1449 1451 This is a much better approximation as the variation of $\rho$ with depth (and thus pressure) 1450 is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation. 1451 This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and 1452 the slopes of neutral surfaces (\autoref{sec:LDF_slp}). 1453 1454 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, 1455 1458 both averaging and differencing operators appear. 1456 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: 1457 1460 in contrast to diffusion and pressure gradient computations, 1458 1461 no correction for partial steps is applied for advection. 1459 1462 The main motivation is to preserve the domain averaged mean variance of the advected field when 1460 1463 using the $2^{nd}$ order centred scheme. 1461 Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of 1462 partial cells should be further investigated in the near future. 1463 %%% 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. 1464 1466 \gmcomment{gm : this last remark has to be done} 1465 %%% 1466 1467 \biblio 1468 1469 \pindex 1467 1468 \onlyinsubfile{\input{../../global/epilogue}} 1470 1469 1471 1470 \end{document}
Note: See TracChangeset
for help on using the changeset viewer.