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NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex
r10146 r10354 21 21 \label{sec:ZDF_zdf} 22 22 23 The discrete form of the ocean subgrid scale physics has been presented in 24 \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}. At the surface and bottom boundaries, 25 the turbulent fluxes of momentum, heat and salt have to be defined. At the 26 surface they are prescribed from the surface forcing (see \autoref{chap:SBC}), 27 while at the bottom they are set to zero for heat and salt, unless a geothermal 28 flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} 29 defined, see \autoref{subsec:TRA_bbc}), and specified through a bottom friction 30 parameterisation for momentum (see \autoref{sec:ZDF_bfr}). 31 32 In this section we briefly discuss the various choices offered to compute 33 the vertical eddy viscosity and diffusivity coefficients, $A_u^{vm}$ , 34 $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- 35 points, respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}). These 36 coefficients can be assumed to be either constant, or a function of the local 37 Richardson number, or computed from a turbulent closure model (either 38 TKE or GLS formulation). The computation of these coefficients is initialized 39 in the \mdl{zdfini} module and performed in the \mdl{zdfric}, \mdl{zdftke} or 40 \mdl{zdfgls} modules. The trends due to the vertical momentum and tracer 41 diffusion, including the surface forcing, are computed and added to the 42 general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively. 43 These trends can be computed using either a forward time stepping scheme 44 (namelist parameter \np{ln\_zdfexp}\forcode{ = .true.}) or a backward time stepping 45 scheme (\np{ln\_zdfexp}\forcode{ = .false.}) depending on the magnitude of the mixing 46 coefficients, and thus of the formulation used (see \autoref{chap:STP}). 23 The discrete form of the ocean subgrid scale physics has been presented in 24 \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}. 25 At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt have to be defined. 26 At the surface they are prescribed from the surface forcing (see \autoref{chap:SBC}), 27 while at the bottom they are set to zero for heat and salt, 28 unless a geothermal flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} defined, 29 see \autoref{subsec:TRA_bbc}), and specified through a bottom friction parameterisation for momentum 30 (see \autoref{sec:ZDF_bfr}). 31 32 In this section we briefly discuss the various choices offered to compute the vertical eddy viscosity and 33 diffusivity coefficients, $A_u^{vm}$ , $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- points, 34 respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}). 35 These coefficients can be assumed to be either constant, or a function of the local Richardson number, 36 or computed from a turbulent closure model (either TKE or GLS formulation). 37 The computation of these coefficients is initialized in the \mdl{zdfini} module and performed in 38 the \mdl{zdfric}, \mdl{zdftke} or \mdl{zdfgls} modules. 39 The trends due to the vertical momentum and tracer diffusion, including the surface forcing, 40 are computed and added to the general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively. 41 These trends can be computed using either a forward time stepping scheme 42 (namelist parameter \np{ln\_zdfexp}\forcode{ = .true.}) or a backward time stepping scheme 43 (\np{ln\_zdfexp}\forcode{ = .false.}) depending on the magnitude of the mixing coefficients, 44 and thus of the formulation used (see \autoref{chap:STP}). 47 45 48 46 % ------------------------------------------------------------------------------------------------------------- … … 56 54 %-------------------------------------------------------------------------------------------------------------- 57 55 58 Options are defined through the \ngn{namzdf} namelist variables. 59 When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients 60 are set to constant values over the whole ocean. This is the crudest way to define 61 the vertical ocean physics. It is recommended that this option is only used in 62 process studies, not in basin scale simulations. Typical values used in this case are: 56 Options are defined through the \ngn{namzdf} namelist variables. 57 When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients are set to 58 constant values over the whole ocean. 59 This is the crudest way to define the vertical ocean physics. 60 It is recommended that this option is only used in process studies, not in basin scale simulations. 61 Typical values used in this case are: 63 62 \begin{align*} 64 63 A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1} \\ … … 67 66 68 67 These values are set through the \np{rn\_avm0} and \np{rn\_avt0} namelist parameters. 69 In all cases, do not use values smaller that those associated with the molecular 70 viscosity and diffusivity, that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, 71 $\sim10^{- 7}~m^2.s^{-1}$ for temperature and $\sim10^{-9}~m^2.s^{-1}$ for salinity.68 In all cases, do not use values smaller that those associated with the molecular viscosity and diffusivity, 69 that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and 70 $\sim10^{-9}~m^2.s^{-1}$ for salinity. 72 71 73 72 … … 83 82 %-------------------------------------------------------------------------------------------------------------- 84 83 85 When \key{zdfric} is defined, a local Richardson number dependent formulation 86 for the vertical momentum and tracer eddy coefficients is set through the \ngn{namzdf\_ric} 87 namelist variables.The vertical mixing 88 coefficients are diagnosed from the large scale variables computed by the model. 89 \textit{In situ} measurements have been used to link vertical turbulent activity to 90 large scale ocean structures. The hypothesis of a mixing mainly maintained by the 91 growth of Kelvin-Helmholtz like instabilities leads to a dependency between the 92 vertical eddy coefficients and the local Richardson number ($i.e.$ the 93 ratio of stratification to vertical shear). Following \citet{Pacanowski_Philander_JPO81}, the following 94 formulation has been implemented: 84 When \key{zdfric} is defined, a local Richardson number dependent formulation for the vertical momentum and 85 tracer eddy coefficients is set through the \ngn{namzdf\_ric} namelist variables. 86 The vertical mixing coefficients are diagnosed from the large scale variables computed by the model. 87 \textit{In situ} measurements have been used to link vertical turbulent activity to large scale ocean structures. 88 The hypothesis of a mixing mainly maintained by the growth of Kelvin-Helmholtz like instabilities leads to 89 a dependency between the vertical eddy coefficients and the local Richardson number 90 ($i.e.$ the ratio of stratification to vertical shear). 91 Following \citet{Pacanowski_Philander_JPO81}, the following formulation has been implemented: 95 92 \begin{equation} \label{eq:zdfric} 96 93 \left\{ \begin{aligned} … … 99 96 \end{aligned} \right. 100 97 \end{equation} 101 where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson 102 number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 103 $A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the 104 constant case (see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ 105 is the maximum value that can be reached by the coefficient when $Ri\leq 0$, 106 $a=5$ and $n=2$. The last three values can be modified by setting the 107 \np{rn\_avmri}, \np{rn\_alp} and \np{nn\_ric} namelist parameters, respectively. 108 109 A simple mixing-layer model to transfer and dissipate the atmospheric 110 forcings (wind-stress and buoyancy fluxes) can be activated setting 111 the \np{ln\_mldw}\forcode{ = .true.} in the namelist. 112 113 In this case, the local depth of turbulent wind-mixing or "Ekman depth" 114 $h_{e}(x,y,t)$ is evaluated and the vertical eddy coefficients prescribed within this layer. 98 where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number, 99 $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 100 $A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the constant case 101 (see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ is the maximum value that 102 can be reached by the coefficient when $Ri\leq 0$, $a=5$ and $n=2$. 103 The last three values can be modified by setting the \np{rn\_avmri}, \np{rn\_alp} and 104 \np{nn\_ric} namelist parameters, respectively. 105 106 A simple mixing-layer model to transfer and dissipate the atmospheric forcings 107 (wind-stress and buoyancy fluxes) can be activated setting the \np{ln\_mldw}\forcode{ = .true.} in the namelist. 108 109 In this case, the local depth of turbulent wind-mixing or "Ekman depth" $h_{e}(x,y,t)$ is evaluated and 110 the vertical eddy coefficients prescribed within this layer. 115 111 116 112 This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation: 117 113 \begin{equation} 118 h_{e} = Ek \frac {u^{*}} {f_{0}} \\ 119 \end{equation} 120 where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis 121 parameter. 114 h_{e} = Ek \frac {u^{*}} {f_{0}} 115 \end{equation} 116 where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter. 122 117 123 118 In this similarity height relationship, the turbulent friction velocity: 124 119 \begin{equation} 125 u^{*} = \sqrt \frac {|\tau|} {\rho_o} \\ 126 \end{equation} 127 120 u^{*} = \sqrt \frac {|\tau|} {\rho_o} 121 \end{equation} 128 122 is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$. 129 123 The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}. 130 Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to 124 Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to 131 125 the empirical values \np{rn\_wtmix} and \np{rn\_wvmix} \citep{Lermusiaux2001}. 132 126 … … 142 136 %-------------------------------------------------------------------------------------------------------------- 143 137 144 The vertical eddy viscosity and diffusivity coefficients are computed from a TKE 145 turbulent closure model based on a prognostic equation for $\bar{e}$, the turbulent 146 kinetic energy, and a closure assumption for the turbulent length scales. This 147 turbulent closure model has been developed by \citet{Bougeault1989} in the 148 atmospheric case, adapted by \citet{Gaspar1990} for the oceanic case, and 149 embedded in OPA, the ancestor of NEMO, by \citet{Blanke1993} for equatorial Atlantic 150 simulations. Since then, significant modifications have been introduced by 151 \citet{Madec1998} in both the implementation and the formulation of the mixing 152 length scale. The time evolution of $\bar{e}$ is the result of the production of 153 $\bar{e}$ through vertical shear, its destruction through stratification, its vertical 154 diffusion, and its dissipation of \citet{Kolmogorov1942} type: 138 The vertical eddy viscosity and diffusivity coefficients are computed from a TKE turbulent closure model based on 139 a prognostic equation for $\bar{e}$, the turbulent kinetic energy, 140 and a closure assumption for the turbulent length scales. 141 This turbulent closure model has been developed by \citet{Bougeault1989} in the atmospheric case, 142 adapted by \citet{Gaspar1990} for the oceanic case, and embedded in OPA, the ancestor of NEMO, 143 by \citet{Blanke1993} for equatorial Atlantic simulations. 144 Since then, significant modifications have been introduced by \citet{Madec1998} in both the implementation and 145 the formulation of the mixing length scale. 146 The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear, 147 its destruction through stratification, its vertical diffusion, and its dissipation of \citet{Kolmogorov1942} type: 155 148 \begin{equation} \label{eq:zdftke_e} 156 149 \frac{\partial \bar{e}}{\partial t} = … … 170 163 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 171 164 $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, 172 $P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity 173 and diffusivity coefficients. The constants $C_k = 0.1$ and $C_\epsilon = \sqrt {2} /2$ 174 $\approx 0.7$ are designed to deal with vertical mixing at any depth \citep{Gaspar1990}. 175 They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}. 176 $P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function 177 of the local Richardson number, $R_i$: 165 $P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity and diffusivity coefficients. 166 The constants $C_k = 0.1$ and $C_\epsilon = \sqrt {2} /2$ $\approx 0.7$ are designed to deal with 167 vertical mixing at any depth \citep{Gaspar1990}. 168 They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}. 169 $P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function of the local Richardson number, $R_i$: 178 170 \begin{align*} \label{eq:prt} 179 171 P_{rt} = \begin{cases} … … 186 178 The choice of $P_{rt}$ is controlled by the \np{nn\_pdl} namelist variable. 187 179 188 At the sea surface, the value of $\bar{e}$ is prescribed from the wind 189 stress field as $\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb} 190 namelist parameter. The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}), 191 however a much larger value can be used when taking into account the 192 surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}). 193 The bottom value of TKE is assumed to be equal to the value of the level just above. 194 The time integration of the $\bar{e}$ equation may formally lead to negative values 195 because the numerical scheme does not ensure its positivity. To overcome this 196 problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} 197 namelist parameter). Following \citet{Gaspar1990}, the cut-off value is set 198 to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. This allows the subsequent formulations 199 to match that of \citet{Gargett1984} for the diffusion in the thermocline and 200 deep ocean : $K_\rho = 10^{-3} / N$. 201 In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical 202 instabilities associated with too weak vertical diffusion. They must be 203 specified at least larger than the molecular values, and are set through 204 \np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \autoref{subsec:ZDF_cst}). 180 At the sea surface, the value of $\bar{e}$ is prescribed from the wind stress field as 181 $\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb} namelist parameter. 182 The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}), however a much larger value can be used when 183 taking into account the surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}). 184 The bottom value of TKE is assumed to be equal to the value of the level just above. 185 The time integration of the $\bar{e}$ equation may formally lead to negative values because 186 the numerical scheme does not ensure its positivity. 187 To overcome this problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} namelist parameter). 188 Following \citet{Gaspar1990}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. 189 This allows the subsequent formulations to match that of \citet{Gargett1984} for the diffusion in 190 the thermocline and deep ocean : $K_\rho = 10^{-3} / N$. 191 In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical instabilities associated with 192 too weak vertical diffusion. 193 They must be specified at least larger than the molecular values, and are set through \np{rn\_avm0} and 194 \np{rn\_avt0} (namzdf namelist, see \autoref{subsec:ZDF_cst}). 205 195 206 196 \subsubsection{Turbulent length scale} 207 For computational efficiency, the original formulation of the turbulent length 208 scales proposed by \citet{Gaspar1990} has been simplified. Four formulations 209 are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist 210 parameter. The first two are based on the following first order approximation 211 \citep{Blanke1993}: 197 For computational efficiency, the original formulation of the turbulent length scales proposed by 198 \citet{Gaspar1990} has been simplified. 199 Four formulations are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist parameter. 200 The first two are based on the following first order approximation \citep{Blanke1993}: 212 201 \begin{equation} \label{eq:tke_mxl0_1} 213 202 l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N 214 203 \end{equation} 215 which is valid in a stable stratified region with constant values of the Brunt- 216 Vais\"{a}l\"{a} frequency. The resulting length scale is bounded by the distance 217 to the surface or to the bottom (\np{nn\_mxl}\forcode{ = 0}) or by the local vertical scale factor 218 (\np{nn\_mxl}\forcode{ = 1}). \citet{Blanke1993} notice that this simplification has two major 219 drawbacks: it makes no sense for locally unstable stratification and the 220 computation no longer uses all the information contained in the vertical density 221 profile. To overcome these drawbacks, \citet{Madec1998} introduces the 222 \np{nn\_mxl}\forcode{ = 2..3} cases, which add an extra assumption concerning the vertical 223 gradient of the computed length scale. So, the length scales are first evaluated 224 as in \autoref{eq:tke_mxl0_1} and then bounded such that: 204 which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency. 205 The resulting length scale is bounded by the distance to the surface or to the bottom 206 (\np{nn\_mxl}\forcode{ = 0}) or by the local vertical scale factor (\np{nn\_mxl}\forcode{ = 1}). 207 \citet{Blanke1993} notice that this simplification has two major drawbacks: 208 it makes no sense for locally unstable stratification and the computation no longer uses all 209 the information contained in the vertical density profile. 210 To overcome these drawbacks, \citet{Madec1998} introduces the \np{nn\_mxl}\forcode{ = 2..3} cases, 211 which add an extra assumption concerning the vertical gradient of the computed length scale. 212 So, the length scales are first evaluated as in \autoref{eq:tke_mxl0_1} and then bounded such that: 225 213 \begin{equation} \label{eq:tke_mxl_constraint} 226 214 \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 227 215 \qquad \text{with }\ l = l_k = l_\epsilon 228 216 \end{equation} 229 \autoref{eq:tke_mxl_constraint} means that the vertical variations of the length 230 scale cannot be larger than the variations of depth. It provides a better 231 approximation of the \citet{Gaspar1990} formulation while being much less 232 time consuming. In particular, it allows the length scale to be limited not only 233 by the distance to the surface or to the ocean bottom but also by the distance 234 to a strongly stratified portion of the water column such as the thermocline 235 (\autoref{fig:mixing_length}). In order to impose the \autoref{eq:tke_mxl_constraint} 236 constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$, 237 the upward and downward length scales, and evaluate the dissipation and 238 mixing length scales as (and note that here we use numerical indexing): 217 \autoref{eq:tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than 218 the variations of depth. 219 It provides a better approximation of the \citet{Gaspar1990} formulation while being much less 220 time consuming. 221 In particular, it allows the length scale to be limited not only by the distance to the surface or 222 to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as 223 the thermocline (\autoref{fig:mixing_length}). 224 In order to impose the \autoref{eq:tke_mxl_constraint} constraint, we introduce two additional length scales: 225 $l_{up}$ and $l_{dwn}$, the upward and downward length scales, and 226 evaluate the dissipation and mixing length scales as 227 (and note that here we use numerical indexing): 239 228 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 240 229 \begin{figure}[!t] \begin{center} … … 253 242 \end{aligned} 254 243 \end{equation} 255 where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, 256 $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 257 258 In the \np{nn\_mxl}\forcode{ = 2} case, the dissipation and mixing length scales take the same 259 value: $ l_k= l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right)$, while in the 260 \np{nn\_mxl}\forcode{ = 3} case, the dissipation and mixing turbulent length scales are give 261 as in \citet{Gaspar1990}: 244 where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 245 246 In the \np{nn\_mxl}\forcode{ = 2} case, the dissipation and mixing length scales take the same value: 247 $ l_k= l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right)$, while in the \np{nn\_mxl}\forcode{ = 3} case, 248 the dissipation and mixing turbulent length scales are give as in \citet{Gaspar1990}: 262 249 \begin{equation} \label{eq:tke_mxl_gaspar} 263 250 \begin{aligned} … … 267 254 \end{equation} 268 255 269 At the ocean surface, a non zero length scale is set through the \np{rn\_mxl0} namelist 270 parameter. Usually the surface scale is given by $l_o = \kappa \,z_o$ 271 where $\kappa = 0.4$ is von Karman's constant and $z_o$ the roughness 272 parameter of the surface. Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94} 273 leads to a 0.04~m, the default value of \np{rn\_mxl0}. In the ocean interior 274 a minimum length scale is set to recover the molecular viscosity when $\bar{e}$ 275 reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ). 256 At the ocean surface, a non zero length scale is set through the \np{rn\_mxl0} namelist parameter. 257 Usually the surface scale is given by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant and 258 $z_o$ the roughness parameter of the surface. 259 Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94} leads to a 0.04~m, the default value of \np{rn\_mxl0}. 260 In the ocean interior a minimum length scale is set to recover the molecular viscosity when 261 $\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ). 276 262 277 263 278 264 \subsubsection{Surface wave breaking parameterization} 279 265 %-----------------------------------------------------------------------% 280 Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified 281 to include the effect of surface wave breaking energetics. This results in a reduction of summertime 282 surface temperature when the mixed layer is relatively shallow. The \citet{Mellor_Blumberg_JPO04} 283 modifications acts on surface length scale and TKE values and air-sea drag coefficient. 266 Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified to 267 include the effect of surface wave breaking energetics. 268 This results in a reduction of summertime surface temperature when the mixed layer is relatively shallow. 269 The \citet{Mellor_Blumberg_JPO04} modifications acts on surface length scale and TKE values and 270 air-sea drag coefficient. 284 271 The latter concerns the bulk formulea and is not discussed here. 285 272 … … 288 275 \bar{e}_o = \frac{1}{2}\,\left( 15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} 289 276 \end{equation} 290 where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality 291 which depends on the ''wave age'', ranging from 57 for mature waves to 146 for 292 younger waves \citep{Mellor_Blumberg_JPO04}. 277 where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality which depends on the ''wave age'', 278 ranging from 57 for mature waves to 146 for younger waves \citep{Mellor_Blumberg_JPO04}. 293 279 The boundary condition on the turbulent length scale follows the Charnock's relation: 294 280 \begin{equation} \label{eq:ZDF_Lsbc} … … 296 282 \end{equation} 297 283 where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant. 298 \citet{Mellor_Blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by \citet{Stacey_JPO99} 299 citing observation evidence, and $\alpha_{CB} = 100$ the Craig and Banner's value. 300 As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, 284 \citet{Mellor_Blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by 285 \citet{Stacey_JPO99} citing observation evidence, and 286 $\alpha_{CB} = 100$ the Craig and Banner's value. 287 As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, 301 288 with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}\forcode{ = 67.83} corresponds 302 to $\alpha_{CB} = 100$. Further setting \np{ln\_mxl0} to true applies \autoref{eq:ZDF_Lsbc} 303 as surface boundary condition on length scale, with $\beta$ hard coded to the Stacey's value. 304 Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) 305 is applied on surface $\bar{e}$ value. 289 to $\alpha_{CB} = 100$. 290 Further setting \np{ln\_mxl0} to true applies \autoref{eq:ZDF_Lsbc} as surface boundary condition on length scale, 291 with $\beta$ hard coded to the Stacey's value. 292 Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) is applied on 293 surface $\bar{e}$ value. 306 294 307 295 308 296 \subsubsection{Langmuir cells} 309 297 %--------------------------------------% 310 Langmuir circulations (LC) can be described as ordered large-scale vertical motions 311 in the surface layer of the oceans. Although LC have nothing to do with convection, 312 the circulation pattern is rather similar to so-called convective rolls in the atmospheric 313 boundary layer. The detailed physics behind LC is described in, for example, 314 \citet{Craik_Leibovich_JFM76}. The prevailing explanation is that LC arise from 315 a nonlinear interaction between the Stokes drift and wind drift currents. 316 317 Here we introduced in the TKE turbulent closure the simple parameterization of 318 Langmuir circulations proposed by \citep{Axell_JGR02} for a $k-\epsilon$ turbulent closure. 319 The parameterization, tuned against large-eddy simulation, includes the whole effect 320 of LC in an extra source terms of TKE, $P_{LC}$. 321 The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled 322 by setting \np{ln\_lc} to \forcode{.true.} in the namtke namelist. 298 Langmuir circulations (LC) can be described as ordered large-scale vertical motions in 299 the surface layer of the oceans. 300 Although LC have nothing to do with convection, the circulation pattern is rather similar to 301 so-called convective rolls in the atmospheric boundary layer. 302 The detailed physics behind LC is described in, for example, \citet{Craik_Leibovich_JFM76}. 303 The prevailing explanation is that LC arise from a nonlinear interaction between the Stokes drift and 304 wind drift currents. 305 306 Here we introduced in the TKE turbulent closure the simple parameterization of Langmuir circulations proposed by 307 \citep{Axell_JGR02} for a $k-\epsilon$ turbulent closure. 308 The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in 309 an extra source terms of TKE, $P_{LC}$. 310 The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to 311 \forcode{.true.} in the namtke namelist. 323 312 324 By making an analogy with the characteristic convective velocity scale 325 ($e.g.$, \citet{D'Alessio_al_JPO98}),$P_{LC}$ is assumed to be :313 By making an analogy with the characteristic convective velocity scale ($e.g.$, \citet{D'Alessio_al_JPO98}), 314 $P_{LC}$ is assumed to be : 326 315 \begin{equation} 327 316 P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}} … … 330 319 With no information about the wave field, $w_{LC}$ is assumed to be proportional to 331 320 the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module 332 \footnote{Following \citet{Li_Garrett_JMR93}, the surface Stoke drift velocity 333 may be expressed as $u_s = 0.016 \,|U_{10m}|$. Assuming an air density of 334 $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of $1.5~10^{-3}$ give the expression 335 used of $u_s$ as a function of the module of surface stress}. 336 For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as 337 a t a finite depth $H_{LC}$ (which is often close to the mixed layer depth), and simply338 varies as a sine function in between (a first-order profile for the Langmuir cell structures).321 \footnote{Following \citet{Li_Garrett_JMR93}, the surface Stoke drift velocity may be expressed as 322 $u_s = 0.016 \,|U_{10m}|$. 323 Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of 324 $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress}. 325 For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at 326 a finite depth $H_{LC}$ (which is often close to the mixed layer depth), 327 and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures). 339 328 The resulting expression for $w_{LC}$ is : 340 329 \begin{equation} … … 344 333 \end{cases} 345 334 \end{equation} 346 where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise 347 to fit LES data. The chosen value yields maximum vertical velocities $w_{LC}$ of the order 348 of a few centimeters per second. The value of $c_{LC}$ is set through the \np{rn\_lc} 349 namelist parameter,having in mind that it should stay between 0.15 and 0.54 \citep{Axell_JGR02}.335 where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise to fit LES data. 336 The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second. 337 The value of $c_{LC}$ is set through the \np{rn\_lc} namelist parameter, 338 having in mind that it should stay between 0.15 and 0.54 \citep{Axell_JGR02}. 350 339 351 340 The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations: 352 $H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift 353 c an reach on its own by converting its kinetic energy to potential energy, according to341 $H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by 342 converting its kinetic energy to potential energy, according to 354 343 \begin{equation} 355 344 - \int_{-H_{LC}}^0 { N^2\;z \;dz} = \frac{1}{2} u_s^2 … … 360 349 %--------------------------------------------------------------% 361 350 362 Vertical mixing parameterizations commonly used in ocean general circulation models 363 tend toproduce mixed-layer depths that are too shallow during summer months and windy conditions.364 This bias is particularly acute over the Southern Ocean. 365 To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme 366 The parameterization is an empirical one, $i.e.$ not derived from theoretical considerations, 351 Vertical mixing parameterizations commonly used in ocean general circulation models tend to 352 produce mixed-layer depths that are too shallow during summer months and windy conditions. 353 This bias is particularly acute over the Southern Ocean. 354 To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{Rodgers_2014}. 355 The parameterization is an empirical one, $i.e.$ not derived from theoretical considerations, 367 356 but rather is meant to account for observed processes that affect the density structure of 368 357 the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme 369 358 ($i.e.$ near-inertial oscillations and ocean swells and waves). 370 359 371 When using this parameterization ($i.e.$ when \np{nn\_etau}\forcode{ = 1}), the TKE input to the ocean ($S$) 372 imposed by the winds in the form of near-inertial oscillations, swell and waves is parameterized 373 by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition, plus a depth depend one given by: 360 When using this parameterization ($i.e.$ when \np{nn\_etau}\forcode{ = 1}), 361 the TKE input to the ocean ($S$) imposed by the winds in the form of near-inertial oscillations, 362 swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition, 363 plus a depth depend one given by: 374 364 \begin{equation} \label{eq:ZDF_Ehtau} 375 365 S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau} 376 366 \end{equation} 377 where 378 $z$ is the depth, 379 $e_s$ is TKE surface boundary condition, 380 $f_r$ is the fraction of the surface TKE that penetrate in the ocean, 381 $h_\tau$ is a vertical mixing length scale that controls exponential shape of the penetration, 382 and $f_i$ is the ice concentration (no penetration if $f_i=1$, that is if the ocean is entirely 383 covered by sea-ice). 384 The value of $f_r$, usually a few percents, is specified through \np{rn\_efr} namelist parameter. 385 The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np{nn\_etau}\forcode{ = 0}) 386 or a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m 387 at high latitudes (\np{nn\_etau}\forcode{ = 1}). 388 389 Note that two other option existe, \np{nn\_etau}\forcode{ = 2..3}. They correspond to applying 390 \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer, or to using the high frequency part 391 of the stress to evaluate the fraction of TKE that penetrate the ocean. 367 where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that 368 penetrate in the ocean, $h_\tau$ is a vertical mixing length scale that controls exponential shape of 369 the penetration, and $f_i$ is the ice concentration 370 (no penetration if $f_i=1$, that is if the ocean is entirely covered by sea-ice). 371 The value of $f_r$, usually a few percents, is specified through \np{rn\_efr} namelist parameter. 372 The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np{nn\_etau}\forcode{ = 0}) or 373 a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m at high latitudes 374 (\np{nn\_etau}\forcode{ = 1}). 375 376 Note that two other option existe, \np{nn\_etau}\forcode{ = 2..3}. 377 They correspond to applying \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer, 378 or to using the high frequency part of the stress to evaluate the fraction of TKE that penetrate the ocean. 392 379 Those two options are obsolescent features introduced for test purposes. 393 380 They will be removed in the next release. … … 420 407 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 421 408 422 The production of turbulence by vertical shear (the first term of the right hand side 423 of \autoref{eq:zdftke_e}) should balance the loss of kinetic energy associated with 424 the vertical momentum diffusion (first line in \autoref{eq:PE_zdf}). To do so a special care 425 have to be taken for both the time and space discretization of the TKE equation 426 \citep{Burchard_OM02,Marsaleix_al_OM08}. 427 428 Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows 429 how the two-level Leap-Frog time stepping of the momentum and tracer equations interplays 430 with the one-level forward time stepping of TKE equation. With this framework, the total loss 431 of kinetic energy (in 1D for the demonstration) due to the vertical momentum diffusion is 432 obtained by multiplying this quantity by $u^t$ and summing the result vertically: 409 The production of turbulence by vertical shear (the first term of the right hand side of 410 \autoref{eq:zdftke_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion 411 (first line in \autoref{eq:PE_zdf}). 412 To do so a special care have to be taken for both the time and space discretization of 413 the TKE equation \citep{Burchard_OM02,Marsaleix_al_OM08}. 414 415 Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows how 416 the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with 417 the one-level forward time stepping of TKE equation. 418 With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to 419 the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and 420 summing the result vertically: 433 421 \begin{equation} \label{eq:energ1} 434 422 \begin{split} … … 438 426 \end{split} 439 427 \end{equation} 440 Here, the vertical diffusion of momentum is discretized backward in time 441 with a coefficient, $K_m$, known at time $t$ (\autoref{fig:TKE_time_scheme}), 442 as it is required when using the TKE scheme (see \autoref{sec:STP_forward_imp}). 443 The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy 444 transfer at the surface (atmospheric forcing) and at the bottom (friction effect). 445 The second term is always negative. It is the dissipation rate of kinetic energy, 446 and thus minus the shear production rate of $\bar{e}$. \autoref{eq:energ1} 447 implies that, to be energetically consistent, the production rate of $\bar{e}$ 448 used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as 449 ${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ (and not by the more straightforward 450 $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$). 451 452 A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification 453 (second term of the right hand side of \autoref{eq:zdftke_e}). This term 454 must balance the input of potential energy resulting from vertical mixing. 455 The rate of change of potential energy (in 1D for the demonstration) due vertical 456 mixing is obtained by multiplying vertical density diffusion 457 tendency by $g\,z$ and and summing the result vertically: 428 Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$, 429 known at time $t$ (\autoref{fig:TKE_time_scheme}), as it is required when using the TKE scheme 430 (see \autoref{sec:STP_forward_imp}). 431 The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy transfer at 432 the surface (atmospheric forcing) and at the bottom (friction effect). 433 The second term is always negative. 434 It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$. 435 \autoref{eq:energ1} implies that, to be energetically consistent, 436 the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as 437 ${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ 438 (and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$). 439 440 A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification 441 (second term of the right hand side of \autoref{eq:zdftke_e}). 442 This term must balance the input of potential energy resulting from vertical mixing. 443 The rate of change of potential energy (in 1D for the demonstration) due vertical mixing is obtained by 444 multiplying vertical density diffusion tendency by $g\,z$ and and summing the result vertically: 458 445 \begin{equation} \label{eq:energ2} 459 446 \begin{split} … … 466 453 \end{equation} 467 454 where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. 468 The first term of the right hand side of \autoref{eq:energ2} is always zero 469 because there is no diffusive flux through the ocean surface and bottom). 470 The second term is minus the destruction rate of $\bar{e}$ due to stratification. 471 Therefore \autoref{eq:energ1} implies that, to be energetically consistent, the product 472 ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e}, the TKE equation. 473 474 Let us now address the space discretization issue. 475 The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity 476 components are in the centre of the side faces of a $t$-box in staggered C-grid 477 (\autoref{fig:cell}). A space averaging is thus required to obtain the shear TKE production term. 478 By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of 479 eddy coefficient by the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. 480 Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into 481 account. 482 483 The above energetic considerations leads to 484 the following final discrete form for the TKE equation: 455 The first term of the right hand side of \autoref{eq:energ2} is always zero because 456 there is no diffusive flux through the ocean surface and bottom). 457 The second term is minus the destruction rate of $\bar{e}$ due to stratification. 458 Therefore \autoref{eq:energ1} implies that, to be energetically consistent, 459 the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e}, the TKE equation. 460 461 Let us now address the space discretization issue. 462 The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in 463 the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:cell}). 464 A space averaging is thus required to obtain the shear TKE production term. 465 By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of eddy coefficient by 466 the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. 467 Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into account. 468 469 The above energetic considerations leads to the following final discrete form for the TKE equation: 485 470 \begin{equation} \label{eq:zdftke_ene} 486 471 \begin{split} … … 500 485 \end{split} 501 486 \end{equation} 502 where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation) 503 are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}). 504 Note that the Kolmogorov term has been linearized in time in order to render 505 the implicit computation possible. The restart of the TKE scheme 506 requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as they all appear in 507 the right hand side of \autoref{eq:zdftke_ene}. For the latter, it is in fact 508 the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 487 where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation) 488 are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}). 489 Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible. 490 The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as 491 they all appear in the right hand side of \autoref{eq:zdftke_ene}. 492 For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 509 493 510 494 % ------------------------------------------------------------------------------------------------------------- … … 519 503 %-------------------------------------------------------------------------------------------------------------- 520 504 521 The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on 522 two prognostic equations: one for the turbulent kinetic energy $\bar {e}$, and another 523 for the generic length scale, $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}. 524 This later variable is defined as : $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, 525 where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover 526 a number of well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982}, 527 $k$-$\epsilon$ \citep{Rodi_1987}, $k$-$\omega$ \citep{Wilcox_1988} 528 among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}). 505 The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on two prognostic equations: 506 one for the turbulent kinetic energy $\bar {e}$, and another for the generic length scale, 507 $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}. 508 This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, 509 where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover a number of 510 well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982}, $k$-$\epsilon$ \citep{Rodi_1987}, 511 $k$-$\omega$ \citep{Wilcox_1988} among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}). 529 512 The GLS scheme is given by the following set of equations: 530 513 \begin{equation} \label{eq:zdfgls_e} … … 558 541 {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 559 542 \end{equation} 560 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) 561 and $\epsilon$ the dissipation rate. 562 The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) 563 depends of the choice of the turbulence model. Four different turbulent models are pre-defined 564 (Tab.\autoref{tab:GLS}). They are made available through the \np{nn\_clo} namelist parameter. 543 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and 544 $\epsilon$ the dissipation rate. 545 The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of 546 the choice of the turbulence model. 547 Four different turbulent models are pre-defined (Tab.\autoref{tab:GLS}). 548 They are made available through the \np{nn\_clo} namelist parameter. 565 549 566 550 %--------------------------------------------------TABLE-------------------------------------------------- … … 584 568 \end{tabular} 585 569 \caption{ \protect\label{tab:GLS} 586 Set of predefined GLS parameters, or equivalently predefined turbulence models available 587 with \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls}.}570 Set of predefined GLS parameters, or equivalently predefined turbulence models available with 571 \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls}.} 588 572 \end{center} \end{table} 589 573 %-------------------------------------------------------------------------------------------------------------- 590 574 591 In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force 592 the convergence of the mixing length towards $K z_b$ ($K$: Kappa and $z_b$: rugosity length) 593 value near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$ 594 are calculated from stability function proposed by \citet{Galperin_al_JAS88}, or by \citet{Kantha_Clayson_1994} 595 or one of the two functions suggested by \citet{Canuto_2001} (\np{nn\_stab\_func}\forcode{ = 0..3}, resp.). 575 In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force the convergence of 576 the mixing length towards $K z_b$ ($K$: Kappa and $z_b$: rugosity length) value near physical boundaries 577 (logarithmic boundary layer law). 578 $C_{\mu}$ and $C_{\mu'}$ are calculated from stability function proposed by \citet{Galperin_al_JAS88}, 579 or by \citet{Kantha_Clayson_1994} or one of the two functions suggested by \citet{Canuto_2001} 580 (\np{nn\_stab\_func}\forcode{ = 0..3}, resp.). 596 581 The value of $C_{0\mu}$ depends of the choice of the stability function. 597 582 598 The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated 599 thanks to Dirichlet or Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp. 600 As for TKE closure , the wave effect on the mixing is considered when \np{ln\_crban}\forcode{ = .true.} 601 \citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}. The \np{rn\_crban} namelist parameter 602 is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}. 603 604 The $\psi$ equation is known to fail in stably stratified flows, and for this reason 605 almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy. 606 With this clipping, the maximum permissible length scale is determined by 607 $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$. A value of $c_{lim} = 0.53$ is often used 608 \citep{Galperin_al_JAS88}. \cite{Umlauf_Burchard_CSR05} show that the value of 609 the clipping factor is of crucial importance for the entrainment depth predicted in 610 stably stratified situations, and that its value has to be chosen in accordance 611 with the algebraic model for the turbulent fluxes. The clipping is only activated 612 if \np{ln\_length\_lim}\forcode{ = .true.}, and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value. 613 614 The time and space discretization of the GLS equations follows the same energetic 615 consideration as for the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{Burchard_OM02}. 583 The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated thanks to Dirichlet or 584 Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp. 585 As for TKE closure, the wave effect on the mixing is considered when 586 \np{ln\_crban}\forcode{ = .true.} \citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}. 587 The \np{rn\_crban} namelist parameter is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and 588 \np{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}. 589 590 The $\psi$ equation is known to fail in stably stratified flows, and for this reason 591 almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy. 592 With this clipping, the maximum permissible length scale is determined by $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$. 593 A value of $c_{lim} = 0.53$ is often used \citep{Galperin_al_JAS88}. 594 \cite{Umlauf_Burchard_CSR05} show that the value of the clipping factor is of crucial importance for 595 the entrainment depth predicted in stably stratified situations, 596 and that its value has to be chosen in accordance with the algebraic model for the turbulent fluxes. 597 The clipping is only activated if \np{ln\_length\_lim}\forcode{ = .true.}, 598 and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value. 599 600 The time and space discretization of the GLS equations follows the same energetic consideration as for 601 the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{Burchard_OM02}. 616 602 Examples of performance of the 4 turbulent closure scheme can be found in \citet{Warner_al_OM05}. 617 603 … … 640 626 %-------------------------------------------------------------------------------------------------------------- 641 627 642 Static instabilities (i.e. light potential densities under heavy ones) may 643 occur at particular ocean grid points. In nature, convective processes 644 quickly re-establish the static stability of the water column. These 645 processes have been removed from the model via the hydrostatic 646 assumption so they must be parameterized. Three parameterisations 647 are available to deal with convective processes: a non-penetrative 648 convective adjustment or an enhanced vertical diffusion, or/and the 649 use of a turbulent closure scheme. 628 Static instabilities (i.e. light potential densities under heavy ones) may occur at particular ocean grid points. 629 In nature, convective processes quickly re-establish the static stability of the water column. 630 These processes have been removed from the model via the hydrostatic assumption so they must be parameterized. 631 Three parameterisations are available to deal with convective processes: 632 a non-penetrative convective adjustment or an enhanced vertical diffusion, 633 or/and the use of a turbulent closure scheme. 650 634 651 635 % ------------------------------------------------------------------------------------------------------------- … … 665 649 \includegraphics[width=0.90\textwidth]{Fig_npc} 666 650 \caption{ \protect\label{fig:npc} 667 Example of an unstable density profile treated by the non penetrative 668 convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from 669 the surface to the bottom. It is found to be unstable between levels 3 and 4. 670 They are mixed. The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 671 are mixed. The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are 672 mixed. The $1^{st}$ step ends since the density profile is then stable below 673 the level 3. $2^{nd}$ step: the new $\rho$ profile is checked following the same 674 procedure as in $1^{st}$ step: levels 2 to 5 are mixed. The new density profile 675 is checked. It is found stable: end of algorithm.} 651 Example of an unstable density profile treated by the non penetrative convective adjustment algorithm. 652 $1^{st}$ step: the initial profile is checked from the surface to the bottom. 653 It is found to be unstable between levels 3 and 4. 654 They are mixed. 655 The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed. 656 The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed. 657 The $1^{st}$ step ends since the density profile is then stable below the level 3. 658 $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step: 659 levels 2 to 5 are mixed. 660 The new density profile is checked. 661 It is found stable: end of algorithm.} 676 662 \end{center} \end{figure} 677 663 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 678 664 679 Options are defined through the \ngn{namzdf} namelist variables. 680 The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}\forcode{ = .true.}. 681 It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously 682 the statically unstable portion of the water column, but only until the density 683 structure becomes neutrally stable ($i.e.$ until the mixed portion of the water 684 column has \textit{exactly} the density of the water just below) \citep{Madec_al_JPO91}. 685 The associated algorithm is an iterative process used in the following way 686 (\autoref{fig:npc}): starting from the top of the ocean, the first instability is 687 found. Assume in the following that the instability is located between levels 688 $k$ and $k+1$. The temperature and salinity in the two levels are 689 vertically mixed, conserving the heat and salt contents of the water column. 690 The new density is then computed by a linear approximation. If the new 691 density profile is still unstable between levels $k+1$ and $k+2$, levels $k$, 692 $k+1$ and $k+2$ are then mixed. This process is repeated until stability is 693 established below the level $k$ (the mixing process can go down to the 694 ocean bottom). The algorithm is repeated to check if the density profile 695 between level $k-1$ and $k$ is unstable and/or if there is no deeper instability. 696 697 This algorithm is significantly different from mixing statically unstable levels 698 two by two. The latter procedure cannot converge with a finite number 699 of iterations for some vertical profiles while the algorithm used in \NEMO 700 converges for any profile in a number of iterations which is less than the 701 number of vertical levels. This property is of paramount importance as 702 pointed out by \citet{Killworth1989}: it avoids the existence of permanent 703 and unrealistic static instabilities at the sea surface. This non-penetrative 704 convective algorithm has been proved successful in studies of the deep 705 water formation in the north-western Mediterranean Sea 706 \citep{Madec_al_JPO91, Madec_al_DAO91, Madec_Crepon_Bk91}. 707 708 The current implementation has been modified in order to deal with any non linear 709 equation of seawater (L. Brodeau, personnal communication). 710 Two main differences have been introduced compared to the original algorithm: 665 Options are defined through the \ngn{namzdf} namelist variables. 666 The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}\forcode{ = .true.}. 667 It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of 668 the water column, but only until the density structure becomes neutrally stable 669 ($i.e.$ until the mixed portion of the water column has \textit{exactly} the density of the water just below) 670 \citep{Madec_al_JPO91}. 671 The associated algorithm is an iterative process used in the following way (\autoref{fig:npc}): 672 starting from the top of the ocean, the first instability is found. 673 Assume in the following that the instability is located between levels $k$ and $k+1$. 674 The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of 675 the water column. 676 The new density is then computed by a linear approximation. 677 If the new density profile is still unstable between levels $k+1$ and $k+2$, 678 levels $k$, $k+1$ and $k+2$ are then mixed. 679 This process is repeated until stability is established below the level $k$ 680 (the mixing process can go down to the ocean bottom). 681 The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or 682 if there is no deeper instability. 683 684 This algorithm is significantly different from mixing statically unstable levels two by two. 685 The latter procedure cannot converge with a finite number of iterations for some vertical profiles while 686 the algorithm used in \NEMO converges for any profile in a number of iterations which is less than 687 the number of vertical levels. 688 This property is of paramount importance as pointed out by \citet{Killworth1989}: 689 it avoids the existence of permanent and unrealistic static instabilities at the sea surface. 690 This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in 691 the north-western Mediterranean Sea \citep{Madec_al_JPO91, Madec_al_DAO91, Madec_Crepon_Bk91}. 692 693 The current implementation has been modified in order to deal with any non linear equation of seawater 694 (L. Brodeau, personnal communication). 695 Two main differences have been introduced compared to the original algorithm: 711 696 $(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency 712 (not the the difference in potential density) ; 713 $(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients 714 are vertically mixed in the same way their temperature and salinity has been mixed. 715 These two modifications allow the algorithm to perform properly and accurately 716 with TEOS10 or EOS-80 without having to recompute the expansion coefficients at each 717 mixing iteration. 697 (not the the difference in potential density); 698 $(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in 699 the same way their temperature and salinity has been mixed. 700 These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without 701 having to recompute the expansion coefficients at each mixing iteration. 718 702 719 703 % ------------------------------------------------------------------------------------------------------------- … … 729 713 730 714 Options are defined through the \ngn{namzdf} namelist variables. 731 The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}\forcode{ = .true.}. 732 In this case, the vertical eddy mixing coefficients are assigned very large values 733 (a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable 734 ($i.e.$ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) 735 \citep{Lazar_PhD97, Lazar_al_JPO99}. This is done either on tracers only 736 (\np{nn\_evdm}\forcode{ = 0}) or on both momentum and tracers (\np{nn\_evdm}\forcode{ = 1}). 737 738 In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and 739 if \np{nn\_evdm}\forcode{ = 1}, the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ 740 values also, are set equal to the namelist parameter \np{rn\_avevd}. A typical value 741 for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$. This parameterisation of 742 convective processes is less time consuming than the convective adjustment 743 algorithm presented above when mixing both tracers and momentum in the 744 case of static instabilities. It requires the use of an implicit time stepping on 745 vertical diffusion terms (i.e. \np{ln\_zdfexp}\forcode{ = .false.}). 746 747 Note that the stability test is performed on both \textit{before} and \textit{now} 748 values of $N^2$. This removes a potential source of divergence of odd and 749 even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \autoref{sec:STP_mLF}). 715 The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}\forcode{ = .true.}. 716 In this case, the vertical eddy mixing coefficients are assigned very large values 717 (a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable 718 ($i.e.$ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{Lazar_PhD97, Lazar_al_JPO99}. 719 This is done either on tracers only (\np{nn\_evdm}\forcode{ = 0}) or 720 on both momentum and tracers (\np{nn\_evdm}\forcode{ = 1}). 721 722 In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np{nn\_evdm}\forcode{ = 1}, 723 the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to 724 the namelist parameter \np{rn\_avevd}. 725 A typical value for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$. 726 This parameterisation of convective processes is less time consuming than 727 the convective adjustment algorithm presented above when mixing both tracers and 728 momentum in the case of static instabilities. 729 It requires the use of an implicit time stepping on vertical diffusion terms 730 (i.e. \np{ln\_zdfexp}\forcode{ = .false.}). 731 732 Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$. 733 This removes a potential source of divergence of odd and even time step in 734 a leapfrog environment \citep{Leclair_PhD2010} (see \autoref{sec:STP_mLF}). 750 735 751 736 % ------------------------------------------------------------------------------------------------------------- … … 755 740 \label{subsec:ZDF_tcs} 756 741 757 The turbulent closure scheme presented in \autoref{subsec:ZDF_tke} and \autoref{subsec:ZDF_gls} 758 (\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically 759 unstable density profiles. In such a case, the term corresponding to the 760 destruction of turbulent kinetic energy through stratification in \autoref{eq:zdftke_e} 761 or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative. 762 It results in large values of $A_T^{vT}$ and $A_T^{vT}$, and also the four neighbouring 763 $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1})$. These large values 764 restore the static stability of the water column in a way similar to that of the 765 enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}). However, 766 in the vicinity of the sea surface (first ocean layer), the eddy coefficients 767 computed by the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$, 768 because the mixing length scale is bounded by the distance to the sea surface. 769 It can thus be useful to combine the enhanced vertical 770 diffusion with the turbulent closure scheme, $i.e.$ setting the \np{ln\_zdfnpc} 771 namelist parameter to true and defining the turbulent closure CPP key all together. 772 773 The KPP turbulent closure scheme already includes enhanced vertical diffusion 774 in the case of convection, as governed by the variables $bvsqcon$ and $difcon$ 775 found in \mdl{zdfkpp}, therefore \np{ln\_zdfevd}\forcode{ = .false.} should be used with the KPP 776 scheme. %gm% + one word on non local flux with KPP scheme trakpp.F90 module... 742 The turbulent closure scheme presented in \autoref{subsec:ZDF_tke} and \autoref{subsec:ZDF_gls} 743 (\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically unstable density profiles. 744 In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in 745 \autoref{eq:zdftke_e} or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative. 746 It results in large values of $A_T^{vT}$ and $A_T^{vT}$, and also the four neighbouring $A_u^{vm} {and}\;A_v^{vm}$ 747 (up to $1\;m^2s^{-1}$). 748 These large values restore the static stability of the water column in a way similar to that of 749 the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}). 750 However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by 751 the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$, 752 because the mixing length scale is bounded by the distance to the sea surface. 753 It can thus be useful to combine the enhanced vertical diffusion with the turbulent closure scheme, 754 $i.e.$ setting the \np{ln\_zdfnpc} namelist parameter to true and 755 defining the turbulent closure CPP key all together. 756 757 The KPP turbulent closure scheme already includes enhanced vertical diffusion in the case of convection, 758 as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp}, 759 therefore \np{ln\_zdfevd}\forcode{ = .false.} should be used with the KPP scheme. 760 % gm% + one word on non local flux with KPP scheme trakpp.F90 module... 777 761 778 762 % ================================================================ … … 788 772 789 773 Options are defined through the \ngn{namzdf\_ddm} namelist variables. 790 Double diffusion occurs when relatively warm, salty water overlies cooler, fresher 791 water, or vice versa. The former condition leads to salt fingering and the latter 792 to diffusive convection. Double-diffusive phenomena contribute to diapycnal 793 mixing in extensive regions of the ocean. \citet{Merryfield1999} include a 794 parameterisation of such phenomena in a global ocean model and show that 795 it leads to relatively minor changes in circulation but exerts significant regional 796 influences on temperature and salinity. This parameterisation has been 797 introduced in \mdl{zdfddm} module and is controlled by the \key{zdfddm} CPP key. 774 Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa. 775 The former condition leads to salt fingering and the latter to diffusive convection. 776 Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean. 777 \citet{Merryfield1999} include a parameterisation of such phenomena in a global ocean model and show that 778 it leads to relatively minor changes in circulation but exerts significant regional influences on 779 temperature and salinity. 780 This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the \key{zdfddm} CPP key. 798 781 799 782 Diapycnal mixing of S and T are described by diapycnal diffusion coefficients … … 802 785 &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} 803 786 \end{align*} 804 where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, 805 and $o$ by processes other than double diffusion. The rates of double-diffusive 806 mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, 807 where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline 808 contraction (see \autoref{subsec:TRA_eos}). To represent mixing of $S$ and $T$ by salt 809 fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981): 787 where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, 788 and $o$ by processes other than double diffusion. 789 The rates of double-diffusive mixing depend on the buoyancy ratio 790 $R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of 791 thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}). 792 To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt 793 (1981): 810 794 \begin{align} \label{eq:zdfddm_f} 811 795 A_f^{vS} &= \begin{cases} … … 821 805 \includegraphics[width=0.99\textwidth]{Fig_zdfddm} 822 806 \caption{ \protect\label{fig:zdfddm} 823 From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ 824 and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy 825 curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves 826 $A^{\ast v} = 10^{-4}~m^2.s^{-1}$ ; (b) diapycnal diffusivities $A_d^{vT}$ and 827 $A_d^{vS}$ for temperature and salt in regions of diffusive convection. Heavy 828 curves denote the Federov parameterisation and thin curves the Kelley 829 parameterisation. The latter is not implemented in \NEMO. } 807 From \citet{Merryfield1999} : 808 (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering. 809 Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$; 810 (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in regions of diffusive convection. 811 Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation. 812 The latter is not implemented in \NEMO. } 830 813 \end{center} \end{figure} 831 814 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 832 815 833 The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio 834 $\alpha F_T /\beta F_S \approx 0.7$ of buoyancy flux of heat to buoyancy 835 flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). Following \citet{Merryfield1999}, 836 we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$. 837 838 To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested byFederov (1988) is used:816 The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx 0.7$ of 817 buoyancy flux of heat to buoyancy flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). 818 Following \citet{Merryfield1999}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$. 819 820 To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested by 821 Federov (1988) is used: 839 822 \begin{align} \label{eq:zdfddm_d} 840 823 A_d^{vT} &= \begin{cases} … … 853 836 \end{align} 854 837 855 The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ 856 are illustrated in \autoref{fig:zdfddm}. Implementing this requires computing 857 $R_\rho$ at each grid point on every time step. This is done in \mdl{eosbn2} at the 858 same time as $N^2$ is computed. This avoids duplication in the computation of 859 $\alpha$ and $\beta$ (which is usually quite expensive).838 The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ are illustrated in 839 \autoref{fig:zdfddm}. 840 Implementing this requires computing $R_\rho$ at each grid point on every time step. 841 This is done in \mdl{eosbn2} at the same time as $N^2$ is computed. 842 This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive). 860 843 861 844 % ================================================================ … … 870 853 %-------------------------------------------------------------------------------------------------------------- 871 854 872 Options to define the top and bottom friction are defined through the 873 The bottom friction represents the friction generated by the bathymetry. 874 The top friction represents the friction generated by the ice shelf/ocean interface. 875 As the friction processes at the top and bottom are treated in similar way, 855 Options to define the top and bottom friction are defined through the \ngn{nambfr} namelist variables. 856 The bottom friction represents the friction generated by the bathymetry. 857 The top friction represents the friction generated by the ice shelf/ocean interface. 858 As the friction processes at the top and bottom are treated in similar way, 876 859 only the bottom friction is described in detail below. 877 860 878 861 879 Both the surface momentum flux (wind stress) and the bottom momentum 880 flux (bottom friction) enter the equations as a condition on the vertical 881 diffusive flux.For the bottom boundary layer, one has:862 Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as 863 a condition on the vertical diffusive flux. 864 For the bottom boundary layer, one has: 882 865 \begin{equation} \label{eq:zdfbfr_flux} 883 866 A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 884 867 \end{equation} 885 where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum 886 outside the logarithmic turbulent boundary layer (thickness of the order of 887 1~m in the ocean). How ${\cal F}_h^{\textbf U}$ influences the interior depends on the 888 vertical resolution of the model near the bottom relative to the Ekman layer 889 depth. For example, in order to obtain an Ekman layer depth 890 $d = \sqrt{2\;A^{vm}} / f = 50$~m, one needs a vertical diffusion coefficient 891 $A^{vm} = 0.125$~m$^2$s$^{-1}$ (for a Coriolis frequency 892 $f = 10^{-4}$~m$^2$s$^{-1}$). With a background diffusion coefficient 893 $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m. 894 When the vertical mixing coefficient is this small, using a flux condition is 895 equivalent to entering the viscous forces (either wind stress or bottom friction) 896 as a body force over the depth of the top or bottom model layer. To illustrate 897 this, consider the equation for $u$ at $k$, the last ocean level: 868 where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside 869 the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean). 870 How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near 871 the bottom relative to the Ekman layer depth. 872 For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m, 873 one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$ 874 (for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$). 875 With a background diffusion coefficient $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m. 876 When the vertical mixing coefficient is this small, using a flux condition is equivalent to 877 entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or 878 bottom model layer. 879 To illustrate this, consider the equation for $u$ at $k$, the last ocean level: 898 880 \begin{equation} \label{eq:zdfbfr_flux2} 899 881 \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}} 900 882 \end{equation} 901 If the bottom layer thickness is 200~m, the Ekman transport will 902 be distributed over that depth. On the other hand, if the vertical resolution 903 is high (1~m or less) and a turbulent closure model is used, the turbulent 904 Ekman layer will be represented explicitly by the model. However, the 905 logarithmic layer is never represented in current primitive equation model 906 applications: it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $. 907 Two choices are available in \NEMO: a linear and a quadratic bottom friction. 908 Note that in both cases, the rotation between the interior velocity and the 909 bottom friction is neglected in the present release of \NEMO. 910 911 In the code, the bottom friction is imposed by adding the trend due to the bottom 912 friction to the general momentum trend in \mdl{dynbfr}. For the time-split surface 913 pressure gradient algorithm, the momentum trend due to the barotropic component 914 needs to be handled separately. For this purpose it is convenient to compute and 915 store coefficients which can be simply combined with bottom velocities and geometric 916 values to provide the momentum trend due to bottom friction. 917 These coefficients are computed in \mdl{zdfbfr} and generally take the form 918 $c_b^{\textbf U}$ where: 883 If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth. 884 On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used, 885 the turbulent Ekman layer will be represented explicitly by the model. 886 However, the logarithmic layer is never represented in current primitive equation model applications: 887 it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $. 888 Two choices are available in \NEMO: a linear and a quadratic bottom friction. 889 Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in 890 the present release of \NEMO. 891 892 In the code, the bottom friction is imposed by adding the trend due to the bottom friction to 893 the general momentum trend in \mdl{dynbfr}. 894 For the time-split surface pressure gradient algorithm, the momentum trend due to 895 the barotropic component needs to be handled separately. 896 For this purpose it is convenient to compute and store coefficients which can be simply combined with 897 bottom velocities and geometric values to provide the momentum trend due to bottom friction. 898 These coefficients are computed in \mdl{zdfbfr} and generally take the form $c_b^{\textbf U}$ where: 919 899 \begin{equation} \label{eq:zdfbfr_bdef} 920 900 \frac{\partial {\textbf U_h}}{\partial t} = … … 929 909 \label{subsec:ZDF_bfr_linear} 930 910 931 The linear bottom friction parameterisation (including the special case 932 of a free-slip condition) assumes that the bottom friction 933 is proportional to the interior velocity (i.e. the velocity of the last 934 model level): 911 The linear bottom friction parameterisation (including the special case of a free-slip condition) assumes that 912 the bottom friction is proportional to the interior velocity (i.e. the velocity of the last model level): 935 913 \begin{equation} \label{eq:zdfbfr_linear} 936 914 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 937 915 \end{equation} 938 where $r$ is a friction coefficient expressed in ms$^{-1}$. 939 This coefficient is generally estimated by setting a typical decay time 940 $\tau$ in the deep ocean, 941 and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted 942 values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}. 943 A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used 944 in quasi-geostrophic models. One may consider the linear friction as an 945 approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982}, 946 Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed 947 of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, and assuming an ocean depth 948 $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$. 949 This is the default value used in \NEMO. It corresponds to a decay time scale 950 of 115~days. It can be changed by specifying \np{rn\_bfri1} (namelist parameter). 951 952 For the linear friction case the coefficients defined in the general 953 expression \autoref{eq:zdfbfr_bdef} are: 916 where $r$ is a friction coefficient expressed in ms$^{-1}$. 917 This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean, 918 and setting $r = H / \tau$, where $H$ is the ocean depth. 919 Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}. 920 A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models. 921 One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ 922 (\citet{Gill1982}, Eq. 9.6.6). 923 For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, 924 and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$. 925 This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days. 926 It can be changed by specifying \np{rn\_bfri1} (namelist parameter). 927 928 For the linear friction case the coefficients defined in the general expression \autoref{eq:zdfbfr_bdef} are: 954 929 \begin{equation} \label{eq:zdfbfr_linbfr_b} 955 930 \begin{split} … … 958 933 \end{split} 959 934 \end{equation} 960 When \np{nn\_botfr}\forcode{ = 1}, the value of $r$ used is \np{rn\_bfri1}. 961 Setting \np{nn\_botfr}\forcode{ = 0} is equivalent to setting $r=0$ and leads to a free-slip 962 bottom boundary condition. These values are assigned in \mdl{zdfbfr}. 963 From v3.2 onwards there is support for local enhancement of these values 964 via an externally defined 2D mask array (\np{ln\_bfr2d}\forcode{ = .true.}) given 965 in the \ifile{bfr\_coef} input NetCDF file. The mask values should vary from 0 to 1. 966 Locations with a non-zero mask value will have the friction coefficient increased 967 by $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri1}. 935 When \np{nn\_botfr}\forcode{ = 1}, the value of $r$ used is \np{rn\_bfri1}. 936 Setting \np{nn\_botfr}\forcode{ = 0} is equivalent to setting $r=0$ and 937 leads to a free-slip bottom boundary condition. 938 These values are assigned in \mdl{zdfbfr}. 939 From v3.2 onwards there is support for local enhancement of these values via an externally defined 2D mask array 940 (\np{ln\_bfr2d}\forcode{ = .true.}) given in the \ifile{bfr\_coef} input NetCDF file. 941 The mask values should vary from 0 to 1. 942 Locations with a non-zero mask value will have the friction coefficient increased by 943 $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri1}. 968 944 969 945 % ------------------------------------------------------------------------------------------------------------- … … 973 949 \label{subsec:ZDF_bfr_nonlinear} 974 950 975 The non-linear bottom friction parameterisation assumes that the bottom 976 friction is quadratic: 951 The non-linear bottom friction parameterisation assumes that the bottom friction is quadratic: 977 952 \begin{equation} \label{eq:zdfbfr_nonlinear} 978 953 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 979 954 }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b 980 955 \end{equation} 981 where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy 982 due to tides, internal waves breaking and other short time scale currents. 983 A typical value of the drag coefficient is $C_D = 10^{-3} $. As an example, 984 the CME experiment \citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and 985 $e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992} 986 uses $C_D = 1.4\;10^{-3}$ and $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$. 987 The CME choices have been set as default values (\np{rn\_bfri2} and \np{rn\_bfeb2} 988 namelist parameters). 989 990 As for the linear case, the bottom friction is imposed in the code by 991 adding the trend due to the bottom friction to the general momentum trend 992 in \mdl{dynbfr}. 993 For the non-linear friction case the terms 994 computed in \mdl{zdfbfr} are: 956 where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy due to tides, 957 internal waves breaking and other short time scale currents. 958 A typical value of the drag coefficient is $C_D = 10^{-3} $. 959 As an example, the CME experiment \citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and 960 $e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992} uses $C_D = 1.4\;10^{-3}$ and 961 $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$. 962 The CME choices have been set as default values (\np{rn\_bfri2} and \np{rn\_bfeb2} namelist parameters). 963 964 As for the linear case, the bottom friction is imposed in the code by adding the trend due to 965 the bottom friction to the general momentum trend in \mdl{dynbfr}. 966 For the non-linear friction case the terms computed in \mdl{zdfbfr} are: 995 967 \begin{equation} \label{eq:zdfbfr_nonlinbfr} 996 968 \begin{split} … … 1000 972 \end{equation} 1001 973 1002 The coefficients that control the strength of the non-linear bottom friction are 1003 initialised as namelist parameters:$C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}.1004 Note for applications which treat tides explicitly a low or even zero value of 1005 \np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$ is possible 1006 via an externally defined 2D mask array (\np{ln\_bfr2d}\forcode{ = .true.}). This works in the same way 1007 as for the linear bottom friction case with non-zero masked locations increased by974 The coefficients that control the strength of the non-linear bottom friction are initialised as namelist parameters: 975 $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}. 976 Note for applications which treat tides explicitly a low or even zero value of \np{rn\_bfeb2} is recommended. 977 From v3.2 onwards a local enhancement of $C_D$ is possible via an externally defined 2D mask array 978 (\np{ln\_bfr2d}\forcode{ = .true.}). 979 This works in the same way as for the linear bottom friction case with non-zero masked locations increased by 1008 980 $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri2}. 1009 981 … … 1015 987 \label{subsec:ZDF_bfr_loglayer} 1016 988 1017 In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally 1018 enhanced using a "law of the wall" scaling. If \np{ln\_loglayer} = .true., $C_D$ is no 1019 longer constant but is related to the thickness of the last wet layer in each column by: 1020 989 In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally enhanced using 990 a "law of the wall" scaling. 991 If \np{ln\_loglayer} = .true., $C_D$ is no longer constant but is related to the thickness of 992 the last wet layer in each column by: 1021 993 \begin{equation} 1022 994 C_D = \left ( {\kappa \over {\rm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2 1023 995 \end{equation} 1024 996 1025 \noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness 1026 length provided viathe namelist.997 \noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness length provided via 998 the namelist. 1027 999 1028 1000 For stability, the drag coefficient is bounded such that it is kept greater or equal to 1029 the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional 1030 namelist parameter: \np{rn\_bfri2\_max}, i.e.: 1031 1001 the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional namelist parameter: 1002 \np{rn\_bfri2\_max}, i.e.: 1032 1003 \begin{equation} 1033 1004 rn\_bfri2 \leq C_D \leq rn\_bfri2\_max 1034 1005 \end{equation} 1035 1006 1036 \noindent Note also that a log-layer enhancement can also be applied to the top boundary 1037 friction if under ice-shelf cavities are in use (\np{ln\_isfcav}\forcode{ = .true.}). In this case, the 1038 relevant namelist parameters are \np{rn\_tfrz0}, \np{rn\_tfri2} 1039 and \np{rn\_tfri2\_max}. 1007 \noindent Note also that a log-layer enhancement can also be applied to the top boundary friction if 1008 under ice-shelf cavities are in use (\np{ln\_isfcav}\forcode{ = .true.}). 1009 In this case, the relevant namelist parameters are \np{rn\_tfrz0}, \np{rn\_tfri2} and \np{rn\_tfri2\_max}. 1040 1010 1041 1011 % ------------------------------------------------------------------------------------------------------------- … … 1045 1015 \label{subsec:ZDF_bfr_stability} 1046 1016 1047 Some care needs to exercised over the choice of parameters to ensure that the 1048 implementation of bottom friction does not induce numerical instability. For 1049 the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2} 1050 is: 1017 Some care needs to exercised over the choice of parameters to ensure that the implementation of 1018 bottom friction does not induce numerical instability. 1019 For the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2} is: 1051 1020 \begin{equation} \label{eq:Eqn_bfrstab} 1052 1021 \begin{split} … … 1055 1024 \end{split} 1056 1025 \end{equation} 1057 \noindent where linear bottom friction and a leapfrog timestep have been assumed. 1026 \noindent where linear bottom friction and a leapfrog timestep have been assumed. 1058 1027 To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have: 1059 1028 \begin{equation} … … 1064 1033 r\frac{2\rdt}{e_{3u}} < 1 \qquad \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ 1065 1034 \end{equation} 1066 This same inequality can also be derived in the non-linear bottom friction case 1067 if a velocity of 1 m.s$^{-1}$ is assumed. Alternatively, this criterion can be 1068 rearranged to suggest a minimum bottom box thickness to ensure stability:1035 This same inequality can also be derived in the non-linear bottom friction case if 1036 a velocity of 1 m.s$^{-1}$ is assumed. 1037 Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability: 1069 1038 \begin{equation} 1070 1039 e_{3u} > 2\;r\;\rdt 1071 1040 \end{equation} 1072 \noindent which it may be necessary to impose if partial steps are being used. 1073 For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then 1074 $e_{3u}$ should be greater than 3.6 m. For most applications, with physically 1075 sensible parameters these restrictions should not be of concern. But 1076 caution may be necessary if attempts are made to locally enhance the bottom 1077 friction parameters. 1078 To ensure stability limits are imposed on the bottom friction coefficients both during 1079 initialisation and at each time step. Checks at initialisation are made in \mdl{zdfbfr} 1080 (assuming a 1 m.s$^{-1}$ velocity in the non-linear case). 1081 The number of breaches of the stability criterion are reported as well as the minimum 1082 and maximum values that have been set. The criterion is also checked at each time step, 1083 using the actual velocity, in \mdl{dynbfr}. Values of the bottom friction coefficient are 1084 reduced as necessary to ensure stability; these changes are not reported. 1041 \noindent which it may be necessary to impose if partial steps are being used. 1042 For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m. 1043 For most applications, with physically sensible parameters these restrictions should not be of concern. 1044 But caution may be necessary if attempts are made to locally enhance the bottom friction parameters. 1045 To ensure stability limits are imposed on the bottom friction coefficients both 1046 during initialisation and at each time step. 1047 Checks at initialisation are made in \mdl{zdfbfr} (assuming a 1 m.s$^{-1}$ velocity in the non-linear case). 1048 The number of breaches of the stability criterion are reported as well as 1049 the minimum and maximum values that have been set. 1050 The criterion is also checked at each time step, using the actual velocity, in \mdl{dynbfr}. 1051 Values of the bottom friction coefficient are reduced as necessary to ensure stability; 1052 these changes are not reported. 1085 1053 1086 1054 Limits on the bottom friction coefficient are not imposed if the user has elected to 1087 handle the bottom friction implicitly (see \autoref{subsec:ZDF_bfr_imp}). The number of potential1088 breaches of the explicit stability criterion are still reported for information purposes.1055 handle the bottom friction implicitly (see \autoref{subsec:ZDF_bfr_imp}). 1056 The number of potential breaches of the explicit stability criterion are still reported for information purposes. 1089 1057 1090 1058 % ------------------------------------------------------------------------------------------------------------- … … 1094 1062 \label{subsec:ZDF_bfr_imp} 1095 1063 1096 An optional implicit form of bottom friction has been implemented to improve 1097 model stability. We recommend this option for shelf sea and coastal ocean applications, especially 1098 for split-explicit time splitting. This option can be invoked by setting \np{ln\_bfrimp} 1099 to \forcode{.true.} in the \textit{nambfr} namelist. This option requires \np{ln\_zdfexp} to be \forcode{.false.} 1100 in the \textit{namzdf} namelist. 1101 1102 This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, the 1103 bottom boundary condition is implemented implicitly. 1064 An optional implicit form of bottom friction has been implemented to improve model stability. 1065 We recommend this option for shelf sea and coastal ocean applications, especially for split-explicit time splitting. 1066 This option can be invoked by setting \np{ln\_bfrimp} to \forcode{.true.} in the \textit{nambfr} namelist. 1067 This option requires \np{ln\_zdfexp} to be \forcode{.false.} in the \textit{namzdf} namelist. 1068 1069 This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, 1070 the bottom boundary condition is implemented implicitly. 1104 1071 1105 1072 \begin{equation} \label{eq:dynzdf_bfr} … … 1108 1075 \end{equation} 1109 1076 1110 where $mbk$ is the layer number of the bottom wet layer. superscript $n+1$ means the velocity used in the1111 friction formula is to be calculated, so, it is implicit.1112 1113 If split-explicit time splitting is used, care must be taken to avoid the double counting of 1114 the bottom friction in the 2-D barotropic momentum equations. As NEMO only updates the barotropic1115 pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, we need to remove 1116 the bottom friction induced by these two terms which has been included in the 3-D momentum trend1117 and update it with the latest value. On the other hand, the bottom friction contributed by the1118 other terms (e.g. the advection term, viscosity term) has been included in the 3-D momentum equations1119 and should not be added in the 2-D barotropic mode. 1120 1121 The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the 1122 following:1077 where $mbk$ is the layer number of the bottom wet layer. 1078 Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so, it is implicit. 1079 1080 If split-explicit time splitting is used, care must be taken to avoid the double counting of the bottom friction in 1081 the 2-D barotropic momentum equations. 1082 As NEMO only updates the barotropic pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, 1083 we need to remove the bottom friction induced by these two terms which has been included in the 3-D momentum trend 1084 and update it with the latest value. 1085 On the other hand, the bottom friction contributed by the other terms 1086 (e.g. the advection term, viscosity term) has been included in the 3-D momentum equations and 1087 should not be added in the 2-D barotropic mode. 1088 1089 The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the following: 1123 1090 1124 1091 \begin{equation} \label{eq:dynspg_ts_bfr1} … … 1132 1099 \end{equation} 1133 1100 1134 where $\textbf{T}$ is the vertical integrated 3-D momentum trend. We assume the leap-frog time-stepping 1135 is used here. $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step. 1136 $c_{b}$ is the friction coefficient. $\eta$ is the sea surface level calculated in the barotropic loops 1137 while $\eta^{'}$ is the sea surface level used in the 3-D baroclinic mode. $\textbf{u}_{b}$ is the bottom 1138 layer horizontal velocity. 1101 where $\textbf{T}$ is the vertical integrated 3-D momentum trend. 1102 We assume the leap-frog time-stepping is used here. 1103 $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step. 1104 $c_{b}$ is the friction coefficient. 1105 $\eta$ is the sea surface level calculated in the barotropic loops while $\eta^{'}$ is the sea surface level used in 1106 the 3-D baroclinic mode. 1107 $\textbf{u}_{b}$ is the bottom layer horizontal velocity. 1139 1108 1140 1109 … … 1148 1117 \label{subsec:ZDF_bfr_ts} 1149 1118 1150 When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the 1151 bottom velocity at the before time step is used. This velocity includes both the 1152 baroclinic and barotropic components which is appropriate when using either the 1153 explicit or filtered surface pressure gradient algorithms (\key{dynspg\_exp} or 1154 \key{dynspg\_flt}). Extra attention is required, however, when using 1155 split-explicit time stepping (\key{dynspg\_ts}). In this case the free surface 1156 equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, while the three 1157 dimensional prognostic variables are solved with the longer time step 1158 of \np{rn\_rdt} seconds. The trend in the barotropic momentum due to bottom 1159 friction appropriate to this method is that given by the selected parameterisation 1160 ($i.e.$ linear or non-linear bottom friction) computed with the evolving velocities 1161 at each barotropic timestep. 1162 1163 In the case of non-linear bottom friction, we have elected to partially linearise 1164 the problem by keeping the coefficients fixed throughout the barotropic 1165 time-stepping to those computed in \mdl{zdfbfr} using the now timestep. 1119 When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, 1120 the bottom velocity at the before time step is used. 1121 This velocity includes both the baroclinic and barotropic components which is appropriate when 1122 using either the explicit or filtered surface pressure gradient algorithms 1123 (\key{dynspg\_exp} or \key{dynspg\_flt}). 1124 Extra attention is required, however, when using split-explicit time stepping (\key{dynspg\_ts}). 1125 In this case the free surface equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, 1126 while the three dimensional prognostic variables are solved with the longer time step of \np{rn\_rdt} seconds. 1127 The trend in the barotropic momentum due to bottom friction appropriate to this method is that given by 1128 the selected parameterisation ($i.e.$ linear or non-linear bottom friction) computed with 1129 the evolving velocities at each barotropic timestep. 1130 1131 In the case of non-linear bottom friction, we have elected to partially linearise the problem by 1132 keeping the coefficients fixed throughout the barotropic time-stepping to those computed in 1133 \mdl{zdfbfr} using the now timestep. 1166 1134 This decision allows an efficient use of the $c_b^{\vect{U}}$ coefficients to: 1167 1135 1168 1136 \begin{enumerate} 1169 \item On entry to \rou{dyn\_spg\_ts}, remove the contribution of the before 1170 barotropic velocity to the bottom friction component of the vertically 1171 integrated momentum trend. Note the same stability check that is carried out 1172 on the bottom friction coefficient in \mdl{dynbfr} has to be applied here to 1173 ensure that the trend removed matches that which was added in \mdl{dynbfr}. 1174 \item At each barotropic step, compute the contribution of the current barotropic 1175 velocity to the trend due to bottom friction. Add this contribution to the 1176 vertically integrated momentum trend. This contribution is handled implicitly which 1177 eliminates the need to impose a stability criteria on the values of the bottom friction 1178 coefficient within the barotropic loop. 1137 \item On entry to \rou{dyn\_spg\_ts}, remove the contribution of the before barotropic velocity to 1138 the bottom friction component of the vertically integrated momentum trend. 1139 Note the same stability check that is carried out on the bottom friction coefficient in \mdl{dynbfr} has to 1140 be applied here to ensure that the trend removed matches that which was added in \mdl{dynbfr}. 1141 \item At each barotropic step, compute the contribution of the current barotropic velocity to 1142 the trend due to bottom friction. 1143 Add this contribution to the vertically integrated momentum trend. 1144 This contribution is handled implicitly which eliminates the need to impose a stability criteria on 1145 the values of the bottom friction coefficient within the barotropic loop. 1179 1146 \end{enumerate} 1180 1147 1181 Note that the use of an implicit formulation within the barotropic loop 1182 for the bottom friction trend means that any limiting of the bottom friction coefficient 1183 in \mdl{dynbfr} does not adversely affect the solution when using split-explicit time 1184 splitting. This is because the major contribution to bottom friction is likely to come from 1185 the barotropic component which uses the unrestricted value of the coefficient. However, if the 1186 limiting is thought to be having a major effect (a more likely prospect in coastal and shelf seas 1187 applications) then the fully implicit form of the bottom friction should be used (see \autoref{subsec:ZDF_bfr_imp} ) 1148 Note that the use of an implicit formulation within the barotropic loop for the bottom friction trend means that 1149 any limiting of the bottom friction coefficient in \mdl{dynbfr} does not adversely affect the solution when 1150 using split-explicit time splitting. 1151 This is because the major contribution to bottom friction is likely to come from the barotropic component which 1152 uses the unrestricted value of the coefficient. 1153 However, if the limiting is thought to be having a major effect 1154 (a more likely prospect in coastal and shelf seas applications) then 1155 the fully implicit form of the bottom friction should be used (see \autoref{subsec:ZDF_bfr_imp}) 1188 1156 which can be selected by setting \np{ln\_bfrimp} $=$ \forcode{.true.}. 1189 1157 … … 1193 1161 \end{equation} 1194 1162 where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height), 1195 $c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and $RHS$ represents 1196 all the components to the vertically integrated momentum trend except for that due to bottom friction. 1163 $c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and 1164 $RHS$ represents all the components to the vertically integrated momentum trend except for 1165 that due to bottom friction. 1197 1166 1198 1167 … … 1218 1187 1219 1188 Options are defined through the \ngn{namzdf\_tmx} namelist variables. 1220 The parameterization of tidal mixing follows the general formulation for 1221 the vertical eddy diffusivity proposed by \citet{St_Laurent_al_GRL02} and 1222 first introduced in an OGCM by \citep{Simmons_al_OM04}. 1223 In this formulation an additional vertical diffusivity resulting from internal tide breaking, 1224 $A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, the energy transfer from barotropic 1225 tides to baroclinic tides : 1189 The parameterization of tidal mixing follows the general formulation for the vertical eddy diffusivity proposed by 1190 \citet{St_Laurent_al_GRL02} and first introduced in an OGCM by \citep{Simmons_al_OM04}. 1191 In this formulation an additional vertical diffusivity resulting from internal tide breaking, 1192 $A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, 1193 the energy transfer from barotropic tides to baroclinic tides: 1226 1194 \begin{equation} \label{eq:Ktides} 1227 1195 A^{vT}_{tides} = q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 } 1228 1196 \end{equation} 1229 where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency 1230 (see \autoref{subsec:TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency,1231 and $F(z)$ the vertical structure function. 1232 1233 The mixing efficiency of turbulence is set by $\Gamma$ (\np{rn\_me} namelist parameter) 1234 and is usually taken to be the canonical value of $\Gamma = 0.2$ (Osborn 1980). 1235 The tidal dissipation efficiency is given by the parameter $q$ (\np{rn\_tfe} namelist parameter) 1236 represents the part of the internal wave energy flux $E(x, y)$ that is dissipated locally, 1237 with the remaining $1-q$ radiating away as low mode internal waves and 1238 contributing to the background internal wave field. A value of $q=1/3$ is typically used 1239 \citet{St_Laurent_al_GRL02}.1240 The vertical structure function $F(z)$ models the distribution of the turbulent mixing in the vertical. 1241 It is implemented as a simple exponential decaying upward away from the bottom, 1242 with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, with atypical value of $500\,m$) \citep{St_Laurent_Nash_DSR04},1197 where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 1198 $\rho$ the density, $q$ the tidal dissipation efficiency, and $F(z)$ the vertical structure function. 1199 1200 The mixing efficiency of turbulence is set by $\Gamma$ (\np{rn\_me} namelist parameter) and 1201 is usually taken to be the canonical value of $\Gamma = 0.2$ (Osborn 1980). 1202 The tidal dissipation efficiency is given by the parameter $q$ (\np{rn\_tfe} namelist parameter) 1203 represents the part of the internal wave energy flux $E(x, y)$ that is dissipated locally, 1204 with the remaining $1-q$ radiating away as low mode internal waves and 1205 contributing to the background internal wave field. 1206 A value of $q=1/3$ is typically used \citet{St_Laurent_al_GRL02}. 1207 The vertical structure function $F(z)$ models the distribution of the turbulent mixing in the vertical. 1208 It is implemented as a simple exponential decaying upward away from the bottom, 1209 with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, 1210 with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04}, 1243 1211 \begin{equation} \label{eq:Fz} 1244 1212 F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) } … … 1246 1214 and is normalized so that vertical integral over the water column is unity. 1247 1215 1248 The associated vertical viscosity is calculated from the vertical 1249 diffusivity assuming a Prandtl number of 1, $i.e.$ $A^{vm}_{tides}=A^{vT}_{tides}$. 1250 In the limit of $N \rightarrow 0$ (or becoming negative), the vertical diffusivity 1251 is capped at $300\,cm^2/s$ and impose a lower limit on $N^2$ of \np{rn\_n2min} 1252 usually set to $10^{-8} s^{-2}$. These bounds are usually rarely encountered. 1253 1254 The internal wave energy map, $E(x, y)$ in \autoref{eq:Ktides}, is derived 1255 from a barotropic model of the tides utilizing a parameterization of the 1256 conversion of barotropic tidal energy into internal waves. 1257 The essential goal of the parameterization is to represent the momentum 1258 exchange between the barotropic tides and the unrepresented internal waves 1259 induced by the tidal flow over rough topography in a stratified ocean. 1260 In the current version of \NEMO, the map is built from the output of 1216 The associated vertical viscosity is calculated from the vertical diffusivity assuming a Prandtl number of 1, 1217 $i.e.$ $A^{vm}_{tides}=A^{vT}_{tides}$. 1218 In the limit of $N \rightarrow 0$ (or becoming negative), the vertical diffusivity is capped at $300\,cm^2/s$ and 1219 impose a lower limit on $N^2$ of \np{rn\_n2min} usually set to $10^{-8} s^{-2}$. 1220 These bounds are usually rarely encountered. 1221 1222 The internal wave energy map, $E(x, y)$ in \autoref{eq:Ktides}, is derived from a barotropic model of 1223 the tides utilizing a parameterization of the conversion of barotropic tidal energy into internal waves. 1224 The essential goal of the parameterization is to represent the momentum exchange between the barotropic tides and 1225 the unrepresented internal waves induced by the tidal flow over rough topography in a stratified ocean. 1226 In the current version of \NEMO, the map is built from the output of 1261 1227 the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}. 1262 This model provides the dissipation associated with internal wave energy for the M2 and K1 1263 tides component (\autoref{fig:ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated 1264 as being $1/4$ of the M2 one. The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$. 1265 Its global mean value is $1.1$ TW, in agreement with independent estimates 1266 \citep{Egbert_Ray_Nat00, Egbert_Ray_JGR01}. 1228 This model provides the dissipation associated with internal wave energy for the M2 and K1 tides component 1229 (\autoref{fig:ZDF_M2_K1_tmx}). 1230 The S2 dissipation is simply approximated as being $1/4$ of the M2 one. 1231 The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$. 1232 Its global mean value is $1.1$ TW, 1233 in agreement with independent estimates \citep{Egbert_Ray_Nat00, Egbert_Ray_JGR01}. 1267 1234 1268 1235 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 1281 1248 1282 1249 When the Indonesian Through Flow (ITF) area is included in the model domain, 1283 a specific treatment of tidal induced mixing in this area can be used. 1284 It is activated through the namelist logical \np{ln\_tmx\_itf}, and the user must provide 1285 an input NetCDF file, \ifile{mask\_itf}, which contains a mask array defining the ITF area 1286 where the specific treatment is applied. 1287 1288 When \np{ln\_tmx\_itf}\forcode{ = .true.}, the two key parameters $q$ and $F(z)$ are adjusted following 1250 a specific treatment of tidal induced mixing in this area can be used. 1251 It is activated through the namelist logical \np{ln\_tmx\_itf}, and the user must provide an input NetCDF file, 1252 \ifile{mask\_itf}, which contains a mask array defining the ITF area where the specific treatment is applied. 1253 1254 When \np{ln\_tmx\_itf}\forcode{ = .true.}, the two key parameters $q$ and $F(z)$ are adjusted following 1289 1255 the parameterisation developed by \citet{Koch-Larrouy_al_GRL07}: 1290 1256 1291 First, the Indonesian archipelago is a complex geographic region 1292 with a series of large, deep, semi-enclosed basins connected via 1293 numerous narrow straits. Once generated, internal tides remain 1294 confined within this semi-enclosed area and hardly radiate away. 1295 Therefore all the internal tides energy is consumed within this area. 1257 First, the Indonesian archipelago is a complex geographic region with a series of 1258 large, deep, semi-enclosed basins connected via numerous narrow straits. 1259 Once generated, internal tides remain confined within this semi-enclosed area and hardly radiate away. 1260 Therefore all the internal tides energy is consumed within this area. 1296 1261 So it is assumed that $q = 1$, $i.e.$ all the energy generated is available for mixing. 1297 Note that for test purposed, the ITF tidal dissipation efficiency is a 1298 namelist parameter (\np{rn\_tfe\_itf}). A value of $1$ or close to is 1299 this recommended for this parameter. 1300 1301 Second, the vertical structure function, $F(z)$, is no more associated 1302 with a bottom intensification of the mixing, but with a maximum of 1303 energy available within the thermocline. \citet{Koch-Larrouy_al_GRL07} 1304 have suggested that the vertical distribution of the energy dissipation 1305 proportional to $N^2$ below the core of the thermocline and to $N$ above. 1262 Note that for test purposed, the ITF tidal dissipation efficiency is a namelist parameter (\np{rn\_tfe\_itf}). 1263 A value of $1$ or close to is this recommended for this parameter. 1264 1265 Second, the vertical structure function, $F(z)$, is no more associated with a bottom intensification of the mixing, 1266 but with a maximum of energy available within the thermocline. 1267 \citet{Koch-Larrouy_al_GRL07} have suggested that the vertical distribution of 1268 the energy dissipation proportional to $N^2$ below the core of the thermocline and to $N$ above. 1306 1269 The resulting $F(z)$ is: 1307 1270 \begin{equation} \label{eq:Fz_itf} … … 1313 1276 1314 1277 Averaged over the ITF area, the resulting tidal mixing coefficient is $1.5\,cm^2/s$, 1315 which agrees with the independent estimates inferred from observations. 1316 Introduced in a regional OGCM, the parameterization improves the water mass 1317 characteristics in the different Indonesian seas, suggesting that the horizontal 1318 and vertical distributions of the mixing are adequately prescribed 1319 \citep{Koch-Larrouy_al_GRL07, Koch-Larrouy_al_OD08a, Koch-Larrouy_al_OD08b}. 1320 Note also that such a parameterisation has a significant impact on the behaviour 1321 of global coupled GCMs \citep{Koch-Larrouy_al_CD10}. 1278 which agrees with the independent estimates inferred from observations. 1279 Introduced in a regional OGCM, the parameterization improves the water mass characteristics in 1280 the different Indonesian seas, suggesting that the horizontal and vertical distributions of 1281 the mixing are adequately prescribed \citep{Koch-Larrouy_al_GRL07, Koch-Larrouy_al_OD08a, Koch-Larrouy_al_OD08b}. 1282 Note also that such a parameterisation has a significant impact on the behaviour of 1283 global coupled GCMs \citep{Koch-Larrouy_al_CD10}. 1322 1284 1323 1285 … … 1333 1295 %-------------------------------------------------------------------------------------------------------------- 1334 1296 1335 The parameterization of mixing induced by breaking internal waves is a generalization 1336 of the approach originally proposed by \citet{St_Laurent_al_GRL02}. 1337 A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed, 1297 The parameterization of mixing induced by breaking internal waves is a generalization of 1298 the approach originally proposed by \citet{St_Laurent_al_GRL02}. 1299 A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed, 1338 1300 and the resulting diffusivity is obtained as 1339 1301 \begin{equation} \label{eq:Kwave} 1340 1302 A^{vT}_{wave} = R_f \,\frac{ \epsilon }{ \rho \, N^2 } 1341 1303 \end{equation} 1342 where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution 1343 of the energy available for mixing. If the \np{ln\_mevar} namelist parameter is set to false, 1344 the mixing efficiency is taken as constant and equal to 1/6 \citep{Osborn_JPO80}. 1345 In the opposite (recommended) case, $R_f$ is instead a function of the turbulence intensity parameter 1346 $Re_b = \frac{ \epsilon}{\nu \, N^2}$, with $\nu$ the molecular viscosity of seawater, 1347 following the model of \cite{Bouffard_Boegman_DAO2013} 1348 and the implementation of \cite{de_lavergne_JPO2016_efficiency}. 1349 Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when the mixing efficiency is constant. 1304 where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of 1305 the energy available for mixing. 1306 If the \np{ln\_mevar} namelist parameter is set to false, the mixing efficiency is taken as constant and 1307 equal to 1/6 \citep{Osborn_JPO80}. 1308 In the opposite (recommended) case, $R_f$ is instead a function of 1309 the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$, 1310 with $\nu$ the molecular viscosity of seawater, following the model of \cite{Bouffard_Boegman_DAO2013} and 1311 the implementation of \cite{de_lavergne_JPO2016_efficiency}. 1312 Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when 1313 the mixing efficiency is constant. 1350 1314 1351 1315 In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary 1352 as a function of $Re_b$ by setting the \np{ln\_tsdiff} parameter to true, a recommended choice ).1353 This parameterization of differential mixing, due to \cite{Jackson_Rehmann_JPO2014}, 1316 as a function of $Re_b$ by setting the \np{ln\_tsdiff} parameter to true, a recommended choice. 1317 This parameterization of differential mixing, due to \cite{Jackson_Rehmann_JPO2014}, 1354 1318 is implemented as in \cite{de_lavergne_JPO2016_efficiency}. 1355 1319 1356 The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$, is constructed1357 from three static maps of column-integrated internal wave energy dissipation, $E_{cri}(i,j)$, 1358 $E_{ pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures1320 The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$, 1321 is constructed from three static maps of column-integrated internal wave energy dissipation, 1322 $E_{cri}(i,j)$, $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures 1359 1323 (de Lavergne et al., in prep): 1360 1324 \begin{align*} … … 1363 1327 F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} } 1364 1328 \end{align*} 1365 In the above formula, $h_{ab}$ denotes the height above bottom, 1329 In the above formula, $h_{ab}$ denotes the height above bottom, 1366 1330 $h_{wkb}$ denotes the WKB-stretched height above bottom, defined by 1367 1331 \begin{equation*} 1368 1332 h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz' } \; , 1369 1333 \end{equation*} 1370 The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist) controls the stratification-dependence of the pycnocline-intensified dissipation. 1334 The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist) 1335 controls the stratification-dependence of the pycnocline-intensified dissipation. 1371 1336 It can take values of 1 (recommended) or 2. 1372 Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of 1373 the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps. 1374 $h_{cri}$ is related to the large-scale topography of the ocean (etopo2) 1375 and $h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of 1337 Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of 1338 the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps. 1339 $h_{cri}$ is related to the large-scale topography of the ocean (etopo2) and 1340 $h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of 1376 1341 the abyssal hill topography \citep{Goff_JGR2010} and the latitude. 1377 1342
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