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branches/2017/dev_merge_2017/DOC/tex_sub/chap_ZDF.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Vertical Ocean Physics (ZDF)} 7 \label{ ZDF}7 \label{chap:ZDF} 8 8 \minitoc 9 9 … … 19 19 % ================================================================ 20 20 \section{Vertical mixing} 21 \label{ ZDF_zdf}21 \label{sec:ZDF_zdf} 22 22 23 23 The discrete form of the ocean subgrid scale physics has been presented in 24 \ S\ref{TRA_zdf} and \S\ref{DYN_zdf}. At the surface and bottom boundaries,24 \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}. At the surface and bottom boundaries, 25 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 Chap.~\ref{SBC}),26 surface they are prescribed from the surface forcing (see \autoref{chap:SBC}), 27 27 while at the bottom they are set to zero for heat and salt, unless a geothermal 28 28 flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} 29 defined, see \ S\ref{TRA_bbc}), and specified through a bottom friction30 parameterisation for momentum (see \ S\ref{ZDF_bfr}).29 defined, see \autoref{subsec:TRA_bbc}), and specified through a bottom friction 30 parameterisation for momentum (see \autoref{sec:ZDF_bfr}). 31 31 32 32 In this section we briefly discuss the various choices offered to compute 33 33 the vertical eddy viscosity and diffusivity coefficients, $A_u^{vm}$ , 34 34 $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- 35 points, respectively (see \ S\ref{TRA_zdf} and \S\ref{DYN_zdf}). These35 points, respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}). These 36 36 coefficients can be assumed to be either constant, or a function of the local 37 37 Richardson number, or computed from a turbulent closure model (either … … 44 44 (namelist parameter \np{ln\_zdfexp}\forcode{ = .true.}) or a backward time stepping 45 45 scheme (\np{ln\_zdfexp}\forcode{ = .false.}) depending on the magnitude of the mixing 46 coefficients, and thus of the formulation used (see \ S\ref{STP}).46 coefficients, and thus of the formulation used (see \autoref{chap:STP}). 47 47 48 48 % ------------------------------------------------------------------------------------------------------------- … … 50 50 % ------------------------------------------------------------------------------------------------------------- 51 51 \subsection{Constant (\protect\key{zdfcst})} 52 \label{ ZDF_cst}52 \label{subsec:ZDF_cst} 53 53 %--------------------------------------------namzdf--------------------------------------------------------- 54 54 \forfile{../namelists/namzdf} … … 75 75 % ------------------------------------------------------------------------------------------------------------- 76 76 \subsection{Richardson number dependent (\protect\key{zdfric})} 77 \label{ ZDF_ric}77 \label{subsec:ZDF_ric} 78 78 79 79 %--------------------------------------------namric--------------------------------------------------------- … … 91 91 ratio of stratification to vertical shear). Following \citet{Pacanowski_Philander_JPO81}, the following 92 92 formulation has been implemented: 93 \begin{equation} \label{ Eq_zdfric}93 \begin{equation} \label{eq:zdfric} 94 94 \left\{ \begin{aligned} 95 95 A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT} \\ … … 98 98 \end{equation} 99 99 where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson 100 number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \ S\ref{TRA_bn2}),100 number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 101 101 $A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the 102 constant case (see \ S\ref{ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$102 constant case (see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ 103 103 is the maximum value that can be reached by the coefficient when $Ri\leq 0$, 104 104 $a=5$ and $n=2$. The last three values can be modified by setting the … … 133 133 % ------------------------------------------------------------------------------------------------------------- 134 134 \subsection{TKE turbulent closure scheme (\protect\key{zdftke})} 135 \label{ ZDF_tke}135 \label{subsec:ZDF_tke} 136 136 137 137 %--------------------------------------------namzdf_tke-------------------------------------------------- … … 150 150 $\bar{e}$ through vertical shear, its destruction through stratification, its vertical 151 151 diffusion, and its dissipation of \citet{Kolmogorov1942} type: 152 \begin{equation} \label{ Eq_zdftke_e}152 \begin{equation} \label{eq:zdftke_e} 153 153 \frac{\partial \bar{e}}{\partial t} = 154 154 \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 … … 159 159 - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon } 160 160 \end{equation} 161 \begin{equation} \label{ Eq_zdftke_kz}161 \begin{equation} \label{eq:zdftke_kz} 162 162 \begin{split} 163 163 K_m &= C_k\ l_k\ \sqrt {\bar{e}\; } \\ … … 165 165 \end{split} 166 166 \end{equation} 167 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \ S\ref{TRA_bn2}),167 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 168 168 $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, 169 169 $P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity … … 173 173 $P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function 174 174 of the local Richardson number, $R_i$: 175 \begin{align*} \label{ Eq_prt}175 \begin{align*} \label{eq:prt} 176 176 P_{rt} = \begin{cases} 177 177 \ \ \ 1 & \text{if $\ R_i \leq 0.2$} \\ … … 187 187 namelist parameter. The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}), 188 188 however a much larger value can be used when taking into account the 189 surface wave breaking (see below Eq. \ eqref{ZDF_Esbc}).189 surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}). 190 190 The bottom value of TKE is assumed to be equal to the value of the level just above. 191 191 The time integration of the $\bar{e}$ equation may formally lead to negative values … … 199 199 instabilities associated with too weak vertical diffusion. They must be 200 200 specified at least larger than the molecular values, and are set through 201 \np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \ S\ref{ZDF_cst}).201 \np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \autoref{subsec:ZDF_cst}). 202 202 203 203 \subsubsection{Turbulent length scale} … … 207 207 parameter. The first two are based on the following first order approximation 208 208 \citep{Blanke1993}: 209 \begin{equation} \label{ Eq_tke_mxl0_1}209 \begin{equation} \label{eq:tke_mxl0_1} 210 210 l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N 211 211 \end{equation} … … 219 219 \np{nn\_mxl}\forcode{ = 2..3} cases, which add an extra assumption concerning the vertical 220 220 gradient of the computed length scale. So, the length scales are first evaluated 221 as in \ eqref{Eq_tke_mxl0_1} and then bounded such that:222 \begin{equation} \label{ Eq_tke_mxl_constraint}221 as in \autoref{eq:tke_mxl0_1} and then bounded such that: 222 \begin{equation} \label{eq:tke_mxl_constraint} 223 223 \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 224 224 \qquad \text{with }\ l = l_k = l_\epsilon 225 225 \end{equation} 226 \ eqref{Eq_tke_mxl_constraint} means that the vertical variations of the length226 \autoref{eq:tke_mxl_constraint} means that the vertical variations of the length 227 227 scale cannot be larger than the variations of depth. It provides a better 228 228 approximation of the \citet{Gaspar1990} formulation while being much less … … 230 230 by the distance to the surface or to the ocean bottom but also by the distance 231 231 to a strongly stratified portion of the water column such as the thermocline 232 ( Fig.~\ref{Fig_mixing_length}). In order to impose the \eqref{Eq_tke_mxl_constraint}232 (\autoref{fig:mixing_length}). In order to impose the \autoref{eq:tke_mxl_constraint} 233 233 constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$, 234 234 the upward and downward length scales, and evaluate the dissipation and … … 237 237 \begin{figure}[!t] \begin{center} 238 238 \includegraphics[width=1.00\textwidth]{Fig_mixing_length} 239 \caption{ \protect\label{ Fig_mixing_length}239 \caption{ \protect\label{fig:mixing_length} 240 240 Illustration of the mixing length computation. } 241 241 \end{center} 242 242 \end{figure} 243 243 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 244 \begin{equation} \label{ Eq_tke_mxl2}244 \begin{equation} \label{eq:tke_mxl2} 245 245 \begin{aligned} 246 246 l_{up\ \ }^{(k)} &= \min \left( l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \; \right) … … 250 250 \end{aligned} 251 251 \end{equation} 252 where $l^{(k)}$ is computed using \ eqref{Eq_tke_mxl0_1},252 where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, 253 253 $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 254 254 … … 257 257 \np{nn\_mxl}\forcode{ = 3} case, the dissipation and mixing turbulent length scales are give 258 258 as in \citet{Gaspar1990}: 259 \begin{equation} \label{ Eq_tke_mxl_gaspar}259 \begin{equation} \label{eq:tke_mxl_gaspar} 260 260 \begin{aligned} 261 261 & l_k = \sqrt{\ l_{up} \ \ l_{dwn}\ } \\ … … 282 282 283 283 Following \citet{Craig_Banner_JPO94}, the boundary condition on surface TKE value is : 284 \begin{equation} \label{ ZDF_Esbc}284 \begin{equation} \label{eq:ZDF_Esbc} 285 285 \bar{e}_o = \frac{1}{2}\,\left( 15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} 286 286 \end{equation} … … 289 289 younger waves \citep{Mellor_Blumberg_JPO04}. 290 290 The boundary condition on the turbulent length scale follows the Charnock's relation: 291 \begin{equation} \label{ ZDF_Lsbc}291 \begin{equation} \label{eq:ZDF_Lsbc} 292 292 l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o} 293 293 \end{equation} … … 297 297 As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, 298 298 with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}\forcode{ = 67.83} corresponds 299 to $\alpha_{CB} = 100$. Further setting \np{ln\_mxl0} to true applies \ eqref{ZDF_Lsbc}299 to $\alpha_{CB} = 100$. Further setting \np{ln\_mxl0} to true applies \autoref{eq:ZDF_Lsbc} 300 300 as surface boundary condition on length scale, with $\beta$ hard coded to the Stacey's value. 301 301 Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) … … 316 316 The parameterization, tuned against large-eddy simulation, includes the whole effect 317 317 of LC in an extra source terms of TKE, $P_{LC}$. 318 The presence of $P_{LC}$ in \ eqref{Eq_zdftke_e}, the TKE equation, is controlled318 The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled 319 319 by setting \np{ln\_lc} to \forcode{.true.} in the namtke namelist. 320 320 … … 368 368 When using this parameterization ($i.e.$ when \np{nn\_etau}\forcode{ = 1}), the TKE input to the ocean ($S$) 369 369 imposed by the winds in the form of near-inertial oscillations, swell and waves is parameterized 370 by \ eqref{ZDF_Esbc} the standard TKE surface boundary condition, plus a depth depend one given by:371 \begin{equation} \label{ ZDF_Ehtau}370 by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition, plus a depth depend one given by: 371 \begin{equation} \label{eq:ZDF_Ehtau} 372 372 S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau} 373 373 \end{equation} … … 385 385 386 386 Note that two other option existe, \np{nn\_etau}\forcode{ = 2..3}. They correspond to applying 387 \ eqref{ZDF_Ehtau} only at the base of the mixed layer, or to using the high frequency part387 \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer, or to using the high frequency part 388 388 of the stress to evaluate the fraction of TKE that penetrate the ocean. 389 389 Those two options are obsolescent features introduced for test purposes. … … 406 406 % ------------------------------------------------------------------------------------------------------------- 407 407 \subsection{TKE discretization considerations (\protect\key{zdftke})} 408 \label{ ZDF_tke_ene}408 \label{subsec:ZDF_tke_ene} 409 409 410 410 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 411 411 \begin{figure}[!t] \begin{center} 412 412 \includegraphics[width=1.00\textwidth]{Fig_ZDF_TKE_time_scheme} 413 \caption{ \protect\label{ Fig_TKE_time_scheme}413 \caption{ \protect\label{fig:TKE_time_scheme} 414 414 Illustration of the TKE time integration and its links to the momentum and tracer time integration. } 415 415 \end{center} … … 418 418 419 419 The production of turbulence by vertical shear (the first term of the right hand side 420 of \ eqref{Eq_zdftke_e}) should balance the loss of kinetic energy associated with421 the vertical momentum diffusion (first line in \ eqref{Eq_PE_zdf}). To do so a special care420 of \autoref{eq:zdftke_e}) should balance the loss of kinetic energy associated with 421 the vertical momentum diffusion (first line in \autoref{eq:PE_zdf}). To do so a special care 422 422 have to be taken for both the time and space discretization of the TKE equation 423 423 \citep{Burchard_OM02,Marsaleix_al_OM08}. 424 424 425 Let us first address the time stepping issue. Fig.~\ref{Fig_TKE_time_scheme} shows425 Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows 426 426 how the two-level Leap-Frog time stepping of the momentum and tracer equations interplays 427 427 with the one-level forward time stepping of TKE equation. With this framework, the total loss 428 428 of kinetic energy (in 1D for the demonstration) due to the vertical momentum diffusion is 429 429 obtained by multiplying this quantity by $u^t$ and summing the result vertically: 430 \begin{equation} \label{ Eq_energ1}430 \begin{equation} \label{eq:energ1} 431 431 \begin{split} 432 432 \int_{-H}^{\eta} u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt} \right) \,dz \\ … … 436 436 \end{equation} 437 437 Here, the vertical diffusion of momentum is discretized backward in time 438 with a coefficient, $K_m$, known at time $t$ ( Fig.~\ref{Fig_TKE_time_scheme}),439 as it is required when using the TKE scheme (see \ S\ref{STP_forward_imp}).440 The first term of the right hand side of \ eqref{Eq_energ1} represents the kinetic energy438 with a coefficient, $K_m$, known at time $t$ (\autoref{fig:TKE_time_scheme}), 439 as it is required when using the TKE scheme (see \autoref{sec:STP_forward_imp}). 440 The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy 441 441 transfer at the surface (atmospheric forcing) and at the bottom (friction effect). 442 442 The second term is always negative. It is the dissipation rate of kinetic energy, 443 and thus minus the shear production rate of $\bar{e}$. \ eqref{Eq_energ1}443 and thus minus the shear production rate of $\bar{e}$. \autoref{eq:energ1} 444 444 implies that, to be energetically consistent, the production rate of $\bar{e}$ 445 445 used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as … … 448 448 449 449 A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification 450 (second term of the right hand side of \ eqref{Eq_zdftke_e}). This term450 (second term of the right hand side of \autoref{eq:zdftke_e}). This term 451 451 must balance the input of potential energy resulting from vertical mixing. 452 452 The rate of change of potential energy (in 1D for the demonstration) due vertical 453 453 mixing is obtained by multiplying vertical density diffusion 454 454 tendency by $g\,z$ and and summing the result vertically: 455 \begin{equation} \label{ Eq_energ2}455 \begin{equation} \label{eq:energ2} 456 456 \begin{split} 457 457 \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt} \right) \,dz \\ … … 463 463 \end{equation} 464 464 where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. 465 The first term of the right hand side of \ eqref{Eq_energ2} is always zero465 The first term of the right hand side of \autoref{eq:energ2} is always zero 466 466 because there is no diffusive flux through the ocean surface and bottom). 467 467 The second term is minus the destruction rate of $\bar{e}$ due to stratification. 468 Therefore \ eqref{Eq_energ1} implies that, to be energetically consistent, the product469 ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \ eqref{Eq_zdftke_e}, the TKE equation.468 Therefore \autoref{eq:energ1} implies that, to be energetically consistent, the product 469 ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e}, the TKE equation. 470 470 471 471 Let us now address the space discretization issue. 472 472 The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity 473 473 components are in the centre of the side faces of a $t$-box in staggered C-grid 474 ( Fig.\ref{Fig_cell}). A space averaging is thus required to obtain the shear TKE production term.475 By redoing the \ eqref{Eq_energ1} in the 3D case, it can be shown that the product of474 (\autoref{fig:cell}). A space averaging is thus required to obtain the shear TKE production term. 475 By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of 476 476 eddy coefficient by the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. 477 477 Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into … … 480 480 The above energetic considerations leads to 481 481 the following final discrete form for the TKE equation: 482 \begin{equation} \label{ Eq_zdftke_ene}482 \begin{equation} \label{eq:zdftke_ene} 483 483 \begin{split} 484 484 \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt} \equiv … … 497 497 \end{split} 498 498 \end{equation} 499 where the last two terms in \ eqref{Eq_zdftke_ene} (vertical diffusion and Kolmogorov dissipation)500 are time stepped using a backward scheme (see\ S\ref{STP_forward_imp}).499 where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation) 500 are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}). 501 501 Note that the Kolmogorov term has been linearized in time in order to render 502 502 the implicit computation possible. The restart of the TKE scheme 503 503 requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as they all appear in 504 the right hand side of \ eqref{Eq_zdftke_ene}. For the latter, it is in fact504 the right hand side of \autoref{eq:zdftke_ene}. For the latter, it is in fact 505 505 the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 506 506 … … 509 509 % ------------------------------------------------------------------------------------------------------------- 510 510 \subsection{GLS: Generic Length Scale (\protect\key{zdfgls})} 511 \label{ ZDF_gls}511 \label{subsec:ZDF_gls} 512 512 513 513 %--------------------------------------------namzdf_gls--------------------------------------------------------- … … 519 519 for the generic length scale, $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}. 520 520 This later variable is defined as : $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, 521 where the triplet $(p, m, n)$ value given in Tab.\ ref{Tab_GLS} allows to recover521 where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover 522 522 a number of well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982}, 523 523 $k$-$\epsilon$ \citep{Rodi_1987}, $k$-$\omega$ \citep{Wilcox_1988} 524 524 among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}). 525 525 The GLS scheme is given by the following set of equations: 526 \begin{equation} \label{ Eq_zdfgls_e}526 \begin{equation} \label{eq:zdfgls_e} 527 527 \frac{\partial \bar{e}}{\partial t} = 528 528 \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 … … 533 533 \end{equation} 534 534 535 \begin{equation} \label{ Eq_zdfgls_psi}535 \begin{equation} \label{eq:zdfgls_psi} 536 536 \begin{split} 537 537 \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ … … 544 544 \end{equation} 545 545 546 \begin{equation} \label{ Eq_zdfgls_kz}546 \begin{equation} \label{eq:zdfgls_kz} 547 547 \begin{split} 548 548 K_m &= C_{\mu} \ \sqrt {\bar{e}} \ l \\ … … 551 551 \end{equation} 552 552 553 \begin{equation} \label{ Eq_zdfgls_eps}553 \begin{equation} \label{eq:zdfgls_eps} 554 554 {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 555 555 \end{equation} 556 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \ S\ref{TRA_bn2})556 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) 557 557 and $\epsilon$ the dissipation rate. 558 558 The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) 559 559 depends of the choice of the turbulence model. Four different turbulent models are pre-defined 560 (Tab.\ ref{Tab_GLS}). They are made available through the \np{nn\_clo} namelist parameter.560 (Tab.\autoref{tab:GLS}). They are made available through the \np{nn\_clo} namelist parameter. 561 561 562 562 %--------------------------------------------------TABLE-------------------------------------------------- … … 579 579 \hline 580 580 \end{tabular} 581 \caption{ \protect\label{ Tab_GLS}581 \caption{ \protect\label{tab:GLS} 582 582 Set of predefined GLS parameters, or equivalently predefined turbulence models available 583 583 with \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls} .} … … 596 596 As for TKE closure , the wave effect on the mixing is considered when \np{ln\_crban}\forcode{ = .true.} 597 597 \citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}. The \np{rn\_crban} namelist parameter 598 is $\alpha_{CB}$ in \ eqref{ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \eqref{ZDF_Lsbc}.598 is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}. 599 599 600 600 The $\psi$ equation is known to fail in stably stratified flows, and for this reason … … 609 609 610 610 The time and space discretization of the GLS equations follows the same energetic 611 consideration as for the TKE case described in \ S\ref{ZDF_tke_ene} \citep{Burchard_OM02}.611 consideration as for the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{Burchard_OM02}. 612 612 Examples of performance of the 4 turbulent closure scheme can be found in \citet{Warner_al_OM05}. 613 613 … … 616 616 % ------------------------------------------------------------------------------------------------------------- 617 617 \subsection{OSM: OSMOSIS boundary layer scheme (\protect\key{zdfosm})} 618 \label{ ZDF_osm}618 \label{subsec:ZDF_osm} 619 619 620 620 %--------------------------------------------namzdf_osm--------------------------------------------------------- … … 628 628 % ================================================================ 629 629 \section{Convection} 630 \label{ ZDF_conv}630 \label{sec:ZDF_conv} 631 631 632 632 %--------------------------------------------namzdf-------------------------------------------------------- … … 648 648 \subsection[Non-penetrative convective adjmt (\protect\np{ln\_tranpc}\forcode{ = .true.})] 649 649 {Non-penetrative convective adjustment (\protect\np{ln\_tranpc}\forcode{ = .true.})} 650 \label{ ZDF_npc}650 \label{subsec:ZDF_npc} 651 651 652 652 %--------------------------------------------namzdf-------------------------------------------------------- … … 657 657 \begin{figure}[!htb] \begin{center} 658 658 \includegraphics[width=0.90\textwidth]{Fig_npc} 659 \caption{ \protect\label{ Fig_npc}659 \caption{ \protect\label{fig:npc} 660 660 Example of an unstable density profile treated by the non penetrative 661 661 convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from … … 677 677 column has \textit{exactly} the density of the water just below) \citep{Madec_al_JPO91}. 678 678 The associated algorithm is an iterative process used in the following way 679 ( Fig. \ref{Fig_npc}): starting from the top of the ocean, the first instability is679 (\autoref{fig:npc}): starting from the top of the ocean, the first instability is 680 680 found. Assume in the following that the instability is located between levels 681 681 $k$ and $k+1$. The temperature and salinity in the two levels are … … 714 714 % ------------------------------------------------------------------------------------------------------------- 715 715 \subsection{Enhanced vertical diffusion (\protect\np{ln\_zdfevd}\forcode{ = .true.})} 716 \label{ ZDF_evd}716 \label{subsec:ZDF_evd} 717 717 718 718 %--------------------------------------------namzdf-------------------------------------------------------- … … 739 739 Note that the stability test is performed on both \textit{before} and \textit{now} 740 740 values of $N^2$. This removes a potential source of divergence of odd and 741 even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \ S\ref{STP_mLF}).741 even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \autoref{sec:STP_mLF}). 742 742 743 743 % ------------------------------------------------------------------------------------------------------------- … … 745 745 % ------------------------------------------------------------------------------------------------------------- 746 746 \subsection[Turbulent closure scheme (\protect\key{zdf}\{tke,gls,osm\})]{Turbulent Closure Scheme (\protect\key{zdftke}, \protect\key{zdfgls} or \protect\key{zdfosm})} 747 \label{ ZDF_tcs}748 749 The turbulent closure scheme presented in \ S\ref{ZDF_tke} and \S\ref{ZDF_gls}747 \label{subsec:ZDF_tcs} 748 749 The turbulent closure scheme presented in \autoref{subsec:ZDF_tke} and \autoref{subsec:ZDF_gls} 750 750 (\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically 751 751 unstable density profiles. In such a case, the term corresponding to the 752 destruction of turbulent kinetic energy through stratification in \ eqref{Eq_zdftke_e}753 or \ eqref{Eq_zdfgls_e} becomes a source term, since $N^2$ is negative.752 destruction of turbulent kinetic energy through stratification in \autoref{eq:zdftke_e} 753 or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative. 754 754 It results in large values of $A_T^{vT}$ and $A_T^{vT}$, and also the four neighbouring 755 755 $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1})$. These large values 756 756 restore the static stability of the water column in a way similar to that of the 757 enhanced vertical diffusion parameterisation (\ S\ref{ZDF_evd}). However,757 enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}). However, 758 758 in the vicinity of the sea surface (first ocean layer), the eddy coefficients 759 759 computed by the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$, … … 772 772 % ================================================================ 773 773 \section{Double diffusion mixing (\protect\key{zdfddm})} 774 \label{ ZDF_ddm}774 \label{sec:ZDF_ddm} 775 775 776 776 %-------------------------------------------namzdf_ddm------------------------------------------------- … … 789 789 790 790 Diapycnal mixing of S and T are described by diapycnal diffusion coefficients 791 \begin{align*} % \label{ Eq_zdfddm_Kz}791 \begin{align*} % \label{eq:zdfddm_Kz} 792 792 &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ 793 793 &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} … … 797 797 mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, 798 798 where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline 799 contraction (see \ S\ref{TRA_eos}). To represent mixing of $S$ and $T$ by salt799 contraction (see \autoref{subsec:TRA_eos}). To represent mixing of $S$ and $T$ by salt 800 800 fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981): 801 \begin{align} \label{ Eq_zdfddm_f}801 \begin{align} \label{eq:zdfddm_f} 802 802 A_f^{vS} &= \begin{cases} 803 803 \frac{A^{\ast v}}{1+(R_\rho / R_c)^n } &\text{if $R_\rho > 1$ and $N^2>0$ } \\ 804 804 0 &\text{otherwise} 805 805 \end{cases} 806 \\ \label{ Eq_zdfddm_f_T}806 \\ \label{eq:zdfddm_f_T} 807 807 A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 808 808 \end{align} … … 811 811 \begin{figure}[!t] \begin{center} 812 812 \includegraphics[width=0.99\textwidth]{Fig_zdfddm} 813 \caption{ \protect\label{ Fig_zdfddm}813 \caption{ \protect\label{fig:zdfddm} 814 814 From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ 815 815 and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy … … 822 822 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 823 823 824 The factor 0.7 in \ eqref{Eq_zdfddm_f_T} reflects the measured ratio824 The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio 825 825 $\alpha F_T /\beta F_S \approx 0.7$ of buoyancy flux of heat to buoyancy 826 826 flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). Following \citet{Merryfield1999}, … … 828 828 829 829 To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested by Federov (1988) is used: 830 \begin{align} \label{ Eq_zdfddm_d}830 \begin{align} \label{eq:zdfddm_d} 831 831 A_d^{vT} &= \begin{cases} 832 832 1.3635 \, \exp{\left( 4.6\, \exp{ \left[ -0.54\,( R_{\rho}^{-1} - 1 ) \right] } \right)} … … 834 834 0 &\text{otherwise} 835 835 \end{cases} 836 \\ \label{ Eq_zdfddm_d_S}836 \\ \label{eq:zdfddm_d_S} 837 837 A_d^{vS} &= \begin{cases} 838 838 A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) … … 844 844 \end{align} 845 845 846 The dependencies of \ eqref{Eq_zdfddm_f} to \eqref{Eq_zdfddm_d_S} on $R_\rho$847 are illustrated in Fig.~\ref{Fig_zdfddm}. Implementing this requires computing846 The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ 847 are illustrated in \autoref{fig:zdfddm}. Implementing this requires computing 848 848 $R_\rho$ at each grid point on every time step. This is done in \mdl{eosbn2} at the 849 849 same time as $N^2$ is computed. This avoids duplication in the computation of … … 854 854 % ================================================================ 855 855 \section{Bottom and top friction (\protect\mdl{zdfbfr})} 856 \label{ ZDF_bfr}856 \label{sec:ZDF_bfr} 857 857 858 858 %--------------------------------------------nambfr-------------------------------------------------------- … … 870 870 flux (bottom friction) enter the equations as a condition on the vertical 871 871 diffusive flux. For the bottom boundary layer, one has: 872 \begin{equation} \label{ Eq_zdfbfr_flux}872 \begin{equation} \label{eq:zdfbfr_flux} 873 873 A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 874 874 \end{equation} … … 886 886 as a body force over the depth of the top or bottom model layer. To illustrate 887 887 this, consider the equation for $u$ at $k$, the last ocean level: 888 \begin{equation} \label{ Eq_zdfbfr_flux2}888 \begin{equation} \label{eq:zdfbfr_flux2} 889 889 \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}} 890 890 \end{equation} … … 907 907 These coefficients are computed in \mdl{zdfbfr} and generally take the form 908 908 $c_b^{\textbf U}$ where: 909 \begin{equation} \label{ Eq_zdfbfr_bdef}909 \begin{equation} \label{eq:zdfbfr_bdef} 910 910 \frac{\partial {\textbf U_h}}{\partial t} = 911 911 - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b … … 917 917 % ------------------------------------------------------------------------------------------------------------- 918 918 \subsection{Linear bottom friction (\protect\np{nn\_botfr}\forcode{ = 0..1})} 919 \label{ ZDF_bfr_linear}919 \label{subsec:ZDF_bfr_linear} 920 920 921 921 The linear bottom friction parameterisation (including the special case … … 923 923 is proportional to the interior velocity (i.e. the velocity of the last 924 924 model level): 925 \begin{equation} \label{ Eq_zdfbfr_linear}925 \begin{equation} \label{eq:zdfbfr_linear} 926 926 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 927 927 \end{equation} … … 941 941 942 942 For the linear friction case the coefficients defined in the general 943 expression \ eqref{Eq_zdfbfr_bdef} are:944 \begin{equation} \label{ Eq_zdfbfr_linbfr_b}943 expression \autoref{eq:zdfbfr_bdef} are: 944 \begin{equation} \label{eq:zdfbfr_linbfr_b} 945 945 \begin{split} 946 946 c_b^u &= - r\\ … … 961 961 % ------------------------------------------------------------------------------------------------------------- 962 962 \subsection{Non-linear bottom friction (\protect\np{nn\_botfr}\forcode{ = 2})} 963 \label{ ZDF_bfr_nonlinear}963 \label{subsec:ZDF_bfr_nonlinear} 964 964 965 965 The non-linear bottom friction parameterisation assumes that the bottom 966 966 friction is quadratic: 967 \begin{equation} \label{ Eq_zdfbfr_nonlinear}967 \begin{equation} \label{eq:zdfbfr_nonlinear} 968 968 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 969 969 }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b … … 983 983 For the non-linear friction case the terms 984 984 computed in \mdl{zdfbfr} are: 985 \begin{equation} \label{ Eq_zdfbfr_nonlinbfr}985 \begin{equation} \label{eq:zdfbfr_nonlinbfr} 986 986 \begin{split} 987 987 c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\ … … 1003 1003 \subsection[Log-layer btm frict enhncmnt (\protect\np{nn\_botfr}\forcode{ = 2}, \protect\np{ln\_loglayer}\forcode{ = .true.})] 1004 1004 {Log-layer bottom friction enhancement (\protect\np{nn\_botfr}\forcode{ = 2}, \protect\np{ln\_loglayer}\forcode{ = .true.})} 1005 \label{ ZDF_bfr_loglayer}1005 \label{subsec:ZDF_bfr_loglayer} 1006 1006 1007 1007 In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally … … 1033 1033 % ------------------------------------------------------------------------------------------------------------- 1034 1034 \subsection{Bottom friction stability considerations} 1035 \label{ ZDF_bfr_stability}1035 \label{subsec:ZDF_bfr_stability} 1036 1036 1037 1037 Some care needs to exercised over the choice of parameters to ensure that the 1038 1038 implementation of bottom friction does not induce numerical instability. For 1039 the purposes of stability analysis, an approximation to \ eqref{Eq_zdfbfr_flux2}1039 the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2} 1040 1040 is: 1041 \begin{equation} \label{ Eqn_bfrstab}1041 \begin{equation} \label{eq:Eqn_bfrstab} 1042 1042 \begin{split} 1043 1043 \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt \\ … … 1050 1050 |\Delta u| < \;|u| 1051 1051 \end{equation} 1052 \noindent which, using \ eqref{Eqn_bfrstab}, gives:1052 \noindent which, using \autoref{eq:Eqn_bfrstab}, gives: 1053 1053 \begin{equation} 1054 1054 r\frac{2\rdt}{e_{3u}} < 1 \qquad \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ … … 1075 1075 1076 1076 Limits on the bottom friction coefficient are not imposed if the user has elected to 1077 handle the bottom friction implicitly (see \ S\ref{ZDF_bfr_imp}). The number of potential1077 handle the bottom friction implicitly (see \autoref{subsec:ZDF_bfr_imp}). The number of potential 1078 1078 breaches of the explicit stability criterion are still reported for information purposes. 1079 1079 … … 1082 1082 % ------------------------------------------------------------------------------------------------------------- 1083 1083 \subsection{Implicit bottom friction (\protect\np{ln\_bfrimp}\forcode{ = .true.})} 1084 \label{ ZDF_bfr_imp}1084 \label{subsec:ZDF_bfr_imp} 1085 1085 1086 1086 An optional implicit form of bottom friction has been implemented to improve … … 1093 1093 bottom boundary condition is implemented implicitly. 1094 1094 1095 \begin{equation} \label{ Eq_dynzdf_bfr}1095 \begin{equation} \label{eq:dynzdf_bfr} 1096 1096 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk} 1097 1097 = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}} … … 1112 1112 following: 1113 1113 1114 \begin{equation} \label{ Eq_dynspg_ts_bfr1}1114 \begin{equation} \label{eq:dynspg_ts_bfr1} 1115 1115 \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b} 1116 1116 \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right) 1117 1117 \end{equation} 1118 \begin{equation} \label{ Eq_dynspg_ts_bfr2}1118 \begin{equation} \label{eq:dynspg_ts_bfr2} 1119 1119 \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+ 1120 1120 \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)- … … 1136 1136 \subsection[Bottom friction w/ split-explicit time splitting (\protect\np{ln\_bfrimp})] 1137 1137 {Bottom friction with split-explicit time splitting (\protect\np{ln\_bfrimp})} 1138 \label{ ZDF_bfr_ts}1138 \label{subsec:ZDF_bfr_ts} 1139 1139 1140 1140 When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the … … 1175 1175 the barotropic component which uses the unrestricted value of the coefficient. However, if the 1176 1176 limiting is thought to be having a major effect (a more likely prospect in coastal and shelf seas 1177 applications) then the fully implicit form of the bottom friction should be used (see \ S\ref{ZDF_bfr_imp} )1177 applications) then the fully implicit form of the bottom friction should be used (see \autoref{subsec:ZDF_bfr_imp} ) 1178 1178 which can be selected by setting \np{ln\_bfrimp} $=$ \forcode{.true.}. 1179 1179 1180 1180 Otherwise, the implicit formulation takes the form: 1181 \begin{equation} \label{ Eq_zdfbfr_implicitts}1181 \begin{equation} \label{eq:zdfbfr_implicitts} 1182 1182 \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ] 1183 1183 \end{equation} … … 1193 1193 % ================================================================ 1194 1194 \section{Tidal mixing (\protect\key{zdftmx})} 1195 \label{ ZDF_tmx}1195 \label{sec:ZDF_tmx} 1196 1196 1197 1197 %--------------------------------------------namzdf_tmx-------------------------------------------------- … … 1204 1204 % ------------------------------------------------------------------------------------------------------------- 1205 1205 \subsection{Bottom intensified tidal mixing} 1206 \label{ ZDF_tmx_bottom}1206 \label{subsec:ZDF_tmx_bottom} 1207 1207 1208 1208 Options are defined through the \ngn{namzdf\_tmx} namelist variables. … … 1213 1213 $A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, the energy transfer from barotropic 1214 1214 tides to baroclinic tides : 1215 \begin{equation} \label{ Eq_Ktides}1215 \begin{equation} \label{eq:Ktides} 1216 1216 A^{vT}_{tides} = q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 } 1217 1217 \end{equation} 1218 1218 where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency 1219 (see \ S\ref{TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency,1219 (see \autoref{subsec:TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency, 1220 1220 and $F(z)$ the vertical structure function. 1221 1221 … … 1230 1230 It is implemented as a simple exponential decaying upward away from the bottom, 1231 1231 with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04}, 1232 \begin{equation} \label{ Eq_Fz}1232 \begin{equation} \label{eq:Fz} 1233 1233 F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) } 1234 1234 \end{equation} … … 1241 1241 usually set to $10^{-8} s^{-2}$. These bounds are usually rarely encountered. 1242 1242 1243 The internal wave energy map, $E(x, y)$ in \ eqref{Eq_Ktides}, is derived1243 The internal wave energy map, $E(x, y)$ in \autoref{eq:Ktides}, is derived 1244 1244 from a barotropic model of the tides utilizing a parameterization of the 1245 1245 conversion of barotropic tidal energy into internal waves. … … 1250 1250 the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}. 1251 1251 This model provides the dissipation associated with internal wave energy for the M2 and K1 1252 tides component ( Fig.~\ref{Fig_ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated1252 tides component (\autoref{fig:ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated 1253 1253 as being $1/4$ of the M2 one. The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$. 1254 1254 Its global mean value is $1.1$ TW, in agreement with independent estimates … … 1258 1258 \begin{figure}[!t] \begin{center} 1259 1259 \includegraphics[width=0.90\textwidth]{Fig_ZDF_M2_K1_tmx} 1260 \caption{ \protect\label{ Fig_ZDF_M2_K1_tmx}1260 \caption{ \protect\label{fig:ZDF_M2_K1_tmx} 1261 1261 (a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). } 1262 1262 \end{center} \end{figure} … … 1267 1267 % ------------------------------------------------------------------------------------------------------------- 1268 1268 \subsection{Indonesian area specific treatment (\protect\np{ln\_zdftmx\_itf})} 1269 \label{ ZDF_tmx_itf}1269 \label{subsec:ZDF_tmx_itf} 1270 1270 1271 1271 When the Indonesian Through Flow (ITF) area is included in the model domain, … … 1294 1294 proportional to $N^2$ below the core of the thermocline and to $N$ above. 1295 1295 The resulting $F(z)$ is: 1296 \begin{equation} \label{ Eq_Fz_itf}1296 \begin{equation} \label{eq:Fz_itf} 1297 1297 F(i,j,k) \sim \left\{ \begin{aligned} 1298 1298 \frac{q\,\Gamma E(i,j) } {\rho N \, \int N dz} \qquad \text{when $\partial_z N < 0$} \\ … … 1315 1315 % ================================================================ 1316 1316 \section{Internal wave-driven mixing (\protect\key{zdftmx\_new})} 1317 \label{ ZDF_tmx_new}1317 \label{sec:ZDF_tmx_new} 1318 1318 1319 1319 %--------------------------------------------namzdf_tmx_new------------------------------------------ … … 1325 1325 A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed, 1326 1326 and the resulting diffusivity is obtained as 1327 \begin{equation} \label{ Eq_Kwave}1327 \begin{equation} \label{eq:Kwave} 1328 1328 A^{vT}_{wave} = R_f \,\frac{ \epsilon }{ \rho \, N^2 } 1329 1329 \end{equation}
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