[707] | 1 | % ================================================================ |
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| 2 | % Chapter Ñ Vertical Ocean Physics (ZDF) |
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| 3 | % ================================================================ |
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| 4 | \chapter{Vertical Ocean Physics (ZDF)} |
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| 5 | \label{ZDF} |
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| 6 | \minitoc |
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| 7 | |
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| 8 | %gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN. |
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[1224] | 9 | \gmcomment{Steven remark (not taken into account : problem here with turbulent vs turbulence. I've changed "turbulent closure" to "turbulence closure", "turbulent mixing length" to "turbulence mixing length", but I've left "turbulent kinetic energy" alone - though I think it is an historical abberation! |
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| 10 | Gurvan : I kept "turbulent closure etc "...} |
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[707] | 11 | |
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[994] | 12 | |
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[707] | 13 | % ================================================================ |
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| 14 | % Vertical Mixing |
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| 15 | % ================================================================ |
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| 16 | \section{Vertical Mixing} |
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| 17 | \label{ZDF_zdf} |
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| 18 | |
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[994] | 19 | The discrete form of the ocean subgrid scale physics has been presented in |
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| 20 | \S\ref{TRA_zdf} and \S\ref{DYN_zdf}. At the surface and bottom boundaries, |
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| 21 | the turbulent fluxes of momentum, heat and salt have to be defined. At the |
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| 22 | surface they are prescribed from the surface forcing (see Chap.~\ref{SBC}), |
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| 23 | while at the bottom they are set to zero for heat and salt, unless a geothermal |
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| 24 | flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} |
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| 25 | defined, see \S\ref{TRA_bbc}), and specified through a bottom friction |
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[1224] | 26 | parameterisation for momentum (see \S\ref{ZDF_bfr}). |
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[707] | 27 | |
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| 28 | In this section we briefly discuss the various choices offered to compute |
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[994] | 29 | the vertical eddy viscosity and diffusivity coefficients, $A_u^{vm}$ , |
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| 30 | $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- |
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| 31 | points, respectively (see \S\ref{TRA_zdf} and \S\ref{DYN_zdf}). These |
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| 32 | coefficients can be assumed to be either constant, or a function of the local |
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| 33 | Richardson number, or computed from a turbulent closure model (either |
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| 34 | TKE or KPP formulation). The computation of these coefficients is initialized |
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| 35 | in the \mdl{zdfini} module and performed in the \mdl{zdfric}, \mdl{zdftke} or |
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| 36 | \mdl{zdfkpp} modules. The trends due to the vertical momentum and tracer |
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| 37 | diffusion, including the surface forcing, are computed and added to the |
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| 38 | general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively. |
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| 39 | These trends can be computed using either a forward time stepping scheme |
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| 40 | (namelist parameter \np{np\_zdfexp}=true) or a backward time stepping |
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| 41 | scheme (\np{np\_zdfexp}=false) depending on the magnitude of the mixing |
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| 42 | coefficients, and thus of the formulation used (see \S\ref{DOM_nxt}). |
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[707] | 43 | |
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| 44 | % ------------------------------------------------------------------------------------------------------------- |
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| 45 | % Constant |
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| 46 | % ------------------------------------------------------------------------------------------------------------- |
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| 47 | \subsection{Constant (\key{zdfcst})} |
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| 48 | \label{ZDF_cst} |
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| 49 | %--------------------------------------------namzdf--------------------------------------------------------- |
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| 50 | \namdisplay{namzdf} |
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| 51 | %-------------------------------------------------------------------------------------------------------------- |
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| 52 | |
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[994] | 53 | When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients |
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| 54 | are set to constant values over the whole ocean. This is the crudest way to define |
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| 55 | the vertical ocean physics. It is recommended that this option is only used in |
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| 56 | process studies, not in basin scale simulations. Typical values used in this case are: |
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[707] | 57 | \begin{align*} |
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| 58 | A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1} \\ |
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| 59 | \\ |
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| 60 | A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1} |
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| 61 | \end{align*} |
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| 62 | |
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[994] | 63 | These values are set through the \np{avm0} and \np{avt0} namelist parameters. |
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| 64 | In all cases, do not use values smaller that those associated with the molecular |
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| 65 | viscosity and diffusivity, that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, |
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| 66 | $\sim10^{-7}~m^2.s^{-1}$ for temperature and $\sim10^{-9}~m^2.s^{-1}$ for salinity. |
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[707] | 67 | |
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| 68 | |
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| 69 | % ------------------------------------------------------------------------------------------------------------- |
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| 70 | % Richardson Number Dependent |
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| 71 | % ------------------------------------------------------------------------------------------------------------- |
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| 72 | \subsection{Richardson Number Dependent (\key{zdfric})} |
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| 73 | \label{ZDF_ric} |
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| 74 | |
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| 75 | %--------------------------------------------namric--------------------------------------------------------- |
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| 76 | \namdisplay{namric} |
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| 77 | %-------------------------------------------------------------------------------------------------------------- |
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| 78 | |
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[994] | 79 | When \key{zdfric} is defined, a local Richardson number dependent formulation |
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| 80 | for the vertical momentum and tracer eddy coefficients is set. The vertical mixing |
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| 81 | coefficients are diagnosed from the large scale variables computed by the model. |
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| 82 | \textit{In situ} measurements have been used to link vertical turbulent activity to |
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| 83 | large scale ocean structures. The hypothesis of a mixing mainly maintained by the |
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| 84 | growth of Kelvin-Helmholtz like instabilities leads to a dependency between the |
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[1224] | 85 | vertical eddy coefficients and the local Richardson number ($i.e.$ the |
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[994] | 86 | ratio of stratification to vertical shear). Following \citet{PacPhil1981}, the following |
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| 87 | formulation has been implemented: |
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[707] | 88 | \begin{equation} \label{Eq_zdfric} |
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| 89 | \left\{ \begin{aligned} |
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| 90 | A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT} \\ |
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| 91 | \\ |
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| 92 | A^{vm} &= \frac{A^{vT} }{\left( 1+ a \;Ri \right) } + A_b^{vm} |
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| 93 | \end{aligned} \right. |
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| 94 | \end{equation} |
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[994] | 95 | where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson |
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| 96 | number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}), |
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| 97 | $A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the |
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| 98 | constant case (see \S\ref{ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ |
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| 99 | is the maximum value that can be reached by the coefficient when $Ri\leq 0$, |
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| 100 | $a=5$ and $n=2$. The last three values can be modified by setting the |
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| 101 | \np{avmri}, \np{alp} and \np{nric} namelist parameters, respectively. |
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[707] | 102 | |
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| 103 | % ------------------------------------------------------------------------------------------------------------- |
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| 104 | % TKE Turbulent Closure Scheme |
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| 105 | % ------------------------------------------------------------------------------------------------------------- |
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| 106 | \subsection{TKE Turbulent Closure Scheme (\key{zdftke})} |
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| 107 | \label{ZDF_tke} |
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| 108 | |
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| 109 | %--------------------------------------------namtke--------------------------------------------------------- |
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[1225] | 110 | \namdisplay{namtke} |
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[707] | 111 | %-------------------------------------------------------------------------------------------------------------- |
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| 112 | |
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[994] | 113 | The vertical eddy viscosity and diffusivity coefficients are computed from a TKE |
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| 114 | turbulent closure model based on a prognostic equation for $\bar {e}$, the turbulent |
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[1224] | 115 | kinetic energy, and a closure assumption for the turbulent length scales. This |
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[994] | 116 | turbulent closure model has been developed by \citet{Bougeault1989} in the |
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| 117 | atmospheric case, adapted by \citet{Gaspar1990} for the oceanic case, and |
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| 118 | embedded in OPA by \citet{Blanke1993} for equatorial Atlantic simulations. Since |
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| 119 | then, significant modifications have been introduced by \citet{Madec1998} in both |
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| 120 | the implementation and the formulation of the mixing length scale. The time |
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| 121 | evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical |
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| 122 | shear, its destruction through stratification, its vertical diffusion, and its dissipation |
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| 123 | of \citet{Kolmogorov1942} type: |
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[707] | 124 | \begin{equation} \label{Eq_zdftke_e} |
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| 125 | \frac{\partial \bar{e}}{\partial t} = |
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| 126 | \frac{A^{vm}}{e_3 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 |
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| 127 | +\left( {\frac{\partial v}{\partial k}} \right)^2} \right] |
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| 128 | -A^{vT}\,N^2 |
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| 129 | +\frac{1}{e_3} \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 } |
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| 130 | \;\frac{\partial \bar{e}}{\partial k}} \right] |
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| 131 | - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon } |
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| 132 | \end{equation} |
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| 133 | \begin{equation} \label{Eq_zdftke_kz} |
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| 134 | \begin{split} |
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[994] | 135 | A^{vm} &= C_k\ l_k\ \sqrt {\bar{e}} \\ |
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[707] | 136 | A^{vT} &= A^{vm} / P_{rt} |
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| 137 | \end{split} |
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| 138 | \end{equation} |
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[994] | 139 | where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}), |
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| 140 | $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, |
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| 141 | $P_{rt} $ is the Prandtl number. The constants $C_k = \sqrt {2} /2$ and |
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| 142 | $C_\epsilon = 0.1$ are designed to deal with vertical mixing at any depth |
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| 143 | \citep{Gaspar1990}. They are set through namelist parameters \np{ediff} |
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| 144 | and \np{ediss}. $P_{rt} $ can be set to unity or, following \citet{Blanke1993}, |
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| 145 | be a function of the local Richardson number, $R_i $: |
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[707] | 146 | \begin{align*} \label{Eq_prt} |
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| 147 | P_{rt} = \begin{cases} |
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| 148 | \ \ \ 1 & \text{if $\ R_i \leq 0.2$} \\ |
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| 149 | 5\,R_i & \text{if $\ 0.2 \leq R_i \leq 2$} \\ |
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| 150 | \ \ 10 & \text{if $\ 2 \leq R_i$} |
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| 151 | \end{cases} |
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| 152 | \end{align*} |
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[994] | 153 | Note that a horizontal Shapiro filter can optionally be applied to $R_i$. |
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| 154 | However it is an obsolescent option that is not recommended. |
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| 155 | The choice of $P_{rt} $ is controlled by the \np{npdl} namelist parameter. |
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[707] | 156 | |
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[1224] | 157 | For computational efficiency, the original formulation of the turbulent length |
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[994] | 158 | scales proposed by \citet{Gaspar1990} has been simplified. Four formulations |
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| 159 | are proposed, the choice of which is controlled by the \np{nmxl} namelist |
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| 160 | parameter. The first two are based on the following first order approximation |
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| 161 | \citep{Blanke1993}: |
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[707] | 162 | \begin{equation} \label{Eq_tke_mxl0_1} |
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| 163 | l_k = l_\epsilon = \sqrt {2 \bar e} / N |
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| 164 | \end{equation} |
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[994] | 165 | which is valid in a stable stratified region with constant values of the brunt- |
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| 166 | Vais\"{a}l\"{a} frequency. The resulting length scale is bounded by the distance |
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| 167 | to the surface or to the bottom (\np{nmxl}=0) or by the local vertical scale factor (\np{nmxl}=1). \citet{Blanke1993} notice that this simplification has two major |
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| 168 | drawbacks: it makes no sense for locally unstable stratification and the |
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| 169 | computation no longer uses all the information contained in the vertical density |
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| 170 | profile. To overcome these drawbacks, \citet{Madec1998} introduces the |
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| 171 | \np{nmxl}=2 or 3 cases, which add an extra assumption concerning the vertical |
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| 172 | gradient of the computed length scale. So, the length scales are first evaluated |
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| 173 | as in \eqref{Eq_tke_mxl0_1} and then bounded such that: |
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[707] | 174 | \begin{equation} \label{Eq_tke_mxl_constraint} |
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| 175 | \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 |
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| 176 | \qquad \text{with }\ l = l_k = l_\epsilon |
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| 177 | \end{equation} |
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[994] | 178 | \eqref{Eq_tke_mxl_constraint} means that the vertical variations of the length |
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| 179 | scale cannot be larger than the variations of depth. It provides a better |
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| 180 | approximation of the \citet{Gaspar1990} formulation while being much less |
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| 181 | time consuming. In particular, it allows the length scale to be limited not only |
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| 182 | by the distance to the surface or to the ocean bottom but also by the distance |
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| 183 | to a strongly stratified portion of the water column such as the thermocline |
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| 184 | (Fig.~\ref{Fig_mixing_length}). In order to impose the \eqref{Eq_tke_mxl_constraint} |
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| 185 | constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$, |
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| 186 | the upward and downward length scales, and evaluate the dissipation and |
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[1224] | 187 | mixing turbulent length scales as (and note that here we use numerical |
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[994] | 188 | indexing): |
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[707] | 189 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 190 | \begin{figure}[!t] \label{Fig_mixing_length} \begin{center} |
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[998] | 191 | \includegraphics[width=1.00\textwidth]{./TexFiles/Figures/Fig_mixing_length.pdf} |
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[707] | 192 | \caption {Illustration of the mixing length computation. } |
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| 193 | \end{center} |
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| 194 | \end{figure} |
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| 195 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 196 | \begin{equation} \label{Eq_tke_mxl2} |
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| 197 | \begin{aligned} |
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[994] | 198 | l_{up\ \ }^{(k)} &= \min \left( l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3T}^{(k)}\ \ \ \; \right) |
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[707] | 199 | \quad &\text{ from $k=1$ to $jpk$ }\ \\ |
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| 200 | l_{dwn}^{(k)} &= \min \left( l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3T}^{(k-1)} \right) |
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| 201 | \quad &\text{ from $k=jpk$ to $1$ }\ \\ |
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| 202 | \end{aligned} |
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| 203 | \end{equation} |
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[994] | 204 | where $l^{(k)}$ is computed using \eqref{Eq_tke_mxl0_1}, |
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| 205 | $i.e.$ $l^{(k)} = \sqrt {2 \bar e^{(k)} / N^{(k)} }$. |
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[707] | 206 | |
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[994] | 207 | In the \np{nmxl}=2 case, the dissipation and mixing length scales take the same |
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| 208 | value: $ l_k= l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right)$, while in the |
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[1224] | 209 | \np{nmxl}=2 case, the dissipation and mixing length scales are give |
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[994] | 210 | as in \citet{Gaspar1990}: |
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[707] | 211 | \begin{equation} \label{Eq_tke_mxl_gaspar} |
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| 212 | \begin{aligned} |
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| 213 | & l_k = \sqrt{\ l_{up} \ \ l_{dwn}\ } \\ |
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| 214 | & l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right) |
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| 215 | \end{aligned} |
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| 216 | \end{equation} |
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| 217 | |
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| 218 | At the sea surface the value of $\bar{e}$ is prescribed from the wind |
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| 219 | stress field: $\bar{e}=ebb\;\left| \tau \right|$ ($ebb=60$ by default) |
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| 220 | with a minimal threshold of $emin0=10^{-4}~m^2.s^{-2}$ (namelist |
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[994] | 221 | parameters). Its value at the bottom of the ocean is assumed to be |
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| 222 | equal to the value of the level just above. The time integration of the |
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| 223 | $\bar{e}$ equation may formally lead to negative values because the |
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| 224 | numerical scheme does not ensure its positivity. To overcome this |
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| 225 | problem, a cut-off in the minimum value of $\bar{e}$ is used. Following |
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| 226 | \citet{Gaspar1990}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. |
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| 227 | This allows the subsequent formulations to match that of\citet{Gargett1984} |
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| 228 | for the diffusion in the thermocline and deep ocean : $(A^{vT} = 10^{-3} / N)$. |
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| 229 | In addition, a cut-off is applied on $A^{vm}$ and $A^{vT}$ to avoid numerical |
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[707] | 230 | instabilities associated with too weak vertical diffusion. They must be |
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| 231 | specified at least larger than the molecular values, and are set through |
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| 232 | \textit{avm0} and \textit{avt0} (\textbf{namelist} parameters). |
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| 233 | |
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| 234 | % ------------------------------------------------------------------------------------------------------------- |
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| 235 | % K Profile Parametrisation (KPP) |
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| 236 | % ------------------------------------------------------------------------------------------------------------- |
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| 237 | \subsection{K Profile Parametrisation (KPP) (\key{zdfkpp}) } |
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| 238 | \label{ZDF_kpp} |
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| 239 | |
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| 240 | %--------------------------------------------namkpp-------------------------------------------------------- |
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| 241 | \namdisplay{namkpp} |
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| 242 | %-------------------------------------------------------------------------------------------------------------- |
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| 243 | |
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[1224] | 244 | The K-Profile Parametrization (KKP) developed by \cite{Large_al_RG94} has been |
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| 245 | implemented in \NEMO by J. Chanut (PhD reference to be added here!). |
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[994] | 246 | |
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[707] | 247 | \colorbox{yellow}{Add a description of KPP here.} |
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| 248 | |
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| 249 | |
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| 250 | % ================================================================ |
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| 251 | % Convection |
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| 252 | % ================================================================ |
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| 253 | \section{Convection} |
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| 254 | \label{ZDF_conv} |
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| 255 | |
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| 256 | %--------------------------------------------namzdf-------------------------------------------------------- |
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| 257 | \namdisplay{namzdf} |
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| 258 | %-------------------------------------------------------------------------------------------------------------- |
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| 259 | |
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| 260 | Static instabilities (i.e. light potential densities under heavy ones) may |
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| 261 | occur at particular ocean grid points. In nature, convective processes |
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| 262 | quickly re-establish the static stability of the water column. These |
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[994] | 263 | processes have been removed from the model via the hydrostatic |
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[1224] | 264 | assumption so they must be parameterized. Three parameterisations |
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[994] | 265 | are available to deal with convective processes: a non-penetrative |
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| 266 | convective adjustment or an enhanced vertical diffusion, or/and the |
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| 267 | use of a turbulent closure scheme. |
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[707] | 268 | |
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| 269 | % ------------------------------------------------------------------------------------------------------------- |
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| 270 | % Non-Penetrative Convective Adjustment |
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| 271 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 272 | \subsection [Non-Penetrative Convective Adjustment (\np{ln\_tranpc}) ] |
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| 273 | {Non-Penetrative Convective Adjustment (\np{ln\_tranpc}=.true.) } |
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[707] | 274 | \label{ZDF_npc} |
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| 275 | |
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| 276 | %--------------------------------------------namnpc-------------------------------------------------------- |
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| 277 | \namdisplay{namnpc} |
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| 278 | %-------------------------------------------------------------------------------------------------------------- |
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| 279 | |
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| 280 | |
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| 281 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[994] | 282 | \begin{figure}[!htb] \label{Fig_npc} \begin{center} |
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[998] | 283 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_npc.pdf} |
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[994] | 284 | \caption {Example of an unstable density profile treated by the non penetrative |
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| 285 | convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from |
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| 286 | the surface to the bottom. It is found to be unstable between levels 3 and 4. |
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| 287 | They are mixed. The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 |
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| 288 | are mixed. The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are |
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| 289 | mixed. The $1^{st}$ step ends since the density profile is then stable below |
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| 290 | the level 3. $2^{nd}$ step: the new $\rho$ profile is checked following the same |
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| 291 | procedure as in $1^{st}$ step: levels 2 to 5 are mixed. The new density profile |
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| 292 | is checked. It is found stable: end of algorithm.} |
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[707] | 293 | \end{center} \end{figure} |
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| 294 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 295 | |
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[994] | 296 | The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}=true. |
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| 297 | It is applied at each \np{nnpc1} time step and mixes downwards instantaneously |
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| 298 | the statically unstable portion of the water column, but only until the density |
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| 299 | structure becomes neutrally stable ($i.e.$ until the mixed portion of the water |
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| 300 | column has \textit{exactly} the density of the water just below) \citep{Madec1991a}. |
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| 301 | The associated algorithm is an iterative process used in the following way |
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| 302 | (Fig. \ref{Fig_npc}): starting from the top of the ocean, the first instability is |
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| 303 | found. Assume in the following that the instability is located between levels |
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| 304 | $k$ and $k+1$. The potential temperature and salinity in the two levels are |
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| 305 | vertically mixed, conserving the heat and salt contents of the water column. |
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| 306 | The new density is then computed by a linear approximation. If the new |
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| 307 | density profile is still unstable between levels $k+1$ and $k+2$, levels $k$, |
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| 308 | $k+1$ and $k+2$ are then mixed. This process is repeated until stability is |
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| 309 | established below the level $k$ (the mixing process can go down to the |
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| 310 | ocean bottom). The algorithm is repeated to check if the density profile |
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[707] | 311 | between level $k-1$ and $k$ is unstable and/or if there is no deeper instability. |
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| 312 | |
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[994] | 313 | This algorithm is significantly different from mixing statically unstable levels |
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| 314 | two by two. The latter procedure cannot converge with a finite number |
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| 315 | of iterations for some vertical profiles while the algorithm used in \NEMO |
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| 316 | converges for any profile in a number of iterations which is less than the |
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| 317 | number of vertical levels. This property is of paramount importance as |
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| 318 | pointed out by \citet{Killworth1989}: it avoids the existence of permanent |
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| 319 | and unrealistic static instabilities at the sea surface. This non-penetrative |
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| 320 | convective algorithm has been proved successful in studies of the deep |
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| 321 | water formation in the north-western Mediterranean Sea |
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| 322 | \citep{Madec1991a, Madec1991b, Madec1991c}. |
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[707] | 323 | |
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[994] | 324 | Note that in the current implementation of this algorithm presents several |
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| 325 | limitations. First, potential density referenced to the sea surface is used to |
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| 326 | check whether the density profile is stable or not. This is a strong |
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| 327 | simplification which leads to large errors for realistic ocean simulations. |
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| 328 | Indeed, many water masses of the world ocean, especially Antarctic Bottom |
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| 329 | Water, are unstable when represented in surface-referenced potential density. |
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| 330 | The scheme will erroneously mix them up. Second, the mixing of potential |
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| 331 | density is assumed to be linear. This assures the convergence of the algorithm |
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| 332 | even when the equation of state is non-linear. Small static instabilities can thus |
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| 333 | persist due to cabbeling: they will be treated at the next time step. |
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| 334 | Third, temperature and salinity, and thus density, are mixed, but the |
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| 335 | corresponding velocity fields remain unchanged. When using a Richardson |
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| 336 | Number dependent eddy viscosity, the mixing of momentum is done through |
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| 337 | the vertical diffusion: after a static adjustment, the Richardson Number is zero |
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| 338 | and thus the eddy viscosity coefficient is at a maximum. When this convective |
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| 339 | adjustment algorithm is used with constant vertical eddy viscosity, spurious |
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| 340 | solutions can occur since the vertical momentum diffusion remains small even |
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| 341 | after a static adjustment. In that case, we recommend the addition of momentum |
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| 342 | mixing in a manner that mimics the mixing in temperature and salinity |
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| 343 | \citep{Speich1992, Speich1996}. |
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[707] | 344 | |
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| 345 | % ------------------------------------------------------------------------------------------------------------- |
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| 346 | % Enhanced Vertical Diffusion |
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| 347 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 348 | \subsection [Enhanced Vertical Diffusion (\np{ln\_zdfevd})] |
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| 349 | {Enhanced Vertical Diffusion (\np{ln\_zdfevd}=.true.)} |
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[707] | 350 | \label{ZDF_evd} |
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| 351 | |
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| 352 | %--------------------------------------------namzdf-------------------------------------------------------- |
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| 353 | \namdisplay{namzdf} |
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| 354 | %-------------------------------------------------------------------------------------------------------------- |
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| 355 | |
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[1224] | 356 | The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}=true. |
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[994] | 357 | In this case, the vertical eddy mixing coefficients are assigned very large values |
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| 358 | (a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable |
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| 359 | ($i.e.$ when the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{Lazar1997, Lazar1999}. |
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| 360 | This is done either on tracers only (\np{n\_evdm}=0) or on both momentum and |
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| 361 | tracers (\np{n\_evdm}=1). |
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[707] | 362 | |
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[994] | 363 | In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and |
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| 364 | if \np{n\_evdm}=1, the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ |
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| 365 | values also, are set equal to the namelist parameter \np{avevd}. A typical value |
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[1224] | 366 | for $avevd$ is between 1 and $100~m^2.s^{-1}$. This parameterisation of |
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[994] | 367 | convective processes is less time consuming than the convective adjustment |
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| 368 | algorithm presented above when mixing both tracers and momentum in the |
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| 369 | case of static instabilities. It requires the use of an implicit time stepping on |
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| 370 | vertical diffusion terms (i.e. \np{ln\_zdfexp}=false). |
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[707] | 371 | |
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| 372 | % ------------------------------------------------------------------------------------------------------------- |
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| 373 | % Turbulent Closure Scheme |
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| 374 | % ------------------------------------------------------------------------------------------------------------- |
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| 375 | \subsection{Turbulent Closure Scheme (\key{zdftke})} |
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| 376 | \label{ZDF_tcs} |
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| 377 | |
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[994] | 378 | The TKE turbulent closure scheme presented in \S\ref{ZDF_tke} and used |
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| 379 | when the \key{zdftke} is defined, in theory solves the problem of statically |
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| 380 | unstable density profiles. In such a case, the term corresponding to the |
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| 381 | destruction of turbulent kinetic energy through stratification in \eqref{Eq_zdftke_e} |
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| 382 | becomes a source term, since $N^2$ is negative. It results in large values of |
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| 383 | $A_T^{vT}$ and $A_T^{vT}$, and also the four neighbouring |
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| 384 | $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1})$. These large values |
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| 385 | restore the static stability of the water column in a way similar to that of the |
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[1224] | 386 | enhanced vertical diffusion parameterisation (\S\ref{ZDF_evd}). However, |
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[994] | 387 | in the vicinity of the sea surface (first ocean layer), the eddy coefficients |
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[1224] | 388 | computed by the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$, |
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[994] | 389 | because the mixing length scale is bounded by the distance to the sea surface |
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| 390 | (see \S\ref{ZDF_tke}). It can thus be useful to combine the enhanced vertical |
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| 391 | diffusion with the turbulent closure scheme, $i.e.$ setting the \np{ln\_zdfnpc} |
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| 392 | namelist parameter to true and defining the \key{zdftke} CPP key all together. |
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[707] | 393 | |
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[994] | 394 | The KPP turbulent closure scheme already includes enhanced vertical diffusion |
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| 395 | in the case of convection, as governed by the variables $bvsqcon$ and $difcon$ |
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| 396 | found in \mdl{zdfkpp}, therefore \np{np\_zdfevd} should not be used with the KPP |
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| 397 | scheme. %gm% + one word on non local flux with KPP scheme trakpp.F90 module... |
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[707] | 398 | |
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| 399 | % ================================================================ |
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| 400 | % Double Diffusion Mixing |
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| 401 | % ================================================================ |
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[817] | 402 | \section [Double Diffusion Mixing (\textit{zdfddm} - \key{zdfddm})] |
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| 403 | {Double Diffusion Mixing (\mdl{zdfddm} module - \key{zdfddm})} |
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[707] | 404 | \label{ZDF_ddm} |
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| 405 | |
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| 406 | %-------------------------------------------namddm-------------------------------------------------------- |
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| 407 | \namdisplay{namddm} |
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| 408 | %-------------------------------------------------------------------------------------------------------------- |
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| 409 | |
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[994] | 410 | Double diffusion occurs when relatively warm, salty water overlies cooler, fresher |
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| 411 | water, or vice versa. The former condition leads to salt fingering and the latter |
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| 412 | to diffusive convection. Double-diffusive phenomena contribute to diapycnal |
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| 413 | mixing in extensive regions of the ocean. \citet{Merryfield1999} include a |
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[1224] | 414 | parameterisation of such phenomena in a global ocean model and show that |
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[994] | 415 | it leads to relatively minor changes in circulation but exerts significant regional |
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| 416 | influences on temperature and salinity. |
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[707] | 417 | |
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| 418 | Diapycnal mixing of S and T are described by diapycnal diffusion coefficients |
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| 419 | \begin{align*} % \label{Eq_zdfddm_Kz} |
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| 420 | &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ |
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| 421 | &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} |
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| 422 | \end{align*} |
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[994] | 423 | where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, |
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[1224] | 424 | and $o$ by processes other than double diffusion. The rates of double-diffusive mixing |
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| 425 | depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, |
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[994] | 426 | where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline |
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| 427 | contraction (see \S\ref{TRA_eos}). To represent mixing of $S$ and $T$ by salt |
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| 428 | fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981): |
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[707] | 429 | \begin{align} \label{Eq_zdfddm_f} |
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| 430 | A_f^{vS} &= \begin{cases} |
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| 431 | \frac{A^{\ast v}}{1+(R_\rho / R_c)^n } &\text{if $R_\rho > 1$ and $N^2>0$ } \\ |
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| 432 | 0 &\text{otherwise} |
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| 433 | \end{cases} |
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[994] | 434 | \\ \label{Eq_zdfddm_f_T} |
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[707] | 435 | A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho |
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| 436 | \end{align} |
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| 437 | |
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| 438 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 439 | \begin{figure}[!t] \label{Fig_zdfddm} \begin{center} |
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[998] | 440 | \includegraphics[width=0.99\textwidth]{./TexFiles/Figures/Fig_zdfddm.pdf} |
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[994] | 441 | \caption {From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ |
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| 442 | and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy |
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| 443 | curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves |
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| 444 | $A^{\ast v} = 10^{-4}~m^2.s^{-1}$ ; (b) diapycnal diffusivities $A_d^{vT}$ and |
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| 445 | $A_d^{vS}$ for temperature and salt in regions of diffusive convection. Heavy |
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[1224] | 446 | curves denote the Federov parameterisation and thin curves the Kelley |
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| 447 | parameterisation. The latter is not implemented in \NEMO. } |
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[707] | 448 | \end{center} \end{figure} |
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| 449 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 450 | |
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[994] | 451 | The factor 0.7 in \eqref{Eq_zdfddm_f_T} reflects the measured ratio |
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| 452 | $\alpha F_T /\beta F_S \approx 0.7$ of buoyancy flux of heat to buoyancy |
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| 453 | flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). Following \citet{Merryfield1999}, |
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| 454 | we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$. |
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[707] | 455 | |
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[1224] | 456 | To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested |
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| 457 | by Federov (1988) is used: |
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[994] | 458 | \begin{align} \label{Eq_zdfddm_d} |
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[707] | 459 | A_d^{vT} &= \begin{cases} |
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| 460 | 1.3635 \, \exp{\left( 4.6\, \exp{ \left[ -0.54\,( R_{\rho}^{-1} - 1 ) \right] } \right)} |
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| 461 | &\text{if $0<R_\rho < 1$ and $N^2>0$ } \\ |
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| 462 | 0 &\text{otherwise} |
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| 463 | \end{cases} |
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[994] | 464 | \\ \label{Eq_zdfddm_d_S} |
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[707] | 465 | A_d^{vS} &= \begin{cases} |
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| 466 | A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) |
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| 467 | &\text{if $0.5 \leq R_\rho<1$ and $N^2>0$ } \\ |
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| 468 | A_d^{vT} \ 0.15 \ R_\rho |
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| 469 | &\text{if $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\ |
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| 470 | 0 &\text{otherwise} |
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| 471 | \end{cases} |
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| 472 | \end{align} |
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| 473 | |
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[994] | 474 | The dependencies of \eqref{Eq_zdfddm_f} to \eqref{Eq_zdfddm_d_S} on $R_\rho$ |
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| 475 | are illustrated in Fig.~\ref{Fig_zdfddm}. Implementing this requires computing |
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| 476 | $R_\rho$ at each grid point on every time step. This is done in \mdl{eosbn2} at the |
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| 477 | same time as $N^2$ is computed. This avoids duplication in the computation of |
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| 478 | $\alpha$ and $\beta$ (which is usually quite expensive). |
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[707] | 479 | |
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| 480 | % ================================================================ |
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| 481 | % Bottom Friction |
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| 482 | % ================================================================ |
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[817] | 483 | \section [Bottom Friction (\textit{zdfbfr})] |
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| 484 | {Bottom Friction (\mdl{zdfbfr} module)} |
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[707] | 485 | \label{ZDF_bfr} |
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| 486 | |
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| 487 | %--------------------------------------------nambfr-------------------------------------------------------- |
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| 488 | \namdisplay{nambfr} |
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| 489 | %-------------------------------------------------------------------------------------------------------------- |
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| 490 | |
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[994] | 491 | Both the surface momentum flux (wind stress) and the bottom momentum |
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| 492 | flux (bottom friction) enter the equations as a condition on the vertical |
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[707] | 493 | diffusive flux. For the bottom boundary layer, one has: |
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| 494 | \begin{equation} \label{Eq_zdfbfr_flux} |
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| 495 | A^{vm} \left( \partial \textbf{U}_h / \partial z \right) = \textbf{F}_h |
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| 496 | \end{equation} |
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| 497 | where $\textbf{F}_h$ is supposed to represent the horizontal momentum flux |
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| 498 | outside the logarithmic turbulent boundary layer (thickness of the order of |
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| 499 | 1~m in the ocean). How $\textbf{F}_h$ influences the interior depends on the |
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| 500 | vertical resolution of the model near the bottom relative to the Ekman layer |
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[994] | 501 | depth. For example, in order to obtain an Ekman layer depth |
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| 502 | $d = \sqrt{2\;A^{vm}} / f = 50$~m, one needs a vertical diffusion coefficient |
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| 503 | $A^{vm} = 0.125$~m$^2$s$^{-1}$ (for a Coriolis frequency |
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| 504 | $f = 10^{-4}$~m$^2$s$^{-1}$). With a background diffusion coefficient |
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| 505 | $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m. |
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| 506 | When the vertical mixing coefficient is this small, using a flux condition is |
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| 507 | equivalent to entering the viscous forces (either wind stress or bottom friction) |
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| 508 | as a body force over the depth of the top or bottom model layer. To illustrate |
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| 509 | this, consider the equation for $u$ at $k$, the last ocean level: |
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[707] | 510 | \begin{equation} \label{Eq_zdfbfr_flux2} |
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| 511 | \frac{\partial u \; (k)}{\partial t} = \frac{1}{e_{3u}} \left[ A^{vm} \; (k) \frac{U \; (k-1) - U \; (k)}{e_{3uw} \; (k-1)} - F_u \right] \approx - \frac{F_u}{e_{3u}} |
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| 512 | \end{equation} |
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[994] | 513 | For example, if the bottom layer thickness is 200~m, the Ekman transport will |
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| 514 | be distributed over that depth. On the other hand, if the vertical resolution |
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[707] | 515 | is high (1~m or less) and a turbulent closure model is used, the turbulent |
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| 516 | Ekman layer will be represented explicitly by the model. However, the |
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| 517 | logarithmic layer is never represented in current primitive equation model |
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[994] | 518 | applications: it is \emph{necessary} to parameterize the flux $\textbf{F}_h $. |
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| 519 | Two choices are available in \NEMO: a linear and a quadratic bottom friction. |
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| 520 | Note that in both cases, the rotation between the interior velocity and the |
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| 521 | bottom friction is neglected in the present release of \NEMO. |
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[707] | 522 | |
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| 523 | % ------------------------------------------------------------------------------------------------------------- |
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| 524 | % Linear Bottom Friction |
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| 525 | % ------------------------------------------------------------------------------------------------------------- |
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[994] | 526 | \subsection{Linear Bottom Friction (\np{nbotfr} = 1) } |
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[707] | 527 | \label{ZDF_bfr_linear} |
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| 528 | |
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[1224] | 529 | The linear bottom friction parameterisation assumes that the bottom friction |
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[994] | 530 | is proportional to the interior velocity (i.e. the velocity of the last model level): |
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[707] | 531 | \begin{equation} \label{Eq_zdfbfr_linear} |
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[994] | 532 | \textbf{F}_h = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b |
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[707] | 533 | \end{equation} |
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[994] | 534 | where $\textbf{U}_h^b$ is the horizontal velocity vector of the bottom ocean |
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| 535 | layer and $r$ is a friction coefficient expressed in m.s$^{-1}$. This coefficient |
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| 536 | is generally estimated by setting a typical decay time $\tau$ in the deep ocean, |
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| 537 | and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted |
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| 538 | values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly1984}. |
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| 539 | A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used |
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| 540 | in quasi-geostrophic models. One may consider the linear friction as an |
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| 541 | approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982}, |
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| 542 | Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed |
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| 543 | of tidal currents of $U_{av} =0.1$~m.s$^{-1}$, and assuming an ocean depth |
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| 544 | $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m.s$^{-1}$. |
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| 545 | This is the default value used in \NEMO. It corresponds to a decay time scale |
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| 546 | of 115~days. It can be changed by specifying \np{bfric1} (namelist parameter). |
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[707] | 547 | |
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[994] | 548 | In the code, the bottom friction is imposed by updating the value of the |
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| 549 | vertical eddy coefficient at the bottom level. Indeed, the discrete formulation |
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| 550 | of (\ref{Eq_zdfbfr_linear}) at the last ocean $T-$level, using the fact that |
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| 551 | $\textbf {U}_h =0$ below the ocean floor, leads to |
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[707] | 552 | \begin{equation} \label{Eq_zdfbfr_linKz} |
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| 553 | \begin{split} |
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| 554 | A_u^{vm} &= r\;e_{3uw}\\ |
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[994] | 555 | A_v^{vm} &= r\;e_{3vw}\\ |
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[707] | 556 | \end{split} |
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| 557 | \end{equation} |
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| 558 | |
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[994] | 559 | This update is done in \mdl{zdfbfr} when \np{nbotfr}=1. The value of $r$ |
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| 560 | used is \np{bfric1}. Setting \np{nbotfr}=3 is equivalent to setting $r=0$ and |
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| 561 | leads to a free-slip bottom boundary condition. Setting \np{nbotfr}=0 sets |
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| 562 | $r=2\;A_{vb}^{\rm {\bf U}} $, where $A_{vb}^{\rm {\bf U}} $ is the background |
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| 563 | vertical eddy coefficient, and a no-slip boundary condition is imposed. |
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[707] | 564 | Note that this latter choice generally leads to an underestimation of the |
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[994] | 565 | bottom friction: for example with a deepest level thickness of $200~m$ |
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| 566 | and $A_{vb}^{\rm {\bf U}} =10^{-4}$m$^2$.s$^{-1}$, the friction coefficient |
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| 567 | is only $r=10^{-6}$m.s$^{-1}$. |
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[707] | 568 | |
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| 569 | % ------------------------------------------------------------------------------------------------------------- |
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| 570 | % Non-Linear Bottom Friction |
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| 571 | % ------------------------------------------------------------------------------------------------------------- |
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[994] | 572 | \subsection{Non-Linear Bottom Friction (\np{nbotfr} = 2)} |
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[707] | 573 | \label{ZDF_bfr_nonlinear} |
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| 574 | |
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[1224] | 575 | The non-linear bottom friction parameterisation assumes that the bottom |
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[707] | 576 | friction is quadratic: |
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| 577 | \begin{equation} \label{Eq_zdfbfr_nonlinear} |
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| 578 | \textbf {F}_h = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h |
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| 579 | }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b |
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| 580 | \end{equation} |
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| 581 | |
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[994] | 582 | with $\textbf{U}_h^b = (u_b\;,\;v_b)$ the horizontal interior velocity ($i.e.$ |
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| 583 | the horizontal velocity of the bottom ocean layer), $C_D$ a drag coefficient, |
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| 584 | and $e_b $ a bottom turbulent kinetic energy due to tides, internal waves |
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| 585 | breaking and other short time scale currents. A typical value of the drag |
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| 586 | coefficient is $C_D = 10^{-3} $. As an example, the CME experiment |
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| 587 | \citep{Treguier1992} uses $C_D = 10^{-3}$ and $e_b = 2.5\;10^{-3}$m$^2$.s$^{-2}$, |
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| 588 | while the FRAM experiment \citep{Killworth1992} uses $e_b =0$ |
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| 589 | and $e_b =2.5\;\;10^{-3}$m$^2$.s$^{-2}$. The FRAM choices have been |
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| 590 | set as default values (\np{bfric2} and \np{bfeb2} namelist parameters). |
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[707] | 591 | |
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[994] | 592 | As for the linear case, the bottom friction is imposed in the code by |
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[707] | 593 | updating the value of the vertical eddy coefficient at the bottom level: |
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| 594 | \begin{equation} \label{Eq_zdfbfr_nonlinKz} |
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| 595 | \begin{split} |
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| 596 | A_u^{vm} &=C_D\; e_{3uw} \left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^ |
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| 597 | {1/2}\\ |
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[994] | 598 | A_v^{vm} &=C_D\; e_{3uw} \left[ \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ |
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[707] | 599 | \end{split} |
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| 600 | \end{equation} |
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| 601 | |
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| 602 | This update is done in \mdl{zdfbfr}. The coefficients that control the strength of the |
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[996] | 603 | non-linear bottom friction are initialized as namelist parameters: $C_D$= \np{bfri2}, |
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| 604 | and $e_b$ =\np{bfeb2}. |
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[707] | 605 | |
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[1225] | 606 | % ================================================================ |
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