[707] | 1 | % ================================================================ |
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[3294] | 2 | % Chapter Vertical Ocean Physics (ZDF) |
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[707] | 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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| 9 | |
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[994] | 10 | |
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[2282] | 11 | \newpage |
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| 12 | $\ $\newline % force a new ligne |
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| 13 | |
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| 14 | |
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[707] | 15 | % ================================================================ |
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| 16 | % Vertical Mixing |
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| 17 | % ================================================================ |
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| 18 | \section{Vertical Mixing} |
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| 19 | \label{ZDF_zdf} |
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| 20 | |
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[994] | 21 | The discrete form of the ocean subgrid scale physics has been presented in |
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| 22 | \S\ref{TRA_zdf} and \S\ref{DYN_zdf}. At the surface and bottom boundaries, |
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| 23 | the turbulent fluxes of momentum, heat and salt have to be defined. At the |
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| 24 | surface they are prescribed from the surface forcing (see Chap.~\ref{SBC}), |
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| 25 | while at the bottom they are set to zero for heat and salt, unless a geothermal |
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| 26 | flux forcing is prescribed as a bottom boundary condition ($i.e.$ \key{trabbl} |
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| 27 | defined, see \S\ref{TRA_bbc}), and specified through a bottom friction |
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[1224] | 28 | parameterisation for momentum (see \S\ref{ZDF_bfr}). |
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[707] | 29 | |
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| 30 | In this section we briefly discuss the various choices offered to compute |
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[994] | 31 | the vertical eddy viscosity and diffusivity coefficients, $A_u^{vm}$ , |
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| 32 | $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- |
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| 33 | points, respectively (see \S\ref{TRA_zdf} and \S\ref{DYN_zdf}). These |
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| 34 | coefficients can be assumed to be either constant, or a function of the local |
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| 35 | Richardson number, or computed from a turbulent closure model (either |
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[6289] | 36 | TKE or GLS formulation). The computation of these coefficients is initialized |
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[994] | 37 | in the \mdl{zdfini} module and performed in the \mdl{zdfric}, \mdl{zdftke} or |
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[6289] | 38 | \mdl{zdfgls} modules. The trends due to the vertical momentum and tracer |
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[994] | 39 | diffusion, including the surface forcing, are computed and added to the |
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| 40 | general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively. |
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| 41 | These trends can be computed using either a forward time stepping scheme |
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[2282] | 42 | (namelist parameter \np{ln\_zdfexp}=true) or a backward time stepping |
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| 43 | scheme (\np{ln\_zdfexp}=false) depending on the magnitude of the mixing |
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| 44 | coefficients, and thus of the formulation used (see \S\ref{STP}). |
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[707] | 45 | |
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| 46 | % ------------------------------------------------------------------------------------------------------------- |
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| 47 | % Constant |
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| 48 | % ------------------------------------------------------------------------------------------------------------- |
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| 49 | \subsection{Constant (\key{zdfcst})} |
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| 50 | \label{ZDF_cst} |
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| 51 | %--------------------------------------------namzdf--------------------------------------------------------- |
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| 52 | \namdisplay{namzdf} |
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| 53 | %-------------------------------------------------------------------------------------------------------------- |
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| 54 | |
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[4147] | 55 | Options are defined through the \ngn{namzdf} namelist variables. |
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[994] | 56 | When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients |
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| 57 | are set to constant values over the whole ocean. This is the crudest way to define |
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| 58 | the vertical ocean physics. It is recommended that this option is only used in |
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| 59 | process studies, not in basin scale simulations. Typical values used in this case are: |
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[707] | 60 | \begin{align*} |
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| 61 | A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1} \\ |
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| 62 | A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1} |
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| 63 | \end{align*} |
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| 64 | |
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[2282] | 65 | These values are set through the \np{rn\_avm0} and \np{rn\_avt0} namelist parameters. |
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[994] | 66 | In all cases, do not use values smaller that those associated with the molecular |
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| 67 | viscosity and diffusivity, that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, |
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| 68 | $\sim10^{-7}~m^2.s^{-1}$ for temperature and $\sim10^{-9}~m^2.s^{-1}$ for salinity. |
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[707] | 69 | |
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| 70 | |
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| 71 | % ------------------------------------------------------------------------------------------------------------- |
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| 72 | % Richardson Number Dependent |
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| 73 | % ------------------------------------------------------------------------------------------------------------- |
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| 74 | \subsection{Richardson Number Dependent (\key{zdfric})} |
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| 75 | \label{ZDF_ric} |
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| 76 | |
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| 77 | %--------------------------------------------namric--------------------------------------------------------- |
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[2282] | 78 | \namdisplay{namzdf_ric} |
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[707] | 79 | %-------------------------------------------------------------------------------------------------------------- |
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| 80 | |
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[994] | 81 | When \key{zdfric} is defined, a local Richardson number dependent formulation |
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[4147] | 82 | for the vertical momentum and tracer eddy coefficients is set through the \ngn{namzdf\_ric} |
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| 83 | namelist variables.The vertical mixing |
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[994] | 84 | coefficients are diagnosed from the large scale variables computed by the model. |
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| 85 | \textit{In situ} measurements have been used to link vertical turbulent activity to |
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| 86 | large scale ocean structures. The hypothesis of a mixing mainly maintained by the |
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| 87 | growth of Kelvin-Helmholtz like instabilities leads to a dependency between the |
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[1224] | 88 | vertical eddy coefficients and the local Richardson number ($i.e.$ the |
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[2282] | 89 | ratio of stratification to vertical shear). Following \citet{Pacanowski_Philander_JPO81}, the following |
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[994] | 90 | formulation has been implemented: |
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[707] | 91 | \begin{equation} \label{Eq_zdfric} |
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| 92 | \left\{ \begin{aligned} |
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| 93 | A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT} \\ |
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| 94 | A^{vm} &= \frac{A^{vT} }{\left( 1+ a \;Ri \right) } + A_b^{vm} |
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| 95 | \end{aligned} \right. |
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| 96 | \end{equation} |
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[994] | 97 | where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson |
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| 98 | number, $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}), |
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| 99 | $A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the |
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| 100 | constant case (see \S\ref{ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ |
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| 101 | is the maximum value that can be reached by the coefficient when $Ri\leq 0$, |
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| 102 | $a=5$ and $n=2$. The last three values can be modified by setting the |
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[2282] | 103 | \np{rn\_avmri}, \np{rn\_alp} and \np{nn\_ric} namelist parameters, respectively. |
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[707] | 104 | |
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[3294] | 105 | A simple mixing-layer model to transfer and dissipate the atmospheric |
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| 106 | forcings (wind-stress and buoyancy fluxes) can be activated setting |
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| 107 | the \np{ln\_mldw} =.true. in the namelist. |
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| 108 | |
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| 109 | In this case, the local depth of turbulent wind-mixing or "Ekman depth" |
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| 110 | $h_{e}(x,y,t)$ is evaluated and the vertical eddy coefficients prescribed within this layer. |
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| 111 | |
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| 112 | This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation: |
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| 113 | \begin{equation} |
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| 114 | h_{e} = Ek \frac {u^{*}} {f_{0}} \\ |
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| 115 | \end{equation} |
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| 116 | where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis |
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| 117 | parameter. |
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| 118 | |
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| 119 | In this similarity height relationship, the turbulent friction velocity: |
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| 120 | \begin{equation} |
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| 121 | u^{*} = \sqrt \frac {|\tau|} {\rho_o} \\ |
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| 122 | \end{equation} |
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| 123 | |
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[3764] | 124 | is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$. |
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[3294] | 125 | The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}. |
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| 126 | Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to |
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| 127 | the empirical values \np{rn\_wtmix} and \np{rn\_wvmix} \citep{Lermusiaux2001}. |
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| 128 | |
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[707] | 129 | % ------------------------------------------------------------------------------------------------------------- |
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| 130 | % TKE Turbulent Closure Scheme |
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| 131 | % ------------------------------------------------------------------------------------------------------------- |
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| 132 | \subsection{TKE Turbulent Closure Scheme (\key{zdftke})} |
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| 133 | \label{ZDF_tke} |
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| 134 | |
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[2282] | 135 | %--------------------------------------------namzdf_tke-------------------------------------------------- |
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| 136 | \namdisplay{namzdf_tke} |
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[707] | 137 | %-------------------------------------------------------------------------------------------------------------- |
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| 138 | |
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[994] | 139 | The vertical eddy viscosity and diffusivity coefficients are computed from a TKE |
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[2282] | 140 | turbulent closure model based on a prognostic equation for $\bar{e}$, the turbulent |
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[1224] | 141 | kinetic energy, and a closure assumption for the turbulent length scales. This |
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[994] | 142 | turbulent closure model has been developed by \citet{Bougeault1989} in the |
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| 143 | atmospheric case, adapted by \citet{Gaspar1990} for the oceanic case, and |
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[2282] | 144 | embedded in OPA, the ancestor of NEMO, by \citet{Blanke1993} for equatorial Atlantic |
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| 145 | simulations. Since then, significant modifications have been introduced by |
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| 146 | \citet{Madec1998} in both the implementation and the formulation of the mixing |
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| 147 | length scale. The time evolution of $\bar{e}$ is the result of the production of |
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| 148 | $\bar{e}$ through vertical shear, its destruction through stratification, its vertical |
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| 149 | diffusion, and its dissipation of \citet{Kolmogorov1942} type: |
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[707] | 150 | \begin{equation} \label{Eq_zdftke_e} |
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| 151 | \frac{\partial \bar{e}}{\partial t} = |
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[2282] | 152 | \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 |
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| 153 | +\left( {\frac{\partial v}{\partial k}} \right)^2} \right] |
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| 154 | -K_\rho\,N^2 |
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[707] | 155 | +\frac{1}{e_3} \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 } |
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| 156 | \;\frac{\partial \bar{e}}{\partial k}} \right] |
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| 157 | - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon } |
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| 158 | \end{equation} |
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| 159 | \begin{equation} \label{Eq_zdftke_kz} |
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| 160 | \begin{split} |
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[2282] | 161 | K_m &= C_k\ l_k\ \sqrt {\bar{e}\; } \\ |
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| 162 | K_\rho &= A^{vm} / P_{rt} |
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[707] | 163 | \end{split} |
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| 164 | \end{equation} |
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[994] | 165 | where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}), |
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| 166 | $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, |
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[2282] | 167 | $P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity |
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| 168 | and diffusivity coefficients. The constants $C_k = 0.1$ and $C_\epsilon = \sqrt {2} /2$ |
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| 169 | $\approx 0.7$ are designed to deal with vertical mixing at any depth \citep{Gaspar1990}. |
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| 170 | They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}. |
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| 171 | $P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function |
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| 172 | of the local Richardson number, $R_i$: |
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[707] | 173 | \begin{align*} \label{Eq_prt} |
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| 174 | P_{rt} = \begin{cases} |
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| 175 | \ \ \ 1 & \text{if $\ R_i \leq 0.2$} \\ |
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| 176 | 5\,R_i & \text{if $\ 0.2 \leq R_i \leq 2$} \\ |
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| 177 | \ \ 10 & \text{if $\ 2 \leq R_i$} |
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| 178 | \end{cases} |
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| 179 | \end{align*} |
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[4147] | 180 | Options are defined through the \ngn{namzdfy\_tke} namelist variables. |
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| 181 | The choice of $P_{rt}$ is controlled by the \np{nn\_pdl} namelist variable. |
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[707] | 182 | |
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[2376] | 183 | At the sea surface, the value of $\bar{e}$ is prescribed from the wind |
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| 184 | stress field as $\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb} |
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| 185 | namelist parameter. The default value of $e_{bb}$ is 3.75. \citep{Gaspar1990}), |
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| 186 | however a much larger value can be used when taking into account the |
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| 187 | surface wave breaking (see below Eq. \eqref{ZDF_Esbc}). |
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| 188 | The bottom value of TKE is assumed to be equal to the value of the level just above. |
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| 189 | The time integration of the $\bar{e}$ equation may formally lead to negative values |
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| 190 | because the numerical scheme does not ensure its positivity. To overcome this |
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| 191 | problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} |
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| 192 | namelist parameter). Following \citet{Gaspar1990}, the cut-off value is set |
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| 193 | to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. This allows the subsequent formulations |
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| 194 | to match that of \citet{Gargett1984} for the diffusion in the thermocline and |
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| 195 | deep ocean : $K_\rho = 10^{-3} / N$. |
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| 196 | In addition, a cut-off is applied on $K_m$ and $K_\rho$ to avoid numerical |
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| 197 | instabilities associated with too weak vertical diffusion. They must be |
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| 198 | specified at least larger than the molecular values, and are set through |
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| 199 | \np{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \S\ref{ZDF_cst}). |
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| 200 | |
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| 201 | \subsubsection{Turbulent length scale} |
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[1224] | 202 | For computational efficiency, the original formulation of the turbulent length |
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[994] | 203 | scales proposed by \citet{Gaspar1990} has been simplified. Four formulations |
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[2282] | 204 | are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist |
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[994] | 205 | parameter. The first two are based on the following first order approximation |
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| 206 | \citep{Blanke1993}: |
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[707] | 207 | \begin{equation} \label{Eq_tke_mxl0_1} |
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[2282] | 208 | l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N |
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[707] | 209 | \end{equation} |
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[2282] | 210 | which is valid in a stable stratified region with constant values of the Brunt- |
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[994] | 211 | Vais\"{a}l\"{a} frequency. The resulting length scale is bounded by the distance |
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[2282] | 212 | to the surface or to the bottom (\np{nn\_mxl} = 0) or by the local vertical scale factor |
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| 213 | (\np{nn\_mxl} = 1). \citet{Blanke1993} notice that this simplification has two major |
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[994] | 214 | drawbacks: it makes no sense for locally unstable stratification and the |
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| 215 | computation no longer uses all the information contained in the vertical density |
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| 216 | profile. To overcome these drawbacks, \citet{Madec1998} introduces the |
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[2282] | 217 | \np{nn\_mxl} = 2 or 3 cases, which add an extra assumption concerning the vertical |
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[994] | 218 | gradient of the computed length scale. So, the length scales are first evaluated |
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| 219 | as in \eqref{Eq_tke_mxl0_1} and then bounded such that: |
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[707] | 220 | \begin{equation} \label{Eq_tke_mxl_constraint} |
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| 221 | \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 |
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| 222 | \qquad \text{with }\ l = l_k = l_\epsilon |
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| 223 | \end{equation} |
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[994] | 224 | \eqref{Eq_tke_mxl_constraint} means that the vertical variations of the length |
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| 225 | scale cannot be larger than the variations of depth. It provides a better |
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| 226 | approximation of the \citet{Gaspar1990} formulation while being much less |
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| 227 | time consuming. In particular, it allows the length scale to be limited not only |
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| 228 | by the distance to the surface or to the ocean bottom but also by the distance |
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| 229 | to a strongly stratified portion of the water column such as the thermocline |
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| 230 | (Fig.~\ref{Fig_mixing_length}). In order to impose the \eqref{Eq_tke_mxl_constraint} |
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| 231 | constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$, |
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| 232 | the upward and downward length scales, and evaluate the dissipation and |
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[2282] | 233 | mixing length scales as (and note that here we use numerical indexing): |
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[707] | 234 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[2376] | 235 | \begin{figure}[!t] \begin{center} |
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[998] | 236 | \includegraphics[width=1.00\textwidth]{./TexFiles/Figures/Fig_mixing_length.pdf} |
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[2376] | 237 | \caption{ \label{Fig_mixing_length} |
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| 238 | Illustration of the mixing length computation. } |
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[707] | 239 | \end{center} |
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| 240 | \end{figure} |
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| 241 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 242 | \begin{equation} \label{Eq_tke_mxl2} |
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| 243 | \begin{aligned} |
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[2282] | 244 | l_{up\ \ }^{(k)} &= \min \left( l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \; \right) |
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[707] | 245 | \quad &\text{ from $k=1$ to $jpk$ }\ \\ |
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[2282] | 246 | l_{dwn}^{(k)} &= \min \left( l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)} \right) |
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[707] | 247 | \quad &\text{ from $k=jpk$ to $1$ }\ \\ |
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| 248 | \end{aligned} |
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| 249 | \end{equation} |
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[994] | 250 | where $l^{(k)}$ is computed using \eqref{Eq_tke_mxl0_1}, |
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[2282] | 251 | $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. |
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[707] | 252 | |
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[2376] | 253 | In the \np{nn\_mxl}~=~2 case, the dissipation and mixing length scales take the same |
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[994] | 254 | value: $ l_k= l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right)$, while in the |
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[2376] | 255 | \np{nn\_mxl}~=~3 case, the dissipation and mixing turbulent length scales are give |
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[994] | 256 | as in \citet{Gaspar1990}: |
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[707] | 257 | \begin{equation} \label{Eq_tke_mxl_gaspar} |
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| 258 | \begin{aligned} |
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| 259 | & l_k = \sqrt{\ l_{up} \ \ l_{dwn}\ } \\ |
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| 260 | & l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right) |
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| 261 | \end{aligned} |
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| 262 | \end{equation} |
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| 263 | |
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[6497] | 264 | At the ocean surface, a non zero length scale is set through the \np{rn\_mxl0} namelist |
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[2376] | 265 | parameter. Usually the surface scale is given by $l_o = \kappa \,z_o$ |
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| 266 | where $\kappa = 0.4$ is von Karman's constant and $z_o$ the roughness |
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| 267 | parameter of the surface. Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94} |
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[6497] | 268 | leads to a 0.04~m, the default value of \np{rn\_mxl0}. In the ocean interior |
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[2376] | 269 | a minimum length scale is set to recover the molecular viscosity when $\bar{e}$ |
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| 270 | reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ). |
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[707] | 271 | |
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[2376] | 272 | |
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| 273 | \subsubsection{Surface wave breaking parameterization} |
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| 274 | %-----------------------------------------------------------------------% |
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| 275 | Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified |
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| 276 | to include the effect of surface wave breaking energetics. This results in a reduction of summertime |
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| 277 | surface temperature when the mixed layer is relatively shallow. The \citet{Mellor_Blumberg_JPO04} |
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| 278 | modifications acts on surface length scale and TKE values and air-sea drag coefficient. |
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| 279 | The latter concerns the bulk formulea and is not discussed here. |
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| 280 | |
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| 281 | Following \citet{Craig_Banner_JPO94}, the boundary condition on surface TKE value is : |
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| 282 | \begin{equation} \label{ZDF_Esbc} |
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| 283 | \bar{e}_o = \frac{1}{2}\,\left( 15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} |
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| 284 | \end{equation} |
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| 285 | where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality |
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| 286 | which depends on the ''wave age'', ranging from 57 for mature waves to 146 for |
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| 287 | younger waves \citep{Mellor_Blumberg_JPO04}. |
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| 288 | The boundary condition on the turbulent length scale follows the Charnock's relation: |
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| 289 | \begin{equation} \label{ZDF_Lsbc} |
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| 290 | l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o} |
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| 291 | \end{equation} |
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| 292 | where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant. |
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| 293 | \citet{Mellor_Blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by \citet{Stacey_JPO99} |
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| 294 | citing observation evidence, and $\alpha_{CB} = 100$ the Craig and Banner's value. |
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| 295 | As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, |
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| 296 | with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}~=~67.83 corresponds |
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[6497] | 297 | to $\alpha_{CB} = 100$. Further setting \np{ln\_mxl0} to true applies \eqref{ZDF_Lsbc} |
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| 298 | as surface boundary condition on length scale, with $\beta$ hard coded to the Stacey's value. |
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[2376] | 299 | Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) |
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| 300 | is applied on surface $\bar{e}$ value. |
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| 301 | |
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| 302 | |
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| 303 | \subsubsection{Langmuir cells} |
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| 304 | %--------------------------------------% |
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| 305 | Langmuir circulations (LC) can be described as ordered large-scale vertical motions |
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| 306 | in the surface layer of the oceans. Although LC have nothing to do with convection, |
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| 307 | the circulation pattern is rather similar to so-called convective rolls in the atmospheric |
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| 308 | boundary layer. The detailed physics behind LC is described in, for example, |
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| 309 | \citet{Craik_Leibovich_JFM76}. The prevailing explanation is that LC arise from |
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| 310 | a nonlinear interaction between the Stokes drift and wind drift currents. |
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| 311 | |
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| 312 | Here we introduced in the TKE turbulent closure the simple parameterization of |
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| 313 | Langmuir circulations proposed by \citep{Axell_JGR02} for a $k-\epsilon$ turbulent closure. |
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| 314 | The parameterization, tuned against large-eddy simulation, includes the whole effect |
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| 315 | of LC in an extra source terms of TKE, $P_{LC}$. |
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| 316 | The presence of $P_{LC}$ in \eqref{Eq_zdftke_e}, the TKE equation, is controlled |
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| 317 | by setting \np{ln\_lc} to \textit{true} in the namtke namelist. |
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| 318 | |
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| 319 | By making an analogy with the characteristic convective velocity scale |
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| 320 | ($e.g.$, \citet{D'Alessio_al_JPO98}), $P_{LC}$ is assumed to be : |
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| 321 | \begin{equation} |
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| 322 | P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}} |
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| 323 | \end{equation} |
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| 324 | where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth. |
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| 325 | With no information about the wave field, $w_{LC}$ is assumed to be proportional to |
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| 326 | the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module |
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| 327 | \footnote{Following \citet{Li_Garrett_JMR93}, the surface Stoke drift velocity |
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| 328 | may be expressed as $u_s = 0.016 \,|U_{10m}|$. Assuming an air density of |
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| 329 | $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of $1.5~10^{-3}$ give the expression |
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| 330 | used of $u_s$ as a function of the module of surface stress}. |
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| 331 | For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as |
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| 332 | at a finite depth $H_{LC}$ (which is often close to the mixed layer depth), and simply |
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| 333 | varies as a sine function in between (a first-order profile for the Langmuir cell structures). |
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| 334 | The resulting expression for $w_{LC}$ is : |
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| 335 | \begin{equation} |
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| 336 | w_{LC} = \begin{cases} |
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| 337 | c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} ) & \text{if $-z \leq H_{LC}$} \\ |
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| 338 | 0 & \text{otherwise} |
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| 339 | \end{cases} |
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| 340 | \end{equation} |
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| 341 | where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise |
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| 342 | to fit LES data. The chosen value yields maximum vertical velocities $w_{LC}$ of the order |
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| 343 | of a few centimeters per second. The value of $c_{LC}$ is set through the \np{rn\_lc} |
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| 344 | namelist parameter, having in mind that it should stay between 0.15 and 0.54 \citep{Axell_JGR02}. |
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| 345 | |
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| 346 | The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations: |
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| 347 | $H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift |
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| 348 | can reach on its own by converting its kinetic energy to potential energy, according to |
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| 349 | \begin{equation} |
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| 350 | - \int_{-H_{LC}}^0 { N^2\;z \;dz} = \frac{1}{2} u_s^2 |
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| 351 | \end{equation} |
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| 352 | |
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| 353 | |
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[2414] | 354 | \subsubsection{Mixing just below the mixed layer} |
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| 355 | %--------------------------------------------------------------% |
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[2376] | 356 | |
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[6289] | 357 | Vertical mixing parameterizations commonly used in ocean general circulation models |
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| 358 | tend to produce mixed-layer depths that are too shallow during summer months and windy conditions. |
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| 359 | This bias is particularly acute over the Southern Ocean. |
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| 360 | To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{Rodgers_2014}. |
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| 361 | The parameterization is an empirical one, $i.e.$ not derived from theoretical considerations, |
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| 362 | but rather is meant to account for observed processes that affect the density structure of |
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| 363 | the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme |
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| 364 | ($i.e.$ near-inertial oscillations and ocean swells and waves). |
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[2376] | 365 | |
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[6289] | 366 | When using this parameterization ($i.e.$ when \np{nn\_etau}~=~1), the TKE input to the ocean ($S$) |
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| 367 | imposed by the winds in the form of near-inertial oscillations, swell and waves is parameterized |
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| 368 | by \eqref{ZDF_Esbc} the standard TKE surface boundary condition, plus a depth depend one given by: |
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| 369 | \begin{equation} \label{ZDF_Ehtau} |
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| 370 | S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau} |
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| 371 | \end{equation} |
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| 372 | where |
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| 373 | $z$ is the depth, |
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| 374 | $e_s$ is TKE surface boundary condition, |
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| 375 | $f_r$ is the fraction of the surface TKE that penetrate in the ocean, |
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| 376 | $h_\tau$ is a vertical mixing length scale that controls exponential shape of the penetration, |
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| 377 | and $f_i$ is the ice concentration (no penetration if $f_i=1$, that is if the ocean is entirely |
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| 378 | covered by sea-ice). |
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| 379 | The value of $f_r$, usually a few percents, is specified through \np{rn\_efr} namelist parameter. |
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| 380 | The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np{nn\_etau}~=~0) |
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| 381 | or a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m |
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| 382 | at high latitudes (\np{nn\_etau}~=~1). |
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| 383 | |
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| 384 | Note that two other option existe, \np{nn\_etau}~=~2, or 3. They correspond to applying |
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| 385 | \eqref{ZDF_Ehtau} only at the base of the mixed layer, or to using the high frequency part |
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| 386 | of the stress to evaluate the fraction of TKE that penetrate the ocean. |
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| 387 | Those two options are obsolescent features introduced for test purposes. |
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| 388 | They will be removed in the next release. |
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| 389 | |
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| 390 | |
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| 391 | |
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[2414] | 392 | % from Burchard et al OM 2008 : |
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[6289] | 393 | % the most critical process not reproduced by statistical turbulence models is the activity of |
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| 394 | % internal waves and their interaction with turbulence. After the Reynolds decomposition, |
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| 395 | % internal waves are in principle included in the RANS equations, but later partially |
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| 396 | % excluded by the hydrostatic assumption and the model resolution. |
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| 397 | % Thus far, the representation of internal wave mixing in ocean models has been relatively crude |
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| 398 | % (e.g. Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002). |
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[2414] | 399 | |
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| 400 | |
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| 401 | |
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[707] | 402 | % ------------------------------------------------------------------------------------------------------------- |
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[2376] | 403 | % TKE discretization considerations |
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[2282] | 404 | % ------------------------------------------------------------------------------------------------------------- |
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| 405 | \subsection{TKE discretization considerations (\key{zdftke})} |
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| 406 | \label{ZDF_tke_ene} |
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| 407 | |
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| 408 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[2376] | 409 | \begin{figure}[!t] \begin{center} |
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[2282] | 410 | \includegraphics[width=1.00\textwidth]{./TexFiles/Figures/Fig_ZDF_TKE_time_scheme.pdf} |
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[2376] | 411 | \caption{ \label{Fig_TKE_time_scheme} |
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| 412 | Illustration of the TKE time integration and its links to the momentum and tracer time integration. } |
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[2282] | 413 | \end{center} |
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| 414 | \end{figure} |
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| 415 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 416 | |
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| 417 | The production of turbulence by vertical shear (the first term of the right hand side |
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| 418 | of \eqref{Eq_zdftke_e}) should balance the loss of kinetic energy associated with |
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| 419 | the vertical momentum diffusion (first line in \eqref{Eq_PE_zdf}). To do so a special care |
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[2541] | 420 | have to be taken for both the time and space discretization of the TKE equation |
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| 421 | \citep{Burchard_OM02,Marsaleix_al_OM08}. |
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[2282] | 422 | |
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| 423 | Let us first address the time stepping issue. Fig.~\ref{Fig_TKE_time_scheme} shows |
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| 424 | how the two-level Leap-Frog time stepping of the momentum and tracer equations interplays |
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| 425 | with the one-level forward time stepping of TKE equation. With this framework, the total loss |
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| 426 | of kinetic energy (in 1D for the demonstration) due to the vertical momentum diffusion is |
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| 427 | obtained by multiplying this quantity by $u^t$ and summing the result vertically: |
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| 428 | \begin{equation} \label{Eq_energ1} |
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| 429 | \begin{split} |
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| 430 | \int_{-H}^{\eta} u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt} \right) \,dz \\ |
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| 431 | &= \Bigl[ u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta} |
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| 432 | - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz } |
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| 433 | \end{split} |
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| 434 | \end{equation} |
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| 435 | Here, the vertical diffusion of momentum is discretized backward in time |
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| 436 | with a coefficient, $K_m$, known at time $t$ (Fig.~\ref{Fig_TKE_time_scheme}), |
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| 437 | as it is required when using the TKE scheme (see \S\ref{STP_forward_imp}). |
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| 438 | The first term of the right hand side of \eqref{Eq_energ1} represents the kinetic energy |
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| 439 | transfer at the surface (atmospheric forcing) and at the bottom (friction effect). |
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| 440 | The second term is always negative. It is the dissipation rate of kinetic energy, |
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| 441 | and thus minus the shear production rate of $\bar{e}$. \eqref{Eq_energ1} |
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| 442 | implies that, to be energetically consistent, the production rate of $\bar{e}$ |
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| 443 | used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as |
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| 444 | ${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ (and not by the more straightforward |
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| 445 | $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$). |
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| 446 | |
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| 447 | A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification |
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| 448 | (second term of the right hand side of \eqref{Eq_zdftke_e}). This term |
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| 449 | must balance the input of potential energy resulting from vertical mixing. |
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| 450 | The rate of change of potential energy (in 1D for the demonstration) due vertical |
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| 451 | mixing is obtained by multiplying vertical density diffusion |
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| 452 | tendency by $g\,z$ and and summing the result vertically: |
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| 453 | \begin{equation} \label{Eq_energ2} |
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| 454 | \begin{split} |
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| 455 | \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt} \right) \,dz \\ |
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| 456 | &= \Bigl[ g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta} |
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| 457 | - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz \\ |
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| 458 | &= - \Bigl[ z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta} |
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| 459 | + \int_{-H}^{\eta}{ \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz } |
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| 460 | \end{split} |
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| 461 | \end{equation} |
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| 462 | where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. |
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| 463 | The first term of the right hand side of \eqref{Eq_energ2} is always zero |
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| 464 | because there is no diffusive flux through the ocean surface and bottom). |
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| 465 | The second term is minus the destruction rate of $\bar{e}$ due to stratification. |
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| 466 | Therefore \eqref{Eq_energ1} implies that, to be energetically consistent, the product |
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| 467 | ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \eqref{Eq_zdftke_e}, the TKE equation. |
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| 468 | |
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| 469 | Let us now address the space discretization issue. |
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| 470 | The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity |
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| 471 | components are in the centre of the side faces of a $t$-box in staggered C-grid |
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| 472 | (Fig.\ref{Fig_cell}). A space averaging is thus required to obtain the shear TKE production term. |
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| 473 | By redoing the \eqref{Eq_energ1} in the 3D case, it can be shown that the product of |
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| 474 | eddy coefficient by the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. |
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| 475 | Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into |
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| 476 | account. |
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| 477 | |
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| 478 | The above energetic considerations leads to |
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| 479 | the following final discrete form for the TKE equation: |
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| 480 | \begin{equation} \label{Eq_zdftke_ene} |
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| 481 | \begin{split} |
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| 482 | \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt} \equiv |
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| 483 | \Biggl\{ \Biggr. |
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| 484 | &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} } |
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| 485 | \ \frac{\delta_{k+1/2}[u^ t ]}{{e_3u}^ t } \right) }^{\,i} \\ |
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| 486 | +&\overline{ \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} } |
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| 487 | \ \frac{\delta_{k+1/2}[v^ t ]}{{e_3v}^ t } \right) }^{\,j} |
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| 488 | \Biggr. \Biggr\} \\ |
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| 489 | % |
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| 490 | - &{K_\rho}^{t-\rdt}\,{(N^2)^t} \\ |
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| 491 | % |
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| 492 | +&\frac{1}{{e_3w}^{t+\rdt}} \;\delta_{k+1/2} \left[ {K_m}^{t-\rdt} \,\frac{\delta_{k}[(\bar{e})^{t+\rdt}]} {{e_3w}^{t+\rdt}} \right] \\ |
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| 493 | % |
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| 494 | - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt} |
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| 495 | \end{split} |
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| 496 | \end{equation} |
---|
| 497 | where the last two terms in \eqref{Eq_zdftke_ene} (vertical diffusion and Kolmogorov dissipation) |
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| 498 | are time stepped using a backward scheme (see\S\ref{STP_forward_imp}). |
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| 499 | Note that the Kolmogorov term has been linearized in time in order to render |
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| 500 | the implicit computation possible. The restart of the TKE scheme |
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| 501 | requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as they all appear in |
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| 502 | the right hand side of \eqref{Eq_zdftke_ene}. For the latter, it is in fact |
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| 503 | the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. |
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| 504 | |
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| 505 | % ------------------------------------------------------------------------------------------------------------- |
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| 506 | % GLS Generic Length Scale Scheme |
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| 507 | % ------------------------------------------------------------------------------------------------------------- |
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| 508 | \subsection{GLS Generic Length Scale (\key{zdfgls})} |
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| 509 | \label{ZDF_gls} |
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| 510 | |
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[2349] | 511 | %--------------------------------------------namzdf_gls--------------------------------------------------------- |
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| 512 | \namdisplay{namzdf_gls} |
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[2282] | 513 | %-------------------------------------------------------------------------------------------------------------- |
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| 514 | |
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| 515 | The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on |
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| 516 | two prognostic equations: one for the turbulent kinetic energy $\bar {e}$, and another |
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| 517 | for the generic length scale, $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}. |
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| 518 | This later variable is defined as : $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, |
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| 519 | where the triplet $(p, m, n)$ value given in Tab.\ref{Tab_GLS} allows to recover |
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| 520 | a number of well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982}, |
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| 521 | $k$-$\epsilon$ \citep{Rodi_1987}, $k$-$\omega$ \citep{Wilcox_1988} |
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| 522 | among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}). |
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| 523 | The GLS scheme is given by the following set of equations: |
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| 524 | \begin{equation} \label{Eq_zdfgls_e} |
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| 525 | \frac{\partial \bar{e}}{\partial t} = |
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| 526 | \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 |
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| 527 | +\left( \frac{\partial v}{\partial k} \right)^2} \right] |
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| 528 | -K_\rho \,N^2 |
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| 529 | +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right] |
---|
| 530 | - \epsilon |
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| 531 | \end{equation} |
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| 532 | |
---|
| 533 | \begin{equation} \label{Eq_zdfgls_psi} |
---|
| 534 | \begin{split} |
---|
| 535 | \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ |
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| 536 | \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 |
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| 537 | +\left( \frac{\partial v}{\partial k} \right)^2} \right] |
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| 538 | - C_3 \,K_\rho\,N^2 - C_2 \,\epsilon \,Fw \right\} \\ |
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| 539 | &+\frac{1}{e_3} \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 } |
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| 540 | \;\frac{\partial \psi}{\partial k}} \right]\; |
---|
| 541 | \end{split} |
---|
| 542 | \end{equation} |
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| 543 | |
---|
| 544 | \begin{equation} \label{Eq_zdfgls_kz} |
---|
| 545 | \begin{split} |
---|
| 546 | K_m &= C_{\mu} \ \sqrt {\bar{e}} \ l \\ |
---|
| 547 | K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l |
---|
| 548 | \end{split} |
---|
| 549 | \end{equation} |
---|
| 550 | |
---|
| 551 | \begin{equation} \label{Eq_zdfgls_eps} |
---|
| 552 | {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; |
---|
| 553 | \end{equation} |
---|
| 554 | where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}) |
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| 555 | and $\epsilon$ the dissipation rate. |
---|
| 556 | The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) |
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| 557 | depends of the choice of the turbulence model. Four different turbulent models are pre-defined |
---|
[2349] | 558 | (Tab.\ref{Tab_GLS}). They are made available through the \np{nn\_clo} namelist parameter. |
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[2282] | 559 | |
---|
| 560 | %--------------------------------------------------TABLE-------------------------------------------------- |
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[2376] | 561 | \begin{table}[htbp] \begin{center} |
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[2282] | 562 | %\begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c} |
---|
| 563 | \begin{tabular}{ccccc} |
---|
| 564 | & $k-kl$ & $k-\epsilon$ & $k-\omega$ & generic \\ |
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| 565 | % & \citep{Mellor_Yamada_1982} & \citep{Rodi_1987} & \citep{Wilcox_1988} & \\ |
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| 566 | \hline \hline |
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| 567 | \np{nn\_clo} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} \\ |
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| 568 | \hline |
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| 569 | $( p , n , m )$ & ( 0 , 1 , 1 ) & ( 3 , 1.5 , -1 ) & ( -1 , 0.5 , -1 ) & ( 2 , 1 , -0.67 ) \\ |
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| 570 | $\sigma_k$ & 2.44 & 1. & 2. & 0.8 \\ |
---|
| 571 | $\sigma_\psi$ & 2.44 & 1.3 & 2. & 1.07 \\ |
---|
| 572 | $C_1$ & 0.9 & 1.44 & 0.555 & 1. \\ |
---|
| 573 | $C_2$ & 0.5 & 1.92 & 0.833 & 1.22 \\ |
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| 574 | $C_3$ & 1. & 1. & 1. & 1. \\ |
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| 575 | $F_{wall}$ & Yes & -- & -- & -- \\ |
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| 576 | \hline |
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| 577 | \hline |
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| 578 | \end{tabular} |
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[2376] | 579 | \caption{ \label{Tab_GLS} |
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| 580 | Set of predefined GLS parameters, or equivalently predefined turbulence models available |
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[4147] | 581 | with \key{zdfgls} and controlled by the \np{nn\_clos} namelist variable in \ngn{namzdf\_gls} .} |
---|
[2376] | 582 | \end{center} \end{table} |
---|
[2282] | 583 | %-------------------------------------------------------------------------------------------------------------- |
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| 584 | |
---|
| 585 | In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force |
---|
[2349] | 586 | the convergence of the mixing length towards $K z_b$ ($K$: Kappa and $z_b$: rugosity length) |
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[2282] | 587 | value near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$ |
---|
| 588 | are calculated from stability function proposed by \citet{Galperin_al_JAS88}, or by \citet{Kantha_Clayson_1994} |
---|
[2376] | 589 | or one of the two functions suggested by \citet{Canuto_2001} (\np{nn\_stab\_func} = 0, 1, 2 or 3, resp.}). |
---|
| 590 | The value of $C_{0\mu}$ depends of the choice of the stability function. |
---|
[2282] | 591 | |
---|
| 592 | The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated |
---|
| 593 | thanks to Dirichlet or Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp. |
---|
[2376] | 594 | As for TKE closure , the wave effect on the mixing is considered when \np{ln\_crban}~=~true |
---|
| 595 | \citep{Craig_Banner_JPO94, Mellor_Blumberg_JPO04}. The \np{rn\_crban} namelist parameter |
---|
| 596 | is $\alpha_{CB}$ in \eqref{ZDF_Esbc} and \np{rn\_charn} provides the value of $\beta$ in \eqref{ZDF_Lsbc}. |
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[2282] | 597 | |
---|
| 598 | The $\psi$ equation is known to fail in stably stratified flows, and for this reason |
---|
| 599 | almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy. |
---|
| 600 | With this clipping, the maximum permissible length scale is determined by |
---|
| 601 | $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$. A value of $c_{lim} = 0.53$ is often used |
---|
| 602 | \citep{Galperin_al_JAS88}. \cite{Umlauf_Burchard_CSR05} show that the value of |
---|
| 603 | the clipping factor is of crucial importance for the entrainment depth predicted in |
---|
| 604 | stably stratified situations, and that its value has to be chosen in accordance |
---|
[3294] | 605 | with the algebraic model for the turbulent fluxes. The clipping is only activated |
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[2349] | 606 | if \np{ln\_length\_lim}=true, and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value. |
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[2282] | 607 | |
---|
[2376] | 608 | The time and space discretization of the GLS equations follows the same energetic |
---|
| 609 | consideration as for the TKE case described in \S\ref{ZDF_tke_ene} \citep{Burchard_OM02}. |
---|
| 610 | Examples of performance of the 4 turbulent closure scheme can be found in \citet{Warner_al_OM05}. |
---|
| 611 | |
---|
[707] | 612 | |
---|
| 613 | % ================================================================ |
---|
| 614 | % Convection |
---|
| 615 | % ================================================================ |
---|
| 616 | \section{Convection} |
---|
| 617 | \label{ZDF_conv} |
---|
| 618 | |
---|
| 619 | %--------------------------------------------namzdf-------------------------------------------------------- |
---|
| 620 | \namdisplay{namzdf} |
---|
| 621 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 622 | |
---|
| 623 | Static instabilities (i.e. light potential densities under heavy ones) may |
---|
| 624 | occur at particular ocean grid points. In nature, convective processes |
---|
| 625 | quickly re-establish the static stability of the water column. These |
---|
[994] | 626 | processes have been removed from the model via the hydrostatic |
---|
[1224] | 627 | assumption so they must be parameterized. Three parameterisations |
---|
[994] | 628 | are available to deal with convective processes: a non-penetrative |
---|
| 629 | convective adjustment or an enhanced vertical diffusion, or/and the |
---|
| 630 | use of a turbulent closure scheme. |
---|
[707] | 631 | |
---|
| 632 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 633 | % Non-Penetrative Convective Adjustment |
---|
| 634 | % ------------------------------------------------------------------------------------------------------------- |
---|
[817] | 635 | \subsection [Non-Penetrative Convective Adjustment (\np{ln\_tranpc}) ] |
---|
| 636 | {Non-Penetrative Convective Adjustment (\np{ln\_tranpc}=.true.) } |
---|
[707] | 637 | \label{ZDF_npc} |
---|
| 638 | |
---|
[2282] | 639 | %--------------------------------------------namzdf-------------------------------------------------------- |
---|
| 640 | \namdisplay{namzdf} |
---|
[707] | 641 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 642 | |
---|
| 643 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[2376] | 644 | \begin{figure}[!htb] \begin{center} |
---|
[998] | 645 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_npc.pdf} |
---|
[2376] | 646 | \caption{ \label{Fig_npc} |
---|
| 647 | Example of an unstable density profile treated by the non penetrative |
---|
[994] | 648 | convective adjustment algorithm. $1^{st}$ step: the initial profile is checked from |
---|
| 649 | the surface to the bottom. It is found to be unstable between levels 3 and 4. |
---|
| 650 | They are mixed. The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 |
---|
| 651 | are mixed. The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are |
---|
| 652 | mixed. The $1^{st}$ step ends since the density profile is then stable below |
---|
| 653 | the level 3. $2^{nd}$ step: the new $\rho$ profile is checked following the same |
---|
| 654 | procedure as in $1^{st}$ step: levels 2 to 5 are mixed. The new density profile |
---|
| 655 | is checked. It is found stable: end of algorithm.} |
---|
[707] | 656 | \end{center} \end{figure} |
---|
| 657 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 658 | |
---|
[4147] | 659 | Options are defined through the \ngn{namzdf} namelist variables. |
---|
[6289] | 660 | The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}~=~\textit{true}. |
---|
[2282] | 661 | It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously |
---|
[994] | 662 | the statically unstable portion of the water column, but only until the density |
---|
| 663 | structure becomes neutrally stable ($i.e.$ until the mixed portion of the water |
---|
[2282] | 664 | column has \textit{exactly} the density of the water just below) \citep{Madec_al_JPO91}. |
---|
[994] | 665 | The associated algorithm is an iterative process used in the following way |
---|
| 666 | (Fig. \ref{Fig_npc}): starting from the top of the ocean, the first instability is |
---|
| 667 | found. Assume in the following that the instability is located between levels |
---|
[6289] | 668 | $k$ and $k+1$. The temperature and salinity in the two levels are |
---|
[994] | 669 | vertically mixed, conserving the heat and salt contents of the water column. |
---|
| 670 | The new density is then computed by a linear approximation. If the new |
---|
| 671 | density profile is still unstable between levels $k+1$ and $k+2$, levels $k$, |
---|
| 672 | $k+1$ and $k+2$ are then mixed. This process is repeated until stability is |
---|
| 673 | established below the level $k$ (the mixing process can go down to the |
---|
| 674 | ocean bottom). The algorithm is repeated to check if the density profile |
---|
[707] | 675 | between level $k-1$ and $k$ is unstable and/or if there is no deeper instability. |
---|
| 676 | |
---|
[994] | 677 | This algorithm is significantly different from mixing statically unstable levels |
---|
| 678 | two by two. The latter procedure cannot converge with a finite number |
---|
| 679 | of iterations for some vertical profiles while the algorithm used in \NEMO |
---|
| 680 | converges for any profile in a number of iterations which is less than the |
---|
| 681 | number of vertical levels. This property is of paramount importance as |
---|
| 682 | pointed out by \citet{Killworth1989}: it avoids the existence of permanent |
---|
| 683 | and unrealistic static instabilities at the sea surface. This non-penetrative |
---|
| 684 | convective algorithm has been proved successful in studies of the deep |
---|
| 685 | water formation in the north-western Mediterranean Sea |
---|
[2282] | 686 | \citep{Madec_al_JPO91, Madec_al_DAO91, Madec_Crepon_Bk91}. |
---|
[707] | 687 | |
---|
[6289] | 688 | The current implementation has been modified in order to deal with any non linear |
---|
| 689 | equation of seawater (L. Brodeau, personnal communication). |
---|
| 690 | Two main differences have been introduced compared to the original algorithm: |
---|
| 691 | $(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency |
---|
| 692 | (not the the difference in potential density) ; |
---|
| 693 | $(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients |
---|
| 694 | are vertically mixed in the same way their temperature and salinity has been mixed. |
---|
| 695 | These two modifications allow the algorithm to perform properly and accurately |
---|
| 696 | with TEOS10 or EOS-80 without having to recompute the expansion coefficients at each |
---|
| 697 | mixing iteration. |
---|
[707] | 698 | |
---|
| 699 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 700 | % Enhanced Vertical Diffusion |
---|
| 701 | % ------------------------------------------------------------------------------------------------------------- |
---|
[817] | 702 | \subsection [Enhanced Vertical Diffusion (\np{ln\_zdfevd})] |
---|
[6289] | 703 | {Enhanced Vertical Diffusion (\np{ln\_zdfevd}=true)} |
---|
[707] | 704 | \label{ZDF_evd} |
---|
| 705 | |
---|
| 706 | %--------------------------------------------namzdf-------------------------------------------------------- |
---|
| 707 | \namdisplay{namzdf} |
---|
| 708 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 709 | |
---|
[4147] | 710 | Options are defined through the \ngn{namzdf} namelist variables. |
---|
[1224] | 711 | The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}=true. |
---|
[994] | 712 | In this case, the vertical eddy mixing coefficients are assigned very large values |
---|
| 713 | (a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable |
---|
[2282] | 714 | ($i.e.$ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) |
---|
| 715 | \citep{Lazar_PhD97, Lazar_al_JPO99}. This is done either on tracers only |
---|
| 716 | (\np{nn\_evdm}=0) or on both momentum and tracers (\np{nn\_evdm}=1). |
---|
[707] | 717 | |
---|
[994] | 718 | In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and |
---|
[2282] | 719 | if \np{nn\_evdm}=1, the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ |
---|
| 720 | values also, are set equal to the namelist parameter \np{rn\_avevd}. A typical value |
---|
| 721 | for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$. This parameterisation of |
---|
[994] | 722 | convective processes is less time consuming than the convective adjustment |
---|
| 723 | algorithm presented above when mixing both tracers and momentum in the |
---|
| 724 | case of static instabilities. It requires the use of an implicit time stepping on |
---|
| 725 | vertical diffusion terms (i.e. \np{ln\_zdfexp}=false). |
---|
[707] | 726 | |
---|
[2282] | 727 | Note that the stability test is performed on both \textit{before} and \textit{now} |
---|
| 728 | values of $N^2$. This removes a potential source of divergence of odd and |
---|
| 729 | even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \S\ref{STP_mLF}). |
---|
| 730 | |
---|
[707] | 731 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 732 | % Turbulent Closure Scheme |
---|
| 733 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2349] | 734 | \subsection{Turbulent Closure Scheme (\key{zdftke} or \key{zdfgls})} |
---|
[707] | 735 | \label{ZDF_tcs} |
---|
| 736 | |
---|
[2349] | 737 | The turbulent closure scheme presented in \S\ref{ZDF_tke} and \S\ref{ZDF_gls} |
---|
| 738 | (\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically |
---|
[994] | 739 | unstable density profiles. In such a case, the term corresponding to the |
---|
| 740 | destruction of turbulent kinetic energy through stratification in \eqref{Eq_zdftke_e} |
---|
[2349] | 741 | or \eqref{Eq_zdfgls_e} becomes a source term, since $N^2$ is negative. |
---|
| 742 | It results in large values of $A_T^{vT}$ and $A_T^{vT}$, and also the four neighbouring |
---|
[994] | 743 | $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1})$. These large values |
---|
| 744 | restore the static stability of the water column in a way similar to that of the |
---|
[1224] | 745 | enhanced vertical diffusion parameterisation (\S\ref{ZDF_evd}). However, |
---|
[994] | 746 | in the vicinity of the sea surface (first ocean layer), the eddy coefficients |
---|
[1224] | 747 | computed by the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$, |
---|
[2349] | 748 | because the mixing length scale is bounded by the distance to the sea surface. |
---|
| 749 | It can thus be useful to combine the enhanced vertical |
---|
[994] | 750 | diffusion with the turbulent closure scheme, $i.e.$ setting the \np{ln\_zdfnpc} |
---|
[2349] | 751 | namelist parameter to true and defining the turbulent closure CPP key all together. |
---|
[707] | 752 | |
---|
[994] | 753 | The KPP turbulent closure scheme already includes enhanced vertical diffusion |
---|
| 754 | in the case of convection, as governed by the variables $bvsqcon$ and $difcon$ |
---|
[2282] | 755 | found in \mdl{zdfkpp}, therefore \np{ln\_zdfevd}=false should be used with the KPP |
---|
[994] | 756 | scheme. %gm% + one word on non local flux with KPP scheme trakpp.F90 module... |
---|
[707] | 757 | |
---|
| 758 | % ================================================================ |
---|
| 759 | % Double Diffusion Mixing |
---|
| 760 | % ================================================================ |
---|
[2349] | 761 | \section [Double Diffusion Mixing (\key{zdfddm})] |
---|
| 762 | {Double Diffusion Mixing (\key{zdfddm})} |
---|
[707] | 763 | \label{ZDF_ddm} |
---|
| 764 | |
---|
[2282] | 765 | %-------------------------------------------namzdf_ddm------------------------------------------------- |
---|
| 766 | \namdisplay{namzdf_ddm} |
---|
[707] | 767 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 768 | |
---|
[4147] | 769 | Options are defined through the \ngn{namzdf\_ddm} namelist variables. |
---|
[994] | 770 | Double diffusion occurs when relatively warm, salty water overlies cooler, fresher |
---|
| 771 | water, or vice versa. The former condition leads to salt fingering and the latter |
---|
| 772 | to diffusive convection. Double-diffusive phenomena contribute to diapycnal |
---|
| 773 | mixing in extensive regions of the ocean. \citet{Merryfield1999} include a |
---|
[1224] | 774 | parameterisation of such phenomena in a global ocean model and show that |
---|
[994] | 775 | it leads to relatively minor changes in circulation but exerts significant regional |
---|
[2349] | 776 | influences on temperature and salinity. This parameterisation has been |
---|
| 777 | introduced in \mdl{zdfddm} module and is controlled by the \key{zdfddm} CPP key. |
---|
[707] | 778 | |
---|
| 779 | Diapycnal mixing of S and T are described by diapycnal diffusion coefficients |
---|
| 780 | \begin{align*} % \label{Eq_zdfddm_Kz} |
---|
| 781 | &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ |
---|
| 782 | &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} |
---|
| 783 | \end{align*} |
---|
[994] | 784 | where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, |
---|
[2349] | 785 | and $o$ by processes other than double diffusion. The rates of double-diffusive |
---|
| 786 | mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, |
---|
[994] | 787 | where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline |
---|
| 788 | contraction (see \S\ref{TRA_eos}). To represent mixing of $S$ and $T$ by salt |
---|
| 789 | fingering, we adopt the diapycnal diffusivities suggested by Schmitt (1981): |
---|
[707] | 790 | \begin{align} \label{Eq_zdfddm_f} |
---|
| 791 | A_f^{vS} &= \begin{cases} |
---|
| 792 | \frac{A^{\ast v}}{1+(R_\rho / R_c)^n } &\text{if $R_\rho > 1$ and $N^2>0$ } \\ |
---|
| 793 | 0 &\text{otherwise} |
---|
| 794 | \end{cases} |
---|
[994] | 795 | \\ \label{Eq_zdfddm_f_T} |
---|
[707] | 796 | A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho |
---|
| 797 | \end{align} |
---|
| 798 | |
---|
| 799 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[2376] | 800 | \begin{figure}[!t] \begin{center} |
---|
[998] | 801 | \includegraphics[width=0.99\textwidth]{./TexFiles/Figures/Fig_zdfddm.pdf} |
---|
[2376] | 802 | \caption{ \label{Fig_zdfddm} |
---|
| 803 | From \citet{Merryfield1999} : (a) Diapycnal diffusivities $A_f^{vT}$ |
---|
[994] | 804 | and $A_f^{vS}$ for temperature and salt in regions of salt fingering. Heavy |
---|
| 805 | curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves |
---|
| 806 | $A^{\ast v} = 10^{-4}~m^2.s^{-1}$ ; (b) diapycnal diffusivities $A_d^{vT}$ and |
---|
| 807 | $A_d^{vS}$ for temperature and salt in regions of diffusive convection. Heavy |
---|
[1224] | 808 | curves denote the Federov parameterisation and thin curves the Kelley |
---|
| 809 | parameterisation. The latter is not implemented in \NEMO. } |
---|
[707] | 810 | \end{center} \end{figure} |
---|
| 811 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 812 | |
---|
[994] | 813 | The factor 0.7 in \eqref{Eq_zdfddm_f_T} reflects the measured ratio |
---|
| 814 | $\alpha F_T /\beta F_S \approx 0.7$ of buoyancy flux of heat to buoyancy |
---|
| 815 | flux of salt ($e.g.$, \citet{McDougall_Taylor_JMR84}). Following \citet{Merryfield1999}, |
---|
| 816 | we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$. |
---|
[707] | 817 | |
---|
[2282] | 818 | To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested by Federov (1988) is used: |
---|
[994] | 819 | \begin{align} \label{Eq_zdfddm_d} |
---|
[707] | 820 | A_d^{vT} &= \begin{cases} |
---|
| 821 | 1.3635 \, \exp{\left( 4.6\, \exp{ \left[ -0.54\,( R_{\rho}^{-1} - 1 ) \right] } \right)} |
---|
| 822 | &\text{if $0<R_\rho < 1$ and $N^2>0$ } \\ |
---|
| 823 | 0 &\text{otherwise} |
---|
| 824 | \end{cases} |
---|
[994] | 825 | \\ \label{Eq_zdfddm_d_S} |
---|
[707] | 826 | A_d^{vS} &= \begin{cases} |
---|
| 827 | A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) |
---|
| 828 | &\text{if $0.5 \leq R_\rho<1$ and $N^2>0$ } \\ |
---|
| 829 | A_d^{vT} \ 0.15 \ R_\rho |
---|
| 830 | &\text{if $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\ |
---|
| 831 | 0 &\text{otherwise} |
---|
| 832 | \end{cases} |
---|
| 833 | \end{align} |
---|
| 834 | |
---|
[994] | 835 | The dependencies of \eqref{Eq_zdfddm_f} to \eqref{Eq_zdfddm_d_S} on $R_\rho$ |
---|
| 836 | are illustrated in Fig.~\ref{Fig_zdfddm}. Implementing this requires computing |
---|
| 837 | $R_\rho$ at each grid point on every time step. This is done in \mdl{eosbn2} at the |
---|
| 838 | same time as $N^2$ is computed. This avoids duplication in the computation of |
---|
| 839 | $\alpha$ and $\beta$ (which is usually quite expensive). |
---|
[707] | 840 | |
---|
| 841 | % ================================================================ |
---|
| 842 | % Bottom Friction |
---|
| 843 | % ================================================================ |
---|
[6320] | 844 | \section [Bottom and Top Friction (\textit{zdfbfr})] {Bottom and Top Friction (\mdl{zdfbfr} module)} |
---|
[707] | 845 | \label{ZDF_bfr} |
---|
| 846 | |
---|
| 847 | %--------------------------------------------nambfr-------------------------------------------------------- |
---|
| 848 | \namdisplay{nambfr} |
---|
| 849 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 850 | |
---|
[5120] | 851 | Options to define the top and bottom friction are defined through the \ngn{nambfr} namelist variables. |
---|
[6320] | 852 | The bottom friction represents the friction generated by the bathymetry. |
---|
| 853 | The top friction represents the friction generated by the ice shelf/ocean interface. |
---|
[6497] | 854 | As the friction processes at the top and bottom are treated in similar way, |
---|
| 855 | only the bottom friction is described in detail below. |
---|
[5120] | 856 | |
---|
[6320] | 857 | |
---|
[994] | 858 | Both the surface momentum flux (wind stress) and the bottom momentum |
---|
| 859 | flux (bottom friction) enter the equations as a condition on the vertical |
---|
[707] | 860 | diffusive flux. For the bottom boundary layer, one has: |
---|
| 861 | \begin{equation} \label{Eq_zdfbfr_flux} |
---|
[2282] | 862 | A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} |
---|
[707] | 863 | \end{equation} |
---|
[2282] | 864 | where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum |
---|
[707] | 865 | outside the logarithmic turbulent boundary layer (thickness of the order of |
---|
[2282] | 866 | 1~m in the ocean). How ${\cal F}_h^{\textbf U}$ influences the interior depends on the |
---|
[707] | 867 | vertical resolution of the model near the bottom relative to the Ekman layer |
---|
[994] | 868 | depth. For example, in order to obtain an Ekman layer depth |
---|
| 869 | $d = \sqrt{2\;A^{vm}} / f = 50$~m, one needs a vertical diffusion coefficient |
---|
| 870 | $A^{vm} = 0.125$~m$^2$s$^{-1}$ (for a Coriolis frequency |
---|
| 871 | $f = 10^{-4}$~m$^2$s$^{-1}$). With a background diffusion coefficient |
---|
| 872 | $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m. |
---|
| 873 | When the vertical mixing coefficient is this small, using a flux condition is |
---|
| 874 | equivalent to entering the viscous forces (either wind stress or bottom friction) |
---|
| 875 | as a body force over the depth of the top or bottom model layer. To illustrate |
---|
| 876 | this, consider the equation for $u$ at $k$, the last ocean level: |
---|
[707] | 877 | \begin{equation} \label{Eq_zdfbfr_flux2} |
---|
[2282] | 878 | \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}} |
---|
[707] | 879 | \end{equation} |
---|
[2282] | 880 | If the bottom layer thickness is 200~m, the Ekman transport will |
---|
[994] | 881 | be distributed over that depth. On the other hand, if the vertical resolution |
---|
[707] | 882 | is high (1~m or less) and a turbulent closure model is used, the turbulent |
---|
| 883 | Ekman layer will be represented explicitly by the model. However, the |
---|
| 884 | logarithmic layer is never represented in current primitive equation model |
---|
[2282] | 885 | applications: it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $. |
---|
[994] | 886 | Two choices are available in \NEMO: a linear and a quadratic bottom friction. |
---|
| 887 | Note that in both cases, the rotation between the interior velocity and the |
---|
| 888 | bottom friction is neglected in the present release of \NEMO. |
---|
[707] | 889 | |
---|
[2282] | 890 | In the code, the bottom friction is imposed by adding the trend due to the bottom |
---|
| 891 | friction to the general momentum trend in \mdl{dynbfr}. For the time-split surface |
---|
| 892 | pressure gradient algorithm, the momentum trend due to the barotropic component |
---|
| 893 | needs to be handled separately. For this purpose it is convenient to compute and |
---|
| 894 | store coefficients which can be simply combined with bottom velocities and geometric |
---|
| 895 | values to provide the momentum trend due to bottom friction. |
---|
| 896 | These coefficients are computed in \mdl{zdfbfr} and generally take the form |
---|
| 897 | $c_b^{\textbf U}$ where: |
---|
| 898 | \begin{equation} \label{Eq_zdfbfr_bdef} |
---|
| 899 | \frac{\partial {\textbf U_h}}{\partial t} = |
---|
| 900 | - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b |
---|
| 901 | \end{equation} |
---|
| 902 | where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity. |
---|
| 903 | |
---|
[707] | 904 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 905 | % Linear Bottom Friction |
---|
| 906 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 907 | \subsection{Linear Bottom Friction (\np{nn\_botfr} = 0 or 1) } |
---|
[707] | 908 | \label{ZDF_bfr_linear} |
---|
| 909 | |
---|
[2282] | 910 | The linear bottom friction parameterisation (including the special case |
---|
| 911 | of a free-slip condition) assumes that the bottom friction |
---|
| 912 | is proportional to the interior velocity (i.e. the velocity of the last |
---|
| 913 | model level): |
---|
[707] | 914 | \begin{equation} \label{Eq_zdfbfr_linear} |
---|
[2282] | 915 | {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b |
---|
[707] | 916 | \end{equation} |
---|
[2282] | 917 | where $r$ is a friction coefficient expressed in ms$^{-1}$. |
---|
| 918 | This coefficient is generally estimated by setting a typical decay time |
---|
| 919 | $\tau$ in the deep ocean, |
---|
[994] | 920 | and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted |
---|
[2282] | 921 | values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}. |
---|
[994] | 922 | A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used |
---|
| 923 | in quasi-geostrophic models. One may consider the linear friction as an |
---|
| 924 | approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982}, |
---|
| 925 | Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed |
---|
[2282] | 926 | of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, and assuming an ocean depth |
---|
| 927 | $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$. |
---|
[994] | 928 | This is the default value used in \NEMO. It corresponds to a decay time scale |
---|
[6497] | 929 | of 115~days. It can be changed by specifying \np{rn\_bfri1} (namelist parameter). |
---|
[707] | 930 | |
---|
[2282] | 931 | For the linear friction case the coefficients defined in the general |
---|
| 932 | expression \eqref{Eq_zdfbfr_bdef} are: |
---|
| 933 | \begin{equation} \label{Eq_zdfbfr_linbfr_b} |
---|
[707] | 934 | \begin{split} |
---|
[2282] | 935 | c_b^u &= - r\\ |
---|
| 936 | c_b^v &= - r\\ |
---|
[707] | 937 | \end{split} |
---|
| 938 | \end{equation} |
---|
[6497] | 939 | When \np{nn\_botfr}=1, the value of $r$ used is \np{rn\_bfri1}. |
---|
[2282] | 940 | Setting \np{nn\_botfr}=0 is equivalent to setting $r=0$ and leads to a free-slip |
---|
| 941 | bottom boundary condition. These values are assigned in \mdl{zdfbfr}. |
---|
| 942 | From v3.2 onwards there is support for local enhancement of these values |
---|
| 943 | via an externally defined 2D mask array (\np{ln\_bfr2d}=true) given |
---|
| 944 | in the \ifile{bfr\_coef} input NetCDF file. The mask values should vary from 0 to 1. |
---|
| 945 | Locations with a non-zero mask value will have the friction coefficient increased |
---|
[6497] | 946 | by $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri1}. |
---|
[707] | 947 | |
---|
| 948 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 949 | % Non-Linear Bottom Friction |
---|
| 950 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 951 | \subsection{Non-Linear Bottom Friction (\np{nn\_botfr} = 2)} |
---|
[707] | 952 | \label{ZDF_bfr_nonlinear} |
---|
| 953 | |
---|
[1224] | 954 | The non-linear bottom friction parameterisation assumes that the bottom |
---|
[707] | 955 | friction is quadratic: |
---|
| 956 | \begin{equation} \label{Eq_zdfbfr_nonlinear} |
---|
[2282] | 957 | {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h |
---|
[707] | 958 | }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b |
---|
| 959 | \end{equation} |
---|
[2282] | 960 | where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy |
---|
| 961 | due to tides, internal waves breaking and other short time scale currents. |
---|
| 962 | A typical value of the drag coefficient is $C_D = 10^{-3} $. As an example, |
---|
| 963 | the CME experiment \citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and |
---|
| 964 | $e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992} |
---|
| 965 | uses $C_D = 1.4\;10^{-3}$ and $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$. |
---|
[6497] | 966 | The CME choices have been set as default values (\np{rn\_bfri2} and \np{rn\_bfeb2} |
---|
[2282] | 967 | namelist parameters). |
---|
[707] | 968 | |
---|
[994] | 969 | As for the linear case, the bottom friction is imposed in the code by |
---|
[2282] | 970 | adding the trend due to the bottom friction to the general momentum trend |
---|
| 971 | in \mdl{dynbfr}. |
---|
| 972 | For the non-linear friction case the terms |
---|
| 973 | computed in \mdl{zdfbfr} are: |
---|
| 974 | \begin{equation} \label{Eq_zdfbfr_nonlinbfr} |
---|
[707] | 975 | \begin{split} |
---|
[2282] | 976 | c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\ |
---|
| 977 | c_b^v &= - \; C_D\;\left[ \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ |
---|
[707] | 978 | \end{split} |
---|
| 979 | \end{equation} |
---|
| 980 | |
---|
[6497] | 981 | The coefficients that control the strength of the non-linear bottom friction are |
---|
| 982 | initialised as namelist parameters: $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}. |
---|
| 983 | Note for applications which treat tides explicitly a low or even zero value of |
---|
| 984 | \np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$ is possible |
---|
| 985 | via an externally defined 2D mask array (\np{ln\_bfr2d}=true). This works in the same way |
---|
| 986 | as for the linear bottom friction case with non-zero masked locations increased by |
---|
| 987 | $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri2}. |
---|
[707] | 988 | |
---|
[2282] | 989 | % ------------------------------------------------------------------------------------------------------------- |
---|
[6497] | 990 | % Bottom Friction Log-layer |
---|
| 991 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 992 | \subsection{Log-layer Bottom Friction enhancement (\np{nn\_botfr} = 2, \np{ln\_loglayer} = .true.)} |
---|
| 993 | \label{ZDF_bfr_loglayer} |
---|
| 994 | |
---|
| 995 | In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally |
---|
| 996 | enhanced using a "law of the wall" scaling. If \np{ln\_loglayer} = .true., $C_D$ is no |
---|
| 997 | longer constant but is related to the thickness of the last wet layer in each column by: |
---|
| 998 | |
---|
| 999 | \begin{equation} |
---|
| 1000 | C_D = \left ( {\kappa \over {\rm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2 |
---|
| 1001 | \end{equation} |
---|
| 1002 | |
---|
| 1003 | \noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness |
---|
| 1004 | length provided via the namelist. |
---|
| 1005 | |
---|
| 1006 | For stability, the drag coefficient is bounded such that it is kept greater or equal to |
---|
| 1007 | the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional |
---|
| 1008 | namelist parameter: \np{rn\_bfri2\_max}, i.e.: |
---|
| 1009 | |
---|
| 1010 | \begin{equation} |
---|
| 1011 | rn\_bfri2 \leq C_D \leq rn\_bfri2\_max |
---|
| 1012 | \end{equation} |
---|
| 1013 | |
---|
| 1014 | \noindent Note also that a log-layer enhancement can also be applied to the top boundary |
---|
| 1015 | friction if under ice-shelf cavities are in use (\np{ln\_isfcav}=.true.). In this case, the |
---|
| 1016 | relevant namelist parameters are \np{rn\_tfrz0}, \np{rn\_tfri2} |
---|
| 1017 | and \np{rn\_tfri2\_max}. |
---|
| 1018 | |
---|
| 1019 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 1020 | % Bottom Friction stability |
---|
| 1021 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1022 | \subsection{Bottom Friction stability considerations} |
---|
| 1023 | \label{ZDF_bfr_stability} |
---|
| 1024 | |
---|
| 1025 | Some care needs to exercised over the choice of parameters to ensure that the |
---|
| 1026 | implementation of bottom friction does not induce numerical instability. For |
---|
| 1027 | the purposes of stability analysis, an approximation to \eqref{Eq_zdfbfr_flux2} |
---|
| 1028 | is: |
---|
| 1029 | \begin{equation} \label{Eqn_bfrstab} |
---|
| 1030 | \begin{split} |
---|
| 1031 | \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt \\ |
---|
| 1032 | &= -\frac{ru}{e_{3u}}\;2\rdt\\ |
---|
| 1033 | \end{split} |
---|
| 1034 | \end{equation} |
---|
| 1035 | \noindent where linear bottom friction and a leapfrog timestep have been assumed. |
---|
| 1036 | To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have: |
---|
| 1037 | \begin{equation} |
---|
| 1038 | |\Delta u| < \;|u| |
---|
| 1039 | \end{equation} |
---|
| 1040 | \noindent which, using \eqref{Eqn_bfrstab}, gives: |
---|
| 1041 | \begin{equation} |
---|
| 1042 | r\frac{2\rdt}{e_{3u}} < 1 \qquad \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ |
---|
| 1043 | \end{equation} |
---|
| 1044 | This same inequality can also be derived in the non-linear bottom friction case |
---|
| 1045 | if a velocity of 1 m.s$^{-1}$ is assumed. Alternatively, this criterion can be |
---|
| 1046 | rearranged to suggest a minimum bottom box thickness to ensure stability: |
---|
| 1047 | \begin{equation} |
---|
| 1048 | e_{3u} > 2\;r\;\rdt |
---|
| 1049 | \end{equation} |
---|
| 1050 | \noindent which it may be necessary to impose if partial steps are being used. |
---|
| 1051 | For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then |
---|
| 1052 | $e_{3u}$ should be greater than 3.6 m. For most applications, with physically |
---|
| 1053 | sensible parameters these restrictions should not be of concern. But |
---|
| 1054 | caution may be necessary if attempts are made to locally enhance the bottom |
---|
| 1055 | friction parameters. |
---|
| 1056 | To ensure stability limits are imposed on the bottom friction coefficients both during |
---|
| 1057 | initialisation and at each time step. Checks at initialisation are made in \mdl{zdfbfr} |
---|
| 1058 | (assuming a 1 m.s$^{-1}$ velocity in the non-linear case). |
---|
| 1059 | The number of breaches of the stability criterion are reported as well as the minimum |
---|
| 1060 | and maximum values that have been set. The criterion is also checked at each time step, |
---|
| 1061 | using the actual velocity, in \mdl{dynbfr}. Values of the bottom friction coefficient are |
---|
| 1062 | reduced as necessary to ensure stability; these changes are not reported. |
---|
| 1063 | |
---|
[3294] | 1064 | Limits on the bottom friction coefficient are not imposed if the user has elected to |
---|
| 1065 | handle the bottom friction implicitly (see \S\ref{ZDF_bfr_imp}). The number of potential |
---|
| 1066 | breaches of the explicit stability criterion are still reported for information purposes. |
---|
| 1067 | |
---|
[2282] | 1068 | % ------------------------------------------------------------------------------------------------------------- |
---|
[3294] | 1069 | % Implicit Bottom Friction |
---|
| 1070 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1071 | \subsection{Implicit Bottom Friction (\np{ln\_bfrimp}$=$\textit{T})} |
---|
| 1072 | \label{ZDF_bfr_imp} |
---|
| 1073 | |
---|
| 1074 | An optional implicit form of bottom friction has been implemented to improve |
---|
| 1075 | model stability. We recommend this option for shelf sea and coastal ocean applications, especially |
---|
| 1076 | for split-explicit time splitting. This option can be invoked by setting \np{ln\_bfrimp} |
---|
| 1077 | to \textit{true} in the \textit{nambfr} namelist. This option requires \np{ln\_zdfexp} to be \textit{false} |
---|
| 1078 | in the \textit{namzdf} namelist. |
---|
| 1079 | |
---|
| 1080 | This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, the |
---|
| 1081 | bottom boundary condition is implemented implicitly. |
---|
| 1082 | |
---|
| 1083 | \begin{equation} \label{Eq_dynzdf_bfr} |
---|
| 1084 | \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk} |
---|
| 1085 | = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}} |
---|
| 1086 | \end{equation} |
---|
| 1087 | |
---|
| 1088 | where $mbk$ is the layer number of the bottom wet layer. superscript $n+1$ means the velocity used in the |
---|
| 1089 | friction formula is to be calculated, so, it is implicit. |
---|
| 1090 | |
---|
| 1091 | If split-explicit time splitting is used, care must be taken to avoid the double counting of |
---|
| 1092 | the bottom friction in the 2-D barotropic momentum equations. As NEMO only updates the barotropic |
---|
| 1093 | pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, we need to remove |
---|
| 1094 | the bottom friction induced by these two terms which has been included in the 3-D momentum trend |
---|
| 1095 | and update it with the latest value. On the other hand, the bottom friction contributed by the |
---|
| 1096 | other terms (e.g. the advection term, viscosity term) has been included in the 3-D momentum equations |
---|
| 1097 | and should not be added in the 2-D barotropic mode. |
---|
| 1098 | |
---|
| 1099 | The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the |
---|
| 1100 | following: |
---|
| 1101 | |
---|
| 1102 | \begin{equation} \label{Eq_dynspg_ts_bfr1} |
---|
| 1103 | \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b} |
---|
| 1104 | \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right) |
---|
| 1105 | \end{equation} |
---|
| 1106 | \begin{equation} \label{Eq_dynspg_ts_bfr2} |
---|
| 1107 | \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+ |
---|
| 1108 | \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)- |
---|
| 1109 | 2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right) |
---|
| 1110 | \end{equation} |
---|
| 1111 | |
---|
| 1112 | where $\textbf{T}$ is the vertical integrated 3-D momentum trend. We assume the leap-frog time-stepping |
---|
| 1113 | is used here. $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step. |
---|
| 1114 | $c_{b}$ is the friction coefficient. $\eta$ is the sea surface level calculated in the barotropic loops |
---|
| 1115 | while $\eta^{'}$ is the sea surface level used in the 3-D baroclinic mode. $\textbf{u}_{b}$ is the bottom |
---|
| 1116 | layer horizontal velocity. |
---|
| 1117 | |
---|
| 1118 | |
---|
| 1119 | |
---|
| 1120 | |
---|
| 1121 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 1122 | % Bottom Friction with split-explicit time splitting |
---|
| 1123 | % ------------------------------------------------------------------------------------------------------------- |
---|
[3294] | 1124 | \subsection{Bottom Friction with split-explicit time splitting (\np{ln\_bfrimp}$=$\textit{F})} |
---|
[2282] | 1125 | \label{ZDF_bfr_ts} |
---|
| 1126 | |
---|
| 1127 | When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the |
---|
| 1128 | bottom velocity at the before time step is used. This velocity includes both the |
---|
| 1129 | baroclinic and barotropic components which is appropriate when using either the |
---|
| 1130 | explicit or filtered surface pressure gradient algorithms (\key{dynspg\_exp} or |
---|
| 1131 | {\key{dynspg\_flt}). Extra attention is required, however, when using |
---|
| 1132 | split-explicit time stepping (\key{dynspg\_ts}). In this case the free surface |
---|
[3294] | 1133 | equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, while the three |
---|
| 1134 | dimensional prognostic variables are solved with the longer time step |
---|
| 1135 | of \np{rn\_rdt} seconds. The trend in the barotropic momentum due to bottom |
---|
[2282] | 1136 | friction appropriate to this method is that given by the selected parameterisation |
---|
| 1137 | ($i.e.$ linear or non-linear bottom friction) computed with the evolving velocities |
---|
| 1138 | at each barotropic timestep. |
---|
| 1139 | |
---|
| 1140 | In the case of non-linear bottom friction, we have elected to partially linearise |
---|
| 1141 | the problem by keeping the coefficients fixed throughout the barotropic |
---|
| 1142 | time-stepping to those computed in \mdl{zdfbfr} using the now timestep. |
---|
| 1143 | This decision allows an efficient use of the $c_b^{\vect{U}}$ coefficients to: |
---|
| 1144 | |
---|
| 1145 | \begin{enumerate} |
---|
| 1146 | \item On entry to \rou{dyn\_spg\_ts}, remove the contribution of the before |
---|
| 1147 | barotropic velocity to the bottom friction component of the vertically |
---|
| 1148 | integrated momentum trend. Note the same stability check that is carried out |
---|
| 1149 | on the bottom friction coefficient in \mdl{dynbfr} has to be applied here to |
---|
| 1150 | ensure that the trend removed matches that which was added in \mdl{dynbfr}. |
---|
| 1151 | \item At each barotropic step, compute the contribution of the current barotropic |
---|
| 1152 | velocity to the trend due to bottom friction. Add this contribution to the |
---|
| 1153 | vertically integrated momentum trend. This contribution is handled implicitly which |
---|
| 1154 | eliminates the need to impose a stability criteria on the values of the bottom friction |
---|
| 1155 | coefficient within the barotropic loop. |
---|
| 1156 | \end{enumerate} |
---|
| 1157 | |
---|
[3294] | 1158 | Note that the use of an implicit formulation within the barotropic loop |
---|
[2282] | 1159 | for the bottom friction trend means that any limiting of the bottom friction coefficient |
---|
| 1160 | in \mdl{dynbfr} does not adversely affect the solution when using split-explicit time |
---|
| 1161 | splitting. This is because the major contribution to bottom friction is likely to come from |
---|
[3294] | 1162 | the barotropic component which uses the unrestricted value of the coefficient. However, if the |
---|
| 1163 | limiting is thought to be having a major effect (a more likely prospect in coastal and shelf seas |
---|
| 1164 | applications) then the fully implicit form of the bottom friction should be used (see \S\ref{ZDF_bfr_imp} ) |
---|
| 1165 | which can be selected by setting \np{ln\_bfrimp} $=$ \textit{true}. |
---|
[2282] | 1166 | |
---|
[3294] | 1167 | Otherwise, the implicit formulation takes the form: |
---|
[2282] | 1168 | \begin{equation} \label{Eq_zdfbfr_implicitts} |
---|
| 1169 | \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ] |
---|
| 1170 | \end{equation} |
---|
| 1171 | where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height), |
---|
| 1172 | $c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and $RHS$ represents |
---|
| 1173 | all the components to the vertically integrated momentum trend except for that due to bottom friction. |
---|
| 1174 | |
---|
| 1175 | |
---|
| 1176 | |
---|
| 1177 | |
---|
[1225] | 1178 | % ================================================================ |
---|
[2282] | 1179 | % Tidal Mixing |
---|
| 1180 | % ================================================================ |
---|
[2349] | 1181 | \section{Tidal Mixing (\key{zdftmx})} |
---|
[2282] | 1182 | \label{ZDF_tmx} |
---|
| 1183 | |
---|
| 1184 | %--------------------------------------------namzdf_tmx-------------------------------------------------- |
---|
| 1185 | \namdisplay{namzdf_tmx} |
---|
| 1186 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 1187 | |
---|
| 1188 | |
---|
| 1189 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1190 | % Bottom intensified tidal mixing |
---|
| 1191 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1192 | \subsection{Bottom intensified tidal mixing} |
---|
| 1193 | \label{ZDF_tmx_bottom} |
---|
| 1194 | |
---|
[4147] | 1195 | Options are defined through the \ngn{namzdf\_tmx} namelist variables. |
---|
[2282] | 1196 | The parameterization of tidal mixing follows the general formulation for |
---|
| 1197 | the vertical eddy diffusivity proposed by \citet{St_Laurent_al_GRL02} and |
---|
| 1198 | first introduced in an OGCM by \citep{Simmons_al_OM04}. |
---|
| 1199 | In this formulation an additional vertical diffusivity resulting from internal tide breaking, |
---|
| 1200 | $A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, the energy transfer from barotropic |
---|
| 1201 | tides to baroclinic tides : |
---|
| 1202 | \begin{equation} \label{Eq_Ktides} |
---|
| 1203 | A^{vT}_{tides} = q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 } |
---|
| 1204 | \end{equation} |
---|
| 1205 | where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency |
---|
| 1206 | (see \S\ref{TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency, |
---|
| 1207 | and $F(z)$ the vertical structure function. |
---|
| 1208 | |
---|
| 1209 | The mixing efficiency of turbulence is set by $\Gamma$ (\np{rn\_me} namelist parameter) |
---|
| 1210 | and is usually taken to be the canonical value of $\Gamma = 0.2$ (Osborn 1980). |
---|
| 1211 | The tidal dissipation efficiency is given by the parameter $q$ (\np{rn\_tfe} namelist parameter) |
---|
| 1212 | represents the part of the internal wave energy flux $E(x, y)$ that is dissipated locally, |
---|
| 1213 | with the remaining $1-q$ radiating away as low mode internal waves and |
---|
| 1214 | contributing to the background internal wave field. A value of $q=1/3$ is typically used |
---|
| 1215 | \citet{St_Laurent_al_GRL02}. |
---|
| 1216 | The vertical structure function $F(z)$ models the distribution of the turbulent mixing in the vertical. |
---|
| 1217 | It is implemented as a simple exponential decaying upward away from the bottom, |
---|
| 1218 | with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04}, |
---|
| 1219 | \begin{equation} \label{Eq_Fz} |
---|
| 1220 | F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) } |
---|
| 1221 | \end{equation} |
---|
| 1222 | and is normalized so that vertical integral over the water column is unity. |
---|
| 1223 | |
---|
| 1224 | The associated vertical viscosity is calculated from the vertical |
---|
| 1225 | diffusivity assuming a Prandtl number of 1, $i.e.$ $A^{vm}_{tides}=A^{vT}_{tides}$. |
---|
| 1226 | In the limit of $N \rightarrow 0$ (or becoming negative), the vertical diffusivity |
---|
| 1227 | is capped at $300\,cm^2/s$ and impose a lower limit on $N^2$ of \np{rn\_n2min} |
---|
| 1228 | usually set to $10^{-8} s^{-2}$. These bounds are usually rarely encountered. |
---|
| 1229 | |
---|
| 1230 | The internal wave energy map, $E(x, y)$ in \eqref{Eq_Ktides}, is derived |
---|
| 1231 | from a barotropic model of the tides utilizing a parameterization of the |
---|
| 1232 | conversion of barotropic tidal energy into internal waves. |
---|
| 1233 | The essential goal of the parameterization is to represent the momentum |
---|
| 1234 | exchange between the barotropic tides and the unrepresented internal waves |
---|
[3294] | 1235 | induced by the tidal flow over rough topography in a stratified ocean. |
---|
[2282] | 1236 | In the current version of \NEMO, the map is built from the output of |
---|
| 1237 | the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}. |
---|
| 1238 | This model provides the dissipation associated with internal wave energy for the M2 and K1 |
---|
| 1239 | tides component (Fig.~\ref{Fig_ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated |
---|
| 1240 | as being $1/4$ of the M2 one. The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$. |
---|
| 1241 | Its global mean value is $1.1$ TW, in agreement with independent estimates |
---|
| 1242 | \citep{Egbert_Ray_Nat00, Egbert_Ray_JGR01}. |
---|
| 1243 | |
---|
| 1244 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[2376] | 1245 | \begin{figure}[!t] \begin{center} |
---|
[2282] | 1246 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_ZDF_M2_K1_tmx.pdf} |
---|
[2376] | 1247 | \caption{ \label{Fig_ZDF_M2_K1_tmx} |
---|
[3764] | 1248 | (a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). } |
---|
[2376] | 1249 | \end{center} \end{figure} |
---|
[2282] | 1250 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 1251 | |
---|
| 1252 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1253 | % Indonesian area specific treatment |
---|
| 1254 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2349] | 1255 | \subsection{Indonesian area specific treatment (\np{ln\_zdftmx\_itf})} |
---|
[2282] | 1256 | \label{ZDF_tmx_itf} |
---|
| 1257 | |
---|
| 1258 | When the Indonesian Through Flow (ITF) area is included in the model domain, |
---|
| 1259 | a specific treatment of tidal induced mixing in this area can be used. |
---|
| 1260 | It is activated through the namelist logical \np{ln\_tmx\_itf}, and the user must provide |
---|
| 1261 | an input NetCDF file, \ifile{mask\_itf}, which contains a mask array defining the ITF area |
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| 1262 | where the specific treatment is applied. |
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| 1263 | |
---|
| 1264 | When \np{ln\_tmx\_itf}=true, the two key parameters $q$ and $F(z)$ are adjusted following |
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[3764] | 1265 | the parameterisation developed by \citet{Koch-Larrouy_al_GRL07}: |
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[2282] | 1266 | |
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| 1267 | First, the Indonesian archipelago is a complex geographic region |
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| 1268 | with a series of large, deep, semi-enclosed basins connected via |
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| 1269 | numerous narrow straits. Once generated, internal tides remain |
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| 1270 | confined within this semi-enclosed area and hardly radiate away. |
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| 1271 | Therefore all the internal tides energy is consumed within this area. |
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| 1272 | So it is assumed that $q = 1$, $i.e.$ all the energy generated is available for mixing. |
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| 1273 | Note that for test purposed, the ITF tidal dissipation efficiency is a |
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| 1274 | namelist parameter (\np{rn\_tfe\_itf}). A value of $1$ or close to is |
---|
| 1275 | this recommended for this parameter. |
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| 1276 | |
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| 1277 | Second, the vertical structure function, $F(z)$, is no more associated |
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| 1278 | with a bottom intensification of the mixing, but with a maximum of |
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[3764] | 1279 | energy available within the thermocline. \citet{Koch-Larrouy_al_GRL07} |
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[2282] | 1280 | have suggested that the vertical distribution of the energy dissipation |
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| 1281 | proportional to $N^2$ below the core of the thermocline and to $N$ above. |
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| 1282 | The resulting $F(z)$ is: |
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| 1283 | \begin{equation} \label{Eq_Fz_itf} |
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| 1284 | F(i,j,k) \sim \left\{ \begin{aligned} |
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| 1285 | \frac{q\,\Gamma E(i,j) } {\rho N \, \int N dz} \qquad \text{when $\partial_z N < 0$} \\ |
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| 1286 | \frac{q\,\Gamma E(i,j) } {\rho \, \int N^2 dz} \qquad \text{when $\partial_z N > 0$} |
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| 1287 | \end{aligned} \right. |
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| 1288 | \end{equation} |
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| 1289 | |
---|
| 1290 | Averaged over the ITF area, the resulting tidal mixing coefficient is $1.5\,cm^2/s$, |
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| 1291 | which agrees with the independent estimates inferred from observations. |
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| 1292 | Introduced in a regional OGCM, the parameterization improves the water mass |
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| 1293 | characteristics in the different Indonesian seas, suggesting that the horizontal |
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| 1294 | and vertical distributions of the mixing are adequately prescribed |
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| 1295 | \citep{Koch-Larrouy_al_GRL07, Koch-Larrouy_al_OD08a, Koch-Larrouy_al_OD08b}. |
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[3764] | 1296 | Note also that such a parameterisation has a significant impact on the behaviour |
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[2282] | 1297 | of global coupled GCMs \citep{Koch-Larrouy_al_CD10}. |
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| 1298 | |
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| 1299 | |
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[3294] | 1300 | % ================================================================ |
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[6497] | 1301 | % Internal wave-driven mixing |
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| 1302 | % ================================================================ |
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| 1303 | \section{Internal wave-driven mixing (\key{zdftmx\_new})} |
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| 1304 | \label{ZDF_tmx_new} |
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| 1305 | |
---|
| 1306 | %--------------------------------------------namzdf_tmx_new------------------------------------------ |
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| 1307 | \namdisplay{namzdf_tmx_new} |
---|
| 1308 | %-------------------------------------------------------------------------------------------------------------- |
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| 1309 | |
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| 1310 | The parameterization of mixing induced by breaking internal waves is a generalization |
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| 1311 | of the approach originally proposed by \citet{St_Laurent_al_GRL02}. |
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| 1312 | A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed, |
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| 1313 | and the resulting diffusivity is obtained as |
---|
| 1314 | \begin{equation} \label{Eq_Kwave} |
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| 1315 | A^{vT}_{wave} = R_f \,\frac{ \epsilon }{ \rho \, N^2 } |
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| 1316 | \end{equation} |
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| 1317 | where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution |
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| 1318 | of the energy available for mixing. If the \np{ln\_mevar} namelist parameter is set to false, |
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| 1319 | the mixing efficiency is taken as constant and equal to 1/6 \citep{Osborn_JPO80}. |
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| 1320 | In the opposite (recommended) case, $R_f$ is instead a function of the turbulence intensity parameter |
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| 1321 | $Re_b = \frac{ \epsilon}{\nu \, N^2}$, with $\nu$ the molecular viscosity of seawater, |
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| 1322 | following the model of \cite{Bouffard_Boegman_DAO2013} |
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| 1323 | and the implementation of \cite{de_lavergne_JPO2016_efficiency}. |
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| 1324 | Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when the mixing efficiency is constant. |
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| 1325 | |
---|
| 1326 | In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary |
---|
| 1327 | as a function of $Re_b$ by setting the \np{ln\_tsdiff} parameter to true, a recommended choice). |
---|
| 1328 | This parameterization of differential mixing, due to \cite{Jackson_Rehmann_JPO2014}, |
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| 1329 | is implemented as in \cite{de_lavergne_JPO2016_efficiency}. |
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| 1330 | |
---|
| 1331 | The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$, is constructed |
---|
| 1332 | from three static maps of column-integrated internal wave energy dissipation, $E_{cri}(i,j)$, |
---|
| 1333 | $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures |
---|
| 1334 | (de Lavergne et al., in prep): |
---|
| 1335 | \begin{align*} |
---|
| 1336 | F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\ |
---|
| 1337 | F_{pyc}(i,j,k) &\propto N^{n\_p}\\ |
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| 1338 | F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} } |
---|
| 1339 | \end{align*} |
---|
| 1340 | In the above formula, $h_{ab}$ denotes the height above bottom, |
---|
| 1341 | $h_{wkb}$ denotes the WKB-stretched height above bottom, defined by |
---|
| 1342 | \begin{equation*} |
---|
| 1343 | h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz' } \; , |
---|
| 1344 | \end{equation*} |
---|
| 1345 | The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist) controls the stratification-dependence of the pycnocline-intensified dissipation. |
---|
| 1346 | It can take values of 1 (recommended) or 2. |
---|
| 1347 | Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of |
---|
| 1348 | the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps. |
---|
| 1349 | $h_{cri}$ is related to the large-scale topography of the ocean (etopo2) |
---|
| 1350 | and $h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of |
---|
| 1351 | the abyssal hill topography \citep{Goff_JGR2010} and the latitude. |
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| 1352 | |
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| 1353 | % ================================================================ |
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| 1354 | |
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| 1355 | |
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| 1356 | |
---|