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