1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | % ================================================================ |
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5 | % Chapter Vertical Ocean Physics (ZDF) |
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6 | % ================================================================ |
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7 | \chapter{Vertical Ocean Physics (ZDF)} |
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8 | \label{chap:ZDF} |
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9 | |
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10 | \minitoc |
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11 | |
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12 | %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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13 | |
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14 | \newpage |
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15 | |
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16 | % ================================================================ |
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17 | % Vertical Mixing |
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18 | % ================================================================ |
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19 | \section{Vertical mixing} |
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20 | \label{sec:ZDF_zdf} |
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21 | |
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22 | The discrete form of the ocean subgrid scale physics has been presented in |
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23 | \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}. |
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24 | At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt have to be defined. |
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25 | At the surface they are prescribed from the surface forcing (see \autoref{chap:SBC}), |
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26 | while at the bottom they are set to zero for heat and salt, |
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27 | unless a geothermal flux forcing is prescribed as a bottom boundary condition (\ie \key{trabbl} defined, |
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28 | see \autoref{subsec:TRA_bbc}), and specified through a bottom friction parameterisation for momentum |
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29 | (see \autoref{sec:ZDF_bfr}). |
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30 | |
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31 | In this section we briefly discuss the various choices offered to compute the vertical eddy viscosity and |
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32 | diffusivity coefficients, $A_u^{vm}$ , $A_v^{vm}$ and $A^{vT}$ ($A^{vS}$), defined at $uw$-, $vw$- and $w$- points, |
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33 | respectively (see \autoref{sec:TRA_zdf} and \autoref{sec:DYN_zdf}). |
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34 | These coefficients can be assumed to be either constant, or a function of the local Richardson number, |
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35 | or computed from a turbulent closure model (either TKE or GLS formulation). |
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36 | The computation of these coefficients is initialized in the \mdl{zdfini} module and performed in |
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37 | the \mdl{zdfric}, \mdl{zdftke} or \mdl{zdfgls} modules. |
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38 | The trends due to the vertical momentum and tracer diffusion, including the surface forcing, |
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39 | are computed and added to the general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively. |
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40 | These trends can be computed using either a forward time stepping scheme |
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41 | (namelist parameter \np{ln\_zdfexp}\forcode{ = .true.}) or a backward time stepping scheme |
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42 | (\np{ln\_zdfexp}\forcode{ = .false.}) depending on the magnitude of the mixing coefficients, |
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43 | and thus of the formulation used (see \autoref{chap:STP}). |
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44 | |
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45 | % ------------------------------------------------------------------------------------------------------------- |
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46 | % Constant |
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47 | % ------------------------------------------------------------------------------------------------------------- |
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48 | \subsection{Constant (\protect\key{zdfcst})} |
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49 | \label{subsec:ZDF_cst} |
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50 | %--------------------------------------------namzdf--------------------------------------------------------- |
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51 | |
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52 | \nlst{namzdf} |
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53 | %-------------------------------------------------------------------------------------------------------------- |
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54 | |
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55 | Options are defined through the \ngn{namzdf} namelist variables. |
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56 | When \key{zdfcst} is defined, the momentum and tracer vertical eddy coefficients are set to |
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57 | constant values over the whole ocean. |
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58 | This is the crudest way to define the vertical ocean physics. |
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59 | It is recommended that this option is only used in process studies, not in basin scale simulations. |
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60 | Typical values used in this case are: |
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61 | \begin{align*} |
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62 | A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1} \\ |
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63 | A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1} |
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64 | \end{align*} |
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65 | |
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66 | These values are set through the \np{rn\_avm0} and \np{rn\_avt0} namelist parameters. |
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67 | In all cases, do not use values smaller that those associated with the molecular viscosity and diffusivity, |
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68 | that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and |
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69 | $\sim10^{-9}~m^2.s^{-1}$ for salinity. |
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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 (\protect\key{zdfric})} |
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75 | \label{subsec:ZDF_ric} |
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76 | |
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77 | %--------------------------------------------namric--------------------------------------------------------- |
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78 | |
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79 | \nlst{namzdf_ric} |
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80 | %-------------------------------------------------------------------------------------------------------------- |
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81 | |
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82 | When \key{zdfric} is defined, a local Richardson number dependent formulation for the vertical momentum and |
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83 | tracer eddy coefficients is set through the \ngn{namzdf\_ric} namelist variables. |
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84 | The vertical mixing 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 large scale ocean structures. |
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86 | The hypothesis of a mixing mainly maintained by the growth of Kelvin-Helmholtz like instabilities leads to |
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87 | a dependency between the vertical eddy coefficients and the local Richardson number |
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88 | (\ie the ratio of stratification to vertical shear). |
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89 | Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented: |
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90 | \[ |
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91 | % \label{eq:zdfric} |
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92 | \left\{ |
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93 | \begin{aligned} |
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94 | A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT} \\ |
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95 | A^{vm} &= \frac{A^{vT} }{\left( 1+ a \;Ri \right) } + A_b^{vm} |
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96 | \end{aligned} |
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97 | \right. |
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98 | \] |
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99 | where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number, |
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100 | $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), |
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101 | $A_b^{vT} $ and $A_b^{vm}$ are the constant background values set as in the constant case |
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102 | (see \autoref{subsec:ZDF_cst}), and $A_{ric}^{vT} = 10^{-4}~m^2.s^{-1}$ is the maximum value that |
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103 | can be reached by the coefficient when $Ri\leq 0$, $a=5$ and $n=2$. |
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104 | The last three values can be modified by setting the \np{rn\_avmri}, \np{rn\_alp} and |
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105 | \np{nn\_ric} namelist parameters, respectively. |
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106 | |
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107 | A simple mixing-layer model to transfer and dissipate the atmospheric forcings |
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108 | (wind-stress and buoyancy fluxes) can be activated setting the \np{ln\_mldw}\forcode{ = .true.} in the namelist. |
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109 | |
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110 | In this case, the local depth of turbulent wind-mixing or "Ekman depth" $h_{e}(x,y,t)$ is evaluated and |
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111 | the vertical eddy coefficients prescribed within this layer. |
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112 | |
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113 | This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation: |
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114 | \[ |
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115 | h_{e} = Ek \frac {u^{*}} {f_{0}} |
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116 | \] |
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117 | where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis 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 | \[ |
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121 | u^{*} = \sqrt \frac {|\tau|} {\rho_o} |
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122 | \] |
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123 | is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$. |
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124 | The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}. |
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125 | Once $h_{e}$ is computed, the vertical eddy coefficients within $h_{e}$ are set to |
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126 | the empirical values \np{rn\_wtmix} and \np{rn\_wvmix} \citep{lermusiaux_JMS01}. |
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127 | |
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128 | % ------------------------------------------------------------------------------------------------------------- |
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129 | % TKE Turbulent Closure Scheme |
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130 | % ------------------------------------------------------------------------------------------------------------- |
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131 | \subsection{TKE turbulent closure scheme (\protect\key{zdftke})} |
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132 | \label{subsec:ZDF_tke} |
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133 | |
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134 | %--------------------------------------------namzdf_tke-------------------------------------------------- |
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135 | |
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136 | \nlst{namzdf_tke} |
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137 | %-------------------------------------------------------------------------------------------------------------- |
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138 | |
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139 | The vertical eddy viscosity and diffusivity coefficients are computed from a TKE turbulent closure model based on |
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140 | a prognostic equation for $\bar{e}$, the turbulent kinetic energy, |
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141 | and a closure assumption for the turbulent length scales. |
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142 | This turbulent closure model has been developed by \citet{bougeault.lacarrere_MWR89} in the atmospheric case, |
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143 | adapted by \citet{gaspar.gregoris.ea_JGR90} for the oceanic case, and embedded in OPA, the ancestor of NEMO, |
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144 | by \citet{blanke.delecluse_JPO93} for equatorial Atlantic simulations. |
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145 | Since then, significant modifications have been introduced by \citet{madec.delecluse.ea_NPM98} in both the implementation and |
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146 | the formulation of the mixing length scale. |
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147 | The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear, |
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148 | its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type: |
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149 | \begin{equation} |
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150 | \label{eq:zdftke_e} |
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151 | \frac{\partial \bar{e}}{\partial t} = |
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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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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 | \[ |
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160 | % \label{eq:zdftke_kz} |
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161 | \begin{split} |
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162 | K_m &= C_k\ l_k\ \sqrt {\bar{e}\; } \\ |
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163 | K_\rho &= A^{vm} / P_{rt} |
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164 | \end{split} |
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165 | \] |
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166 | where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), |
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167 | $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, |
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168 | $P_{rt}$ is the Prandtl number, $K_m$ and $K_\rho$ are the vertical eddy viscosity and diffusivity coefficients. |
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169 | The constants $C_k = 0.1$ and $C_\epsilon = \sqrt {2} /2$ $\approx 0.7$ are designed to deal with |
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170 | vertical mixing at any depth \citep{gaspar.gregoris.ea_JGR90}. |
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171 | They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}. |
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172 | $P_{rt}$ can be set to unity or, following \citet{blanke.delecluse_JPO93}, be a function of the local Richardson number, $R_i$: |
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173 | \begin{align*} |
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174 | % \label{eq:prt} |
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175 | P_{rt} = |
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176 | \begin{cases} |
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177 | \ \ \ 1 & \text{if $\ R_i \leq 0.2$} \\ |
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178 | 5\,R_i & \text{if $\ 0.2 \leq R_i \leq 2$} \\ |
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179 | \ \ 10 & \text{if $\ 2 \leq R_i$} |
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180 | \end{cases} |
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181 | \end{align*} |
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182 | Options are defined through the \ngn{namzdfy\_tke} namelist variables. |
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183 | The choice of $P_{rt}$ is controlled by the \np{nn\_pdl} namelist variable. |
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184 | |
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185 | At the sea surface, the value of $\bar{e}$ is prescribed from the wind stress field as |
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186 | $\bar{e}_o = e_{bb} |\tau| / \rho_o$, with $e_{bb}$ the \np{rn\_ebb} namelist parameter. |
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187 | The default value of $e_{bb}$ is 3.75. \citep{gaspar.gregoris.ea_JGR90}), however a much larger value can be used when |
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188 | taking into account the surface wave breaking (see below Eq. \autoref{eq:ZDF_Esbc}). |
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189 | The bottom value of TKE is assumed to be equal to the value of the level just above. |
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190 | The time integration of the $\bar{e}$ equation may formally lead to negative values because |
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191 | the numerical scheme does not ensure its positivity. |
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192 | To overcome this problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} namelist parameter). |
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193 | Following \citet{gaspar.gregoris.ea_JGR90}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. |
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194 | This allows the subsequent formulations to match that of \citet{gargett_JMR84} for the diffusion in |
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195 | the thermocline and 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 instabilities associated with |
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197 | too weak vertical diffusion. |
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198 | They must be specified at least larger than the molecular values, and are set through \np{rn\_avm0} and |
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199 | \np{rn\_avt0} (namzdf namelist, see \autoref{subsec:ZDF_cst}). |
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200 | |
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201 | \subsubsection{Turbulent length scale} |
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202 | |
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203 | For computational efficiency, the original formulation of the turbulent length scales proposed by |
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204 | \citet{gaspar.gregoris.ea_JGR90} has been simplified. |
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205 | Four formulations are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist parameter. |
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206 | The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}: |
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207 | \begin{equation} |
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208 | \label{eq:tke_mxl0_1} |
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209 | l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N |
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210 | \end{equation} |
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211 | which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency. |
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212 | The resulting length scale is bounded by the distance to the surface or to the bottom |
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213 | (\np{nn\_mxl}\forcode{ = 0}) or by the local vertical scale factor (\np{nn\_mxl}\forcode{ = 1}). |
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214 | \citet{blanke.delecluse_JPO93} notice that this simplification has two major drawbacks: |
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215 | it makes no sense for locally unstable stratification and the computation no longer uses all |
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216 | the information contained in the vertical density profile. |
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217 | To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np{nn\_mxl}\forcode{ = 2..3} cases, |
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218 | which add an extra assumption concerning the vertical gradient of the computed length scale. |
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219 | So, the length scales are first evaluated as in \autoref{eq:tke_mxl0_1} and then bounded such that: |
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220 | \begin{equation} |
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221 | \label{eq:tke_mxl_constraint} |
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222 | \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 |
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223 | \qquad \text{with }\ l = l_k = l_\epsilon |
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224 | \end{equation} |
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225 | \autoref{eq:tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than |
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226 | the variations of depth. |
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227 | It provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less |
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228 | time consuming. |
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229 | In particular, it allows the length scale to be limited not only by the distance to the surface or |
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230 | to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as |
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231 | the thermocline (\autoref{fig:mixing_length}). |
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232 | In order to impose the \autoref{eq:tke_mxl_constraint} constraint, we introduce two additional length scales: |
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233 | $l_{up}$ and $l_{dwn}$, the upward and downward length scales, and |
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234 | evaluate the dissipation and mixing length scales as |
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235 | (and note that here we use numerical indexing): |
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236 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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237 | \begin{figure}[!t] |
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238 | \begin{center} |
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239 | \includegraphics[width=\textwidth]{Fig_mixing_length} |
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240 | \caption{ |
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241 | \protect\label{fig:mixing_length} |
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242 | Illustration of the mixing length computation. |
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243 | } |
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244 | \end{center} |
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245 | \end{figure} |
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246 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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247 | \[ |
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248 | % \label{eq:tke_mxl2} |
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249 | \begin{aligned} |
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250 | l_{up\ \ }^{(k)} &= \min \left( l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \; \right) |
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251 | \quad &\text{ from $k=1$ to $jpk$ }\ \\ |
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252 | l_{dwn}^{(k)} &= \min \left( l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)} \right) |
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253 | \quad &\text{ from $k=jpk$ to $1$ }\ \\ |
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254 | \end{aligned} |
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255 | \] |
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256 | where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, \ie $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. |
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257 | |
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258 | In the \np{nn\_mxl}\forcode{ = 2} case, the dissipation and mixing length scales take the same value: |
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259 | $ l_k= l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right)$, while in the \np{nn\_mxl}\forcode{ = 3} case, |
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260 | the dissipation and mixing turbulent length scales are give as in \citet{gaspar.gregoris.ea_JGR90}: |
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261 | \[ |
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262 | % \label{eq:tke_mxl_gaspar} |
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263 | \begin{aligned} |
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264 | & l_k = \sqrt{\ l_{up} \ \ l_{dwn}\ } \\ |
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265 | & l_\epsilon = \min \left(\ l_{up} \;,\; l_{dwn}\ \right) |
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266 | \end{aligned} |
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267 | \] |
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268 | |
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269 | At the ocean surface, a non zero length scale is set through the \np{rn\_mxl0} namelist parameter. |
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270 | Usually the surface scale is given by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant and |
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271 | $z_o$ the roughness parameter of the surface. |
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272 | Assuming $z_o=0.1$~m \citep{craig.banner_JPO94} leads to a 0.04~m, the default value of \np{rn\_mxl0}. |
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273 | In the ocean interior a minimum length scale is set to recover the molecular viscosity when |
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274 | $\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ). |
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275 | |
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276 | \subsubsection{Surface wave breaking parameterization} |
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277 | %-----------------------------------------------------------------------% |
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278 | |
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279 | Following \citet{mellor.blumberg_JPO04}, the TKE turbulence closure model has been modified to |
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280 | include the effect of surface wave breaking energetics. |
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281 | This results in a reduction of summertime surface temperature when the mixed layer is relatively shallow. |
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282 | The \citet{mellor.blumberg_JPO04} modifications acts on surface length scale and TKE values and |
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283 | air-sea drag coefficient. |
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284 | The latter concerns the bulk formulea and is not discussed here. |
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285 | |
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286 | Following \citet{craig.banner_JPO94}, the boundary condition on surface TKE value is : |
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287 | \begin{equation} |
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288 | \label{eq:ZDF_Esbc} |
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289 | \bar{e}_o = \frac{1}{2}\,\left( 15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} |
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290 | \end{equation} |
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291 | where $\alpha_{CB}$ is the \citet{craig.banner_JPO94} constant of proportionality which depends on the ''wave age'', |
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292 | ranging from 57 for mature waves to 146 for younger waves \citep{mellor.blumberg_JPO04}. |
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293 | The boundary condition on the turbulent length scale follows the Charnock's relation: |
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294 | \begin{equation} |
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295 | \label{eq:ZDF_Lsbc} |
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296 | l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o} |
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297 | \end{equation} |
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298 | where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant. |
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299 | \citet{mellor.blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by |
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300 | \citet{stacey_JPO99} citing observation evidence, and |
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301 | $\alpha_{CB} = 100$ the Craig and Banner's value. |
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302 | As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, |
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303 | with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}\forcode{ = 67.83} corresponds |
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304 | to $\alpha_{CB} = 100$. |
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305 | Further setting \np{ln\_mxl0} to true applies \autoref{eq:ZDF_Lsbc} as surface boundary condition on length scale, |
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306 | with $\beta$ hard coded to the Stacey's value. |
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307 | Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) is applied on |
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308 | surface $\bar{e}$ value. |
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309 | |
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310 | \subsubsection{Langmuir cells} |
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311 | %--------------------------------------% |
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312 | |
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313 | Langmuir circulations (LC) can be described as ordered large-scale vertical motions in |
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314 | the surface layer of the oceans. |
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315 | Although LC have nothing to do with convection, the circulation pattern is rather similar to |
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316 | so-called convective rolls in the atmospheric boundary layer. |
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317 | The detailed physics behind LC is described in, for example, \citet{craik.leibovich_JFM76}. |
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318 | The prevailing explanation is that LC arise from a nonlinear interaction between the Stokes drift and |
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319 | wind drift currents. |
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320 | |
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321 | Here we introduced in the TKE turbulent closure the simple parameterization of Langmuir circulations proposed by |
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322 | \citep{axell_JGR02} for a $k-\epsilon$ turbulent closure. |
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323 | The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in |
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324 | an extra source terms of TKE, $P_{LC}$. |
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325 | The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to |
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326 | \forcode{.true.} in the namtke namelist. |
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327 | |
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328 | By making an analogy with the characteristic convective velocity scale (\eg, \citet{dalessio.abdella.ea_JPO98}), |
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329 | $P_{LC}$ is assumed to be : |
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330 | \[ |
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331 | P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}} |
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332 | \] |
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333 | where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth. |
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334 | With no information about the wave field, $w_{LC}$ is assumed to be proportional to |
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335 | the Stokes drift $u_s = 0.377\,\,|\tau|^{1/2}$, where $|\tau|$ is the surface wind stress module |
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336 | \footnote{Following \citet{li.garrett_JMR93}, the surface Stoke drift velocity may be expressed as |
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337 | $u_s = 0.016 \,|U_{10m}|$. |
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338 | Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of |
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339 | $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress |
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340 | }. |
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341 | For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at |
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342 | a finite depth $H_{LC}$ (which is often close to the mixed layer depth), |
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343 | and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures). |
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344 | The resulting expression for $w_{LC}$ is : |
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345 | \[ |
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346 | w_{LC} = |
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347 | \begin{cases} |
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348 | c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} ) & \text{if $-z \leq H_{LC}$} \\ |
---|
349 | 0 & \text{otherwise} |
---|
350 | \end{cases} |
---|
351 | \] |
---|
352 | where $c_{LC} = 0.15$ has been chosen by \citep{axell_JGR02} as a good compromise to fit LES data. |
---|
353 | The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second. |
---|
354 | The value of $c_{LC}$ is set through the \np{rn\_lc} namelist parameter, |
---|
355 | having in mind that it should stay between 0.15 and 0.54 \citep{axell_JGR02}. |
---|
356 | |
---|
357 | The $H_{LC}$ is estimated in a similar way as the turbulent length scale of TKE equations: |
---|
358 | $H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by |
---|
359 | converting its kinetic energy to potential energy, according to |
---|
360 | \[ |
---|
361 | - \int_{-H_{LC}}^0 { N^2\;z \;dz} = \frac{1}{2} u_s^2 |
---|
362 | \] |
---|
363 | |
---|
364 | \subsubsection{Mixing just below the mixed layer} |
---|
365 | %--------------------------------------------------------------% |
---|
366 | |
---|
367 | Vertical mixing parameterizations commonly used in ocean general circulation models tend to |
---|
368 | produce mixed-layer depths that are too shallow during summer months and windy conditions. |
---|
369 | This bias is particularly acute over the Southern Ocean. |
---|
370 | To overcome this systematic bias, an ad hoc parameterization is introduced into the TKE scheme \cite{rodgers.aumont.ea_B14}. |
---|
371 | The parameterization is an empirical one, \ie not derived from theoretical considerations, |
---|
372 | but rather is meant to account for observed processes that affect the density structure of |
---|
373 | the ocean’s planetary boundary layer that are not explicitly captured by default in the TKE scheme |
---|
374 | (\ie near-inertial oscillations and ocean swells and waves). |
---|
375 | |
---|
376 | When using this parameterization (\ie when \np{nn\_etau}\forcode{ = 1}), |
---|
377 | the TKE input to the ocean ($S$) imposed by the winds in the form of near-inertial oscillations, |
---|
378 | swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition, |
---|
379 | plus a depth depend one given by: |
---|
380 | \begin{equation} |
---|
381 | \label{eq:ZDF_Ehtau} |
---|
382 | S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau} |
---|
383 | \end{equation} |
---|
384 | where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that |
---|
385 | penetrate in the ocean, $h_\tau$ is a vertical mixing length scale that controls exponential shape of |
---|
386 | the penetration, and $f_i$ is the ice concentration |
---|
387 | (no penetration if $f_i=1$, that is if the ocean is entirely covered by sea-ice). |
---|
388 | The value of $f_r$, usually a few percents, is specified through \np{rn\_efr} namelist parameter. |
---|
389 | The vertical mixing length scale, $h_\tau$, can be set as a 10~m uniform value (\np{nn\_etau}\forcode{ = 0}) or |
---|
390 | a latitude dependent value (varying from 0.5~m at the Equator to a maximum value of 30~m at high latitudes |
---|
391 | (\np{nn\_etau}\forcode{ = 1}). |
---|
392 | |
---|
393 | Note that two other option existe, \np{nn\_etau}\forcode{ = 2..3}. |
---|
394 | They correspond to applying \autoref{eq:ZDF_Ehtau} only at the base of the mixed layer, |
---|
395 | or to using the high frequency part of the stress to evaluate the fraction of TKE that penetrate the ocean. |
---|
396 | Those two options are obsolescent features introduced for test purposes. |
---|
397 | They will be removed in the next release. |
---|
398 | |
---|
399 | % from Burchard et al OM 2008 : |
---|
400 | % the most critical process not reproduced by statistical turbulence models is the activity of |
---|
401 | % internal waves and their interaction with turbulence. After the Reynolds decomposition, |
---|
402 | % internal waves are in principle included in the RANS equations, but later partially |
---|
403 | % excluded by the hydrostatic assumption and the model resolution. |
---|
404 | % Thus far, the representation of internal wave mixing in ocean models has been relatively crude |
---|
405 | % (\eg Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002). |
---|
406 | |
---|
407 | % ------------------------------------------------------------------------------------------------------------- |
---|
408 | % TKE discretization considerations |
---|
409 | % ------------------------------------------------------------------------------------------------------------- |
---|
410 | \subsection{TKE discretization considerations (\protect\key{zdftke})} |
---|
411 | \label{subsec:ZDF_tke_ene} |
---|
412 | |
---|
413 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
414 | \begin{figure}[!t] |
---|
415 | \begin{center} |
---|
416 | \includegraphics[width=\textwidth]{Fig_ZDF_TKE_time_scheme} |
---|
417 | \caption{ |
---|
418 | \protect\label{fig:TKE_time_scheme} |
---|
419 | Illustration of the TKE time integration and its links to the momentum and tracer time integration. |
---|
420 | } |
---|
421 | \end{center} |
---|
422 | \end{figure} |
---|
423 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
424 | |
---|
425 | The production of turbulence by vertical shear (the first term of the right hand side of |
---|
426 | \autoref{eq:zdftke_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion |
---|
427 | (first line in \autoref{eq:PE_zdf}). |
---|
428 | To do so a special care have to be taken for both the time and space discretization of |
---|
429 | the TKE equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}. |
---|
430 | |
---|
431 | Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows how |
---|
432 | the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with |
---|
433 | the one-level forward time stepping of TKE equation. |
---|
434 | With this framework, the total loss of kinetic energy (in 1D for the demonstration) due to |
---|
435 | the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and |
---|
436 | summing the result vertically: |
---|
437 | \begin{equation} |
---|
438 | \label{eq:energ1} |
---|
439 | \begin{split} |
---|
440 | \int_{-H}^{\eta} u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt} \right) \,dz \\ |
---|
441 | &= \Bigl[ u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta} |
---|
442 | - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz } |
---|
443 | \end{split} |
---|
444 | \end{equation} |
---|
445 | Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$, |
---|
446 | known at time $t$ (\autoref{fig:TKE_time_scheme}), as it is required when using the TKE scheme |
---|
447 | (see \autoref{sec:STP_forward_imp}). |
---|
448 | The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy transfer at |
---|
449 | the surface (atmospheric forcing) and at the bottom (friction effect). |
---|
450 | The second term is always negative. |
---|
451 | It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$. |
---|
452 | \autoref{eq:energ1} implies that, to be energetically consistent, |
---|
453 | the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as |
---|
454 | ${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ |
---|
455 | (and not by the more straightforward $K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$). |
---|
456 | |
---|
457 | A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification |
---|
458 | (second term of the right hand side of \autoref{eq:zdftke_e}). |
---|
459 | This term must balance the input of potential energy resulting from vertical mixing. |
---|
460 | The rate of change of potential energy (in 1D for the demonstration) due vertical mixing is obtained by |
---|
461 | multiplying vertical density diffusion tendency by $g\,z$ and and summing the result vertically: |
---|
462 | \begin{equation} |
---|
463 | \label{eq:energ2} |
---|
464 | \begin{split} |
---|
465 | \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt} \right) \,dz \\ |
---|
466 | &= \Bigl[ g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta} |
---|
467 | - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz \\ |
---|
468 | &= - \Bigl[ z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta} |
---|
469 | + \int_{-H}^{\eta}{ \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz } |
---|
470 | \end{split} |
---|
471 | \end{equation} |
---|
472 | where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. |
---|
473 | The first term of the right hand side of \autoref{eq:energ2} is always zero because |
---|
474 | there is no diffusive flux through the ocean surface and bottom). |
---|
475 | The second term is minus the destruction rate of $\bar{e}$ due to stratification. |
---|
476 | Therefore \autoref{eq:energ1} implies that, to be energetically consistent, |
---|
477 | the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e}, the TKE equation. |
---|
478 | |
---|
479 | Let us now address the space discretization issue. |
---|
480 | The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in |
---|
481 | the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:cell}). |
---|
482 | A space averaging is thus required to obtain the shear TKE production term. |
---|
483 | By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of eddy coefficient by |
---|
484 | the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. |
---|
485 | Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into account. |
---|
486 | |
---|
487 | The above energetic considerations leads to the following final discrete form for the TKE equation: |
---|
488 | \begin{equation} |
---|
489 | \label{eq:zdftke_ene} |
---|
490 | \begin{split} |
---|
491 | \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt} \equiv |
---|
492 | \Biggl\{ \Biggr. |
---|
493 | &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} } |
---|
494 | \ \frac{\delta_{k+1/2}[u^ t ]}{{e_3u}^ t } \right) }^{\,i} \\ |
---|
495 | +&\overline{ \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} } |
---|
496 | \ \frac{\delta_{k+1/2}[v^ t ]}{{e_3v}^ t } \right) }^{\,j} |
---|
497 | \Biggr. \Biggr\} \\ |
---|
498 | % |
---|
499 | - &{K_\rho}^{t-\rdt}\,{(N^2)^t} \\ |
---|
500 | % |
---|
501 | +&\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] \\ |
---|
502 | % |
---|
503 | - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt} |
---|
504 | \end{split} |
---|
505 | \end{equation} |
---|
506 | where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation) |
---|
507 | are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}). |
---|
508 | Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible. |
---|
509 | The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as |
---|
510 | they all appear in the right hand side of \autoref{eq:zdftke_ene}. |
---|
511 | For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. |
---|
512 | |
---|
513 | % ------------------------------------------------------------------------------------------------------------- |
---|
514 | % GLS Generic Length Scale Scheme |
---|
515 | % ------------------------------------------------------------------------------------------------------------- |
---|
516 | \subsection{GLS: Generic Length Scale (\protect\key{zdfgls})} |
---|
517 | \label{subsec:ZDF_gls} |
---|
518 | |
---|
519 | %--------------------------------------------namzdf_gls--------------------------------------------------------- |
---|
520 | |
---|
521 | \nlst{namzdf_gls} |
---|
522 | %-------------------------------------------------------------------------------------------------------------- |
---|
523 | |
---|
524 | The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on two prognostic equations: |
---|
525 | one for the turbulent kinetic energy $\bar {e}$, and another for the generic length scale, |
---|
526 | $\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}. |
---|
527 | This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, |
---|
528 | where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover a number of |
---|
529 | well-known turbulent closures ($k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87}, |
---|
530 | $k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}). |
---|
531 | The GLS scheme is given by the following set of equations: |
---|
532 | \begin{equation} |
---|
533 | \label{eq:zdfgls_e} |
---|
534 | \frac{\partial \bar{e}}{\partial t} = |
---|
535 | \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 |
---|
536 | +\left( \frac{\partial v}{\partial k} \right)^2} \right] |
---|
537 | -K_\rho \,N^2 |
---|
538 | +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right] |
---|
539 | - \epsilon |
---|
540 | \end{equation} |
---|
541 | |
---|
542 | \[ |
---|
543 | % \label{eq:zdfgls_psi} |
---|
544 | \begin{split} |
---|
545 | \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ |
---|
546 | \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 |
---|
547 | +\left( \frac{\partial v}{\partial k} \right)^2} \right] |
---|
548 | - C_3 \,K_\rho\,N^2 - C_2 \,\epsilon \,Fw \right\} \\ |
---|
549 | &+\frac{1}{e_3} \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 } |
---|
550 | \;\frac{\partial \psi}{\partial k}} \right]\; |
---|
551 | \end{split} |
---|
552 | \] |
---|
553 | |
---|
554 | \[ |
---|
555 | % \label{eq:zdfgls_kz} |
---|
556 | \begin{split} |
---|
557 | K_m &= C_{\mu} \ \sqrt {\bar{e}} \ l \\ |
---|
558 | K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l |
---|
559 | \end{split} |
---|
560 | \] |
---|
561 | |
---|
562 | \[ |
---|
563 | % \label{eq:zdfgls_eps} |
---|
564 | {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; |
---|
565 | \] |
---|
566 | where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and |
---|
567 | $\epsilon$ the dissipation rate. |
---|
568 | The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of |
---|
569 | the choice of the turbulence model. |
---|
570 | Four different turbulent models are pre-defined (Tab.\autoref{tab:GLS}). |
---|
571 | They are made available through the \np{nn\_clo} namelist parameter. |
---|
572 | |
---|
573 | %--------------------------------------------------TABLE-------------------------------------------------- |
---|
574 | \begin{table}[htbp] |
---|
575 | \begin{center} |
---|
576 | % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c} |
---|
577 | \begin{tabular}{ccccc} |
---|
578 | & $k-kl$ & $k-\epsilon$ & $k-\omega$ & generic \\ |
---|
579 | % & \citep{mellor.yamada_RG82} & \citep{rodi_JGR87} & \citep{wilcox_AJ88} & \\ |
---|
580 | \hline |
---|
581 | \hline |
---|
582 | \np{nn\_clo} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} \\ |
---|
583 | \hline |
---|
584 | $( p , n , m )$ & ( 0 , 1 , 1 ) & ( 3 , 1.5 , -1 ) & ( -1 , 0.5 , -1 ) & ( 2 , 1 , -0.67 ) \\ |
---|
585 | $\sigma_k$ & 2.44 & 1. & 2. & 0.8 \\ |
---|
586 | $\sigma_\psi$ & 2.44 & 1.3 & 2. & 1.07 \\ |
---|
587 | $C_1$ & 0.9 & 1.44 & 0.555 & 1. \\ |
---|
588 | $C_2$ & 0.5 & 1.92 & 0.833 & 1.22 \\ |
---|
589 | $C_3$ & 1. & 1. & 1. & 1. \\ |
---|
590 | $F_{wall}$ & Yes & -- & -- & -- \\ |
---|
591 | \hline |
---|
592 | \hline |
---|
593 | \end{tabular} |
---|
594 | \caption{ |
---|
595 | \protect\label{tab:GLS} |
---|
596 | Set of predefined GLS parameters, or equivalently predefined turbulence models available with |
---|
597 | \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls}. |
---|
598 | } |
---|
599 | \end{center} |
---|
600 | \end{table} |
---|
601 | %-------------------------------------------------------------------------------------------------------------- |
---|
602 | |
---|
603 | In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force the convergence of |
---|
604 | the mixing length towards $K z_b$ ($K$: Kappa and $z_b$: rugosity length) value near physical boundaries |
---|
605 | (logarithmic boundary layer law). |
---|
606 | $C_{\mu}$ and $C_{\mu'}$ are calculated from stability function proposed by \citet{galperin.kantha.ea_JAS88}, |
---|
607 | or by \citet{kantha.clayson_JGR94} or one of the two functions suggested by \citet{canuto.howard.ea_JPO01} |
---|
608 | (\np{nn\_stab\_func}\forcode{ = 0..3}, resp.). |
---|
609 | The value of $C_{0\mu}$ depends of the choice of the stability function. |
---|
610 | |
---|
611 | The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated thanks to Dirichlet or |
---|
612 | Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp. |
---|
613 | As for TKE closure, the wave effect on the mixing is considered when |
---|
614 | \np{ln\_crban}\forcode{ = .true.} \citep{craig.banner_JPO94, mellor.blumberg_JPO04}. |
---|
615 | The \np{rn\_crban} namelist parameter is $\alpha_{CB}$ in \autoref{eq:ZDF_Esbc} and |
---|
616 | \np{rn\_charn} provides the value of $\beta$ in \autoref{eq:ZDF_Lsbc}. |
---|
617 | |
---|
618 | The $\psi$ equation is known to fail in stably stratified flows, and for this reason |
---|
619 | almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy. |
---|
620 | With this clipping, the maximum permissible length scale is determined by $l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$. |
---|
621 | A value of $c_{lim} = 0.53$ is often used \citep{galperin.kantha.ea_JAS88}. |
---|
622 | \cite{umlauf.burchard_CSR05} show that the value of the clipping factor is of crucial importance for |
---|
623 | the entrainment depth predicted in stably stratified situations, |
---|
624 | and that its value has to be chosen in accordance with the algebraic model for the turbulent fluxes. |
---|
625 | The clipping is only activated if \np{ln\_length\_lim}\forcode{ = .true.}, |
---|
626 | and the $c_{lim}$ is set to the \np{rn\_clim\_galp} value. |
---|
627 | |
---|
628 | The time and space discretization of the GLS equations follows the same energetic consideration as for |
---|
629 | the TKE case described in \autoref{subsec:ZDF_tke_ene} \citep{burchard_OM02}. |
---|
630 | Examples of performance of the 4 turbulent closure scheme can be found in \citet{warner.sherwood.ea_OM05}. |
---|
631 | |
---|
632 | % ------------------------------------------------------------------------------------------------------------- |
---|
633 | % OSM OSMOSIS BL Scheme |
---|
634 | % ------------------------------------------------------------------------------------------------------------- |
---|
635 | \subsection{OSM: OSMOSIS boundary layer scheme (\protect\key{zdfosm})} |
---|
636 | \label{subsec:ZDF_osm} |
---|
637 | |
---|
638 | %--------------------------------------------namzdf_osm--------------------------------------------------------- |
---|
639 | |
---|
640 | \nlst{namzdf_osm} |
---|
641 | %-------------------------------------------------------------------------------------------------------------- |
---|
642 | |
---|
643 | The OSMOSIS turbulent closure scheme is based on...... TBC |
---|
644 | |
---|
645 | % ================================================================ |
---|
646 | % Convection |
---|
647 | % ================================================================ |
---|
648 | \section{Convection} |
---|
649 | \label{sec:ZDF_conv} |
---|
650 | |
---|
651 | %--------------------------------------------namzdf-------------------------------------------------------- |
---|
652 | |
---|
653 | \nlst{namzdf} |
---|
654 | %-------------------------------------------------------------------------------------------------------------- |
---|
655 | |
---|
656 | Static instabilities (\ie light potential densities under heavy ones) may occur at particular ocean grid points. |
---|
657 | In nature, convective processes quickly re-establish the static stability of the water column. |
---|
658 | These processes have been removed from the model via the hydrostatic assumption so they must be parameterized. |
---|
659 | Three parameterisations are available to deal with convective processes: |
---|
660 | a non-penetrative convective adjustment or an enhanced vertical diffusion, |
---|
661 | or/and the use of a turbulent closure scheme. |
---|
662 | |
---|
663 | % ------------------------------------------------------------------------------------------------------------- |
---|
664 | % Non-Penetrative Convective Adjustment |
---|
665 | % ------------------------------------------------------------------------------------------------------------- |
---|
666 | \subsection[Non-penetrative convective adjmt (\protect\np{ln\_tranpc}\forcode{ = .true.})] |
---|
667 | {Non-penetrative convective adjustment (\protect\np{ln\_tranpc}\forcode{ = .true.})} |
---|
668 | \label{subsec:ZDF_npc} |
---|
669 | |
---|
670 | %--------------------------------------------namzdf-------------------------------------------------------- |
---|
671 | |
---|
672 | \nlst{namzdf} |
---|
673 | %-------------------------------------------------------------------------------------------------------------- |
---|
674 | |
---|
675 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
676 | \begin{figure}[!htb] |
---|
677 | \begin{center} |
---|
678 | \includegraphics[width=\textwidth]{Fig_npc} |
---|
679 | \caption{ |
---|
680 | \protect\label{fig:npc} |
---|
681 | Example of an unstable density profile treated by the non penetrative convective adjustment algorithm. |
---|
682 | $1^{st}$ step: the initial profile is checked from the surface to the bottom. |
---|
683 | It is found to be unstable between levels 3 and 4. |
---|
684 | They are mixed. |
---|
685 | The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed. |
---|
686 | The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed. |
---|
687 | The $1^{st}$ step ends since the density profile is then stable below the level 3. |
---|
688 | $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step: |
---|
689 | levels 2 to 5 are mixed. |
---|
690 | The new density profile is checked. |
---|
691 | It is found stable: end of algorithm. |
---|
692 | } |
---|
693 | \end{center} |
---|
694 | \end{figure} |
---|
695 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
696 | |
---|
697 | Options are defined through the \ngn{namzdf} namelist variables. |
---|
698 | The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}\forcode{ = .true.}. |
---|
699 | It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously the statically unstable portion of |
---|
700 | the water column, but only until the density structure becomes neutrally stable |
---|
701 | (\ie until the mixed portion of the water column has \textit{exactly} the density of the water just below) |
---|
702 | \citep{madec.delecluse.ea_JPO91}. |
---|
703 | The associated algorithm is an iterative process used in the following way (\autoref{fig:npc}): |
---|
704 | starting from the top of the ocean, the first instability is found. |
---|
705 | Assume in the following that the instability is located between levels $k$ and $k+1$. |
---|
706 | The temperature and salinity in the two levels are vertically mixed, conserving the heat and salt contents of |
---|
707 | the water column. |
---|
708 | The new density is then computed by a linear approximation. |
---|
709 | If the new density profile is still unstable between levels $k+1$ and $k+2$, |
---|
710 | levels $k$, $k+1$ and $k+2$ are then mixed. |
---|
711 | This process is repeated until stability is established below the level $k$ |
---|
712 | (the mixing process can go down to the ocean bottom). |
---|
713 | The algorithm is repeated to check if the density profile between level $k-1$ and $k$ is unstable and/or |
---|
714 | if there is no deeper instability. |
---|
715 | |
---|
716 | This algorithm is significantly different from mixing statically unstable levels two by two. |
---|
717 | The latter procedure cannot converge with a finite number of iterations for some vertical profiles while |
---|
718 | the algorithm used in \NEMO converges for any profile in a number of iterations which is less than |
---|
719 | the number of vertical levels. |
---|
720 | This property is of paramount importance as pointed out by \citet{killworth_iprc89}: |
---|
721 | it avoids the existence of permanent and unrealistic static instabilities at the sea surface. |
---|
722 | This non-penetrative convective algorithm has been proved successful in studies of the deep water formation in |
---|
723 | the north-western Mediterranean Sea \citep{madec.delecluse.ea_JPO91, madec.chartier.ea_DAO91, madec.crepon_iprc91}. |
---|
724 | |
---|
725 | The current implementation has been modified in order to deal with any non linear equation of seawater |
---|
726 | (L. Brodeau, personnal communication). |
---|
727 | Two main differences have been introduced compared to the original algorithm: |
---|
728 | $(i)$ the stability is now checked using the Brunt-V\"{a}is\"{a}l\"{a} frequency |
---|
729 | (not the the difference in potential density); |
---|
730 | $(ii)$ when two levels are found unstable, their thermal and haline expansion coefficients are vertically mixed in |
---|
731 | the same way their temperature and salinity has been mixed. |
---|
732 | These two modifications allow the algorithm to perform properly and accurately with TEOS10 or EOS-80 without |
---|
733 | having to recompute the expansion coefficients at each mixing iteration. |
---|
734 | |
---|
735 | % ------------------------------------------------------------------------------------------------------------- |
---|
736 | % Enhanced Vertical Diffusion |
---|
737 | % ------------------------------------------------------------------------------------------------------------- |
---|
738 | \subsection{Enhanced vertical diffusion (\protect\np{ln\_zdfevd}\forcode{ = .true.})} |
---|
739 | \label{subsec:ZDF_evd} |
---|
740 | |
---|
741 | %--------------------------------------------namzdf-------------------------------------------------------- |
---|
742 | |
---|
743 | \nlst{namzdf} |
---|
744 | %-------------------------------------------------------------------------------------------------------------- |
---|
745 | |
---|
746 | Options are defined through the \ngn{namzdf} namelist variables. |
---|
747 | The enhanced vertical diffusion parameterisation is used when \np{ln\_zdfevd}\forcode{ = .true.}. |
---|
748 | In this case, the vertical eddy mixing coefficients are assigned very large values |
---|
749 | (a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable |
---|
750 | (\ie when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{lazar_phd97, lazar.madec.ea_JPO99}. |
---|
751 | This is done either on tracers only (\np{nn\_evdm}\forcode{ = 0}) or |
---|
752 | on both momentum and tracers (\np{nn\_evdm}\forcode{ = 1}). |
---|
753 | |
---|
754 | In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and if \np{nn\_evdm}\forcode{ = 1}, |
---|
755 | the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ values also, are set equal to |
---|
756 | the namelist parameter \np{rn\_avevd}. |
---|
757 | A typical value for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$. |
---|
758 | This parameterisation of convective processes is less time consuming than |
---|
759 | the convective adjustment algorithm presented above when mixing both tracers and |
---|
760 | momentum in the case of static instabilities. |
---|
761 | It requires the use of an implicit time stepping on vertical diffusion terms |
---|
762 | (\ie np{ln\_zdfexp}\forcode{ = .false.}). |
---|
763 | |
---|
764 | Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$. |
---|
765 | This removes a potential source of divergence of odd and even time step in |
---|
766 | a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:STP_mLF}). |
---|
767 | |
---|
768 | % ------------------------------------------------------------------------------------------------------------- |
---|
769 | % Turbulent Closure Scheme |
---|
770 | % ------------------------------------------------------------------------------------------------------------- |
---|
771 | \subsection[Turbulent closure scheme (\protect\key{zdf}\{tke,gls,osm\})]{Turbulent Closure Scheme (\protect\key{zdftke}, \protect\key{zdfgls} or \protect\key{zdfosm})} |
---|
772 | \label{subsec:ZDF_tcs} |
---|
773 | |
---|
774 | The turbulent closure scheme presented in \autoref{subsec:ZDF_tke} and \autoref{subsec:ZDF_gls} |
---|
775 | (\key{zdftke} or \key{zdftke} is defined) in theory solves the problem of statically unstable density profiles. |
---|
776 | In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in |
---|
777 | \autoref{eq:zdftke_e} or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative. |
---|
778 | It results in large values of $A_T^{vT}$ and $A_T^{vT}$, and also the four neighbouring $A_u^{vm} {and}\;A_v^{vm}$ |
---|
779 | (up to $1\;m^2s^{-1}$). |
---|
780 | These large values restore the static stability of the water column in a way similar to that of |
---|
781 | the enhanced vertical diffusion parameterisation (\autoref{subsec:ZDF_evd}). |
---|
782 | However, in the vicinity of the sea surface (first ocean layer), the eddy coefficients computed by |
---|
783 | the turbulent closure scheme do not usually exceed $10^{-2}m.s^{-1}$, |
---|
784 | because the mixing length scale is bounded by the distance to the sea surface. |
---|
785 | It can thus be useful to combine the enhanced vertical diffusion with the turbulent closure scheme, |
---|
786 | \ie setting the \np{ln\_zdfnpc} namelist parameter to true and |
---|
787 | defining the turbulent closure CPP key all together. |
---|
788 | |
---|
789 | The KPP turbulent closure scheme already includes enhanced vertical diffusion in the case of convection, |
---|
790 | as governed by the variables $bvsqcon$ and $difcon$ found in \mdl{zdfkpp}, |
---|
791 | therefore \np{ln\_zdfevd}\forcode{ = .false.} should be used with the KPP scheme. |
---|
792 | % gm% + one word on non local flux with KPP scheme trakpp.F90 module... |
---|
793 | |
---|
794 | % ================================================================ |
---|
795 | % Double Diffusion Mixing |
---|
796 | % ================================================================ |
---|
797 | \section{Double diffusion mixing (\protect\key{zdfddm})} |
---|
798 | \label{sec:ZDF_ddm} |
---|
799 | |
---|
800 | %-------------------------------------------namzdf_ddm------------------------------------------------- |
---|
801 | % |
---|
802 | %\nlst{namzdf_ddm} |
---|
803 | %-------------------------------------------------------------------------------------------------------------- |
---|
804 | |
---|
805 | Options are defined through the \ngn{namzdf\_ddm} namelist variables. |
---|
806 | Double diffusion occurs when relatively warm, salty water overlies cooler, fresher water, or vice versa. |
---|
807 | The former condition leads to salt fingering and the latter to diffusive convection. |
---|
808 | Double-diffusive phenomena contribute to diapycnal mixing in extensive regions of the ocean. |
---|
809 | \citet{merryfield.holloway.ea_JPO99} include a parameterisation of such phenomena in a global ocean model and show that |
---|
810 | it leads to relatively minor changes in circulation but exerts significant regional influences on |
---|
811 | temperature and salinity. |
---|
812 | This parameterisation has been introduced in \mdl{zdfddm} module and is controlled by the \key{zdfddm} CPP key. |
---|
813 | |
---|
814 | Diapycnal mixing of S and T are described by diapycnal diffusion coefficients |
---|
815 | \begin{align*} |
---|
816 | % \label{eq:zdfddm_Kz} |
---|
817 | &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ |
---|
818 | &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} |
---|
819 | \end{align*} |
---|
820 | where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, |
---|
821 | and $o$ by processes other than double diffusion. |
---|
822 | The rates of double-diffusive mixing depend on the buoyancy ratio |
---|
823 | $R_\rho = \alpha \partial_z T / \beta \partial_z S$, where $\alpha$ and $\beta$ are coefficients of |
---|
824 | thermal expansion and saline contraction (see \autoref{subsec:TRA_eos}). |
---|
825 | To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt |
---|
826 | (1981): |
---|
827 | \begin{align} |
---|
828 | \label{eq:zdfddm_f} |
---|
829 | A_f^{vS} &= |
---|
830 | \begin{cases} |
---|
831 | \frac{A^{\ast v}}{1+(R_\rho / R_c)^n } &\text{if $R_\rho > 1$ and $N^2>0$ } \\ |
---|
832 | 0 &\text{otherwise} |
---|
833 | \end{cases} |
---|
834 | \\ \label{eq:zdfddm_f_T} |
---|
835 | A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho |
---|
836 | \end{align} |
---|
837 | |
---|
838 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
839 | \begin{figure}[!t] |
---|
840 | \begin{center} |
---|
841 | \includegraphics[width=\textwidth]{Fig_zdfddm} |
---|
842 | \caption{ |
---|
843 | \protect\label{fig:zdfddm} |
---|
844 | From \citet{merryfield.holloway.ea_JPO99} : |
---|
845 | (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering. |
---|
846 | Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$; |
---|
847 | (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in regions of |
---|
848 | diffusive convection. |
---|
849 | Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation. |
---|
850 | The latter is not implemented in \NEMO. |
---|
851 | } |
---|
852 | \end{center} |
---|
853 | \end{figure} |
---|
854 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
855 | |
---|
856 | The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx 0.7$ of |
---|
857 | buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}). |
---|
858 | Following \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$. |
---|
859 | |
---|
860 | To represent mixing of S and T by diffusive layering, the diapycnal diffusivities suggested by |
---|
861 | Federov (1988) is used: |
---|
862 | \begin{align} |
---|
863 | % \label{eq:zdfddm_d} |
---|
864 | A_d^{vT} &= |
---|
865 | \begin{cases} |
---|
866 | 1.3635 \, \exp{\left( 4.6\, \exp{ \left[ -0.54\,( R_{\rho}^{-1} - 1 ) \right] } \right)} |
---|
867 | &\text{if $0<R_\rho < 1$ and $N^2>0$ } \\ |
---|
868 | 0 &\text{otherwise} |
---|
869 | \end{cases} |
---|
870 | \nonumber \\ |
---|
871 | \label{eq:zdfddm_d_S} |
---|
872 | A_d^{vS} &= |
---|
873 | \begin{cases} |
---|
874 | A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if $0.5 \leq R_\rho<1$ and $N^2>0$ } \\ |
---|
875 | A_d^{vT} \ 0.15 \ R_\rho &\text{if $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\ |
---|
876 | 0 &\text{otherwise} |
---|
877 | \end{cases} |
---|
878 | \end{align} |
---|
879 | |
---|
880 | The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ are illustrated in |
---|
881 | \autoref{fig:zdfddm}. |
---|
882 | Implementing this requires computing $R_\rho$ at each grid point on every time step. |
---|
883 | This is done in \mdl{eosbn2} at the same time as $N^2$ is computed. |
---|
884 | This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive). |
---|
885 | |
---|
886 | % ================================================================ |
---|
887 | % Bottom Friction |
---|
888 | % ================================================================ |
---|
889 | \section{Bottom and top friction (\protect\mdl{zdfbfr})} |
---|
890 | \label{sec:ZDF_bfr} |
---|
891 | |
---|
892 | %--------------------------------------------nambfr-------------------------------------------------------- |
---|
893 | % |
---|
894 | %\nlst{nambfr} |
---|
895 | %-------------------------------------------------------------------------------------------------------------- |
---|
896 | |
---|
897 | Options to define the top and bottom friction are defined through the \ngn{nambfr} namelist variables. |
---|
898 | The bottom friction represents the friction generated by the bathymetry. |
---|
899 | The top friction represents the friction generated by the ice shelf/ocean interface. |
---|
900 | As the friction processes at the top and bottom are treated in similar way, |
---|
901 | only the bottom friction is described in detail below. |
---|
902 | |
---|
903 | |
---|
904 | Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as |
---|
905 | a condition on the vertical diffusive flux. |
---|
906 | For the bottom boundary layer, one has: |
---|
907 | \[ |
---|
908 | % \label{eq:zdfbfr_flux} |
---|
909 | A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} |
---|
910 | \] |
---|
911 | where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside |
---|
912 | the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean). |
---|
913 | How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near |
---|
914 | the bottom relative to the Ekman layer depth. |
---|
915 | For example, in order to obtain an Ekman layer depth $d = \sqrt{2\;A^{vm}} / f = 50$~m, |
---|
916 | one needs a vertical diffusion coefficient $A^{vm} = 0.125$~m$^2$s$^{-1}$ |
---|
917 | (for a Coriolis frequency $f = 10^{-4}$~m$^2$s$^{-1}$). |
---|
918 | With a background diffusion coefficient $A^{vm} = 10^{-4}$~m$^2$s$^{-1}$, the Ekman layer depth is only 1.4~m. |
---|
919 | When the vertical mixing coefficient is this small, using a flux condition is equivalent to |
---|
920 | entering the viscous forces (either wind stress or bottom friction) as a body force over the depth of the top or |
---|
921 | bottom model layer. |
---|
922 | To illustrate this, consider the equation for $u$ at $k$, the last ocean level: |
---|
923 | \begin{equation} |
---|
924 | \label{eq:zdfbfr_flux2} |
---|
925 | \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}} |
---|
926 | \end{equation} |
---|
927 | If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth. |
---|
928 | On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used, |
---|
929 | the turbulent Ekman layer will be represented explicitly by the model. |
---|
930 | However, the logarithmic layer is never represented in current primitive equation model applications: |
---|
931 | it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $. |
---|
932 | Two choices are available in \NEMO: a linear and a quadratic bottom friction. |
---|
933 | Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in |
---|
934 | the present release of \NEMO. |
---|
935 | |
---|
936 | In the code, the bottom friction is imposed by adding the trend due to the bottom friction to |
---|
937 | the general momentum trend in \mdl{dynbfr}. |
---|
938 | For the time-split surface pressure gradient algorithm, the momentum trend due to |
---|
939 | the barotropic component needs to be handled separately. |
---|
940 | For this purpose it is convenient to compute and store coefficients which can be simply combined with |
---|
941 | bottom velocities and geometric values to provide the momentum trend due to bottom friction. |
---|
942 | These coefficients are computed in \mdl{zdfbfr} and generally take the form $c_b^{\textbf U}$ where: |
---|
943 | \begin{equation} |
---|
944 | \label{eq:zdfbfr_bdef} |
---|
945 | \frac{\partial {\textbf U_h}}{\partial t} = |
---|
946 | - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b |
---|
947 | \end{equation} |
---|
948 | where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity. |
---|
949 | |
---|
950 | % ------------------------------------------------------------------------------------------------------------- |
---|
951 | % Linear Bottom Friction |
---|
952 | % ------------------------------------------------------------------------------------------------------------- |
---|
953 | \subsection{Linear bottom friction (\protect\np{nn\_botfr}\forcode{ = 0..1})} |
---|
954 | \label{subsec:ZDF_bfr_linear} |
---|
955 | |
---|
956 | The linear bottom friction parameterisation (including the special case of a free-slip condition) assumes that |
---|
957 | the bottom friction is proportional to the interior velocity (\ie the velocity of the last model level): |
---|
958 | \[ |
---|
959 | % \label{eq:zdfbfr_linear} |
---|
960 | {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b |
---|
961 | \] |
---|
962 | where $r$ is a friction coefficient expressed in ms$^{-1}$. |
---|
963 | This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean, |
---|
964 | and setting $r = H / \tau$, where $H$ is the ocean depth. |
---|
965 | Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{weatherly_JMR84}. |
---|
966 | A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models. |
---|
967 | One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ |
---|
968 | (\citet{gill_bk82}, Eq. 9.6.6). |
---|
969 | For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, |
---|
970 | and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$. |
---|
971 | This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days. |
---|
972 | It can be changed by specifying \np{rn\_bfri1} (namelist parameter). |
---|
973 | |
---|
974 | For the linear friction case the coefficients defined in the general expression \autoref{eq:zdfbfr_bdef} are: |
---|
975 | \[ |
---|
976 | % \label{eq:zdfbfr_linbfr_b} |
---|
977 | \begin{split} |
---|
978 | c_b^u &= - r\\ |
---|
979 | c_b^v &= - r\\ |
---|
980 | \end{split} |
---|
981 | \] |
---|
982 | When \np{nn\_botfr}\forcode{ = 1}, the value of $r$ used is \np{rn\_bfri1}. |
---|
983 | Setting \np{nn\_botfr}\forcode{ = 0} is equivalent to setting $r=0$ and |
---|
984 | leads to a free-slip bottom boundary condition. |
---|
985 | These values are assigned in \mdl{zdfbfr}. |
---|
986 | From v3.2 onwards there is support for local enhancement of these values via an externally defined 2D mask array |
---|
987 | (\np{ln\_bfr2d}\forcode{ = .true.}) given in the \ifile{bfr\_coef} input NetCDF file. |
---|
988 | The mask values should vary from 0 to 1. |
---|
989 | Locations with a non-zero mask value will have the friction coefficient increased by |
---|
990 | $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri1}. |
---|
991 | |
---|
992 | % ------------------------------------------------------------------------------------------------------------- |
---|
993 | % Non-Linear Bottom Friction |
---|
994 | % ------------------------------------------------------------------------------------------------------------- |
---|
995 | \subsection{Non-linear bottom friction (\protect\np{nn\_botfr}\forcode{ = 2})} |
---|
996 | \label{subsec:ZDF_bfr_nonlinear} |
---|
997 | |
---|
998 | The non-linear bottom friction parameterisation assumes that the bottom friction is quadratic: |
---|
999 | \[ |
---|
1000 | % \label{eq:zdfbfr_nonlinear} |
---|
1001 | {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h |
---|
1002 | }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b |
---|
1003 | \] |
---|
1004 | where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy due to tides, |
---|
1005 | internal waves breaking and other short time scale currents. |
---|
1006 | A typical value of the drag coefficient is $C_D = 10^{-3} $. |
---|
1007 | As an example, the CME experiment \citep{treguier_JGR92} uses $C_D = 10^{-3}$ and |
---|
1008 | $e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{killworth_JPO92} uses $C_D = 1.4\;10^{-3}$ and |
---|
1009 | $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$. |
---|
1010 | The CME choices have been set as default values (\np{rn\_bfri2} and \np{rn\_bfeb2} namelist parameters). |
---|
1011 | |
---|
1012 | As for the linear case, the bottom friction is imposed in the code by adding the trend due to |
---|
1013 | the bottom friction to the general momentum trend in \mdl{dynbfr}. |
---|
1014 | For the non-linear friction case the terms computed in \mdl{zdfbfr} are: |
---|
1015 | \[ |
---|
1016 | % \label{eq:zdfbfr_nonlinbfr} |
---|
1017 | \begin{split} |
---|
1018 | c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\ |
---|
1019 | c_b^v &= - \; C_D\;\left[ \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ |
---|
1020 | \end{split} |
---|
1021 | \] |
---|
1022 | |
---|
1023 | The coefficients that control the strength of the non-linear bottom friction are initialised as namelist parameters: |
---|
1024 | $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}. |
---|
1025 | Note for applications which treat tides explicitly a low or even zero value of \np{rn\_bfeb2} is recommended. |
---|
1026 | From v3.2 onwards a local enhancement of $C_D$ is possible via an externally defined 2D mask array |
---|
1027 | (\np{ln\_bfr2d}\forcode{ = .true.}). |
---|
1028 | This works in the same way as for the linear bottom friction case with non-zero masked locations increased by |
---|
1029 | $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfri2}. |
---|
1030 | |
---|
1031 | % ------------------------------------------------------------------------------------------------------------- |
---|
1032 | % Bottom Friction Log-layer |
---|
1033 | % ------------------------------------------------------------------------------------------------------------- |
---|
1034 | \subsection[Log-layer btm frict enhncmnt (\protect\np{nn\_botfr}\forcode{ = 2}, \protect\np{ln\_loglayer}\forcode{ = .true.})] |
---|
1035 | {Log-layer bottom friction enhancement (\protect\np{nn\_botfr}\forcode{ = 2}, \protect\np{ln\_loglayer}\forcode{ = .true.})} |
---|
1036 | \label{subsec:ZDF_bfr_loglayer} |
---|
1037 | |
---|
1038 | In the non-linear bottom friction case, the drag coefficient, $C_D$, can be optionally enhanced using |
---|
1039 | a "law of the wall" scaling. |
---|
1040 | If \np{ln\_loglayer} = .true., $C_D$ is no longer constant but is related to the thickness of |
---|
1041 | the last wet layer in each column by: |
---|
1042 | \[ |
---|
1043 | C_D = \left ( {\kappa \over {\mathrm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2 |
---|
1044 | \] |
---|
1045 | |
---|
1046 | \noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness length provided via |
---|
1047 | the namelist. |
---|
1048 | |
---|
1049 | For stability, the drag coefficient is bounded such that it is kept greater or equal to |
---|
1050 | the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional namelist parameter: |
---|
1051 | \np{rn\_bfri2\_max}, \ie |
---|
1052 | \[ |
---|
1053 | rn\_bfri2 \leq C_D \leq rn\_bfri2\_max |
---|
1054 | \] |
---|
1055 | |
---|
1056 | \noindent Note also that a log-layer enhancement can also be applied to the top boundary friction if |
---|
1057 | under ice-shelf cavities are in use (\np{ln\_isfcav}\forcode{ = .true.}). |
---|
1058 | In this case, the relevant namelist parameters are \np{rn\_tfrz0}, \np{rn\_tfri2} and \np{rn\_tfri2\_max}. |
---|
1059 | |
---|
1060 | % ------------------------------------------------------------------------------------------------------------- |
---|
1061 | % Bottom Friction stability |
---|
1062 | % ------------------------------------------------------------------------------------------------------------- |
---|
1063 | \subsection{Bottom friction stability considerations} |
---|
1064 | \label{subsec:ZDF_bfr_stability} |
---|
1065 | |
---|
1066 | Some care needs to exercised over the choice of parameters to ensure that the implementation of |
---|
1067 | bottom friction does not induce numerical instability. |
---|
1068 | For the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2} is: |
---|
1069 | \begin{equation} |
---|
1070 | \label{eq:Eqn_bfrstab} |
---|
1071 | \begin{split} |
---|
1072 | \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt \\ |
---|
1073 | &= -\frac{ru}{e_{3u}}\;2\rdt\\ |
---|
1074 | \end{split} |
---|
1075 | \end{equation} |
---|
1076 | \noindent where linear bottom friction and a leapfrog timestep have been assumed. |
---|
1077 | To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have: |
---|
1078 | \[ |
---|
1079 | |\Delta u| < \;|u| |
---|
1080 | \] |
---|
1081 | \noindent which, using \autoref{eq:Eqn_bfrstab}, gives: |
---|
1082 | \[ |
---|
1083 | r\frac{2\rdt}{e_{3u}} < 1 \qquad \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ |
---|
1084 | \] |
---|
1085 | This same inequality can also be derived in the non-linear bottom friction case if |
---|
1086 | a velocity of 1 m.s$^{-1}$ is assumed. |
---|
1087 | Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability: |
---|
1088 | \[ |
---|
1089 | e_{3u} > 2\;r\;\rdt |
---|
1090 | \] |
---|
1091 | \noindent which it may be necessary to impose if partial steps are being used. |
---|
1092 | For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m. |
---|
1093 | For most applications, with physically sensible parameters these restrictions should not be of concern. |
---|
1094 | But caution may be necessary if attempts are made to locally enhance the bottom friction parameters. |
---|
1095 | To ensure stability limits are imposed on the bottom friction coefficients both |
---|
1096 | during initialisation and at each time step. |
---|
1097 | Checks at initialisation are made in \mdl{zdfbfr} (assuming a 1 m.s$^{-1}$ velocity in the non-linear case). |
---|
1098 | The number of breaches of the stability criterion are reported as well as |
---|
1099 | the minimum and maximum values that have been set. |
---|
1100 | The criterion is also checked at each time step, using the actual velocity, in \mdl{dynbfr}. |
---|
1101 | Values of the bottom friction coefficient are reduced as necessary to ensure stability; |
---|
1102 | these changes are not reported. |
---|
1103 | |
---|
1104 | Limits on the bottom friction coefficient are not imposed if the user has elected to |
---|
1105 | handle the bottom friction implicitly (see \autoref{subsec:ZDF_bfr_imp}). |
---|
1106 | The number of potential breaches of the explicit stability criterion are still reported for information purposes. |
---|
1107 | |
---|
1108 | % ------------------------------------------------------------------------------------------------------------- |
---|
1109 | % Implicit Bottom Friction |
---|
1110 | % ------------------------------------------------------------------------------------------------------------- |
---|
1111 | \subsection{Implicit bottom friction (\protect\np{ln\_bfrimp}\forcode{ = .true.})} |
---|
1112 | \label{subsec:ZDF_bfr_imp} |
---|
1113 | |
---|
1114 | An optional implicit form of bottom friction has been implemented to improve model stability. |
---|
1115 | We recommend this option for shelf sea and coastal ocean applications, especially for split-explicit time splitting. |
---|
1116 | This option can be invoked by setting \np{ln\_bfrimp} to \forcode{.true.} in the \textit{nambfr} namelist. |
---|
1117 | This option requires \np{ln\_zdfexp} to be \forcode{.false.} in the \textit{namzdf} namelist. |
---|
1118 | |
---|
1119 | This implementation is realised in \mdl{dynzdf\_imp} and \mdl{dynspg\_ts}. In \mdl{dynzdf\_imp}, |
---|
1120 | the bottom boundary condition is implemented implicitly. |
---|
1121 | |
---|
1122 | \[ |
---|
1123 | % \label{eq:dynzdf_bfr} |
---|
1124 | \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk} |
---|
1125 | = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}} |
---|
1126 | \] |
---|
1127 | |
---|
1128 | where $mbk$ is the layer number of the bottom wet layer. |
---|
1129 | Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so, it is implicit. |
---|
1130 | |
---|
1131 | If split-explicit time splitting is used, care must be taken to avoid the double counting of the bottom friction in |
---|
1132 | the 2-D barotropic momentum equations. |
---|
1133 | As NEMO only updates the barotropic pressure gradient and Coriolis' forcing terms in the 2-D barotropic calculation, |
---|
1134 | we need to remove the bottom friction induced by these two terms which has been included in the 3-D momentum trend |
---|
1135 | and update it with the latest value. |
---|
1136 | On the other hand, the bottom friction contributed by the other terms |
---|
1137 | (\eg the advection term, viscosity term) has been included in the 3-D momentum equations and |
---|
1138 | should not be added in the 2-D barotropic mode. |
---|
1139 | |
---|
1140 | The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the following: |
---|
1141 | |
---|
1142 | \[ |
---|
1143 | % \label{eq:dynspg_ts_bfr1} |
---|
1144 | \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b} |
---|
1145 | \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right) |
---|
1146 | \] |
---|
1147 | \[ |
---|
1148 | \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+ |
---|
1149 | \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)- |
---|
1150 | 2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right) |
---|
1151 | \] |
---|
1152 | |
---|
1153 | where $\textbf{T}$ is the vertical integrated 3-D momentum trend. |
---|
1154 | We assume the leap-frog time-stepping is used here. |
---|
1155 | $\Delta t$ is the barotropic mode time step and $\Delta t_{bc}$ is the baroclinic mode time step. |
---|
1156 | $c_{b}$ is the friction coefficient. |
---|
1157 | $\eta$ is the sea surface level calculated in the barotropic loops while $\eta^{'}$ is the sea surface level used in |
---|
1158 | the 3-D baroclinic mode. |
---|
1159 | $\textbf{u}_{b}$ is the bottom layer horizontal velocity. |
---|
1160 | |
---|
1161 | % ------------------------------------------------------------------------------------------------------------- |
---|
1162 | % Bottom Friction with split-explicit time splitting |
---|
1163 | % ------------------------------------------------------------------------------------------------------------- |
---|
1164 | \subsection[Bottom friction w/ split-explicit time splitting (\protect\np{ln\_bfrimp})] |
---|
1165 | {Bottom friction with split-explicit time splitting (\protect\np{ln\_bfrimp})} |
---|
1166 | \label{subsec:ZDF_bfr_ts} |
---|
1167 | |
---|
1168 | When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, |
---|
1169 | the bottom velocity at the before time step is used. |
---|
1170 | This velocity includes both the baroclinic and barotropic components which is appropriate when |
---|
1171 | using either the explicit or filtered surface pressure gradient algorithms |
---|
1172 | (\key{dynspg\_exp} or \key{dynspg\_flt}). |
---|
1173 | Extra attention is required, however, when using split-explicit time stepping (\key{dynspg\_ts}). |
---|
1174 | In this case the free surface equation is solved with a small time step \np{rn\_rdt}/\np{nn\_baro}, |
---|
1175 | while the three dimensional prognostic variables are solved with the longer time step of \np{rn\_rdt} seconds. |
---|
1176 | The trend in the barotropic momentum due to bottom friction appropriate to this method is that given by |
---|
1177 | the selected parameterisation (\ie linear or non-linear bottom friction) computed with |
---|
1178 | the evolving velocities at each barotropic timestep. |
---|
1179 | |
---|
1180 | In the case of non-linear bottom friction, we have elected to partially linearise the problem by |
---|
1181 | keeping the coefficients fixed throughout the barotropic time-stepping to those computed in |
---|
1182 | \mdl{zdfbfr} using the now timestep. |
---|
1183 | This decision allows an efficient use of the $c_b^{\vect{U}}$ coefficients to: |
---|
1184 | |
---|
1185 | \begin{enumerate} |
---|
1186 | \item On entry to \rou{dyn\_spg\_ts}, remove the contribution of the before barotropic velocity to |
---|
1187 | the bottom friction component of the vertically integrated momentum trend. |
---|
1188 | Note the same stability check that is carried out on the bottom friction coefficient in \mdl{dynbfr} has to |
---|
1189 | be applied here to ensure that the trend removed matches that which was added in \mdl{dynbfr}. |
---|
1190 | \item At each barotropic step, compute the contribution of the current barotropic velocity to |
---|
1191 | the trend due to bottom friction. |
---|
1192 | Add this contribution to the vertically integrated momentum trend. |
---|
1193 | This contribution is handled implicitly which eliminates the need to impose a stability criteria on |
---|
1194 | the values of the bottom friction coefficient within the barotropic loop. |
---|
1195 | \end{enumerate} |
---|
1196 | |
---|
1197 | Note that the use of an implicit formulation within the barotropic loop for the bottom friction trend means that |
---|
1198 | any limiting of the bottom friction coefficient in \mdl{dynbfr} does not adversely affect the solution when |
---|
1199 | using split-explicit time splitting. |
---|
1200 | This is because the major contribution to bottom friction is likely to come from the barotropic component which |
---|
1201 | uses the unrestricted value of the coefficient. |
---|
1202 | However, if the limiting is thought to be having a major effect |
---|
1203 | (a more likely prospect in coastal and shelf seas applications) then |
---|
1204 | the fully implicit form of the bottom friction should be used (see \autoref{subsec:ZDF_bfr_imp}) |
---|
1205 | which can be selected by setting \np{ln\_bfrimp} $=$ \forcode{.true.}. |
---|
1206 | |
---|
1207 | Otherwise, the implicit formulation takes the form: |
---|
1208 | \[ |
---|
1209 | % \label{eq:zdfbfr_implicitts} |
---|
1210 | \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ] |
---|
1211 | \] |
---|
1212 | where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height), |
---|
1213 | $c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and |
---|
1214 | $RHS$ represents all the components to the vertically integrated momentum trend except for |
---|
1215 | that due to bottom friction. |
---|
1216 | |
---|
1217 | % ================================================================ |
---|
1218 | % Tidal Mixing |
---|
1219 | % ================================================================ |
---|
1220 | \section{Tidal mixing (\protect\key{zdftmx})} |
---|
1221 | \label{sec:ZDF_tmx} |
---|
1222 | |
---|
1223 | %--------------------------------------------namzdf_tmx-------------------------------------------------- |
---|
1224 | % |
---|
1225 | %\nlst{namzdf_tmx} |
---|
1226 | %-------------------------------------------------------------------------------------------------------------- |
---|
1227 | |
---|
1228 | |
---|
1229 | % ------------------------------------------------------------------------------------------------------------- |
---|
1230 | % Bottom intensified tidal mixing |
---|
1231 | % ------------------------------------------------------------------------------------------------------------- |
---|
1232 | \subsection{Bottom intensified tidal mixing} |
---|
1233 | \label{subsec:ZDF_tmx_bottom} |
---|
1234 | |
---|
1235 | Options are defined through the \ngn{namzdf\_tmx} namelist variables. |
---|
1236 | The parameterization of tidal mixing follows the general formulation for the vertical eddy diffusivity proposed by |
---|
1237 | \citet{st-laurent.simmons.ea_GRL02} and first introduced in an OGCM by \citep{simmons.jayne.ea_OM04}. |
---|
1238 | In this formulation an additional vertical diffusivity resulting from internal tide breaking, |
---|
1239 | $A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, |
---|
1240 | the energy transfer from barotropic tides to baroclinic tides: |
---|
1241 | \begin{equation} |
---|
1242 | \label{eq:Ktides} |
---|
1243 | A^{vT}_{tides} = q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 } |
---|
1244 | \end{equation} |
---|
1245 | where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), |
---|
1246 | $\rho$ the density, $q$ the tidal dissipation efficiency, and $F(z)$ the vertical structure function. |
---|
1247 | |
---|
1248 | The mixing efficiency of turbulence is set by $\Gamma$ (\np{rn\_me} namelist parameter) and |
---|
1249 | is usually taken to be the canonical value of $\Gamma = 0.2$ (Osborn 1980). |
---|
1250 | The tidal dissipation efficiency is given by the parameter $q$ (\np{rn\_tfe} namelist parameter) |
---|
1251 | represents the part of the internal wave energy flux $E(x, y)$ that is dissipated locally, |
---|
1252 | with the remaining $1-q$ radiating away as low mode internal waves and |
---|
1253 | contributing to the background internal wave field. |
---|
1254 | A value of $q=1/3$ is typically used \citet{st-laurent.simmons.ea_GRL02}. |
---|
1255 | The vertical structure function $F(z)$ models the distribution of the turbulent mixing in the vertical. |
---|
1256 | It is implemented as a simple exponential decaying upward away from the bottom, |
---|
1257 | with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, |
---|
1258 | with a typical value of $500\,m$) \citep{st-laurent.nash_DSR04}, |
---|
1259 | \[ |
---|
1260 | % \label{eq:Fz} |
---|
1261 | F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) } |
---|
1262 | \] |
---|
1263 | and is normalized so that vertical integral over the water column is unity. |
---|
1264 | |
---|
1265 | The associated vertical viscosity is calculated from the vertical diffusivity assuming a Prandtl number of 1, |
---|
1266 | \ie $A^{vm}_{tides}=A^{vT}_{tides}$. |
---|
1267 | In the limit of $N \rightarrow 0$ (or becoming negative), the vertical diffusivity is capped at $300\,cm^2/s$ and |
---|
1268 | impose a lower limit on $N^2$ of \np{rn\_n2min} usually set to $10^{-8} s^{-2}$. |
---|
1269 | These bounds are usually rarely encountered. |
---|
1270 | |
---|
1271 | The internal wave energy map, $E(x, y)$ in \autoref{eq:Ktides}, is derived from a barotropic model of |
---|
1272 | the tides utilizing a parameterization of the conversion of barotropic tidal energy into internal waves. |
---|
1273 | The essential goal of the parameterization is to represent the momentum exchange between the barotropic tides and |
---|
1274 | the unrepresented internal waves induced by the tidal flow over rough topography in a stratified ocean. |
---|
1275 | In the current version of \NEMO, the map is built from the output of |
---|
1276 | the barotropic global ocean tide model MOG2D-G \citep{carrere.lyard_GRL03}. |
---|
1277 | This model provides the dissipation associated with internal wave energy for the M2 and K1 tides component |
---|
1278 | (\autoref{fig:ZDF_M2_K1_tmx}). |
---|
1279 | The S2 dissipation is simply approximated as being $1/4$ of the M2 one. |
---|
1280 | The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$. |
---|
1281 | Its global mean value is $1.1$ TW, |
---|
1282 | in agreement with independent estimates \citep{egbert.ray_N00, egbert.ray_JGR01}. |
---|
1283 | |
---|
1284 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1285 | \begin{figure}[!t] |
---|
1286 | \begin{center} |
---|
1287 | \includegraphics[width=\textwidth]{Fig_ZDF_M2_K1_tmx} |
---|
1288 | \caption{ |
---|
1289 | \protect\label{fig:ZDF_M2_K1_tmx} |
---|
1290 | (a) M2 and (b) K1 internal wave drag energy from \citet{carrere.lyard_GRL03} ($W/m^2$). |
---|
1291 | } |
---|
1292 | \end{center} |
---|
1293 | \end{figure} |
---|
1294 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1295 | |
---|
1296 | % ------------------------------------------------------------------------------------------------------------- |
---|
1297 | % Indonesian area specific treatment |
---|
1298 | % ------------------------------------------------------------------------------------------------------------- |
---|
1299 | \subsection{Indonesian area specific treatment (\protect\np{ln\_zdftmx\_itf})} |
---|
1300 | \label{subsec:ZDF_tmx_itf} |
---|
1301 | |
---|
1302 | When the Indonesian Through Flow (ITF) area is included in the model domain, |
---|
1303 | a specific treatment of tidal induced mixing in this area can be used. |
---|
1304 | It is activated through the namelist logical \np{ln\_tmx\_itf}, and the user must provide an input NetCDF file, |
---|
1305 | \ifile{mask\_itf}, which contains a mask array defining the ITF area where the specific treatment is applied. |
---|
1306 | |
---|
1307 | When \np{ln\_tmx\_itf}\forcode{ = .true.}, the two key parameters $q$ and $F(z)$ are adjusted following |
---|
1308 | the parameterisation developed by \citet{koch-larrouy.madec.ea_GRL07}: |
---|
1309 | |
---|
1310 | First, the Indonesian archipelago is a complex geographic region with a series of |
---|
1311 | large, deep, semi-enclosed basins connected via numerous narrow straits. |
---|
1312 | Once generated, internal tides remain confined within this semi-enclosed area and hardly radiate away. |
---|
1313 | Therefore all the internal tides energy is consumed within this area. |
---|
1314 | So it is assumed that $q = 1$, \ie all the energy generated is available for mixing. |
---|
1315 | Note that for test purposed, the ITF tidal dissipation efficiency is a namelist parameter (\np{rn\_tfe\_itf}). |
---|
1316 | A value of $1$ or close to is this recommended for this parameter. |
---|
1317 | |
---|
1318 | Second, the vertical structure function, $F(z)$, is no more associated with a bottom intensification of the mixing, |
---|
1319 | but with a maximum of energy available within the thermocline. |
---|
1320 | \citet{koch-larrouy.madec.ea_GRL07} have suggested that the vertical distribution of |
---|
1321 | the energy dissipation proportional to $N^2$ below the core of the thermocline and to $N$ above. |
---|
1322 | The resulting $F(z)$ is: |
---|
1323 | \[ |
---|
1324 | % \label{eq:Fz_itf} |
---|
1325 | F(i,j,k) \sim \left\{ |
---|
1326 | \begin{aligned} |
---|
1327 | \frac{q\,\Gamma E(i,j) } {\rho N \, \int N dz} \qquad \text{when $\partial_z N < 0$} \\ |
---|
1328 | \frac{q\,\Gamma E(i,j) } {\rho \, \int N^2 dz} \qquad \text{when $\partial_z N > 0$} |
---|
1329 | \end{aligned} |
---|
1330 | \right. |
---|
1331 | \] |
---|
1332 | |
---|
1333 | Averaged over the ITF area, the resulting tidal mixing coefficient is $1.5\,cm^2/s$, |
---|
1334 | which agrees with the independent estimates inferred from observations. |
---|
1335 | Introduced in a regional OGCM, the parameterization improves the water mass characteristics in |
---|
1336 | the different Indonesian seas, suggesting that the horizontal and vertical distributions of |
---|
1337 | the mixing are adequately prescribed \citep{koch-larrouy.madec.ea_GRL07, koch-larrouy.madec.ea_OD08*a, koch-larrouy.madec.ea_OD08*b}. |
---|
1338 | Note also that such a parameterisation has a significant impact on the behaviour of |
---|
1339 | global coupled GCMs \citep{koch-larrouy.lengaigne.ea_CD10}. |
---|
1340 | |
---|
1341 | % ================================================================ |
---|
1342 | % Internal wave-driven mixing |
---|
1343 | % ================================================================ |
---|
1344 | \section{Internal wave-driven mixing (\protect\key{zdftmx\_new})} |
---|
1345 | \label{sec:ZDF_tmx_new} |
---|
1346 | |
---|
1347 | %--------------------------------------------namzdf_tmx_new------------------------------------------ |
---|
1348 | % |
---|
1349 | %\nlst{namzdf_tmx_new} |
---|
1350 | %-------------------------------------------------------------------------------------------------------------- |
---|
1351 | |
---|
1352 | The parameterization of mixing induced by breaking internal waves is a generalization of |
---|
1353 | the approach originally proposed by \citet{st-laurent.simmons.ea_GRL02}. |
---|
1354 | A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed, |
---|
1355 | and the resulting diffusivity is obtained as |
---|
1356 | \[ |
---|
1357 | % \label{eq:Kwave} |
---|
1358 | A^{vT}_{wave} = R_f \,\frac{ \epsilon }{ \rho \, N^2 } |
---|
1359 | \] |
---|
1360 | where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of |
---|
1361 | the energy available for mixing. |
---|
1362 | If the \np{ln\_mevar} namelist parameter is set to false, the mixing efficiency is taken as constant and |
---|
1363 | equal to 1/6 \citep{osborn_JPO80}. |
---|
1364 | In the opposite (recommended) case, $R_f$ is instead a function of |
---|
1365 | the turbulence intensity parameter $Re_b = \frac{ \epsilon}{\nu \, N^2}$, |
---|
1366 | with $\nu$ the molecular viscosity of seawater, following the model of \cite{bouffard.boegman_DAO13} and |
---|
1367 | the implementation of \cite{de-lavergne.madec.ea_JPO16}. |
---|
1368 | Note that $A^{vT}_{wave}$ is bounded by $10^{-2}\,m^2/s$, a limit that is often reached when |
---|
1369 | the mixing efficiency is constant. |
---|
1370 | |
---|
1371 | In addition to the mixing efficiency, the ratio of salt to heat diffusivities can chosen to vary |
---|
1372 | as a function of $Re_b$ by setting the \np{ln\_tsdiff} parameter to true, a recommended choice. |
---|
1373 | This parameterization of differential mixing, due to \cite{jackson.rehmann_JPO14}, |
---|
1374 | is implemented as in \cite{de-lavergne.madec.ea_JPO16}. |
---|
1375 | |
---|
1376 | The three-dimensional distribution of the energy available for mixing, $\epsilon(i,j,k)$, |
---|
1377 | is constructed from three static maps of column-integrated internal wave energy dissipation, |
---|
1378 | $E_{cri}(i,j)$, $E_{pyc}(i,j)$, and $E_{bot}(i,j)$, combined to three corresponding vertical structures |
---|
1379 | (de Lavergne et al., in prep): |
---|
1380 | \begin{align*} |
---|
1381 | F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\ |
---|
1382 | F_{pyc}(i,j,k) &\propto N^{n\_p}\\ |
---|
1383 | F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} } |
---|
1384 | \end{align*} |
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1385 | In the above formula, $h_{ab}$ denotes the height above bottom, |
---|
1386 | $h_{wkb}$ denotes the WKB-stretched height above bottom, defined by |
---|
1387 | \[ |
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1388 | h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz' } \; , |
---|
1389 | \] |
---|
1390 | The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist) |
---|
1391 | controls the stratification-dependence of the pycnocline-intensified dissipation. |
---|
1392 | It can take values of 1 (recommended) or 2. |
---|
1393 | Finally, the vertical structures $F_{cri}$ and $F_{bot}$ require the specification of |
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1394 | the decay scales $h_{cri}(i,j)$ and $h_{bot}(i,j)$, which are defined by two additional input maps. |
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1395 | $h_{cri}$ is related to the large-scale topography of the ocean (etopo2) and |
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1396 | $h_{bot}$ is a function of the energy flux $E_{bot}$, the characteristic horizontal scale of |
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1397 | the abyssal hill topography \citep{goff_JGR10} and the latitude. |
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1398 | |
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1399 | % ================================================================ |
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1400 | |
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1401 | \biblio |
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1402 | |
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1403 | \pindex |
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1404 | |
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1405 | \end{document} |
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