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{Diurnal SST Models (DIU)} |
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6 | \label{chap:DIU} |
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7 | |
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8 | \thispagestyle{plain} |
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9 | |
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10 | \chaptertoc |
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11 | |
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12 | \paragraph{Changes record} ~\\ |
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13 | |
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14 | {\footnotesize |
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15 | \begin{tabularx}{\textwidth}{l||X|X} |
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16 | Release & Author(s) & Modifications \\ |
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17 | \hline |
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18 | {\em 4.0} & {\em ...} & {\em ...} \\ |
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19 | {\em 3.6} & {\em ...} & {\em ...} \\ |
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20 | {\em 3.4} & {\em ...} & {\em ...} \\ |
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21 | {\em <=3.4} & {\em ...} & {\em ...} |
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22 | \end{tabularx} |
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23 | } |
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24 | |
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25 | \clearpage |
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26 | |
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27 | Code to produce an estimate of the diurnal warming and cooling of the sea surface skin |
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28 | temperature (skin SST) is found in the DIU directory. |
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29 | The skin temperature can be split into three parts: |
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30 | \begin{itemize} |
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31 | \item A foundation SST which is free from diurnal warming. |
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32 | \item A warm layer, typically ~3\,m thick, |
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33 | where heating from solar radiation can cause a warm stably stratified layer during the daytime |
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34 | \item A cool skin, a thin layer, approximately ~1\, mm thick, |
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35 | where long wave cooling is dominant and cools the immediate ocean surface. |
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36 | \end{itemize} |
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37 | |
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38 | Models are provided for both the warm layer, \mdl{diurnal\_bulk}, and the cool skin, \mdl{cool\_skin}. |
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39 | Foundation SST is not considered as it can be obtained either from the main \NEMO\ model |
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40 | (\ie\ from the temperature of the top few model levels) or from some other source. |
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41 | It must be noted that both the cool skin and warm layer models produce estimates of the change in temperature |
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42 | ($\Delta T_{\mathrm{cs}}$ and $\Delta T_{\mathrm{wl}}$) and |
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43 | both must be added to a foundation SST to obtain the true skin temperature. |
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44 | |
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45 | Both the cool skin and warm layer models are controlled through the namelist \nam{diu}{diu}: |
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46 | |
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47 | \begin{listing} |
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48 | \nlst{namdiu} |
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49 | \caption{\forcode{&namdiu}} |
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50 | \label{lst:namdiu} |
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51 | \end{listing} |
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52 | |
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53 | This namelist contains only two variables: |
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54 | \begin{description} |
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55 | \item [{\np{ln_diurnal}{ln\_diurnal}}] A logical switch for turning on/off both the cool skin and warm layer. |
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56 | \item [{\np{ln_diurnal_only}{ln\_diurnal\_only}}] A logical switch which if \forcode{.true.} will run the diurnal model without the other dynamical parts of \NEMO. |
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57 | \np{ln_diurnal_only}{ln\_diurnal\_only} must be \forcode{.false.} if \np{ln_diurnal}{ln\_diurnal} is \forcode{.false.}. |
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58 | \end{description} |
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59 | |
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60 | Output for the diurnal model is through the variables `sst\_wl' (warm\_layer) and `sst\_cs' (cool skin). |
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61 | These are 2-D variables which will be included in the model output if they are specified in the iodef.xml file. |
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62 | |
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63 | Initialisation is through the restart file. |
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64 | Specifically the code will expect the presence of the 2-D variable ``Dsst'' to initialise the warm layer. |
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65 | The cool skin model, which is determined purely by the instantaneous fluxes, has no initialisation variable. |
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66 | |
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67 | %% ================================================================================================= |
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68 | \section{Warm layer model} |
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69 | \label{sec:DIU_warm_layer_sec} |
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70 | |
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71 | The warm layer is calculated using the model of \citet{takaya.bidlot.ea_JGR10} (TAKAYA10 model hereafter). |
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72 | This is a simple flux based model that is defined by the equations |
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73 | \begin{align} |
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74 | \frac{\partial{\Delta T_{\mathrm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p |
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75 | \nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,} |
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76 | \label{eq:DIU_ecmwf1} \\ |
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77 | L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:DIU_ecmwf2} |
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78 | \end{align} |
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79 | where $\Delta T_{\mathrm{wl}}$ is the temperature difference between the top of the warm layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal. |
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80 | In equation (\autoref{eq:DIU_ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water, |
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81 | $\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat capacity at constant pressure of sea water, |
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82 | $\rho_w$ is the water density, and $L$ is the Monin-Obukhov length. |
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83 | The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via |
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84 | $T(z) = T(0) - \left( \frac{z}{D_T} \right)^\nu \Delta T_{\mathrm{wl}}$, |
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85 | where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer. |
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86 | The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$, |
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87 | which can be related to the 10\,m wind speed $u_{10}$ through |
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88 | the relationship $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is the drag coefficient, |
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89 | and $\rho_a$ is the density of air. |
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90 | The symbol $Q$ in equation (\autoref{eq:DIU_ecmwf1}) is the instantaneous total thermal energy flux into |
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91 | the diurnal layer, \ie |
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92 | \[ |
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93 | Q = Q_{\mathrm{sol}} + Q_{\mathrm{lw}} + Q_{\mathrm{h}}\mbox{,} |
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94 | % \label{eq:DIU_e_flux_eqn} |
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95 | \] |
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96 | where $Q_{\mathrm{h}}$ is the sensible and latent heat flux, $Q_{\mathrm{lw}}$ is the long wave flux, |
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97 | and $Q_{\mathrm{sol}}$ is the solar flux absorbed within the diurnal warm layer. |
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98 | For $Q_{\mathrm{sol}}$ the 9 term representation of \citet{gentemann.minnett.ea_JGR09} is used. |
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99 | In equation \autoref{eq:DIU_ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, |
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100 | where $L_a=0.3$\footnote{ |
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101 | This is a global average value, more accurately $L_a$ could be computed as $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$, |
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102 | where $u_s$ is the stokes drift, but this is not currently done |
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103 | } is the turbulent Langmuir number and is a parametrization of the effect of waves. |
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104 | The function $\Phi\!\left(\frac{D_T}{L}\right)$ is the similarity function that |
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105 | parametrizes the stability of the water column and is given by: |
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106 | \begin{equation} |
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107 | \Phi(\zeta) = \left\{ \begin{array}{cc} 1 + \frac{5\zeta + |
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108 | 4\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\ |
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109 | (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,} |
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110 | \end{array} \right. \label{eq:DIU_stab_func_eqn} |
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111 | \end{equation} |
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112 | where $\zeta=\frac{D_T}{L}$. It is clear that the first derivative of (\autoref{eq:DIU_stab_func_eqn}), |
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113 | and thus of (\autoref{eq:DIU_ecmwf1}), is discontinuous at $\zeta=0$ (\ie\ $Q\rightarrow0$ in |
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114 | equation (\autoref{eq:DIU_ecmwf2})). |
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115 | |
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116 | The two terms on the right hand side of (\autoref{eq:DIU_ecmwf1}) represent different processes. |
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117 | The first term is simply the diabatic heating or cooling of the diurnal warm layer due to |
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118 | thermal energy fluxes into and out of the layer. |
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119 | The second term parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind induced mixing. |
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120 | In practice the second term acts as a relaxation on the temperature. |
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121 | |
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122 | %% ================================================================================================= |
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123 | \section{Cool skin model} |
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124 | \label{sec:DIU_cool_skin_sec} |
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125 | |
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126 | The cool skin is modelled using the framework of \citet{saunders_JAS67} who used a formulation of the near surface temperature difference based upon the heat flux and the friction velocity $u^*_{w}$. |
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127 | As the cool skin is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the Saunders equation for the cool skin temperature difference $\Delta T_{\mathrm{cs}}$ becomes |
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128 | \[ |
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129 | % \label{eq:DIU_sunders_eqn} |
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130 | \Delta T_{\mathrm{cs}}=\frac{Q_{\mathrm{ns}}\delta}{k_t} \mbox{,} |
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131 | \] |
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132 | where $Q_{\mathrm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and |
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133 | $k_t$ is the thermal conductivity of sea water. |
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134 | $\delta$ is the thickness of the skin layer and is given by |
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135 | \begin{equation} |
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136 | \label{eq:DIU_sunders_thick_eqn} |
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137 | \delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,} |
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138 | \end{equation} |
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139 | where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of proportionality which |
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140 | \citet{saunders_JAS67} suggested varied between 5 and 10. |
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141 | |
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142 | The value of $\lambda$ used in equation (\autoref{eq:DIU_sunders_thick_eqn}) is that of \citet{artale.iudicone.ea_JGR02}, |
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143 | which is shown in \citet{tu.tsuang_GRL05} to outperform a number of other parametrisations at |
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144 | both low and high wind speeds. |
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145 | Specifically, |
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146 | \[ |
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147 | % \label{eq:DIU_artale_lambda_eqn} |
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148 | \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} |
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149 | \] |
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150 | where $h=10$\,m is a reference depth and |
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151 | $\gamma$ is a dimensionless function of wind speed $u$: |
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152 | \[ |
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153 | % \label{eq:DIU_artale_gamma_eqn} |
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154 | \gamma = |
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155 | \begin{cases} |
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156 | 0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\ |
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157 | 1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\ |
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158 | 6\mbox{,} & u \ge 10\,\mbox{ms}^{-1} \\ |
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159 | \end{cases} |
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160 | \] |
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161 | |
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162 | \onlyinsubfile{\input{../../global/epilogue}} |
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163 | |
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164 | \end{document} |
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