1 | \documentclass[NEMO_book]{subfiles} |
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2 | \begin{document} |
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3 | % ================================================================ |
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4 | % Diurnal SST models (DIU) |
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5 | % Edited by James While |
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6 | % ================================================================ |
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7 | \chapter{Diurnal SST models (DIU)} |
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8 | \label{DIU} |
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9 | |
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10 | \minitoc |
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11 | |
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12 | |
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13 | \newpage |
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14 | $\ $\newline % force a new line |
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15 | |
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16 | Code to produce an estimate of the diurnal warming and cooling of the sea surface skin |
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17 | temperature (skin SST) is found in the DIU directory. |
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18 | The skin temperature can be split into three parts: |
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19 | \begin{itemize} |
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20 | \item A foundation SST which is free from diurnal warming. |
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21 | \item A warm layer, typically ~3\,m thick, where heating from solar radiation can |
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22 | cause a warm stably stratified layer during the daytime |
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23 | \item A cool skin, a thin layer, approximately ~1\,mm thick, where long wave cooling |
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24 | is dominant and cools the immediate ocean surface. |
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25 | \end{itemize} |
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26 | |
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27 | Models are provided for both the warm layer, diurnal\_bulk.F90, and the cool skin, |
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28 | cool\_skin.F90. Foundation SST is not considered as it can be obtained |
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29 | either from the main NEMO model ($i.e.$ from the temperature of the top few model levels) |
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30 | or from some other source. |
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31 | It must be noted that both the cool skin and warm layer models produce estimates of |
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32 | the change in temperature ($\Delta T_{\rm{cs}}$ and $\Delta T_{\rm{wl}}$) |
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33 | and both must be added to a foundation SST to obtain the true skin temperature. |
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34 | |
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35 | Both the cool skin and warm layer models are controlled through the namelist \ngn{namdiu}: |
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36 | \fortranfile{namelists/namdiu} |
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37 | This namelist contains only two variables: |
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38 | \begin{description} |
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39 | \item[\np{ln\_diurnal}] A logical switch for turning on/off both the cool skin and warm layer. |
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40 | \item[\np{ln\_diurnal\_only}] A logical switch which if .TRUE. will run the diurnal model |
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41 | without the other dynamical parts of NEMO. |
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42 | \np{ln\_diurnal\_only} must be .FALSE. if \np{ln\_diurnal} is .FALSE. |
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43 | \end{description} |
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44 | |
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45 | Output for the diurnal model is through the variables `sst\_wl' (warm\_layer) and |
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46 | `sst\_cs' (cool skin). These are 2-D variables which will be included in the model |
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47 | output if they are specified in the iodef.xml file. |
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48 | |
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49 | Initialisation is through the restart file. Specifically the code will expect |
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50 | the presence of the 2-D variable ``Dsst'' to initialise the warm layer. |
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51 | The cool skin model, which is determined purely by the instantaneous fluxes, |
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52 | has no initialisation variable. |
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53 | |
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54 | %=============================================================== |
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55 | \section{Warm Layer model} |
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56 | \label{warm_layer_sec} |
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57 | %=============================================================== |
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58 | |
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59 | The warm layer is calculated using the model of \citet{Takaya_al_JGR10} (TAKAYA10 model |
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60 | hereafter). This is a simple flux based model that is defined by the equations |
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61 | \begin{eqnarray} |
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62 | \frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p |
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63 | \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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64 | \label{ecmwf1} \\ |
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65 | L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{ecmwf2} |
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66 | \end{eqnarray} |
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67 | where $\Delta T_{\rm{wl}}$ is the temperature difference between the top of the warm |
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68 | layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal. In |
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69 | equation (\ref{ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion |
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70 | coefficient of water, $\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat |
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71 | capacity at constant pressure of sea water, $\rho_w$ is the |
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72 | water density, and $L$ is the Monin-Obukhov length. The tunable |
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73 | variable $\nu$ is a shape parameter that defines the expected |
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74 | subskin temperature profile via $T(z)=T(0)-\left(\frac{z}{D_T}\right)^\nu\Delta |
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75 | T_{\rm{wl}}$, |
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76 | where $T$ is the absolute temperature and $z\le D_T$ is the depth |
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77 | below the top of the warm layer. |
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78 | The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity |
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79 | of the water |
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80 | $u^*_{w}$, which can be related to the 10\,m wind speed $u_{10}$ through the relationship |
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81 | $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is |
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82 | the drag coefficient, and $\rho_a$ is the density of air. The symbol $Q$ in equation |
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83 | (\ref{ecmwf1}) is the instantaneous total thermal energy |
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84 | flux into |
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85 | the diurnal layer, $i.e.$ |
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86 | \begin{equation} |
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87 | Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{e_flux_eqn} |
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88 | \end{equation} |
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89 | where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long |
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90 | wave flux, and $Q_{\rm{sol}}$ is the solar flux absorbed |
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91 | within the diurnal warm layer. For $Q_{\rm{sol}}$ the 9 term |
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92 | representation of \citet{Gentemann_al_JGR09} is used. In equation \ref{ecmwf1} |
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93 | the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, where $L_a=0.3$\footnote{This |
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94 | is a global average value, more accurately $L_a$ could be computed as |
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95 | $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$, where $u_s$ is the stokes drift, but this is not |
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96 | currently done} is the turbulent Langmuir number and is a |
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97 | parametrization of the effect of waves. |
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98 | The function $\Phi\!\left(\frac{D_T}{L}\right)$ is the similarity function that |
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99 | parametrizes the stability of the water column and |
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100 | is given by: |
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101 | \begin{equation} |
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102 | \Phi(\zeta) = \left\{ \begin{array}{cc} 1 + \frac{5\zeta + |
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103 | 4\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\ |
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104 | (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,} |
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105 | \end{array} \right. \label{stab_func_eqn} |
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106 | \end{equation} |
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107 | where $\zeta=\frac{D_T}{L}$. It is clear that the first derivative of |
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108 | (\ref{stab_func_eqn}), and thus of (\ref{ecmwf1}), |
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109 | is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in equation (\ref{ecmwf2})). |
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110 | |
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111 | The two terms on the right hand side of (\ref{ecmwf1}) represent different processes. |
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112 | The first term is simply the diabatic heating or cooling of the |
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113 | diurnal warm |
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114 | layer due to thermal energy |
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115 | fluxes into and out of the layer. The second term |
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116 | parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind |
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117 | induced mixing. In practice the second term acts as a relaxation |
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118 | on the temperature. |
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119 | |
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120 | %=============================================================== |
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121 | |
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122 | \section{Cool Skin model} |
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123 | \label{cool_skin_sec} |
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124 | |
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125 | %=============================================================== |
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126 | |
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127 | The cool skin is modelled using the framework of \citet{Saunders_JAS82} who used a |
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128 | formulation of the near surface temperature difference based upon the heat flux and |
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129 | the friction velocity $u^*_{w}$. As the cool skin |
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130 | is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the |
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131 | Saunders equation for the cool skin temperature difference $\Delta T_{\rm{cs}}$ becomes |
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132 | \begin{equation} |
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133 | \label{sunders_eqn} |
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134 | \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,} |
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135 | \end{equation} |
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136 | where $Q_{\rm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and |
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137 | $k_t$ is the thermal conductivity of sea water. $\delta$ is the thickness of the |
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138 | skin layer and is given by |
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139 | \begin{equation} |
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140 | \label{sunders_thick_eqn} |
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141 | \delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,} |
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142 | \end{equation} |
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143 | where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of |
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144 | proportionality which \citet{Saunders_JAS82} suggested varied between 5 and 10. |
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145 | |
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146 | The value of $\lambda$ used in equation (\ref{sunders_thick_eqn}) is that of |
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147 | \citet{Artale_al_JGR02}, |
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148 | which is shown in \citet{Tu_Tsuang_GRL05} to outperform a number of other |
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149 | parametrisations at both low and high wind speeds. Specifically, |
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150 | \begin{equation} |
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151 | \label{artale_lambda_eqn} |
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152 | \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} |
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153 | \end{equation} |
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154 | where $h=10$\,m is a reference depth and |
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155 | $\gamma$ is a dimensionless function of wind speed $u$: |
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156 | \begin{equation} |
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157 | \label{artale_gamma_eqn} |
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158 | \gamma = \left\{ \begin{matrix} |
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159 | 0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\ |
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160 | 1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\ |
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161 | 6\mbox{,} & \ge 10\,\mbox{ms}^{-1} \\ |
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162 | \end{matrix} |
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163 | \right. |
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164 | \end{equation} |
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165 | |
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166 | \end{document} |
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