Changeset 9407 for branches/2017/dev_merge_2017/DOC/tex_sub/chap_DIU.tex
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branches/2017/dev_merge_2017/DOC/tex_sub/chap_DIU.tex
r9394 r9407 6 6 % ================================================================ 7 7 \chapter{Diurnal SST Models (DIU)} 8 \label{ DIU}8 \label{chap:DIU} 9 9 10 10 \minitoc … … 54 54 %=============================================================== 55 55 \section{Warm layer model} 56 \label{ warm_layer_sec}56 \label{sec:warm_layer_sec} 57 57 %=============================================================== 58 58 … … 62 62 \frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p 63 63 \nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,} 64 \label{e cmwf1} \\65 L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{e cmwf2}64 \label{eq:ecmwf1} \\ 65 L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2} 66 66 \end{eqnarray} 67 67 where $\Delta T_{\rm{wl}}$ is the temperature difference between the top of the warm 68 68 layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal. In 69 equation (\ ref{ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion69 equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion 70 70 coefficient of water, $\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat 71 71 capacity at constant pressure of sea water, $\rho_w$ is the … … 81 81 $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is 82 82 the drag coefficient, and $\rho_a$ is the density of air. The symbol $Q$ in equation 83 (\ ref{ecmwf1}) is the instantaneous total thermal energy83 (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy 84 84 flux into 85 85 the diurnal layer, $i.e.$ 86 86 \begin{equation} 87 Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{e _flux_eqn}87 Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{eq:e_flux_eqn} 88 88 \end{equation} 89 89 where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long 90 90 wave flux, and $Q_{\rm{sol}}$ is the solar flux absorbed 91 91 within the diurnal warm layer. For $Q_{\rm{sol}}$ the 9 term 92 representation of \citet{Gentemann_al_JGR09} is used. In equation \ ref{ecmwf1}92 representation of \citet{Gentemann_al_JGR09} is used. In equation \autoref{eq:ecmwf1} 93 93 the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, where $L_a=0.3$\footnote{This 94 94 is a global average value, more accurately $L_a$ could be computed as … … 103 103 4\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\ 104 104 (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,} 105 \end{array} \right. \label{ stab_func_eqn}105 \end{array} \right. \label{eq:stab_func_eqn} 106 106 \end{equation} 107 107 where $\zeta=\frac{D_T}{L}$. It is clear that the first derivative of 108 (\ ref{stab_func_eqn}), and thus of (\ref{ecmwf1}),109 is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in equation (\ ref{ecmwf2})).108 (\autoref{eq:stab_func_eqn}), and thus of (\autoref{eq:ecmwf1}), 109 is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in equation (\autoref{eq:ecmwf2})). 110 110 111 The two terms on the right hand side of (\ ref{ecmwf1}) represent different processes.111 The two terms on the right hand side of (\autoref{eq:ecmwf1}) represent different processes. 112 112 The first term is simply the diabatic heating or cooling of the 113 113 diurnal warm … … 121 121 122 122 \section{Cool skin model} 123 \label{ cool_skin_sec}123 \label{sec:cool_skin_sec} 124 124 125 125 %=============================================================== … … 131 131 Saunders equation for the cool skin temperature difference $\Delta T_{\rm{cs}}$ becomes 132 132 \begin{equation} 133 \label{ sunders_eqn}133 \label{eq:sunders_eqn} 134 134 \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,} 135 135 \end{equation} … … 138 138 skin layer and is given by 139 139 \begin{equation} 140 \label{ sunders_thick_eqn}140 \label{eq:sunders_thick_eqn} 141 141 \delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,} 142 142 \end{equation} … … 144 144 proportionality which \citet{Saunders_JAS82} suggested varied between 5 and 10. 145 145 146 The value of $\lambda$ used in equation (\ ref{sunders_thick_eqn}) is that of146 The value of $\lambda$ used in equation (\autoref{eq:sunders_thick_eqn}) is that of 147 147 \citet{Artale_al_JGR02}, 148 148 which is shown in \citet{Tu_Tsuang_GRL05} to outperform a number of other 149 149 parametrisations at both low and high wind speeds. Specifically, 150 150 \begin{equation} 151 \label{ artale_lambda_eqn}151 \label{eq:artale_lambda_eqn} 152 152 \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} 153 153 \end{equation} … … 155 155 $\gamma$ is a dimensionless function of wind speed $u$: 156 156 \begin{equation} 157 \label{ artale_gamma_eqn}157 \label{eq:artale_gamma_eqn} 158 158 \gamma = \left\{ \begin{matrix} 159 159 0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\
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