Changeset 10419 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIU.tex
- Timestamp:
- 2018-12-19T20:46:30+01:00 (5 years ago)
- Location:
- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex
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NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIU.tex
r10368 r10419 1 \documentclass[../tex_main/NEMO_manual]{subfiles} 1 \documentclass[../main/NEMO_manual]{subfiles} 2 2 3 \begin{document} 3 4 % ================================================================ … … 28 29 \end{itemize} 29 30 30 Models are provided for both the warm layer, \md fl{diurnal_bulk}, and the cool skin, \mdl{cool_skin}.31 Models are provided for both the warm layer, \mdl{diurnal\_bulk}, and the cool skin, \mdl{cool\_skin}. 31 32 Foundation SST is not considered as it can be obtained either from the main NEMO model 32 33 ($i.e.$ from the temperature of the top few model levels) or from some other source. … … 61 62 The warm layer is calculated using the model of \citet{Takaya_al_JGR10} (TAKAYA10 model hereafter). 62 63 This is a simple flux based model that is defined by the equations 63 \begin{ eqnarray}64 \begin{align} 64 65 \frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p 65 66 \nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,} 66 67 \label{eq:ecmwf1} \\ 67 68 L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2} 68 \end{ eqnarray}69 \end{align} 69 70 where $\Delta T_{\rm{wl}}$ 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. 70 71 In equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water, … … 72 73 $\rho_w$ is the water density, and $L$ is the Monin-Obukhov length. 73 74 The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via 74 $T(z) =T(0)-\left(\frac{z}{D_T}\right)^\nu\DeltaT_{\rm{wl}}$,75 $T(z) = T(0) - \left( \frac{z}{D_T} \right)^\nu \Delta T_{\rm{wl}}$, 75 76 where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer. 76 77 The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$, … … 80 81 The symbol $Q$ in equation (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy flux into 81 82 the diurnal layer, $i.e.$ 82 \begin{equation} 83 Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{eq:e_flux_eqn} 84 \end{equation} 83 \[ 84 Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} 85 % \label{eq:e_flux_eqn} 86 \] 85 87 where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long wave flux, 86 88 and $Q_{\rm{sol}}$ is the solar flux absorbed within the diurnal warm layer. … … 118 120 The cool skin is modelled using the framework of \citet{Saunders_JAS82} who used a formulation of the near surface temperature difference based upon the heat flux and the friction velocity $u^*_{w}$. 119 121 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 $\Delta T_{\rm{cs}}$ becomes 120 \ begin{equation}121 \label{eq:sunders_eqn}122 \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,}123 \ end{equation}122 \[ 123 % \label{eq:sunders_eqn} 124 \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,} 125 \] 124 126 where $Q_{\rm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and 125 127 $k_t$ is the thermal conductivity of sea water. … … 136 138 both low and high wind speeds. 137 139 Specifically, 138 \ begin{equation}139 \label{eq:artale_lambda_eqn}140 \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,}141 \ end{equation}140 \[ 141 % \label{eq:artale_lambda_eqn} 142 \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} 143 \] 142 144 where $h=10$\,m is a reference depth and 143 145 $\gamma$ is a dimensionless function of wind speed $u$: 144 \begin{equation} 145 \label{eq:artale_gamma_eqn} 146 \gamma = \left\{ \begin{matrix} 147 0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\ 148 1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\ 149 6\mbox{,} & \ge 10\,\mbox{ms}^{-1} \\ 150 \end{matrix} 151 \right. 152 \end{equation} 146 \[ 147 % \label{eq:artale_gamma_eqn} 148 \gamma = 149 \begin{cases} 150 0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\ 151 1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\ 152 6\mbox{,} & u \ge 10\,\mbox{ms}^{-1} \\ 153 \end{cases} 154 \] 155 156 \biblio 153 157 154 158 \end{document}
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