Changeset 11015 for NEMO/trunk/doc/latex/SI3/subfiles/chap_model_basics.tex
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 20190520T20:57:09+02:00 (18 months ago)
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 NEMO/trunk/doc/latex/SI3/subfiles
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NEMO/trunk/doc/latex/SI3/subfiles/chap_model_basics.tex
r9974 r11015 1 1 2 \documentclass[../ ../tex_main/NEMO_manual]{subfiles}2 \documentclass[../main/SI3_manual]{subfiles} 3 3 4 4 \begin{document} … … 41 41 \begin{center} 42 42 \vspace{0cm} 43 \includegraphics[height=10cm,angle=00]{ ../Figures/ice_scheme.png}44 \caption{Representation of the ice pack, using multiple categories with specific ice concentration ($a_l, l=1, 2, ..., L$), thickness ($h^i_l$), snow depth ($h^s_l$), vertical temperature and salinity profiles ($T^i_{kl}$, $S^{*}_{kl}$) and a single ice velocity vector ($\ bm{u}$).}43 \includegraphics[height=10cm,angle=00]{ice_scheme} 44 \caption{Representation of the ice pack, using multiple categories with specific ice concentration ($a_l, l=1, 2, ..., L$), thickness ($h^i_l$), snow depth ($h^s_l$), vertical temperature and salinity profiles ($T^i_{kl}$, $S^{*}_{kl}$) and a single ice velocity vector ($\mathbf{u}$).} 45 45 \label{ice_scheme} 46 46 \end{center} … … 162 162 % 163 163 164 We first present the essentials of the thickness distribution framework \citep{Thorndikeetal75}. Consider a given region of area $R$ centered at spatial coordinates $(\ bm{x})$ at a given time $t$. $R$ could be e.g. a model grid cell. The ice thickness distribution $g(\mathbf{x},t, h)$ is introduced as follows:164 We first present the essentials of the thickness distribution framework \citep{Thorndikeetal75}. Consider a given region of area $R$ centered at spatial coordinates $(\mathbf{x})$ at a given time $t$. $R$ could be e.g. a model grid cell. The ice thickness distribution $g(\mathbf{x},t, h)$ is introduced as follows: 165 165 \begin{linenomath} 166 166 \begin{align} … … 184 184 \begin{center} 185 185 \vspace{0cm} 186 \includegraphics[height=6cm,angle=00]{ ../Figures/g_h.png}186 \includegraphics[height=6cm,angle=00]{g_h} 187 187 \caption{Representation of the relation between real thickness profiles and the ice thickness distribution function $g(h)$} 188 188 \label{fig_g_h} … … 202 202 \begin{linenomath} 203 203 \begin{align} 204 \frac{\partial a_l}{\partial t} =  \ bm{\nabla} \cdot (a_l \mathbf{u}) + \Theta^a_l + \int_{H^*_{l1}}^{H^*_l} dh \psi.204 \frac{\partial a_l}{\partial t} =  \mathbf{\nabla} \cdot (a_l \mathbf{u}) + \Theta^a_l + \int_{H^*_{l1}}^{H^*_l} dh \psi. 205 205 \label{eq:gt} 206 206 \end{align} … … 211 211 \begin{linenomath} 212 212 \begin{align} 213 A(\ bm{x},t) &=\int_{0^+}^{\infty} dh \cdot g(h,\bm{x},t) \sim A_{ij} = \sum_{l=1}^L a_{ijl}, & \\214 V_i(\ bm{x},t)&=\int_{0}^{\infty} dh \cdot g(h,\bm{x},t) \cdot h \sim V^i_{ij} = \sum_{l=1}^L v^i_{ijl}. & \\213 A(\mathbf{x},t) &=\int_{0^+}^{\infty} dh \cdot g(h,\mathbf{x},t) \sim A_{ij} = \sum_{l=1}^L a_{ijl}, & \\ 214 V_i(\mathbf{x},t)&=\int_{0}^{\infty} dh \cdot g(h,\mathbf{x},t) \cdot h \sim V^i_{ij} = \sum_{l=1}^L v^i_{ijl}. & \\ 215 215 \end{align} 216 216 \end{linenomath} … … 228 228 \begin{linenomath} 229 229 \begin{align} 230 m \frac{\partial \ bm{u}} {\partial t} & = \bm{\nabla}\cdot\bm{\sigma} +A \left(\bm{\tau}_{a}+\bm{\tau}_{w}\right)  m f \bm{k} \times \bm{u}  m g \bm{\nabla}{\eta},230 m \frac{\partial \mathbf{u}} {\partial t} & = \mathbf{\nabla}\cdot\mathbf{\sigma} +A \left(\mathbf{\tau}_{a}+\mathbf{\tau}_{w}\right)  m f \mathbf{k} \times \mathbf{u}  m g \mathbf{\nabla}{\eta}, 231 231 \label{a} 232 232 \end{align} 233 233 \end{linenomath} 234 where $m=\rho_i V_i + \rho_s V_s $ is the ice and snow mass per unit area, $\ bm{u}$ is the ice velocity, $\bm{\sigma}$ is the internal stress tensor, $\bm{\tau}_a$ and $\bm{\tau}_w$ are the air and ocean stresses, respectively, $f$ is the Coriolis parameter, $\bm{k}$ is a unit vector pointing upwards, $g$ is the gravity acceleration and $\eta$ is the ocean surface elevation. The EVP approach used in LIM \citep{Bouillonetal13} gives the stress tensor as a function of the strain rate tensor $\dot{\bm{\epsilon}}$ and some of the sea ice state variables:235 \begin{linenomath} 236 \begin{align} 237 \ bm{\sigma} & = \bm{\sigma} (\dot{ \bm{\epsilon}}, \text{ice state}).234 where $m=\rho_i V_i + \rho_s V_s $ is the ice and snow mass per unit area, $\mathbf{u}$ is the ice velocity, $\mathbf{\sigma}$ is the internal stress tensor, $\mathbf{\tau}_a$ and $\mathbf{\tau}_w$ are the air and ocean stresses, respectively, $f$ is the Coriolis parameter, $\mathbf{k}$ is a unit vector pointing upwards, $g$ is the gravity acceleration and $\eta$ is the ocean surface elevation. The EVP approach used in LIM \citep{Bouillonetal13} gives the stress tensor as a function of the strain rate tensor $\dot{\mathbf{\epsilon}}$ and some of the sea ice state variables: 235 \begin{linenomath} 236 \begin{align} 237 \mathbf{\sigma} & = \mathbf{\sigma} (\dot{ \mathbf{\epsilon}}, \text{ice state}). 238 238 \end{align} 239 239 \end{linenomath} … … 245 245 \end{align} 246 246 \end{linenomath} 247 including the effets of transport, thermodynamics ($\Theta^X$) and mechanical redistribution ($\Psi^X$). Solving these $jpl.(4+2.jpk)$ equations gives the temporal evolution of $\ bm{u}$, $\bm{\sigma}$ and the rest of the global (extensive) variables listed in Table \ref{GVariables_table}.247 including the effets of transport, thermodynamics ($\Theta^X$) and mechanical redistribution ($\Psi^X$). Solving these $jpl.(4+2.jpk)$ equations gives the temporal evolution of $\mathbf{u}$, $\mathbf{\sigma}$ and the rest of the global (extensive) variables listed in Table \ref{GVariables_table}. 248 248 249 249 \section{Ice Dynamics} … … 272 272 \begin{center} 273 273 \vspace{0cm} 274 \includegraphics[height=6cm,angle=00]{ ../Figures/yield_curve.png}274 \includegraphics[height=6cm,angle=00]{yield_curve} 275 275 \caption{Elliptical yield curve used in the VP rheologies, drawn in the space of the principal components of the stress tensor ($\sigma_1$ and $\sigma_2$).} 276 276 \label{fig_yield} … … 383 383 \begin{center} 384 384 \vspace{0cm} 385 \includegraphics[height=8cm,angle=00]{ ../Figures/Thermal_properties.png}385 \includegraphics[height=8cm,angle=00]{Thermal_properties} 386 386 \caption{Thermal properties of sea ice vs temperature for different bulk salinities: brine fraction, specific enthalpy, thermal conductivity, and effective specific heat.} 387 387 \label{fig_thermal_properties}
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