# Changeset 11015 for NEMO/trunk/doc/latex/SI3/subfiles/chap_model_basics.tex

Ignore:
Timestamp:
2019-05-20T20:57:09+02:00 (18 months ago)
Message:

Modification of the content to be in line with the NEMO manual
SI3 manual can now be build like the NEMO manual with ./manual_build.sh SI3

• Mimick the directory organisation with main and subfiles folders.
• Regarding the particular case of namelists
• Remove the duplicates already contained in the global namelists folder at 1st level of ./doc
• Keep the namelists sub-folder only for namdyn_adv & namsbc which already exist in ocean namelists
• Rewriting of SI3_manual.tex with NEMO_manual.tex as template to easily highlight differences
• Updating of several paths for figures/namelists inclusion or LaTeX files referencing
• LaTeX source:
• Replacement of \forfile command for namelists with pre-configured \nlst alias
• " "" \bm with \mathbf (save installation of an extra package)
Location:
NEMO/trunk/doc/latex/SI3/subfiles
Files:
1 edited
1 moved

### Legend:

Unmodified
 r9974 \documentclass[../../tex_main/NEMO_manual]{subfiles} \documentclass[../main/SI3_manual]{subfiles} \begin{document} \begin{center} \vspace{0cm} \includegraphics[height=10cm,angle=-00]{../Figures/ice_scheme.png} \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 ($\bm{u}$).} \includegraphics[height=10cm,angle=-00]{ice_scheme} \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 ($\mathbf{u}$).} \label{ice_scheme} \end{center} %------------------------------------------------------------------------------------------------------------------------- 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: 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: \begin{linenomath} \begin{align} \begin{center} \vspace{0cm} \includegraphics[height=6cm,angle=-00]{../Figures/g_h.png} \includegraphics[height=6cm,angle=-00]{g_h} \caption{Representation of the relation between real thickness profiles and the ice thickness distribution function $g(h)$} \label{fig_g_h} \begin{linenomath} \begin{align} \frac{\partial a_l}{\partial t} = - \bm{\nabla} \cdot (a_l \mathbf{u}) + \Theta^a_l + \int_{H^*_{l-1}}^{H^*_l} dh \psi. \frac{\partial a_l}{\partial t} = - \mathbf{\nabla} \cdot (a_l \mathbf{u}) + \Theta^a_l + \int_{H^*_{l-1}}^{H^*_l} dh \psi. \label{eq:gt} \end{align} \begin{linenomath} \begin{align} A(\bm{x},t) &=\int_{0^+}^{\infty} dh \cdot g(h,\bm{x},t) \sim A_{ij} = \sum_{l=1}^L a_{ijl}, & \\ 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}. & \\ A(\mathbf{x},t) &=\int_{0^+}^{\infty} dh \cdot g(h,\mathbf{x},t) \sim A_{ij} = \sum_{l=1}^L a_{ijl}, & \\ 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}. & \\ \end{align} \end{linenomath} \begin{linenomath} \begin{align} 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}, 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}, \label{a} \end{align} \end{linenomath} 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: \begin{linenomath} \begin{align} \bm{\sigma} & = \bm{\sigma} (\dot{ \bm{\epsilon}}, \text{ice state}). 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: \begin{linenomath} \begin{align} \mathbf{\sigma} & = \mathbf{\sigma} (\dot{ \mathbf{\epsilon}}, \text{ice state}). \end{align} \end{linenomath} \end{align} \end{linenomath} 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}. 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}. \section{Ice Dynamics} \begin{center} \vspace{0cm} \includegraphics[height=6cm,angle=-00]{../Figures/yield_curve.png} \includegraphics[height=6cm,angle=-00]{yield_curve} \caption{Elliptical yield curve used in the VP rheologies, drawn in the space of the principal components of the stress tensor ($\sigma_1$ and $\sigma_2$).} \label{fig_yield} \begin{center} \vspace{0cm} \includegraphics[height=8cm,angle=-00]{../Figures/Thermal_properties.png} \includegraphics[height=8cm,angle=-00]{Thermal_properties} \caption{Thermal properties of sea ice vs temperature for different bulk salinities: brine fraction, specific enthalpy, thermal conductivity, and effective specific heat.} \label{fig_thermal_properties}