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Changeset 9983 for NEMO/trunk/doc/si3_doc/tex_sub/chap_domain.tex – NEMO

# Changeset 9983 for NEMO/trunk/doc/si3_doc/tex_sub/chap_domain.tex

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Timestamp:
2018-07-20T15:45:54+02:00 (6 years ago)
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SI3 doc update

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 r9974 \newpage $\$\newline    % force a new line Excel Having defined the model equations in previous Chapter, we need now to choose the numerical discretization.  In the present chapter, we provide a general description of the SI$^3$ discretization strategy, in terms of time, space and thickness, which is considered as an extra independent variable. Sea ice state variables are typically expressed as: X(ji,jj,\textcolor{gray}{jk},jl). $ji$ and $jj$ are x-y spatial indices, as in the ocean. $jk=1, ..., nlay\_i$ corresponds to the vertical coordinate system in sea ice (ice layers), and only applies to vertically-resolved quantities (ice enthalpy and salinity). $jl=1, ..., jpl$ corresponds to the ice categories, discretizing thickness space. \section{Time domain} Time stepping. Dynamics then thermodynamics. nn\_fsbc. EVP subcycles. %-------------------------------------------------------------------------------------------------------------------- % % FIG x : Time Stepping % \begin{figure}[ht] \begin{center} \vspace{0cm} \includegraphics[height=6cm,angle=-00]{../Figures/time_stepping.png} \caption{Schematic representation of time stepping in SI$^3$, assuming $nn\_fsbc=5$.} \label{ice_scheme} \end{center} \end{figure} % %-------------------------------------------------------------------------------------------------------------------- The sea ice time stepping is synchronized with that of the ocean. Because of the potentially large numerical cost of sea ice physics, in particular rheology, SI$^3$ can be called every nn\_fsbc time steps (namsbc in \textit{namelist\_ref}). The sea ice time step is therefore $rdt\_ice = rdt * nn\_fsbc$. In terms of quality, the best value for \textit{nn\_fsbc} is 1, providing full consistency between sea ice and oceanic fields. Larger values (typically 2 to 5) can be used but numerical instabilities can appear because of the progressive decoupling between the state of sea ice and that of the ocean, hence changing $nn\_fsbc$ must be done carefully. Ice dynamics (rheology, advection, ridging/rafting) and thermodynamics are called successively. To avoid pathological situations, thermodynamics were chosen to be applied on fields that have been updated by dynamics, in a somehow semi-implicit procedure. There are a few iterative / subcycling procedures throughout the code, notably for rheology, advection, ridging/ rafting and the diffusion of heat. In some cases, the arrays at the beginning of the sea ice time step are required. Those are referred to as $X\_b$. \section{Spatial domain} Not much to say about domain. Handled by NEMO. C-grid. Scale factors. %-------------------------------------------------------------------------------------------------------------------- % % FIGx : Vertical grid % % \begin{figure}[!ht] \begin{center} \vspace{0cm} \includegraphics[height=10cm,angle=-00]{../Figures/thermogrid.eps} \caption{\footnotesize{Vertical grid of the model, used to resolve vertical temperature and salinity profiles}}\label{fig_dom_icelayers} \end{center} \end{figure} % %-------------------------------------------------------------------------------------------------------------------- Vertical layers (nlay\_i, nlay\_s) The horizontal indices $ji$ and $jj$ are handled as for the ocean in NEMO, assuming C-grid discretization and in most cases a finite difference expression for scale factors. \section{Thickness category boundaries} The vertical index $jk=1, ..., nlay\_i$ is used for enthalpy (temperature) and salinity. In each ice category, the temperature and salinity profiles are vertically resolved over $nlay\_i$ equally-spaced layers. The number of snow layers can currently only be set to $nlay\_s=1$ (Fig. \ref{fig_dom_icelayers}). [ jpl, nn\_virtual\_itd ] To increase numerical efficiency of the code, the two horizontal dimensions of an array $X(ji,jj,jk,jl)$ are collapsed into one (array $X\_1d(ji,jk,jl)$) for thermodynamic computations, and re-expanded afterwards. 2 formulations to describe \forfile{../namelists/nampar} [ ln\_cat\_hfn (function), rn\_himean ] \section{Thickness space domain} ln\_cat\_usr (user defined), rn\_catbnd, rn\_himin Categories: boundary definitions. See doc 2.0, there are commented bits of text in the tex file. \forfile{../namelists/namitd} Recall recommendations from Francois's, Antoine et al's paper. Thickness space is discretized using $jl=1, ..., jpl$ thickness categories, with prescribed boundaries $hi\_max(jl-1),hi\_max(jl)$. Following \cite{Lipscomb01}, ice thickness can freely evolve between these boundaries. The number of ice categories $jpl$ can be adjusted from the namelist ($nampar$). %%-------------------------------------------------------------------------------------------------------------------- %% %% FIGx : Ice categories %% %% %\begin{figure}[ht] %\begin{center} %\vspace{0cm} %\includegraphics[height=6cm,angle=-00]{./Figures/ice_cats_new.eps} %\caption{\footnotesize{Boundaries of the model ice thickness categories (m) for varying number of categories, prescribed mean thickness ($\overline h$ and formulation}}\label{ice_cats} %\end{center} %\end{figure} %% %%-------------------------------------------------------------------------------------------------------------------- There are two means to specify the position of the thickness boundaries of ice categories. The first option (ln\_cat\_hfn) is to use a fitting function that places the category boundaries between 0 and 3$\overline h$, with $\overline h$ the expected mean ice thickness over the domain (namelist parameter rn\_himean), and with a greater resolution for thin ice (Fig. \ref{fig_dom_icecats}). More specifically, the upper limits for ice in category $jl=1, ..., jpl-1$ are: \begin{eqnarray} hi\_max(jl) = \biggr ( \frac{jl \cdot (3\overline h + 1 )^{\alpha}}{ (jpl-jl)(3 \overline h + 1)^{\alpha} + jl }\biggr )^{\alpha^{-1}} - 1, \end{eqnarray} with $hi\_max(0)$=0 m and $\alpha = 0.05$. The last category has no upper boundary, so that it can contain arbitrarily thick ice. %-------------------------------------------------------------------------------------------------------------------- % %The thickness distribution function $g(h)$ is numerically discretized into several ice thickness categories. The numerical formulation of the thickness categories follows Bitz et al. (2001) and Lipscomb (2001). A fixed number $L$ of thickness categories with a typical value of $L=5$ is imposed. For some variables, sea ice in each category is further divided into N vertical layers of ice and one layer of snow. In the remainder of the text, the $l=1, ..., L$ index runs for ice thickness categories and $k=1, ..., N$ for the vertical ice layers. % FIGx : Ice categories % %Each thickness category has a mean thickness $h^i_l$ ranging over $[H^*_{l-1}$, $H^*_{l}$]. $H^*_{0}=0$, while the other boundaries are typically chosen with greater resolution for thin ice. % %There are two options for discretization in $h$-space, illustrated in Fig. \ref{ice_cats}. \begin{figure}[!ht] \begin{center} \vspace{0cm} \includegraphics[height=6cm,angle=-00]{../Figures/ice_cats.eps} \caption{\footnotesize{Boundaries of the model ice thickness categories (m) for varying number of categories and prescribed mean thickness ($\overline h$). The formerly used $tanh$ formulation is also depicted.}}\label{fig_dom_icecats} \end{center} \end{figure} % %\textbf{1.} The tanh hyperbolic formulation from CICE. %\begin{linenomath} %\begin{align} %H^*_l &= H^*_{l-1} + \frac{3}{L} + \frac{30}{L} \biggr [ 1 + tanh \biggr ( \frac{3l - 3 - 3L}{L} \biggr ) \biggr] \quad (l=1, ..., L-1). %\end{align} %\label{eq_301} %\end{linenomath} %The upper boundary $H^*_L$ is set to a very high value (99.). % %\textbf{2.} An adjustable home-made $1/h^\alpha$ formulation. % %To construct the discretization in $h$-space, we first prescribe $H^*_0$ and $H^*_L=H_{max}$. We then introduce a fitting function $f$, defined over $[0,\infty]$, stricly positive and decreasing. We impose that the $H^*_l$'s must be such that  their images in the $f$-space ($f_l = f(H^*_l)$) are equally spaced. In mathematical terms: %\begin{eqnarray} %f_l & = & f_0 - l \Delta f  \qquad (l = 2, ..., L-1), %\label{eq_fl} %\end{eqnarray} %where $\Delta f = \frac{f_0 - f_L}{L}$. % %Let us now construct a discretization in $h$-space. We use the function $f(h)=1/(h+1)^\alpha$, where $\alpha$ is strictly positive;  and impose that $H^*_{max}=3\overline h$, where $\overline h$ is the mean thickness in the domain $\overline h$. Replacing in $\ref{eq_fl}$, we get: %\begin{eqnarray} %H^*_l = \left ( \frac{ L ( H^*_L + 1 ) ^\alpha}{(L-l)( H^*_L + 1 ) ^\alpha + l} \right ) ^{1/\alpha} - 1 %\end{eqnarray} %\label{intro} %There are two parameters to tune: $\overline h$ and $\alpha$ (typically 0.05, used for Fig. \ref{ice_cats}). % %Each ice category has its own set of global state variables %-------------------------------------------------------------------------------------------------------------------- The other option (ln\_cat\_usr) is to specify category boundaries by hand using rn\_catbnd. The first category must always be thickner than rn\_himin (0.1 m by default). The choice of ice categories is important, because it constraints the ability of the model to resolve the ice thickness distribution. The latest study \citep{Massonnetetal18b} recommends to use at least 5 categories, which should include one thick ice with lower bounds at $\sim$4 m and $\sim$2 m for the Arctic and Antarctic, respectively, for allowing the storage of deformed ice. With a fixed number of cores, the cost of the model linearly increases with the number of ice categories. Using $jpl=1$ single ice category is also much cheaper than with 5 categories, but seriously deteriorates the ability of the model to grow and melt ice. Indeed, thin ice thicknes faster than thick ice, and shrinks more rapidly as well. When nn\_virtual\_itd=1 ($jpl$ = 1 only), two parameterizations are activated to compensate for these shortcomings. Heat conduction and areal decay of melting ice are adjusted to closely approach the 5 categories case. \end{document}