[9974] | 1 | |
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| 2 | \documentclass[../../tex_main/NEMO_manual]{subfiles} |
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| 3 | |
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| 4 | \begin{document} |
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| 5 | |
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| 6 | % ================================================================ |
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| 7 | % Chapter 2 Ñ Domain |
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| 8 | % ================================================================ |
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| 9 | |
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| 10 | \chapter{Time, space and thickness space domain} |
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| 11 | \label{chap:DOM} |
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| 12 | \minitoc |
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| 13 | |
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| 14 | \newpage |
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| 15 | $\ $\newline % force a new line |
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[9983] | 16 | |
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| 17 | Having defined the model equations in previous Chapter, we need now to choose the numerical discretization. In the present chapter, we provide a general description of the SI$^3$ discretization strategy, in terms of time, space and thickness, which is considered as an extra independent variable. |
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| 18 | |
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| 19 | Sea ice state variables are typically expressed as: |
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| 20 | \begin{equation} |
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| 21 | X(ji,jj,\textcolor{gray}{jk},jl). |
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| 22 | \end{equation} |
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| 23 | $ji$ 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. |
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| 24 | |
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[9974] | 25 | \section{Time domain} |
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| 26 | |
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[9983] | 27 | %-------------------------------------------------------------------------------------------------------------------- |
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| 28 | % |
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| 29 | % FIG x : Time Stepping |
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| 30 | % |
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| 31 | \begin{figure}[ht] |
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| 32 | \begin{center} |
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| 33 | \vspace{0cm} |
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| 34 | \includegraphics[height=6cm,angle=-00]{../Figures/time_stepping.png} |
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| 35 | \caption{Schematic representation of time stepping in SI$^3$, assuming $nn\_fsbc=5$.} |
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| 36 | \label{ice_scheme} |
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| 37 | \end{center} |
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| 38 | \end{figure} |
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| 39 | % |
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| 40 | %-------------------------------------------------------------------------------------------------------------------- |
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[9974] | 41 | |
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[9983] | 42 | 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. |
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| 43 | |
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| 44 | 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. |
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| 45 | |
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| 46 | 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 $X\_b$. |
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| 47 | |
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[9974] | 48 | \section{Spatial domain} |
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| 49 | |
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[9983] | 50 | %-------------------------------------------------------------------------------------------------------------------- |
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| 51 | % |
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| 52 | % FIGx : Vertical grid |
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| 53 | % |
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| 54 | % |
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| 55 | \begin{figure}[!ht] |
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| 56 | \begin{center} |
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| 57 | \vspace{0cm} |
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| 58 | \includegraphics[height=10cm,angle=-00]{../Figures/thermogrid.eps} |
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| 59 | \caption{\footnotesize{Vertical grid of the model, used to resolve vertical temperature and salinity profiles}}\label{fig_dom_icelayers} |
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| 60 | \end{center} |
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| 61 | \end{figure} |
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| 62 | % |
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| 63 | %-------------------------------------------------------------------------------------------------------------------- |
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[9974] | 64 | |
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[9983] | 65 | 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. |
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[9974] | 66 | |
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[9983] | 67 | 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}). |
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[9974] | 68 | |
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[9983] | 69 | 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. |
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[9974] | 70 | |
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[9983] | 71 | \forfile{../namelists/nampar} |
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[9974] | 72 | |
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[9983] | 73 | \section{Thickness space domain} |
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[9974] | 74 | |
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[9983] | 75 | \forfile{../namelists/namitd} |
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[9974] | 76 | |
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[9983] | 77 | 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 ($nampar$). |
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[9974] | 78 | |
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[9983] | 79 | 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: |
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| 80 | \begin{eqnarray} |
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| 81 | hi\_max(jl) = \biggr ( \frac{jl \cdot (3\overline h + 1 )^{\alpha}}{ (jpl-jl)(3 \overline h + 1)^{\alpha} + jl }\biggr )^{\alpha^{-1}} - 1, |
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| 82 | \end{eqnarray} |
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| 83 | 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. |
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| 84 | |
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| 85 | %-------------------------------------------------------------------------------------------------------------------- |
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[9974] | 86 | % |
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[9983] | 87 | % FIGx : Ice categories |
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[9974] | 88 | % |
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| 89 | % |
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[9983] | 90 | \begin{figure}[!ht] |
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| 91 | \begin{center} |
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| 92 | \vspace{0cm} |
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| 93 | \includegraphics[height=6cm,angle=-00]{../Figures/ice_cats.eps} |
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| 94 | \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} |
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| 95 | \end{center} |
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| 96 | \end{figure} |
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[9974] | 97 | % |
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[9983] | 98 | %-------------------------------------------------------------------------------------------------------------------- |
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[9974] | 99 | |
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[9983] | 100 | 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). |
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| 101 | |
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| 102 | 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. |
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| 103 | |
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| 104 | 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. |
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| 105 | |
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[9974] | 106 | \end{document} |
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