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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 | $\ $\newline % force a new line |
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16 | |
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17 | Having defined the model equations in previous Chapter, we need now to choose the numerical discretization. In the present chapter, we provide a general description of the SI$^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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25 | \section{Time domain} |
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26 | |
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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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41 | |
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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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48 | \section{Spatial domain} |
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49 | |
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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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64 | |
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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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66 | |
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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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68 | |
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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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70 | |
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71 | \forfile{../namelists/nampar} |
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72 | |
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73 | \section{Thickness space domain} |
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74 | |
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75 | \forfile{../namelists/namitd} |
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76 | |
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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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78 | |
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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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86 | % |
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87 | % FIGx : Ice categories |
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88 | % |
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89 | % |
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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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97 | % |
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98 | %-------------------------------------------------------------------------------------------------------------------- |
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99 | |
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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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106 | \end{document} |
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