1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | % ================================================================ |
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5 | % Chapter Assimilation increments (ASM) |
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6 | % ================================================================ |
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7 | \chapter{Apply Assimilation Increments (ASM)} |
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8 | \label{chap:ASM} |
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9 | |
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10 | Authors: D. Lea, M. Martin, K. Mogensen, A. Weaver, ... % do we keep |
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11 | |
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12 | \minitoc |
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13 | |
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14 | \newpage |
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15 | |
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16 | The ASM code adds the functionality to apply increments to the model variables: temperature, salinity, |
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17 | sea surface height, velocity and sea ice concentration. |
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18 | These are read into the model from a NetCDF file which may be produced by separate data assimilation code. |
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19 | The code can also output model background fields which are used as an input to data assimilation code. |
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20 | This is all controlled by the namelist \textit{\ngn{nam\_asminc} }. |
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21 | There is a brief description of all the namelist options provided. |
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22 | To build the ASM code \key{asminc} must be set. |
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23 | |
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24 | %=============================================================== |
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25 | |
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26 | \section{Direct initialization} |
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27 | \label{sec:ASM_DI} |
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28 | |
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29 | Direct initialization (DI) refers to the instantaneous correction of the model background state using |
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30 | the analysis increment. |
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31 | DI is used when \np{ln\_asmdin} is set to true. |
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32 | |
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33 | \section{Incremental analysis updates} |
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34 | \label{sec:ASM_IAU} |
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35 | |
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36 | Rather than updating the model state directly with the analysis increment, |
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37 | it may be preferable to introduce the increment gradually into the ocean model in order to |
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38 | minimize spurious adjustment processes. |
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39 | This technique is referred to as Incremental Analysis Updates (IAU) \citep{bloom.takacs.ea_MWR96}. |
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40 | IAU is a common technique used with 3D assimilation methods such as 3D-Var or OI. |
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41 | IAU is used when \np{ln\_asmiau} is set to true. |
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42 | |
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43 | With IAU, the model state trajectory ${\mathbf x}$ in the assimilation window ($t_{0} \leq t_{i} \leq t_{N}$) |
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44 | is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as |
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45 | additional tendency terms to the prognostic equations: |
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46 | \begin{align*} |
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47 | % \label{eq:wa_traj_iau} |
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48 | {\mathbf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\mathbf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\mathbf x}^{a} |
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49 | \end{align*} |
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50 | where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\mathbf x}^{a}$ defined such that |
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51 | $\sum_{i=1}^{N} F_{i}=1$. |
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52 | ${\mathbf x}^b$ denotes the model initial state and ${\mathbf x}^a$ is the model state after the increments are applied. |
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53 | To control the adjustment time of the model to the increment, |
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54 | the increment can be applied over an arbitrary sub-window, $t_{m} \leq t_{i} \leq t_{n}$, |
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55 | of the main assimilation window, where $t_{0} \leq t_{m} \leq t_{i}$ and $t_{i} \leq t_{n} \leq t_{N}$. |
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56 | Typically the increments are spread evenly over the full window. |
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57 | In addition, two different weighting functions have been implemented. |
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58 | The first function employs constant weights, |
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59 | \begin{align} |
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60 | \label{eq:F1_i} |
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61 | F^{(1)}_{i} |
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62 | =\left\{ |
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63 | \begin{array}{ll} |
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64 | 0 & {\mathrm if} \; \; \; t_{i} < t_{m} \\ |
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65 | 1/M & {\mathrm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\ |
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66 | 0 & {\mathrm if} \; \; \; t_{i} > t_{n} |
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67 | \end{array} |
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68 | \right. |
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69 | \end{align} |
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70 | where $M = m-n$. |
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71 | The second function employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window, |
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72 | with the weighting reduced linearly to a small value at the window end-points: |
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73 | \begin{align} |
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74 | \label{eq:F2_i} |
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75 | F^{(2)}_{i} |
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76 | =\left\{ |
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77 | \begin{array}{ll} |
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78 | 0 & {\mathrm if} \; \; \; t_{i} < t_{m} \\ |
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79 | \alpha \, i & {\mathrm if} \; \; \; t_{m} \leq t_{i} \leq t_{M/2} \\ |
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80 | \alpha \, (M - i +1) & {\mathrm if} \; \; \; t_{M/2} < t_{i} \leq t_{n} \\ |
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81 | 0 & {\mathrm if} \; \; \; t_{i} > t_{n} |
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82 | \end{array} |
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83 | \right. |
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84 | \end{align} |
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85 | where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. |
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86 | The weights described by \autoref{eq:F2_i} provide a smoother transition of the analysis trajectory from |
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87 | one assimilation cycle to the next than that described by \autoref{eq:F1_i}. |
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88 | |
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89 | %========================================================================== |
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90 | % Divergence damping description %%% |
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91 | \section{Divergence damping initialisation} |
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92 | \label{sec:ASM_div_dmp} |
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93 | |
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94 | The velocity increments may be initialized by the iterative application of a divergence damping operator. |
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95 | In iteration step $n$ new estimates of velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by: |
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96 | \begin{equation} |
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97 | \label{eq:asm_dmp} |
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98 | \left\{ |
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99 | \begin{aligned} |
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100 | u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D |
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101 | \;\chi^{n-1}_I } \right) \\ \\ |
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102 | v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D |
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103 | \;\chi^{n-1}_I } \right) \\ |
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104 | \end{aligned} |
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105 | \right., |
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106 | \end{equation} |
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107 | where |
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108 | \[ |
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109 | % \label{eq:asm_div} |
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110 | \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } |
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111 | \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] |
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112 | +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). |
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113 | \] |
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114 | By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered in each iteration, |
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115 | and the vorticity is left unchanged. |
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116 | In the presence of coastal boundaries with zero velocity increments perpendicular to the coast |
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117 | the divergence is strongly damped. |
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118 | This type of the initialisation reduces the vertical velocity magnitude and |
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119 | alleviates the problem of the excessive unphysical vertical mixing in the first steps of the model integration |
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120 | \citep{talagrand_JAS72, dobricic.pinardi.ea_OS07}. |
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121 | Diffusion coefficients are defined as $A_D = \alpha e_{1t} e_{2t}$, where $\alpha = 0.2$. |
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122 | The divergence damping is activated by assigning to \np{nn\_divdmp} in the \textit{nam\_asminc} namelist |
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123 | a value greater than zero. |
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124 | By choosing this value to be of the order of 100 the increments in |
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125 | the vertical velocity will be significantly reduced. |
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126 | |
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127 | |
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128 | %========================================================================== |
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129 | |
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130 | \section{Implementation details} |
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131 | \label{sec:ASM_details} |
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132 | |
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133 | Here we show an example \ngn{namasm} namelist and the header of an example assimilation increments file on |
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134 | the ORCA2 grid. |
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135 | |
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136 | %------------------------------------------namasm----------------------------------------------------- |
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137 | % |
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138 | \nlst{nam_asminc} |
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139 | %------------------------------------------------------------------------------------------------------------- |
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140 | |
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141 | The header of an assimilation increments file produced using the NetCDF tool |
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142 | \mbox{\textit{ncdump~-h}} is shown below |
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143 | |
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144 | \begin{clines} |
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145 | netcdf assim_background_increments { |
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146 | dimensions: |
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147 | x = 182 ; |
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148 | y = 149 ; |
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149 | z = 31 ; |
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150 | t = UNLIMITED ; // (1 currently) |
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151 | variables: |
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152 | float nav_lon(y, x) ; |
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153 | float nav_lat(y, x) ; |
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154 | float nav_lev(z) ; |
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155 | double time_counter(t) ; |
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156 | double time ; |
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157 | double z_inc_dateb ; |
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158 | double z_inc_datef ; |
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159 | double bckint(t, z, y, x) ; |
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160 | double bckins(t, z, y, x) ; |
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161 | double bckinu(t, z, y, x) ; |
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162 | double bckinv(t, z, y, x) ; |
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163 | double bckineta(t, y, x) ; |
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164 | |
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165 | // global attributes: |
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166 | :DOMAIN_number_total = 1 ; |
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167 | :DOMAIN_number = 0 ; |
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168 | :DOMAIN_dimensions_ids = 1, 2 ; |
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169 | :DOMAIN_size_global = 182, 149 ; |
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170 | :DOMAIN_size_local = 182, 149 ; |
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171 | :DOMAIN_position_first = 1, 1 ; |
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172 | :DOMAIN_position_last = 182, 149 ; |
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173 | :DOMAIN_halo_size_start = 0, 0 ; |
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174 | :DOMAIN_halo_size_end = 0, 0 ; |
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175 | :DOMAIN_type = "BOX" ; |
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176 | } |
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177 | \end{clines} |
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178 | |
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179 | \biblio |
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180 | |
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181 | \pindex |
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182 | |
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183 | \end{document} |
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