1 | \documentclass[NEMO_book]{subfiles} |
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2 | \begin{document} |
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3 | % ================================================================ |
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4 | % Chapter Assimilation increments (ASM) |
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5 | % ================================================================ |
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6 | \chapter{Apply assimilation increments (ASM)} |
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7 | \label{ASM} |
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8 | |
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9 | Authors: D. Lea, M. Martin, K. Mogensen, A. Weaver, ... % do we keep |
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10 | |
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11 | \minitoc |
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12 | |
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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 | The ASM code adds the functionality to apply increments to the model variables: |
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18 | temperature, salinity, sea surface height, velocity and sea ice concentration. |
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19 | These are read into the model from a NetCDF file which may be produced by separate data |
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20 | assimilation code. The code can also output model background fields which are used |
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21 | as an input to data assimilation code. This is all controlled by the namelist |
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22 | \textit{\ngn{nam\_asminc} }. There is a brief description of all the namelist options |
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23 | provided. To build the ASM code \key{asminc} must be set. |
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24 | |
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25 | %=============================================================== |
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26 | |
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27 | \section{Direct initialization} |
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28 | \label{ASM_DI} |
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29 | |
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30 | Direct initialization (DI) refers to the instantaneous correction |
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31 | of the model background state using the analysis increment. |
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32 | DI is used when \np{ln\_asmdin} is set to true. |
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33 | |
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34 | \section{Incremental Analysis Updates} |
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35 | \label{ASM_IAU} |
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36 | |
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37 | Rather than updating the model state directly with the analysis increment, |
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38 | it may be preferable to introduce the increment gradually into the ocean |
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39 | model in order to minimize spurious adjustment processes. This technique |
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40 | is referred to as Incremental Analysis Updates (IAU) \citep{Bloom_al_MWR96}. |
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41 | IAU is a common technique used with 3D assimilation methods such as 3D-Var or OI. |
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42 | IAU is used when \np{ln\_asmiau} is set to true. |
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43 | |
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44 | With IAU, the model state trajectory ${\bf x}$ in the assimilation window |
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45 | ($t_{0} \leq t_{i} \leq t_{N}$) |
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46 | is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH |
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47 | as additional tendency terms to the prognostic equations: |
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48 | \begin{eqnarray} \label{eq:wa_traj_iau} |
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49 | {\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})] |
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50 | \; + \; F_{i} \delta \tilde{\bf x}^{a} |
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51 | \end{eqnarray} |
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52 | where $F_{i}$ is a weighting function for applying the increments $\delta |
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53 | \tilde{\bf x}^{a}$ defined such that $\sum_{i=1}^{N} F_{i}=1$. |
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54 | ${\bf x}^b$ denotes the model initial state and ${\bf x}^a$ is the model state |
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55 | after the increments are applied. |
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56 | To control the adjustment time of the model to the increment, |
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57 | the increment can be applied over an arbitrary sub-window, |
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58 | $t_{m} \leq t_{i} \leq t_{n}$, of the main assimilation window, |
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59 | where $t_{0} \leq t_{m} \leq t_{i}$ and $t_{i} \leq t_{n} \leq t_{N}$, |
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60 | Typically the increments are spread evenly over the full window. |
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61 | In addition, two different weighting functions have been implemented. |
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62 | The first function employs constant weights, |
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63 | \begin{eqnarray} \label{eq:F1_i} |
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64 | F^{(1)}_{i} |
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65 | =\left\{ \begin{array}{ll} |
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66 | 0 & {\rm if} \; \; \; t_{i} < t_{m} \\ |
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67 | 1/M & {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\ |
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68 | 0 & {\rm if} \; \; \; t_{i} > t_{n} |
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69 | \end{array} \right. |
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70 | \end{eqnarray} |
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71 | where $M = m-n$. |
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72 | The second function employs peaked hat-like weights in order to give maximum |
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73 | weight in the centre of the sub-window, with the weighting reduced |
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74 | linearly to a small value at the window end-points: |
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75 | \begin{eqnarray} \label{eq:F2_i} |
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76 | F^{(2)}_{i} |
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77 | =\left\{ \begin{array}{ll} |
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78 | 0 & {\rm if} \; \; \; t_{i} < t_{m} \\ |
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79 | \alpha \, i & {\rm if} \; \; \; t_{m} \leq t_{i} \leq t_{M/2} \\ |
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80 | \alpha \, (M - i +1) & {\rm if} \; \; \; t_{M/2} < t_{i} \leq t_{n} \\ |
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81 | 0 & {\rm if} \; \; \; t_{i} > t_{n} |
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82 | \end{array} \right. |
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83 | \end{eqnarray} |
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84 | where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. |
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85 | The weights described by \eqref{eq:F2_i} provide a |
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86 | smoother transition of the analysis trajectory from one assimilation cycle |
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87 | to the next than that described by \eqref{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{ASM_details} |
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93 | |
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94 | The velocity increments may be initialized by the iterative application of |
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95 | a divergence damping operator. In iteration step $n$ new estimates of |
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96 | velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by: |
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97 | \begin{equation} \label{eq:asm_dmp} |
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98 | \left\{ \begin{aligned} |
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99 | u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta _{i+1/2} \left( {A_D |
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100 | \;\chi^{n-1}_I } \right) \\ |
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101 | \\ |
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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} \right., |
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105 | \end{equation} |
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106 | where |
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107 | \begin{equation} \label{eq:asm_div} |
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108 | \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } |
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109 | \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] |
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110 | +\delta _j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). |
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111 | \end{equation} |
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112 | By the application of \eqref{eq:asm_dmp} and \eqref{eq:asm_dmp} the divergence is filtered |
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113 | in each iteration, and the vorticity is left unchanged. In the presence of coastal boundaries |
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114 | with zero velocity increments perpendicular to the coast the divergence is strongly damped. |
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115 | This type of the initialisation reduces the vertical velocity magnitude and alleviates the |
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116 | problem of the excessive unphysical vertical mixing in the first steps of the model |
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117 | integration \citep{Talagrand_JAS72, Dobricic_al_OS07}. Diffusion coefficients are defined as |
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118 | $A_D = \alpha e_{1t} e_{2t}$, where $\alpha = 0.2$. The divergence damping is activated by |
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119 | assigning to \np{nn\_divdmp} in the \textit{nam\_asminc} namelist a value greater than zero. |
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120 | By choosing this value to be of the order of 100 the increments in the vertical velocity will |
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121 | be significantly reduced. |
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122 | |
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123 | |
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124 | %========================================================================== |
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125 | |
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126 | \section{Implementation details} |
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127 | \label{ASM_details} |
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128 | |
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129 | Here we show an example \ngn{namasm} namelist and the header of an example assimilation |
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130 | increments file on the ORCA2 grid. |
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131 | |
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132 | %------------------------------------------namasm----------------------------------------------------- |
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133 | \namdisplay{namasm} |
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134 | %------------------------------------------------------------------------------------------------------------- |
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135 | |
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136 | The header of an assimilation increments file produced using the NetCDF tool |
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137 | \mbox{\textit{ncdump~-h}} is shown below |
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138 | |
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139 | \begin{alltt} |
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140 | \tiny |
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141 | \begin{verbatim} |
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142 | netcdf assim_background_increments { |
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143 | dimensions: |
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144 | x = 182 ; |
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145 | y = 149 ; |
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146 | z = 31 ; |
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147 | t = UNLIMITED ; // (1 currently) |
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148 | variables: |
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149 | float nav_lon(y, x) ; |
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150 | float nav_lat(y, x) ; |
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151 | float nav_lev(z) ; |
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152 | double time_counter(t) ; |
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153 | double time ; |
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154 | double z_inc_dateb ; |
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155 | double z_inc_datef ; |
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156 | double bckint(t, z, y, x) ; |
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157 | double bckins(t, z, y, x) ; |
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158 | double bckinu(t, z, y, x) ; |
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159 | double bckinv(t, z, y, x) ; |
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160 | double bckineta(t, y, x) ; |
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161 | |
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162 | // global attributes: |
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163 | :DOMAIN_number_total = 1 ; |
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164 | :DOMAIN_number = 0 ; |
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165 | :DOMAIN_dimensions_ids = 1, 2 ; |
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166 | :DOMAIN_size_global = 182, 149 ; |
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167 | :DOMAIN_size_local = 182, 149 ; |
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168 | :DOMAIN_position_first = 1, 1 ; |
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169 | :DOMAIN_position_last = 182, 149 ; |
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170 | :DOMAIN_halo_size_start = 0, 0 ; |
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171 | :DOMAIN_halo_size_end = 0, 0 ; |
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172 | :DOMAIN_type = "BOX" ; |
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173 | } |
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174 | \end{verbatim} |
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175 | \end{alltt} |
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176 | \end{document} |
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