1 | % ================================================================ |
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2 | % Chapter � Miscellaneous Topics |
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3 | % ================================================================ |
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4 | \chapter{Miscellaneous Topics} |
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5 | \label{MISC} |
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6 | \minitoc |
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7 | |
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8 | \newpage |
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9 | $\ $\newline % force a new ligne |
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10 | |
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11 | % ================================================================ |
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12 | % Representation of Unresolved Straits |
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13 | % ================================================================ |
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14 | \section{Representation of Unresolved Straits} |
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15 | \label{MISC_strait} |
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16 | |
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17 | In climate modeling, it often occurs that a crucial connections between water masses |
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18 | is broken as the grid mesh is too coarse to resolve narrow straits. For example, coarse |
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19 | grid spacing typically closes off the Mediterranean from the Atlantic at the Strait of |
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20 | Gibraltar. In this case, it is important for climate models to include the effects of salty |
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21 | water entering the Atlantic from the Mediterranean. Likewise, it is important for the |
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22 | Mediterranean to replenish its supply of water from the Atlantic to balance the net |
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23 | evaporation occurring over the Mediterranean region. This problem occurs even in |
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24 | eddy permitting simulations. For example, in ORCA 1/4\deg several straits of the Indonesian |
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25 | archipelago (Ombai, Lombok...) are much narrow than even a single ocean grid-point. |
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26 | |
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27 | We describe briefly here the three methods that can be used in \NEMO to handle |
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28 | such improperly resolved straits. The first two consist of opening the strait by hand |
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29 | while ensuring that the mass exchanges through the strait are not too large by |
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30 | either artificially reducing the surface of the strait grid-cells or, locally increasing |
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31 | the lateral friction. In the third one, the strait is closed but exchanges of mass, |
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32 | heat and salt across the land are allowed. |
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33 | Note that such modifications are so specific to a given configuration that no attempt |
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34 | has been made to set them in a generic way. However, examples of how |
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35 | they can be set up is given in the ORCA 2\deg and 0.5\deg configurations. For example, |
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36 | for details of implementation in ORCA2, search: |
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37 | \vspace{-10pt} |
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38 | \begin{alltt} |
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39 | \tiny |
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40 | \begin{verbatim} |
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41 | IF( cp_cfg == "orca" .AND. jp_cfg == 2 ) |
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42 | \end{verbatim} |
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43 | \end{alltt} |
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44 | |
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45 | % ------------------------------------------------------------------------------------------------------------- |
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46 | % Hand made geometry changes |
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47 | % ------------------------------------------------------------------------------------------------------------- |
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48 | \subsection{Hand made geometry changes} |
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49 | \label{MISC_strait_hand} |
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50 | |
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51 | $\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement |
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52 | with the true mean width of the strait. (Fig.~\ref{Fig_MISC_strait_hand}). |
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53 | This technique is sometime called "partially open face" or "partially closed cells". |
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54 | The key issue here is only to reduce the faces of $T$-cell ($i.e.$ change the value |
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55 | of the horizontal scale factors at $u$- or $v$-point) but not the volume of the $T$-cell. |
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56 | Indeed, reducing the volume of strait $T$-cell can easily produce a numerical |
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57 | instability at that grid point that would require a reduction of the model time step. |
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58 | The changes associated with strait management are done in \mdl{domhgr}, |
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59 | just after the definition or reading of the horizontal scale factors. |
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60 | |
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61 | $\bullet$ increase of the viscous boundary layer thickness by local increase of the |
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62 | fmask value at the coast (Fig.~\ref{Fig_MISC_strait_hand}). This is done in |
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63 | \mdl{dommsk} together with the setting of the coastal value of fmask |
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64 | (see Section \ref{LBC_coast}) |
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65 | |
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66 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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67 | \begin{figure}[!tbp] \begin{center} |
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68 | \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar.pdf} |
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69 | \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar2.pdf} |
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70 | \caption{ \label{Fig_MISC_strait_hand} |
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71 | Example of the Gibraltar strait defined in a $1\deg \times 1\deg$ mesh. |
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72 | \textit{Top}: using partially open cells. The meridional scale factor at $v$-point |
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73 | is reduced on both sides of the strait to account for the real width of the strait |
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74 | (about 20 km). Note that the scale factors of the strait $T$-point remains unchanged. |
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75 | \textit{Bottom}: using viscous boundary layers. The four fmask parameters |
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76 | along the strait coastlines are set to a value larger than 4, $i.e.$ "strong" no-slip |
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77 | case (see Fig.\ref{Fig_LBC_shlat}) creating a large viscous boundary layer |
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78 | that allows a reduced transport through the strait.} |
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79 | \end{center} \end{figure} |
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80 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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81 | |
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82 | % ------------------------------------------------------------------------------------------------------------- |
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83 | % Cross Land Advection |
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84 | % ------------------------------------------------------------------------------------------------------------- |
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85 | \subsection{Cross Land Advection (\mdl{tracla})} |
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86 | \label{MISC_strait_cla} |
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87 | %--------------------------------------------namcla-------------------------------------------------------- |
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88 | \namdisplay{namcla} |
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89 | %-------------------------------------------------------------------------------------------------------------- |
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90 | |
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91 | \colorbox{yellow}{Add a short description of CLA staff here or in lateral boundary condition chapter?} |
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92 | Options are defined through the \ngn{namcla} namelist variables. |
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93 | |
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94 | %The problem is resolved here by allowing the mixing of tracers and mass/volume between non-adjacent water columns at nominated regions within the model. Momentum is not mixed. The scheme conserves total tracer content, and total volume (the latter in $z*$- or $s*$-coordinate), and maintains compatibility between the tracer and mass/volume budgets. |
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95 | |
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96 | % ================================================================ |
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97 | % Closed seas |
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98 | % ================================================================ |
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99 | \section{Closed seas (\mdl{closea})} |
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100 | \label{MISC_closea} |
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101 | |
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102 | \colorbox{yellow}{Add here a short description of the way closed seas are managed} |
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103 | |
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104 | |
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105 | % ================================================================ |
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106 | % Sub-Domain Functionality (\textit{nizoom, njzoom}, namelist parameters) |
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107 | % ================================================================ |
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108 | \section{Sub-Domain Functionality (\np{jpizoom}, \np{jpjzoom})} |
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109 | \label{MISC_zoom} |
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110 | |
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111 | The sub-domain functionality, also improperly called the zoom option |
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112 | (improperly because it is not associated with a change in model resolution) |
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113 | is a quite simple function that allows a simulation over a sub-domain of an |
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114 | already defined configuration ($i.e.$ without defining a new mesh, initial |
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115 | state and forcings). This option can be useful for testing the user settings |
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116 | of surface boundary conditions, or the initial ocean state of a huge ocean |
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117 | model configuration while having a small computer memory requirement. |
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118 | It can also be used to easily test specific physics in a sub-domain (for example, |
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119 | see \citep{Madec_al_JPO96} for a test of the coupling used in the global ocean |
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120 | version of OPA between sea-ice and ocean model over the Arctic or Antarctic |
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121 | ocean, using a sub-domain). In the standard model, this option does not |
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122 | include any specific treatment for the ocean boundaries of the sub-domain: |
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123 | they are considered as artificial vertical walls. Nevertheless, it is quite easy |
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124 | to add a restoring term toward a climatology in the vicinity of such boundaries |
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125 | (see \S\ref{TRA_dmp}). |
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126 | |
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127 | In order to easily define a sub-domain over which the computation can be |
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128 | performed, the dimension of all input arrays (ocean mesh, bathymetry, |
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129 | forcing, initial state, ...) are defined as \np{jpidta}, \np{jpjdta} and \np{jpkdta} |
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130 | ( in \ngn{namcfg} namelist), while the computational domain is defined through |
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131 | \np{jpiglo}, \np{jpjglo} and \jp{jpk} (\ngn{namcfg} namelist). When running the |
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132 | model over the whole domain, the user sets \np{jpiglo}=\np{jpidta} \np{jpjglo}=\np{jpjdta} |
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133 | and \jp{jpk}=\jp{jpkdta}. When running the model over a sub-domain, the user |
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134 | has to provide the size of the sub-domain, (\np{jpiglo}, \np{jpjglo}, \np{jpkglo}), |
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135 | and the indices of the south western corner as \np{jpizoom} and \np{jpjzoom} in |
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136 | the \ngn{namcfg} namelist (Fig.~\ref{Fig_LBC_zoom}). |
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137 | |
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138 | Note that a third set of dimensions exist, \jp{jpi}, \jp{jpj} and \jp{jpk} which is |
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139 | actually used to perform the computation. It is set by default to \jp{jpi}=\np{jpjglo} |
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140 | and \jp{jpj}=\np{jpjglo}, except for massively parallel computing where the |
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141 | computational domain is laid out on local processor memories following a 2D |
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142 | horizontal splitting. % (see {\S}IV.2-c) ref to the section to be updated |
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143 | |
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144 | \subsection{Simple subsetting of input files via netCDF attributes} |
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145 | |
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146 | The extended grids for use with the under-shelf ice cavities will result in redundant rows |
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147 | around Antarctica if the ice cavities are not active. A simple mechanism for subsetting |
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148 | input files associated with the extended domains has been implemented to avoid the need to |
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149 | maintain different sets of input fields for use with or without active ice cavities. The |
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150 | existing 'zoom' options are overly complex for this task and marked for deletion anyway. |
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151 | This alternative subsetting operates for the j-direction only and works by optionally |
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152 | looking for and using a global file attribute (named: \np{open\_ocean\_jstart}) to |
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153 | determine the starting j-row for input. The use of this option is best explained with an |
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154 | example: Consider an ORCA1 configuration using the extended grid bathymetry and coordinate |
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155 | files: |
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156 | \vspace{-10pt} |
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157 | \begin{alltt} |
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158 | \tiny |
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159 | \begin{verbatim} |
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160 | eORCA1_bathymetry_v2.nc |
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161 | eORCA1_coordinates.nc |
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162 | \end{verbatim} |
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163 | \end{alltt} |
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164 | \noindent These files define a horizontal domain of 362x332. Assuming the first row with |
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165 | open ocean wet points in the non-isf bathymetry for this set is row 42 (Fortran indexing) |
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166 | then the formally correct setting for \np{open\_ocean\_jstart} is 41. Using this value as the |
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167 | first row to be read will result in a 362x292 domain which is the same size as the original |
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168 | ORCA1 domain. Thus the extended coordinates and bathymetry files can be used with all the |
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169 | original input files for ORCA1 if the ice cavities are not active (\np{ln\_isfcav = |
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170 | .false.}). Full instructions for achieving this are: |
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171 | |
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172 | \noindent Add the new attribute to any input files requiring a j-row offset, i.e: |
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173 | \vspace{-10pt} |
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174 | \begin{alltt} |
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175 | \tiny |
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176 | \begin{verbatim} |
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177 | ncatted -a open_ocean_jstart,global,a,d,41 eORCA1_coordinates.nc |
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178 | ncatted -a open_ocean_jstart,global,a,d,41 eORCA1_bathymetry_v2.nc |
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179 | \end{verbatim} |
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180 | \end{alltt} |
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181 | |
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182 | \noindent Add the logical switch to \ngn{namcfg} in the configuration namelist and set true: |
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183 | %--------------------------------------------namcfg-------------------------------------------------------- |
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184 | \namdisplay{namcfg_orca1} |
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185 | %-------------------------------------------------------------------------------------------------------------- |
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186 | |
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187 | \noindent Note the j-size of the global domain is the (extended j-size minus |
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188 | \np{open\_ocean\_jstart} + 1 ) and this must match the size of all datasets other than |
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189 | bathymetry and coordinates currently. However the option can be extended to any global, 2D |
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190 | and 3D, netcdf, input field by adding the: |
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191 | \vspace{-10pt} |
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192 | \begin{alltt} |
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193 | \tiny |
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194 | \begin{verbatim} |
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195 | lrowattr=ln_use_jattr |
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196 | \end{verbatim} |
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197 | \end{alltt} |
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198 | optional argument to the appropriate \np{iom\_get} call and the \np{open\_ocean\_jstart} attribute to the corresponding input files. It remains the users responsibility to set \np{jpjdta} and \np{jpjglo} values in the \np{namelist\_cfg} file according to their needs. |
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199 | |
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200 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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201 | \begin{figure}[!ht] \begin{center} |
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202 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_zoom.pdf} |
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203 | \caption{ \label{Fig_LBC_zoom} |
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204 | Position of a model domain compared to the data input domain when the zoom functionality is used.} |
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205 | \end{center} \end{figure} |
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206 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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207 | |
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208 | |
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209 | % ================================================================ |
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210 | % Accelerating the Convergence |
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211 | % ================================================================ |
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212 | \section{Accelerating the Convergence (\np{nn\_acc} = 1)} |
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213 | \label{MISC_acc} |
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214 | %--------------------------------------------namdom------------------------------------------------------- |
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215 | \namdisplay{namdom} |
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216 | %-------------------------------------------------------------------------------------------------------------- |
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217 | |
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218 | Searching an equilibrium state with an global ocean model requires a very long time |
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219 | integration period (a few thousand years for a global model). Due to the size of |
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220 | the time step required for numerical stability (less than a few hours), |
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221 | this usually requires a large elapsed time. In order to overcome this problem, |
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222 | \citet{Bryan1984} introduces a technique that is intended to accelerate |
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223 | the spin up to equilibrium. It uses a larger time step in |
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224 | the tracer evolution equations than in the momentum evolution |
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225 | equations. It does not affect the equilibrium solution but modifies the |
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226 | trajectory to reach it. |
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227 | |
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228 | Options are defined through the \ngn{namdom} namelist variables. |
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229 | The acceleration of convergence option is used when \np{nn\_acc}=1. In that case, |
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230 | $\rdt=rn\_rdt$ is the time step of dynamics while $\widetilde{\rdt}=rdttra$ is the |
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231 | tracer time-step. the former is set from the \np{rn\_rdt} namelist parameter while the latter |
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232 | is computed using a hyperbolic tangent profile and the following namelist parameters : |
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233 | \np{rn\_rdtmin}, \np{rn\_rdtmax} and \np{rn\_rdth}. Those three parameters correspond |
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234 | to the surface value the deep ocean value and the depth at which the transition occurs, respectively. |
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235 | The set of prognostic equations to solve becomes: |
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236 | \begin{equation} \label{Eq_acc} |
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237 | \begin{split} |
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238 | \frac{\partial \textbf{U}_h }{\partial t} |
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239 | &\equiv \frac{\textbf{U}_h ^{t+1}-\textbf{U}_h^{t-1} }{2\rdt} = \ldots \\ |
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240 | \frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ |
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241 | \frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ |
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242 | \end{split} |
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243 | \end{equation} |
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244 | |
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245 | \citet{Bryan1984} has examined the consequences of this distorted physics. |
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246 | Free waves have a slower phase speed, their meridional structure is slightly |
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247 | modified, and the growth rate of baroclinically unstable waves is reduced |
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248 | but with a wider range of instability. This technique is efficient for |
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249 | searching for an equilibrium state in coarse resolution models. However its |
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250 | application is not suitable for many oceanic problems: it cannot be used for |
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251 | transient or time evolving problems (in particular, it is very questionable |
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252 | to use this technique when there is a seasonal cycle in the forcing fields), |
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253 | and it cannot be used in high-resolution models where baroclinically |
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254 | unstable processes are important. Moreover, the vertical variation of |
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255 | $\widetilde{ \rdt}$ implies that the heat and salt contents are no longer |
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256 | conserved due to the vertical coupling of the ocean level through both |
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257 | advection and diffusion. Therefore \np{rn\_rdtmin} = \np{rn\_rdtmax} should be |
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258 | a more clever choice. |
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259 | |
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260 | |
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261 | % ================================================================ |
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262 | % Accuracy and Reproducibility |
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263 | % ================================================================ |
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264 | \section{Accuracy and Reproducibility (\mdl{lib\_fortran})} |
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265 | \label{MISC_fortran} |
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266 | |
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267 | \subsection{Issues with intrinsinc SIGN function (\key{nosignedzero})} |
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268 | \label{MISC_sign} |
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269 | |
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270 | The SIGN(A, B) is the \textsc {Fortran} intrinsic function delivers the magnitude |
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271 | of A with the sign of B. For example, SIGN(-3.0,2.0) has the value 3.0. |
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272 | The problematic case is when the second argument is zero, because, on platforms |
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273 | that support IEEE arithmetic, zero is actually a signed number. |
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274 | There is a positive zero and a negative zero. |
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275 | |
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276 | In \textsc{Fortran}~90, the processor was required always to deliver a positive result for SIGN(A, B) |
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277 | if B was zero. Nevertheless, in \textsc{Fortran}~95, the processor is allowed to do the correct thing |
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278 | and deliver ABS(A) when B is a positive zero and -ABS(A) when B is a negative zero. |
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279 | This change in the specification becomes apparent only when B is of type real, and is zero, |
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280 | and the processor is capable of distinguishing between positive and negative zero, |
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281 | and B is negative real zero. Then SIGN delivers a negative result where, under \textsc{Fortran}~90 |
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282 | rules, it used to return a positive result. |
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283 | This change may be especially sensitive for the ice model, so we overwrite the intrinsinc |
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284 | function with our own function simply performing : \\ |
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285 | \verb? IF( B >= 0.e0 ) THEN ; SIGN(A,B) = ABS(A) ? \\ |
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286 | \verb? ELSE ; SIGN(A,B) =-ABS(A) ? \\ |
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287 | \verb? ENDIF ? \\ |
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288 | This feature can be found in \mdl{lib\_fortran} module and is effective when \key{nosignedzero} |
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289 | is defined. We use a CPP key as the overwritting of a intrinsic function can present |
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290 | performance issues with some computers/compilers. |
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291 | |
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292 | |
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293 | \subsection{MPP reproducibility} |
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294 | \label{MISC_glosum} |
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295 | |
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296 | The numerical reproducibility of simulations on distributed memory parallel computers |
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297 | is a critical issue. In particular, within NEMO global summation of distributed arrays |
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298 | is most susceptible to rounding errors, and their propagation and accumulation cause |
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299 | uncertainty in final simulation reproducibility on different numbers of processors. |
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300 | To avoid so, based on \citet{He_Ding_JSC01} review of different technics, |
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301 | we use a so called self-compensated summation method. The idea is to estimate |
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302 | the roundoff error, store it in a buffer, and then add it back in the next addition. |
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303 | |
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304 | Suppose we need to calculate $b = a_1 + a_2 + a_3$. The following algorithm |
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305 | will allow to split the sum in two ($sum_1 = a_{1} + a_{2}$ and $b = sum_2 = sum_1 + a_3$) |
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306 | with exactly the same rounding errors as the sum performed all at once. |
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307 | \begin{align*} |
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308 | sum_1 \ \ &= a_1 + a_2 \\ |
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309 | error_1 &= a_2 + ( a_1 - sum_1 ) \\ |
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310 | sum_2 \ \ &= sum_1 + a_3 + error_1 \\ |
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311 | error_2 &= a_3 + error_1 + ( sum_1 - sum_2 ) \\ |
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312 | b \qquad \ &= sum_2 \\ |
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313 | \end{align*} |
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314 | This feature can be found in \mdl{lib\_fortran} module and is effective when \key{mpp\_rep}. |
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315 | In that case, all calls to glob\_sum function (summation over the entire basin excluding |
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316 | duplicated rows and columns due to cyclic or north fold boundary condition as well as |
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317 | overlap MPP areas). |
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318 | Note this implementation may be sensitive to the optimization level. |
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319 | |
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320 | \subsection{MPP scalability} |
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321 | \label{MISC_mppsca} |
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322 | |
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323 | The default method of communicating values across the north-fold in distributed memory applications |
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324 | (\key{mpp\_mpi}) uses a \textsc{MPI\_ALLGATHER} function to exchange values from each processing |
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325 | region in the northern row with every other processing region in the northern row. This enables a |
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326 | global width array containing the top 4 rows to be collated on every northern row processor and then |
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327 | folded with a simple algorithm. Although conceptually simple, this "All to All" communication will |
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328 | hamper performance scalability for large numbers of northern row processors. From version 3.4 |
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329 | onwards an alternative method is available which only performs direct "Peer to Peer" communications |
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330 | between each processor and its immediate "neighbours" across the fold line. This is achieved by |
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331 | using the default \textsc{MPI\_ALLGATHER} method during initialisation to help identify the "active" |
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332 | neighbours. Stored lists of these neighbours are then used in all subsequent north-fold exchanges to |
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333 | restrict exchanges to those between associated regions. The collated global width array for each |
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334 | region is thus only partially filled but is guaranteed to be set at all the locations actually |
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335 | required by each individual for the fold operation. This alternative method should give identical |
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336 | results to the default \textsc{ALLGATHER} method and is recommended for large values of \np{jpni}. |
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337 | The new method is activated by setting \np{ln\_nnogather} to be true ({\bf nammpp}). The |
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338 | reproducibility of results using the two methods should be confirmed for each new, non-reference |
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339 | configuration. |
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340 | |
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341 | % ================================================================ |
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342 | % Model optimisation, Control Print and Benchmark |
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343 | % ================================================================ |
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344 | \section{Model Optimisation, Control Print and Benchmark} |
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345 | \label{MISC_opt} |
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346 | %--------------------------------------------namctl------------------------------------------------------- |
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347 | \namdisplay{namctl} |
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348 | %-------------------------------------------------------------------------------------------------------------- |
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349 | |
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350 | \gmcomment{why not make these bullets into subsections?} |
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351 | Options are defined through the \ngn{namctl} namelist variables. |
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352 | |
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353 | $\bullet$ Vector optimisation: |
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354 | |
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355 | \key{vectopt\_loop} enables the internal loops to collapse. This is very |
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356 | a very efficient way to increase the length of vector calculations and thus |
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357 | to speed up the model on vector computers. |
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358 | |
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359 | % Add here also one word on NPROMA technique that has been found useless, since compiler have made significant progress during the last decade. |
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360 | |
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361 | % Add also one word on NEC specific optimisation (Novercheck option for example) |
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362 | |
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363 | $\bullet$ Control print %: describe here 4 things: |
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364 | |
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365 | 1- \np{ln\_ctl} : compute and print the trends averaged over the interior domain |
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366 | in all TRA, DYN, LDF and ZDF modules. This option is very helpful when |
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367 | diagnosing the origin of an undesired change in model results. |
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368 | |
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369 | 2- also \np{ln\_ctl} but using the nictl and njctl namelist parameters to check |
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370 | the source of differences between mono and multi processor runs. |
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371 | |
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372 | 3- \key{esopa} (to be rename key\_nemo) : which is another option for model |
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373 | management. When defined, this key forces the activation of all options and |
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374 | CPP keys. For example, all tracer and momentum advection schemes are called! |
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375 | Therefore the model results have no physical meaning. |
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376 | However, this option forces both the compiler and the model to run through |
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377 | all the \textsc{Fortran} lines of the model. This allows the user to check for obvious |
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378 | compilation or execution errors with all CPP options, and errors in namelist options. |
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379 | |
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380 | 4- last digit comparison (\np{nn\_bit\_cmp}). In an MPP simulation, the computation of |
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381 | a sum over the whole domain is performed as the summation over all processors of |
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382 | each of their sums over their interior domains. This double sum never gives exactly |
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383 | the same result as a single sum over the whole domain, due to truncation differences. |
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384 | The "bit comparison" option has been introduced in order to be able to check that |
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385 | mono-processor and multi-processor runs give exactly the same results. |
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386 | %THIS is to be updated with the mpp_sum_glo introduced in v3.3 |
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387 | % nn_bit_cmp today only check that the nn_cla = 0 (no cross land advection) |
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388 | |
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389 | $\bullet$ Benchmark (\np{nn\_bench}). This option defines a benchmark run based on |
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390 | a GYRE configuration (see \S\ref{CFG_gyre}) in which the resolution remains the same |
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391 | whatever the domain size. This allows a very large model domain to be used, just by |
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392 | changing the domain size (\jp{jpiglo}, \jp{jpjglo}) and without adjusting either the time-step |
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393 | or the physical parameterisations. |
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394 | |
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395 | |
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396 | % ================================================================ |
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397 | % Elliptic solvers (SOL) |
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398 | % ================================================================ |
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399 | \section{Elliptic solvers (SOL)} |
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400 | \label{MISC_sol} |
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401 | %--------------------------------------------namdom------------------------------------------------------- |
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402 | \namdisplay{namsol} |
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403 | %-------------------------------------------------------------------------------------------------------------- |
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404 | |
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405 | When the filtered sea surface height option is used, the surface pressure gradient is |
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406 | computed in \mdl{dynspg\_flt}. The force added in the momentum equation is solved implicitely. |
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407 | It is thus solution of an elliptic equation \eqref{Eq_PE_flt} for which two solvers are available: |
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408 | a Successive-Over-Relaxation scheme (SOR) and a preconditioned conjugate gradient |
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409 | scheme(PCG) \citep{Madec_al_OM88, Madec_PhD90}. The solver is selected trough the |
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410 | the value of \np{nn\_solv} \ngn{namsol} namelist variable. |
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411 | |
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412 | The PCG is a very efficient method for solving elliptic equations on vector computers. |
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413 | It is a fast and rather easy method to use; which are attractive features for a large |
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414 | number of ocean situations (variable bottom topography, complex coastal geometry, |
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415 | variable grid spacing, open or cyclic boundaries, etc ...). It does not require |
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416 | a search for an optimal parameter as in the SOR method. However, the SOR has |
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417 | been retained because it is a linear solver, which is a very useful property when |
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418 | using the adjoint model of \NEMO. |
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419 | |
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420 | At each time step, the time derivative of the sea surface height at time step $t+1$ |
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421 | (or equivalently the divergence of the \textit{after} barotropic transport) that appears |
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422 | in the filtering forced is the solution of the elliptic equation obtained from the horizontal |
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423 | divergence of the vertical summation of \eqref{Eq_PE_flt}. |
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424 | Introducing the following coefficients: |
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425 | \begin{equation} \label{Eq_sol_matrix} |
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426 | \begin{aligned} |
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427 | &c_{i,j}^{NS} &&= {2 \rdt }^2 \; \frac{H_v (i,j) \; e_{1v} (i,j)}{e_{2v}(i,j)} \\ |
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428 | &c_{i,j}^{EW} &&= {2 \rdt }^2 \; \frac{H_u (i,j) \; e_{2u} (i,j)}{e_{1u}(i,j)} \\ |
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429 | &b_{i,j} &&= \delta_i \left[ e_{2u}M_u \right] - \delta_j \left[ e_{1v}M_v \right]\ , \\ |
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430 | \end{aligned} |
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431 | \end{equation} |
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432 | the resulting five-point finite difference equation is given by: |
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433 | \begin{equation} \label{Eq_solmat} |
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434 | \begin{split} |
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435 | c_{i+1,j}^{NS} D_{i+1,j} + \; c_{i,j+1}^{EW} D_{i,j+1} |
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436 | + c_{i,j} ^{NS} D_{i-1,j} + \; c_{i,j} ^{EW} D_{i,j-1} & \\ |
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437 | - \left(c_{i+1,j}^{NS} + c_{i,j+1}^{EW} + c_{i,j}^{NS} + c_{i,j}^{EW} \right) D_{i,j} &= b_{i,j} |
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438 | \end{split} |
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439 | \end{equation} |
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440 | \eqref{Eq_solmat} is a linear symmetric system of equations. All the elements of |
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441 | the corresponding matrix \textbf{A} vanish except those of five diagonals. With |
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442 | the natural ordering of the grid points (i.e. from west to east and from |
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443 | south to north), the structure of \textbf{A} is block-tridiagonal with |
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444 | tridiagonal or diagonal blocks. \textbf{A} is a positive-definite symmetric |
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445 | matrix of size $(jpi \cdot jpj)^2$, and \textbf{B}, the right hand side of |
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446 | \eqref{Eq_solmat}, is a vector. |
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447 | |
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448 | Note that in the linear free surface case, the depth that appears in \eqref{Eq_sol_matrix} |
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449 | does not vary with time, and thus the matrix can be computed once for all. In non-linear free surface |
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450 | (\key{vvl} defined) the matrix have to be updated at each time step. |
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451 | |
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452 | % ------------------------------------------------------------------------------------------------------------- |
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453 | % Successive Over Relaxation |
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454 | % ------------------------------------------------------------------------------------------------------------- |
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455 | \subsection{Successive Over Relaxation (\np{nn\_solv}=2, \mdl{solsor})} |
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456 | \label{MISC_solsor} |
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457 | |
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458 | Let us introduce the four cardinal coefficients: |
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459 | \begin{align*} |
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460 | a_{i,j}^S &= c_{i,j }^{NS}/d_{i,j} &\qquad a_{i,j}^W &= c_{i,j}^{EW}/d_{i,j} \\ |
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461 | a_{i,j}^E &= c_{i,j+1}^{EW}/d_{i,j} &\qquad a_{i,j}^N &= c_{i+1,j}^{NS}/d_{i,j} |
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462 | \end{align*} |
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463 | where $d_{i,j} = c_{i,j}^{NS}+ c_{i+1,j}^{NS} + c_{i,j}^{EW} + c_{i,j+1}^{EW}$ |
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464 | (i.e. the diagonal of the matrix). \eqref{Eq_solmat} can be rewritten as: |
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465 | \begin{equation} \label{Eq_solmat_p} |
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466 | \begin{split} |
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467 | a_{i,j}^{N} D_{i+1,j} +\,a_{i,j}^{E} D_{i,j+1} +\, a_{i,j}^{S} D_{i-1,j} +\,a_{i,j}^{W} D_{i,j-1} - D_{i,j} = \tilde{b}_{i,j} |
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468 | \end{split} |
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469 | \end{equation} |
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470 | with $\tilde b_{i,j} = b_{i,j}/d_{i,j}$. \eqref{Eq_solmat_p} is the equation actually solved |
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471 | with the SOR method. This method used is an iterative one. Its algorithm can be |
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472 | summarised as follows (see \citet{Haltiner1980} for a further discussion): |
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473 | |
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474 | initialisation (evaluate a first guess from previous time step computations) |
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475 | \begin{equation} |
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476 | D_{i,j}^0 = 2 \, D_{i,j}^t - D_{i,j}^{t-1} |
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477 | \end{equation} |
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478 | iteration $n$, from $n=0$ until convergence, do : |
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479 | \begin{equation} \label{Eq_sor_algo} |
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480 | \begin{split} |
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481 | R_{i,j}^n = &a_{i,j}^{N} D_{i+1,j}^n +\,a_{i,j}^{E} D_{i,j+1} ^n |
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482 | +\, a_{i,j}^{S} D_{i-1,j} ^{n+1}+\,a_{i,j}^{W} D_{i,j-1} ^{n+1} |
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483 | - D_{i,j}^n - \tilde{b}_{i,j} \\ |
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484 | D_{i,j} ^{n+1} = &D_{i,j} ^{n} + \omega \;R_{i,j}^n |
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485 | \end{split} |
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486 | \end{equation} |
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487 | where \textit{$\omega $ }satisfies $1\leq \omega \leq 2$. An optimal value exists for |
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488 | \textit{$\omega$} which significantly accelerates the convergence, but it has to be |
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489 | adjusted empirically for each model domain (except for a uniform grid where an |
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490 | analytical expression for \textit{$\omega$} can be found \citep{Richtmyer1967}). |
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491 | The value of $\omega$ is set using \np{rn\_sor}, a \textbf{namelist} parameter. |
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492 | The convergence test is of the form: |
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493 | \begin{equation} |
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494 | \delta = \frac{\sum\limits_{i,j}{R_{i,j}^n}{R_{i,j}^n}} |
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495 | {\sum\limits_{i,j}{ \tilde{b}_{i,j}^n}{\tilde{b}_{i,j}^n}} \leq \epsilon |
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496 | \end{equation} |
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497 | where $\epsilon$ is the absolute precision that is required. It is recommended |
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498 | that a value smaller or equal to $10^{-6}$ is used for $\epsilon$ since larger |
---|
499 | values may lead to numerically induced basin scale barotropic oscillations. |
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500 | The precision is specified by setting \np{rn\_eps} (\textbf{namelist} parameter). |
---|
501 | In addition, two other tests are used to halt the iterative algorithm. They involve |
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502 | the number of iterations and the modulus of the right hand side. If the former |
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503 | exceeds a specified value, \np{nn\_max} (\textbf{namelist} parameter), |
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504 | or the latter is greater than $10^{15}$, the whole model computation is stopped |
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505 | and the last computed time step fields are saved in a abort.nc NetCDF file. |
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506 | In both cases, this usually indicates that there is something wrong in the model |
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507 | configuration (an error in the mesh, the initial state, the input forcing, |
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508 | or the magnitude of the time step or of the mixing coefficients). A typical value of |
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509 | $nn\_max$ is a few hundred when $\epsilon = 10^{-6}$, increasing to a few |
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510 | thousand when $\epsilon = 10^{-12}$. |
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511 | The vectorization of the SOR algorithm is not straightforward. The scheme |
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512 | contains two linear recurrences on $i$ and $j$. This inhibits the vectorisation. |
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513 | \eqref{Eq_sor_algo} can be been rewritten as: |
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514 | \begin{equation} |
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515 | \begin{split} |
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516 | R_{i,j}^n |
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517 | = &a_{i,j}^{N} D_{i+1,j}^n +\,a_{i,j}^{E} D_{i,j+1} ^n |
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518 | +\,a_{i,j}^{S} D_{i-1,j} ^{n}+\,_{i,j}^{W} D_{i,j-1} ^{n} - D_{i,j}^n - \tilde{b}_{i,j} \\ |
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519 | R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{S}\; R_{i,j-1}^n \\ |
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520 | R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{W}\; R_{i-1,j}^n |
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521 | \end{split} |
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522 | \end{equation} |
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523 | This technique slightly increases the number of iteration required to reach the convergence, |
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524 | but this is largely compensated by the gain obtained by the suppression of the recurrences. |
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525 | |
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526 | Another technique have been chosen, the so-called red-black SOR. It consist in solving successively |
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527 | \eqref{Eq_sor_algo} for odd and even grid points. It also slightly reduced the convergence rate |
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528 | but allows the vectorisation. In addition, and this is the reason why it has been chosen, it is able to handle the north fold boundary condition used in ORCA configuration ($i.e.$ tri-polar global ocean mesh). |
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529 | |
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530 | The SOR method is very flexible and can be used under a wide range of conditions, |
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531 | including irregular boundaries, interior boundary points, etc. Proofs of convergence, etc. |
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532 | may be found in the standard numerical methods texts for partial differential equations. |
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533 | |
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534 | % ------------------------------------------------------------------------------------------------------------- |
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535 | % Preconditioned Conjugate Gradient |
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536 | % ------------------------------------------------------------------------------------------------------------- |
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537 | \subsection{Preconditioned Conjugate Gradient (\np{nn\_solv}=1, \mdl{solpcg}) } |
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538 | \label{MISC_solpcg} |
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539 | |
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540 | \textbf{A} is a definite positive symmetric matrix, thus solving the linear |
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541 | system \eqref{Eq_solmat} is equivalent to the minimisation of a quadratic |
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542 | functional: |
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543 | \begin{equation*} |
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544 | \textbf{Ax} = \textbf{b} \leftrightarrow \textbf{x} =\text{inf}_{y} \,\phi (\textbf{y}) |
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545 | \quad , \qquad |
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546 | \phi (\textbf{y}) = 1/2 \langle \textbf{Ay},\textbf{y}\rangle - \langle \textbf{b},\textbf{y} \rangle |
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547 | \end{equation*} |
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548 | where $\langle , \rangle$ is the canonical dot product. The idea of the |
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549 | conjugate gradient method is to search for the solution in the following |
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550 | iterative way: assuming that $\textbf{x}^n$ has been obtained, $\textbf{x}^{n+1}$ |
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551 | is found from $\textbf {x}^{n+1}={\textbf {x}}^n+\alpha^n{\textbf {d}}^n$ which satisfies: |
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552 | \begin{equation*} |
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553 | {\textbf{ x}}^{n+1}=\text{inf} _{{\textbf{ y}}={\textbf{ x}}^n+\alpha^n \,{\textbf{ d}}^n} \,\phi ({\textbf{ y}})\;\;\Leftrightarrow \;\;\frac{d\phi }{d\alpha}=0 |
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554 | \end{equation*} |
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555 | and expressing $\phi (\textbf{y})$ as a function of \textit{$\alpha $}, we obtain the |
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556 | value that minimises the functional: |
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557 | \begin{equation*} |
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558 | \alpha ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle / \langle {\textbf{ A d}^n, \textbf{d}^n} \rangle |
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559 | \end{equation*} |
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560 | where $\textbf{r}^n = \textbf{b}-\textbf{A x}^n = \textbf{A} (\textbf{x}-\textbf{x}^n)$ |
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561 | is the error at rank $n$. The descent vector $\textbf{d}^n$ s chosen to be dependent |
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562 | on the error: $\textbf{d}^n = \textbf{r}^n + \beta^n \,\textbf{d}^{n-1}$. $\beta ^n$ |
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563 | is searched such that the descent vectors form an orthogonal basis for the dot |
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564 | product linked to \textbf{A}. Expressing the condition |
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565 | $\langle \textbf{A d}^n, \textbf{d}^{n-1} \rangle = 0$ the value of $\beta ^n$ is found: |
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566 | $\beta ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle / \langle {\textbf{r}^{n-1}, \textbf{r}^{n-1}} \rangle$. |
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567 | As a result, the errors $ \textbf{r}^n$ form an orthogonal |
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568 | base for the canonic dot product while the descent vectors $\textbf{d}^n$ form |
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569 | an orthogonal base for the dot product linked to \textbf{A}. The resulting |
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570 | algorithm is thus the following one: |
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571 | |
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572 | initialisation : |
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573 | \begin{equation*} |
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574 | \begin{split} |
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575 | \textbf{x}^0 &= D_{i,j}^0 = 2 D_{i,j}^t - D_{i,j}^{t-1} \quad, \text{the initial guess } \\ |
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576 | \textbf{r}^0 &= \textbf{d}^0 = \textbf{b} - \textbf{A x}^0 \\ |
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577 | \gamma_0 &= \langle{ \textbf{r}^0 , \textbf{r}^0} \rangle |
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578 | \end{split} |
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579 | \end{equation*} |
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580 | |
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581 | iteration $n,$ from $n=0$ until convergence, do : |
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582 | \begin{equation} |
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583 | \begin{split} |
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584 | \text{z}^n& = \textbf{A d}^n \\ |
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585 | \alpha_n &= \gamma_n / \langle{ \textbf{z}^n , \textbf{d}^n} \rangle \\ |
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586 | \textbf{x}^{n+1} &= \textbf{x}^n + \alpha_n \,\textbf{d}^n \\ |
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587 | \textbf{r}^{n+1} &= \textbf{r}^n - \alpha_n \,\textbf{z}^n \\ |
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588 | \gamma_{n+1} &= \langle{ \textbf{r}^{n+1} , \textbf{r}^{n+1}} \rangle \\ |
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589 | \beta_{n+1} &= \gamma_{n+1}/\gamma_{n} \\ |
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590 | \textbf{d}^{n+1} &= \textbf{r}^{n+1} + \beta_{n+1}\; \textbf{d}^{n}\\ |
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591 | \end{split} |
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592 | \end{equation} |
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593 | |
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594 | |
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595 | The convergence test is: |
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596 | \begin{equation} |
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597 | \delta = \gamma_{n}\; / \langle{ \textbf{b} , \textbf{b}} \rangle \leq \epsilon |
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598 | \end{equation} |
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599 | where $\epsilon $ is the absolute precision that is required. As for the SOR algorithm, |
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600 | the whole model computation is stopped when the number of iterations, \np{nn\_max}, or |
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601 | the modulus of the right hand side of the convergence equation exceeds a |
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602 | specified value (see \S\ref{MISC_solsor} for a further discussion). The required |
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603 | precision and the maximum number of iterations allowed are specified by setting |
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604 | \np{rn\_eps} and \np{nn\_max} (\textbf{namelist} parameters). |
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605 | |
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606 | It can be demonstrated that the above algorithm is optimal, provides the exact |
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607 | solution in a number of iterations equal to the size of the matrix, and that |
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608 | the convergence rate is faster as the matrix is closer to the identity matrix, |
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609 | $i.e.$ its eigenvalues are closer to 1. Therefore, it is more efficient to solve |
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610 | a better conditioned system which has the same solution. For that purpose, |
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611 | we introduce a preconditioning matrix \textbf{Q} which is an approximation |
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612 | of \textbf{A} but much easier to invert than \textbf{A}, and solve the system: |
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613 | \begin{equation} \label{Eq_pmat} |
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614 | \textbf{Q}^{-1} \textbf{A x} = \textbf{Q}^{-1} \textbf{b} |
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615 | \end{equation} |
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616 | |
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617 | The same algorithm can be used to solve \eqref{Eq_pmat} if instead of the |
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618 | canonical dot product the following one is used: |
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619 | ${\langle{ \textbf{a} , \textbf{b}} \rangle}_Q = \langle{ \textbf{a} , \textbf{Q b}} \rangle$, and |
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620 | if $\textbf{\~{b}} = \textbf{Q}^{-1}\;\textbf{b}$ and $\textbf{\~{A}} = \textbf{Q}^{-1}\;\textbf{A}$ |
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621 | are substituted to \textbf{b} and \textbf{A} \citep{Madec_al_OM88}. |
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622 | In \NEMO, \textbf{Q} is chosen as the diagonal of \textbf{ A}, i.e. the simplest form for |
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623 | \textbf{Q} so that it can be easily inverted. In this case, the discrete formulation of |
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624 | \eqref{Eq_pmat} is in fact given by \eqref{Eq_solmat_p} and thus the matrix and |
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625 | right hand side are computed independently from the solver used. |
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626 | |
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627 | % ================================================================ |
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628 | |
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629 | |
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630 | |
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631 | |
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632 | |
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633 | |
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634 | |
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635 | |
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636 | |
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637 | |
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638 | |
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639 | |
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