1 | % ================================================================ |
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2 | % Chapter Ñ 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 (search for |
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36 | \key{orca\_r2} or \key{orca\_r05} in the code). |
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37 | |
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38 | % ------------------------------------------------------------------------------------------------------------- |
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39 | % Hand made geometry changes |
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40 | % ------------------------------------------------------------------------------------------------------------- |
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41 | \subsection{Hand made geometry changes} |
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42 | \label{MISC_strait_hand} |
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43 | |
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44 | $\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement |
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45 | with the true mean width of the strait. (Fig.~\ref{Fig_MISC_strait_hand}). |
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46 | This technique is sometime called "partially open face" or "partially closed cells". |
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47 | The key issue here is only to reduce the faces of $T$-cell ($i.e.$ change the value |
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48 | of the horizontal scale factors at $u$- or $v$-point) but not the volume of the $T$-cell. |
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49 | Indeed, reducing the volume of strait $T$-cell can easily produce a numerical |
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50 | instability at that grid point that would require a reduction of the model time step. |
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51 | The changes associated with strait management are done in \mdl{domhgr}, |
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52 | just after the definition or reading of the horizontal scale factors. |
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53 | |
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54 | $\bullet$ increase of the viscous boundary layer thickness by local increase of the |
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55 | fmask value at the coast (Fig.~\ref{Fig_MISC_strait_hand}). This is done in |
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56 | \mdl{dommsk} together with the setting of the coastal value of fmask |
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57 | (see Section \ref{LBC_coast}) |
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58 | |
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59 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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60 | \begin{figure}[!tbp] \label{Fig_MISC_strait_hand} \begin{center} |
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61 | \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar.pdf} |
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62 | \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar2.pdf} |
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63 | \caption {Example of the Gibraltar strait defined in a $1\deg \times 1\deg$ mesh. |
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64 | \textit{Top}: using partially open cells. The meridional scale factor at $v$-point |
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65 | is reduced on both sides of the strait to account for the real width of the strait |
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66 | (about 20 km). Note that the scale factors of the strait $T$-point remains unchanged. |
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67 | \textit{Bottom}: using viscous boundary layers. The four fmask parameters |
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68 | along the strait coastlines are set to a value larger than 4, $i.e.$ "strong" no-slip |
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69 | case (see Fig.\ref{Fig_LBC_shlat}) creating a large viscous boundary layer |
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70 | that allows a reduced transport through the strait.} |
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71 | \end{center} \end{figure} |
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72 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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73 | |
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74 | % ------------------------------------------------------------------------------------------------------------- |
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75 | % Cross Land Advection |
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76 | % ------------------------------------------------------------------------------------------------------------- |
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77 | \subsection{Cross Land Advection (\mdl{tracla})} |
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78 | \label{MISC_strait_cla} |
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79 | %--------------------------------------------namcla-------------------------------------------------------- |
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80 | \namdisplay{namcla} |
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81 | %-------------------------------------------------------------------------------------------------------------- |
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82 | |
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83 | \colorbox{yellow}{Add a short description of CLA staff here or in lateral boundary condition chapter?} |
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84 | |
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85 | %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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86 | |
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87 | % ================================================================ |
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88 | % Closed seas |
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89 | % ================================================================ |
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90 | \section{Closed seas (\mdl{closea})} |
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91 | \label{MISC_closea} |
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92 | |
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93 | \colorbox{yellow}{Add here a short description of the way closed seas are managed} |
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94 | |
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95 | |
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96 | % ================================================================ |
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97 | % Sub-Domain Functionality (\textit{nizoom, njzoom}, namelist parameters) |
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98 | % ================================================================ |
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99 | \section{Sub-Domain Functionality (\jp{jpizoom}, \jp{jpjzoom})} |
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100 | \label{MISC_zoom} |
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101 | |
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102 | The sub-domain functionality, also improperly called the zoom option |
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103 | (improperly because it is not associated with a change in model resolution) |
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104 | is a quite simple function that allows a simulation over a sub-domain of an |
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105 | already defined configuration ($i.e.$ without defining a new mesh, initial |
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106 | state and forcings). This option can be useful for testing the user settings |
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107 | of surface boundary conditions, or the initial ocean state of a huge ocean |
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108 | model configuration while having a small computer memory requirement. |
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109 | It can also be used to easily test specific physics in a sub-domain (for example, |
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110 | see \citep{Madec_al_JPO96} for a test of the coupling used in the global ocean |
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111 | version of OPA between sea-ice and ocean model over the Arctic or Antarctic |
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112 | ocean, using a sub-domain). In the standard model, this option does not |
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113 | include any specific treatment for the ocean boundaries of the sub-domain: |
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114 | they are considered as artificial vertical walls. Nevertheless, it is quite easy |
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115 | to add a restoring term toward a climatology in the vicinity of such boundaries |
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116 | (see \S\ref{TRA_dmp}). |
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117 | |
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118 | In order to easily define a sub-domain over which the computation can be |
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119 | performed, the dimension of all input arrays (ocean mesh, bathymetry, |
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120 | forcing, initial state, ...) are defined as \jp{jpidta}, \jp{jpjdta} and \jp{jpkdta} |
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121 | (\mdl{par\_oce} module), while the computational domain is defined through |
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122 | \jp{jpiglo}, \jp{jpjglo} and \jp{jpk} (\mdl{par\_oce} module). When running the |
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123 | model over the whole domain, the user sets \jp{jpiglo}=\jp{jpidta} \jp{jpjglo}=\jp{jpjdta} |
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124 | and \jp{jpk}=\jp{jpkdta}. When running the model over a sub-domain, the user |
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125 | has to provide the size of the sub-domain, (\jp{jpiglo}, \jp{jpjglo}, \jp{jpkglo}), |
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126 | and the indices of the south western corner as \jp{jpizoom} and \jp{jpjzoom} in |
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127 | the \mdl{par\_oce} module (Fig.~\ref{Fig_LBC_zoom}). |
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128 | |
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129 | Note that a third set of dimensions exist, \jp{jpi}, \jp{jpj} and \jp{jpk} which is |
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130 | actually used to perform the computation. It is set by default to \jp{jpi}=\jp{jpjglo} |
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131 | and \jp{jpj}=\jp{jpjglo}, except for massively parallel computing where the |
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132 | computational domain is laid out on local processor memories following a 2D |
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133 | horizontal splitting. % (see {\S}IV.2-c) ref to the section to be updated |
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134 | |
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135 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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136 | \begin{figure}[!ht] \label{Fig_LBC_zoom} \begin{center} |
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137 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_zoom.pdf} |
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138 | \caption {Position of a model domain compared to the data input domain when the zoom functionality is used.} |
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139 | \end{center} \end{figure} |
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140 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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141 | |
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142 | |
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143 | % ================================================================ |
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144 | % 1D model functionality |
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145 | % ================================================================ |
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146 | \section{Water column model: 1D model (\key{c1d})} |
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147 | \label{MISC_1D} |
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148 | |
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149 | The 1D model option simulates a stand alone water column within the 3D \NEMO system. |
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150 | It can be applied to the ocean alone or to the ocean-ice system and can include passive tracers |
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151 | or a biogeochemical model. It is set up by defining the \key{c1d} CPP key. |
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152 | The 1D model is a very useful tool |
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153 | \textit{(a)} to learn about the physics and numerical treatment of vertical mixing processes ; |
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154 | \textit{(b)} to investigate suitable parameterisations of unresolved turbulence (wind steering, |
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155 | langmuir circulation, skin layers) ; |
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156 | \textit{(c)} to compare the behaviour of different vertical mixing schemes ; |
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157 | \textit{(d)} to perform sensitivity studies on the vertical diffusion at a particular point of an ocean domain ; |
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158 | \textit{(d)} to produce extra diagnostics, without the large memory requirement of the full 3D model. |
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159 | |
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160 | The methodology is based on the use of the zoom functionality over the smallest possible |
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161 | domain : a 3 x 3 domain centred on the grid point of interest (see \S\ref{MISC_zoom}), |
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162 | with some extra routines. There is no need to define a new mesh, bathymetry, |
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163 | initial state or forcing, since the 1D model will use those of the configuration it is a zoom of. |
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164 | The chosen grid point is set in par\_oce.F90 module by setting the jpizoom and jpjzoom |
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165 | parameters to the indices of the location of the chosen grid point. |
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166 | |
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167 | % ================================================================ |
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168 | % Accelerating the Convergence |
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169 | % ================================================================ |
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170 | \section{Accelerating the Convergence (\np{nn\_acc} = 1)} |
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171 | \label{MISC_acc} |
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172 | %--------------------------------------------namdom------------------------------------------------------- |
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173 | \namdisplay{namdom} |
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174 | %-------------------------------------------------------------------------------------------------------------- |
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175 | |
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176 | Searching an equilibrium state with an global ocean model requires a very long time |
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177 | integration period (a few thousand years for a global model). Due to the size of |
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178 | the time step required for numerical stability (less than a few hours), |
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179 | this usually requires a large elapsed time. In order to overcome this problem, |
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180 | \citet{Bryan1984} introduces a technique that is intended to accelerate |
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181 | the spin up to equilibrium. It uses a larger time step in |
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182 | the tracer evolution equations than in the momentum evolution |
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183 | equations. It does not affect the equilibrium solution but modifies the |
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184 | trajectory to reach it. |
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185 | |
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186 | The acceleration of convergence option is used when \np{nn\_acc}=1. In that case, |
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187 | $\rdt=rn\_rdt$ is the time step of dynamics while $\widetilde{\rdt}=rdttra$ is the |
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188 | tracer time-step. the former is set from the \np{rn\_rdt} namelist parameter while the latter |
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189 | is computed using a hyperbolic tangent profile and the following namelist parameters : |
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190 | \np{rn\_rdtmin}, \np{rn\_rdtmax} and \np{rn\_rdth}. Those three parameters correspond |
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191 | to the surface value the deep ocean value and the depth at which the transition occurs, respectively. |
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192 | The set of prognostic equations to solve becomes: |
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193 | \begin{equation} \label{Eq_acc} |
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194 | \begin{split} |
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195 | \frac{\partial \textbf{U}_h }{\partial t} |
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196 | &\equiv \frac{\textbf{U}_h ^{t+1}-\textbf{U}_h^{t-1} }{2\rdt} = \ldots \\ |
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197 | \frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ |
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198 | \frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ |
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199 | \end{split} |
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200 | \end{equation} |
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201 | |
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202 | \citet{Bryan1984} has examined the consequences of this distorted physics. |
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203 | Free waves have a slower phase speed, their meridional structure is slightly |
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204 | modified, and the growth rate of baroclinically unstable waves is reduced |
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205 | but with a wider range of instability. This technique is efficient for |
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206 | searching for an equilibrium state in coarse resolution models. However its |
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207 | application is not suitable for many oceanic problems: it cannot be used for |
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208 | transient or time evolving problems (in particular, it is very questionable |
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209 | to use this technique when there is a seasonal cycle in the forcing fields), |
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210 | and it cannot be used in high-resolution models where baroclinically |
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211 | unstable processes are important. Moreover, the vertical variation of |
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212 | $\widetilde{ \rdt}$ implies that the heat and salt contents are no longer |
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213 | conserved due to the vertical coupling of the ocean level through both |
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214 | advection and diffusion. Therefore \np{rn\_rdtmin} = \np{rn\_rdtmax} should be |
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215 | a more clever choice. |
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216 | |
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217 | % ================================================================ |
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218 | % Model optimisation, Control Print and Benchmark |
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219 | % ================================================================ |
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220 | \section{Model Optimisation, Control Print and Benchmark} |
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221 | \label{MISC_opt} |
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222 | %--------------------------------------------namctl------------------------------------------------------- |
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223 | \namdisplay{namctl} |
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224 | %-------------------------------------------------------------------------------------------------------------- |
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225 | |
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226 | % \gmcomment{why not make these bullets into subsections?} |
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227 | |
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228 | |
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229 | $\bullet$ Vector optimisation: |
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230 | |
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231 | \key{vectopt\_loop} enables the internal loops to collapse. This is very |
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232 | a very efficient way to increase the length of vector calculations and thus |
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233 | to speed up the model on vector computers. |
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234 | |
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235 | % 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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236 | |
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237 | % Add also one word on NEC specific optimisation (Novercheck option for example) |
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238 | |
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239 | $\bullet$ Control print %: describe here 4 things: |
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240 | |
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241 | 1- \np{ln\_ctl} : compute and print the trends averaged over the interior domain |
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242 | in all TRA, DYN, LDF and ZDF modules. This option is very helpful when |
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243 | diagnosing the origin of an undesired change in model results. |
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244 | |
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245 | 2- also \np{ln\_ctl} but using the nictl and njctl namelist parameters to check |
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246 | the source of differences between mono and multi processor runs. |
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247 | |
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248 | 3- \key{esopa} (to be rename key\_nemo) : which is another option for model |
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249 | management. When defined, this key forces the activation of all options and |
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250 | CPP keys. For example, all tracer and momentum advection schemes are called! |
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251 | Therefore the model results have no physical meaning. |
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252 | However, this option forces both the compiler and the model to run through |
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253 | all the \textsc{Fortran} lines of the model. This allows the user to check for obvious |
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254 | compilation or execution errors with all CPP options, and errors in namelist options. |
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255 | |
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256 | 4- last digit comparison (\np{nn\_bit\_cmp}). In an MPP simulation, the computation of |
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257 | a sum over the whole domain is performed as the summation over all processors of |
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258 | each of their sums over their interior domains. This double sum never gives exactly |
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259 | the same result as a single sum over the whole domain, due to truncation differences. |
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260 | The "bit comparison" option has been introduced in order to be able to check that |
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261 | mono-processor and multi-processor runs give exactly the same results. |
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262 | |
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263 | $\bullet$ Benchmark (\np{nn\_bench}). This option defines a benchmark run based on |
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264 | a GYRE configuration in which the resolution remains the same whatever the domain |
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265 | size. This allows a very large model domain to be used, just by changing the domain |
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266 | size (\jp{jpiglo}, \jp{jpjglo}) and without adjusting either the time-step or the physical |
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267 | parameterisations. |
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268 | |
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269 | |
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270 | % ================================================================ |
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271 | % Elliptic solvers (SOL) |
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272 | % ================================================================ |
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273 | \section{Elliptic solvers (SOL)} |
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274 | \label{MISC_sol} |
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275 | %--------------------------------------------namdom------------------------------------------------------- |
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276 | \namdisplay{namsol} |
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277 | %-------------------------------------------------------------------------------------------------------------- |
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278 | |
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279 | When the filtered sea surface height option is used, the surface pressure gradient is |
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280 | computed in \mdl{dynspg\_flt}. The force added in the momentum equation is solved implicitely. |
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281 | It is thus solution of an elliptic equation \eqref{Eq_PE_flt} for which two solvers are available: |
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282 | a Successive-Over-Relaxation scheme (SOR) and a preconditioned conjugate gradient |
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283 | scheme(PCG) \citep{Madec_al_OM88, Madec_PhD90}. The solver is selected trough the |
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284 | the value of \np{nn\_solv} (namelist parameter). |
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285 | |
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286 | The PCG is a very efficient method for solving elliptic equations on vector computers. |
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287 | It is a fast and rather easy method to use; which are attractive features for a large |
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288 | number of ocean situations (variable bottom topography, complex coastal geometry, |
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289 | variable grid spacing, islands, open or cyclic boundaries, etc ...). It does not require |
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290 | a search for an optimal parameter as in the SOR method. However, the SOR has |
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291 | been retained because it is a linear solver, which is a very useful property when |
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292 | using the adjoint model of \NEMO. |
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293 | |
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294 | At each time step, the time derivative of the sea surface height at time step $t+1$ |
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295 | (or equivalently the divergence of the \textit{after} barotropic transport) that appears |
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296 | in the filtering forced is the solution of the elliptic equation obtained from the horizontal |
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297 | divergence of the vertical summation of \eqref{Eq_PE_flt}. |
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298 | Introducing the following coefficients: |
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299 | \begin{equation} \label{Eq_sol_matrix} |
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300 | \begin{aligned} |
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301 | &c_{i,j}^{NS} &&= {2 \rdt }^2 \; \frac{H_v (i,j) \; e_{1v} (i,j)}{e_{2v}(i,j)} \\ |
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302 | &c_{i,j}^{EW} &&= {2 \rdt }^2 \; \frac{H_u (i,j) \; e_{2u} (i,j)}{e_{1u}(i,j)} \\ |
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303 | &b_{i,j} &&= \delta_i \left[ e_{2u}M_u \right] - \delta_j \left[ e_{1v}M_v \right]\ , \\ |
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304 | \end{aligned} |
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305 | \end{equation} |
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306 | the five-point finite difference equation \eqref{Eq_psi_total} can be rewritten as: |
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307 | \begin{equation} \label{Eq_solmat} |
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308 | \begin{split} |
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309 | c_{i+1,j}^{NS} D_{i+1,j} + \; c_{i,j+1}^{EW} D_{i,j+1} |
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310 | + c_{i,j} ^{NS} D_{i-1,j} + \; c_{i,j} ^{EW} D_{i,j-1} & \\ |
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311 | - \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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312 | \end{split} |
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313 | \end{equation} |
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314 | \eqref{Eq_solmat} is a linear symmetric system of equations. All the elements of |
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315 | the corresponding matrix \textbf{A} vanish except those of five diagonals. With |
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316 | the natural ordering of the grid points (i.e. from west to east and from |
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317 | south to north), the structure of \textbf{A} is block-tridiagonal with |
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318 | tridiagonal or diagonal blocks. \textbf{A} is a positive-definite symmetric |
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319 | matrix of size $(jpi \cdot jpj)^2$, and \textbf{B}, the right hand side of |
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320 | \eqref{Eq_solmat}, is a vector. |
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321 | |
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322 | Note that in the linear free surface case, the depth that appears in \eqref{Eq_sol_matrix} |
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323 | does not vary with time, and thus the matrix can be computed once for all. In non-linear free surface |
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324 | (\key{vvl} defined) the matrix have to be updated at each time step. |
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325 | |
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326 | % ------------------------------------------------------------------------------------------------------------- |
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327 | % Successive Over Relaxation |
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328 | % ------------------------------------------------------------------------------------------------------------- |
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329 | \subsection{Successive Over Relaxation (\np{nn\_solv}=2, \mdl{solsor})} |
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330 | \label{MISC_solsor} |
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331 | |
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332 | Let us introduce the four cardinal coefficients: |
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333 | \begin{align*} |
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334 | 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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335 | 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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336 | \end{align*} |
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337 | 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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338 | (i.e. the diagonal of the matrix). \eqref{Eq_solmat} can be rewritten as: |
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339 | \begin{equation} \label{Eq_solmat_p} |
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340 | \begin{split} |
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341 | 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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342 | \end{split} |
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343 | \end{equation} |
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344 | with $\tilde b_{i,j} = b_{i,j}/d_{i,j}$. \eqref{Eq_solmat_p} is the equation actually solved |
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345 | with the SOR method. This method used is an iterative one. Its algorithm can be |
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346 | summarised as follows (see \citet{Haltiner1980} for a further discussion): |
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347 | |
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348 | initialisation (evaluate a first guess from previous time step computations) |
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349 | \begin{equation} |
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350 | D_{i,j}^0 = 2 \, D_{i,j}^t - D_{i,j}^{t-1} |
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351 | \end{equation} |
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352 | iteration $n$, from $n=0$ until convergence, do : |
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353 | \begin{equation} \label{Eq_sor_algo} |
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354 | \begin{split} |
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355 | 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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356 | +\, a_{i,j}^{S} D_{i-1,j} ^{n+1}+\,a_{i,j}^{W} D_{i,j-1} ^{n+1} |
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357 | - D_{i,j}^n - \tilde{b}_{i,j} \\ |
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358 | D_{i,j} ^{n+1} = &D_{i,j} ^{n} + \omega \;R_{i,j}^n |
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359 | \end{split} |
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360 | \end{equation} |
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361 | where \textit{$\omega $ }satisfies $1\leq \omega \leq 2$. An optimal value exists for |
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362 | \textit{$\omega$} which significantly accelerates the convergence, but it has to be |
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363 | adjusted empirically for each model domain (except for a uniform grid where an |
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364 | analytical expression for \textit{$\omega$} can be found \citep{Richtmyer1967}). |
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365 | The value of $\omega$ is set using \np{rn\_sor}, a \textbf{namelist} parameter. |
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366 | The convergence test is of the form: |
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367 | \begin{equation} |
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368 | \delta = \frac{\sum\limits_{i,j}{R_{i,j}^n}{R_{i,j}^n}} |
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369 | {\sum\limits_{i,j}{ \tilde{b}_{i,j}^n}{\tilde{b}_{i,j}^n}} \leq \epsilon |
---|
370 | \end{equation} |
---|
371 | where $\epsilon$ is the absolute precision that is required. It is recommended |
---|
372 | that a value smaller or equal to $10^{-6}$ is used for $\epsilon$ since larger |
---|
373 | values may lead to numerically induced basin scale barotropic oscillations. |
---|
374 | The precision is specified by setting \np{rn\_eps} (\textbf{namelist} parameter). |
---|
375 | In addition, two other tests are used to halt the iterative algorithm. They involve |
---|
376 | the number of iterations and the modulus of the right hand side. If the former |
---|
377 | exceeds a specified value, \np{nn\_max} (\textbf{namelist} parameter), |
---|
378 | or the latter is greater than $10^{15}$, the whole model computation is stopped |
---|
379 | and the last computed time step fields are saved in a abort.nc NetCDF file. |
---|
380 | In both cases, this usually indicates that there is something wrong in the model |
---|
381 | configuration (an error in the mesh, the initial state, the input forcing, |
---|
382 | or the magnitude of the time step or of the mixing coefficients). A typical value of |
---|
383 | $nn\_max$ is a few hundred when $\epsilon = 10^{-6}$, increasing to a few |
---|
384 | thousand when $\epsilon = 10^{-12}$. |
---|
385 | The vectorization of the SOR algorithm is not straightforward. The scheme |
---|
386 | contains two linear recurrences on $i$ and $j$. This inhibits the vectorisation. |
---|
387 | \eqref{Eq_sor_algo} can be been rewritten as: |
---|
388 | \begin{equation} |
---|
389 | \begin{split} |
---|
390 | R_{i,j}^n |
---|
391 | = &a_{i,j}^{N} D_{i+1,j}^n +\,a_{i,j}^{E} D_{i,j+1} ^n |
---|
392 | +\,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} \\ |
---|
393 | R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{S}\; R_{i,j-1}^n \\ |
---|
394 | R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{W}\; R_{i-1,j}^n |
---|
395 | \end{split} |
---|
396 | \end{equation} |
---|
397 | This technique slightly increases the number of iteration required to reach the convergence, |
---|
398 | but this is largely compensated by the gain obtained by the suppression of the recurrences. |
---|
399 | |
---|
400 | Another technique have been chosen, the so-called red-black SOR. It consist in solving successively |
---|
401 | \eqref{Eq_sor_algo} for odd and even grid points. It also slightly reduced the convergence rate |
---|
402 | 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). |
---|
403 | |
---|
404 | The SOR method is very flexible and can be used under a wide range of conditions, |
---|
405 | including irregular boundaries, interior boundary points, etc. Proofs of convergence, etc. |
---|
406 | may be found in the standard numerical methods texts for partial differential equations. |
---|
407 | |
---|
408 | % ------------------------------------------------------------------------------------------------------------- |
---|
409 | % Preconditioned Conjugate Gradient |
---|
410 | % ------------------------------------------------------------------------------------------------------------- |
---|
411 | \subsection{Preconditioned Conjugate Gradient (\np{nn\_solv}=1, \mdl{solpcg}) } |
---|
412 | \label{MISC_solpcg} |
---|
413 | |
---|
414 | \textbf{A} is a definite positive symmetric matrix, thus solving the linear |
---|
415 | system \eqref{Eq_solmat} is equivalent to the minimisation of a quadratic |
---|
416 | functional: |
---|
417 | \begin{equation*} |
---|
418 | \textbf{Ax} = \textbf{b} \leftrightarrow \textbf{x} =\text{inf}_{y} \,\phi (\textbf{y}) |
---|
419 | \quad , \qquad |
---|
420 | \phi (\textbf{y}) = 1/2 \langle \textbf{Ay},\textbf{y}\rangle - \langle \textbf{b},\textbf{y} \rangle |
---|
421 | \end{equation*} |
---|
422 | where $\langle , \rangle$ is the canonical dot product. The idea of the |
---|
423 | conjugate gradient method is to search for the solution in the following |
---|
424 | iterative way: assuming that $\textbf{x}^n$ has been obtained, $\textbf{x}^{n+1}$ |
---|
425 | is found from $\textbf {x}^{n+1}={\textbf {x}}^n+\alpha^n{\textbf {d}}^n$ which satisfies: |
---|
426 | \begin{equation*} |
---|
427 | {\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 |
---|
428 | \end{equation*} |
---|
429 | and expressing $\phi (\textbf{y})$ as a function of \textit{$\alpha $}, we obtain the |
---|
430 | value that minimises the functional: |
---|
431 | \begin{equation*} |
---|
432 | \alpha ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle / \langle {\textbf{ A d}^n, \textbf{d}^n} \rangle |
---|
433 | \end{equation*} |
---|
434 | where $\textbf{r}^n = \textbf{b}-\textbf{A x}^n = \textbf{A} (\textbf{x}-\textbf{x}^n)$ |
---|
435 | is the error at rank $n$. The descent vector $\textbf{d}^n$ s chosen to be dependent |
---|
436 | on the error: $\textbf{d}^n = \textbf{r}^n + \beta^n \,\textbf{d}^{n-1}$. $\beta ^n$ |
---|
437 | is searched such that the descent vectors form an orthogonal basis for the dot |
---|
438 | product linked to \textbf{A}. Expressing the condition |
---|
439 | $\langle \textbf{A d}^n, \textbf{d}^{n-1} \rangle = 0$ the value of $\beta ^n$ is found: |
---|
440 | $\beta ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle / \langle {\textbf{r}^{n-1}, \textbf{r}^{n-1}} \rangle$. |
---|
441 | As a result, the errors $ \textbf{r}^n$ form an orthogonal |
---|
442 | base for the canonic dot product while the descent vectors $\textbf{d}^n$ form |
---|
443 | an orthogonal base for the dot product linked to \textbf{A}. The resulting |
---|
444 | algorithm is thus the following one: |
---|
445 | |
---|
446 | initialisation : |
---|
447 | \begin{equation*} |
---|
448 | \begin{split} |
---|
449 | \textbf{x}^0 &= D_{i,j}^0 = 2 D_{i,j}^t - D_{i,j}^{t-1} \quad, \text{the initial guess } \\ |
---|
450 | \textbf{r}^0 &= \textbf{d}^0 = \textbf{b} - \textbf{A x}^0 \\ |
---|
451 | \gamma_0 &= \langle{ \textbf{r}^0 , \textbf{r}^0} \rangle |
---|
452 | \end{split} |
---|
453 | \end{equation*} |
---|
454 | |
---|
455 | iteration $n,$ from $n=0$ until convergence, do : |
---|
456 | \begin{equation} |
---|
457 | \begin{split} |
---|
458 | \text{z}^n& = \textbf{A d}^n \\ |
---|
459 | \alpha_n &= \gamma_n / \langle{ \textbf{z}^n , \textbf{d}^n} \rangle \\ |
---|
460 | \textbf{x}^{n+1} &= \textbf{x}^n + \alpha_n \,\textbf{d}^n \\ |
---|
461 | \textbf{r}^{n+1} &= \textbf{r}^n - \alpha_n \,\textbf{z}^n \\ |
---|
462 | \gamma_{n+1} &= \langle{ \textbf{r}^{n+1} , \textbf{r}^{n+1}} \rangle \\ |
---|
463 | \beta_{n+1} &= \gamma_{n+1}/\gamma_{n} \\ |
---|
464 | \textbf{d}^{n+1} &= \textbf{r}^{n+1} + \beta_{n+1}\; \textbf{d}^{n}\\ |
---|
465 | \end{split} |
---|
466 | \end{equation} |
---|
467 | |
---|
468 | |
---|
469 | The convergence test is: |
---|
470 | \begin{equation} |
---|
471 | \delta = \gamma_{n}\; / \langle{ \textbf{b} , \textbf{b}} \rangle \leq \epsilon |
---|
472 | \end{equation} |
---|
473 | where $\epsilon $ is the absolute precision that is required. As for the SOR algorithm, |
---|
474 | the whole model computation is stopped when the number of iterations, \np{nn\_max}, or |
---|
475 | the modulus of the right hand side of the convergence equation exceeds a |
---|
476 | specified value (see \S\ref{MISC_solsor} for a further discussion). The required |
---|
477 | precision and the maximum number of iterations allowed are specified by setting |
---|
478 | \np{rn\_eps} and \np{nn\_max} (\textbf{namelist} parameters). |
---|
479 | |
---|
480 | It can be demonstrated that the above algorithm is optimal, provides the exact |
---|
481 | solution in a number of iterations equal to the size of the matrix, and that |
---|
482 | the convergence rate is faster as the matrix is closer to the identity matrix, |
---|
483 | $i.e.$ its eigenvalues are closer to 1. Therefore, it is more efficient to solve |
---|
484 | a better conditioned system which has the same solution. For that purpose, |
---|
485 | we introduce a preconditioning matrix \textbf{Q} which is an approximation |
---|
486 | of \textbf{A} but much easier to invert than \textbf{A}, and solve the system: |
---|
487 | \begin{equation} \label{Eq_pmat} |
---|
488 | \textbf{Q}^{-1} \textbf{A x} = \textbf{Q}^{-1} \textbf{b} |
---|
489 | \end{equation} |
---|
490 | |
---|
491 | The same algorithm can be used to solve \eqref{Eq_pmat} if instead of the |
---|
492 | canonical dot product the following one is used: |
---|
493 | ${\langle{ \textbf{a} , \textbf{b}} \rangle}_Q = \langle{ \textbf{a} , \textbf{Q b}} \rangle$, and |
---|
494 | if $\textbf{\~{b}} = \textbf{Q}^{-1}\;\textbf{b}$ and $\textbf{\~{A}} = \textbf{Q}^{-1}\;\textbf{A}$ |
---|
495 | are substituted to \textbf{b} and \textbf{A} \citep{Madec_al_OM88}. |
---|
496 | In \NEMO, \textbf{Q} is chosen as the diagonal of \textbf{ A}, i.e. the simplest form for |
---|
497 | \textbf{Q} so that it can be easily inverted. In this case, the discrete formulation of |
---|
498 | \eqref{Eq_pmat} is in fact given by \eqref{Eq_solmat_p} and thus the matrix and |
---|
499 | right hand side are computed independently from the solver used. |
---|
500 | |
---|
501 | % ================================================================ |
---|
502 | % Diagnostics |
---|
503 | % ================================================================ |
---|
504 | \section{Diagnostics (DIA, IOM, TRD, FLO)} |
---|
505 | \label{MISC_diag} |
---|
506 | |
---|
507 | % ------------------------------------------------------------------------------------------------------------- |
---|
508 | % Standard Model Output |
---|
509 | % ------------------------------------------------------------------------------------------------------------- |
---|
510 | \subsection{Model Output (default or \key{iomput} or \key{dimgout})} |
---|
511 | \label{MISC_iom} |
---|
512 | |
---|
513 | %to be updated with Seb documentation on the IO |
---|
514 | |
---|
515 | The model outputs are of three types: the restart file, the output listing, |
---|
516 | and the output file(s). The restart file is used internally by the code when |
---|
517 | the user wants to start the model with initial conditions defined by a |
---|
518 | previous simulation. It contains all the information that is necessary in |
---|
519 | order for there to be no changes in the model results (even at the computer |
---|
520 | precision) between a run performed with several restarts and the same run |
---|
521 | performed in one step. It should be noted that this requires that the restart file |
---|
522 | contain two consecutive time steps for all the prognostic variables, and |
---|
523 | that it is saved in the same binary format as the one used by the computer |
---|
524 | that is to read it (in particular, 32 bits binary IEEE format must not be used for |
---|
525 | this file). The output listing and file(s) are predefined but should be checked |
---|
526 | and eventually adapted to the user's needs. The output listing is stored in |
---|
527 | the $ocean.output$ file. The information is printed from within the code on the |
---|
528 | logical unit $numout$. To locate these prints, use the UNIX command |
---|
529 | "\textit{grep -i numout}" in the source code directory. |
---|
530 | |
---|
531 | In the standard configuration, the user will find the model results in |
---|
532 | NetCDF files containing mean values (or instantaneous values if |
---|
533 | \key{diainstant} is defined) for every time-step where output is demanded. |
---|
534 | These outputs are defined in the \mdl{diawri} module. |
---|
535 | When defining \key{dimgout}, the output are written in DIMG format, |
---|
536 | an IEEE output format. |
---|
537 | |
---|
538 | Since version 3.2, an I/O server has been added which provides more |
---|
539 | flexibility in the choice of the fields to be output as well as how the |
---|
540 | writing work is distributed over the processors in massively parallel |
---|
541 | computing. It is activated when \key{iomput} is defined. |
---|
542 | |
---|
543 | % ------------------------------------------------------------------------------------------------------------- |
---|
544 | % Tracer/Dynamics Trends |
---|
545 | % ------------------------------------------------------------------------------------------------------------- |
---|
546 | \subsection[Tracer/Dynamics Trends (TRD)] |
---|
547 | {Tracer/Dynamics Trends (\key{trdmld}, \key{trdtra}, \key{trddyn}, \key{trdmld\_trc})} |
---|
548 | \label{MISC_tratrd} |
---|
549 | |
---|
550 | %------------------------------------------namtrd---------------------------------------------------- |
---|
551 | \namdisplay{namtrd} |
---|
552 | %------------------------------------------------------------------------------------------------------------- |
---|
553 | |
---|
554 | When \key{trddyn} and/or \key{trddyn} CPP variables are defined, each |
---|
555 | trend of the dynamics and/or temperature and salinity time evolution equations |
---|
556 | is stored in three-dimensional arrays just after their computation ($i.e.$ at the end |
---|
557 | of each $dyn\cdots.F90$ and/or $tra\cdots.F90$ routines). These trends are then |
---|
558 | used in \mdl{trdmod} (see TRD directory) every \textit{nn\_trd } time-steps. |
---|
559 | |
---|
560 | What is done depends on the CPP keys defined: |
---|
561 | \begin{description} |
---|
562 | \item[\key{trddyn}, \key{trdtra}] : a check of the basin averaged properties of the momentum |
---|
563 | and/or tracer equations is performed ; |
---|
564 | \item[\key{trdvor}] : a vertical summation of the moment tendencies is performed, |
---|
565 | then the curl is computed to obtain the barotropic vorticity tendencies which are output ; |
---|
566 | \item[\key{trdmld}] : output of the tracer tendencies averaged vertically |
---|
567 | either over the mixed layer (\np{nn\_ctls}=0), |
---|
568 | or over a fixed number of model levels (\np{nn\_ctls}$>$1 provides the number of level), |
---|
569 | or over a spatially varying but temporally fixed number of levels (typically the base |
---|
570 | of the winter mixed layer) read in \ifile{ctlsurf\_idx} (\np{nn\_ctls}=1). |
---|
571 | \end{description} |
---|
572 | |
---|
573 | The units in the output file can be changed using the \np{nn\_ucf} namelist parameter. |
---|
574 | For example, in case of salinity tendency the units are given by PSU/s/\np{nn\_ucf}. |
---|
575 | Setting \np{nn\_ucf}=86400 provides the tendencies in PSU/d. |
---|
576 | |
---|
577 | When \key{trdmld} is defined, two time averaging procedure are proposed. |
---|
578 | Setting \np{ln\_trdmld\_instant} to \textit{true}, a simple time averaging is performed, |
---|
579 | so that the resulting tendency is the contribution to the change of a quantity between |
---|
580 | the two instantaneous values taken at the extremities of the time averaging period. |
---|
581 | Setting \np{ln\_trdmld\_instant} to \textit{false}, a double time averaging is performed, |
---|
582 | so that the resulting tendency is the contribution to the change of a quantity between |
---|
583 | two \textit{time mean} values. The later option requires the use of an extra file, \ifile{restart\_mld} |
---|
584 | (\np{ln\_trdmld\_restart}=true), to restart a run. |
---|
585 | |
---|
586 | |
---|
587 | Note that the mixed layer tendency diagnostic can also be used on biogeochemical models |
---|
588 | via Êthe \key{trdtrc} and \key{trdmld\_trc} CPP keys. |
---|
589 | |
---|
590 | % ------------------------------------------------------------------------------------------------------------- |
---|
591 | % On-line Floats trajectories |
---|
592 | % ------------------------------------------------------------------------------------------------------------- |
---|
593 | \subsection{On-line Floats trajectories (FLO) (\key{floats}} |
---|
594 | \label{FLO} |
---|
595 | %--------------------------------------------namflo------------------------------------------------------- |
---|
596 | \namdisplay{namflo} |
---|
597 | %-------------------------------------------------------------------------------------------------------------- |
---|
598 | |
---|
599 | The on-line computation of floats advected either by the three dimensional velocity |
---|
600 | field or constraint to remain at a given depth ($w = 0$ in the computation) have been |
---|
601 | introduced in the system during the CLIPPER project. The algorithm used is based |
---|
602 | either on the work of \cite{Blanke_Raynaud_JPO97} (default option), or on a $4^th$ |
---|
603 | Runge-Hutta algorithm (\np{ln\_flork4}=true). Note that the \cite{Blanke_Raynaud_JPO97} |
---|
604 | algorithm have the advantage of providing trajectories which are consistent with the |
---|
605 | numeric of the code, so that the trajectories never intercept the bathymetry. |
---|
606 | |
---|
607 | See also the web site describing the off-line use of this marvellous diagnostic tool |
---|
608 | (http://stockage.univ-brest.fr/~grima/Ariane/). |
---|
609 | |
---|
610 | % ------------------------------------------------------------------------------------------------------------- |
---|
611 | % Other Diagnostics |
---|
612 | % ------------------------------------------------------------------------------------------------------------- |
---|
613 | \subsection{Other Diagnostics (\key{diahth}, \key{diaar5})} |
---|
614 | \label{MISC_diag_others} |
---|
615 | |
---|
616 | |
---|
617 | Aside from the standard model variables, other diagnostics can be computed |
---|
618 | on-line. The available ready-to-add diagnostics routines can be found in directory DIA. |
---|
619 | Among the available diagnostics the following ones are obtained when defining |
---|
620 | the \key{diahth} CPP key: |
---|
621 | |
---|
622 | - the mixed layer depth (based on a density criterion, \citet{de_Boyer_Montegut_al_JGR04}) (\mdl{diahth}) |
---|
623 | |
---|
624 | - the turbocline depth (based on a turbulent mixing coefficient criterion) (\mdl{diahth}) |
---|
625 | |
---|
626 | - the depth of the 20\deg C isotherm (\mdl{diahth}) |
---|
627 | |
---|
628 | - the depth of the thermocline (maximum of the vertical temperature gradient) (\mdl{diahth}) |
---|
629 | |
---|
630 | The poleward heat and salt transports, their advective and diffusive component, and |
---|
631 | the meriodional stream function can be computed on-line in \mdl{diaptr} by setting |
---|
632 | \np{ln\_diaptr} to true (see the \textit{namptr} namelist below). |
---|
633 | When \np{ln\_subbas}~=~true, transports and stream function are computed |
---|
634 | for the Atlantic, Indian, Pacific and Indo-Pacific Oceans (defined north of 30\deg S) |
---|
635 | as well as for the World Ocean. The sub-basin decomposition requires an input file |
---|
636 | (\ifile{subbasins}) which contains three 2D mask arrays, the Indo-Pacific mask |
---|
637 | been deduced from the sum of the Indian and Pacific mask (Fig~\ref{Fig_mask_subasins}). |
---|
638 | |
---|
639 | %------------------------------------------namptr---------------------------------------------------- |
---|
640 | \namdisplay{namptr} |
---|
641 | %------------------------------------------------------------------------------------------------------------- |
---|
642 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
643 | \begin{figure}[!t] \label{Fig_mask_subasins} \begin{center} |
---|
644 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_mask_subasins.pdf} |
---|
645 | \caption {Decomposition of the World Ocean (here ORCA2) into sub-basin used in to compute |
---|
646 | the heat and salt transports as well as the meridional stream-function: Atlantic basin (red), |
---|
647 | Pacific basin (green), Indian basin (bleue), Indo-Pacific basin (bleue+green). |
---|
648 | Note that semi-enclosed seas (Red, Med and Baltic seas) as well as Hudson Bay |
---|
649 | are removed from the sub-basin. Note also that the Arctic Ocean has been split |
---|
650 | into Atlantic and Pacific basins along the North fold line. } |
---|
651 | \end{center} \end{figure} |
---|
652 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
653 | |
---|
654 | In addition, a series of diagnostics has been added in the \mdl{diaar5}. |
---|
655 | They corresponds to outputs that are required for AR5 simulations |
---|
656 | (see Section \ref{MISC_steric} below for one of them). |
---|
657 | Activating those outputs requires to define the \key{diaar5} CPP key. |
---|
658 | |
---|
659 | |
---|
660 | |
---|
661 | % ================================================================ |
---|
662 | % Steric effect in sea surface height |
---|
663 | % ================================================================ |
---|
664 | \section{Steric effect in sea surface height} |
---|
665 | \label{MISC_steric} |
---|
666 | |
---|
667 | |
---|
668 | Changes in steric sea level are caused when changes in the density of the water |
---|
669 | column imply an expansion or contraction of the column. It is essentially produced |
---|
670 | through surface heating/cooling and to a lesser extent through non-linear effects of |
---|
671 | the equation of state (cabbeling, thermobaricity...). |
---|
672 | Non-Boussinesq models contain all ocean effects within the ocean acting |
---|
673 | on the sea level. In particular, they include the steric effect. In contrast, |
---|
674 | Boussinesq models, such as \NEMO, conserve volume, rather than mass, |
---|
675 | and so do not properly represent expansion or contraction. The steric effect is |
---|
676 | therefore not explicitely represented. |
---|
677 | This approximation does not represent a serious error with respect to the flow field |
---|
678 | calculated by the model \citep{Greatbatch_JGR94}, but extra attention is required |
---|
679 | when investigating sea level, as steric changes are an important |
---|
680 | contribution to local changes in sea level on seasonal and climatic time scales. |
---|
681 | This is especially true for investigation into sea level rise due to global warming. |
---|
682 | |
---|
683 | Fortunately, the steric contribution to the sea level consists of a spatially uniform |
---|
684 | component that can be diagnosed by considering the mass budget of the world |
---|
685 | ocean \citep{Greatbatch_JGR94}. |
---|
686 | In order to better understand how global mean sea level evolves and thus how |
---|
687 | the steric sea level can be diagnosed, we compare, in the following, the |
---|
688 | non-Boussinesq and Boussinesq cases. |
---|
689 | |
---|
690 | Let denote |
---|
691 | $\mathcal{M}$ the total mass of liquid seawater ($\mathcal{M}=\int_D \rho dv$), |
---|
692 | $\mathcal{V}$ the total volume of seawater ($\mathcal{V}=\int_D dv$), |
---|
693 | $\mathcal{A}$ the total surface of the ocean ($\mathcal{A}=\int_S ds$), |
---|
694 | $\bar{\rho}$ the global mean seawater (\textit{in situ}) density ($\bar{\rho}= 1/\mathcal{V} \int_D \rho \,dv$), and |
---|
695 | $\bar{\eta}$ the global mean sea level ($\bar{\eta}=1/\mathcal{A}\int_S \eta \,ds$). |
---|
696 | |
---|
697 | A non-Boussinesq fluid conserves mass. It satisfies the following relations: |
---|
698 | \begin{equation} \label{Eq_MV_nBq} |
---|
699 | \begin{split} |
---|
700 | \mathcal{M} &= \mathcal{V} \;\bar{\rho} \\ |
---|
701 | \mathcal{V} &= \mathcal{A} \;\bar{\eta} |
---|
702 | \end{split} |
---|
703 | \end{equation} |
---|
704 | Temporal changes in total mass is obtained from the density conservation equation : |
---|
705 | \begin{equation} \label{Eq_Co_nBq} |
---|
706 | \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface} |
---|
707 | \end{equation} |
---|
708 | where $\rho$ is the \textit{in situ} density, and \textit{emp} the surface mass |
---|
709 | exchanges with the other media of the Earth system (atmosphere, sea-ice, land). |
---|
710 | Its global averaged leads to the total mass change |
---|
711 | \begin{equation} \label{Eq_Mass_nBq} |
---|
712 | \partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}} |
---|
713 | \end{equation} |
---|
714 | where $\overline{\textit{emp}}=\int_S \textit{emp}\,ds$ is the net mass flux |
---|
715 | through the ocean surface. |
---|
716 | Bringing \eqref{Eq_Mass_nBq} and the time derivative of \eqref{Eq_MV_nBq} |
---|
717 | together leads to the evolution equation of the mean sea level |
---|
718 | \begin{equation} \label{Eq_ssh_nBq} |
---|
719 | \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}} |
---|
720 | - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}} |
---|
721 | \end{equation} |
---|
722 | The first term in equation \eqref{Eq_ssh_nBq} alters sea level by adding or |
---|
723 | subtracting mass from the ocean. |
---|
724 | The second term arises from temporal changes in the global mean |
---|
725 | density; $i.e.$ from steric effects. |
---|
726 | |
---|
727 | In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when $\rho$ |
---|
728 | appears multiplied by the gravity ($i.e.$ in the hydrostatic balance of the primitive Equations). |
---|
729 | In particular, the mass conservation equation, \eqref{Eq_Co_nBq}, degenerates into |
---|
730 | the incompressibility equation: |
---|
731 | \begin{equation} \label{Eq_Co_Bq} |
---|
732 | \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface} |
---|
733 | \end{equation} |
---|
734 | and the global average of this equation now gives the temporal change of the total volume, |
---|
735 | \begin{equation} \label{Eq_V_Bq} |
---|
736 | \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o} |
---|
737 | \end{equation} |
---|
738 | Only the volume is conserved, not mass, or, more precisely, the mass which is conserved is the |
---|
739 | Boussinesq mass, $\mathcal{M}_o = \rho_o \mathcal{V}$. The total volume (or equivalently |
---|
740 | the global mean sea level) is altered only by net volume fluxes across the ocean surface, |
---|
741 | not by changes in mean mass of the ocean: the steric effect is missing in a Boussinesq fluid. |
---|
742 | |
---|
743 | Nevertheless, following \citep{Greatbatch_JGR94}, the steric effect on the volume can be |
---|
744 | diagnosed by considering the mass budget of the ocean. |
---|
745 | The apparent changes in $\mathcal{M}$, mass of the ocean, which are not induced by surface |
---|
746 | mass flux must be compensated by a spatially uniform change in the mean sea level due to |
---|
747 | expansion/contraction of the ocean \citep{Greatbatch_JGR94}. In others words, the Boussinesq |
---|
748 | mass, $\mathcal{M}_o$, can be related to $\mathcal{M}$, the total mass of the ocean seen |
---|
749 | by the Boussinesq model, via the steric contribution to the sea level, $\eta_s$, a spatially |
---|
750 | uniform variable, as follows: |
---|
751 | \begin{equation} \label{Eq_M_Bq} |
---|
752 | \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A} |
---|
753 | \end{equation} |
---|
754 | Any change in $\mathcal{M}$ which cannot be explained by the net mass flux through |
---|
755 | the ocean surface is converted into a mean change in sea level. Introducing the total density |
---|
756 | anomaly, $\mathcal{D}= \int_D d_a \,dv$, where $d_a= (\rho -\rho_o ) / \rho_o$ |
---|
757 | is the density anomaly used in \NEMO (cf. \S\ref{TRA_eos}) in \eqref{Eq_M_Bq} |
---|
758 | leads to a very simple form for the steric height: |
---|
759 | \begin{equation} \label{Eq_steric_Bq} |
---|
760 | \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D} |
---|
761 | \end{equation} |
---|
762 | |
---|
763 | The above formulation of the steric height of a Boussinesq ocean requires four remarks. |
---|
764 | First, one can be tempted to define $\rho_o$ as the initial value of $\mathcal{M}/\mathcal{V}$, |
---|
765 | $i.e.$ set $\mathcal{D}_{t=0}=0$, so that the initial steric height is zero. We do not |
---|
766 | recommend that. Indeed, in this case $\rho_o$ depends on the initial state of the ocean. |
---|
767 | Since $\rho_o$ has a direct effect on the dynamics of the ocean (it appears in the pressure |
---|
768 | gradient term of the momentum equation) it is definitively not a good idea when |
---|
769 | inter-comparing experiments. |
---|
770 | We better recommend to fixe once for all $\rho_o$ to $1035\;Kg\,m^{-3}$. This value is a |
---|
771 | sensible choice for the reference density used in a Boussinesq ocean climate model since, |
---|
772 | with the exception of only a small percentage of the ocean, density in the World Ocean |
---|
773 | varies by no more than 2$\%$ from this value (\cite{Gill1982}, page 47). |
---|
774 | |
---|
775 | Second, we have assumed here that the total ocean surface, $\mathcal{A}$, does not |
---|
776 | change when the sea level is changing as it is the case in all global ocean GCMs |
---|
777 | (wetting and drying of grid point is not allowed). |
---|
778 | |
---|
779 | Third, the discretisation of \eqref{Eq_steric_Bq} depends on the type of free surface |
---|
780 | which is considered. In the non linear free surface case, $i.e.$ \key{vvl} defined, it is |
---|
781 | given by |
---|
782 | \begin{equation} \label{Eq_discrete_steric_Bq} |
---|
783 | \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} } |
---|
784 | { \sum_{i,\,j,\,k} e_{1t} e_{2t} e_{3t} } |
---|
785 | \end{equation} |
---|
786 | whereas in the linear free surface, the volume above the \textit{z=0} surface must be explicitly taken |
---|
787 | into account to better approximate the total ocean mass and thus the steric sea level: |
---|
788 | \begin{equation} \label{Eq_discrete_steric_Bq} |
---|
789 | \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta } |
---|
790 | {\sum_{i,\,j,\,k} e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} e_{1t}e_{2t} \eta } |
---|
791 | \end{equation} |
---|
792 | |
---|
793 | The fourth and last remark concerns the effective sea level and the presence of sea-ice. |
---|
794 | In the real ocean, sea ice (and snow above it) depresses the liquid seawater through |
---|
795 | its mass loading. This depression is a result of the mass of sea ice/snow system acting |
---|
796 | on the liquid ocean. There is, however, no dynamical effect associated with these depressions |
---|
797 | in the liquid ocean sea level, so that there are no associated ocean currents. Hence, the |
---|
798 | dynamically relevant sea level is the effective sea level, $i.e.$ the sea level as if sea ice |
---|
799 | (and snow) were converted to liquid seawater \citep{Campin_al_OM08}. However, |
---|
800 | in the current version of \NEMO the sea-ice is levitating above the ocean without |
---|
801 | mass exchanges between ice and ocean. Therefore the model effective sea level |
---|
802 | is always given by $\eta + \eta_s$, whether or not there is sea ice present. |
---|
803 | |
---|
804 | In AR5 outputs, the thermosteric sea level is demanded. It is steric sea level due to |
---|
805 | changes in ocean density arising just from changes in temperature. It is given by: |
---|
806 | \begin{equation} \label{Eq_thermosteric_Bq} |
---|
807 | \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv |
---|
808 | \end{equation} |
---|
809 | where $S_o$ and $p_o$ are the initial salinity and pressure, respectively. |
---|
810 | |
---|
811 | Both steric and thermosteric sea level are computed in \mdl{diaar5} which needs |
---|
812 | the \key{diaar5} defined to be called. |
---|
813 | |
---|
814 | |
---|
815 | % ================================================================ |
---|
816 | % predefined configurations |
---|
817 | % ================================================================ |
---|
818 | \section{predefined configurations} |
---|
819 | \label{MISC_config} |
---|
820 | |
---|
821 | There is several predefined ocean configuration which use is controlled by a specific CPP key. |
---|
822 | |
---|
823 | The key set the domain sizes (\jp{jpiglo}, \jp{jpjglo}, \jp{jpk}), the mesh and the bathymetry, |
---|
824 | and, in some cases, add to the model physics some specific treatments. |
---|
825 | |
---|
826 | % ------------------------------------------------------------------------------------------------------------- |
---|
827 | % ORCA family configurations |
---|
828 | % ------------------------------------------------------------------------------------------------------------- |
---|
829 | \subsection{ORCA family: global ocean with tripolar grid} |
---|
830 | \label{MISC_config_orca} |
---|
831 | |
---|
832 | The NEMO system is provided with four built-in ORCA configurations which differ in the |
---|
833 | horizontal resolution used: |
---|
834 | \begin{description} |
---|
835 | \item[\key{orca\_r4}] \jp{cp\_cfg}~=~orca ; \jp{jp\_cfg}~=~4 |
---|
836 | \item[\key{orca\_r2}] \jp{cp\_cfg}~=~orca ; \jp{jp\_cfg}~=~2 |
---|
837 | \item[\key{orca\_r1}] \jp{cp\_cfg}~=~orca ; \jp{jp\_cfg}~=~1 |
---|
838 | \item[\key{orca\_r05}] \jp{cp\_cfg}~=~orca ; \jp{jp\_cfg}~=~05 |
---|
839 | \item[\key{orca\_r025}] \jp{cp\_cfg}~=~orca ; \jp{jp\_cfg}~=~025 |
---|
840 | \end{description} |
---|
841 | |
---|
842 | \subsubsection{ORCA mesh} |
---|
843 | |
---|
844 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
845 | \begin{figure}[!t] \label{Fig_MISC_ORCA_msh} \begin{center} |
---|
846 | \includegraphics[width=0.98\textwidth]{./TexFiles/Figures/Fig_ORCA_NH_mesh.pdf} |
---|
847 | \caption {ORCA mesh conception. The departure from an isotropic Mercator grid start poleward of 20\deg N. |
---|
848 | The two "north pole" are the foci of a series of embedded ellipses (blue curves) |
---|
849 | which are determined analytically and form the i-lines of the ORCA mesh (pseudo latitudes). |
---|
850 | Then, following \citet{Madec_Imbard_CD96}, the normal to the series of ellipses (red curves) is computed |
---|
851 | which provide the j-lines of the mesh (pseudo longitudes). |
---|
852 | } |
---|
853 | \end{center} \end{figure} |
---|
854 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
855 | |
---|
856 | |
---|
857 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
858 | \begin{figure}[!tbp] \label{Fig_MISC_ORCA_e1e2} \begin{center} |
---|
859 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_ORCA_NH_msh05_e1_e2.pdf} |
---|
860 | \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_ORCA_aniso.pdf} |
---|
861 | \caption {\textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and |
---|
862 | \textit{Bottom}: ratio of anisotropy ($e_1 / e_2$) |
---|
863 | for ORCA 0.5\deg ~mesh. South of 20\deg N a Mercator grid is used ($e_1 = e_2$) |
---|
864 | so that the anisotropy ratio is 1. Poleward of 20\deg N, the two "north pole" |
---|
865 | introduce a weak anisotropy over the ocean areas ($< 1.2$) except in vicinity of Victoria Island |
---|
866 | (Canadian Arctic Archipelago). } |
---|
867 | \end{center} \end{figure} |
---|
868 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
869 | |
---|
870 | %--------------------------------------------------TABLE-------------------------------------------------- |
---|
871 | \begin{table}[htbp] \label{Tab_ORCA} |
---|
872 | \begin{center} |
---|
873 | \begin{tabular}{ccccc} |
---|
874 | key & \jp{jp\_cfg} & \jp{jpiglo} & \jp{jpiglo} & \\ |
---|
875 | \hline \hline |
---|
876 | \key{orca\_r4} & 4 & 92 & 76 & \\ |
---|
877 | \key{orca\_r2} & 2 & 182 & 149 & \\ |
---|
878 | %\key{orca\_r1} & 1 & 362 & 511 & \\ |
---|
879 | \key{orca\_r05} & 05 & 722 & 261 & \\ |
---|
880 | \key{orca\_r025} & 025 & 1442 & 1021 & \\ |
---|
881 | %\key{orca\_r8} & 8 & 2882 & 2042 & \\ |
---|
882 | %\key{orca\_r12} & 12 & 4322 & 3062 & \\ |
---|
883 | \hline |
---|
884 | \hline |
---|
885 | \end{tabular} |
---|
886 | \caption {Set of predefined ORCA parameters. } |
---|
887 | \end{center} |
---|
888 | \end{table} |
---|
889 | %-------------------------------------------------------------------------------------------------------------- |
---|
890 | |
---|
891 | The tripolar grid used in ORCA configuration .... |
---|
892 | |
---|
893 | NB: the two north poles position has been chosen to minimise the anisotropy ratio in |
---|
894 | the Gulf Stream and kuroshio areas, two highly turbulent regions. |
---|
895 | |
---|
896 | ORCA~2 : a $2\deg$ zonal resolution, and a meridional resolution varying from $0.5\deg$ at the |
---|
897 | equator to $2\deg cos\phi$ south of $20\deg$S (Fig. 1). The grid features two points of convergence in the |
---|
898 | Northern Hemisphere, both situated on continents. Minimum resolution in high latitudes is about |
---|
899 | 65~km in the Arctic and 50~km in the Antarctic. Local mesh refinements are applied to the |
---|
900 | Mediterranean, Red, Black and Caspian Seas. None of them appears to be of particular |
---|
901 | importance for the study of high latitude climate, but the fine resolution is needed in order to have |
---|
902 | their local circulation and their role in the World Ocean's circulation considered correctly. |
---|
903 | |
---|
904 | |
---|
905 | |
---|
906 | ORCA2-LIM (global ocean sea-ice configuration \citep{Timmermann_al_OM05}. |
---|
907 | The horizontal mesh is based on a $2\deg \times 2\deg$ Mercator grid ($i.e.$ same zonal and |
---|
908 | meridional grid spacing) which has been modified poleward |
---|
909 | of $20\deg$N in order to include two numerical inland poles \citep{Murray_JCP96}. |
---|
910 | This modification is semi-analytical \citep{Madec_Imbard_CD96} |
---|
911 | and based on a series of embedded ellipses. It insures that the mesh remains |
---|
912 | close to isotropy and that the smallest grid cell is along Antarctica. |
---|
913 | In order to refine the meridional resolution up to $0.5\deg$ at the equator, |
---|
914 | additional local transformations were applied with in the Tropics. |
---|
915 | Local mesh refinements are also applied to the Mediterranean, Red, Black |
---|
916 | and Caspian Seas so that the resolution is $1\deg \time 1\deg$ there. |
---|
917 | There are 31 levels in the vertical, with the highest resolution (10m) |
---|
918 | in the upper 150m. The bottom topography and the coastlines are derived |
---|
919 | from the global atlas of Smith and Sandwell (1997). |
---|
920 | |
---|
921 | \key{orca\_lev10} 10 time more vertical levels |
---|
922 | |
---|
923 | \key{agrif} : ORCA2-LIM plus an AGRIF zoom over the Agulhas current area |
---|
924 | |
---|
925 | \key{arctic}, \key{antarctic} (not used in ORCA\_R4) |
---|
926 | |
---|
927 | |
---|
928 | We thus only provide a brief introduction in this chapter. |
---|
929 | The global coupled ocean-ice configuration is very similar to that used as part of the climate |
---|
930 | model developed at GFDL for the 4th IPCC assessment of climate change (Griffies et al., 2005; |
---|
931 | Gnanadesikan et al., 2006). |
---|
932 | The ORCA2-LIM configuration is also the basis for the \NEMO contribution to the |
---|
933 | Coordinate Ocean-ice Reference Experiments (COREs) documented in \citet{Griffies_al_OM09}. |
---|
934 | These experiments employ the boundary forcing from \citet{Large_Yeager_Rep04} (see \S\ref{SBC_blk_core}), |
---|
935 | which was developed for the purpose of running global coupled ocean-ice simulations without an |
---|
936 | interactive atmosphere. This \citet{Large_Yeager_Rep04} dataset is available through the GFDL web |
---|
937 | site \footnote{http://nomads.gfdl.noaa.gov/nomads/forms/mom4/CORE.html}. |
---|
938 | The "normal year" of \citet{Large_Yeager_Rep04} has been chosen of the \NEMO distribution |
---|
939 | since release v3.3. |
---|
940 | |
---|
941 | % ------------------------------------------------------------------------------------------------------------- |
---|
942 | % GYRE family configuration |
---|
943 | % ------------------------------------------------------------------------------------------------------------- |
---|
944 | \subsection{GYRE family: double gyre basin (\key{gyre})} |
---|
945 | \label{MISC_config_gyre} |
---|
946 | |
---|
947 | The GYRE configuration \citep{Levy_al_OM10} have been built to simulated |
---|
948 | the seasonal cycle of a double-gyre box model. It consist in an idealized domain |
---|
949 | similar to that used in the studies of \citet{Drijfhout_JPO94} and \citet{Hazeleger_Drijfhout_JPO98, |
---|
950 | Hazeleger_Drijfhout_JPO99, Hazeleger_Drijfhout_JGR00, Hazeleger_Drijfhout_JPO00}, |
---|
951 | over which an analytical seasonal forcing is applied. This allows to investigate the |
---|
952 | spontaneous generation of a large number of interacting, transient mesoscale eddies |
---|
953 | and their contribution to the large scale circulation. |
---|
954 | |
---|
955 | The domain geometry is a closed rectangular basin on the $\beta$-plane centred |
---|
956 | at $\sim 30\deg$N and rotated by 45\deg, 3180~km long, 2120~km wide |
---|
957 | and 4~km deep (Fig.~\ref{Fig_MISC_strait_hand}). |
---|
958 | The domain is bounded by vertical walls and by a ßat bottom. The configuration is |
---|
959 | meant to represent an idealized North Atlantic or North Pacific basin. |
---|
960 | The circulation is forced by analytical profiles of wind and buoyancy ßuxes. |
---|
961 | The applied forcings vary seasonally in a sinusoidal manner between winter |
---|
962 | and summer extrema \citep{Levy_al_OM10}. |
---|
963 | The wind stress is zonal and its curl changes sign at 22\deg N and 36\deg N. |
---|
964 | It forces a subpolar gyre in the north, a subtropical gyre in the wider part of the domain |
---|
965 | and a small recirculation gyre in the southern corner. |
---|
966 | The net heat ßux takes the form of a restoring toward a zonal apparent air |
---|
967 | temperature profile. A portion of the net heat ßux which comes from the solar radiation |
---|
968 | is allowed to penetrate within the water column. |
---|
969 | The fresh water ßux is also prescribed and varies zonally. |
---|
970 | It is determined such as, at each time step, the basin-integrated ßux is zero. |
---|
971 | The basin is initialised at rest with vertical profiles of temperature and salinity |
---|
972 | uniformly applied to the whole domain. |
---|
973 | |
---|
974 | The GYRE configuration is set through the \key{gyre} CPP key. Its horizontal resolution |
---|
975 | (and thus the size of the domain) is determined by setting \jp{jp\_cfg} in \hf{par\_GYRE} file: \\ |
---|
976 | \jp{jpiglo} $= 30 \times$ \jp{jp\_cfg} + 2 \\ |
---|
977 | \jp{jpjglo} $= 20 \times$ \jp{jp\_cfg} + 2 \\ |
---|
978 | Obviously, the namelist parameters have to be adjusted to the chosen resolution. |
---|
979 | In the vertical, GYRE uses the default 30 ocean levels (\jp{jpk}=31) (Fig.~\ref{Fig_zgr}). |
---|
980 | |
---|
981 | The GYRE configuration is also used in benchmark test as it is very simple to increase |
---|
982 | its resolution and as it does not requires any input file. For example, keeping a same model size |
---|
983 | on each processor while increasing the number of processor used is very easy, even though the |
---|
984 | physical integrity of the solution can be compromised. |
---|
985 | |
---|
986 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
987 | \begin{figure}[!t] \label{Fig_GYRE} \begin{center} |
---|
988 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_GYRE.pdf} |
---|
989 | \caption {Snapshot of relative vorticity at the surface of the model domain |
---|
990 | in GYRE R9, R27 and R54. From \citet{Levy_al_OM10}.} |
---|
991 | \end{center} \end{figure} |
---|
992 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
993 | |
---|
994 | % ------------------------------------------------------------------------------------------------------------- |
---|
995 | % EEL family configuration |
---|
996 | % ------------------------------------------------------------------------------------------------------------- |
---|
997 | \subsection{EEL family: periodic channel} |
---|
998 | \label{MISC_config_EEL} |
---|
999 | |
---|
1000 | \begin{description} |
---|
1001 | \item[\key{eel\_r2}] |
---|
1002 | \item[\key{eel\_r5}] |
---|
1003 | \item[\key{eel\_r6}] |
---|
1004 | \end{description} |
---|
1005 | |
---|
1006 | % ------------------------------------------------------------------------------------------------------------- |
---|
1007 | % POMME configuration |
---|
1008 | % ------------------------------------------------------------------------------------------------------------- |
---|
1009 | \subsection{POMME: mid-latitude sub-domain} |
---|
1010 | \label{MISC_config_POMME} |
---|
1011 | |
---|
1012 | |
---|
1013 | \key{pomme\_r025} |
---|
1014 | |
---|
1015 | |
---|
1016 | |
---|