1 | % ================================================================ |
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2 | % Chapter Ñ Ocean Dynamics (DYN) |
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3 | % ================================================================ |
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4 | \chapter{Ocean Dynamics (DYN)} |
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5 | \label{DYN} |
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6 | \minitoc |
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7 | |
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8 | % add a figure for dynvor ens, ene latices |
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9 | |
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10 | %\vspace{2.cm} |
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11 | $\ $\newline %force an empty line |
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12 | |
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13 | Using the representation described in Chap.\ref{DOM}, several semi-discrete |
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14 | space forms of the dynamical equations are available depending on the vertical |
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15 | coordinate used and on the conservation properties of the vorticity term. In all |
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16 | the equations presented here, the masking has been omitted for simplicity. |
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17 | One must be aware that all the quantities are masked fields and that each time a |
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18 | average or difference operator is used, the resulting field is multiplied by a mask. |
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19 | |
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20 | The prognostic ocean dynamics equation can be summarized as follows: |
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21 | \begin{equation*} |
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22 | \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } |
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23 | {\text{COR} + \text{ADV} } |
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24 | + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} |
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25 | \end{equation*} |
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26 | |
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27 | NXT stands for next, referring to the time-stepping. The first group of terms on |
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28 | the rhs of the momentum equations corresponds to the Coriolis and advection |
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29 | terms that are decomposed into a vorticity part (VOR), a kinetic energy part (KEG) |
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30 | and, a vertical advection part (ZAD) in the vector invariant formulation or a Coriolis |
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31 | and advection part(COR+ADV) in the flux formulation. The terms following these |
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32 | are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient, |
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33 | and SPG, Surface Pressure Gradient); and contributions from lateral diffusion |
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34 | (LDF) and vertical diffusion (ZDF), which are added to the rhs in the \mdl{dynldf} |
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35 | and \mdl{dynzdf} modules. The vertical diffusion term includes the surface and |
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36 | bottom stresses. The external forcings and parameterisations require complex |
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37 | inputs (surface wind stress calculation using bulk formulae, estimation of mixing |
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38 | coefficients) that are carried out in modules SBC, LDF and ZDF and are described |
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39 | in Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively. |
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40 | |
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41 | In the present chapter we also describe the diagnostic equations used to compute |
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42 | the horizontal divergence and curl of the velocities (\emph{divcur} module) as well |
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43 | as the vertical velocity (\emph{wzvmod} module). |
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44 | |
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45 | The different options available to the user are managed by namelist variables. |
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46 | For equation term \textit{ttt}, the logical namelist variables are \textit{ln\_dynttt\_xxx}, |
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47 | where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. |
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48 | If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding |
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49 | code can be found in the \textit{dynttt\_xxx} module in the DYN directory, and it is |
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50 | usually computed in the \textit{dyn\_ttt\_xxx} subroutine. |
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51 | |
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52 | The user has the option of extracting each tendency term of both the rhs of the |
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53 | 3D momentum equation (\key{trddyn} defined) for output, as described in |
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54 | Chap.\ref{MISC}. Furthermore, the tendency terms associated to the 2D |
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55 | barotropic vorticity balance (\key{trdvor} defined) can be derived on-line from the |
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56 | 3D terms. |
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57 | %%% |
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58 | \gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does |
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59 | MISC correspond to "extracting tendency terms" or "vorticity balance"?} |
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60 | |
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61 | $\ $\newline % force a new ligne |
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62 | % ================================================================ |
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63 | % Coriolis and Advection terms: vector invariant form |
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64 | % ================================================================ |
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65 | \section{Coriolis and Advection: vector invariant form} |
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66 | \label{DYN_adv_cor_vect} |
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67 | %-----------------------------------------nam_dynadv---------------------------------------------------- |
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68 | \namdisplay{nam_dynadv} |
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69 | %------------------------------------------------------------------------------------------------------------- |
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70 | |
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71 | The vector invariant form of the momentum equations is the one most |
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72 | often used in applications of \NEMO ocean model. The flux form option |
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73 | (see next section) has been introduced since version $2$. |
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74 | Coriolis and momentum advection terms are evaluated using a leapfrog |
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75 | scheme, $i.e.$ the velocity appearing in these expressions is centred in |
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76 | time (\textit{now} velocity). |
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77 | At the lateral boundaries either free slip, no slip or partial slip boundary |
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78 | conditions are applied following Chap.\ref{LBC}. |
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79 | |
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80 | % ------------------------------------------------------------------------------------------------------------- |
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81 | % Vorticity term |
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82 | % ------------------------------------------------------------------------------------------------------------- |
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83 | \subsection [Vorticity term (\textit{dynvor}) ] |
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84 | {Vorticity term (\mdl{dynvor})} |
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85 | \label{DYN_vor} |
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86 | %------------------------------------------nam_dynvor---------------------------------------------------- |
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87 | \namdisplay{nam_dynvor} |
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88 | %------------------------------------------------------------------------------------------------------------- |
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89 | |
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90 | Different discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=.true.) are |
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91 | available: conserving potential enstrophy of horizontally non-divergent flow ; |
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92 | conserving horizontal kinetic energy ; or conserving potential enstrophy for the |
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93 | relative vorticity term and horizontal kinetic energy for the planetary vorticity |
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94 | term (see Appendix~\ref{Apdx_C}). The vorticity terms are given below for the |
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95 | general case, but note that in the full step $z$-coordinate (\key{zco} is defined), |
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96 | $e_{3u} =e_{3v} =e_{3f}$ so that the vertical scale factors disappear. They are |
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97 | all computed in dedicated routines that can be found in the \mdl{dynvor} module. |
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98 | |
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99 | %------------------------------------------------------------- |
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100 | % enstrophy conserving scheme |
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101 | %------------------------------------------------------------- |
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102 | \subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=.true.)} |
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103 | \label{DYN_vor_ens} |
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104 | |
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105 | In the enstrophy conserving case (ENS scheme), the discrete formulation of the |
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106 | vorticity term provides a global conservation of the enstrophy |
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107 | ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent |
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108 | flow ($i.e.$ $\chi=0$), but does not conserve the total kinetic energy. It is given by: |
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109 | \begin{equation} \label{Eq_dynvor_ens} |
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110 | \left\{ |
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111 | \begin{aligned} |
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112 | {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} |
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113 | & {\overline{\overline {\left( {e_{1v} e_{3v} v} \right)}} }^{\,i, j+1/2} \\ |
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114 | {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} |
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115 | & {\overline{\overline {\left( {e_{2u} e_{3u} u} \right)}} }^{\,i+1/2, j} |
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116 | \end{aligned} |
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117 | \right. |
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118 | \end{equation} |
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119 | |
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120 | %------------------------------------------------------------- |
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121 | % energy conserving scheme |
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122 | %------------------------------------------------------------- |
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123 | \subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=.true.)} |
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124 | \label{DYN_vor_ene} |
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125 | |
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126 | The kinetic energy conserving scheme (ENE scheme) conserves the global |
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127 | kinetic energy but not the global enstrophy. It is given by: |
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128 | \begin{equation} \label{Eq_dynvor_ene} |
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129 | \left\{ \begin{aligned} |
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130 | {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) |
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131 | \; \overline {\left( {e_{1v} e_{3v} v} \right)} ^{\,i+1/2}} }^{\,j} } \\ |
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132 | {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) |
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133 | \; \overline {\left( {e_{2u} e_{3u} u} \right)} ^{\,j+1/2}} }^{\,i} } |
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134 | \end{aligned} \right. |
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135 | \end{equation} |
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136 | |
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137 | %------------------------------------------------------------- |
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138 | % mix energy/enstrophy conserving scheme |
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139 | %------------------------------------------------------------- |
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140 | \subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=.true.) } |
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141 | \label{DYN_vor_mix} |
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142 | |
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143 | The mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the |
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144 | two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens}) |
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145 | to the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied |
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146 | to the planetary vorticity term. |
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147 | \begin{equation} \label{Eq_dynvor_mix} |
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148 | \left\{ { |
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149 | \begin{aligned} |
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150 | {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} |
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151 | \; {\overline{\overline {\left( {e_{1v} \; e_{3v} \ v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} } |
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152 | \; {\overline {\left( {\frac{f}{e_{3f} }} \right) |
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153 | \;\overline {\left( {e_{1v} \; e_{3v} \ v} \right)} ^{\,i+1/2}} }^{\,j} } \\ |
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154 | {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j |
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155 | \; {\overline{\overline {\left( {e_{2u} \; e_{3u} \ u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} } |
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156 | \; {\overline {\left( {\frac{f}{e_{3f} }} \right) |
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157 | \;\overline {\left( {e_{2u}\; e_{3u} \ u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill |
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158 | \end{aligned} |
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159 | } \right. |
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160 | \end{equation} |
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161 | |
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162 | %------------------------------------------------------------- |
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163 | % energy and enstrophy conserving scheme |
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164 | %------------------------------------------------------------- |
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165 | \subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=.true.) } |
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166 | \label{DYN_vor_een} |
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167 | |
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168 | In the energy and enstrophy conserving scheme (EEN scheme), the vorticity term |
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169 | is evaluated using the vorticity advection scheme of \citet{Arakawa1990}. |
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170 | This scheme conserves both total energy and potential enstrophy in the limit of |
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171 | horizontally nondivergent flow ($i.e. \ \chi=0$). While EEN is more complicated |
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172 | than ENS or ENE and does not conserve potential enstrophy and total energy in |
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173 | general flow, it tolerates arbitrarily thin layers. This feature is essential for |
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174 | $z$-coordinate with partial step. |
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175 | %%% |
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176 | \gmcomment{gm : it actually conserve kinetic energy ! show that in appendix C } |
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177 | %%% |
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178 | |
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179 | The \citet{Arakawa1990} vorticity advection scheme for a single layer is modified |
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180 | for spherical coordinates as described by \citet{Arakawa1981} to obtain the EEN |
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181 | scheme. The potential vorticity, defined at an $f$-point, is: |
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182 | \begin{equation} \label{Eq_pot_vor} |
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183 | q_f = \frac{\zeta +f} {e_{3f} } |
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184 | \end{equation} |
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185 | where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter |
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186 | is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: |
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187 | \begin{equation} \label{Eq_een_e3f} |
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188 | e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} |
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189 | \end{equation} |
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190 | |
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191 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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192 | \begin{figure}[!ht] \label{Fig_DYN_een_triad} |
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193 | \begin{center} |
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194 | \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_DYN_een_triad.pdf} |
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195 | \caption{Triads used in the energy and enstrophy conserving scheme (een) for |
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196 | $u$-component (upper panel) and $v$-component (lower panel).} |
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197 | \end{center} |
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198 | \end{figure} |
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199 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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200 | |
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201 | Note that a key point in \eqref{Eq_een_e3f} is that the averaging in \textbf{i}- and |
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202 | \textbf{j}- directions uses the masked vertical scale factor but is always divided by |
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203 | $4$, not by the sum of the mask at $T$-point. This preserves the continuity of |
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204 | $e_{3f}$ when one or more of the neighbouring $e_{3T}$ tends to zero and |
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205 | extends by continuity the value of $e_{3f}$ in the land areas. |
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206 | %%% |
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207 | \gmcomment{this has to be further investigate in case of several step topography} |
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208 | %%% |
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209 | |
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210 | The vorticity terms are represented as: |
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211 | \begin{equation} \label{Eq_dynvor_een} |
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212 | \left\{ { |
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213 | \begin{aligned} |
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214 | +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \left[ |
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215 | {{\begin{array}{*{20}c} |
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216 | {\,\ \ a_{j+1/2}^{i } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i+1/2} } |
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217 | {\,+\,b_{j+1/2}^{i } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i-1/2} } \\ |
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218 | \\ |
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219 | { +\,c_{j-1/2}^{i } \left( {e_{1v} e_{3v} \ v} \right)_{j }^{i+1/2} } |
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220 | {\,+\,d_{j+1/2}^{i } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i+1/2} } \\ |
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221 | \end{array} }} \right] \\ |
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222 | \\ |
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223 | -q\,e_3 \,u &\equiv -\frac{1}{e_{2v} } \left[ |
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224 | {{\begin{array}{*{20}c} |
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225 | {\,\ \ a_{j-1/2}^{i } \left( {e_{2u} e_{3u} \ u} \right)_{j+1}^{i+1/2} } |
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226 | {\,+\,b_{j-1/2}^{i+1} \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i+1} } \\ |
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227 | \\ |
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228 | { +\,c_{j+1/2}^{i+1} \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i+1} } |
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229 | {\,+\,d_{j+1/2}^{i } \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i } } \\ |
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230 | \end{array} }} \right] |
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231 | \end{aligned} |
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232 | } \right. |
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233 | \end{equation} |
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234 | where $a$, $b$, $c$ and $d$ are the following triad combinations of the |
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235 | neighbouring potential vorticities (Fig.~\ref{Fig_DYN_een_triad}): |
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236 | \begin{equation} \label{Eq_een_triads} |
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237 | \left\{ |
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238 | \begin{aligned} |
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239 | a_{\,j+1/2}^i & = \frac{1} {12} \left( q_{j+1/2}^{i+1} + q_{j+1 /2}^i + q_{j-1/2}^i \right) \\ |
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240 | \\ |
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241 | b_{\,j+1/2}^i & = \frac{1} {12} \left( q_{j+1/2}^{i-1} +q_{j+1/2}^i +q_{j-1/2}^i \right) \\ |
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242 | \\ |
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243 | c_{\,j+1/2}^i & = \frac{1} {12} \left( q_{j-1/2}^{i-1} +q_{j+1/2}^i +q_{j-1/2}^i \right) \\ |
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244 | \\ |
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245 | d_{\,j+1/2}^i & = \frac{1} {12} \left( q_{j-1/2}^{i+1} +q_{j+1/2}^i +q_{j-1/2}^i \right) \\ |
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246 | \end{aligned} |
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247 | \right. |
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248 | \end{equation} |
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249 | |
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250 | %-------------------------------------------------------------------------------------------------------------- |
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251 | % Kinetic Energy Gradient term |
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252 | %-------------------------------------------------------------------------------------------------------------- |
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253 | \subsection [Kinetic Energy Gradient term (\textit{dynkeg})] |
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254 | {Kinetic Energy Gradient term (\mdl{dynkeg})} |
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255 | \label{DYN_keg} |
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256 | |
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257 | As demonstarted in Appendix~\ref{Apdx_C}, there is a single discrete formulation |
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258 | of the kinetic energy gradient term that, together with the formulation chosen for |
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259 | the vertical advection (see below), conserves the total kinetic energy: |
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260 | \begin{equation} \label{Eq_dynkeg} |
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261 | \left\{ \begin{aligned} |
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262 | -\frac{1}{2 \; e_{1u} } |
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263 | & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] \\ |
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264 | -\frac{1}{2 \; e_{2v} } |
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265 | & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] |
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266 | \end{aligned} \right. |
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267 | \end{equation} |
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268 | |
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269 | %-------------------------------------------------------------------------------------------------------------- |
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270 | % Vertical advection term |
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271 | %-------------------------------------------------------------------------------------------------------------- |
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272 | \subsection [Vertical advection term (\textit{dynzad}) ] |
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273 | {Vertical advection term (\mdl{dynzad}) } |
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274 | \label{DYN_zad} |
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275 | |
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276 | The discrete formulation of the vertical advection, together with the formulation |
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277 | chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic |
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278 | energy. Indeed, the change of KE due to the vertical advection is exactly |
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279 | balanced by the change of KE due to the gradient of KE (see Appendix~\ref{Apdx_C}). |
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280 | \begin{equation} \label{Eq_dynzad} |
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281 | \left\{ \begin{aligned} |
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282 | -\frac{1} { e_{1u}\,e_{2u}\,e_{3u} } & |
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283 | \ {\overline {\overline{ e_{1T}\,e_{2T}\,w } ^{\,i+1/2} \;\delta _{k+1/2} \left[ u \right] }^{\,k} } \\ |
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284 | -\frac{1} { e_{1v}\,e_{2v}\,e_{3v} } & |
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285 | \ {\overline {\overline{ e_{1T}\,e_{2T}\,w } ^{\,j+1/2} \;\delta _{k+1/2} \left[ u \right] }^{\,k} } |
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286 | \end{aligned} \right. |
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287 | \end{equation} |
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288 | |
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289 | % ================================================================ |
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290 | % Coriolis and Advection : flux form |
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291 | % ================================================================ |
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292 | \section{Coriolis and Advection: flux form} |
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293 | \label{DYN_adv_cor_flux} |
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294 | %------------------------------------------nam_dynadv---------------------------------------------------- |
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295 | \namdisplay{nam_dynadv} |
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296 | %------------------------------------------------------------------------------------------------------------- |
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297 | |
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298 | In the flux form (as in the vector invariant form), the Coriolis and momentum |
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299 | advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity |
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300 | appearing in their expressions is centred in time (\textit{now} velocity). At the |
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301 | lateral boundaries either free slip, no slip or partial slip boundary conditions |
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302 | are applied following Chap.\ref{LBC}. |
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303 | |
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304 | |
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305 | %-------------------------------------------------------------------------------------------------------------- |
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306 | % Coriolis plus curvature metric terms |
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307 | %-------------------------------------------------------------------------------------------------------------- |
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308 | \subsection [Coriolis plus curvature metric terms (\textit{dynvor}) ] |
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309 | {Coriolis plus curvature metric terms (\mdl{dynvor}) } |
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310 | \label{DYN_cor_flux} |
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311 | |
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312 | In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis |
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313 | parameter has been modified to account for the "metric" term. This altered |
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314 | Coriolis parameter is thus discretised at $f$-points. It is given by: |
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315 | \begin{multline} \label{Eq_dyncor_metric} |
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316 | f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} - u\frac{\partial e_1 }{\partial j}} \right) \\ |
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317 | \equiv f + \frac{1}{e_{1f} e_{2f} } |
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318 | \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right] |
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319 | - \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right] } \ \right) |
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320 | \end{multline} |
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321 | |
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322 | Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een}) |
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323 | schemes can be used to compute the product of the Coriolis parameter and the |
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324 | vorticity. However, the energy-conserving scheme (\ref{Eq_dynvor_een}) has |
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325 | exclusively been used to date. This term is evaluated using a leapfrog scheme, |
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326 | $i.e.$ the velocity is centred in time (\textit{now} velocity). |
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327 | |
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328 | %-------------------------------------------------------------------------------------------------------------- |
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329 | % Flux form Advection term |
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330 | %-------------------------------------------------------------------------------------------------------------- |
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331 | \subsection [Flux form Advection term (\textit{dynadv}) ] |
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332 | {Flux form Advection term (\mdl{dynadv}) } |
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333 | \label{DYN_adv_flux} |
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334 | |
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335 | The discrete expression of the advection term is given by : |
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336 | \begin{equation} \label{Eq_dynadv} |
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337 | \left\{ |
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338 | \begin{aligned} |
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339 | \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} |
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340 | \left( \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\ u }^{i } \ u_T \right] |
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341 | + \delta _{j } \left[ \overline{e_{1u}\,e_{3u}\ v }^{i+1/2} \ u_F \right] \right. \ \; \\ |
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342 | \left. + \delta _{k } \left[ \overline{e_{1w}\,e_{2w} w}^{i+1/2} \ u_{uw} \right] \right) \\ |
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343 | \\ |
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344 | \frac{1}{e_{1v}\,e_{2v}\,e_{3v}} |
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345 | \left( \delta _{i } \left[ \overline{e_{2u}\,e_{3u } \ u }^{j+1/2} \ v_F \right] |
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346 | + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u } \ v }^{i } \ v_T \right] \right. \ \, \\ |
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347 | \left. + \delta _{k } \left[ \overline{e_{1w}\,e_{2w} \ w}^{j+1/2} \ v_{vw} \right] \right) \\ |
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348 | \end{aligned} |
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349 | \right. |
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350 | \end{equation} |
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351 | |
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352 | Two advection schemes are available: a $2^{nd}$ order centered finite |
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353 | difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS. |
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354 | The latter is described in \citet{Sacha2005}. The schemes are selected using |
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355 | the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. In flux |
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356 | form, the schemes differ by the choice of a space and time interpolation to |
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357 | define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells, |
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358 | $i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and |
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359 | $vw$-points for $v$. |
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360 | |
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361 | %------------------------------------------------------------- |
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362 | % 2nd order centred scheme |
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363 | %------------------------------------------------------------- |
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364 | \subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=.true.)} |
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365 | \label{DYN_adv_cen2} |
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366 | |
---|
367 | In the centered $2^{nd}$ order formulation, the velocity is evaluated as the |
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368 | mean of the two neighbouring points : |
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369 | \begin{equation} \label{Eq_dynadv_cen2} |
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370 | \left\{ \begin{aligned} |
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371 | u_T^{cen2} &=\overline u^{i } \quad & |
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372 | u_F^{cen2} &=\overline u^{j+1/2} \quad & |
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373 | u_{uw}^{cen2} &=\overline u^{k+1/2} \\ |
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374 | v_F^{cen2} &=\overline v ^{i+1/2} \quad & |
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375 | v_F^{cen2} &=\overline v^j \quad & |
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376 | v_{vw}^{cen2} &=\overline v ^{k+1/2} \\ |
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377 | \end{aligned} \right. |
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378 | \end{equation} |
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379 | |
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380 | The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive |
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381 | ($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be |
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382 | used in conjunction with an explicit diffusion operator to produce a sensible solution. |
---|
383 | The associated time-stepping is performed using a leapfrog scheme in conjunction |
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384 | with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities. |
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385 | |
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386 | %------------------------------------------------------------- |
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387 | % UBS scheme |
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388 | %------------------------------------------------------------- |
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389 | \subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=.true.)} |
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390 | \label{DYN_adv_ubs} |
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391 | |
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392 | The UBS advection scheme is an upstream biased third order scheme based on |
---|
393 | an upstream-biased parabolic interpolation. For example, the evaluation of |
---|
394 | $u_T^{ubs} $ is done as follows: |
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395 | \begin{equation} \label{Eq_dynadv_ubs} |
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396 | u_T^{ubs} =\overline u ^i-\;\frac{1}{6} \begin{cases} |
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397 | u"_{i-1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i \geqslant 0$ } \\ |
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398 | u"_{i+1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i < 0$ } |
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399 | \end{cases} |
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400 | \end{equation} |
---|
401 | where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results |
---|
402 | in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Sacha2005}. |
---|
403 | The overall performance of the advection scheme is similar to that reported in |
---|
404 | \citet{Farrow1995}. It is a relatively good compromise between accuracy and |
---|
405 | smoothness. It is not a \emph{positive} scheme, meaning that false extrema are |
---|
406 | permitted. But the amplitudes of the false extrema are significantly reduced over |
---|
407 | those in the centred second order method. |
---|
408 | |
---|
409 | The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ |
---|
410 | order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and |
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411 | $u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is |
---|
412 | associated with vertical mixing of momentum. \gmcomment{ gm pursue the |
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413 | sentence:Since vertical mixing of momentum is a source term of the TKE equation... } |
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414 | |
---|
415 | For stability reasons, the first term in (\ref{Eq_dynadv_ubs}), which corresponds |
---|
416 | to a second order centred scheme, is evaluated using the \textit{now} velocity |
---|
417 | (centred in time), while the second term, which is the diffusive part of the scheme, |
---|
418 | is evaluated using the \textit{before} velocity (forward in time). This is discussed |
---|
419 | by \citet{Webb1998} in the context of the Quick advection scheme. |
---|
420 | |
---|
421 | Note that the UBS and Quadratic Upstream Interpolation for Convective Kinematics |
---|
422 | (QUICK) schemes only differ by one coefficient. Substituting $1/6$ with $1/8$ in |
---|
423 | (\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. |
---|
424 | This option is not available through a namelist parameter, since the $1/6$ coefficient |
---|
425 | is hard coded. Nevertheless it is quite easy to make the substitution in |
---|
426 | \mdl{dynadv\_ubs} module and obtain a QUICK scheme. |
---|
427 | |
---|
428 | Note also that in the current version of \mdl{dynadv\_ubs}, there is also the |
---|
429 | possibility of using a $4^{th}$ order evaluation of the advective velocity as in |
---|
430 | ROMS. This is an error and should be suppressed soon. |
---|
431 | %%% |
---|
432 | \gmcomment{action : this have to be done} |
---|
433 | %%% |
---|
434 | |
---|
435 | % ================================================================ |
---|
436 | % Hydrostatic pressure gradient term |
---|
437 | % ================================================================ |
---|
438 | \section [Hydrostatic pressure gradient (\textit{dynhpg})] |
---|
439 | {Hydrostatic pressure gradient (\mdl{dynhpg})} |
---|
440 | \label{DYN_hpg} |
---|
441 | %------------------------------------------nam_dynhpg--------------------------------------------------- |
---|
442 | \namdisplay{nam_dynhpg} |
---|
443 | \namdisplay{namflg} |
---|
444 | %------------------------------------------------------------------------------------------------------------- |
---|
445 | %%% |
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446 | \gmcomment{Suppress the namflg namelist and incorporate it in the namhpg namelist} |
---|
447 | %%% |
---|
448 | |
---|
449 | The key distinction between the different algorithms used for the hydrostatic |
---|
450 | pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal} |
---|
451 | pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any |
---|
452 | tilt of the surface of the computational levels will require a specific treatment to |
---|
453 | compute the hydrostatic pressure gradient. |
---|
454 | |
---|
455 | The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, |
---|
456 | $i.e.$ the density appearing in its expression is centred in time (\emph{now} rho), or |
---|
457 | a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip |
---|
458 | boundary conditions are applied. |
---|
459 | |
---|
460 | %-------------------------------------------------------------------------------------------------------------- |
---|
461 | % z-coordinate with full step |
---|
462 | %-------------------------------------------------------------------------------------------------------------- |
---|
463 | \subsection [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ] |
---|
464 | {$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=.true.)} |
---|
465 | \label{DYN_hpg_zco} |
---|
466 | |
---|
467 | The hydrostatic pressure can be obtained by integrating the hydrostatic equation |
---|
468 | vertically from the surface. However, the pressure is large at great depth while its |
---|
469 | horizontal gradient is several orders of magnitude smaller. This may lead to large |
---|
470 | truncation errors in the pressure gradient terms. Thus, the two horizontal components |
---|
471 | of the hydrostatic pressure gradient are computed directly as follows: |
---|
472 | |
---|
473 | for $k=km$ (surface layer, $jk=1$ in the code) |
---|
474 | \begin{equation} \label{Eq_dynhpg_zco_surf} |
---|
475 | \left\{ \begin{aligned} |
---|
476 | \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k=km} |
---|
477 | &= \frac{1}{2} g \ \left. \delta _{i+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\ |
---|
478 | \left. \delta _{j+1/2} \left[ p^h \right] \right|_{k=km} |
---|
479 | &= \frac{1}{2} g \ \left. \delta _{j+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\ |
---|
480 | \end{aligned} \right. |
---|
481 | \end{equation} |
---|
482 | |
---|
483 | for $1<k<km$ (interior layer) |
---|
484 | \begin{equation} \label{Eq_dynhpg_zco} |
---|
485 | \left\{ \begin{aligned} |
---|
486 | \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k} |
---|
487 | &= \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k-1} |
---|
488 | + \frac{1}{2}\;g\; \left. \delta _{i+1/2} \left[ e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k} \\ |
---|
489 | \left. \delta _{j+1/2} \left[ p^h \right] \right|_{k} |
---|
490 | &= \left. \delta _{j+1/2} \left[ p^h \right] \right|_{k-1} |
---|
491 | + \frac{1}{2}\;g\; \left. \delta _{j+1/2} \left[ e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k} \\ |
---|
492 | \end{aligned} \right. |
---|
493 | \end{equation} |
---|
494 | |
---|
495 | Note that the $1/2$ factor in (\ref{Eq_dynhpg_zco_surf}) is adequate because of |
---|
496 | the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface |
---|
497 | level ($z=0)$. |
---|
498 | |
---|
499 | %-------------------------------------------------------------------------------------------------------------- |
---|
500 | % z-coordinate with partial step |
---|
501 | %-------------------------------------------------------------------------------------------------------------- |
---|
502 | \subsection [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})] |
---|
503 | {$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=.true.)} |
---|
504 | \label{DYN_hpg_zps} |
---|
505 | |
---|
506 | With partial bottom cells, tracers in horizontally adjacent cells generally live at |
---|
507 | different depths. Before taking horizontal gradients between these tracer points, |
---|
508 | a linear interpolation is used to approximate the deeper tracer as if it actually lived |
---|
509 | at the depth of the shallower tracer point. |
---|
510 | |
---|
511 | Apart from this modification, the horizontal hydrostatic pressure gradient evaluated |
---|
512 | in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case. |
---|
513 | As explained in detail in section \S\ref{TRA_zpshde}, the nonlinearity of pressure |
---|
514 | effects in the equation of state is such that it is better to interpolate temperature and |
---|
515 | salinity vertically before computing the density. Horizontal gradients of temperature |
---|
516 | and salinity are needed for the TRA modules, which is the reason why the horizontal |
---|
517 | gradients of density at the deepest model level are computed in module \mdl{zpsdhe} |
---|
518 | located in the TRA directory and described in \S\ref{TRA_zpshde}. |
---|
519 | |
---|
520 | %-------------------------------------------------------------------------------------------------------------- |
---|
521 | % s- and s-z-coordinates |
---|
522 | %-------------------------------------------------------------------------------------------------------------- |
---|
523 | \subsection{$s$- and $z$-$s$-coordinates} |
---|
524 | \label{DYN_hpg_sco} |
---|
525 | |
---|
526 | Pressure gradient formulations in $s$-coordinate have been the subject of a vast |
---|
527 | literature ($e.g.$, \citet{Song1998, Sacha2003}). A number of different pressure |
---|
528 | gradient options are coded, but they are not yet fully documented or tested. |
---|
529 | |
---|
530 | $\bullet$ Traditional coding (see for example \citet{Madec1996}: (\np{ln\_dynhpg\_sco}=.true., |
---|
531 | \np{ln\_dynhpg\_hel}=.true.) |
---|
532 | \begin{equation} \label{Eq_dynhpg_sco} |
---|
533 | \left\{ \begin{aligned} |
---|
534 | - \frac{1} {\rho_o \, e_{1u}} \; \delta _{i+1/2} \left[ p^h \right] |
---|
535 | + \frac{g\; \overline {\rho}^{i+1/2}} {\rho_o \, e_{1u}} \; \delta _{i+1/2} \left[ z_T \right] \\ |
---|
536 | - \frac{1} {\rho_o \, e_{2v}} \; \delta _{j+1/2} \left[ p^h \right] |
---|
537 | + \frac{g\; \overline {\rho}^{j+1/2}} {\rho_o \, e_{2v}} \; \delta _{j+1/2} \left[ z_T \right] \\ |
---|
538 | \end{aligned} \right. |
---|
539 | \end{equation} |
---|
540 | |
---|
541 | Where the first term is the pressure gradient along coordinates, computed as in |
---|
542 | \eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of |
---|
543 | the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point |
---|
544 | ($e_{3w}$). The version \np{ln\_dynhpg\_hel}=.true. has been added by Aike |
---|
545 | Beckmann and involves a redefinition of the relative position of $T$-points relative |
---|
546 | to $w$-points. |
---|
547 | |
---|
548 | $\bullet$ Weighted density Jacobian (WDJ) \citep{Song1998} (\np{ln\_dynhpg\_wdj}=.true.) |
---|
549 | |
---|
550 | $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Sacha2003} |
---|
551 | (\np{ln\_dynhpg\_djc}=.true.) |
---|
552 | |
---|
553 | $\bullet$ Rotated axes scheme (rot) \citep{Thiem2006} (\np{ln\_dynhpg\_rot}=.true.) |
---|
554 | |
---|
555 | Note that expression \eqref{Eq_dynhpg_sco} is used when the variable volume |
---|
556 | formulation is activated (\key{vvl}) because in that case, even with a flat bottom, |
---|
557 | the coordinate surfaces are not horizontal but follow the free surface |
---|
558 | \citep{Levier2007}. The other pressure gradient options are not yet available. |
---|
559 | |
---|
560 | %-------------------------------------------------------------------------------------------------------------- |
---|
561 | % Time-scheme |
---|
562 | %-------------------------------------------------------------------------------------------------------------- |
---|
563 | \subsection [Time-scheme (\np{ln\_dynhpg\_imp}) ] |
---|
564 | {Time-scheme (\np{ln\_dynhpg\_imp}=.true./.false.)} |
---|
565 | \label{DYN_hpg_imp} |
---|
566 | |
---|
567 | The default time differencing scheme used for the horizontal pressure gradient is |
---|
568 | a leapfrog scheme and therefore the density used in all discrete expressions given |
---|
569 | above is the \textit{now} density, computed from the \textit{now} temperature and |
---|
570 | salinity. In some specific cases (usually high resolution simulations over an ocean |
---|
571 | domain which includes weakly stratified regions) the physical phenomenum that |
---|
572 | controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for |
---|
573 | doubling the stability limit associated with IGWs can be used \citep{Brown1978, |
---|
574 | Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an |
---|
575 | average over the three time levels $t-\Delta t$, $t$, and $t+\Delta t$ ($i.e.$ |
---|
576 | \textit{before}, \textit{now} and \textit{after} time-steps), rather than at central |
---|
577 | time level $t$ only, as in the standard leapfrog scheme. |
---|
578 | |
---|
579 | $\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=.true.): |
---|
580 | |
---|
581 | \begin{equation} \label{Eq_dynhpg_lf} |
---|
582 | \frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t} |
---|
583 | =\;\cdots \;-\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right] |
---|
584 | \end{equation} |
---|
585 | |
---|
586 | $\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=.true.): |
---|
587 | \begin{equation} \label{Eq_dynhpg_imp} |
---|
588 | \frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t} |
---|
589 | =\;\cdots \;-\frac{1}{\rho _o \,e_{1u} } \delta _{i+1/2} \left[ \frac{ p_h^{t+\Delta t} +2p_h^t |
---|
590 | +p_h^{t-\Delta t} } { 4 } \right] |
---|
591 | \end{equation} |
---|
592 | |
---|
593 | The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without |
---|
594 | significant additional computation since the density can be updated to time level |
---|
595 | $t+\Delta t$ before computing the horizontal hydrostatic pressure gradient. It can |
---|
596 | be easily shown that the stability limit associated with the hydrostatic pressure |
---|
597 | gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the |
---|
598 | standard leapfrog scheme \eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp} |
---|
599 | is equivalent to applying a time filter to the pressure gradient to eliminate high |
---|
600 | frequency IGWs. Obviously, when using \eqref{Eq_dynhpg_imp}, the doubling of |
---|
601 | the time-step is achievable only if no other factors control the time-step, such as |
---|
602 | the stability limits associated with advection or diffusion. |
---|
603 | |
---|
604 | In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=.true.. |
---|
605 | In this case, we choose to apply the time filter to temperature and salinity used in |
---|
606 | the equation of state, instead of applying it to the hydrostatic pressure or to the |
---|
607 | density, so that no additional storage array has to be defined. The density used to |
---|
608 | compute the hydrostatic pressure gradient (whatever the formulation) is evaluated |
---|
609 | as follows: |
---|
610 | \begin{equation} \label{Eq_rho_flt} |
---|
611 | \rho^t = \rho( \widetilde{T},\widetilde {S},z_T) |
---|
612 | \quad \text{with} \quad |
---|
613 | \widetilde{\,\cdot\,} = \frac{ \,\cdot\,^{t+\Delta t} +2 \,\,\cdot\,^t + \,\cdot\,^{t-\Delta t} } {4} |
---|
614 | \end{equation} |
---|
615 | \gmcomment{STEVEN: bullets look odd in this, could use X} |
---|
616 | |
---|
617 | Note that in the semi-implicit case, it is necessary to save the filtered density, an |
---|
618 | extra three-dimensional field, in the restart file to restart the model with exact |
---|
619 | reproducibility. This option is controlled by the namelist parameter |
---|
620 | \np{nn\_dynhpg\_rst}=.true.. |
---|
621 | |
---|
622 | % ================================================================ |
---|
623 | % Surface Pressure Gradient |
---|
624 | % ================================================================ |
---|
625 | \section [Surface pressure gradient (\textit{dynspg}) ] |
---|
626 | {Surface pressure gradient (\mdl{dynspg})} |
---|
627 | \label{DYN_hpg_spg} |
---|
628 | %-----------------------------------------nam_dynspg---------------------------------------------------- |
---|
629 | \namdisplay{nam_dynspg} |
---|
630 | %------------------------------------------------------------------------------------------------------------ |
---|
631 | |
---|
632 | The form of the surface pressure gradient term is dependent on the representation |
---|
633 | of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed |
---|
634 | volume case (linear free surface or rigid lid) and the variable volume case |
---|
635 | (nonlinear free surface, \key{vvl} is defined). In the linear free surface case |
---|
636 | (\S\ref{PE_free_surface}) and the rigid lid case (\S\ref{PE_rigid_lid}), the vertical |
---|
637 | scale factors $e_{3}$ are fixed in time, whilst in the nonlinear case |
---|
638 | (\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear |
---|
639 | free surface, external gravity waves are allowed in the equations, which imposes |
---|
640 | a very small time step when an explicit time stepping is used. Two methods are |
---|
641 | proposed to allow a longer time step for the three-dimensional equations: the |
---|
642 | filtered free surface method, which involves a modification of the continuous |
---|
643 | equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface method |
---|
644 | described below. The extra term introduced in the filtered method is calculated |
---|
645 | implicitly, so that the update of the $next$ velocities is done in module |
---|
646 | \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
---|
647 | |
---|
648 | %-------------------------------------------------------------------------------------------------------------- |
---|
649 | % Linear free surface formulation |
---|
650 | %-------------------------------------------------------------------------------------------------------------- |
---|
651 | \subsection{Linear free surface formulation (\key{exp} or \textbf{\_ts} or \textbf{\_flt})} |
---|
652 | \label{DYN_spg_linear} |
---|
653 | |
---|
654 | In the linear free surface formulation, the sea surface height is assumed to be |
---|
655 | small compared to the thickness of the ocean levels, so that $(a)$ the time |
---|
656 | evolution of the sea surface height becomes a linear equation, and $(b)$ the |
---|
657 | thickness of the ocean levels is assumed to be constant with time. |
---|
658 | As mentioned in (\S\ref{PE_free_surface}) the linearization affects the |
---|
659 | conservation of tracers. |
---|
660 | |
---|
661 | %------------------------------------------------------------- |
---|
662 | % Explicit |
---|
663 | %------------------------------------------------------------- |
---|
664 | \subsubsection{Explicit (\key{dynspg\_exp})} |
---|
665 | \label{DYN_spg_exp} |
---|
666 | |
---|
667 | In the explicit free surface formulation, the model time step is chosen to be |
---|
668 | small enough to describe the external gravity waves (typically a few tens of |
---|
669 | seconds). The sea surface height is given by : |
---|
670 | \begin{equation} \label{Eq_dynspg_ssh} |
---|
671 | \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} |
---|
672 | e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} |
---|
673 | \right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)} |
---|
674 | \end{equation} |
---|
675 | where EMP is the surface freshwater budget, expressed in Kg/m$^2$/s |
---|
676 | (which is equal to mm/s), and $\rho _w$=1,000~Kg/m$^3$ is the volumic |
---|
677 | mass of pure water. If river runoff is expressed as a surface freshwater flux |
---|
678 | (see \S\ref{SBC}) then EMP can be written as the evaporation minus |
---|
679 | precipitation, minus the river runoff. The sea-surface height is evaluated |
---|
680 | using a leapfrog scheme in combination with an Asselin time filter, $i.e.$ |
---|
681 | the velocity appearing in \eqref{Eq_dynspg_ssh} is centred in time |
---|
682 | (\textit{now} velocity). |
---|
683 | |
---|
684 | The surface pressure gradient, also evaluated using a leap-frog scheme, is |
---|
685 | then simply given by : |
---|
686 | \begin{equation} \label{Eq_dynspg_exp} |
---|
687 | \left\{ \begin{aligned} |
---|
688 | - \frac{1}{e_{1u}} \; \delta _{i+1/2} \left[ \,\eta\, \right] \\ |
---|
689 | - \frac{1}{e_{2v}} \; \delta _{j+1/2} \left[ \,\eta\, \right] |
---|
690 | \end{aligned} \right. |
---|
691 | \end{equation} |
---|
692 | |
---|
693 | Consistent with the linearization, a factor of $\left. \rho \right|_{k=1} / \rho _o$ |
---|
694 | is omitted in \eqref{Eq_dynspg_exp}. |
---|
695 | |
---|
696 | %------------------------------------------------------------- |
---|
697 | % Split-explicit time-stepping |
---|
698 | %------------------------------------------------------------- |
---|
699 | \subsubsection{Split-explicit time-stepping (\key{dynspg\_ts})} |
---|
700 | \label{DYN_spg_ts} |
---|
701 | %--------------------------------------------namdom---------------------------------------------------- |
---|
702 | \namdisplay{namdom} |
---|
703 | %-------------------------------------------------------------------------------------------------------------- |
---|
704 | |
---|
705 | The split-explicit free surface formulation used in \NEMO follows the one |
---|
706 | proposed by \citet{Griffies2004}. The general idea is to solve the free surface |
---|
707 | equation with a small time step \np{rdtbt}, while the three dimensional |
---|
708 | prognostic variables are solved with a longer time step that is a multiple of |
---|
709 | \np{rdtbt} (Fig.\ref {Fig_DYN_dynspg_ts}). |
---|
710 | |
---|
711 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
---|
712 | \begin{figure}[!t] \label{Fig_DYN_dynspg_ts} |
---|
713 | \begin{center} |
---|
714 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf} |
---|
715 | \caption{Schematic of the split-explicit time stepping scheme for the external |
---|
716 | and internal modes. Time increases to the right. |
---|
717 | Internal mode time steps (which are also the model time steps) are denoted |
---|
718 | by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. |
---|
719 | The curved line represents a leap-frog time step, and the smaller time |
---|
720 | steps $N \Delta t_e=\frac{3}{2}\Delta t$ are denoted by the zig-zag line. |
---|
721 | The vertically integrated forcing \textbf{M}(t) computed at the model time step $t$ |
---|
722 | represents the interaction between the external and internal motions. |
---|
723 | While keeping \textbf{M} and freshwater forcing field fixed, a leap-frog |
---|
724 | integration carries the external mode variables (surface height and vertically |
---|
725 | integrated velocity) from $t$ to $t+\frac{3}{2} \Delta t$ using N external time |
---|
726 | steps of length $\Delta t_e$. Time averaging the external fields over the |
---|
727 | $\frac{2}{3}N+1$ time steps (endpoints included) centers the vertically integrated |
---|
728 | velocity and the sea surface height at the model timestep $t+\Delta t$. |
---|
729 | These averaged values are used to update \textbf{M}(t) with both the surface |
---|
730 | pressure gradient and the Coriolis force, therefore providing the $t+\Delta t$ |
---|
731 | velocity. The model time stepping scheme can then be achieved by a baroclinic |
---|
732 | leap-frog time step that carries the surface height from $t-\Delta t$ to $t+\Delta t$. } |
---|
733 | \end{center} |
---|
734 | \end{figure} |
---|
735 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
---|
736 | |
---|
737 | The split-explicit formulation has a damping effect on external gravity waves, |
---|
738 | which is weaker damping than for the filtered free surface but still significant as |
---|
739 | shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. |
---|
740 | |
---|
741 | %------------------------------------------------------------- |
---|
742 | % Filtered formulation |
---|
743 | %------------------------------------------------------------- |
---|
744 | \subsubsection{Filtered formulation (\key{dynspg\_flt})} |
---|
745 | \label{DYN_spg_flt} |
---|
746 | |
---|
747 | The filtered formulation follows the \citet{Roullet2000} implementation. The extra |
---|
748 | term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic |
---|
749 | solvers available in the code are documented in \S\ref{MISC}. The amplitude of |
---|
750 | the extra term is given by the namelist variable \np{rnu}. The default value is 1, |
---|
751 | as recommended by \citet{Roullet2000} |
---|
752 | |
---|
753 | \gmcomment{\np{rnu}=1 to be suppressed from namelist !} |
---|
754 | |
---|
755 | %------------------------------------------------------------- |
---|
756 | % Non-linear free surface formulation |
---|
757 | %------------------------------------------------------------- |
---|
758 | \subsection{Non-linear free surface formulation (\key{vvl})} |
---|
759 | \label{DYN_spg_vvl} |
---|
760 | |
---|
761 | In the non-linear free surface formulation, the variations of volume are fully |
---|
762 | taken into account. This option is presented in a report \citep{Levier2007} |
---|
763 | available on the \NEMO web site. The three time-stepping methods (explicit, |
---|
764 | split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except |
---|
765 | that the ocean depth is now time-dependent. In particular, this means that |
---|
766 | in the filtered case, the matrix to be inverted has to be recomputed at each |
---|
767 | time-step. |
---|
768 | |
---|
769 | %-------------------------------------------------------------------------------------------------------------- |
---|
770 | % Rigid-lid formulation |
---|
771 | %-------------------------------------------------------------------------------------------------------------- |
---|
772 | \subsection{Rigid-lid formulation (\key{dynspg\_rl})} |
---|
773 | \label{DYN_spg_rl} |
---|
774 | |
---|
775 | With the rigid lid formulation, an elliptic equation has to be solved for |
---|
776 | the barotropic streamfunction. For consistency this equation is obtained by |
---|
777 | taking the discrete curl of the discrete vertical sum of the discrete |
---|
778 | momentum equation: |
---|
779 | \begin{equation}\label{Eq_dynspg_rl} |
---|
780 | \frac{1}{\rho _o }\nabla _h p_s \equiv \left( {{\begin{array}{*{20}c} |
---|
781 | {\overline M_u +\frac{1}{H\;e_2 } \delta_ j \left[ \partial_t \psi \right]} \\ |
---|
782 | \\ |
---|
783 | {\overline M_v -\frac{1}{H\;e_1 } \delta_i \left[ \partial_t \psi \right]} \\ |
---|
784 | \end{array} }} \right) |
---|
785 | \end{equation} |
---|
786 | |
---|
787 | Here ${\rm {\bf M}}= \left( M_u,M_v \right)$ represents the collected |
---|
788 | contributions of nonlinear, viscous and hydrostatic pressure gradient terms in |
---|
789 | \eqref{Eq_PE_dyn} and the overbar indicates a vertical average over the |
---|
790 | whole water column (i.e. from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid). |
---|
791 | The time derivative of $\psi$ is the solution of an elliptic equation: |
---|
792 | \begin{multline} \label{Eq_bsf} |
---|
793 | \delta_{i+1/2} \left[ \frac{e_{2v}}{H_v\;e_{1v}} \delta_{i} \left[ \partial_t \psi \right] \right] |
---|
794 | + \delta_{j+1/2} \left[ \frac{e_{1u}}{H_u\;e_{2u}} \delta_{j} \left[ \partial_t \psi \right] \right] |
---|
795 | \\ = |
---|
796 | \delta_{i+1/2} \left[ e_{2v} M_v \right] |
---|
797 | - \delta_{j+1/2} \left[ e_{1u} M_u \right] |
---|
798 | \end{multline} |
---|
799 | |
---|
800 | The elliptic solvers available in the code are documented in \S\ref{MISC}). |
---|
801 | The boundary conditions must be given on all separate landmasses (islands), |
---|
802 | which is done by integrating the vorticity equation around each island. This |
---|
803 | requires identifying each island in the bathymetry file, a cumbersome task. |
---|
804 | This explains why the rigid lid option is not recommended with complex |
---|
805 | domains such as the global ocean. Parameters jpisl (number of islands) and |
---|
806 | jpnisl (maximum number of points per island) of the \hf{par\_oce} file are |
---|
807 | related to this option. |
---|
808 | |
---|
809 | |
---|
810 | % ================================================================ |
---|
811 | % Lateral diffusion term |
---|
812 | % ================================================================ |
---|
813 | \section [Lateral diffusion term (\textit{dynldf})] |
---|
814 | {Lateral diffusion term (\mdl{dynldf})} |
---|
815 | \label{DYN_ldf} |
---|
816 | %------------------------------------------nam_dynldf---------------------------------------------------- |
---|
817 | \namdisplay{nam_dynldf} |
---|
818 | %------------------------------------------------------------------------------------------------------------- |
---|
819 | |
---|
820 | The options available for lateral diffusion are for the choice of laplacian |
---|
821 | (rotated or not) or biharmonic operators. The coefficients may be constant |
---|
822 | or spatially variable; the description of the coefficients is found in the chapter |
---|
823 | on lateralphysics (Chap.\ref{LDF}). The lateral diffusion of momentum is |
---|
824 | evaluated using a forward scheme, i.e. the velocity appearing in its expression |
---|
825 | is the \textit{before} velocity in time, except for the pure vertical component |
---|
826 | that appears when a tensor of rotation is used. This latter term is solved |
---|
827 | implicitly together with the vertical diffusion term (see \S\ref{DOM_nxt}) |
---|
828 | |
---|
829 | At the lateral boundaries either free slip, no slip or partial slip boundary |
---|
830 | conditions are applied according to the user's choice (see Chap.\ref{LBC}). |
---|
831 | |
---|
832 | % ================================================================ |
---|
833 | \subsection [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ] |
---|
834 | {Iso-level laplacian operator (\np{ln\_dynldf\_lap}=.true.)} |
---|
835 | \label{DYN_ldf_lap} |
---|
836 | |
---|
837 | For lateral iso-level diffusion, the discrete operator is: |
---|
838 | \begin{equation} \label{Eq_dynldf_lap} |
---|
839 | \left\{ \begin{aligned} |
---|
840 | D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} |
---|
841 | \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[ |
---|
842 | {A_f^{lm} \;e_{3f} \zeta } \right] \\ |
---|
843 | \\ |
---|
844 | D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} |
---|
845 | \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[ |
---|
846 | {A_f^{lm} \;e_{3f} \zeta } \right] \\ |
---|
847 | \end{aligned} \right. |
---|
848 | \end{equation} |
---|
849 | |
---|
850 | As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence |
---|
851 | and curl of the vorticity) preserves symmetry and ensures a complete |
---|
852 | separation between the vorticity and divergence parts. Note that in the full step |
---|
853 | $z$-coordinate (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$ so that they |
---|
854 | cancel in the rotational part of \eqref{Eq_dynldf_lap}. |
---|
855 | |
---|
856 | %-------------------------------------------------------------------------------------------------------------- |
---|
857 | % Rotated laplacian operator |
---|
858 | %-------------------------------------------------------------------------------------------------------------- |
---|
859 | \subsection [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ] |
---|
860 | {Rotated laplacian operator (\np{ln\_dynldf\_iso}=.true.)} |
---|
861 | \label{DYN_ldf_iso} |
---|
862 | |
---|
863 | A rotation of the lateral momentum diffusive operator is needed in several cases: |
---|
864 | for iso-neutral diffusion in $z$-coordinate (\np{ln\_dynldf\_iso}=.true.) and for |
---|
865 | either iso-neutral (\np{ln\_dynldf\_iso}=.true.) or geopotential |
---|
866 | (\np{ln\_dynldf\_hor}=.true.) diffusion in $s$-coordinate. In the partial step |
---|
867 | case, coordinates are horizontal excepted at the deepest level and no |
---|
868 | rotation is performed when \np{ln\_dynldf\_hor}=.true.. The diffusive operator |
---|
869 | is defined simply as the divergence of down gradient momentum fluxes on each |
---|
870 | momentum component. It must be emphasized that this formulation ignores |
---|
871 | constraints on the stress tensor such as symmetry. The resulting discrete |
---|
872 | representation is: |
---|
873 | \begin{equation} \label{Eq_dyn_ldf_iso} |
---|
874 | \begin{split} |
---|
875 | D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ |
---|
876 | & \left\{\quad {\delta _{i+1/2} \left[ {A_T^{lm} \left( |
---|
877 | {\frac{e_{2T} \; e_{3T} }{e_{1T} } \,\delta _{i}[u] |
---|
878 | -e_{2T} \; r_{1T} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}} |
---|
879 | \right)} \right]} \right. |
---|
880 | \\ |
---|
881 | & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} |
---|
882 | }\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} |
---|
883 | \,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} |
---|
884 | \right)} \right] |
---|
885 | \\ |
---|
886 | &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline |
---|
887 | {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } |
---|
888 | \right.} \right. |
---|
889 | \\ |
---|
890 | & \ \qquad \qquad \qquad \quad\ |
---|
891 | - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} |
---|
892 | \\ |
---|
893 | & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ |
---|
894 | +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} |
---|
895 | \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} |
---|
896 | \\ |
---|
897 | \\ |
---|
898 | D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\ |
---|
899 | & \left\{\quad {\delta _{i+1/2} \left[ {A_f^{lm} \left( |
---|
900 | {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v] |
---|
901 | -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}} |
---|
902 | \right)} \right]} \right. |
---|
903 | \\ |
---|
904 | & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1T}\,e_{3T} }{e_{2T} |
---|
905 | }\,\delta _{j} [v] - e_{1T}\, r_{2T} |
---|
906 | \,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} |
---|
907 | \right)} \right] |
---|
908 | \\ |
---|
909 | & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline |
---|
910 | {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. |
---|
911 | \\ |
---|
912 | & \ \qquad \qquad \qquad \quad\ |
---|
913 | - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} |
---|
914 | \\ |
---|
915 | & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ |
---|
916 | +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} |
---|
917 | \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} |
---|
918 | \end{split} |
---|
919 | \end{equation} |
---|
920 | where $r_1$ and $r_2$ are the slopes between the surface along which the |
---|
921 | diffusive operator acts and the surface of computation ($z$- or $s$-surfaces). |
---|
922 | The way these slopes are evaluated is given in the lateral physics chapter |
---|
923 | (Chap.\ref{LDF}). |
---|
924 | |
---|
925 | %-------------------------------------------------------------------------------------------------------------- |
---|
926 | % Iso-level bilaplacian operator |
---|
927 | %-------------------------------------------------------------------------------------------------------------- |
---|
928 | \subsection [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})] |
---|
929 | {Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=.true.)} |
---|
930 | \label{DYN_ldf_bilap} |
---|
931 | |
---|
932 | The lateral fourth order operator formulation on momentum is obtained by |
---|
933 | applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on |
---|
934 | boundary conditions: the first derivative term normal to the coast depends on |
---|
935 | the free or no-slip lateral boundary conditions chosen, while the third |
---|
936 | derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}). |
---|
937 | %%% |
---|
938 | \gmcomment{add a remark on the the change in the position of the coefficient} |
---|
939 | %%% |
---|
940 | |
---|
941 | % ================================================================ |
---|
942 | % Vertical diffusion term |
---|
943 | % ================================================================ |
---|
944 | \section [Vertical diffusion term (\mdl{dynzdf})] |
---|
945 | {Vertical diffusion term (\mdl{dynzdf})} |
---|
946 | \label{DYN_zdf} |
---|
947 | %----------------------------------------------namzdf------------------------------------------------------ |
---|
948 | \namdisplay{namzdf} |
---|
949 | %------------------------------------------------------------------------------------------------------------- |
---|
950 | |
---|
951 | The large vertical diffusion coefficient found in the surface mixed layer together |
---|
952 | with high vertical resolution implies that in the case of explicit time stepping there |
---|
953 | would be too restrictive a constraint on the time step. Two time stepping schemes |
---|
954 | can be used for the vertical diffusion term : $(a)$ a forward time differencing |
---|
955 | scheme (\np{ln\_zdfexp}=.true.) using a time splitting technique |
---|
956 | (\np{n\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme |
---|
957 | (\np{ln\_zdfexp}=.false.) (see \S\ref{DOM_nxt}). Note that namelist variables |
---|
958 | \np{ln\_zdfexp} and \np{n\_zdfexp} apply to both tracers and dynamics. |
---|
959 | |
---|
960 | The formulation of the vertical subgrid scale physics is the same whatever |
---|
961 | the vertical coordinate is. The vertical diffusion operators given by |
---|
962 | \eqref{Eq_PE_zdf} take the following semi-discrete space form: |
---|
963 | \begin{equation} \label{Eq_dynzdf} |
---|
964 | \left\{ \begin{aligned} |
---|
965 | D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } |
---|
966 | \ \delta _{k+1/2} [\,u\,] \right] \\ |
---|
967 | \\ |
---|
968 | D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } |
---|
969 | \ \delta _{k+1/2} [\,v\,] \right] |
---|
970 | \end{aligned} \right. |
---|
971 | \end{equation} |
---|
972 | where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and |
---|
973 | diffusivity coefficients. The way these coefficients are evaluated |
---|
974 | depends on the vertical physics used (see \S\ref{ZDF}). |
---|
975 | |
---|
976 | The surface boundary condition on momentum is given by the stress exerted by |
---|
977 | the wind. At the surface, the momentum fluxes are prescribed as the boundary |
---|
978 | condition on the vertical turbulent momentum fluxes, |
---|
979 | \begin{equation} \label{Eq_dynzdf_sbc} |
---|
980 | \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} |
---|
981 | = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v } |
---|
982 | \end{equation} |
---|
983 | where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress |
---|
984 | vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients |
---|
985 | in the surface mixed layer ensure that the surface wind stress is distributed in |
---|
986 | the vertical over the mixed layer depth. If the vertical mixing coefficient |
---|
987 | is small (when no mixed layer scheme is used) the surface stress enters only |
---|
988 | the top model level, as a body force. The surface wind stress is calculated |
---|
989 | in the surface module routines (SBC, see Chap.\ref{SBC}) |
---|
990 | |
---|
991 | The turbulent flux of momentum at the bottom of the ocean is specified through |
---|
992 | a bottom friction parameterisation (see \S\ref{ZDF_bfr}) |
---|
993 | |
---|
994 | % ================================================================ |
---|
995 | % External Forcing |
---|
996 | % ================================================================ |
---|
997 | \section{External Forcings} |
---|
998 | \label{DYN_forcing} |
---|
999 | |
---|
1000 | Besides the surface and bottom stresses (see the above section) which are |
---|
1001 | introduced as boundary conditions on the vertical mixing, two other forcings |
---|
1002 | enter the dynamical equations. |
---|
1003 | |
---|
1004 | One is the effect of atmospheric pressure on the ocean dynamics (to be |
---|
1005 | introduced later). |
---|
1006 | |
---|
1007 | Another forcing term is the tidal potential, which will be introduced in the |
---|
1008 | reference version soon. |
---|
1009 | |
---|
1010 | % ================================================================ |
---|
1011 | % Time evolution term |
---|
1012 | % ================================================================ |
---|
1013 | \section [Time evolution term (\textit{dynnxt})] |
---|
1014 | {Time evolution term (\mdl{dynnxt})} |
---|
1015 | \label{DYN_nxt} |
---|
1016 | |
---|
1017 | %----------------------------------------------namdom---------------------------------------------------- |
---|
1018 | \namdisplay{namdom} |
---|
1019 | %------------------------------------------------------------------------------------------------------------- |
---|
1020 | |
---|
1021 | The general framework for dynamics time stepping is a leap-frog scheme, |
---|
1022 | $i.e.$ a three level centred time scheme associated with an Asselin time filter |
---|
1023 | (cf. \S\ref{DOM_nxt}) |
---|
1024 | \begin{equation} \label{Eq_dynnxt} |
---|
1025 | \begin{split} |
---|
1026 | &u^{t+\Delta t} = u^{t-\Delta t} + 2 \, \Delta t \ \text{RHS}_u^t \\ |
---|
1027 | \\ |
---|
1028 | &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\Delta t} -2u^t+u^{t+\Delta t}} \right] |
---|
1029 | \end{split} |
---|
1030 | \end{equation} |
---|
1031 | where RHS is the right hand side of the momentum equation, the subscript $f$ |
---|
1032 | denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is |
---|
1033 | initialized as \np{atfp} (namelist parameter). Its default value is \np{atfp} = 0.1. |
---|
1034 | |
---|
1035 | Note that whith the filtered free surface, the update of the \textit{next} velocities |
---|
1036 | is done in the \mdl{dynsp\_flt} module, and only the swap of arrays |
---|
1037 | and Asselin filtering is done in \mdl{dynnxt.} |
---|
1038 | |
---|
1039 | % ================================================================ |
---|
1040 | % Diagnostic variables |
---|
1041 | % ================================================================ |
---|
1042 | \section{Diagnostic variables ($\zeta$, $\chi$, $w$)} |
---|
1043 | \label{DYN_divcur_wzv} |
---|
1044 | |
---|
1045 | %-------------------------------------------------------------------------------------------------------------- |
---|
1046 | % Horizontal divergence and relative vorticity |
---|
1047 | %-------------------------------------------------------------------------------------------------------------- |
---|
1048 | \subsection [Horizontal divergence and relative vorticity (\textit{divcur})] |
---|
1049 | {Horizontal divergence and relative vorticity (\mdl{divcur})} |
---|
1050 | \label{DYN_divcur} |
---|
1051 | |
---|
1052 | The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows: |
---|
1053 | \begin{equation} \label{Eq_divcur_cur} |
---|
1054 | \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right] |
---|
1055 | -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) |
---|
1056 | \end{equation} |
---|
1057 | |
---|
1058 | The horizontal divergence is defined at a $T$-point. It is given by: |
---|
1059 | \begin{equation} \label{Eq_divcur_div} |
---|
1060 | \chi =\frac{1}{e_{1T}\,e_{2T}\,e_{3T} } |
---|
1061 | \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right] |
---|
1062 | +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right) |
---|
1063 | \end{equation} |
---|
1064 | |
---|
1065 | Note that in the $z$-coordinate with full step (\key{zco} is defined), |
---|
1066 | $e_{3u} =e_{3v} =e_{3f}$ so that they cancel in \eqref{Eq_divcur_div}. |
---|
1067 | |
---|
1068 | Note also that whereas the vorticity have the same discrete expression in $z$- |
---|
1069 | and $s$-coordinate, its physical meaning is not identical. $\zeta$ is a pseudo |
---|
1070 | vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along |
---|
1071 | geopotential surfaces, but are no more necessary defined at the same depth). |
---|
1072 | |
---|
1073 | The vorticity and divergence at the \textit{before} step are used in the computation |
---|
1074 | of the horizontal diffusion of momentum. Note that because they have been |
---|
1075 | calculated prior to the Asselin filtering of the \textit{before} velocities, the |
---|
1076 | \textit{before} vorticity and divergence arrays must be included in the restart file |
---|
1077 | to ensure perfect restartability. The vorticity and divergence at the \textit{now} |
---|
1078 | time step are used for the computation of the nonlinear advection and of the |
---|
1079 | vertical velocity respectively. |
---|
1080 | |
---|
1081 | %-------------------------------------------------------------------------------------------------------------- |
---|
1082 | % Vertical Velocity |
---|
1083 | %-------------------------------------------------------------------------------------------------------------- |
---|
1084 | \subsection [Vertical velocity (\textit{wzvmod})] |
---|
1085 | {Vertical velocity (\mdl{wzvmod})} |
---|
1086 | \label{DYN_wzv} |
---|
1087 | |
---|
1088 | The vertical velocity is computed by an upward integration of the horizontal |
---|
1089 | divergence from the bottom : |
---|
1090 | |
---|
1091 | \begin{equation} \label{Eq_wzv} |
---|
1092 | \left\{ \begin{aligned} |
---|
1093 | &\left. w \right|_{3/2} \quad= 0 \\ |
---|
1094 | \\ |
---|
1095 | &\left. w \right|_{k+1/2} = \left. w \right|_{k+1/2} + e_{3t}\; \left. \chi \right|_k |
---|
1096 | \end{aligned} \right. |
---|
1097 | \end{equation} |
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1098 | |
---|
1099 | With a free surface, the top vertical velocity is non-zero, due to the |
---|
1100 | freshwater forcing and the variations of the free surface elevation. With a |
---|
1101 | linear free surface or with a rigid lid, the upper boundary condition |
---|
1102 | applies at a fixed level $z=0$. Note that in the rigid-lid case (\key{dynspg\_rl} |
---|
1103 | is defined), the surface boundary condition ($\left. w \right|_\text{surface}=0)$ is |
---|
1104 | automatically achieved at least at computer accuracy, due to the the way the |
---|
1105 | surface pressure gradient is expressed in discrete form (Appendix~\ref{Apdx_C}). |
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1106 | |
---|
1107 | Note also that whereas the vertical velocity has the same discrete |
---|
1108 | expression in $z$- and $s$-coordinate, its physical meaning is not the same: |
---|
1109 | in the second case, $w$ is the velocity normal to the $s$-surfaces. |
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1110 | |
---|
1111 | With the variable volume option, the calculation of the vertical velocity is |
---|
1112 | modified (see \citet{Levier2007}, report available on the \NEMO web site). |
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1113 | |
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1114 | % ================================================================ |
---|