1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | % ================================================================ |
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5 | % Chapter ——— Ocean Dynamics (DYN) |
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6 | % ================================================================ |
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7 | \chapter{Ocean Dynamics (DYN)} |
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8 | \label{chap:DYN} |
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9 | |
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10 | \minitoc |
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11 | |
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12 | Using the representation described in \autoref{chap:DOM}, |
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13 | several semi-discrete space forms of the dynamical equations are available depending on |
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14 | the vertical coordinate used and on the conservation properties of the vorticity term. |
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15 | In all the equations presented here, the masking has been omitted for simplicity. |
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16 | One must be aware that all the quantities are masked fields and |
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17 | that each time an average or difference operator is used, the resulting field is multiplied by a mask. |
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18 | |
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19 | The prognostic ocean dynamics equation can be summarized as follows: |
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20 | \[ |
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21 | \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } |
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22 | {\text{COR} + \text{ADV} } |
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23 | + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} |
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24 | \] |
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25 | NXT stands for next, referring to the time-stepping. |
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26 | The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that |
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27 | are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) and |
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28 | a vertical advection part (ZAD) in the vector invariant formulation, |
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29 | or a Coriolis and advection part (COR+ADV) in the flux formulation. |
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30 | The terms following these are the pressure gradient contributions |
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31 | (HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient); |
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32 | and contributions from lateral diffusion (LDF) and vertical diffusion (ZDF), |
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33 | which are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules. |
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34 | The vertical diffusion term includes the surface and bottom stresses. |
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35 | The external forcings and parameterisations require complex inputs |
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36 | (surface wind stress calculation using bulk formulae, estimation of mixing coefficients) |
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37 | that are carried out in modules SBC, LDF and ZDF and are described in |
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38 | \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. |
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39 | |
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40 | In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence, |
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41 | curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module). |
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42 | |
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43 | The different options available to the user are managed by namelist variables. |
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44 | For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx}, |
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45 | where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. |
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46 | If a CPP key is used for this term its name is \key{ttt}. |
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47 | The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory, |
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48 | and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine. |
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49 | |
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50 | The user has the option of extracting and outputting each tendency term from the 3D momentum equations |
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51 | (\key{trddyn} defined), as described in \autoref{chap:MISC}. |
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52 | Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \key{trdvor} is defined) |
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53 | can be derived from the 3D terms. |
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54 | %%% |
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55 | \gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does |
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56 | MISC correspond to "extracting tendency terms" or "vorticity balance"?} |
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57 | |
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58 | % ================================================================ |
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59 | % Sea Surface Height evolution & Diagnostics variables |
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60 | % ================================================================ |
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61 | \section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)} |
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62 | \label{sec:DYN_divcur_wzv} |
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63 | |
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64 | %-------------------------------------------------------------------------------------------------------------- |
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65 | % Horizontal divergence and relative vorticity |
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66 | %-------------------------------------------------------------------------------------------------------------- |
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67 | \subsection{Horizontal divergence and relative vorticity (\protect\mdl{divcur})} |
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68 | \label{subsec:DYN_divcur} |
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69 | |
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70 | The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows: |
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71 | \begin{equation} |
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72 | \label{eq:divcur_cur} |
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73 | \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right] |
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74 | -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) |
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75 | \end{equation} |
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76 | |
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77 | The horizontal divergence is defined at a $T$-point. |
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78 | It is given by: |
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79 | \[ |
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80 | % \label{eq:divcur_div} |
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81 | \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } |
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82 | \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right] |
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83 | +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right) |
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84 | \] |
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85 | |
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86 | Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates, |
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87 | its physical meaning is not identical. |
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88 | $\zeta$ is a pseudo vorticity along $s$-surfaces |
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89 | (only pseudo because $(u,v)$ are still defined along geopotential surfaces, |
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90 | but are not necessarily defined at the same depth). |
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91 | |
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92 | The vorticity and divergence at the \textit{before} step are used in the computation of |
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93 | the horizontal diffusion of momentum. |
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94 | Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities, |
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95 | the \textit{before} vorticity and divergence arrays must be included in the restart file to |
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96 | ensure perfect restartability. |
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97 | The vorticity and divergence at the \textit{now} time step are used for the computation of |
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98 | the nonlinear advection and of the vertical velocity respectively. |
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99 | |
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100 | %-------------------------------------------------------------------------------------------------------------- |
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101 | % Sea Surface Height evolution |
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102 | %-------------------------------------------------------------------------------------------------------------- |
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103 | \subsection{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} |
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104 | \label{subsec:DYN_sshwzv} |
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105 | |
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106 | The sea surface height is given by: |
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107 | \begin{equation} |
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108 | \label{eq:dynspg_ssh} |
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109 | \begin{aligned} |
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110 | \frac{\partial \eta }{\partial t} |
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111 | &\equiv \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{ \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right] |
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112 | +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right] \right\} } |
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113 | - \frac{\textit{emp}}{\rho_w } \\ |
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114 | &\equiv \sum\limits_k {\chi \ e_{3t}} - \frac{\textit{emp}}{\rho_w } |
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115 | \end{aligned} |
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116 | \end{equation} |
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117 | where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), |
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118 | expressed in Kg/m$^2$/s (which is equal to mm/s), |
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119 | and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation). |
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120 | If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then |
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121 | \textit{emp} can be written as the evaporation minus precipitation, minus the river runoff. |
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122 | The sea-surface height is evaluated using exactly the same time stepping scheme as |
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123 | the tracer equation \autoref{eq:tra_nxt}: |
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124 | a leapfrog scheme in combination with an Asselin time filter, |
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125 | $i.e.$ the velocity appearing in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity). |
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126 | This is of paramount importance. |
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127 | Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to |
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128 | the sea surface height equation otherwise tracer content will not be conserved |
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129 | \citep{Griffies_al_MWR01, Leclair_Madec_OM09}. |
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130 | |
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131 | The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom, |
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132 | taking into account the change of the thickness of the levels: |
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133 | \begin{equation} |
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134 | \label{eq:wzv} |
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135 | \left\{ |
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136 | \begin{aligned} |
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137 | &\left. w \right|_{k_b-1/2} \quad= 0 \qquad \text{where } k_b \text{ is the level just above the sea floor } \\ |
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138 | &\left. w \right|_{k+1/2} = \left. w \right|_{k-1/2} + \left. e_{3t} \right|_{k}\; \left. \chi \right|_k |
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139 | - \frac{1} {2 \rdt} \left( \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right) |
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140 | \end{aligned} |
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141 | \right. |
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142 | \end{equation} |
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143 | |
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144 | In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$, |
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145 | as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses, |
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146 | re-orientated downward. |
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147 | \gmcomment{not sure of this... to be modified with the change in emp setting} |
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148 | In the case of a linear free surface, the time derivative in \autoref{eq:wzv} disappears. |
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149 | The upper boundary condition applies at a fixed level $z=0$. |
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150 | The top vertical velocity is thus equal to the divergence of the barotropic transport |
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151 | ($i.e.$ the first term in the right-hand-side of \autoref{eq:dynspg_ssh}). |
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152 | |
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153 | Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates, |
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154 | its physical meaning is not the same: |
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155 | in the second case, $w$ is the velocity normal to the $s$-surfaces. |
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156 | Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared to |
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157 | the indexing used in the semi-discrete equations such as \autoref{eq:wzv} |
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158 | (see \autoref{subsec:DOM_Num_Index_vertical}). |
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159 | |
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160 | |
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161 | % ================================================================ |
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162 | % Coriolis and Advection terms: vector invariant form |
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163 | % ================================================================ |
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164 | \section{Coriolis and advection: vector invariant form} |
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165 | \label{sec:DYN_adv_cor_vect} |
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166 | %-----------------------------------------nam_dynadv---------------------------------------------------- |
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167 | |
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168 | \nlst{namdyn_adv} |
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169 | %------------------------------------------------------------------------------------------------------------- |
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170 | |
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171 | The vector invariant form of the momentum equations is the one most often used in |
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172 | applications of the \NEMO ocean model. |
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173 | The flux form option (see next section) has been present since version $2$. |
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174 | Options are defined through the \ngn{namdyn\_adv} namelist variables Coriolis and |
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175 | momentum advection terms are evaluated using a leapfrog scheme, |
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176 | $i.e.$ the velocity appearing in these expressions is centred in time (\textit{now} velocity). |
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177 | At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following |
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178 | \autoref{chap:LBC}. |
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179 | |
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180 | % ------------------------------------------------------------------------------------------------------------- |
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181 | % Vorticity term |
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182 | % ------------------------------------------------------------------------------------------------------------- |
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183 | \subsection{Vorticity term (\protect\mdl{dynvor})} |
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184 | \label{subsec:DYN_vor} |
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185 | %------------------------------------------nam_dynvor---------------------------------------------------- |
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186 | |
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187 | \nlst{namdyn_vor} |
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188 | %------------------------------------------------------------------------------------------------------------- |
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189 | |
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190 | Options are defined through the \ngn{namdyn\_vor} namelist variables. |
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191 | Four discretisations of the vorticity term (\np{ln\_dynvor\_xxx}\forcode{ = .true.}) are available: |
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192 | conserving potential enstrophy of horizontally non-divergent flow (ENS scheme); |
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193 | conserving horizontal kinetic energy (ENE scheme); |
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194 | conserving potential enstrophy for the relative vorticity term and |
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195 | horizontal kinetic energy for the planetary vorticity term (MIX scheme); |
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196 | or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy |
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197 | (EEN scheme) (see \autoref{subsec:C_vorEEN}). |
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198 | In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of |
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199 | vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{ = .true.}). |
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200 | The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module. |
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201 | |
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202 | %------------------------------------------------------------- |
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203 | % enstrophy conserving scheme |
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204 | %------------------------------------------------------------- |
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205 | \subsubsection{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} |
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206 | \label{subsec:DYN_vor_ens} |
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207 | |
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208 | In the enstrophy conserving case (ENS scheme), |
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209 | the discrete formulation of the vorticity term provides a global conservation of the enstrophy |
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210 | ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow ($i.e.$ $\chi$=$0$), |
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211 | but does not conserve the total kinetic energy. |
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212 | It is given by: |
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213 | \begin{equation} |
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214 | \label{eq:dynvor_ens} |
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215 | \left\{ |
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216 | \begin{aligned} |
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217 | {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} |
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218 | & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2} \\ |
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219 | {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} |
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220 | & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j} |
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221 | \end{aligned} |
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222 | \right. |
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223 | \end{equation} |
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224 | |
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225 | %------------------------------------------------------------- |
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226 | % energy conserving scheme |
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227 | %------------------------------------------------------------- |
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228 | \subsubsection{Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} |
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229 | \label{subsec:DYN_vor_ene} |
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230 | |
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231 | The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy. |
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232 | It is given by: |
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233 | \begin{equation} |
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234 | \label{eq:dynvor_ene} |
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235 | \left\{ |
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236 | \begin{aligned} |
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237 | {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) |
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238 | \; \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\ |
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239 | {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) |
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240 | \; \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } |
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241 | \end{aligned} |
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242 | \right. |
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243 | \end{equation} |
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244 | |
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245 | %------------------------------------------------------------- |
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246 | % mix energy/enstrophy conserving scheme |
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247 | %------------------------------------------------------------- |
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248 | \subsubsection{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{ = .true.}) } |
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249 | \label{subsec:DYN_vor_mix} |
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250 | |
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251 | For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used. |
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252 | It consists of the ENS scheme (\autoref{eq:dynvor_ens}) for the relative vorticity term, |
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253 | and of the ENE scheme (\autoref{eq:dynvor_ene}) applied to the planetary vorticity term. |
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254 | \[ |
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255 | % \label{eq:dynvor_mix} |
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256 | \left\{ { |
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257 | \begin{aligned} |
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258 | {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} |
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259 | \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} } |
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260 | \; {\overline {\left( {\frac{f}{e_{3f} }} \right) |
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261 | \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\ |
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262 | {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j |
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263 | \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} } |
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264 | \; {\overline {\left( {\frac{f}{e_{3f} }} \right) |
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265 | \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill |
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266 | \end{aligned} |
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267 | } \right. |
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268 | \] |
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269 | |
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270 | %------------------------------------------------------------- |
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271 | % energy and enstrophy conserving scheme |
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272 | %------------------------------------------------------------- |
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273 | \subsubsection{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.}) } |
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274 | \label{subsec:DYN_vor_een} |
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275 | |
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276 | In both the ENS and ENE schemes, |
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277 | it is apparent that the combination of $i$ and $j$ averages of the velocity allows for |
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278 | the presence of grid point oscillation structures that will be invisible to the operator. |
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279 | These structures are \textit{computational modes} that will be at least partly damped by |
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280 | the momentum diffusion operator ($i.e.$ the subgrid-scale advection), but not by the resolved advection term. |
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281 | The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field. |
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282 | Such noise would result in more noise in the vertical velocity field, an undesirable feature. |
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283 | This is a well-known characteristic of $C$-grid discretization where |
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284 | $u$ and $v$ are located at different grid points, |
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285 | a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid. |
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286 | \gmcomment{ To circumvent this, Adcroft (ADD REF HERE) |
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287 | Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....} |
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288 | |
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289 | A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}. |
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290 | The idea is to get rid of the double averaging by considering triad combinations of vorticity. |
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291 | It is noteworthy that this solution is conceptually quite similar to the one proposed by |
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292 | \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}). |
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293 | |
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294 | The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified |
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295 | for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme. |
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296 | First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: |
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297 | \[ |
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298 | % \label{eq:pot_vor} |
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299 | q = \frac{\zeta +f} {e_{3f} } |
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300 | \] |
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301 | where the relative vorticity is defined by (\autoref{eq:divcur_cur}), |
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302 | the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: |
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303 | \begin{equation} |
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304 | \label{eq:een_e3f} |
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305 | e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} |
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306 | \end{equation} |
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307 | |
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308 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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309 | \begin{figure}[!ht] |
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310 | \begin{center} |
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311 | \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad} |
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312 | \caption{ |
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313 | \protect\label{fig:DYN_een_triad} |
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314 | Triads used in the energy and enstrophy conserving scheme (een) for |
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315 | $u$-component (upper panel) and $v$-component (lower panel). |
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316 | } |
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317 | \end{center} |
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318 | \end{figure} |
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319 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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320 | |
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321 | A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. |
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322 | It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks |
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323 | (\np{nn\_een\_e3f}\forcode{ = 1}), or just by $4$ (\np{nn\_een\_e3f}\forcode{ = .true.}). |
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324 | The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and |
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325 | extends by continuity the value of $e_{3f}$ into the land areas. |
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326 | This case introduces a sub-grid-scale topography at f-points |
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327 | (with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry) |
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328 | that tends to reinforce the topostrophy of the flow |
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329 | ($i.e.$ the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}. |
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330 | |
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331 | Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as |
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332 | the following triad combinations of the neighbouring potential vorticities defined at f-points |
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333 | (\autoref{fig:DYN_een_triad}): |
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334 | \begin{equation} |
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335 | \label{eq:Q_triads} |
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336 | _i^j \mathbb{Q}^{i_p}_{j_p} |
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337 | = \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right) |
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338 | \end{equation} |
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339 | where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$. |
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340 | |
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341 | Finally, the vorticity terms are represented as: |
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342 | \begin{equation} |
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343 | \label{eq:dynvor_een} |
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344 | \left\{ { |
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345 | \begin{aligned} |
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346 | +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} |
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347 | {^{i+1/2-i_p}_j} \mathbb{Q}^{i_p}_{j_p} \left( e_{1v}\,e_{3v} \;v \right)^{i+1/2-i_p}_{j+j_p} \\ |
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348 | - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} |
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349 | {^i_{j+1/2-j_p}} \mathbb{Q}^{i_p}_{j_p} \left( e_{2u}\,e_{3u} \;u \right)^{i+i_p}_{j+1/2-j_p} \\ |
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350 | \end{aligned} |
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351 | } \right. |
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352 | \end{equation} |
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353 | |
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354 | This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. |
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355 | It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow |
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356 | ($i.e.$ $\chi$=$0$) (see \autoref{subsec:C_vorEEN}). |
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357 | Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of |
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358 | the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. |
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359 | Furthermore, used in combination with a partial steps representation of bottom topography, |
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360 | it improves the interaction between current and topography, |
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361 | leading to a larger topostrophy of the flow \citep{Barnier_al_OD06, Penduff_al_OS07}. |
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362 | |
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363 | %-------------------------------------------------------------------------------------------------------------- |
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364 | % Kinetic Energy Gradient term |
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365 | %-------------------------------------------------------------------------------------------------------------- |
---|
366 | \subsection{Kinetic energy gradient term (\protect\mdl{dynkeg})} |
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367 | \label{subsec:DYN_keg} |
---|
368 | |
---|
369 | As demonstrated in \autoref{apdx:C}, |
---|
370 | there is a single discrete formulation of the kinetic energy gradient term that, |
---|
371 | together with the formulation chosen for the vertical advection (see below), |
---|
372 | conserves the total kinetic energy: |
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373 | \[ |
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374 | % \label{eq:dynkeg} |
---|
375 | \left\{ |
---|
376 | \begin{aligned} |
---|
377 | -\frac{1}{2 \; e_{1u} } & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] \\ |
---|
378 | -\frac{1}{2 \; e_{2v} } & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] |
---|
379 | \end{aligned} |
---|
380 | \right. |
---|
381 | \] |
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382 | |
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383 | %-------------------------------------------------------------------------------------------------------------- |
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384 | % Vertical advection term |
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385 | %-------------------------------------------------------------------------------------------------------------- |
---|
386 | \subsection{Vertical advection term (\protect\mdl{dynzad}) } |
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387 | \label{subsec:DYN_zad} |
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388 | |
---|
389 | The discrete formulation of the vertical advection, t |
---|
390 | ogether with the formulation chosen for the gradient of kinetic energy (KE) term, |
---|
391 | conserves the total kinetic energy. |
---|
392 | Indeed, the change of KE due to the vertical advection is exactly balanced by |
---|
393 | the change of KE due to the gradient of KE (see \autoref{apdx:C}). |
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394 | \[ |
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395 | % \label{eq:dynzad} |
---|
396 | \left\{ |
---|
397 | \begin{aligned} |
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398 | -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2} \;\delta_{k+1/2} \left[ u \right]\ }^{\,k} \\ |
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399 | -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2} \;\delta_{k+1/2} \left[ u \right]\ }^{\,k} |
---|
400 | \end{aligned} |
---|
401 | \right. |
---|
402 | \] |
---|
403 | When \np{ln\_dynzad\_zts}\forcode{ = .true.}, |
---|
404 | a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term. |
---|
405 | This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. |
---|
406 | Note that in this case, |
---|
407 | a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability, |
---|
408 | an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}). |
---|
409 | |
---|
410 | |
---|
411 | % ================================================================ |
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412 | % Coriolis and Advection : flux form |
---|
413 | % ================================================================ |
---|
414 | \section{Coriolis and advection: flux form} |
---|
415 | \label{sec:DYN_adv_cor_flux} |
---|
416 | %------------------------------------------nam_dynadv---------------------------------------------------- |
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417 | |
---|
418 | \nlst{namdyn_adv} |
---|
419 | %------------------------------------------------------------------------------------------------------------- |
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420 | |
---|
421 | Options are defined through the \ngn{namdyn\_adv} namelist variables. |
---|
422 | In the flux form (as in the vector invariant form), |
---|
423 | the Coriolis and momentum advection terms are evaluated using a leapfrog scheme, |
---|
424 | $i.e.$ the velocity appearing in their expressions is centred in time (\textit{now} velocity). |
---|
425 | At the lateral boundaries either free slip, |
---|
426 | no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}. |
---|
427 | |
---|
428 | |
---|
429 | %-------------------------------------------------------------------------------------------------------------- |
---|
430 | % Coriolis plus curvature metric terms |
---|
431 | %-------------------------------------------------------------------------------------------------------------- |
---|
432 | \subsection{Coriolis plus curvature metric terms (\protect\mdl{dynvor}) } |
---|
433 | \label{subsec:DYN_cor_flux} |
---|
434 | |
---|
435 | In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term. |
---|
436 | This altered Coriolis parameter is thus discretised at $f$-points. |
---|
437 | It is given by: |
---|
438 | \begin{multline*} |
---|
439 | % \label{eq:dyncor_metric} |
---|
440 | f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} - u\frac{\partial e_1 }{\partial j}} \right) \\ |
---|
441 | \equiv f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right] |
---|
442 | - \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right] } \ \right) |
---|
443 | \end{multline*} |
---|
444 | |
---|
445 | Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to |
---|
446 | compute the product of the Coriolis parameter and the vorticity. |
---|
447 | However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has exclusively been used to date. |
---|
448 | This term is evaluated using a leapfrog scheme, $i.e.$ the velocity is centred in time (\textit{now} velocity). |
---|
449 | |
---|
450 | %-------------------------------------------------------------------------------------------------------------- |
---|
451 | % Flux form Advection term |
---|
452 | %-------------------------------------------------------------------------------------------------------------- |
---|
453 | \subsection{Flux form advection term (\protect\mdl{dynadv}) } |
---|
454 | \label{subsec:DYN_adv_flux} |
---|
455 | |
---|
456 | The discrete expression of the advection term is given by: |
---|
457 | \[ |
---|
458 | % \label{eq:dynadv} |
---|
459 | \left\{ |
---|
460 | \begin{aligned} |
---|
461 | \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} |
---|
462 | \left( \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i } \ u_t \right] |
---|
463 | + \delta_{j } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2} \ u_f \right] \right. \ \; \\ |
---|
464 | \left. + \delta_{k } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2} \ u_{uw} \right] \right) \\ |
---|
465 | \\ |
---|
466 | \frac{1}{e_{1v}\,e_{2v}\,e_{3v}} |
---|
467 | \left( \delta_{i } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f \right] |
---|
468 | + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i } \ v_t \right] \right. \ \, \, \\ |
---|
469 | \left. + \delta_{k } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw} \right] \right) \\ |
---|
470 | \end{aligned} |
---|
471 | \right. |
---|
472 | \] |
---|
473 | |
---|
474 | Two advection schemes are available: |
---|
475 | a $2^{nd}$ order centered finite difference scheme, CEN2, |
---|
476 | or a $3^{rd}$ order upstream biased scheme, UBS. |
---|
477 | The latter is described in \citet{Shchepetkin_McWilliams_OM05}. |
---|
478 | The schemes are selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. |
---|
479 | In flux form, the schemes differ by the choice of a space and time interpolation to define the value of |
---|
480 | $u$ and $v$ at the centre of each face of $u$- and $v$-cells, $i.e.$ at the $T$-, $f$-, |
---|
481 | and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$. |
---|
482 | |
---|
483 | %------------------------------------------------------------- |
---|
484 | % 2nd order centred scheme |
---|
485 | %------------------------------------------------------------- |
---|
486 | \subsubsection{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{ = .true.})} |
---|
487 | \label{subsec:DYN_adv_cen2} |
---|
488 | |
---|
489 | In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points: |
---|
490 | \begin{equation} |
---|
491 | \label{eq:dynadv_cen2} |
---|
492 | \left\{ |
---|
493 | \begin{aligned} |
---|
494 | u_T^{cen2} &=\overline u^{i } \quad & u_F^{cen2} &=\overline u^{j+1/2} \quad & u_{uw}^{cen2} &=\overline u^{k+1/2} \\ |
---|
495 | v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j \quad & v_{vw}^{cen2} &=\overline v ^{k+1/2} \\ |
---|
496 | \end{aligned} |
---|
497 | \right. |
---|
498 | \end{equation} |
---|
499 | |
---|
500 | The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive ($i.e.$ it may create false extrema). |
---|
501 | It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to |
---|
502 | produce a sensible solution. |
---|
503 | The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, |
---|
504 | so $u$ and $v$ are the \emph{now} velocities. |
---|
505 | |
---|
506 | %------------------------------------------------------------- |
---|
507 | % UBS scheme |
---|
508 | %------------------------------------------------------------- |
---|
509 | \subsubsection{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{ = .true.})} |
---|
510 | \label{subsec:DYN_adv_ubs} |
---|
511 | |
---|
512 | The UBS advection scheme is an upstream biased third order scheme based on |
---|
513 | an upstream-biased parabolic interpolation. |
---|
514 | For example, the evaluation of $u_T^{ubs} $ is done as follows: |
---|
515 | \begin{equation} |
---|
516 | \label{eq:dynadv_ubs} |
---|
517 | u_T^{ubs} =\overline u ^i-\;\frac{1}{6} |
---|
518 | \begin{cases} |
---|
519 | u"_{i-1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i \geqslant 0$ } \\ |
---|
520 | u"_{i+1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i < 0$ } |
---|
521 | \end{cases} |
---|
522 | \end{equation} |
---|
523 | where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$. |
---|
524 | This results in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error |
---|
525 | \citep{Shchepetkin_McWilliams_OM05}. |
---|
526 | The overall performance of the advection scheme is similar to that reported in \citet{Farrow1995}. |
---|
527 | It is a relatively good compromise between accuracy and smoothness. |
---|
528 | It is not a \emph{positive} scheme, meaning that false extrema are permitted. |
---|
529 | But the amplitudes of the false extrema are significantly reduced over those in the centred second order method. |
---|
530 | As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum |
---|
531 | ($i.e.$ \np{ln\_dynldf\_lap}\forcode{ = }\np{ln\_dynldf\_bilap}\forcode{ = .false.}), |
---|
532 | and it is recommended to do so. |
---|
533 | |
---|
534 | The UBS scheme is not used in all directions. |
---|
535 | In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and |
---|
536 | $u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used. |
---|
537 | UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm pursue the |
---|
538 | sentence:Since vertical mixing of momentum is a source term of the TKE equation... } |
---|
539 | |
---|
540 | For stability reasons, the first term in (\autoref{eq:dynadv_ubs}), |
---|
541 | which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time), |
---|
542 | while the second term, which is the diffusion part of the scheme, |
---|
543 | is evaluated using the \textit{before} velocity (forward in time). |
---|
544 | This is discussed by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. |
---|
545 | |
---|
546 | Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by |
---|
547 | one coefficient. |
---|
548 | Replacing $1/6$ by $1/8$ in (\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. |
---|
549 | This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded. |
---|
550 | Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme. |
---|
551 | |
---|
552 | Note also that in the current version of \mdl{dynadv\_ubs}, |
---|
553 | there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS. |
---|
554 | This is an error and should be suppressed soon. |
---|
555 | %%% |
---|
556 | \gmcomment{action : this have to be done} |
---|
557 | %%% |
---|
558 | |
---|
559 | % ================================================================ |
---|
560 | % Hydrostatic pressure gradient term |
---|
561 | % ================================================================ |
---|
562 | \section{Hydrostatic pressure gradient (\protect\mdl{dynhpg})} |
---|
563 | \label{sec:DYN_hpg} |
---|
564 | %------------------------------------------nam_dynhpg--------------------------------------------------- |
---|
565 | |
---|
566 | \nlst{namdyn_hpg} |
---|
567 | %------------------------------------------------------------------------------------------------------------- |
---|
568 | |
---|
569 | Options are defined through the \ngn{namdyn\_hpg} namelist variables. |
---|
570 | The key distinction between the different algorithms used for |
---|
571 | the hydrostatic pressure gradient is the vertical coordinate used, |
---|
572 | since HPG is a \emph{horizontal} pressure gradient, $i.e.$ computed along geopotential surfaces. |
---|
573 | As a result, any tilt of the surface of the computational levels will require a specific treatment to |
---|
574 | compute the hydrostatic pressure gradient. |
---|
575 | |
---|
576 | The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, |
---|
577 | $i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$), |
---|
578 | or a semi-implcit scheme. |
---|
579 | At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied. |
---|
580 | |
---|
581 | %-------------------------------------------------------------------------------------------------------------- |
---|
582 | % z-coordinate with full step |
---|
583 | %-------------------------------------------------------------------------------------------------------------- |
---|
584 | \subsection{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{ = .true.})} |
---|
585 | \label{subsec:DYN_hpg_zco} |
---|
586 | |
---|
587 | The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface. |
---|
588 | However, the pressure is large at great depth while its horizontal gradient is several orders of magnitude smaller. |
---|
589 | This may lead to large truncation errors in the pressure gradient terms. |
---|
590 | Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows: |
---|
591 | |
---|
592 | for $k=km$ (surface layer, $jk=1$ in the code) |
---|
593 | \begin{equation} |
---|
594 | \label{eq:dynhpg_zco_surf} |
---|
595 | \left\{ |
---|
596 | \begin{aligned} |
---|
597 | \left. \delta_{i+1/2} \left[ p^h \right] \right|_{k=km} |
---|
598 | &= \frac{1}{2} g \ \left. \delta_{i+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\ |
---|
599 | \left. \delta_{j+1/2} \left[ p^h \right] \right|_{k=km} |
---|
600 | &= \frac{1}{2} g \ \left. \delta_{j+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\ |
---|
601 | \end{aligned} |
---|
602 | \right. |
---|
603 | \end{equation} |
---|
604 | |
---|
605 | for $1<k<km$ (interior layer) |
---|
606 | \begin{equation} |
---|
607 | \label{eq:dynhpg_zco} |
---|
608 | \left\{ |
---|
609 | \begin{aligned} |
---|
610 | \left. \delta_{i+1/2} \left[ p^h \right] \right|_{k} |
---|
611 | &= \left. \delta_{i+1/2} \left[ p^h \right] \right|_{k-1} |
---|
612 | + \frac{1}{2}\;g\; \left. \delta_{i+1/2} \left[ e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k} \\ |
---|
613 | \left. \delta_{j+1/2} \left[ p^h \right] \right|_{k} |
---|
614 | &= \left. \delta_{j+1/2} \left[ p^h \right] \right|_{k-1} |
---|
615 | + \frac{1}{2}\;g\; \left. \delta_{j+1/2} \left[ e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k} \\ |
---|
616 | \end{aligned} |
---|
617 | \right. |
---|
618 | \end{equation} |
---|
619 | |
---|
620 | Note that the $1/2$ factor in (\autoref{eq:dynhpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as |
---|
621 | the vertical derivative of the scale factor at the surface level ($z=0$). |
---|
622 | Note also that in case of variable volume level (\key{vvl} defined), |
---|
623 | the surface pressure gradient is included in \autoref{eq:dynhpg_zco_surf} and |
---|
624 | \autoref{eq:dynhpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$. |
---|
625 | |
---|
626 | %-------------------------------------------------------------------------------------------------------------- |
---|
627 | % z-coordinate with partial step |
---|
628 | %-------------------------------------------------------------------------------------------------------------- |
---|
629 | \subsection{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{ = .true.})} |
---|
630 | \label{subsec:DYN_hpg_zps} |
---|
631 | |
---|
632 | With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths. |
---|
633 | Before taking horizontal gradients between these tracer points, |
---|
634 | a linear interpolation is used to approximate the deeper tracer as if |
---|
635 | it actually lived at the depth of the shallower tracer point. |
---|
636 | |
---|
637 | Apart from this modification, |
---|
638 | the horizontal hydrostatic pressure gradient evaluated in the $z$-coordinate with partial step is exactly as in |
---|
639 | the pure $z$-coordinate case. |
---|
640 | As explained in detail in section \autoref{sec:TRA_zpshde}, |
---|
641 | the nonlinearity of pressure effects in the equation of state is such that |
---|
642 | it is better to interpolate temperature and salinity vertically before computing the density. |
---|
643 | Horizontal gradients of temperature and salinity are needed for the TRA modules, |
---|
644 | which is the reason why the horizontal gradients of density at the deepest model level are computed in |
---|
645 | module \mdl{zpsdhe} located in the TRA directory and described in \autoref{sec:TRA_zpshde}. |
---|
646 | |
---|
647 | %-------------------------------------------------------------------------------------------------------------- |
---|
648 | % s- and s-z-coordinates |
---|
649 | %-------------------------------------------------------------------------------------------------------------- |
---|
650 | \subsection{$S$- and $Z$-$S$-coordinates} |
---|
651 | \label{subsec:DYN_hpg_sco} |
---|
652 | |
---|
653 | Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers |
---|
654 | ($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}). |
---|
655 | A number of different pressure gradient options are coded but the ROMS-like, |
---|
656 | density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation. |
---|
657 | |
---|
658 | $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.}) |
---|
659 | \begin{equation} |
---|
660 | \label{eq:dynhpg_sco} |
---|
661 | \left\{ |
---|
662 | \begin{aligned} |
---|
663 | - \frac{1} {\rho_o \, e_{1u}} \; \delta_{i+1/2} \left[ p^h \right] |
---|
664 | + \frac{g\; \overline {\rho}^{i+1/2}} {\rho_o \, e_{1u}} \; \delta_{i+1/2} \left[ z_t \right] \\ |
---|
665 | - \frac{1} {\rho_o \, e_{2v}} \; \delta_{j+1/2} \left[ p^h \right] |
---|
666 | + \frac{g\; \overline {\rho}^{j+1/2}} {\rho_o \, e_{2v}} \; \delta_{j+1/2} \left[ z_t \right] \\ |
---|
667 | \end{aligned} |
---|
668 | \right. |
---|
669 | \end{equation} |
---|
670 | |
---|
671 | Where the first term is the pressure gradient along coordinates, |
---|
672 | computed as in \autoref{eq:dynhpg_zco_surf} - \autoref{eq:dynhpg_zco}, |
---|
673 | and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point |
---|
674 | ($e_{3w}$). |
---|
675 | |
---|
676 | $\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}\forcode{ = .true.}). |
---|
677 | This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}\forcode{ = .true.}). |
---|
678 | |
---|
679 | $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) |
---|
680 | |
---|
681 | $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} |
---|
682 | (\np{ln\_dynhpg\_djc}\forcode{ = .true.}) (currently disabled; under development) |
---|
683 | |
---|
684 | Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is activated |
---|
685 | (\key{vvl}) because in that case, even with a flat bottom, |
---|
686 | the coordinate surfaces are not horizontal but follow the free surface \citep{Levier2007}. |
---|
687 | The pressure jacobian scheme (\np{ln\_dynhpg\_prj}\forcode{ = .true.}) is available as |
---|
688 | an improved option to \np{ln\_dynhpg\_sco}\forcode{ = .true.} when \key{vvl} is active. |
---|
689 | The pressure Jacobian scheme uses a constrained cubic spline to |
---|
690 | reconstruct the density profile across the water column. |
---|
691 | This method maintains the monotonicity between the density nodes. |
---|
692 | The pressure can be calculated by analytical integration of the density profile and |
---|
693 | a pressure Jacobian method is used to solve the horizontal pressure gradient. |
---|
694 | This method can provide a more accurate calculation of the horizontal pressure gradient than the standard scheme. |
---|
695 | |
---|
696 | \subsection{Ice shelf cavity} |
---|
697 | \label{subsec:DYN_hpg_isf} |
---|
698 | Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and |
---|
699 | the pressure gradient due to the ocean load. |
---|
700 | If cavity opened (\np{ln\_isfcav}\forcode{ = .true.}) these 2 terms can be calculated by |
---|
701 | setting \np{ln\_dynhpg\_isf}\forcode{ = .true.}. |
---|
702 | No other scheme are working with the ice shelf.\\ |
---|
703 | |
---|
704 | $\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium. |
---|
705 | The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile |
---|
706 | (prescribed as density of a water at 34.4 PSU and -1.9\degC) and |
---|
707 | corresponds to the water replaced by the ice shelf. |
---|
708 | This top pressure is constant over time. |
---|
709 | A detailed description of this method is described in \citet{Losch2008}.\\ |
---|
710 | |
---|
711 | $\bullet$ The ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in |
---|
712 | \autoref{subsec:DYN_hpg_sco}. |
---|
713 | |
---|
714 | %-------------------------------------------------------------------------------------------------------------- |
---|
715 | % Time-scheme |
---|
716 | %-------------------------------------------------------------------------------------------------------------- |
---|
717 | \subsection{Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{ = .true./.false.})} |
---|
718 | \label{subsec:DYN_hpg_imp} |
---|
719 | |
---|
720 | The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and |
---|
721 | therefore the density used in all discrete expressions given above is the \textit{now} density, |
---|
722 | computed from the \textit{now} temperature and salinity. |
---|
723 | In some specific cases |
---|
724 | (usually high resolution simulations over an ocean domain which includes weakly stratified regions) |
---|
725 | the physical phenomenon that controls the time-step is internal gravity waves (IGWs). |
---|
726 | A semi-implicit scheme for doubling the stability limit associated with IGWs can be used |
---|
727 | \citep{Brown_Campana_MWR78, Maltrud1998}. |
---|
728 | It involves the evaluation of the hydrostatic pressure gradient as |
---|
729 | an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ |
---|
730 | ($i.e.$ \textit{before}, \textit{now} and \textit{after} time-steps), |
---|
731 | rather than at the central time level $t$ only, as in the standard leapfrog scheme. |
---|
732 | |
---|
733 | $\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): |
---|
734 | |
---|
735 | \begin{equation} |
---|
736 | \label{eq:dynhpg_lf} |
---|
737 | \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; |
---|
738 | -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right] |
---|
739 | \end{equation} |
---|
740 | |
---|
741 | $\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): |
---|
742 | \begin{equation} |
---|
743 | \label{eq:dynhpg_imp} |
---|
744 | \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; |
---|
745 | -\frac{1}{4\,\rho_o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt} \right] |
---|
746 | \end{equation} |
---|
747 | |
---|
748 | The semi-implicit time scheme \autoref{eq:dynhpg_imp} is made possible without |
---|
749 | significant additional computation since the density can be updated to time level $t+\rdt$ before |
---|
750 | computing the horizontal hydrostatic pressure gradient. |
---|
751 | It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using |
---|
752 | \autoref{eq:dynhpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:dynhpg_lf}. |
---|
753 | Note that \autoref{eq:dynhpg_imp} is equivalent to applying a time filter to the pressure gradient to |
---|
754 | eliminate high frequency IGWs. |
---|
755 | Obviously, when using \autoref{eq:dynhpg_imp}, |
---|
756 | the doubling of the time-step is achievable only if no other factors control the time-step, |
---|
757 | such as the stability limits associated with advection or diffusion. |
---|
758 | |
---|
759 | In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}\forcode{ = .true.}. |
---|
760 | In this case, we choose to apply the time filter to temperature and salinity used in the equation of state, |
---|
761 | instead of applying it to the hydrostatic pressure or to the density, |
---|
762 | so that no additional storage array has to be defined. |
---|
763 | The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows: |
---|
764 | \[ |
---|
765 | % \label{eq:rho_flt} |
---|
766 | \rho^t = \rho( \widetilde{T},\widetilde {S},z_t) |
---|
767 | \quad \text{with} \quad |
---|
768 | \widetilde{X} = 1 / 4 \left( X^{t+\rdt} +2 \,X^t + X^{t-\rdt} \right) |
---|
769 | \] |
---|
770 | |
---|
771 | Note that in the semi-implicit case, it is necessary to save the filtered density, |
---|
772 | an extra three-dimensional field, in the restart file to restart the model with exact reproducibility. |
---|
773 | This option is controlled by \np{nn\_dynhpg\_rst}, a namelist parameter. |
---|
774 | |
---|
775 | % ================================================================ |
---|
776 | % Surface Pressure Gradient |
---|
777 | % ================================================================ |
---|
778 | \section{Surface pressure gradient (\protect\mdl{dynspg})} |
---|
779 | \label{sec:DYN_spg} |
---|
780 | %-----------------------------------------nam_dynspg---------------------------------------------------- |
---|
781 | |
---|
782 | \nlst{namdyn_spg} |
---|
783 | %------------------------------------------------------------------------------------------------------------ |
---|
784 | |
---|
785 | Options are defined through the \ngn{namdyn\_spg} namelist variables. |
---|
786 | The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). |
---|
787 | The main distinction is between the fixed volume case (linear free surface) and |
---|
788 | the variable volume case (nonlinear free surface, \key{vvl} is defined). |
---|
789 | In the linear free surface case (\autoref{subsec:PE_free_surface}) |
---|
790 | the vertical scale factors $e_{3}$ are fixed in time, |
---|
791 | while they are time-dependent in the nonlinear case (\autoref{subsec:PE_free_surface}). |
---|
792 | With both linear and nonlinear free surface, external gravity waves are allowed in the equations, |
---|
793 | which imposes a very small time step when an explicit time stepping is used. |
---|
794 | Two methods are proposed to allow a longer time step for the three-dimensional equations: |
---|
795 | the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}), |
---|
796 | and the split-explicit free surface described below. |
---|
797 | The extra term introduced in the filtered method is calculated implicitly, |
---|
798 | so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
---|
799 | |
---|
800 | |
---|
801 | The form of the surface pressure gradient term depends on how the user wants to |
---|
802 | handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:PE_hor_pg}). |
---|
803 | Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx): |
---|
804 | an explicit formulation which requires a small time step; |
---|
805 | a filtered free surface formulation which allows a larger time step by |
---|
806 | adding a filtering term into the momentum equation; |
---|
807 | and a split-explicit free surface formulation, described below, which also allows a larger time step. |
---|
808 | |
---|
809 | The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it. |
---|
810 | As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
---|
811 | |
---|
812 | |
---|
813 | %-------------------------------------------------------------------------------------------------------------- |
---|
814 | % Explicit free surface formulation |
---|
815 | %-------------------------------------------------------------------------------------------------------------- |
---|
816 | \subsection{Explicit free surface (\protect\key{dynspg\_exp})} |
---|
817 | \label{subsec:DYN_spg_exp} |
---|
818 | |
---|
819 | In the explicit free surface formulation (\key{dynspg\_exp} defined), |
---|
820 | the model time step is chosen to be small enough to resolve the external gravity waves |
---|
821 | (typically a few tens of seconds). |
---|
822 | The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time), |
---|
823 | is thus simply given by : |
---|
824 | \begin{equation} |
---|
825 | \label{eq:dynspg_exp} |
---|
826 | \left\{ |
---|
827 | \begin{aligned} |
---|
828 | - \frac{1}{e_{1u}\,\rho_o} \; \delta_{i+1/2} \left[ \,\rho \,\eta\, \right] \\ |
---|
829 | - \frac{1}{e_{2v}\,\rho_o} \; \delta_{j+1/2} \left[ \,\rho \,\eta\, \right] |
---|
830 | \end{aligned} |
---|
831 | \right. |
---|
832 | \end{equation} |
---|
833 | |
---|
834 | Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), |
---|
835 | the surface pressure gradient is already included in the momentum tendency through |
---|
836 | the level thickness variation allowed in the computation of the hydrostatic pressure gradient. |
---|
837 | Thus, nothing is done in the \mdl{dynspg\_exp} module. |
---|
838 | |
---|
839 | %-------------------------------------------------------------------------------------------------------------- |
---|
840 | % Split-explict free surface formulation |
---|
841 | %-------------------------------------------------------------------------------------------------------------- |
---|
842 | \subsection{Split-explicit free surface (\protect\key{dynspg\_ts})} |
---|
843 | \label{subsec:DYN_spg_ts} |
---|
844 | %------------------------------------------namsplit----------------------------------------------------------- |
---|
845 | % |
---|
846 | %\nlst{namsplit} |
---|
847 | %------------------------------------------------------------------------------------------------------------- |
---|
848 | |
---|
849 | The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined), |
---|
850 | also called the time-splitting formulation, follows the one proposed by \citet{Shchepetkin_McWilliams_OM05}. |
---|
851 | The general idea is to solve the free surface equation and the associated barotropic velocity equations with |
---|
852 | a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables |
---|
853 | (\autoref{fig:DYN_dynspg_ts}). |
---|
854 | The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through |
---|
855 | the \np{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$. |
---|
856 | This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}\forcode{ = .true.}) considering that |
---|
857 | the stability of the barotropic system is essentially controled by external waves propagation. |
---|
858 | Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry. |
---|
859 | Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}. |
---|
860 | |
---|
861 | %%% |
---|
862 | The barotropic mode solves the following equations: |
---|
863 | % \begin{subequations} |
---|
864 | % \label{eq:BT} |
---|
865 | \begin{equation} |
---|
866 | \label{eq:BT_dyn} |
---|
867 | \frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}= |
---|
868 | -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h} |
---|
869 | -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}} |
---|
870 | \end{equation} |
---|
871 | \[ |
---|
872 | % \label{eq:BT_ssh} |
---|
873 | \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E |
---|
874 | \] |
---|
875 | % \end{subequations} |
---|
876 | where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes, |
---|
877 | surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. |
---|
878 | The third term on the right hand side of \autoref{eq:BT_dyn} represents the bottom stress |
---|
879 | (see section \autoref{sec:ZDF_bfr}), explicitly accounted for at each barotropic iteration. |
---|
880 | Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm |
---|
881 | detailed in \citet{Shchepetkin_McWilliams_OM05}. |
---|
882 | AB3-AM4 coefficients used in \NEMO follow the second-order accurate, |
---|
883 | "multi-purpose" stability compromise as defined in \citet{Shchepetkin_McWilliams_Bk08} |
---|
884 | (see their figure 12, lower left). |
---|
885 | |
---|
886 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
---|
887 | \begin{figure}[!t] |
---|
888 | \begin{center} |
---|
889 | \includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts} |
---|
890 | \caption{ |
---|
891 | \protect\label{fig:DYN_dynspg_ts} |
---|
892 | Schematic of the split-explicit time stepping scheme for the external and internal modes. |
---|
893 | Time increases to the right. In this particular exemple, |
---|
894 | a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$. |
---|
895 | Internal mode time steps (which are also the model time steps) are denoted by $t-\rdt$, $t$ and $t+\rdt$. |
---|
896 | Variables with $k$ superscript refer to instantaneous barotropic variables, |
---|
897 | $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and |
---|
898 | secondary weights (blue vertical bars). |
---|
899 | The former are used to obtain time filtered quantities at $t+\rdt$ while |
---|
900 | the latter are used to obtain time averaged transports to advect tracers. |
---|
901 | a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.}, |
---|
902 | \protect\np{ln\_bt\_av}\forcode{ = .true.}. |
---|
903 | b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.}, |
---|
904 | \protect\np{ln\_bt\_av}\forcode{ = .true.}. |
---|
905 | c) Forward time integration with no time filtering (POM-like scheme): |
---|
906 | \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}. |
---|
907 | } |
---|
908 | \end{center} |
---|
909 | \end{figure} |
---|
910 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
---|
911 | |
---|
912 | In the default case (\np{ln\_bt\_fw}\forcode{ = .true.}), |
---|
913 | the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps |
---|
914 | (\autoref{fig:DYN_dynspg_ts}a). |
---|
915 | To avoid aliasing of fast barotropic motions into three dimensional equations, |
---|
916 | time filtering is eventually applied on barotropic quantities (\np{ln\_bt\_av}\forcode{ = .true.}). |
---|
917 | In that case, the integration is extended slightly beyond \textit{after} time step to |
---|
918 | provide time filtered quantities. |
---|
919 | These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. |
---|
920 | Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, |
---|
921 | asselin filtering is not applied to barotropic quantities.\\ |
---|
922 | Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step |
---|
923 | (\np{ln\_bt\_fw}\forcode{ = .false.}). |
---|
924 | Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary), |
---|
925 | the baroclinic to barotropic forcing term given at \textit{now} time step become centred in |
---|
926 | the middle of the integration window. |
---|
927 | It can easily be shown that this property removes part of splitting errors between modes, |
---|
928 | which increases the overall numerical robustness. |
---|
929 | %references to Patrick Marsaleix' work here. Also work done by SHOM group. |
---|
930 | |
---|
931 | %%% |
---|
932 | |
---|
933 | As far as tracer conservation is concerned, |
---|
934 | barotropic velocities used to advect tracers must also be updated at \textit{now} time step. |
---|
935 | This implies to change the traditional order of computations in \NEMO: |
---|
936 | most of momentum trends (including the barotropic mode calculation) updated first, tracers' after. |
---|
937 | This \textit{de facto} makes semi-implicit hydrostatic pressure gradient |
---|
938 | (see section \autoref{subsec:DYN_hpg_imp}) |
---|
939 | and time splitting not compatible. |
---|
940 | Advective barotropic velocities are obtained by using a secondary set of filtering weights, |
---|
941 | uniquely defined from the filter coefficients used for the time averaging (\citet{Shchepetkin_McWilliams_OM05}). |
---|
942 | Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to |
---|
943 | obtain exact conservation. |
---|
944 | |
---|
945 | %%% |
---|
946 | |
---|
947 | One can eventually choose to feedback instantaneous values by not using any time filter |
---|
948 | (\np{ln\_bt\_av}\forcode{ = .false.}). |
---|
949 | In that case, external mode equations are continuous in time, |
---|
950 | $i.e.$ they are not re-initialized when starting a new sub-stepping sequence. |
---|
951 | This is the method used so far in the POM model, the stability being maintained by |
---|
952 | refreshing at (almost) each barotropic time step advection and horizontal diffusion terms. |
---|
953 | Since the latter terms have not been added in \NEMO for computational efficiency, |
---|
954 | removing time filtering is not recommended except for debugging purposes. |
---|
955 | This may be used for instance to appreciate the damping effect of the standard formulation on |
---|
956 | external gravity waves in idealized or weakly non-linear cases. |
---|
957 | Although the damping is lower than for the filtered free surface, |
---|
958 | it is still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. |
---|
959 | |
---|
960 | %>>>>>=============== |
---|
961 | \gmcomment{ %%% copy from griffies Book |
---|
962 | |
---|
963 | \textbf{title: Time stepping the barotropic system } |
---|
964 | |
---|
965 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. |
---|
966 | Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using |
---|
967 | the small barotropic time step $\rdt$. |
---|
968 | We have |
---|
969 | |
---|
970 | \[ |
---|
971 | % \label{eq:DYN_spg_ts_eta} |
---|
972 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
---|
973 | = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
---|
974 | \] |
---|
975 | \begin{multline*} |
---|
976 | % \label{eq:DYN_spg_ts_u} |
---|
977 | \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ |
---|
978 | = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) |
---|
979 | - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] |
---|
980 | \end{multline*} |
---|
981 | \ |
---|
982 | |
---|
983 | In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with |
---|
984 | the barotropic time steps. |
---|
985 | The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which |
---|
986 | the vertically integrated forcing $\textbf{M}(\tau)$ |
---|
987 | (note that this forcing includes the surface freshwater forcing), |
---|
988 | the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, |
---|
989 | and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over |
---|
990 | a single cycle. |
---|
991 | This is also the time that sets the barotropic time steps via |
---|
992 | \[ |
---|
993 | % \label{eq:DYN_spg_ts_t} |
---|
994 | t_n=\tau+n\rdt |
---|
995 | \] |
---|
996 | with $n$ an integer. |
---|
997 | The density scaled surface pressure is evaluated via |
---|
998 | \[ |
---|
999 | % \label{eq:DYN_spg_ts_ps} |
---|
1000 | p_s^{(b)}(\tau,t_{n}) = |
---|
1001 | \begin{cases} |
---|
1002 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
---|
1003 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
---|
1004 | \end{cases} |
---|
1005 | \] |
---|
1006 | To get started, we assume the following initial conditions |
---|
1007 | \[ |
---|
1008 | % \label{eq:DYN_spg_ts_eta} |
---|
1009 | \begin{split} |
---|
1010 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ |
---|
1011 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} |
---|
1012 | \end{split} |
---|
1013 | \] |
---|
1014 | with |
---|
1015 | \[ |
---|
1016 | % \label{eq:DYN_spg_ts_etaF} |
---|
1017 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n}) |
---|
1018 | \] |
---|
1019 | the time averaged surface height taken from the previous barotropic cycle. |
---|
1020 | Likewise, |
---|
1021 | \[ |
---|
1022 | % \label{eq:DYN_spg_ts_u} |
---|
1023 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ |
---|
1024 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} |
---|
1025 | \] |
---|
1026 | with |
---|
1027 | \[ |
---|
1028 | % \label{eq:DYN_spg_ts_u} |
---|
1029 | \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n}) |
---|
1030 | \] |
---|
1031 | the time averaged vertically integrated transport. |
---|
1032 | Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. |
---|
1033 | |
---|
1034 | Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ , |
---|
1035 | the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at |
---|
1036 | baroclinic time $\tau + \rdt \tau$ |
---|
1037 | \[ |
---|
1038 | % \label{eq:DYN_spg_ts_u} |
---|
1039 | \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
---|
1040 | \] |
---|
1041 | The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using |
---|
1042 | the following form |
---|
1043 | |
---|
1044 | \begin{equation} |
---|
1045 | \label{eq:DYN_spg_ts_ssh} |
---|
1046 | \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] |
---|
1047 | \end{equation} |
---|
1048 | |
---|
1049 | The use of this "big-leap-frog" scheme for the surface height ensures compatibility between |
---|
1050 | the mass/volume budgets and the tracer budgets. |
---|
1051 | More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). |
---|
1052 | |
---|
1053 | In general, some form of time filter is needed to maintain integrity of the surface height field due to |
---|
1054 | the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. |
---|
1055 | We have tried various forms of such filtering, |
---|
1056 | with the following method discussed in \cite{Griffies_al_MWR01} chosen due to |
---|
1057 | its stability and reasonably good maintenance of tracer conservation properties (see ??). |
---|
1058 | |
---|
1059 | \begin{equation} |
---|
1060 | \label{eq:DYN_spg_ts_sshf} |
---|
1061 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
---|
1062 | \end{equation} |
---|
1063 | Another approach tried was |
---|
1064 | |
---|
1065 | \[ |
---|
1066 | % \label{eq:DYN_spg_ts_sshf2} |
---|
1067 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
---|
1068 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt) |
---|
1069 | + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right] |
---|
1070 | \] |
---|
1071 | |
---|
1072 | which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. |
---|
1073 | This isolation allows for an easy check that tracer conservation is exact when |
---|
1074 | eliminating tracer and surface height time filtering (see ?? for more complete discussion). |
---|
1075 | However, in the general case with a non-zero $\alpha$, |
---|
1076 | the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. |
---|
1077 | |
---|
1078 | } %%end gm comment (copy of griffies book) |
---|
1079 | |
---|
1080 | %>>>>>=============== |
---|
1081 | |
---|
1082 | |
---|
1083 | %-------------------------------------------------------------------------------------------------------------- |
---|
1084 | % Filtered free surface formulation |
---|
1085 | %-------------------------------------------------------------------------------------------------------------- |
---|
1086 | \subsection{Filtered free surface (\protect\key{dynspg\_flt})} |
---|
1087 | \label{subsec:DYN_spg_fltp} |
---|
1088 | |
---|
1089 | The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation. |
---|
1090 | The extra term introduced in the equations (see \autoref{subsec:PE_free_surface}) is solved implicitly. |
---|
1091 | The elliptic solvers available in the code are documented in \autoref{chap:MISC}. |
---|
1092 | |
---|
1093 | %% gm %%======>>>> given here the discrete eqs provided to the solver |
---|
1094 | \gmcomment{ %%% copy from chap-model basics |
---|
1095 | \[ |
---|
1096 | % \label{eq:spg_flt} |
---|
1097 | \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}} |
---|
1098 | - g \nabla \left( \tilde{\rho} \ \eta \right) |
---|
1099 | - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) |
---|
1100 | \] |
---|
1101 | where $T_c$, is a parameter with dimensions of time which characterizes the force, |
---|
1102 | $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, |
---|
1103 | and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, |
---|
1104 | non-linear and viscous terms in \autoref{eq:PE_dyn}. |
---|
1105 | } %end gmcomment |
---|
1106 | |
---|
1107 | Note that in the linear free surface formulation (\key{vvl} not defined), |
---|
1108 | the ocean depth is time-independent and so is the matrix to be inverted. |
---|
1109 | It is computed once and for all and applies to all ocean time steps. |
---|
1110 | |
---|
1111 | % ================================================================ |
---|
1112 | % Lateral diffusion term |
---|
1113 | % ================================================================ |
---|
1114 | \section{Lateral diffusion term and operators (\protect\mdl{dynldf})} |
---|
1115 | \label{sec:DYN_ldf} |
---|
1116 | %------------------------------------------nam_dynldf---------------------------------------------------- |
---|
1117 | |
---|
1118 | \nlst{namdyn_ldf} |
---|
1119 | %------------------------------------------------------------------------------------------------------------- |
---|
1120 | |
---|
1121 | Options are defined through the \ngn{namdyn\_ldf} namelist variables. |
---|
1122 | The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators. |
---|
1123 | The coefficients may be constant or spatially variable; |
---|
1124 | the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}). |
---|
1125 | The lateral diffusion of momentum is evaluated using a forward scheme, |
---|
1126 | $i.e.$ the velocity appearing in its expression is the \textit{before} velocity in time, |
---|
1127 | except for the pure vertical component that appears when a tensor of rotation is used. |
---|
1128 | This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). |
---|
1129 | |
---|
1130 | At the lateral boundaries either free slip, |
---|
1131 | no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}). |
---|
1132 | |
---|
1133 | \gmcomment{ |
---|
1134 | Hyperviscous operators are frequently used in the simulation of turbulent flows to |
---|
1135 | control the dissipation of unresolved small scale features. |
---|
1136 | Their primary role is to provide strong dissipation at the smallest scale supported by |
---|
1137 | the grid while minimizing the impact on the larger scale features. |
---|
1138 | Hyperviscous operators are thus designed to be more scale selective than the traditional, |
---|
1139 | physically motivated Laplace operator. |
---|
1140 | In finite difference methods, |
---|
1141 | the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since |
---|
1142 | its damping time ($i.e.$ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$ |
---|
1143 | (so that short waves damped more rapidelly than long ones), |
---|
1144 | whereas the Laplace operator damping time scales only like $\lambda^{-2}$. |
---|
1145 | } |
---|
1146 | |
---|
1147 | % ================================================================ |
---|
1148 | \subsection[Iso-level laplacian (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})] |
---|
1149 | {Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})} |
---|
1150 | \label{subsec:DYN_ldf_lap} |
---|
1151 | |
---|
1152 | For lateral iso-level diffusion, the discrete operator is: |
---|
1153 | \begin{equation} |
---|
1154 | \label{eq:dynldf_lap} |
---|
1155 | \left\{ |
---|
1156 | \begin{aligned} |
---|
1157 | D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm} |
---|
1158 | \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[ |
---|
1159 | {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\ |
---|
1160 | D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm} |
---|
1161 | \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[ |
---|
1162 | {A_f^{lm} \;e_{3f} \zeta } \right] |
---|
1163 | \end{aligned} |
---|
1164 | \right. |
---|
1165 | \end{equation} |
---|
1166 | |
---|
1167 | As explained in \autoref{subsec:PE_ldf}, |
---|
1168 | this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and |
---|
1169 | ensures a complete separation between the vorticity and divergence parts of the momentum diffusion. |
---|
1170 | |
---|
1171 | %-------------------------------------------------------------------------------------------------------------- |
---|
1172 | % Rotated laplacian operator |
---|
1173 | %-------------------------------------------------------------------------------------------------------------- |
---|
1174 | \subsection[Rotated laplacian (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})] |
---|
1175 | {Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})} |
---|
1176 | \label{subsec:DYN_ldf_iso} |
---|
1177 | |
---|
1178 | A rotation of the lateral momentum diffusion operator is needed in several cases: |
---|
1179 | for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}\forcode{ = .true.}) and |
---|
1180 | for either iso-neutral (\np{ln\_dynldf\_iso}\forcode{ = .true.}) or |
---|
1181 | geopotential (\np{ln\_dynldf\_hor}\forcode{ = .true.}) diffusion in the $s$-coordinate. |
---|
1182 | In the partial step case, coordinates are horizontal except at the deepest level and |
---|
1183 | no rotation is performed when \np{ln\_dynldf\_hor}\forcode{ = .true.}. |
---|
1184 | The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on |
---|
1185 | each momentum component. |
---|
1186 | It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry. |
---|
1187 | The resulting discrete representation is: |
---|
1188 | \begin{equation} |
---|
1189 | \label{eq:dyn_ldf_iso} |
---|
1190 | \begin{split} |
---|
1191 | D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ |
---|
1192 | & \left\{\quad {\delta_{i+1/2} \left[ {A_T^{lm} \left( |
---|
1193 | {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u] |
---|
1194 | -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}} |
---|
1195 | \right)} \right]} \right. \\ |
---|
1196 | & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} |
---|
1197 | }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f} |
---|
1198 | \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}} |
---|
1199 | \right)} \right] \\ |
---|
1200 | &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline |
---|
1201 | {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } |
---|
1202 | \right.} \right. \\ |
---|
1203 | & \ \qquad \qquad \qquad \quad\ |
---|
1204 | - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\ |
---|
1205 | & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ |
---|
1206 | +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} |
---|
1207 | \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\ |
---|
1208 | D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\ |
---|
1209 | & \left\{\quad {\delta_{i+1/2} \left[ {A_f^{lm} \left( |
---|
1210 | {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v] |
---|
1211 | -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}} |
---|
1212 | \right)} \right]} \right. \\ |
---|
1213 | & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} |
---|
1214 | }\,\delta_{j} [v] - e_{1t}\, r_{2t} |
---|
1215 | \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}} |
---|
1216 | \right)} \right] \\ |
---|
1217 | & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline |
---|
1218 | {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\ |
---|
1219 | & \ \qquad \qquad \qquad \quad\ |
---|
1220 | - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\ |
---|
1221 | & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ |
---|
1222 | +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} |
---|
1223 | \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} |
---|
1224 | \end{split} |
---|
1225 | \end{equation} |
---|
1226 | where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and |
---|
1227 | the surface of computation ($z$- or $s$-surfaces). |
---|
1228 | The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}). |
---|
1229 | |
---|
1230 | %-------------------------------------------------------------------------------------------------------------- |
---|
1231 | % Iso-level bilaplacian operator |
---|
1232 | %-------------------------------------------------------------------------------------------------------------- |
---|
1233 | \subsection[Iso-level bilaplacian (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})] |
---|
1234 | {Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})} |
---|
1235 | \label{subsec:DYN_ldf_bilap} |
---|
1236 | |
---|
1237 | The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:dynldf_lap} twice. |
---|
1238 | It requires an additional assumption on boundary conditions: |
---|
1239 | the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen, |
---|
1240 | while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}). |
---|
1241 | %%% |
---|
1242 | \gmcomment{add a remark on the the change in the position of the coefficient} |
---|
1243 | %%% |
---|
1244 | |
---|
1245 | % ================================================================ |
---|
1246 | % Vertical diffusion term |
---|
1247 | % ================================================================ |
---|
1248 | \section{Vertical diffusion term (\protect\mdl{dynzdf})} |
---|
1249 | \label{sec:DYN_zdf} |
---|
1250 | %----------------------------------------------namzdf------------------------------------------------------ |
---|
1251 | |
---|
1252 | \nlst{namzdf} |
---|
1253 | %------------------------------------------------------------------------------------------------------------- |
---|
1254 | |
---|
1255 | Options are defined through the \ngn{namzdf} namelist variables. |
---|
1256 | The large vertical diffusion coefficient found in the surface mixed layer together with high vertical resolution implies that in the case of explicit time stepping there would be too restrictive a constraint on the time step. |
---|
1257 | Two time stepping schemes can be used for the vertical diffusion term: |
---|
1258 | $(a)$ a forward time differencing scheme |
---|
1259 | (\np{ln\_zdfexp}\forcode{ = .true.}) using a time splitting technique (\np{nn\_zdfexp} $>$ 1) or |
---|
1260 | $(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{ = .false.}) |
---|
1261 | (see \autoref{chap:STP}). |
---|
1262 | Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. |
---|
1263 | |
---|
1264 | The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is. |
---|
1265 | The vertical diffusion operators given by \autoref{eq:PE_zdf} take the following semi-discrete space form: |
---|
1266 | \[ |
---|
1267 | % \label{eq:dynzdf} |
---|
1268 | \left\{ |
---|
1269 | \begin{aligned} |
---|
1270 | D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } |
---|
1271 | \ \delta_{k+1/2} [\,u\,] \right] \\ |
---|
1272 | \\ |
---|
1273 | D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } |
---|
1274 | \ \delta_{k+1/2} [\,v\,] \right] |
---|
1275 | \end{aligned} |
---|
1276 | \right. |
---|
1277 | \] |
---|
1278 | where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients. |
---|
1279 | The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}). |
---|
1280 | |
---|
1281 | The surface boundary condition on momentum is the stress exerted by the wind. |
---|
1282 | At the surface, the momentum fluxes are prescribed as the boundary condition on |
---|
1283 | the vertical turbulent momentum fluxes, |
---|
1284 | \begin{equation} |
---|
1285 | \label{eq:dynzdf_sbc} |
---|
1286 | \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} |
---|
1287 | = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v } |
---|
1288 | \end{equation} |
---|
1289 | where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in |
---|
1290 | the (\textbf{i},\textbf{j}) coordinate system. |
---|
1291 | The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in |
---|
1292 | the vertical over the mixed layer depth. |
---|
1293 | If the vertical mixing coefficient is small (when no mixed layer scheme is used) |
---|
1294 | the surface stress enters only the top model level, as a body force. |
---|
1295 | The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}). |
---|
1296 | |
---|
1297 | The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation |
---|
1298 | (see \autoref{sec:ZDF_bfr}) |
---|
1299 | |
---|
1300 | % ================================================================ |
---|
1301 | % External Forcing |
---|
1302 | % ================================================================ |
---|
1303 | \section{External forcings} |
---|
1304 | \label{sec:DYN_forcing} |
---|
1305 | |
---|
1306 | Besides the surface and bottom stresses (see the above section) |
---|
1307 | which are introduced as boundary conditions on the vertical mixing, |
---|
1308 | three other forcings may enter the dynamical equations by affecting the surface pressure gradient. |
---|
1309 | |
---|
1310 | (1) When \np{ln\_apr\_dyn}\forcode{ = .true.} (see \autoref{sec:SBC_apr}), |
---|
1311 | the atmospheric pressure is taken into account when computing the surface pressure gradient. |
---|
1312 | |
---|
1313 | (2) When \np{ln\_tide\_pot}\forcode{ = .true.} and \np{ln\_tide}\forcode{ = .true.} (see \autoref{sec:SBC_tide}), |
---|
1314 | the tidal potential is taken into account when computing the surface pressure gradient. |
---|
1315 | |
---|
1316 | (3) When \np{nn\_ice\_embd}\forcode{ = 2} and LIM or CICE is used |
---|
1317 | ($i.e.$ when the sea-ice is embedded in the ocean), |
---|
1318 | the snow-ice mass is taken into account when computing the surface pressure gradient. |
---|
1319 | |
---|
1320 | |
---|
1321 | \gmcomment{ missing : the lateral boundary condition !!! another external forcing |
---|
1322 | } |
---|
1323 | |
---|
1324 | % ================================================================ |
---|
1325 | % Time evolution term |
---|
1326 | % ================================================================ |
---|
1327 | \section{Time evolution term (\protect\mdl{dynnxt})} |
---|
1328 | \label{sec:DYN_nxt} |
---|
1329 | |
---|
1330 | %----------------------------------------------namdom---------------------------------------------------- |
---|
1331 | |
---|
1332 | \nlst{namdom} |
---|
1333 | %------------------------------------------------------------------------------------------------------------- |
---|
1334 | |
---|
1335 | Options are defined through the \ngn{namdom} namelist variables. |
---|
1336 | The general framework for dynamics time stepping is a leap-frog scheme, |
---|
1337 | $i.e.$ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:STP}). |
---|
1338 | The scheme is applied to the velocity, except when |
---|
1339 | using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) |
---|
1340 | in the variable volume case (\key{vvl} defined), |
---|
1341 | where it has to be applied to the thickness weighted velocity (see \autoref{sec:A_momentum}) |
---|
1342 | |
---|
1343 | $\bullet$ vector invariant form or linear free surface |
---|
1344 | (\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \key{vvl} not defined): |
---|
1345 | \[ |
---|
1346 | % \label{eq:dynnxt_vec} |
---|
1347 | \left\{ |
---|
1348 | \begin{aligned} |
---|
1349 | &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt \ \text{RHS}_u^t \\ |
---|
1350 | &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right] |
---|
1351 | \end{aligned} |
---|
1352 | \right. |
---|
1353 | \] |
---|
1354 | |
---|
1355 | $\bullet$ flux form and nonlinear free surface |
---|
1356 | (\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \key{vvl} defined): |
---|
1357 | \[ |
---|
1358 | % \label{eq:dynnxt_flux} |
---|
1359 | \left\{ |
---|
1360 | \begin{aligned} |
---|
1361 | &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t \\ |
---|
1362 | &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t |
---|
1363 | +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right] |
---|
1364 | \end{aligned} |
---|
1365 | \right. |
---|
1366 | \] |
---|
1367 | where RHS is the right hand side of the momentum equation, |
---|
1368 | the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient. |
---|
1369 | $\gamma$ is initialized as \np{nn\_atfp} (namelist parameter). |
---|
1370 | Its default value is \np{nn\_atfp}\forcode{ = 10.e-3}. |
---|
1371 | In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for |
---|
1372 | the momentum equations. |
---|
1373 | |
---|
1374 | Note that with the filtered free surface, |
---|
1375 | the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module, |
---|
1376 | and only array swapping and Asselin filtering is done in \mdl{dynnxt}. |
---|
1377 | |
---|
1378 | % ================================================================ |
---|
1379 | \biblio |
---|
1380 | |
---|
1381 | \end{document} |
---|