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 | \label{chap:DYN} |
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7 | |
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8 | \thispagestyle{plain} |
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9 | |
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10 | \chaptertoc |
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11 | |
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12 | \paragraph{Changes record} ~\\ |
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13 | |
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14 | {\footnotesize |
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15 | \begin{tabularx}{\textwidth}{l||X|X} |
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16 | Release & Author(s) & Modifications \\ |
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17 | \hline |
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18 | {\em 4.0} & {\em ...} & {\em ...} \\ |
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19 | {\em 3.6} & {\em ...} & {\em ...} \\ |
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20 | {\em 3.4} & {\em ...} & {\em ...} \\ |
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21 | {\em <=3.4} & {\em ...} & {\em ...} |
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22 | \end{tabularx} |
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23 | } |
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24 | |
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25 | \clearpage |
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26 | |
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27 | Using the representation described in \autoref{chap:DOM}, |
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28 | several semi-discrete space forms of the dynamical equations are available depending on |
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29 | the vertical coordinate used and on the conservation properties of the vorticity term. |
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30 | In all the equations presented here, the masking has been omitted for simplicity. |
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31 | One must be aware that all the quantities are masked fields and |
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32 | that each time an average or difference operator is used, the resulting field is multiplied by a mask. |
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33 | |
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34 | The prognostic ocean dynamics equation can be summarized as follows: |
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35 | \[ |
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36 | \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } |
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37 | {\text{COR} + \text{ADV} } |
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38 | + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} |
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39 | \] |
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40 | NXT stands for next, referring to the time-stepping. |
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41 | The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that |
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42 | are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) and |
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43 | a vertical advection part (ZAD) in the vector invariant formulation, |
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44 | or a Coriolis and advection part (COR+ADV) in the flux formulation. |
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45 | The terms following these are the pressure gradient contributions |
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46 | (HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient); |
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47 | and contributions from lateral diffusion (LDF) and vertical diffusion (ZDF), |
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48 | which are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules. |
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49 | The vertical diffusion term includes the surface and bottom stresses. |
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50 | The external forcings and parameterisations require complex inputs |
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51 | (surface wind stress calculation using bulk formulae, estimation of mixing coefficients) |
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52 | that are carried out in modules SBC, LDF and ZDF and are described in |
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53 | \autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively. |
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54 | |
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55 | In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence, |
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56 | curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module). |
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57 | |
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58 | The different options available to the user are managed by namelist variables. |
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59 | For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx}, |
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60 | where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. |
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61 | %If a CPP key is used for this term its name is \key{ttt}. |
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62 | The corresponding code can be found in the \textit{dynttt\_xxx} module in the DYN directory, |
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63 | and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine. |
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64 | |
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65 | The user has the option of extracting and outputting each tendency term from the 3D momentum equations |
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66 | (\texttt{trddyn?} defined), as described in \autoref{chap:MISC}. |
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67 | Furthermore, the tendency terms associated with the 2D barotropic vorticity balance (when \texttt{trdvor?} is defined) |
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68 | can be derived from the 3D terms. |
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69 | \gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does |
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70 | MISC correspond to "extracting tendency terms" or "vorticity balance"?} |
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71 | |
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72 | %% ================================================================================================= |
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73 | \section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)} |
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74 | \label{sec:DYN_divcur_wzv} |
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75 | |
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76 | %% ================================================================================================= |
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77 | \subsection[Horizontal divergence and relative vorticity (\textit{divcur.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{divcur})} |
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78 | \label{subsec:DYN_divcur} |
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79 | |
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80 | The vorticity is defined at an $f$-point (\ie\ corner point) as follows: |
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81 | \begin{equation} |
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82 | \label{eq:DYN_divcur_cur} |
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83 | \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right] |
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84 | -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) |
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85 | \end{equation} |
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86 | |
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87 | The horizontal divergence is defined at a $T$-point. |
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88 | It is given by: |
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89 | \[ |
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90 | % \label{eq:DYN_divcur_div} |
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91 | \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } |
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92 | \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right] |
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93 | +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right) |
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94 | \] |
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95 | |
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96 | Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates, |
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97 | its physical meaning is not identical. |
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98 | $\zeta$ is a pseudo vorticity along $s$-surfaces |
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99 | (only pseudo because $(u,v)$ are still defined along geopotential surfaces, |
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100 | but are not necessarily defined at the same depth). |
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101 | |
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102 | The vorticity and divergence at the \textit{before} step are used in the computation of |
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103 | the horizontal diffusion of momentum. |
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104 | Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities, |
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105 | the \textit{before} vorticity and divergence arrays must be included in the restart file to |
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106 | ensure perfect restartability. |
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107 | The vorticity and divergence at the \textit{now} time step are used for the computation of |
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108 | the nonlinear advection and of the vertical velocity respectively. |
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109 | |
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110 | %% ================================================================================================= |
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111 | \subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} |
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112 | \label{subsec:DYN_sshwzv} |
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113 | |
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114 | The sea surface height is given by: |
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115 | \begin{equation} |
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116 | \label{eq:DYN_spg_ssh} |
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117 | \begin{aligned} |
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118 | \frac{\partial \eta }{\partial t} |
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119 | &\equiv \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{ \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right] |
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120 | +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right] \right\} } |
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121 | - \frac{\textit{emp}}{\rho_w } \\ |
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122 | &\equiv \sum\limits_k {\chi \ e_{3t}} - \frac{\textit{emp}}{\rho_w } |
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123 | \end{aligned} |
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124 | \end{equation} |
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125 | where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), |
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126 | expressed in Kg/m$^2$/s (which is equal to mm/s), |
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127 | and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation). |
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128 | If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then |
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129 | \textit{emp} can be written as the evaporation minus precipitation, minus the river runoff. |
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130 | The sea-surface height is evaluated using exactly the same time stepping scheme as |
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131 | the tracer equation \autoref{eq:TRA_nxt}: |
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132 | a leapfrog scheme in combination with an Asselin time filter, |
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133 | \ie\ the velocity appearing in \autoref{eq:DYN_spg_ssh} is centred in time (\textit{now} velocity). |
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134 | This is of paramount importance. |
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135 | Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to |
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136 | the sea surface height equation otherwise tracer content will not be conserved |
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137 | \citep{griffies.pacanowski.ea_MWR01, leclair.madec_OM09}. |
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138 | |
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139 | The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom, |
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140 | taking into account the change of the thickness of the levels: |
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141 | \begin{equation} |
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142 | \label{eq:DYN_wzv} |
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143 | \left\{ |
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144 | \begin{aligned} |
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145 | &\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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146 | &\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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147 | - \frac{1} {2 \rdt} \left( \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right) |
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148 | \end{aligned} |
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149 | \right. |
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150 | \end{equation} |
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151 | |
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152 | In the case of a non-linear free surface (\texttt{vvl?}), the top vertical velocity is $-\textit{emp}/\rho_w$, |
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153 | as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses, |
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154 | re-orientated downward. |
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155 | \gmcomment{not sure of this... to be modified with the change in emp setting} |
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156 | In the case of a linear free surface, the time derivative in \autoref{eq:DYN_wzv} disappears. |
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157 | The upper boundary condition applies at a fixed level $z=0$. |
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158 | The top vertical velocity is thus equal to the divergence of the barotropic transport |
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159 | (\ie\ the first term in the right-hand-side of \autoref{eq:DYN_spg_ssh}). |
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160 | |
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161 | Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates, |
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162 | its physical meaning is not the same: |
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163 | in the second case, $w$ is the velocity normal to the $s$-surfaces. |
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164 | Note also that the $k$-axis is re-orientated downwards in the \fortran\ code compared to |
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165 | the indexing used in the semi-discrete equations such as \autoref{eq:DYN_wzv} |
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166 | (see \autoref{subsec:DOM_Num_Index_vertical}). |
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167 | |
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168 | %% ================================================================================================= |
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169 | \section{Coriolis and advection: vector invariant form} |
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170 | \label{sec:DYN_adv_cor_vect} |
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171 | |
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172 | \begin{listing} |
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173 | \nlst{namdyn_adv} |
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174 | \caption{\forcode{&namdyn_adv}} |
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175 | \label{lst:namdyn_adv} |
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176 | \end{listing} |
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177 | |
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178 | The vector invariant form of the momentum equations is the one most often used in |
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179 | applications of the \NEMO\ ocean model. |
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180 | The flux form option (see next section) has been present since version $2$. |
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181 | Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables Coriolis and |
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182 | momentum advection terms are evaluated using a leapfrog scheme, |
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183 | \ie\ the velocity appearing in these expressions is centred in time (\textit{now} velocity). |
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184 | At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied following |
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185 | \autoref{chap:LBC}. |
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186 | |
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187 | %% ================================================================================================= |
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188 | \subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})} |
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189 | \label{subsec:DYN_vor} |
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190 | |
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191 | \begin{listing} |
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192 | \nlst{namdyn_vor} |
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193 | \caption{\forcode{&namdyn_vor}} |
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194 | \label{lst:namdyn_vor} |
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195 | \end{listing} |
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196 | |
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197 | Options are defined through the \nam{dyn_vor}{dyn\_vor} namelist variables. |
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198 | Four discretisations of the vorticity term (\texttt{ln\_dynvor\_xxx}\forcode{=.true.}) are available: |
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199 | conserving potential enstrophy of horizontally non-divergent flow (ENS scheme); |
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200 | conserving horizontal kinetic energy (ENE scheme); |
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201 | conserving potential enstrophy for the relative vorticity term and |
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202 | horizontal kinetic energy for the planetary vorticity term (MIX scheme); |
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203 | or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy |
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204 | (EEN scheme) (see \autoref{subsec:INVARIANTS_vorEEN}). |
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205 | 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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206 | vorticity term with analytical equations (\np[=.true.]{ln_dynvor_con}{ln\_dynvor\_con}). |
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207 | The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module. |
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208 | |
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209 | % enstrophy conserving scheme |
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210 | %% ================================================================================================= |
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211 | \subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens})]{Enstrophy conserving scheme (\protect\np{ln_dynvor_ens}{ln\_dynvor\_ens})} |
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212 | \label{subsec:DYN_vor_ens} |
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213 | |
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214 | In the enstrophy conserving case (ENS scheme), |
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215 | the discrete formulation of the vorticity term provides a global conservation of the enstrophy |
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216 | ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (\ie\ $\chi$=$0$), |
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217 | but does not conserve the total kinetic energy. |
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218 | It is given by: |
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219 | \begin{equation} |
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220 | \label{eq:DYN_vor_ens} |
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221 | \left\{ |
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222 | \begin{aligned} |
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223 | {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} |
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224 | & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2} \\ |
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225 | {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} |
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226 | & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j} |
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227 | \end{aligned} |
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228 | \right. |
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229 | \end{equation} |
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230 | |
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231 | % energy conserving scheme |
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232 | %% ================================================================================================= |
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233 | \subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene})]{Energy conserving scheme (\protect\np{ln_dynvor_ene}{ln\_dynvor\_ene})} |
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234 | \label{subsec:DYN_vor_ene} |
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235 | |
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236 | The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy. |
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237 | It is given by: |
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238 | \begin{equation} |
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239 | \label{eq:DYN_vor_ene} |
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240 | \left\{ |
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241 | \begin{aligned} |
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242 | {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) |
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243 | \; \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\ |
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244 | {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) |
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245 | \; \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } |
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246 | \end{aligned} |
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247 | \right. |
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248 | \end{equation} |
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249 | |
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250 | % mix energy/enstrophy conserving scheme |
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251 | %% ================================================================================================= |
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252 | \subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix})]{Mixed energy/enstrophy conserving scheme (\protect\np{ln_dynvor_mix}{ln\_dynvor\_mix})} |
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253 | \label{subsec:DYN_vor_mix} |
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254 | |
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255 | For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used. |
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256 | It consists of the ENS scheme (\autoref{eq:DYN_vor_ens}) for the relative vorticity term, |
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257 | and of the ENE scheme (\autoref{eq:DYN_vor_ene}) applied to the planetary vorticity term. |
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258 | \[ |
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259 | % \label{eq:DYN_vor_mix} |
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260 | \left\{ { |
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261 | \begin{aligned} |
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262 | {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} |
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263 | \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} } |
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264 | \; {\overline {\left( {\frac{f}{e_{3f} }} \right) |
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265 | \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\ |
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266 | {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j |
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267 | \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} } |
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268 | \; {\overline {\left( {\frac{f}{e_{3f} }} \right) |
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269 | \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill |
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270 | \end{aligned} |
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271 | } \right. |
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272 | \] |
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273 | |
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274 | % energy and enstrophy conserving scheme |
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275 | %% ================================================================================================= |
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276 | \subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een})]{Energy and enstrophy conserving scheme (\protect\np{ln_dynvor_een}{ln\_dynvor\_een})} |
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277 | \label{subsec:DYN_vor_een} |
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278 | |
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279 | In both the ENS and ENE schemes, |
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280 | it is apparent that the combination of $i$ and $j$ averages of the velocity allows for |
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281 | the presence of grid point oscillation structures that will be invisible to the operator. |
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282 | These structures are \textit{computational modes} that will be at least partly damped by |
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283 | the momentum diffusion operator (\ie\ the subgrid-scale advection), but not by the resolved advection term. |
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284 | The ENS and ENE schemes therefore do not contribute to dump any grid point noise in the horizontal velocity field. |
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285 | Such noise would result in more noise in the vertical velocity field, an undesirable feature. |
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286 | This is a well-known characteristic of $C$-grid discretization where |
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287 | $u$ and $v$ are located at different grid points, |
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288 | a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid. |
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289 | \gmcomment{ To circumvent this, Adcroft (ADD REF HERE) |
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290 | Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....} |
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291 | |
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292 | A very nice solution to the problem of double averaging was proposed by \citet{arakawa.hsu_MWR90}. |
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293 | The idea is to get rid of the double averaging by considering triad combinations of vorticity. |
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294 | It is noteworthy that this solution is conceptually quite similar to the one proposed by |
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295 | \citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:INVARIANTS}). |
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296 | |
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297 | The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified |
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298 | for spherical coordinates as described by \citet{arakawa.lamb_MWR81} to obtain the EEN scheme. |
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299 | First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: |
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300 | \[ |
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301 | % \label{eq:DYN_pot_vor} |
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302 | q = \frac{\zeta +f} {e_{3f} } |
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303 | \] |
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304 | where the relative vorticity is defined by (\autoref{eq:DYN_divcur_cur}), |
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305 | the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: |
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306 | \begin{equation} |
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307 | \label{eq:DYN_een_e3f} |
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308 | e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} |
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309 | \end{equation} |
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310 | |
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311 | \begin{figure}[!ht] |
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312 | \centering |
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313 | \includegraphics[width=0.66\textwidth]{DYN_een_triad} |
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314 | \caption[Triads used in the energy and enstrophy conserving scheme (EEN)]{ |
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315 | Triads used in the energy and enstrophy conserving scheme (EEN) for |
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316 | $u$-component (upper panel) and $v$-component (lower panel).} |
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317 | \label{fig:DYN_een_triad} |
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318 | \end{figure} |
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319 | |
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320 | A key point in \autoref{eq:DYN_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. |
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321 | 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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322 | (\np[=1]{nn_een_e3f}{nn\_een\_e3f}), or just by $4$ (\np[=.true.]{nn_een_e3f}{nn\_een\_e3f}). |
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323 | 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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324 | extends by continuity the value of $e_{3f}$ into the land areas. |
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325 | This case introduces a sub-grid-scale topography at f-points |
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326 | (with a systematic reduction of $e_{3f}$ when a model level intercept the bathymetry) |
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327 | that tends to reinforce the topostrophy of the flow |
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328 | (\ie\ the tendency of the flow to follow the isobaths) \citep{penduff.le-sommer.ea_OS07}. |
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329 | |
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330 | Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as |
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331 | the following triad combinations of the neighbouring potential vorticities defined at f-points |
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332 | (\autoref{fig:DYN_een_triad}): |
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333 | \begin{equation} |
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334 | \label{eq:DYN_Q_triads} |
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335 | _i^j \mathbb{Q}^{i_p}_{j_p} |
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336 | = \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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337 | \end{equation} |
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338 | 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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339 | |
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340 | Finally, the vorticity terms are represented as: |
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341 | \begin{equation} |
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342 | \label{eq:DYN_vor_een} |
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343 | \left\{ { |
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344 | \begin{aligned} |
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345 | +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} |
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346 | {^{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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347 | - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} |
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348 | {^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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349 | \end{aligned} |
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350 | } \right. |
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351 | \end{equation} |
---|
352 | |
---|
353 | This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. |
---|
354 | It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow |
---|
355 | (\ie\ $\chi$=$0$) (see \autoref{subsec:INVARIANTS_vorEEN}). |
---|
356 | Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of |
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357 | the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}. |
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358 | Furthermore, used in combination with a partial steps representation of bottom topography, |
---|
359 | it improves the interaction between current and topography, |
---|
360 | leading to a larger topostrophy of the flow \citep{barnier.madec.ea_OD06, penduff.le-sommer.ea_OS07}. |
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361 | |
---|
362 | %% ================================================================================================= |
---|
363 | \subsection[Kinetic energy gradient term (\textit{dynkeg.F90})]{Kinetic energy gradient term (\protect\mdl{dynkeg})} |
---|
364 | \label{subsec:DYN_keg} |
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365 | |
---|
366 | As demonstrated in \autoref{apdx:INVARIANTS}, |
---|
367 | there is a single discrete formulation of the kinetic energy gradient term that, |
---|
368 | together with the formulation chosen for the vertical advection (see below), |
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369 | conserves the total kinetic energy: |
---|
370 | \[ |
---|
371 | % \label{eq:DYN_keg} |
---|
372 | \left\{ |
---|
373 | \begin{aligned} |
---|
374 | -\frac{1}{2 \; e_{1u} } & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] \\ |
---|
375 | -\frac{1}{2 \; e_{2v} } & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] |
---|
376 | \end{aligned} |
---|
377 | \right. |
---|
378 | \] |
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379 | |
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380 | %% ================================================================================================= |
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381 | \subsection[Vertical advection term (\textit{dynzad.F90})]{Vertical advection term (\protect\mdl{dynzad})} |
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382 | \label{subsec:DYN_zad} |
---|
383 | |
---|
384 | The discrete formulation of the vertical advection, t |
---|
385 | ogether with the formulation chosen for the gradient of kinetic energy (KE) term, |
---|
386 | conserves the total kinetic energy. |
---|
387 | Indeed, the change of KE due to the vertical advection is exactly balanced by |
---|
388 | the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}). |
---|
389 | \[ |
---|
390 | % \label{eq:DYN_zad} |
---|
391 | \left\{ |
---|
392 | \begin{aligned} |
---|
393 | -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2} \;\delta_{k+1/2} \left[ u \right]\ }^{\,k} \\ |
---|
394 | -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2} \;\delta_{k+1/2} \left[ u \right]\ }^{\,k} |
---|
395 | \end{aligned} |
---|
396 | \right. |
---|
397 | \] |
---|
398 | When \np[=.true.]{ln_dynzad_zts}{ln\_dynzad\_zts}, |
---|
399 | a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term. |
---|
400 | This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}. |
---|
401 | Note that in this case, |
---|
402 | a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability, |
---|
403 | an option which is only available with a TVD scheme (see \np{ln_traadv_tvd_zts}{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}). |
---|
404 | |
---|
405 | %% ================================================================================================= |
---|
406 | \section{Coriolis and advection: flux form} |
---|
407 | \label{sec:DYN_adv_cor_flux} |
---|
408 | |
---|
409 | Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables. |
---|
410 | In the flux form (as in the vector invariant form), |
---|
411 | the Coriolis and momentum advection terms are evaluated using a leapfrog scheme, |
---|
412 | \ie\ the velocity appearing in their expressions is centred in time (\textit{now} velocity). |
---|
413 | At the lateral boundaries either free slip, |
---|
414 | no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}. |
---|
415 | |
---|
416 | %% ================================================================================================= |
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417 | \subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})} |
---|
418 | \label{subsec:DYN_cor_flux} |
---|
419 | |
---|
420 | 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. |
---|
421 | This altered Coriolis parameter is thus discretised at $f$-points. |
---|
422 | It is given by: |
---|
423 | \begin{multline*} |
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424 | % \label{eq:DYN_cor_metric} |
---|
425 | f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} - u\frac{\partial e_1 }{\partial j}} \right) \\ |
---|
426 | \equiv f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right] |
---|
427 | - \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right] } \ \right) |
---|
428 | \end{multline*} |
---|
429 | |
---|
430 | Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}) and (\autoref{eq:DYN_vor_een}) schemes can be used to |
---|
431 | compute the product of the Coriolis parameter and the vorticity. |
---|
432 | However, the energy-conserving scheme (\autoref{eq:DYN_vor_een}) has exclusively been used to date. |
---|
433 | This term is evaluated using a leapfrog scheme, \ie\ the velocity is centred in time (\textit{now} velocity). |
---|
434 | |
---|
435 | %% ================================================================================================= |
---|
436 | \subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})} |
---|
437 | \label{subsec:DYN_adv_flux} |
---|
438 | |
---|
439 | The discrete expression of the advection term is given by: |
---|
440 | \[ |
---|
441 | % \label{eq:DYN_adv} |
---|
442 | \left\{ |
---|
443 | \begin{aligned} |
---|
444 | \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} |
---|
445 | \left( \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i } \ u_t \right] |
---|
446 | + \delta_{j } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2} \ u_f \right] \right. \ \; \\ |
---|
447 | \left. + \delta_{k } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2} \ u_{uw} \right] \right) \\ |
---|
448 | \\ |
---|
449 | \frac{1}{e_{1v}\,e_{2v}\,e_{3v}} |
---|
450 | \left( \delta_{i } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f \right] |
---|
451 | + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i } \ v_t \right] \right. \ \, \, \\ |
---|
452 | \left. + \delta_{k } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw} \right] \right) \\ |
---|
453 | \end{aligned} |
---|
454 | \right. |
---|
455 | \] |
---|
456 | |
---|
457 | Two advection schemes are available: |
---|
458 | a $2^{nd}$ order centered finite difference scheme, CEN2, |
---|
459 | or a $3^{rd}$ order upstream biased scheme, UBS. |
---|
460 | The latter is described in \citet{shchepetkin.mcwilliams_OM05}. |
---|
461 | The schemes are selected using the namelist logicals \np{ln_dynadv_cen2}{ln\_dynadv\_cen2} and \np{ln_dynadv_ubs}{ln\_dynadv\_ubs}. |
---|
462 | In flux form, the schemes differ by the choice of a space and time interpolation to define the value of |
---|
463 | $u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-, |
---|
464 | and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$. |
---|
465 | |
---|
466 | % 2nd order centred scheme |
---|
467 | %% ================================================================================================= |
---|
468 | \subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln_dynadv_cen2}{ln\_dynadv\_cen2})} |
---|
469 | \label{subsec:DYN_adv_cen2} |
---|
470 | |
---|
471 | In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points: |
---|
472 | \begin{equation} |
---|
473 | \label{eq:DYN_adv_cen2} |
---|
474 | \left\{ |
---|
475 | \begin{aligned} |
---|
476 | u_T^{cen2} &=\overline u^{i } \quad & u_F^{cen2} &=\overline u^{j+1/2} \quad & u_{uw}^{cen2} &=\overline u^{k+1/2} \\ |
---|
477 | v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j \quad & v_{vw}^{cen2} &=\overline v ^{k+1/2} \\ |
---|
478 | \end{aligned} |
---|
479 | \right. |
---|
480 | \end{equation} |
---|
481 | |
---|
482 | The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema). |
---|
483 | It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to |
---|
484 | produce a sensible solution. |
---|
485 | The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, |
---|
486 | so $u$ and $v$ are the \emph{now} velocities. |
---|
487 | |
---|
488 | % UBS scheme |
---|
489 | %% ================================================================================================= |
---|
490 | \subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs})]{UBS: Upstream Biased Scheme (\protect\np{ln_dynadv_ubs}{ln\_dynadv\_ubs})} |
---|
491 | \label{subsec:DYN_adv_ubs} |
---|
492 | |
---|
493 | The UBS advection scheme is an upstream biased third order scheme based on |
---|
494 | an upstream-biased parabolic interpolation. |
---|
495 | For example, the evaluation of $u_T^{ubs} $ is done as follows: |
---|
496 | \begin{equation} |
---|
497 | \label{eq:DYN_adv_ubs} |
---|
498 | u_T^{ubs} =\overline u ^i-\;\frac{1}{6} |
---|
499 | \begin{cases} |
---|
500 | u"_{i-1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i \geqslant 0$ } \\ |
---|
501 | u"_{i+1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i < 0$ } |
---|
502 | \end{cases} |
---|
503 | \end{equation} |
---|
504 | where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$. |
---|
505 | This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error |
---|
506 | \citep{shchepetkin.mcwilliams_OM05}. |
---|
507 | The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}. |
---|
508 | It is a relatively good compromise between accuracy and smoothness. |
---|
509 | It is not a \emph{positive} scheme, meaning that false extrema are permitted. |
---|
510 | But the amplitudes of the false extrema are significantly reduced over those in the centred second order method. |
---|
511 | As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum |
---|
512 | (\ie\ \np[=]{ln_dynldf_lap}{ln\_dynldf\_lap}\np[=.false.]{ln_dynldf_bilap}{ln\_dynldf\_bilap}), |
---|
513 | and it is recommended to do so. |
---|
514 | |
---|
515 | The UBS scheme is not used in all directions. |
---|
516 | In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and |
---|
517 | $u_{vw}^{ubs}$ in \autoref{eq:DYN_adv_cen2} are used. |
---|
518 | UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm pursue the |
---|
519 | sentence:Since vertical mixing of momentum is a source term of the TKE equation... } |
---|
520 | |
---|
521 | For stability reasons, the first term in (\autoref{eq:DYN_adv_ubs}), |
---|
522 | which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time), |
---|
523 | while the second term, which is the diffusion part of the scheme, |
---|
524 | is evaluated using the \textit{before} velocity (forward in time). |
---|
525 | This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme. |
---|
526 | |
---|
527 | Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by |
---|
528 | one coefficient. |
---|
529 | Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. |
---|
530 | This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded. |
---|
531 | Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme. |
---|
532 | |
---|
533 | Note also that in the current version of \mdl{dynadv\_ubs}, |
---|
534 | there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS. |
---|
535 | This is an error and should be suppressed soon. |
---|
536 | \gmcomment{action : this have to be done} |
---|
537 | |
---|
538 | %% ================================================================================================= |
---|
539 | \section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})} |
---|
540 | \label{sec:DYN_hpg} |
---|
541 | |
---|
542 | \begin{listing} |
---|
543 | \nlst{namdyn_hpg} |
---|
544 | \caption{\forcode{&namdyn_hpg}} |
---|
545 | \label{lst:namdyn_hpg} |
---|
546 | \end{listing} |
---|
547 | |
---|
548 | Options are defined through the \nam{dyn_hpg}{dyn\_hpg} namelist variables. |
---|
549 | The key distinction between the different algorithms used for |
---|
550 | the hydrostatic pressure gradient is the vertical coordinate used, |
---|
551 | since HPG is a \emph{horizontal} pressure gradient, \ie\ computed along geopotential surfaces. |
---|
552 | As a result, any tilt of the surface of the computational levels will require a specific treatment to |
---|
553 | compute the hydrostatic pressure gradient. |
---|
554 | |
---|
555 | The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, |
---|
556 | \ie\ the density appearing in its expression is centred in time (\emph{now} $\rho$), |
---|
557 | or a semi-implcit scheme. |
---|
558 | At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied. |
---|
559 | |
---|
560 | %% ================================================================================================= |
---|
561 | \subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco})]{Full step $Z$-coordinate (\protect\np{ln_dynhpg_zco}{ln\_dynhpg\_zco})} |
---|
562 | \label{subsec:DYN_hpg_zco} |
---|
563 | |
---|
564 | The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface. |
---|
565 | However, the pressure is large at great depth while its horizontal gradient is several orders of magnitude smaller. |
---|
566 | This may lead to large truncation errors in the pressure gradient terms. |
---|
567 | Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows: |
---|
568 | |
---|
569 | for $k=km$ (surface layer, $jk=1$ in the code) |
---|
570 | \begin{equation} |
---|
571 | \label{eq:DYN_hpg_zco_surf} |
---|
572 | \left\{ |
---|
573 | \begin{aligned} |
---|
574 | \left. \delta_{i+1/2} \left[ p^h \right] \right|_{k=km} |
---|
575 | &= \frac{1}{2} g \ \left. \delta_{i+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\ |
---|
576 | \left. \delta_{j+1/2} \left[ p^h \right] \right|_{k=km} |
---|
577 | &= \frac{1}{2} g \ \left. \delta_{j+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\ |
---|
578 | \end{aligned} |
---|
579 | \right. |
---|
580 | \end{equation} |
---|
581 | |
---|
582 | for $1<k<km$ (interior layer) |
---|
583 | \begin{equation} |
---|
584 | \label{eq:DYN_hpg_zco} |
---|
585 | \left\{ |
---|
586 | \begin{aligned} |
---|
587 | \left. \delta_{i+1/2} \left[ p^h \right] \right|_{k} |
---|
588 | &= \left. \delta_{i+1/2} \left[ p^h \right] \right|_{k-1} |
---|
589 | + \frac{1}{2}\;g\; \left. \delta_{i+1/2} \left[ e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k} \\ |
---|
590 | \left. \delta_{j+1/2} \left[ p^h \right] \right|_{k} |
---|
591 | &= \left. \delta_{j+1/2} \left[ p^h \right] \right|_{k-1} |
---|
592 | + \frac{1}{2}\;g\; \left. \delta_{j+1/2} \left[ e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k} \\ |
---|
593 | \end{aligned} |
---|
594 | \right. |
---|
595 | \end{equation} |
---|
596 | |
---|
597 | Note that the $1/2$ factor in (\autoref{eq:DYN_hpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as |
---|
598 | the vertical derivative of the scale factor at the surface level ($z=0$). |
---|
599 | Note also that in case of variable volume level (\texttt{vvl?} defined), |
---|
600 | the surface pressure gradient is included in \autoref{eq:DYN_hpg_zco_surf} and |
---|
601 | \autoref{eq:DYN_hpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$. |
---|
602 | |
---|
603 | %% ================================================================================================= |
---|
604 | \subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps})]{Partial step $Z$-coordinate (\protect\np{ln_dynhpg_zps}{ln\_dynhpg\_zps})} |
---|
605 | \label{subsec:DYN_hpg_zps} |
---|
606 | |
---|
607 | With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths. |
---|
608 | Before taking horizontal gradients between these tracer points, |
---|
609 | a linear interpolation is used to approximate the deeper tracer as if |
---|
610 | it actually lived at the depth of the shallower tracer point. |
---|
611 | |
---|
612 | Apart from this modification, |
---|
613 | the horizontal hydrostatic pressure gradient evaluated in the $z$-coordinate with partial step is exactly as in |
---|
614 | the pure $z$-coordinate case. |
---|
615 | As explained in detail in section \autoref{sec:TRA_zpshde}, |
---|
616 | the nonlinearity of pressure effects in the equation of state is such that |
---|
617 | it is better to interpolate temperature and salinity vertically before computing the density. |
---|
618 | Horizontal gradients of temperature and salinity are needed for the TRA modules, |
---|
619 | which is the reason why the horizontal gradients of density at the deepest model level are computed in |
---|
620 | module \mdl{zpsdhe} located in the TRA directory and described in \autoref{sec:TRA_zpshde}. |
---|
621 | |
---|
622 | %% ================================================================================================= |
---|
623 | \subsection{$S$- and $Z$-$S$-coordinates} |
---|
624 | \label{subsec:DYN_hpg_sco} |
---|
625 | |
---|
626 | Pressure gradient formulations in an $s$-coordinate have been the subject of a vast number of papers |
---|
627 | (\eg, \citet{song_MWR98, shchepetkin.mcwilliams_OM05}). |
---|
628 | A number of different pressure gradient options are coded but the ROMS-like, |
---|
629 | density Jacobian with cubic polynomial method is currently disabled whilst known bugs are under investigation. |
---|
630 | |
---|
631 | $\bullet$ Traditional coding (see for example \citet{madec.delecluse.ea_JPO96}: (\np[=.true.]{ln_dynhpg_sco}{ln\_dynhpg\_sco}) |
---|
632 | \begin{equation} |
---|
633 | \label{eq:DYN_hpg_sco} |
---|
634 | \left\{ |
---|
635 | \begin{aligned} |
---|
636 | - \frac{1} {\rho_o \, e_{1u}} \; \delta_{i+1/2} \left[ p^h \right] |
---|
637 | + \frac{g\; \overline {\rho}^{i+1/2}} {\rho_o \, e_{1u}} \; \delta_{i+1/2} \left[ z_t \right] \\ |
---|
638 | - \frac{1} {\rho_o \, e_{2v}} \; \delta_{j+1/2} \left[ p^h \right] |
---|
639 | + \frac{g\; \overline {\rho}^{j+1/2}} {\rho_o \, e_{2v}} \; \delta_{j+1/2} \left[ z_t \right] \\ |
---|
640 | \end{aligned} |
---|
641 | \right. |
---|
642 | \end{equation} |
---|
643 | |
---|
644 | Where the first term is the pressure gradient along coordinates, |
---|
645 | computed as in \autoref{eq:DYN_hpg_zco_surf} - \autoref{eq:DYN_hpg_zco}, |
---|
646 | and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point |
---|
647 | ($e_{3w}$). |
---|
648 | |
---|
649 | $\bullet$ Traditional coding with adaptation for ice shelf cavities (\np[=.true.]{ln_dynhpg_isf}{ln\_dynhpg\_isf}). |
---|
650 | This scheme need the activation of ice shelf cavities (\np[=.true.]{ln_isfcav}{ln\_isfcav}). |
---|
651 | |
---|
652 | $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np[=.true.]{ln_dynhpg_prj}{ln\_dynhpg\_prj}) |
---|
653 | |
---|
654 | $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{shchepetkin.mcwilliams_OM05} |
---|
655 | (\np[=.true.]{ln_dynhpg_djc}{ln\_dynhpg\_djc}) (currently disabled; under development) |
---|
656 | |
---|
657 | Note that expression \autoref{eq:DYN_hpg_sco} is commonly used when the variable volume formulation is activated |
---|
658 | (\texttt{vvl?}) because in that case, even with a flat bottom, |
---|
659 | the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}. |
---|
660 | The pressure jacobian scheme (\np[=.true.]{ln_dynhpg_prj}{ln\_dynhpg\_prj}) is available as |
---|
661 | an improved option to \np[=.true.]{ln_dynhpg_sco}{ln\_dynhpg\_sco} when \texttt{vvl?} is active. |
---|
662 | The pressure Jacobian scheme uses a constrained cubic spline to |
---|
663 | reconstruct the density profile across the water column. |
---|
664 | This method maintains the monotonicity between the density nodes. |
---|
665 | The pressure can be calculated by analytical integration of the density profile and |
---|
666 | a pressure Jacobian method is used to solve the horizontal pressure gradient. |
---|
667 | This method can provide a more accurate calculation of the horizontal pressure gradient than the standard scheme. |
---|
668 | |
---|
669 | %% ================================================================================================= |
---|
670 | \subsection{Ice shelf cavity} |
---|
671 | \label{subsec:DYN_hpg_isf} |
---|
672 | |
---|
673 | Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and |
---|
674 | the pressure gradient due to the ocean load (\np[=.true.]{ln_dynhpg_isf}{ln\_dynhpg\_isf}).\\ |
---|
675 | |
---|
676 | The main hypothesis to compute the ice shelf load is that the ice shelf is in an isostatic equilibrium. |
---|
677 | The top pressure is computed integrating from surface to the base of the ice shelf a reference density profile |
---|
678 | (prescribed as density of a water at 34.4 PSU and -1.9\deg{C}) and |
---|
679 | corresponds to the water replaced by the ice shelf. |
---|
680 | This top pressure is constant over time. |
---|
681 | A detailed description of this method is described in \citet{losch_JGR08}.\\ |
---|
682 | |
---|
683 | The pressure gradient due to ocean load is computed using the expression \autoref{eq:DYN_hpg_sco} described in |
---|
684 | \autoref{subsec:DYN_hpg_sco}. |
---|
685 | |
---|
686 | %% ================================================================================================= |
---|
687 | \subsection[Time-scheme (\forcode{ln_dynhpg_imp})]{Time-scheme (\protect\np{ln_dynhpg_imp}{ln\_dynhpg\_imp})} |
---|
688 | \label{subsec:DYN_hpg_imp} |
---|
689 | |
---|
690 | The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and |
---|
691 | therefore the density used in all discrete expressions given above is the \textit{now} density, |
---|
692 | computed from the \textit{now} temperature and salinity. |
---|
693 | In some specific cases |
---|
694 | (usually high resolution simulations over an ocean domain which includes weakly stratified regions) |
---|
695 | the physical phenomenon that controls the time-step is internal gravity waves (IGWs). |
---|
696 | A semi-implicit scheme for doubling the stability limit associated with IGWs can be used |
---|
697 | \citep{brown.campana_MWR78, maltrud.smith.ea_JGR98}. |
---|
698 | It involves the evaluation of the hydrostatic pressure gradient as |
---|
699 | an average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ |
---|
700 | (\ie\ \textit{before}, \textit{now} and \textit{after} time-steps), |
---|
701 | rather than at the central time level $t$ only, as in the standard leapfrog scheme. |
---|
702 | |
---|
703 | $\bullet$ leapfrog scheme (\np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}): |
---|
704 | |
---|
705 | \begin{equation} |
---|
706 | \label{eq:DYN_hpg_lf} |
---|
707 | \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; |
---|
708 | -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right] |
---|
709 | \end{equation} |
---|
710 | |
---|
711 | $\bullet$ semi-implicit scheme (\np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}): |
---|
712 | \begin{equation} |
---|
713 | \label{eq:DYN_hpg_imp} |
---|
714 | \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; |
---|
715 | -\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] |
---|
716 | \end{equation} |
---|
717 | |
---|
718 | The semi-implicit time scheme \autoref{eq:DYN_hpg_imp} is made possible without |
---|
719 | significant additional computation since the density can be updated to time level $t+\rdt$ before |
---|
720 | computing the horizontal hydrostatic pressure gradient. |
---|
721 | It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using |
---|
722 | \autoref{eq:DYN_hpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:DYN_hpg_lf}. |
---|
723 | Note that \autoref{eq:DYN_hpg_imp} is equivalent to applying a time filter to the pressure gradient to |
---|
724 | eliminate high frequency IGWs. |
---|
725 | Obviously, when using \autoref{eq:DYN_hpg_imp}, |
---|
726 | the doubling of the time-step is achievable only if no other factors control the time-step, |
---|
727 | such as the stability limits associated with advection or diffusion. |
---|
728 | |
---|
729 | In practice, the semi-implicit scheme is used when \np[=.true.]{ln_dynhpg_imp}{ln\_dynhpg\_imp}. |
---|
730 | In this case, we choose to apply the time filter to temperature and salinity used in the equation of state, |
---|
731 | instead of applying it to the hydrostatic pressure or to the density, |
---|
732 | so that no additional storage array has to be defined. |
---|
733 | The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows: |
---|
734 | \[ |
---|
735 | % \label{eq:DYN_rho_flt} |
---|
736 | \rho^t = \rho( \widetilde{T},\widetilde {S},z_t) |
---|
737 | \quad \text{with} \quad |
---|
738 | \widetilde{X} = 1 / 4 \left( X^{t+\rdt} +2 \,X^t + X^{t-\rdt} \right) |
---|
739 | \] |
---|
740 | |
---|
741 | Note that in the semi-implicit case, it is necessary to save the filtered density, |
---|
742 | an extra three-dimensional field, in the restart file to restart the model with exact reproducibility. |
---|
743 | This option is controlled by \np{nn_dynhpg_rst}{nn\_dynhpg\_rst}, a namelist parameter. |
---|
744 | |
---|
745 | %% ================================================================================================= |
---|
746 | \section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})} |
---|
747 | \label{sec:DYN_spg} |
---|
748 | |
---|
749 | \begin{listing} |
---|
750 | \nlst{namdyn_spg} |
---|
751 | \caption{\forcode{&namdyn_spg}} |
---|
752 | \label{lst:namdyn_spg} |
---|
753 | \end{listing} |
---|
754 | |
---|
755 | Options are defined through the \nam{dyn_spg}{dyn\_spg} namelist variables. |
---|
756 | The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). |
---|
757 | The main distinction is between the fixed volume case (linear free surface) and |
---|
758 | the variable volume case (nonlinear free surface, \texttt{vvl?} is defined). |
---|
759 | In the linear free surface case (\autoref{subsec:MB_free_surface}) |
---|
760 | the vertical scale factors $e_{3}$ are fixed in time, |
---|
761 | while they are time-dependent in the nonlinear case (\autoref{subsec:MB_free_surface}). |
---|
762 | With both linear and nonlinear free surface, external gravity waves are allowed in the equations, |
---|
763 | which imposes a very small time step when an explicit time stepping is used. |
---|
764 | Two methods are proposed to allow a longer time step for the three-dimensional equations: |
---|
765 | the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:MB_flt?}), |
---|
766 | and the split-explicit free surface described below. |
---|
767 | The extra term introduced in the filtered method is calculated implicitly, |
---|
768 | so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
---|
769 | |
---|
770 | The form of the surface pressure gradient term depends on how the user wants to |
---|
771 | handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:MB_hor_pg}). |
---|
772 | Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx): |
---|
773 | an explicit formulation which requires a small time step; |
---|
774 | a filtered free surface formulation which allows a larger time step by |
---|
775 | adding a filtering term into the momentum equation; |
---|
776 | and a split-explicit free surface formulation, described below, which also allows a larger time step. |
---|
777 | |
---|
778 | The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it. |
---|
779 | As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
---|
780 | |
---|
781 | %% ================================================================================================= |
---|
782 | \subsection[Explicit free surface (\forcode{ln_dynspg_exp})]{Explicit free surface (\protect\np{ln_dynspg_exp}{ln\_dynspg\_exp})} |
---|
783 | \label{subsec:DYN_spg_exp} |
---|
784 | |
---|
785 | In the explicit free surface formulation (\np{ln_dynspg_exp}{ln\_dynspg\_exp} set to true), |
---|
786 | the model time step is chosen to be small enough to resolve the external gravity waves |
---|
787 | (typically a few tens of seconds). |
---|
788 | The surface pressure gradient, evaluated using a leap-frog scheme (\ie\ centered in time), |
---|
789 | is thus simply given by : |
---|
790 | \begin{equation} |
---|
791 | \label{eq:DYN_spg_exp} |
---|
792 | \left\{ |
---|
793 | \begin{aligned} |
---|
794 | - \frac{1}{e_{1u}\,\rho_o} \; \delta_{i+1/2} \left[ \,\rho \,\eta\, \right] \\ |
---|
795 | - \frac{1}{e_{2v}\,\rho_o} \; \delta_{j+1/2} \left[ \,\rho \,\eta\, \right] |
---|
796 | \end{aligned} |
---|
797 | \right. |
---|
798 | \end{equation} |
---|
799 | |
---|
800 | Note that in the non-linear free surface case (\ie\ \texttt{vvl?} defined), |
---|
801 | the surface pressure gradient is already included in the momentum tendency through |
---|
802 | the level thickness variation allowed in the computation of the hydrostatic pressure gradient. |
---|
803 | Thus, nothing is done in the \mdl{dynspg\_exp} module. |
---|
804 | |
---|
805 | %% ================================================================================================= |
---|
806 | \subsection[Split-explicit free surface (\forcode{ln_dynspg_ts})]{Split-explicit free surface (\protect\np{ln_dynspg_ts}{ln\_dynspg\_ts})} |
---|
807 | \label{subsec:DYN_spg_ts} |
---|
808 | %\nlst{namsplit} |
---|
809 | |
---|
810 | The split-explicit free surface formulation used in \NEMO\ (\np{ln_dynspg_ts}{ln\_dynspg\_ts} set to true), |
---|
811 | also called the time-splitting formulation, follows the one proposed by \citet{shchepetkin.mcwilliams_OM05}. |
---|
812 | The general idea is to solve the free surface equation and the associated barotropic velocity equations with |
---|
813 | a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables |
---|
814 | (\autoref{fig:DYN_spg_ts}). |
---|
815 | The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through |
---|
816 | the \np{nn_baro}{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$. |
---|
817 | This parameter can be optionally defined automatically (\np[=.true.]{ln_bt_nn_auto}{ln\_bt\_nn\_auto}) considering that |
---|
818 | the stability of the barotropic system is essentially controled by external waves propagation. |
---|
819 | Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry. |
---|
820 | Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn_bt_cmax}{rn\_bt\_cmax}. |
---|
821 | |
---|
822 | The barotropic mode solves the following equations: |
---|
823 | % \begin{subequations} |
---|
824 | % \label{eq:DYN_BT} |
---|
825 | \begin{equation} |
---|
826 | \label{eq:DYN_BT_dyn} |
---|
827 | \frac{\partial {\mathrm \overline{{\mathbf U}}_h} }{\partial t}= |
---|
828 | -f\;{\mathrm {\mathbf k}}\times {\mathrm \overline{{\mathbf U}}_h} |
---|
829 | -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \mathrm {\overline{{\mathbf U}}_h} + \mathrm {\overline{\mathbf G}} |
---|
830 | \end{equation} |
---|
831 | \[ |
---|
832 | % \label{eq:DYN_BT_ssh} |
---|
833 | \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\mathrm{\mathbf \overline{U}}}_h \,} \right]+P-E |
---|
834 | \] |
---|
835 | % \end{subequations} |
---|
836 | where $\mathrm {\overline{\mathbf G}}$ is a forcing term held constant, containing coupling term between modes, |
---|
837 | surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. |
---|
838 | The third term on the right hand side of \autoref{eq:DYN_BT_dyn} represents the bottom stress |
---|
839 | (see section \autoref{sec:ZDF_drg}), explicitly accounted for at each barotropic iteration. |
---|
840 | Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm |
---|
841 | detailed in \citet{shchepetkin.mcwilliams_OM05}. |
---|
842 | AB3-AM4 coefficients used in \NEMO\ follow the second-order accurate, |
---|
843 | "multi-purpose" stability compromise as defined in \citet{shchepetkin.mcwilliams_ibk09} |
---|
844 | (see their figure 12, lower left). |
---|
845 | |
---|
846 | \begin{figure}[!t] |
---|
847 | \centering |
---|
848 | \includegraphics[width=0.66\textwidth]{DYN_dynspg_ts} |
---|
849 | \caption[Split-explicit time stepping scheme for the external and internal modes]{ |
---|
850 | Schematic of the split-explicit time stepping scheme for the external and internal modes. |
---|
851 | Time increases to the right. |
---|
852 | In this particular exemple, |
---|
853 | a boxcar averaging window over \np{nn_baro}{nn\_baro} barotropic time steps is used |
---|
854 | (\np[=1]{nn_bt_flt}{nn\_bt\_flt}) and \np[=5]{nn_baro}{nn\_baro}. |
---|
855 | Internal mode time steps (which are also the model time steps) are denoted by |
---|
856 | $t-\rdt$, $t$ and $t+\rdt$. |
---|
857 | Variables with $k$ superscript refer to instantaneous barotropic variables, |
---|
858 | $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary |
---|
859 | (red vertical bars) and secondary weights (blue vertical bars). |
---|
860 | The former are used to obtain time filtered quantities at $t+\rdt$ while |
---|
861 | the latter are used to obtain time averaged transports to advect tracers. |
---|
862 | a) Forward time integration: |
---|
863 | \protect\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}, \protect\np[=.true.]{ln_bt_av}{ln\_bt\_av}. |
---|
864 | b) Centred time integration: |
---|
865 | \protect\np[=.false.]{ln_bt_fw}{ln\_bt\_fw}, \protect\np[=.true.]{ln_bt_av}{ln\_bt\_av}. |
---|
866 | c) Forward time integration with no time filtering (POM-like scheme): |
---|
867 | \protect\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}, \protect\np[=.false.]{ln_bt_av}{ln\_bt\_av}.} |
---|
868 | \label{fig:DYN_spg_ts} |
---|
869 | \end{figure} |
---|
870 | |
---|
871 | In the default case (\np[=.true.]{ln_bt_fw}{ln\_bt\_fw}), |
---|
872 | the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps |
---|
873 | (\autoref{fig:DYN_spg_ts}a). |
---|
874 | To avoid aliasing of fast barotropic motions into three dimensional equations, |
---|
875 | time filtering is eventually applied on barotropic quantities (\np[=.true.]{ln_bt_av}{ln\_bt\_av}). |
---|
876 | In that case, the integration is extended slightly beyond \textit{after} time step to |
---|
877 | provide time filtered quantities. |
---|
878 | These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. |
---|
879 | Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, |
---|
880 | asselin filtering is not applied to barotropic quantities.\\ |
---|
881 | Alternatively, one can choose to integrate barotropic equations starting from \textit{before} time step |
---|
882 | (\np[=.false.]{ln_bt_fw}{ln\_bt\_fw}). |
---|
883 | Although more computationaly expensive ( \np{nn_baro}{nn\_baro} additional iterations are indeed necessary), |
---|
884 | the baroclinic to barotropic forcing term given at \textit{now} time step become centred in |
---|
885 | the middle of the integration window. |
---|
886 | It can easily be shown that this property removes part of splitting errors between modes, |
---|
887 | which increases the overall numerical robustness. |
---|
888 | %references to Patrick Marsaleix' work here. Also work done by SHOM group. |
---|
889 | |
---|
890 | |
---|
891 | As far as tracer conservation is concerned, |
---|
892 | barotropic velocities used to advect tracers must also be updated at \textit{now} time step. |
---|
893 | This implies to change the traditional order of computations in \NEMO: |
---|
894 | most of momentum trends (including the barotropic mode calculation) updated first, tracers' after. |
---|
895 | This \textit{de facto} makes semi-implicit hydrostatic pressure gradient |
---|
896 | (see section \autoref{subsec:DYN_hpg_imp}) |
---|
897 | and time splitting not compatible. |
---|
898 | Advective barotropic velocities are obtained by using a secondary set of filtering weights, |
---|
899 | uniquely defined from the filter coefficients used for the time averaging (\citet{shchepetkin.mcwilliams_OM05}). |
---|
900 | Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to |
---|
901 | obtain exact conservation. |
---|
902 | |
---|
903 | |
---|
904 | One can eventually choose to feedback instantaneous values by not using any time filter |
---|
905 | (\np[=.false.]{ln_bt_av}{ln\_bt\_av}). |
---|
906 | In that case, external mode equations are continuous in time, |
---|
907 | \ie\ they are not re-initialized when starting a new sub-stepping sequence. |
---|
908 | This is the method used so far in the POM model, the stability being maintained by |
---|
909 | refreshing at (almost) each barotropic time step advection and horizontal diffusion terms. |
---|
910 | Since the latter terms have not been added in \NEMO\ for computational efficiency, |
---|
911 | removing time filtering is not recommended except for debugging purposes. |
---|
912 | This may be used for instance to appreciate the damping effect of the standard formulation on |
---|
913 | external gravity waves in idealized or weakly non-linear cases. |
---|
914 | Although the damping is lower than for the filtered free surface, |
---|
915 | it is still significant as shown by \citet{levier.treguier.ea_rpt07} in the case of an analytical barotropic Kelvin wave. |
---|
916 | |
---|
917 | \gmcomment{ %%% copy from griffies Book |
---|
918 | |
---|
919 | \textbf{title: Time stepping the barotropic system } |
---|
920 | |
---|
921 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. |
---|
922 | Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using |
---|
923 | the small barotropic time step $\rdt$. |
---|
924 | We have |
---|
925 | |
---|
926 | \[ |
---|
927 | % \label{eq:DYN_spg_ts_eta} |
---|
928 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
---|
929 | = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
---|
930 | \] |
---|
931 | \begin{multline*} |
---|
932 | % \label{eq:DYN_spg_ts_u} |
---|
933 | \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ |
---|
934 | = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) |
---|
935 | - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] |
---|
936 | \end{multline*} |
---|
937 | \ |
---|
938 | |
---|
939 | In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with |
---|
940 | the barotropic time steps. |
---|
941 | The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which |
---|
942 | the vertically integrated forcing $\textbf{M}(\tau)$ |
---|
943 | (note that this forcing includes the surface freshwater forcing), |
---|
944 | the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, |
---|
945 | and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over |
---|
946 | a single cycle. |
---|
947 | This is also the time that sets the barotropic time steps via |
---|
948 | \[ |
---|
949 | % \label{eq:DYN_spg_ts_t} |
---|
950 | t_n=\tau+n\rdt |
---|
951 | \] |
---|
952 | with $n$ an integer. |
---|
953 | The density scaled surface pressure is evaluated via |
---|
954 | \[ |
---|
955 | % \label{eq:DYN_spg_ts_ps} |
---|
956 | p_s^{(b)}(\tau,t_{n}) = |
---|
957 | \begin{cases} |
---|
958 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
---|
959 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
---|
960 | \end{cases} |
---|
961 | \] |
---|
962 | To get started, we assume the following initial conditions |
---|
963 | \[ |
---|
964 | % \label{eq:DYN_spg_ts_eta} |
---|
965 | \begin{split} |
---|
966 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ |
---|
967 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} |
---|
968 | \end{split} |
---|
969 | \] |
---|
970 | with |
---|
971 | \[ |
---|
972 | % \label{eq:DYN_spg_ts_etaF} |
---|
973 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n}) |
---|
974 | \] |
---|
975 | the time averaged surface height taken from the previous barotropic cycle. |
---|
976 | Likewise, |
---|
977 | \[ |
---|
978 | % \label{eq:DYN_spg_ts_u} |
---|
979 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ |
---|
980 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} |
---|
981 | \] |
---|
982 | with |
---|
983 | \[ |
---|
984 | % \label{eq:DYN_spg_ts_u} |
---|
985 | \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n}) |
---|
986 | \] |
---|
987 | the time averaged vertically integrated transport. |
---|
988 | Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. |
---|
989 | |
---|
990 | Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ , |
---|
991 | the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at |
---|
992 | baroclinic time $\tau + \rdt \tau$ |
---|
993 | \[ |
---|
994 | % \label{eq:DYN_spg_ts_u} |
---|
995 | \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
---|
996 | \] |
---|
997 | The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using |
---|
998 | the following form |
---|
999 | |
---|
1000 | \begin{equation} |
---|
1001 | \label{eq:DYN_spg_ts_ssh} |
---|
1002 | \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] |
---|
1003 | \end{equation} |
---|
1004 | |
---|
1005 | The use of this "big-leap-frog" scheme for the surface height ensures compatibility between |
---|
1006 | the mass/volume budgets and the tracer budgets. |
---|
1007 | More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). |
---|
1008 | |
---|
1009 | In general, some form of time filter is needed to maintain integrity of the surface height field due to |
---|
1010 | the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. |
---|
1011 | We have tried various forms of such filtering, |
---|
1012 | with the following method discussed in \cite{griffies.pacanowski.ea_MWR01} chosen due to |
---|
1013 | its stability and reasonably good maintenance of tracer conservation properties (see ??). |
---|
1014 | |
---|
1015 | \begin{equation} |
---|
1016 | \label{eq:DYN_spg_ts_sshf} |
---|
1017 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
---|
1018 | \end{equation} |
---|
1019 | Another approach tried was |
---|
1020 | |
---|
1021 | \[ |
---|
1022 | % \label{eq:DYN_spg_ts_sshf2} |
---|
1023 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
---|
1024 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt) |
---|
1025 | + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right] |
---|
1026 | \] |
---|
1027 | |
---|
1028 | which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. |
---|
1029 | This isolation allows for an easy check that tracer conservation is exact when |
---|
1030 | eliminating tracer and surface height time filtering (see ?? for more complete discussion). |
---|
1031 | However, in the general case with a non-zero $\alpha$, |
---|
1032 | the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. |
---|
1033 | |
---|
1034 | } %%end gm comment (copy of griffies book) |
---|
1035 | |
---|
1036 | %% ================================================================================================= |
---|
1037 | \subsection{Filtered free surface (\forcode{dynspg_flt?})} |
---|
1038 | \label{subsec:DYN_spg_fltp} |
---|
1039 | |
---|
1040 | The filtered formulation follows the \citet{roullet.madec_JGR00} implementation. |
---|
1041 | The extra term introduced in the equations (see \autoref{subsec:MB_free_surface}) is solved implicitly. |
---|
1042 | The elliptic solvers available in the code are documented in \autoref{chap:MISC}. |
---|
1043 | |
---|
1044 | %% gm %%======>>>> given here the discrete eqs provided to the solver |
---|
1045 | \gmcomment{ %%% copy from chap-model basics |
---|
1046 | \[ |
---|
1047 | % \label{eq:DYN_spg_flt} |
---|
1048 | \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= {\mathrm {\mathbf M}} |
---|
1049 | - g \nabla \left( \tilde{\rho} \ \eta \right) |
---|
1050 | - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) |
---|
1051 | \] |
---|
1052 | where $T_c$, is a parameter with dimensions of time which characterizes the force, |
---|
1053 | $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, |
---|
1054 | and $\mathrm {\mathbf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, |
---|
1055 | non-linear and viscous terms in \autoref{eq:MB_dyn}. |
---|
1056 | } %end gmcomment |
---|
1057 | |
---|
1058 | Note that in the linear free surface formulation (\texttt{vvl?} not defined), |
---|
1059 | the ocean depth is time-independent and so is the matrix to be inverted. |
---|
1060 | It is computed once and for all and applies to all ocean time steps. |
---|
1061 | |
---|
1062 | %% ================================================================================================= |
---|
1063 | \section[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})} |
---|
1064 | \label{sec:DYN_ldf} |
---|
1065 | |
---|
1066 | \begin{listing} |
---|
1067 | \nlst{namdyn_ldf} |
---|
1068 | \caption{\forcode{&namdyn_ldf}} |
---|
1069 | \label{lst:namdyn_ldf} |
---|
1070 | \end{listing} |
---|
1071 | |
---|
1072 | Options are defined through the \nam{dyn_ldf}{dyn\_ldf} namelist variables. |
---|
1073 | The options available for lateral diffusion are to use either laplacian (rotated or not) or biharmonic operators. |
---|
1074 | The coefficients may be constant or spatially variable; |
---|
1075 | the description of the coefficients is found in the chapter on lateral physics (\autoref{chap:LDF}). |
---|
1076 | The lateral diffusion of momentum is evaluated using a forward scheme, |
---|
1077 | \ie\ the velocity appearing in its expression is the \textit{before} velocity in time, |
---|
1078 | except for the pure vertical component that appears when a tensor of rotation is used. |
---|
1079 | This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}). |
---|
1080 | |
---|
1081 | At the lateral boundaries either free slip, |
---|
1082 | no slip or partial slip boundary conditions are applied according to the user's choice (see \autoref{chap:LBC}). |
---|
1083 | |
---|
1084 | \gmcomment{ |
---|
1085 | Hyperviscous operators are frequently used in the simulation of turbulent flows to |
---|
1086 | control the dissipation of unresolved small scale features. |
---|
1087 | Their primary role is to provide strong dissipation at the smallest scale supported by |
---|
1088 | the grid while minimizing the impact on the larger scale features. |
---|
1089 | Hyperviscous operators are thus designed to be more scale selective than the traditional, |
---|
1090 | physically motivated Laplace operator. |
---|
1091 | In finite difference methods, |
---|
1092 | the biharmonic operator is frequently the method of choice to achieve this scale selective dissipation since |
---|
1093 | its damping time (\ie\ its spin down time) scale like $\lambda^{-4}$ for disturbances of wavelength $\lambda$ |
---|
1094 | (so that short waves damped more rapidelly than long ones), |
---|
1095 | whereas the Laplace operator damping time scales only like $\lambda^{-2}$. |
---|
1096 | } |
---|
1097 | |
---|
1098 | %% ================================================================================================= |
---|
1099 | \subsection[Iso-level laplacian (\forcode{ln_dynldf_lap})]{Iso-level laplacian operator (\protect\np{ln_dynldf_lap}{ln\_dynldf\_lap})} |
---|
1100 | \label{subsec:DYN_ldf_lap} |
---|
1101 | |
---|
1102 | For lateral iso-level diffusion, the discrete operator is: |
---|
1103 | \begin{equation} |
---|
1104 | \label{eq:DYN_ldf_lap} |
---|
1105 | \left\{ |
---|
1106 | \begin{aligned} |
---|
1107 | D_u^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm} |
---|
1108 | \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[ |
---|
1109 | {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\ |
---|
1110 | D_v^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm} |
---|
1111 | \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[ |
---|
1112 | {A_f^{lm} \;e_{3f} \zeta } \right] |
---|
1113 | \end{aligned} |
---|
1114 | \right. |
---|
1115 | \end{equation} |
---|
1116 | |
---|
1117 | As explained in \autoref{subsec:MB_ldf}, |
---|
1118 | this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and |
---|
1119 | ensures a complete separation between the vorticity and divergence parts of the momentum diffusion. |
---|
1120 | |
---|
1121 | %% ================================================================================================= |
---|
1122 | \subsection[Rotated laplacian (\forcode{ln_dynldf_iso})]{Rotated laplacian operator (\protect\np{ln_dynldf_iso}{ln\_dynldf\_iso})} |
---|
1123 | \label{subsec:DYN_ldf_iso} |
---|
1124 | |
---|
1125 | A rotation of the lateral momentum diffusion operator is needed in several cases: |
---|
1126 | for iso-neutral diffusion in the $z$-coordinate (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) and |
---|
1127 | for either iso-neutral (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) or |
---|
1128 | geopotential (\np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}) diffusion in the $s$-coordinate. |
---|
1129 | In the partial step case, coordinates are horizontal except at the deepest level and |
---|
1130 | no rotation is performed when \np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}. |
---|
1131 | The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on |
---|
1132 | each momentum component. |
---|
1133 | It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry. |
---|
1134 | The resulting discrete representation is: |
---|
1135 | \begin{equation} |
---|
1136 | \label{eq:DYN_ldf_iso} |
---|
1137 | \begin{split} |
---|
1138 | D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ |
---|
1139 | & \left\{\quad {\delta_{i+1/2} \left[ {A_T^{lm} \left( |
---|
1140 | {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u] |
---|
1141 | -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}} |
---|
1142 | \right)} \right]} \right. \\ |
---|
1143 | & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} |
---|
1144 | }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f} |
---|
1145 | \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}} |
---|
1146 | \right)} \right] \\ |
---|
1147 | &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline |
---|
1148 | {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } |
---|
1149 | \right.} \right. \\ |
---|
1150 | & \ \qquad \qquad \qquad \quad\ |
---|
1151 | - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\ |
---|
1152 | & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ |
---|
1153 | +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} |
---|
1154 | \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\ |
---|
1155 | D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\ |
---|
1156 | & \left\{\quad {\delta_{i+1/2} \left[ {A_f^{lm} \left( |
---|
1157 | {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v] |
---|
1158 | -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}} |
---|
1159 | \right)} \right]} \right. \\ |
---|
1160 | & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} |
---|
1161 | }\,\delta_{j} [v] - e_{1t}\, r_{2t} |
---|
1162 | \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}} |
---|
1163 | \right)} \right] \\ |
---|
1164 | & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline |
---|
1165 | {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\ |
---|
1166 | & \ \qquad \qquad \qquad \quad\ |
---|
1167 | - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\ |
---|
1168 | & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ |
---|
1169 | +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} |
---|
1170 | \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} |
---|
1171 | \end{split} |
---|
1172 | \end{equation} |
---|
1173 | where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and |
---|
1174 | the surface of computation ($z$- or $s$-surfaces). |
---|
1175 | The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}). |
---|
1176 | |
---|
1177 | %% ================================================================================================= |
---|
1178 | \subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap})]{Iso-level bilaplacian operator (\protect\np{ln_dynldf_bilap}{ln\_dynldf\_bilap})} |
---|
1179 | \label{subsec:DYN_ldf_bilap} |
---|
1180 | |
---|
1181 | The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice. |
---|
1182 | It requires an additional assumption on boundary conditions: |
---|
1183 | the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen, |
---|
1184 | while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}). |
---|
1185 | \gmcomment{add a remark on the the change in the position of the coefficient} |
---|
1186 | |
---|
1187 | %% ================================================================================================= |
---|
1188 | \section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})} |
---|
1189 | \label{sec:DYN_zdf} |
---|
1190 | |
---|
1191 | Options are defined through the \nam{zdf}{zdf} namelist variables. |
---|
1192 | 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. |
---|
1193 | Two time stepping schemes can be used for the vertical diffusion term: |
---|
1194 | $(a)$ a forward time differencing scheme |
---|
1195 | (\np[=.true.]{ln_zdfexp}{ln\_zdfexp}) using a time splitting technique (\np{nn_zdfexp}{nn\_zdfexp} $>$ 1) or |
---|
1196 | $(b)$ a backward (or implicit) time differencing scheme (\np[=.false.]{ln_zdfexp}{ln\_zdfexp}) |
---|
1197 | (see \autoref{chap:TD}). |
---|
1198 | Note that namelist variables \np{ln_zdfexp}{ln\_zdfexp} and \np{nn_zdfexp}{nn\_zdfexp} apply to both tracers and dynamics. |
---|
1199 | |
---|
1200 | The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is. |
---|
1201 | The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form: |
---|
1202 | \[ |
---|
1203 | % \label{eq:DYN_zdf} |
---|
1204 | \left\{ |
---|
1205 | \begin{aligned} |
---|
1206 | D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } |
---|
1207 | \ \delta_{k+1/2} [\,u\,] \right] \\ |
---|
1208 | \\ |
---|
1209 | D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } |
---|
1210 | \ \delta_{k+1/2} [\,v\,] \right] |
---|
1211 | \end{aligned} |
---|
1212 | \right. |
---|
1213 | \] |
---|
1214 | where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients. |
---|
1215 | The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}). |
---|
1216 | |
---|
1217 | The surface boundary condition on momentum is the stress exerted by the wind. |
---|
1218 | At the surface, the momentum fluxes are prescribed as the boundary condition on |
---|
1219 | the vertical turbulent momentum fluxes, |
---|
1220 | \begin{equation} |
---|
1221 | \label{eq:DYN_zdf_sbc} |
---|
1222 | \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} |
---|
1223 | = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v } |
---|
1224 | \end{equation} |
---|
1225 | where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in |
---|
1226 | the (\textbf{i},\textbf{j}) coordinate system. |
---|
1227 | The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in |
---|
1228 | the vertical over the mixed layer depth. |
---|
1229 | If the vertical mixing coefficient is small (when no mixed layer scheme is used) |
---|
1230 | the surface stress enters only the top model level, as a body force. |
---|
1231 | The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}). |
---|
1232 | |
---|
1233 | The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation |
---|
1234 | (see \autoref{sec:ZDF_drg}) |
---|
1235 | |
---|
1236 | %% ================================================================================================= |
---|
1237 | \section{External forcings} |
---|
1238 | \label{sec:DYN_forcing} |
---|
1239 | |
---|
1240 | Besides the surface and bottom stresses (see the above section) |
---|
1241 | which are introduced as boundary conditions on the vertical mixing, |
---|
1242 | three other forcings may enter the dynamical equations by affecting the surface pressure gradient. |
---|
1243 | |
---|
1244 | (1) When \np[=.true.]{ln_apr_dyn}{ln\_apr\_dyn} (see \autoref{sec:SBC_apr}), |
---|
1245 | the atmospheric pressure is taken into account when computing the surface pressure gradient. |
---|
1246 | |
---|
1247 | (2) When \np[=.true.]{ln_tide_pot}{ln\_tide\_pot} and \np[=.true.]{ln_tide}{ln\_tide} (see \autoref{sec:SBC_tide}), |
---|
1248 | the tidal potential is taken into account when computing the surface pressure gradient. |
---|
1249 | |
---|
1250 | (3) When \np[=2]{nn_ice_embd}{nn\_ice\_embd} and LIM or CICE is used |
---|
1251 | (\ie\ when the sea-ice is embedded in the ocean), |
---|
1252 | the snow-ice mass is taken into account when computing the surface pressure gradient. |
---|
1253 | |
---|
1254 | \gmcomment{ missing : the lateral boundary condition !!! another external forcing |
---|
1255 | } |
---|
1256 | |
---|
1257 | %% ================================================================================================= |
---|
1258 | \section{Wetting and drying } |
---|
1259 | \label{sec:DYN_wetdry} |
---|
1260 | |
---|
1261 | There are two main options for wetting and drying code (wd): |
---|
1262 | (a) an iterative limiter (il) and (b) a directional limiter (dl). |
---|
1263 | The directional limiter is based on the scheme developed by \cite{warner.defne.ea_CG13} for RO |
---|
1264 | MS |
---|
1265 | which was in turn based on ideas developed for POM by \cite{oey_OM06}. The iterative |
---|
1266 | limiter is a new scheme. The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$ |
---|
1267 | and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated |
---|
1268 | by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$. |
---|
1269 | |
---|
1270 | \begin{listing} |
---|
1271 | \nlst{namwad} |
---|
1272 | \caption{\forcode{&namwad}} |
---|
1273 | \label{lst:namwad} |
---|
1274 | \end{listing} |
---|
1275 | |
---|
1276 | The following terminology is used. The depth of the topography (positive downwards) |
---|
1277 | at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the \NEMO\ code. |
---|
1278 | The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign |
---|
1279 | conventions used, the water depth, $h$, is the height of the free surface plus the depth of the |
---|
1280 | topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$). |
---|
1281 | |
---|
1282 | Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be |
---|
1283 | covered by water. They require the topography specified with a model |
---|
1284 | configuration to have negative depths at points where the land is higher than the |
---|
1285 | topography's reference sea-level. The vertical grid in \NEMO\ is normally computed relative to an |
---|
1286 | initial state with zero sea surface height elevation. |
---|
1287 | The user can choose to compute the vertical grid and heights in the model relative to |
---|
1288 | a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths |
---|
1289 | (i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays). |
---|
1290 | |
---|
1291 | Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''. |
---|
1292 | $\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies |
---|
1293 | with very steep slopes require larger values for normal choices of time-step. Surface fluxes |
---|
1294 | are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers. |
---|
1295 | The fluxes are tappered down using a $\mathrm{tanh}$ weighting function |
---|
1296 | to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow. |
---|
1297 | The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$. |
---|
1298 | The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$. |
---|
1299 | |
---|
1300 | Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration |
---|
1301 | and in ``realistic'' configurations covering parts of the north-west European shelf. |
---|
1302 | All these configurations have used pure sigma coordinates. It is expected that |
---|
1303 | the wetting and drying code will work in domains with more general s-coordinates provided |
---|
1304 | the coordinates are pure sigma in the region where wetting and drying actually occurs. |
---|
1305 | |
---|
1306 | The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter. |
---|
1307 | The final sub-section covers some additional considerations that are relevant to both schemes. |
---|
1308 | |
---|
1309 | % Iterative limiters |
---|
1310 | %% ================================================================================================= |
---|
1311 | \subsection[Directional limiter (\textit{wet\_dry.F90})]{Directional limiter (\mdl{wet\_dry})} |
---|
1312 | \label{subsec:DYN_wd_directional_limiter} |
---|
1313 | |
---|
1314 | The principal idea of the directional limiter is that |
---|
1315 | water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn_wdmin1}{rn\_wdmin1}). |
---|
1316 | |
---|
1317 | All the changes associated with this option are made to the barotropic solver for the non-linear |
---|
1318 | free surface code within dynspg\_ts. |
---|
1319 | On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells |
---|
1320 | and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells |
---|
1321 | whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells |
---|
1322 | to be at least the minimum depth and hence is able to conserve mass / volume. |
---|
1323 | |
---|
1324 | The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj). |
---|
1325 | If the user sets \np[=.false.]{ln_wd_dl_ramp}{ln\_wd\_dl\_ramp} then zuwdmask is 1 when the |
---|
1326 | flux is from a cell with water depth greater than \np{rn_wdmin1}{rn\_wdmin1} and 0 otherwise. If the user sets |
---|
1327 | \np[=.true.]{ln_wd_dl_ramp}{ln\_wd\_dl\_ramp} the flux across the face is ramped down as the water depth decreases |
---|
1328 | from 2 * \np{rn_wdmin1}{rn\_wdmin1} to \np{rn_wdmin1}{rn\_wdmin1}. The use of this ramp reduced grid-scale noise in idealised test cases. |
---|
1329 | |
---|
1330 | At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen |
---|
1331 | also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have |
---|
1332 | chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells. |
---|
1333 | The rationale for setting the velocity to zero is that it is the momentum equations that are being solved |
---|
1334 | and the total momentum of the upstream cell (treating it as a finite volume) should be considered |
---|
1335 | to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its |
---|
1336 | treatment in the calculation of the flux of mass across the cell face. |
---|
1337 | |
---|
1338 | \cite{warner.defne.ea_CG13} state that in their scheme the velocity masks at the cell faces for the baroclinic |
---|
1339 | timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than |
---|
1340 | or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer |
---|
1341 | fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because |
---|
1342 | the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts |
---|
1343 | to equal their mean value during the barotropic steps. If the user sets \np[=.true.]{ln_wd_dl_bc}{ln\_wd\_dl\_bc}, the |
---|
1344 | baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask. |
---|
1345 | |
---|
1346 | % Iterative limiters |
---|
1347 | |
---|
1348 | %% ================================================================================================= |
---|
1349 | \subsection[Iterative limiter (\textit{wet\_dry.F90})]{Iterative limiter (\mdl{wet\_dry})} |
---|
1350 | \label{subsec:DYN_wd_iterative_limiter} |
---|
1351 | |
---|
1352 | %% ================================================================================================= |
---|
1353 | \subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})]{Iterative flux limiter (\mdl{wet\_dry})} |
---|
1354 | \label{subsec:DYN_wd_il_spg_limiter} |
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1355 | |
---|
1356 | The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry'' |
---|
1357 | or may become dry within the next time-step using an iterative method. |
---|
1358 | |
---|
1359 | The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows: |
---|
1360 | |
---|
1361 | The continuity equation for the total water depth in a column |
---|
1362 | \begin{equation} |
---|
1363 | \label{eq:DYN_wd_continuity} |
---|
1364 | \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . |
---|
1365 | \end{equation} |
---|
1366 | can be written in discrete form as |
---|
1367 | |
---|
1368 | \begin{align} |
---|
1369 | \label{eq:DYN_wd_continuity_2} |
---|
1370 | \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) |
---|
1371 | &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j} + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ |
---|
1372 | &= \mathrm{zzflx}_{i,j} . |
---|
1373 | \end{align} |
---|
1374 | |
---|
1375 | In the above $h$ is the depth of the water in the column at point $(i,j)$, |
---|
1376 | $\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and |
---|
1377 | $\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is |
---|
1378 | the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t = |
---|
1379 | t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and |
---|
1380 | $\mathrm{zzflx}$ is the sum of the fluxes through all the faces. |
---|
1381 | |
---|
1382 | The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell |
---|
1383 | (zzflxp) and fluxes that are into the cell (zzflxn). Clearly |
---|
1384 | |
---|
1385 | \begin{equation} |
---|
1386 | \label{eq:DYN_wd_zzflx_p_n_1} |
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1387 | \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . |
---|
1388 | \end{equation} |
---|
1389 | |
---|
1390 | The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until |
---|
1391 | none of the cells will ``dry out''. To be precise the fluxes are limited until none of the |
---|
1392 | cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$. |
---|
1393 | |
---|
1394 | Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and |
---|
1395 | $\mathrm{flxv}^{(m)}$. Then the adjustment is achieved by seeking a set of coefficients, |
---|
1396 | $\mathrm{zcoef}_{i,j}^{(m)}$ such that: |
---|
1397 | |
---|
1398 | \begin{equation} |
---|
1399 | \label{eq:DYN_wd_continuity_coef} |
---|
1400 | \begin{split} |
---|
1401 | \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ |
---|
1402 | \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} |
---|
1403 | \end{split} |
---|
1404 | \end{equation} |
---|
1405 | |
---|
1406 | where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around |
---|
1407 | cells that would otherwise dry. |
---|
1408 | |
---|
1409 | The iteration is initialised by setting |
---|
1410 | |
---|
1411 | \begin{equation} |
---|
1412 | \label{eq:DYN_wd_zzflx_initial} |
---|
1413 | \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . |
---|
1414 | \end{equation} |
---|
1415 | |
---|
1416 | The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the |
---|
1417 | cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell |
---|
1418 | times the timestep divided by the cell area. Using (\autoref{eq:DYN_wd_continuity_2}) this |
---|
1419 | condition is |
---|
1420 | |
---|
1421 | \begin{equation} |
---|
1422 | \label{eq:DYN_wd_continuity_if} |
---|
1423 | h_{i,j}(t_e) - \mathrm{rn\_wdmin1} < \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) . |
---|
1424 | \end{equation} |
---|
1425 | |
---|
1426 | Rearranging (\autoref{eq:DYN_wd_continuity_if}) we can obtain an expression for the maximum |
---|
1427 | outward flux that can be allowed and still maintain the minimum wet depth: |
---|
1428 | |
---|
1429 | \begin{equation} |
---|
1430 | \label{eq:DYN_wd_max_flux} |
---|
1431 | \begin{split} |
---|
1432 | \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ |
---|
1433 | \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] |
---|
1434 | \end{split} |
---|
1435 | \end{equation} |
---|
1436 | |
---|
1437 | Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\itshape [Q: Why is |
---|
1438 | this necessary/desirable?]}. Substituting from (\autoref{eq:DYN_wd_continuity_coef}) gives an |
---|
1439 | expression for the coefficient needed to multiply the outward flux at this cell in order |
---|
1440 | to avoid drying. |
---|
1441 | |
---|
1442 | \begin{equation} |
---|
1443 | \label{eq:DYN_wd_continuity_nxtcoef} |
---|
1444 | \begin{split} |
---|
1445 | \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ |
---|
1446 | \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } |
---|
1447 | \end{split} |
---|
1448 | \end{equation} |
---|
1449 | |
---|
1450 | Only the outward flux components are altered but, of course, outward fluxes from one cell |
---|
1451 | are inward fluxes to adjacent cells and the balance in these cells may need subsequent |
---|
1452 | adjustment; hence the iterative nature of this scheme. Note, for example, that the flux |
---|
1453 | across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration |
---|
1454 | if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then |
---|
1455 | the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by |
---|
1456 | the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across |
---|
1457 | the faces of the cells do not ``compete'' (they do not over-write each other) and one |
---|
1458 | would expect the scheme to converge relatively quickly. The scheme is flux based so |
---|
1459 | conserves mass. It also conserves constant tracers for the same reason that the |
---|
1460 | directional limiter does. |
---|
1461 | |
---|
1462 | % Surface pressure gradients |
---|
1463 | %% ================================================================================================= |
---|
1464 | \subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})]{Modification of surface pressure gradients (\mdl{dynhpg})} |
---|
1465 | \label{subsec:DYN_wd_il_spg} |
---|
1466 | |
---|
1467 | At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the |
---|
1468 | topography is sloping at these points the sea-surface will have a similar slope and there |
---|
1469 | will hence be very large horizontal pressure gradients at these points. The WAD modifies |
---|
1470 | the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such |
---|
1471 | points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie |
---|
1472 | between $0$ and $1$. |
---|
1473 | |
---|
1474 | We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated |
---|
1475 | at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the |
---|
1476 | neighbouring $(i+1,j)$ and $(i,j)$ tracer points. zcpx is calculated using two logicals |
---|
1477 | variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid |
---|
1478 | column. The three possible combinations are illustrated in \autoref{fig:DYN_WAD_dynhpg}. |
---|
1479 | |
---|
1480 | \begin{figure}[!ht] |
---|
1481 | \centering |
---|
1482 | \includegraphics[width=0.66\textwidth]{DYN_WAD_dynhpg} |
---|
1483 | \caption[Combinations controlling the limiting of the horizontal pressure gradient in |
---|
1484 | wetting and drying regimes]{ |
---|
1485 | Three possible combinations of the logical variables controlling the |
---|
1486 | limiting of the horizontal pressure gradient in wetting and drying regimes} |
---|
1487 | \label{fig:DYN_WAD_dynhpg} |
---|
1488 | \end{figure} |
---|
1489 | |
---|
1490 | The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at |
---|
1491 | both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and |
---|
1492 | the minimum height of the sea surface at the two points is greater than the maximum height |
---|
1493 | of the topography at the two points: |
---|
1494 | |
---|
1495 | \begin{equation} |
---|
1496 | \label{eq:DYN_ll_tmp1} |
---|
1497 | \begin{split} |
---|
1498 | \mathrm{ll\_tmp1} = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ |
---|
1499 | & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\ .and.} \\ |
---|
1500 | & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ |
---|
1501 | & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ |
---|
1502 | & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } |
---|
1503 | \end{split} |
---|
1504 | \end{equation} |
---|
1505 | |
---|
1506 | The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height |
---|
1507 | of the sea surface at the two points is greater than the maximum height of the topography |
---|
1508 | at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ |
---|
1509 | |
---|
1510 | \begin{equation} |
---|
1511 | \label{eq:DYN_ll_tmp2} |
---|
1512 | \begin{split} |
---|
1513 | \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ |
---|
1514 | & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ |
---|
1515 | & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . |
---|
1516 | \end{split} |
---|
1517 | \end{equation} |
---|
1518 | |
---|
1519 | If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$ |
---|
1520 | point is unmodified. If both logicals are false zhpi is set to zero. |
---|
1521 | |
---|
1522 | If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure |
---|
1523 | gradient is multiplied through by zcpx which is the absolute value of the difference in |
---|
1524 | the water depths at the two points divided by the difference in the surface heights at the |
---|
1525 | two points. Thus the sign of the sea surface height gradient is retained but the magnitude |
---|
1526 | of the pressure force is determined by the difference in water depths rather than the |
---|
1527 | difference in surface height between the two points. Note that dividing by the difference |
---|
1528 | between the sea surface heights can be problematic if the heights approach parity. An |
---|
1529 | additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such |
---|
1530 | conditions. |
---|
1531 | |
---|
1532 | %% ================================================================================================= |
---|
1533 | \subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})]{Additional considerations (\mdl{usrdef\_zgr})} |
---|
1534 | \label{subsec:DYN_WAD_additional} |
---|
1535 | |
---|
1536 | In the very shallow water where wetting and drying occurs the parametrisation of |
---|
1537 | bottom drag is clearly very important. In order to promote stability |
---|
1538 | it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach. |
---|
1539 | |
---|
1540 | Suitable specifcation of the surface heat flux in wetting and drying domains in forced and |
---|
1541 | coupled simulations needs further consideration. In order to prevent freezing or boiling |
---|
1542 | in uncoupled integrations the net surface heat fluxes need to be appropriately limited. |
---|
1543 | |
---|
1544 | % The WAD test cases |
---|
1545 | %% ================================================================================================= |
---|
1546 | \subsection[The WAD test cases (\textit{usrdef\_zgr.F90})]{The WAD test cases (\mdl{usrdef\_zgr})} |
---|
1547 | \label{subsec:DYN_WAD_test_cases} |
---|
1548 | |
---|
1549 | See the WAD tests MY\_DOC documention for details of the WAD test cases. |
---|
1550 | |
---|
1551 | %% ================================================================================================= |
---|
1552 | \section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})} |
---|
1553 | \label{sec:DYN_nxt} |
---|
1554 | |
---|
1555 | Options are defined through the \nam{dom}{dom} namelist variables. |
---|
1556 | The general framework for dynamics time stepping is a leap-frog scheme, |
---|
1557 | \ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}). |
---|
1558 | The scheme is applied to the velocity, except when |
---|
1559 | using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) |
---|
1560 | in the variable volume case (\texttt{vvl?} defined), |
---|
1561 | where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum}) |
---|
1562 | |
---|
1563 | $\bullet$ vector invariant form or linear free surface |
---|
1564 | (\np[=.true.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} ; \texttt{vvl?} not defined): |
---|
1565 | \[ |
---|
1566 | % \label{eq:DYN_nxt_vec} |
---|
1567 | \left\{ |
---|
1568 | \begin{aligned} |
---|
1569 | &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt \ \text{RHS}_u^t \\ |
---|
1570 | &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right] |
---|
1571 | \end{aligned} |
---|
1572 | \right. |
---|
1573 | \] |
---|
1574 | |
---|
1575 | $\bullet$ flux form and nonlinear free surface |
---|
1576 | (\np[=.false.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} ; \texttt{vvl?} defined): |
---|
1577 | \[ |
---|
1578 | % \label{eq:DYN_nxt_flux} |
---|
1579 | \left\{ |
---|
1580 | \begin{aligned} |
---|
1581 | &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t \\ |
---|
1582 | &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t |
---|
1583 | +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right] |
---|
1584 | \end{aligned} |
---|
1585 | \right. |
---|
1586 | \] |
---|
1587 | where RHS is the right hand side of the momentum equation, |
---|
1588 | the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient. |
---|
1589 | $\gamma$ is initialized as \np{nn_atfp}{nn\_atfp} (namelist parameter). |
---|
1590 | Its default value is \np[=10.e-3]{nn_atfp}{nn\_atfp}. |
---|
1591 | In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for |
---|
1592 | the momentum equations. |
---|
1593 | |
---|
1594 | Note that with the filtered free surface, |
---|
1595 | the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module, |
---|
1596 | and only array swapping and Asselin filtering is done in \mdl{dynnxt}. |
---|
1597 | |
---|
1598 | \onlyinsubfile{\input{../../global/epilogue}} |
---|
1599 | |
---|
1600 | \end{document} |
---|