1 | % ================================================================ |
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2 | % Chapter � Ocean Dynamics (DYN) |
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3 | % ================================================================ |
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4 | \chapter{Ocean Dynamics (DYN)} |
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5 | \label{DYN} |
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6 | \minitoc |
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7 | |
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8 | % add a figure for dynvor ens, ene latices |
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9 | |
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10 | %\vspace{2.cm} |
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11 | $\ $\newline %force an empty line |
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12 | |
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13 | Using the representation described in Chapter \ref{DOM}, several semi-discrete |
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14 | space forms of the dynamical equations are available depending on the vertical |
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15 | coordinate used and on the conservation properties of the vorticity term. In all |
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16 | the equations presented here, the masking has been omitted for simplicity. |
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17 | One must be aware that all the quantities are masked fields and that each time an |
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18 | average or difference operator is used, the resulting field is multiplied by a mask. |
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19 | |
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20 | The prognostic ocean dynamics equation can be summarized as follows: |
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21 | \begin{equation*} |
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22 | \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } |
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23 | {\text{COR} + \text{ADV} } |
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24 | + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} |
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25 | \end{equation*} |
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26 | NXT stands for next, referring to the time-stepping. The first group of terms on |
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27 | the rhs of this equation corresponds to the Coriolis and advection |
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28 | terms that are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG) |
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29 | and a vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis |
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30 | and advection part (COR+ADV) in the flux formulation. The terms following these |
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31 | are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient, |
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32 | and SPG, Surface Pressure Gradient); and contributions from lateral diffusion |
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33 | (LDF) and vertical diffusion (ZDF), which are added to the rhs in the \mdl{dynldf} |
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34 | and \mdl{dynzdf} modules. The vertical diffusion term includes the surface and |
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35 | bottom stresses. The external forcings and parameterisations require complex |
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36 | inputs (surface wind stress calculation using bulk formulae, estimation of mixing |
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37 | coefficients) that are carried out in modules SBC, LDF and ZDF and are described |
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38 | in Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively. |
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39 | |
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40 | In the present chapter we also describe the diagnostic equations used to compute |
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41 | the horizontal divergence, curl of the velocities (\emph{divcur} module) and |
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42 | the vertical velocity (\emph{wzvmod} module). |
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43 | |
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44 | The different options available to the user are managed by namelist variables. |
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45 | For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx}, |
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46 | where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. |
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47 | If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding |
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48 | code can be found in the \textit{dynttt\_xxx} module in the DYN directory, and it is |
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49 | usually computed in the \textit{dyn\_ttt\_xxx} subroutine. |
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50 | |
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51 | The user has the option of extracting and outputting each tendency term from the |
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52 | 3D momentum equations (\key{trddyn} defined), as described in |
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53 | Chap.\ref{MISC}. Furthermore, the tendency terms associated with the 2D |
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54 | barotropic vorticity balance (when \key{trdvor} is defined) can be derived from the |
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55 | 3D terms. |
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56 | %%% |
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57 | \gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does |
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58 | MISC correspond to "extracting tendency terms" or "vorticity balance"?} |
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59 | |
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60 | $\ $\newline % force a new ligne |
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61 | |
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62 | % ================================================================ |
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63 | % Sea Surface Height evolution & Diagnostics variables |
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64 | % ================================================================ |
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65 | \section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)} |
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66 | \label{DYN_divcur_wzv} |
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67 | |
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68 | %-------------------------------------------------------------------------------------------------------------- |
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69 | % Horizontal divergence and relative vorticity |
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70 | %-------------------------------------------------------------------------------------------------------------- |
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71 | \subsection [Horizontal divergence and relative vorticity (\textit{divcur})] |
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72 | {Horizontal divergence and relative vorticity (\mdl{divcur})} |
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73 | \label{DYN_divcur} |
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74 | |
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75 | The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows: |
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76 | \begin{equation} \label{Eq_divcur_cur} |
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77 | \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right] |
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78 | -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) |
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79 | \end{equation} |
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80 | |
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81 | The horizontal divergence is defined at a $T$-point. It is given by: |
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82 | \begin{equation} \label{Eq_divcur_div} |
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83 | \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } |
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84 | \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right] |
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85 | +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right) |
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86 | \end{equation} |
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87 | |
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88 | Note that although the vorticity has the same discrete expression in $z$- |
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89 | and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo |
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90 | vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along |
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91 | geopotential surfaces, but are not necessarily defined at the same depth). |
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92 | |
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93 | The vorticity and divergence at the \textit{before} step are used in the computation |
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94 | of the horizontal diffusion of momentum. Note that because they have been |
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95 | calculated prior to the Asselin filtering of the \textit{before} velocities, the |
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96 | \textit{before} vorticity and divergence arrays must be included in the restart file |
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97 | to ensure perfect restartability. The vorticity and divergence at the \textit{now} |
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98 | time step are used for the computation of the nonlinear advection and of the |
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99 | vertical velocity respectively. |
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100 | |
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101 | %-------------------------------------------------------------------------------------------------------------- |
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102 | % Sea Surface Height evolution |
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103 | %-------------------------------------------------------------------------------------------------------------- |
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104 | \subsection [Sea surface height evolution and vertical velocity (\textit{sshwzv})] |
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105 | {Horizontal divergence and relative vorticity (\mdl{sshwzv})} |
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106 | \label{DYN_sshwzv} |
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107 | |
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108 | The sea surface height is given by : |
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109 | \begin{equation} \label{Eq_dynspg_ssh} |
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110 | \begin{aligned} |
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111 | \frac{\partial \eta }{\partial t} |
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112 | &\equiv \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{ \delta _i \left[ {e_{2u}\,e_{3u}\;u} \right] |
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113 | +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right] \right\} } |
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114 | - \frac{\textit{emp}}{\rho _w } \\ |
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115 | &\equiv \sum\limits_k {\chi \ e_{3t}} - \frac{\textit{emp}}{\rho _w } |
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116 | \end{aligned} |
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117 | \end{equation} |
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118 | where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), |
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119 | expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,035~Kg/m$^3$ |
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120 | is the reference density of sea water (Boussinesq approximation). If river runoff is |
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121 | expressed as a surface freshwater flux (see \S\ref{SBC}) then \textit{emp} can be |
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122 | written as the evaporation minus precipitation, minus the river runoff. |
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123 | The sea-surface height is evaluated using exactly the same time stepping scheme |
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124 | as the tracer equation \eqref{Eq_tra_nxt}: |
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125 | a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing |
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126 | in \eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity). |
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127 | This is of paramount importance. Replacing $T$ by the number $1$ in the tracer equation and summing |
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128 | over the water column must lead to the sea surface height equation otherwise tracer content |
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129 | will not be conserved \citep{Griffies_al_MWR01, Leclair_Madec_OM09}. |
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130 | |
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131 | The vertical velocity is computed by an upward integration of the horizontal |
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132 | divergence starting at the bottom, taking into account the change of the thickness of the levels : |
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133 | \begin{equation} \label{Eq_wzv} |
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134 | \left\{ \begin{aligned} |
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135 | &\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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136 | &\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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137 | - \frac{1} {2 \rdt} \left( \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right) |
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138 | \end{aligned} \right. |
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139 | \end{equation} |
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140 | |
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141 | In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$, |
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142 | as changes in the divergence of the barotropic transport are absorbed into the change |
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143 | of the level thicknesses, re-orientated downward. |
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144 | \gmcomment{not sure of this... to be modified with the change in emp setting} |
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145 | In the case of a linear free surface, the time derivative in \eqref{Eq_wzv} disappears. |
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146 | The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity |
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147 | is thus equal to the divergence of the barotropic transport ($i.e.$ the first term in the |
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148 | right-hand-side of \eqref{Eq_dynspg_ssh}). |
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149 | |
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150 | Note also that whereas the vertical velocity has the same discrete |
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151 | expression in $z$- and $s$-coordinates, its physical meaning is not the same: |
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152 | in the second case, $w$ is the velocity normal to the $s$-surfaces. |
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153 | Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared |
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154 | to the indexing used in the semi-discrete equations such as \eqref{Eq_wzv} |
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155 | (see \S\ref{DOM_Num_Index_vertical}). |
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156 | |
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157 | |
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158 | % ================================================================ |
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159 | % Coriolis and Advection terms: vector invariant form |
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160 | % ================================================================ |
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161 | \section{Coriolis and Advection: vector invariant form} |
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162 | \label{DYN_adv_cor_vect} |
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163 | %-----------------------------------------nam_dynadv---------------------------------------------------- |
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164 | \namdisplay{namdyn_adv} |
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165 | %------------------------------------------------------------------------------------------------------------- |
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166 | |
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167 | The vector invariant form of the momentum equations is the one most |
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168 | often used in applications of the \NEMO ocean model. The flux form option |
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169 | (see next section) has been present since version $2$. Options are defined |
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170 | through the \ngn{namdyn\_adv} namelist variables |
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171 | Coriolis and momentum advection terms are evaluated using a leapfrog |
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172 | scheme, $i.e.$ the velocity appearing in these expressions is centred in |
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173 | time (\textit{now} velocity). |
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174 | At the lateral boundaries either free slip, no slip or partial slip boundary |
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175 | conditions are applied following Chap.\ref{LBC}. |
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176 | |
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177 | % ------------------------------------------------------------------------------------------------------------- |
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178 | % Vorticity term |
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179 | % ------------------------------------------------------------------------------------------------------------- |
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180 | \subsection [Vorticity term (\textit{dynvor}) ] |
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181 | {Vorticity term (\mdl{dynvor})} |
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182 | \label{DYN_vor} |
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183 | %------------------------------------------nam_dynvor---------------------------------------------------- |
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184 | \namdisplay{namdyn_vor} |
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185 | %------------------------------------------------------------------------------------------------------------- |
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186 | |
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187 | Options are defined through the \ngn{namdyn\_vor} namelist variables. |
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188 | Four discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=true) are available: |
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189 | conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ; |
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190 | conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for |
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191 | the relative vorticity term and horizontal kinetic energy for the planetary vorticity |
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192 | term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent |
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193 | flow and horizontal kinetic energy (EEN scheme) (see Appendix~\ref{Apdx_C_vorEEN}). In the |
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194 | case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the |
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195 | consistency of vorticity term with analytical equations (\textit{ln\_dynvor\_con}=true). |
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196 | The vorticity terms are all computed in dedicated routines that can be found in |
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197 | the \mdl{dynvor} module. |
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198 | |
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199 | %------------------------------------------------------------- |
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200 | % enstrophy conserving scheme |
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201 | %------------------------------------------------------------- |
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202 | \subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=true)} |
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203 | \label{DYN_vor_ens} |
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204 | |
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205 | In the enstrophy conserving case (ENS scheme), the discrete formulation of the |
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206 | vorticity term provides a global conservation of the enstrophy |
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207 | ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent |
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208 | flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by: |
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209 | \begin{equation} \label{Eq_dynvor_ens} |
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210 | \left\{ |
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211 | \begin{aligned} |
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212 | {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} |
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213 | & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2} \\ |
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214 | {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} |
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215 | & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j} |
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216 | \end{aligned} |
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217 | \right. |
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218 | \end{equation} |
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219 | |
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220 | %------------------------------------------------------------- |
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221 | % energy conserving scheme |
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222 | %------------------------------------------------------------- |
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223 | \subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=true)} |
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224 | \label{DYN_vor_ene} |
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225 | |
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226 | The kinetic energy conserving scheme (ENE scheme) conserves the global |
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227 | kinetic energy but not the global enstrophy. It is given by: |
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228 | \begin{equation} \label{Eq_dynvor_ene} |
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229 | \left\{ \begin{aligned} |
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230 | {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) |
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231 | \; \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\ |
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232 | {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) |
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233 | \; \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } |
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234 | \end{aligned} \right. |
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235 | \end{equation} |
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236 | |
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237 | %------------------------------------------------------------- |
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238 | % mix energy/enstrophy conserving scheme |
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239 | %------------------------------------------------------------- |
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240 | \subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=true) } |
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241 | \label{DYN_vor_mix} |
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242 | |
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243 | For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the |
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244 | two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens}) |
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245 | for the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied |
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246 | to the planetary vorticity term. |
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247 | \begin{equation} \label{Eq_dynvor_mix} |
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248 | \left\{ { \begin{aligned} |
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249 | {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} |
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250 | \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} } |
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251 | \; {\overline {\left( {\frac{f}{e_{3f} }} \right) |
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252 | \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\ |
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253 | {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j |
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254 | \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} } |
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255 | \; {\overline {\left( {\frac{f}{e_{3f} }} \right) |
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256 | \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill |
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257 | \end{aligned} } \right. |
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258 | \end{equation} |
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259 | |
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260 | %------------------------------------------------------------- |
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261 | % energy and enstrophy conserving scheme |
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262 | %------------------------------------------------------------- |
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263 | \subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=true) } |
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264 | \label{DYN_vor_een} |
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265 | |
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266 | In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$ |
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267 | averages of the velocity allows for the presence of grid point oscillation structures |
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268 | that will be invisible to the operator. These structures are \textit{computational modes} |
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269 | that will be at least partly damped by the momentum diffusion operator ($i.e.$ the |
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270 | subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes |
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271 | therefore do not contribute to dump any grid point noise in the horizontal velocity field. |
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272 | Such noise would result in more noise in the vertical velocity field, an undesirable feature. |
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273 | This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located |
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274 | at different grid points, a price worth paying to avoid a double averaging in the pressure |
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275 | gradient term as in the $B$-grid. |
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276 | \gmcomment{ To circumvent this, Adcroft (ADD REF HERE) |
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277 | Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....} |
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278 | |
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279 | A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}. |
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280 | The idea is to get rid of the double averaging by considering triad combinations of vorticity. |
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281 | It is noteworthy that this solution is conceptually quite similar to the one proposed by |
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282 | \citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see App.\ref{Apdx_C}). |
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283 | |
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284 | The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified |
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285 | for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme. |
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286 | First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: |
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287 | \begin{equation} \label{Eq_pot_vor} |
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288 | q = \frac{\zeta +f} {e_{3f} } |
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289 | \end{equation} |
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290 | where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter |
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291 | is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: |
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292 | \begin{equation} \label{Eq_een_e3f} |
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293 | e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} |
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294 | \end{equation} |
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295 | |
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296 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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297 | \begin{figure}[!ht] \begin{center} |
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298 | \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_DYN_een_triad.pdf} |
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299 | \caption{ \label{Fig_DYN_een_triad} |
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300 | Triads used in the energy and enstrophy conserving scheme (een) for |
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301 | $u$-component (upper panel) and $v$-component (lower panel).} |
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302 | \end{center} \end{figure} |
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303 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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304 | |
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305 | Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and |
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306 | \textbf{j}- directions uses the masked vertical scale factor but is always divided by |
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307 | $4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of |
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308 | $e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and |
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309 | extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for |
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310 | the $z$-coordinate with partial steps. |
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311 | |
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312 | Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as |
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313 | the following triad combinations of the neighbouring potential vorticities defined at f-points |
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314 | (Fig.~\ref{Fig_DYN_een_triad}): |
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315 | \begin{equation} \label{Q_triads} |
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316 | _i^j \mathbb{Q}^{i_p}_{j_p} |
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317 | = \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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318 | \end{equation} |
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319 | 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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320 | |
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321 | Finally, the vorticity terms are represented as: |
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322 | \begin{equation} \label{Eq_dynvor_een} |
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323 | \left\{ { |
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324 | \begin{aligned} |
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325 | +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}} |
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326 | {^{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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327 | - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}} |
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328 | {^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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329 | \end{aligned} |
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330 | } \right. |
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331 | \end{equation} |
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332 | |
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333 | This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. |
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334 | It conserves both total energy and potential enstrophy in the limit of horizontally |
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335 | nondivergent flow ($i.e.$ $\chi$=$0$) (see Appendix~\ref{Apdx_C_vorEEN}). |
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336 | Applied to a realistic ocean configuration, it has been shown that it leads to a significant |
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337 | reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. |
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338 | Furthermore, used in combination with a partial steps representation of bottom topography, |
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339 | it improves the interaction between current and topography, leading to a larger |
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340 | topostrophy of the flow \citep{Barnier_al_OD06, Penduff_al_OS07}. |
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341 | |
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342 | %-------------------------------------------------------------------------------------------------------------- |
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343 | % Kinetic Energy Gradient term |
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344 | %-------------------------------------------------------------------------------------------------------------- |
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345 | \subsection [Kinetic Energy Gradient term (\textit{dynkeg})] |
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346 | {Kinetic Energy Gradient term (\mdl{dynkeg})} |
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347 | \label{DYN_keg} |
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348 | |
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349 | As demonstrated in Appendix~\ref{Apdx_C}, there is a single discrete formulation |
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350 | of the kinetic energy gradient term that, together with the formulation chosen for |
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351 | the vertical advection (see below), conserves the total kinetic energy: |
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352 | \begin{equation} \label{Eq_dynkeg} |
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353 | \left\{ \begin{aligned} |
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354 | -\frac{1}{2 \; e_{1u} } & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] \\ |
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355 | -\frac{1}{2 \; e_{2v} } & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] |
---|
356 | \end{aligned} \right. |
---|
357 | \end{equation} |
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358 | |
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359 | %-------------------------------------------------------------------------------------------------------------- |
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360 | % Vertical advection term |
---|
361 | %-------------------------------------------------------------------------------------------------------------- |
---|
362 | \subsection [Vertical advection term (\textit{dynzad}) ] |
---|
363 | {Vertical advection term (\mdl{dynzad}) } |
---|
364 | \label{DYN_zad} |
---|
365 | |
---|
366 | The discrete formulation of the vertical advection, together with the formulation |
---|
367 | chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic |
---|
368 | energy. Indeed, the change of KE due to the vertical advection is exactly |
---|
369 | balanced by the change of KE due to the gradient of KE (see Appendix~\ref{Apdx_C}). |
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370 | \begin{equation} \label{Eq_dynzad} |
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371 | \left\{ \begin{aligned} |
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372 | -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2} \;\delta _{k+1/2} \left[ u \right]\ }^{\,k} \\ |
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373 | -\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} |
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374 | \end{aligned} \right. |
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375 | \end{equation} |
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376 | |
---|
377 | % ================================================================ |
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378 | % Coriolis and Advection : flux form |
---|
379 | % ================================================================ |
---|
380 | \section{Coriolis and Advection: flux form} |
---|
381 | \label{DYN_adv_cor_flux} |
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382 | %------------------------------------------nam_dynadv---------------------------------------------------- |
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383 | \namdisplay{namdyn_adv} |
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384 | %------------------------------------------------------------------------------------------------------------- |
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385 | |
---|
386 | Options are defined through the \ngn{namdyn\_adv} namelist variables. |
---|
387 | In the flux form (as in the vector invariant form), the Coriolis and momentum |
---|
388 | advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity |
---|
389 | appearing in their expressions is centred in time (\textit{now} velocity). At the |
---|
390 | lateral boundaries either free slip, no slip or partial slip boundary conditions |
---|
391 | are applied following Chap.\ref{LBC}. |
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392 | |
---|
393 | |
---|
394 | %-------------------------------------------------------------------------------------------------------------- |
---|
395 | % Coriolis plus curvature metric terms |
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396 | %-------------------------------------------------------------------------------------------------------------- |
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397 | \subsection [Coriolis plus curvature metric terms (\textit{dynvor}) ] |
---|
398 | {Coriolis plus curvature metric terms (\mdl{dynvor}) } |
---|
399 | \label{DYN_cor_flux} |
---|
400 | |
---|
401 | In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis |
---|
402 | parameter has been modified to account for the "metric" term. This altered |
---|
403 | Coriolis parameter is thus discretised at $f$-points. It is given by: |
---|
404 | \begin{multline} \label{Eq_dyncor_metric} |
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405 | f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} - u\frac{\partial e_1 }{\partial j}} \right) \\ |
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406 | \equiv f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right] |
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407 | - \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right] } \ \right) |
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408 | \end{multline} |
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409 | |
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410 | Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een}) |
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411 | schemes can be used to compute the product of the Coriolis parameter and the |
---|
412 | vorticity. However, the energy-conserving scheme (\ref{Eq_dynvor_een}) has |
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413 | exclusively been used to date. This term is evaluated using a leapfrog scheme, |
---|
414 | $i.e.$ the velocity is centred in time (\textit{now} velocity). |
---|
415 | |
---|
416 | %-------------------------------------------------------------------------------------------------------------- |
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417 | % Flux form Advection term |
---|
418 | %-------------------------------------------------------------------------------------------------------------- |
---|
419 | \subsection [Flux form Advection term (\textit{dynadv}) ] |
---|
420 | {Flux form Advection term (\mdl{dynadv}) } |
---|
421 | \label{DYN_adv_flux} |
---|
422 | |
---|
423 | The discrete expression of the advection term is given by : |
---|
424 | \begin{equation} \label{Eq_dynadv} |
---|
425 | \left\{ |
---|
426 | \begin{aligned} |
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427 | \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} |
---|
428 | \left( \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i } \ u_t \right] |
---|
429 | + \delta _{j } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2} \ u_f \right] \right. \ \; \\ |
---|
430 | \left. + \delta _{k } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2} \ u_{uw} \right] \right) \\ |
---|
431 | \\ |
---|
432 | \frac{1}{e_{1v}\,e_{2v}\,e_{3v}} |
---|
433 | \left( \delta _{i } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f \right] |
---|
434 | + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i } \ v_t \right] \right. \ \, \, \\ |
---|
435 | \left. + \delta _{k } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw} \right] \right) \\ |
---|
436 | \end{aligned} |
---|
437 | \right. |
---|
438 | \end{equation} |
---|
439 | |
---|
440 | Two advection schemes are available: a $2^{nd}$ order centered finite |
---|
441 | difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS. |
---|
442 | The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are |
---|
443 | selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. |
---|
444 | In flux form, the schemes differ by the choice of a space and time interpolation to |
---|
445 | define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells, |
---|
446 | $i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and |
---|
447 | $vw$-points for $v$. |
---|
448 | |
---|
449 | %------------------------------------------------------------- |
---|
450 | % 2nd order centred scheme |
---|
451 | %------------------------------------------------------------- |
---|
452 | \subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=true)} |
---|
453 | \label{DYN_adv_cen2} |
---|
454 | |
---|
455 | In the centered $2^{nd}$ order formulation, the velocity is evaluated as the |
---|
456 | mean of the two neighbouring points : |
---|
457 | \begin{equation} \label{Eq_dynadv_cen2} |
---|
458 | \left\{ \begin{aligned} |
---|
459 | u_T^{cen2} &=\overline u^{i } \quad & u_F^{cen2} &=\overline u^{j+1/2} \quad & u_{uw}^{cen2} &=\overline u^{k+1/2} \\ |
---|
460 | v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j \quad & v_{vw}^{cen2} &=\overline v ^{k+1/2} \\ |
---|
461 | \end{aligned} \right. |
---|
462 | \end{equation} |
---|
463 | |
---|
464 | The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive |
---|
465 | ($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be |
---|
466 | used in conjunction with an explicit diffusion operator to produce a sensible solution. |
---|
467 | The associated time-stepping is performed using a leapfrog scheme in conjunction |
---|
468 | with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities. |
---|
469 | |
---|
470 | %------------------------------------------------------------- |
---|
471 | % UBS scheme |
---|
472 | %------------------------------------------------------------- |
---|
473 | \subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=true)} |
---|
474 | \label{DYN_adv_ubs} |
---|
475 | |
---|
476 | The UBS advection scheme is an upstream biased third order scheme based on |
---|
477 | an upstream-biased parabolic interpolation. For example, the evaluation of |
---|
478 | $u_T^{ubs} $ is done as follows: |
---|
479 | \begin{equation} \label{Eq_dynadv_ubs} |
---|
480 | u_T^{ubs} =\overline u ^i-\;\frac{1}{6} \begin{cases} |
---|
481 | u"_{i-1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i \geqslant 0$ } \\ |
---|
482 | u"_{i+1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i < 0$ } |
---|
483 | \end{cases} |
---|
484 | \end{equation} |
---|
485 | where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results |
---|
486 | in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Shchepetkin_McWilliams_OM05}. |
---|
487 | The overall performance of the advection scheme is similar to that reported in |
---|
488 | \citet{Farrow1995}. It is a relatively good compromise between accuracy and |
---|
489 | smoothness. It is not a \emph{positive} scheme, meaning that false extrema are |
---|
490 | permitted. But the amplitudes of the false extrema are significantly reduced over |
---|
491 | those in the centred second order method. As the scheme already includes |
---|
492 | a diffusion component, it can be used without explicit lateral diffusion on momentum |
---|
493 | ($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so. |
---|
494 | |
---|
495 | The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ |
---|
496 | order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and |
---|
497 | $u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is |
---|
498 | associated with vertical mixing of momentum. \gmcomment{ gm pursue the |
---|
499 | sentence:Since vertical mixing of momentum is a source term of the TKE equation... } |
---|
500 | |
---|
501 | For stability reasons, the first term in (\ref{Eq_dynadv_ubs}), which corresponds |
---|
502 | to a second order centred scheme, is evaluated using the \textit{now} velocity |
---|
503 | (centred in time), while the second term, which is the diffusion part of the scheme, |
---|
504 | is evaluated using the \textit{before} velocity (forward in time). This is discussed |
---|
505 | by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. |
---|
506 | |
---|
507 | Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) |
---|
508 | schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in |
---|
509 | (\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. |
---|
510 | This option is not available through a namelist parameter, since the $1/6$ coefficient |
---|
511 | is hard coded. Nevertheless it is quite easy to make the substitution in the |
---|
512 | \mdl{dynadv\_ubs} module and obtain a QUICK scheme. |
---|
513 | |
---|
514 | Note also that in the current version of \mdl{dynadv\_ubs}, there is also the |
---|
515 | possibility of using a $4^{th}$ order evaluation of the advective velocity as in |
---|
516 | ROMS. This is an error and should be suppressed soon. |
---|
517 | %%% |
---|
518 | \gmcomment{action : this have to be done} |
---|
519 | %%% |
---|
520 | |
---|
521 | % ================================================================ |
---|
522 | % Hydrostatic pressure gradient term |
---|
523 | % ================================================================ |
---|
524 | \section [Hydrostatic pressure gradient (\textit{dynhpg})] |
---|
525 | {Hydrostatic pressure gradient (\mdl{dynhpg})} |
---|
526 | \label{DYN_hpg} |
---|
527 | %------------------------------------------nam_dynhpg--------------------------------------------------- |
---|
528 | \namdisplay{namdyn_hpg} |
---|
529 | %------------------------------------------------------------------------------------------------------------- |
---|
530 | |
---|
531 | Options are defined through the \ngn{namdyn\_hpg} namelist variables. |
---|
532 | The key distinction between the different algorithms used for the hydrostatic |
---|
533 | pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal} |
---|
534 | pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any |
---|
535 | tilt of the surface of the computational levels will require a specific treatment to |
---|
536 | compute the hydrostatic pressure gradient. |
---|
537 | |
---|
538 | The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, |
---|
539 | $i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$), or |
---|
540 | a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip |
---|
541 | boundary conditions are applied. |
---|
542 | |
---|
543 | %-------------------------------------------------------------------------------------------------------------- |
---|
544 | % z-coordinate with full step |
---|
545 | %-------------------------------------------------------------------------------------------------------------- |
---|
546 | \subsection [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ] |
---|
547 | {$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=true)} |
---|
548 | \label{DYN_hpg_zco} |
---|
549 | |
---|
550 | The hydrostatic pressure can be obtained by integrating the hydrostatic equation |
---|
551 | vertically from the surface. However, the pressure is large at great depth while its |
---|
552 | horizontal gradient is several orders of magnitude smaller. This may lead to large |
---|
553 | truncation errors in the pressure gradient terms. Thus, the two horizontal components |
---|
554 | of the hydrostatic pressure gradient are computed directly as follows: |
---|
555 | |
---|
556 | for $k=km$ (surface layer, $jk=1$ in the code) |
---|
557 | \begin{equation} \label{Eq_dynhpg_zco_surf} |
---|
558 | \left\{ \begin{aligned} |
---|
559 | \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k=km} |
---|
560 | &= \frac{1}{2} g \ \left. \delta _{i+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\ |
---|
561 | \left. \delta _{j+1/2} \left[ p^h \right] \right|_{k=km} |
---|
562 | &= \frac{1}{2} g \ \left. \delta _{j+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\ |
---|
563 | \end{aligned} \right. |
---|
564 | \end{equation} |
---|
565 | |
---|
566 | for $1<k<km$ (interior layer) |
---|
567 | \begin{equation} \label{Eq_dynhpg_zco} |
---|
568 | \left\{ \begin{aligned} |
---|
569 | \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k} |
---|
570 | &= \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k-1} |
---|
571 | + \frac{1}{2}\;g\; \left. \delta _{i+1/2} \left[ e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k} \\ |
---|
572 | \left. \delta _{j+1/2} \left[ p^h \right] \right|_{k} |
---|
573 | &= \left. \delta _{j+1/2} \left[ p^h \right] \right|_{k-1} |
---|
574 | + \frac{1}{2}\;g\; \left. \delta _{j+1/2} \left[ e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k} \\ |
---|
575 | \end{aligned} \right. |
---|
576 | \end{equation} |
---|
577 | |
---|
578 | Note that the $1/2$ factor in (\ref{Eq_dynhpg_zco_surf}) is adequate because of |
---|
579 | the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface |
---|
580 | level ($z=0$). Note also that in case of variable volume level (\key{vvl} defined), the |
---|
581 | surface pressure gradient is included in \eqref{Eq_dynhpg_zco_surf} and \eqref{Eq_dynhpg_zco} |
---|
582 | through the space and time variations of the vertical scale factor $e_{3w}$. |
---|
583 | |
---|
584 | %-------------------------------------------------------------------------------------------------------------- |
---|
585 | % z-coordinate with partial step |
---|
586 | %-------------------------------------------------------------------------------------------------------------- |
---|
587 | \subsection [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})] |
---|
588 | {$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=true)} |
---|
589 | \label{DYN_hpg_zps} |
---|
590 | |
---|
591 | With partial bottom cells, tracers in horizontally adjacent cells generally live at |
---|
592 | different depths. Before taking horizontal gradients between these tracer points, |
---|
593 | a linear interpolation is used to approximate the deeper tracer as if it actually lived |
---|
594 | at the depth of the shallower tracer point. |
---|
595 | |
---|
596 | Apart from this modification, the horizontal hydrostatic pressure gradient evaluated |
---|
597 | in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case. |
---|
598 | As explained in detail in section \S\ref{TRA_zpshde}, the nonlinearity of pressure |
---|
599 | effects in the equation of state is such that it is better to interpolate temperature and |
---|
600 | salinity vertically before computing the density. Horizontal gradients of temperature |
---|
601 | and salinity are needed for the TRA modules, which is the reason why the horizontal |
---|
602 | gradients of density at the deepest model level are computed in module \mdl{zpsdhe} |
---|
603 | located in the TRA directory and described in \S\ref{TRA_zpshde}. |
---|
604 | |
---|
605 | %-------------------------------------------------------------------------------------------------------------- |
---|
606 | % s- and s-z-coordinates |
---|
607 | %-------------------------------------------------------------------------------------------------------------- |
---|
608 | \subsection{$s$- and $z$-$s$-coordinates} |
---|
609 | \label{DYN_hpg_sco} |
---|
610 | |
---|
611 | Pressure gradient formulations in an $s$-coordinate have been the subject of a vast |
---|
612 | number of papers ($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}). |
---|
613 | A number of different pressure gradient options are coded but the ROMS-like, density Jacobian with |
---|
614 | cubic polynomial method is currently disabled whilst known bugs are under investigation. |
---|
615 | |
---|
616 | $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}=true) |
---|
617 | \begin{equation} \label{Eq_dynhpg_sco} |
---|
618 | \left\{ \begin{aligned} |
---|
619 | - \frac{1} {\rho_o \, e_{1u}} \; \delta _{i+1/2} \left[ p^h \right] |
---|
620 | + \frac{g\; \overline {\rho}^{i+1/2}} {\rho_o \, e_{1u}} \; \delta _{i+1/2} \left[ z_t \right] \\ |
---|
621 | - \frac{1} {\rho_o \, e_{2v}} \; \delta _{j+1/2} \left[ p^h \right] |
---|
622 | + \frac{g\; \overline {\rho}^{j+1/2}} {\rho_o \, e_{2v}} \; \delta _{j+1/2} \left[ z_t \right] \\ |
---|
623 | \end{aligned} \right. |
---|
624 | \end{equation} |
---|
625 | |
---|
626 | Where the first term is the pressure gradient along coordinates, computed as in |
---|
627 | \eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of |
---|
628 | the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point |
---|
629 | ($e_{3w}$). |
---|
630 | |
---|
631 | $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}=true) |
---|
632 | |
---|
633 | $\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} |
---|
634 | (\np{ln\_dynhpg\_djc}=true) (currently disabled; under development) |
---|
635 | |
---|
636 | Note that expression \eqref{Eq_dynhpg_sco} is commonly used when the variable volume formulation is |
---|
637 | activated (\key{vvl}) because in that case, even with a flat bottom, the coordinate surfaces are not |
---|
638 | horizontal but follow the free surface \citep{Levier2007}. The pressure jacobian scheme |
---|
639 | (\np{ln\_dynhpg\_prj}=true) is available as an improved option to \np{ln\_dynhpg\_sco}=true when |
---|
640 | \key{vvl} is active. The pressure Jacobian scheme uses a constrained cubic spline to reconstruct |
---|
641 | the density profile across the water column. This method maintains the monotonicity between the |
---|
642 | density nodes The pressure can be calculated by analytical integration of the density profile and a |
---|
643 | pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide |
---|
644 | a more accurate calculation of the horizontal pressure gradient than the standard scheme. |
---|
645 | |
---|
646 | %-------------------------------------------------------------------------------------------------------------- |
---|
647 | % Time-scheme |
---|
648 | %-------------------------------------------------------------------------------------------------------------- |
---|
649 | \subsection [Time-scheme (\np{ln\_dynhpg\_imp}) ] |
---|
650 | {Time-scheme (\np{ln\_dynhpg\_imp}= true/false)} |
---|
651 | \label{DYN_hpg_imp} |
---|
652 | |
---|
653 | The default time differencing scheme used for the horizontal pressure gradient is |
---|
654 | a leapfrog scheme and therefore the density used in all discrete expressions given |
---|
655 | above is the \textit{now} density, computed from the \textit{now} temperature and |
---|
656 | salinity. In some specific cases (usually high resolution simulations over an ocean |
---|
657 | domain which includes weakly stratified regions) the physical phenomenon that |
---|
658 | controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for |
---|
659 | doubling the stability limit associated with IGWs can be used \citep{Brown_Campana_MWR78, |
---|
660 | Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an |
---|
661 | average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ ($i.e.$ |
---|
662 | \textit{before}, \textit{now} and \textit{after} time-steps), rather than at the central |
---|
663 | time level $t$ only, as in the standard leapfrog scheme. |
---|
664 | |
---|
665 | $\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=true): |
---|
666 | |
---|
667 | \begin{equation} \label{Eq_dynhpg_lf} |
---|
668 | \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; |
---|
669 | -\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right] |
---|
670 | \end{equation} |
---|
671 | |
---|
672 | $\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=true): |
---|
673 | \begin{equation} \label{Eq_dynhpg_imp} |
---|
674 | \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; |
---|
675 | -\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] |
---|
676 | \end{equation} |
---|
677 | |
---|
678 | The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without |
---|
679 | significant additional computation since the density can be updated to time level |
---|
680 | $t+\rdt$ before computing the horizontal hydrostatic pressure gradient. It can |
---|
681 | be easily shown that the stability limit associated with the hydrostatic pressure |
---|
682 | gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the |
---|
683 | standard leapfrog scheme \eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp} |
---|
684 | is equivalent to applying a time filter to the pressure gradient to eliminate high |
---|
685 | frequency IGWs. Obviously, when using \eqref{Eq_dynhpg_imp}, the doubling of |
---|
686 | the time-step is achievable only if no other factors control the time-step, such as |
---|
687 | the stability limits associated with advection or diffusion. |
---|
688 | |
---|
689 | In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=true. |
---|
690 | In this case, we choose to apply the time filter to temperature and salinity used in |
---|
691 | the equation of state, instead of applying it to the hydrostatic pressure or to the |
---|
692 | density, so that no additional storage array has to be defined. The density used to |
---|
693 | compute the hydrostatic pressure gradient (whatever the formulation) is evaluated |
---|
694 | as follows: |
---|
695 | \begin{equation} \label{Eq_rho_flt} |
---|
696 | \rho^t = \rho( \widetilde{T},\widetilde {S},z_t) |
---|
697 | \quad \text{with} \quad |
---|
698 | \widetilde{X} = 1 / 4 \left( X^{t+\rdt} +2 \,X^t + X^{t-\rdt} \right) |
---|
699 | \end{equation} |
---|
700 | |
---|
701 | Note that in the semi-implicit case, it is necessary to save the filtered density, an |
---|
702 | extra three-dimensional field, in the restart file to restart the model with exact |
---|
703 | reproducibility. This option is controlled by \np{nn\_dynhpg\_rst}, a namelist parameter. |
---|
704 | |
---|
705 | % ================================================================ |
---|
706 | % Surface Pressure Gradient |
---|
707 | % ================================================================ |
---|
708 | \section [Surface pressure gradient (\textit{dynspg}) ] |
---|
709 | {Surface pressure gradient (\mdl{dynspg})} |
---|
710 | \label{DYN_spg} |
---|
711 | %-----------------------------------------nam_dynspg---------------------------------------------------- |
---|
712 | \namdisplay{namdyn_spg} |
---|
713 | %------------------------------------------------------------------------------------------------------------ |
---|
714 | |
---|
715 | $\ $\newline %force an empty line |
---|
716 | |
---|
717 | %%% |
---|
718 | Options are defined through the \ngn{namdyn\_spg} namelist variables. |
---|
719 | The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface) and the variable volume case (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\S\ref{PE_free_surface}) the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case (\S\ref{PE_free_surface}). With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
---|
720 | |
---|
721 | %%% |
---|
722 | |
---|
723 | |
---|
724 | The form of the surface pressure gradient term depends on how the user wants to handle |
---|
725 | the fast external gravity waves that are a solution of the analytical equation (\S\ref{PE_hor_pg}). |
---|
726 | Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx): |
---|
727 | an explicit formulation which requires a small time step ; |
---|
728 | a filtered free surface formulation which allows a larger time step by adding a filtering |
---|
729 | term into the momentum equation ; |
---|
730 | and a split-explicit free surface formulation, described below, which also allows a larger time step. |
---|
731 | |
---|
732 | The extra term introduced in the filtered method is calculated |
---|
733 | implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ |
---|
734 | velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. |
---|
735 | |
---|
736 | |
---|
737 | |
---|
738 | %-------------------------------------------------------------------------------------------------------------- |
---|
739 | % Explicit free surface formulation |
---|
740 | %-------------------------------------------------------------------------------------------------------------- |
---|
741 | \subsection{Explicit free surface (\key{dynspg\_exp})} |
---|
742 | \label{DYN_spg_exp} |
---|
743 | |
---|
744 | In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step |
---|
745 | is chosen to be small enough to resolve the external gravity waves (typically a few tens of seconds). |
---|
746 | The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time), |
---|
747 | is thus simply given by : |
---|
748 | \begin{equation} \label{Eq_dynspg_exp} |
---|
749 | \left\{ \begin{aligned} |
---|
750 | - \frac{1}{e_{1u}\,\rho_o} \; \delta _{i+1/2} \left[ \,\rho \,\eta\, \right] \\ |
---|
751 | - \frac{1}{e_{2v}\,\rho_o} \; \delta _{j+1/2} \left[ \,\rho \,\eta\, \right] |
---|
752 | \end{aligned} \right. |
---|
753 | \end{equation} |
---|
754 | |
---|
755 | Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), the surface pressure |
---|
756 | gradient is already included in the momentum tendency through the level thickness variation |
---|
757 | allowed in the computation of the hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module. |
---|
758 | |
---|
759 | %-------------------------------------------------------------------------------------------------------------- |
---|
760 | % Split-explict free surface formulation |
---|
761 | %-------------------------------------------------------------------------------------------------------------- |
---|
762 | \subsection{Split-Explicit free surface (\key{dynspg\_ts})} |
---|
763 | \label{DYN_spg_ts} |
---|
764 | |
---|
765 | The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined), |
---|
766 | also called the time-splitting formulation, follows the one |
---|
767 | proposed by \citet{Griffies_Bk04}. The general idea is to solve the free surface |
---|
768 | equation and the associated barotropic velocity equations with a smaller time |
---|
769 | step than $\rdt$, the time step used for the three dimensional prognostic |
---|
770 | variables (Fig.~\ref{Fig_DYN_dynspg_ts}). |
---|
771 | The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) |
---|
772 | is provided through the \np{nn\_baro} namelist parameter as: |
---|
773 | $\rdt_e = \rdt / nn\_baro$. |
---|
774 | |
---|
775 | |
---|
776 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
---|
777 | \begin{figure}[!t] \begin{center} |
---|
778 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf} |
---|
779 | \caption{ \label{Fig_DYN_dynspg_ts} |
---|
780 | Schematic of the split-explicit time stepping scheme for the external |
---|
781 | and internal modes. Time increases to the right. |
---|
782 | Internal mode time steps (which are also the model time steps) are denoted |
---|
783 | by $t-\rdt$, $t, t+\rdt$, and $t+2\rdt$. |
---|
784 | The curved line represents a leap-frog time step, and the smaller time |
---|
785 | steps $N \rdt_e=\frac{3}{2}\rdt$ are denoted by the zig-zag line. |
---|
786 | The vertically integrated forcing \textbf{M}(t) computed at the model time step $t$ |
---|
787 | represents the interaction between the external and internal motions. |
---|
788 | While keeping \textbf{M} and freshwater forcing field fixed, a leap-frog |
---|
789 | integration carries the external mode variables (surface height and vertically |
---|
790 | integrated velocity) from $t$ to $t+\frac{3}{2} \rdt$ using N external time |
---|
791 | steps of length $\rdt_e$. Time averaging the external fields over the |
---|
792 | $\frac{2}{3}N+1$ time steps (endpoints included) centers the vertically integrated |
---|
793 | velocity and the sea surface height at the model timestep $t+\rdt$. |
---|
794 | These averaged values are used to update \textbf{M}(t) with both the surface |
---|
795 | pressure gradient and the Coriolis force, therefore providing the $t+\rdt$ |
---|
796 | velocity. The model time stepping scheme can then be achieved by a baroclinic |
---|
797 | leap-frog time step that carries the surface height from $t-\rdt$ to $t+\rdt$. } |
---|
798 | \end{center} \end{figure} |
---|
799 | %> > > > > > > > > > > > > > > > > > > > > > > > > > > > |
---|
800 | |
---|
801 | The split-explicit formulation has a damping effect on external gravity waves, |
---|
802 | which is weaker damping than that for the filtered free surface but still significant, as |
---|
803 | shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. |
---|
804 | |
---|
805 | %>>>>>=============== |
---|
806 | \gmcomment{ %%% copy from griffies Book |
---|
807 | |
---|
808 | \textbf{title: Time stepping the barotropic system } |
---|
809 | |
---|
810 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence, |
---|
811 | we can update the surface height and vertically integrated velocity with a leap-frog |
---|
812 | scheme using the small barotropic time step $\rdt$. We have |
---|
813 | |
---|
814 | \begin{equation} \label{DYN_spg_ts_eta} |
---|
815 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
---|
816 | = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
---|
817 | \end{equation} |
---|
818 | \begin{multline} \label{DYN_spg_ts_u} |
---|
819 | \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\ |
---|
820 | = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) |
---|
821 | - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] |
---|
822 | \end{multline} |
---|
823 | \ |
---|
824 | |
---|
825 | In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ |
---|
826 | and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time |
---|
827 | that sets the barotropic time steps via |
---|
828 | \begin{equation} \label{DYN_spg_ts_t} |
---|
829 | t_n=\tau+n\rdt |
---|
830 | \end{equation} |
---|
831 | with $n$ an integer. The density scaled surface pressure is evaluated via |
---|
832 | \begin{equation} \label{DYN_spg_ts_ps} |
---|
833 | p_s^{(b)}(\tau,t_{n}) = \begin{cases} |
---|
834 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
---|
835 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
---|
836 | \end{cases} |
---|
837 | \end{equation} |
---|
838 | To get started, we assume the following initial conditions |
---|
839 | \begin{equation} \label{DYN_spg_ts_eta} |
---|
840 | \begin{split} |
---|
841 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} |
---|
842 | \\ |
---|
843 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} |
---|
844 | \end{split} |
---|
845 | \end{equation} |
---|
846 | with |
---|
847 | \begin{equation} \label{DYN_spg_ts_etaF} |
---|
848 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n}) |
---|
849 | \end{equation} |
---|
850 | the time averaged surface height taken from the previous barotropic cycle. Likewise, |
---|
851 | \begin{equation} \label{DYN_spg_ts_u} |
---|
852 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ |
---|
853 | \\ |
---|
854 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} |
---|
855 | \end{equation} |
---|
856 | with |
---|
857 | \begin{equation} \label{DYN_spg_ts_u} |
---|
858 | \overline{\textbf{U}^{(b)}(\tau)} |
---|
859 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n}) |
---|
860 | \end{equation} |
---|
861 | the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. |
---|
862 | |
---|
863 | Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \rdt \tau$ |
---|
864 | \begin{equation} \label{DYN_spg_ts_u} |
---|
865 | \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} |
---|
866 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
---|
867 | \end{equation} |
---|
868 | The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form |
---|
869 | |
---|
870 | \begin{equation} \label{DYN_spg_ts_ssh} |
---|
871 | \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] |
---|
872 | \end{equation} |
---|
873 | |
---|
874 | The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). |
---|
875 | |
---|
876 | In general, some form of time filter is needed to maintain integrity of the surface |
---|
877 | height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We |
---|
878 | have tried various forms of such filtering, with the following method discussed in |
---|
879 | \cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of |
---|
880 | tracer conservation properties (see Section ??) |
---|
881 | |
---|
882 | \begin{equation} \label{DYN_spg_ts_sshf} |
---|
883 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
---|
884 | \end{equation} |
---|
885 | Another approach tried was |
---|
886 | |
---|
887 | \begin{equation} \label{DYN_spg_ts_sshf2} |
---|
888 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
---|
889 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt) |
---|
890 | + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right] |
---|
891 | \end{equation} |
---|
892 | |
---|
893 | which is useful since it isolates all the time filtering aspects into the term multiplied |
---|
894 | by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when |
---|
895 | eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. |
---|
896 | |
---|
897 | } %%end gm comment (copy of griffies book) |
---|
898 | |
---|
899 | %>>>>>=============== |
---|
900 | |
---|
901 | |
---|
902 | %-------------------------------------------------------------------------------------------------------------- |
---|
903 | % Filtered free surface formulation |
---|
904 | %-------------------------------------------------------------------------------------------------------------- |
---|
905 | \subsection{Filtered free surface (\key{dynspg\_flt})} |
---|
906 | \label{DYN_spg_fltp} |
---|
907 | |
---|
908 | The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation. |
---|
909 | The extra term introduced in the equations (see \S\ref{PE_free_surface}) is solved implicitly. |
---|
910 | The elliptic solvers available in the code are documented in \S\ref{MISC}. |
---|
911 | |
---|
912 | %% gm %%======>>>> given here the discrete eqs provided to the solver |
---|
913 | \gmcomment{ %%% copy from chap-model basics |
---|
914 | \begin{equation} \label{Eq_spg_flt} |
---|
915 | \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}} |
---|
916 | - g \nabla \left( \tilde{\rho} \ \eta \right) |
---|
917 | - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) |
---|
918 | \end{equation} |
---|
919 | where $T_c$, is a parameter with dimensions of time which characterizes the force, |
---|
920 | $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ |
---|
921 | represents the collected contributions of the Coriolis, hydrostatic pressure gradient, |
---|
922 | non-linear and viscous terms in \eqref{Eq_PE_dyn}. |
---|
923 | } %end gmcomment |
---|
924 | |
---|
925 | Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth |
---|
926 | is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps. |
---|
927 | |
---|
928 | % ================================================================ |
---|
929 | % Lateral diffusion term |
---|
930 | % ================================================================ |
---|
931 | \section [Lateral diffusion term (\textit{dynldf})] |
---|
932 | {Lateral diffusion term (\mdl{dynldf})} |
---|
933 | \label{DYN_ldf} |
---|
934 | %------------------------------------------nam_dynldf---------------------------------------------------- |
---|
935 | \namdisplay{namdyn_ldf} |
---|
936 | %------------------------------------------------------------------------------------------------------------- |
---|
937 | |
---|
938 | Options are defined through the \ngn{namdyn\_ldf} namelist variables. |
---|
939 | The options available for lateral diffusion are to use either laplacian |
---|
940 | (rotated or not) or biharmonic operators. The coefficients may be constant |
---|
941 | or spatially variable; the description of the coefficients is found in the chapter |
---|
942 | on lateral physics (Chap.\ref{LDF}). The lateral diffusion of momentum is |
---|
943 | evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression |
---|
944 | is the \textit{before} velocity in time, except for the pure vertical component |
---|
945 | that appears when a tensor of rotation is used. This latter term is solved |
---|
946 | implicitly together with the vertical diffusion term (see \S\ref{STP}) |
---|
947 | |
---|
948 | At the lateral boundaries either free slip, no slip or partial slip boundary |
---|
949 | conditions are applied according to the user's choice (see Chap.\ref{LBC}). |
---|
950 | |
---|
951 | % ================================================================ |
---|
952 | \subsection [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ] |
---|
953 | {Iso-level laplacian operator (\np{ln\_dynldf\_lap}=true)} |
---|
954 | \label{DYN_ldf_lap} |
---|
955 | |
---|
956 | For lateral iso-level diffusion, the discrete operator is: |
---|
957 | \begin{equation} \label{Eq_dynldf_lap} |
---|
958 | \left\{ \begin{aligned} |
---|
959 | D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} |
---|
960 | \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[ |
---|
961 | {A_f^{lm} \;e_{3f} \zeta } \right] \\ |
---|
962 | \\ |
---|
963 | D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} |
---|
964 | \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[ |
---|
965 | {A_f^{lm} \;e_{3f} \zeta } \right] \\ |
---|
966 | \end{aligned} \right. |
---|
967 | \end{equation} |
---|
968 | |
---|
969 | As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence |
---|
970 | and curl of the vorticity) preserves symmetry and ensures a complete |
---|
971 | separation between the vorticity and divergence parts of the momentum diffusion. |
---|
972 | |
---|
973 | %-------------------------------------------------------------------------------------------------------------- |
---|
974 | % Rotated laplacian operator |
---|
975 | %-------------------------------------------------------------------------------------------------------------- |
---|
976 | \subsection [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ] |
---|
977 | {Rotated laplacian operator (\np{ln\_dynldf\_iso}=true)} |
---|
978 | \label{DYN_ldf_iso} |
---|
979 | |
---|
980 | A rotation of the lateral momentum diffusion operator is needed in several cases: |
---|
981 | for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for |
---|
982 | either iso-neutral (\np{ln\_dynldf\_iso}=true) or geopotential |
---|
983 | (\np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step |
---|
984 | case, coordinates are horizontal except at the deepest level and no |
---|
985 | rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusion operator |
---|
986 | is defined simply as the divergence of down gradient momentum fluxes on each |
---|
987 | momentum component. It must be emphasized that this formulation ignores |
---|
988 | constraints on the stress tensor such as symmetry. The resulting discrete |
---|
989 | representation is: |
---|
990 | \begin{equation} \label{Eq_dyn_ldf_iso} |
---|
991 | \begin{split} |
---|
992 | D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ |
---|
993 | & \left\{\quad {\delta _{i+1/2} \left[ {A_T^{lm} \left( |
---|
994 | {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u] |
---|
995 | -e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}} |
---|
996 | \right)} \right]} \right. |
---|
997 | \\ |
---|
998 | & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} |
---|
999 | }\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} |
---|
1000 | \,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} |
---|
1001 | \right)} \right] |
---|
1002 | \\ |
---|
1003 | &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline |
---|
1004 | {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } |
---|
1005 | \right.} \right. |
---|
1006 | \\ |
---|
1007 | & \ \qquad \qquad \qquad \quad\ |
---|
1008 | - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} |
---|
1009 | \\ |
---|
1010 | & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ |
---|
1011 | +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} |
---|
1012 | \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} |
---|
1013 | \\ |
---|
1014 | \\ |
---|
1015 | D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\ |
---|
1016 | & \left\{\quad {\delta _{i+1/2} \left[ {A_f^{lm} \left( |
---|
1017 | {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v] |
---|
1018 | -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}} |
---|
1019 | \right)} \right]} \right. |
---|
1020 | \\ |
---|
1021 | & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} |
---|
1022 | }\,\delta _{j} [v] - e_{1t}\, r_{2t} |
---|
1023 | \,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} |
---|
1024 | \right)} \right] |
---|
1025 | \\ |
---|
1026 | & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline |
---|
1027 | {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. |
---|
1028 | \\ |
---|
1029 | & \ \qquad \qquad \qquad \quad\ |
---|
1030 | - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} |
---|
1031 | \\ |
---|
1032 | & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ |
---|
1033 | +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} |
---|
1034 | \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} |
---|
1035 | \end{split} |
---|
1036 | \end{equation} |
---|
1037 | where $r_1$ and $r_2$ are the slopes between the surface along which the |
---|
1038 | diffusion operator acts and the surface of computation ($z$- or $s$-surfaces). |
---|
1039 | The way these slopes are evaluated is given in the lateral physics chapter |
---|
1040 | (Chap.\ref{LDF}). |
---|
1041 | |
---|
1042 | %-------------------------------------------------------------------------------------------------------------- |
---|
1043 | % Iso-level bilaplacian operator |
---|
1044 | %-------------------------------------------------------------------------------------------------------------- |
---|
1045 | \subsection [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})] |
---|
1046 | {Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=true)} |
---|
1047 | \label{DYN_ldf_bilap} |
---|
1048 | |
---|
1049 | The lateral fourth order operator formulation on momentum is obtained by |
---|
1050 | applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on |
---|
1051 | boundary conditions: the first derivative term normal to the coast depends on |
---|
1052 | the free or no-slip lateral boundary conditions chosen, while the third |
---|
1053 | derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}). |
---|
1054 | %%% |
---|
1055 | \gmcomment{add a remark on the the change in the position of the coefficient} |
---|
1056 | %%% |
---|
1057 | |
---|
1058 | % ================================================================ |
---|
1059 | % Vertical diffusion term |
---|
1060 | % ================================================================ |
---|
1061 | \section [Vertical diffusion term (\mdl{dynzdf})] |
---|
1062 | {Vertical diffusion term (\mdl{dynzdf})} |
---|
1063 | \label{DYN_zdf} |
---|
1064 | %----------------------------------------------namzdf------------------------------------------------------ |
---|
1065 | \namdisplay{namzdf} |
---|
1066 | %------------------------------------------------------------------------------------------------------------- |
---|
1067 | |
---|
1068 | Options are defined through the \ngn{namzdf} namelist variables. |
---|
1069 | The large vertical diffusion coefficient found in the surface mixed layer together |
---|
1070 | with high vertical resolution implies that in the case of explicit time stepping there |
---|
1071 | would be too restrictive a constraint on the time step. Two time stepping schemes |
---|
1072 | can be used for the vertical diffusion term : $(a)$ a forward time differencing |
---|
1073 | scheme (\np{ln\_zdfexp}=true) using a time splitting technique |
---|
1074 | (\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme |
---|
1075 | (\np{ln\_zdfexp}=false) (see \S\ref{STP}). Note that namelist variables |
---|
1076 | \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. |
---|
1077 | |
---|
1078 | The formulation of the vertical subgrid scale physics is the same whatever |
---|
1079 | the vertical coordinate is. The vertical diffusion operators given by |
---|
1080 | \eqref{Eq_PE_zdf} take the following semi-discrete space form: |
---|
1081 | \begin{equation} \label{Eq_dynzdf} |
---|
1082 | \left\{ \begin{aligned} |
---|
1083 | D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } |
---|
1084 | \ \delta _{k+1/2} [\,u\,] \right] \\ |
---|
1085 | \\ |
---|
1086 | D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } |
---|
1087 | \ \delta _{k+1/2} [\,v\,] \right] |
---|
1088 | \end{aligned} \right. |
---|
1089 | \end{equation} |
---|
1090 | where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and |
---|
1091 | diffusivity coefficients. The way these coefficients are evaluated |
---|
1092 | depends on the vertical physics used (see \S\ref{ZDF}). |
---|
1093 | |
---|
1094 | The surface boundary condition on momentum is the stress exerted by |
---|
1095 | the wind. At the surface, the momentum fluxes are prescribed as the boundary |
---|
1096 | condition on the vertical turbulent momentum fluxes, |
---|
1097 | \begin{equation} \label{Eq_dynzdf_sbc} |
---|
1098 | \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} |
---|
1099 | = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v } |
---|
1100 | \end{equation} |
---|
1101 | where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress |
---|
1102 | vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients |
---|
1103 | in the surface mixed layer ensure that the surface wind stress is distributed in |
---|
1104 | the vertical over the mixed layer depth. If the vertical mixing coefficient |
---|
1105 | is small (when no mixed layer scheme is used) the surface stress enters only |
---|
1106 | the top model level, as a body force. The surface wind stress is calculated |
---|
1107 | in the surface module routines (SBC, see Chap.\ref{SBC}) |
---|
1108 | |
---|
1109 | The turbulent flux of momentum at the bottom of the ocean is specified through |
---|
1110 | a bottom friction parameterisation (see \S\ref{ZDF_bfr}) |
---|
1111 | |
---|
1112 | % ================================================================ |
---|
1113 | % External Forcing |
---|
1114 | % ================================================================ |
---|
1115 | \section{External Forcings} |
---|
1116 | \label{DYN_forcing} |
---|
1117 | |
---|
1118 | Besides the surface and bottom stresses (see the above section) which are |
---|
1119 | introduced as boundary conditions on the vertical mixing, two other forcings |
---|
1120 | enter the dynamical equations. |
---|
1121 | |
---|
1122 | One is the effect of atmospheric pressure on the ocean dynamics. |
---|
1123 | Another forcing term is the tidal potential. |
---|
1124 | Both of which will be introduced into the reference version soon. |
---|
1125 | |
---|
1126 | \gmcomment{atmospheric pressure is there!!!! include its description } |
---|
1127 | |
---|
1128 | % ================================================================ |
---|
1129 | % Time evolution term |
---|
1130 | % ================================================================ |
---|
1131 | \section [Time evolution term (\textit{dynnxt})] |
---|
1132 | {Time evolution term (\mdl{dynnxt})} |
---|
1133 | \label{DYN_nxt} |
---|
1134 | |
---|
1135 | %----------------------------------------------namdom---------------------------------------------------- |
---|
1136 | \namdisplay{namdom} |
---|
1137 | %------------------------------------------------------------------------------------------------------------- |
---|
1138 | |
---|
1139 | Options are defined through the \ngn{namdom} namelist variables. |
---|
1140 | The general framework for dynamics time stepping is a leap-frog scheme, |
---|
1141 | $i.e.$ a three level centred time scheme associated with an Asselin time filter |
---|
1142 | (cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using |
---|
1143 | the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in the variable |
---|
1144 | volume case (\key{vvl} defined), where it has to be applied to the thickness |
---|
1145 | weighted velocity (see \S\ref{Apdx_A_momentum}) |
---|
1146 | |
---|
1147 | $\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=true ; \key{vvl} not defined): |
---|
1148 | \begin{equation} \label{Eq_dynnxt_vec} |
---|
1149 | \left\{ \begin{aligned} |
---|
1150 | &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt \ \text{RHS}_u^t \\ |
---|
1151 | &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right] |
---|
1152 | \end{aligned} \right. |
---|
1153 | \end{equation} |
---|
1154 | |
---|
1155 | $\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}=false ; \key{vvl} defined): |
---|
1156 | \begin{equation} \label{Eq_dynnxt_flux} |
---|
1157 | \left\{ \begin{aligned} |
---|
1158 | &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t \\ |
---|
1159 | &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t |
---|
1160 | +\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] |
---|
1161 | \end{aligned} \right. |
---|
1162 | \end{equation} |
---|
1163 | where RHS is the right hand side of the momentum equation, the subscript $f$ |
---|
1164 | denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is |
---|
1165 | initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$. |
---|
1166 | In both cases, the modified Asselin filter is not applied since perfect conservation |
---|
1167 | is not an issue for the momentum equations. |
---|
1168 | |
---|
1169 | Note that with the filtered free surface, the update of the \textit{after} velocities |
---|
1170 | is done in the \mdl{dynsp\_flt} module, and only array swapping |
---|
1171 | and Asselin filtering is done in \mdl{dynnxt}. |
---|
1172 | |
---|
1173 | % ================================================================ |
---|
1174 | % Neptune effect |
---|
1175 | % ================================================================ |
---|
1176 | \section [Neptune effect (\textit{dynnept})] |
---|
1177 | {Neptune effect (\mdl{dynnept})} |
---|
1178 | \label{DYN_nept} |
---|
1179 | |
---|
1180 | The "Neptune effect" (thus named in \citep{HollowayOM86}) is a |
---|
1181 | parameterisation of the potentially large effect of topographic form stress |
---|
1182 | (caused by eddies) in driving the ocean circulation. Originally developed for |
---|
1183 | low-resolution models, in which it was applied via a Laplacian (second-order) |
---|
1184 | diffusion-like term in the momentum equation, it can also be applied in eddy |
---|
1185 | permitting or resolving models, in which a more scale-selective bilaplacian |
---|
1186 | (fourth-order) implementation is preferred. This mechanism has a |
---|
1187 | significant effect on boundary currents (including undercurrents), and the |
---|
1188 | upwelling of deep water near continental shelves. |
---|
1189 | |
---|
1190 | The theoretical basis for the method can be found in |
---|
1191 | \citep{HollowayJPO92}, including the explanation of why form stress is not |
---|
1192 | necessarily a drag force, but may actually drive the flow. |
---|
1193 | \citep{HollowayJPO94} demonstrate the effects of the parameterisation in |
---|
1194 | the GFDL-MOM model, at a horizontal resolution of about 1.8 degrees. |
---|
1195 | \citep{HollowayOM08} demonstrate the biharmonic version of the |
---|
1196 | parameterisation in a global run of the POP model, with an average horizontal |
---|
1197 | grid spacing of about 32km. |
---|
1198 | |
---|
1199 | The NEMO implementation is a simplified form of that supplied by |
---|
1200 | Greg Holloway, the testing of which was described in \citep{HollowayJGR09}. |
---|
1201 | The major simplification is that a time invariant Neptune velocity |
---|
1202 | field is assumed. This is computed only once, during start-up, and |
---|
1203 | made available to the rest of the code via a module. Vertical |
---|
1204 | diffusive terms are also ignored, and the model topography itself |
---|
1205 | is used, rather than a separate topographic dataset as in |
---|
1206 | \citep{HollowayOM08}. This implementation is only in the iso-level |
---|
1207 | formulation, as is the case anyway for the bilaplacian operator. |
---|
1208 | |
---|
1209 | The velocity field is derived from a transport stream function given by: |
---|
1210 | |
---|
1211 | \begin{equation} \label{Eq_dynnept_sf} |
---|
1212 | \psi = -fL^2H |
---|
1213 | \end{equation} |
---|
1214 | |
---|
1215 | where $L$ is a latitude-dependant length scale given by: |
---|
1216 | |
---|
1217 | \begin{equation} \label{Eq_dynnept_ls} |
---|
1218 | L = l_1 + (l_2 -l_1)\left ( {1 + \cos 2\phi \over 2 } \right ) |
---|
1219 | \end{equation} |
---|
1220 | |
---|
1221 | where $\phi$ is latitude and $l_1$ and $l_2$ are polar and equatorial length scales respectively. |
---|
1222 | Neptune velocity components, $u^*$, $v^*$ are derived from the stremfunction as: |
---|
1223 | |
---|
1224 | \begin{equation} \label{Eq_dynnept_vel} |
---|
1225 | u^* = -{1\over H} {\partial \psi \over \partial y}\ \ \ ,\ \ \ v^* = {1\over H} {\partial \psi \over \partial x} |
---|
1226 | \end{equation} |
---|
1227 | |
---|
1228 | \smallskip |
---|
1229 | %----------------------------------------------namdom---------------------------------------------------- |
---|
1230 | \namdisplay{namdyn_nept} |
---|
1231 | %-------------------------------------------------------------------------------------------------------- |
---|
1232 | \smallskip |
---|
1233 | |
---|
1234 | The Neptune effect is enabled when \np{ln\_neptsimp}=true (default=false). |
---|
1235 | \np{ln\_smooth\_neptvel} controls whether a scale-selective smoothing is applied |
---|
1236 | to the Neptune effect flow field (default=false) (this smoothing method is as |
---|
1237 | used by Holloway). \np{rn\_tslse} and \np{rn\_tslsp} are the equatorial and |
---|
1238 | polar values respectively of the length-scale parameter $L$ used in determining |
---|
1239 | the Neptune stream function \eqref{Eq_dynnept_sf} and \eqref{Eq_dynnept_ls}. |
---|
1240 | Values at intermediate latitudes are given by a cosine fit, mimicking the |
---|
1241 | variation of the deformation radius with latitude. The default values of 12km |
---|
1242 | and 3km are those given in \citep{HollowayJPO94}, appropriate for a coarse |
---|
1243 | resolution model. The finer resolution study of \citep{HollowayOM08} increased |
---|
1244 | the values of L by a factor of $\sqrt 2$ to 17km and 4.2km, thus doubling the |
---|
1245 | stream function for a given topography. |
---|
1246 | |
---|
1247 | The simple formulation for ($u^*$, $v^*$) can give unacceptably large velocities |
---|
1248 | in shallow water, and \citep{HollowayOM08} add an offset to the depth in the |
---|
1249 | denominator to control this problem. In this implementation we offer instead (at |
---|
1250 | the suggestion of G. Madec) the option of ramping down the Neptune flow field to |
---|
1251 | zero over a finite depth range. The switch \np{ln\_neptramp} activates this |
---|
1252 | option (default=false), in which case velocities at depths greater than |
---|
1253 | \np{rn\_htrmax} are unaltered, but ramp down linearly with depth to zero at a |
---|
1254 | depth of \np{rn\_htrmin} (and shallower). |
---|
1255 | |
---|
1256 | % ================================================================ |
---|