1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | |
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5 | % ================================================================ |
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6 | % Chapter Lateral Ocean Physics (LDF) |
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7 | % ================================================================ |
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8 | \chapter{Lateral Ocean Physics (LDF)} |
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9 | \label{chap:LDF} |
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10 | |
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11 | \chaptertoc |
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12 | |
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13 | \newpage |
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14 | |
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15 | The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and |
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16 | their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). |
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17 | In this section we further discuss each lateral physics option. |
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18 | Choosing one lateral physics scheme means for the user defining, |
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19 | (1) the type of operator used (laplacian or bilaplacian operators, or no lateral mixing term); |
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20 | (2) the direction along which the lateral diffusive fluxes are evaluated |
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21 | (model level, geopotential or isopycnal surfaces); and |
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22 | (3) the space and time variations of the eddy coefficients. |
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23 | These three aspects of the lateral diffusion are set through namelist parameters |
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24 | (see the \nam{tra_ldf}{tra\_ldf} and \nam{dyn_ldf}{dyn\_ldf} below). |
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25 | Note that this chapter describes the standard implementation of iso-neutral tracer mixing. |
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26 | Griffies's implementation, which is used if \np{ln_traldf_triad}{ln\_traldf\_triad}\forcode{=.true.}, |
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27 | is described in \autoref{apdx:TRIADS} |
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28 | |
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29 | %-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- |
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30 | |
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31 | %-------------------------------------------------------------------------------------------------------------- |
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32 | |
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33 | % ================================================================ |
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34 | % Lateral Mixing Operator |
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35 | % ================================================================ |
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36 | \section[Lateral mixing operators]{Lateral mixing operators} |
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37 | \label{sec:LDF_op} |
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38 | We remind here the different lateral mixing operators that can be used. Further details can be found in \autoref{subsec:TRA_ldf_op} and \autoref{sec:DYN_ldf}. |
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39 | |
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40 | \subsection[No lateral mixing (\forcode{ln_traldf_OFF} \& \forcode{ln_dynldf_OFF})]{No lateral mixing (\protect\np{ln_traldf_OFF}{ln\_traldf\_OFF} \& \protect\np{ln_dynldf_OFF}{ln\_dynldf\_OFF})} |
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41 | |
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42 | It is possible to run without explicit lateral diffusion on tracers (\protect\np{ln_traldf_OFF}{ln\_traldf\_OFF}\forcode{=.true.}) and/or |
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43 | momentum (\protect\np{ln_dynldf_OFF}{ln\_dynldf\_OFF}\forcode{=.true.}). The latter option is even recommended if using the |
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44 | UBS advection scheme on momentum (\np{ln_dynadv_ubs}{ln\_dynadv\_ubs}\forcode{=.true.}, |
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45 | see \autoref{subsec:DYN_adv_ubs}) and can be useful for testing purposes. |
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46 | |
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47 | \subsection[Laplacian mixing (\forcode{ln_traldf_lap} \& \forcode{ln_dynldf_lap})]{Laplacian mixing (\protect\np{ln_traldf_lap}{ln\_traldf\_lap} \& \protect\np{ln_dynldf_lap}{ln\_dynldf\_lap})} |
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48 | Setting \protect\np{ln_traldf_lap}{ln\_traldf\_lap}\forcode{=.true.} and/or \protect\np{ln_dynldf_lap}{ln\_dynldf\_lap}\forcode{=.true.} enables |
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49 | a second order diffusion on tracers and momentum respectively. Note that in \NEMO\ 4, one can not combine |
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50 | Laplacian and Bilaplacian operators for the same variable. |
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51 | |
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52 | \subsection[Bilaplacian mixing (\forcode{ln_traldf_blp} \& \forcode{ln_dynldf_blp})]{Bilaplacian mixing (\protect\np{ln_traldf_blp}{ln\_traldf\_blp} \& \protect\np{ln_dynldf_blp}{ln\_dynldf\_blp})} |
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53 | Setting \protect\np{ln_traldf_blp}{ln\_traldf\_blp}\forcode{=.true.} and/or \protect\np{ln_dynldf_blp}{ln\_dynldf\_blp}\forcode{=.true.} enables |
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54 | a fourth order diffusion on tracers and momentum respectively. It is implemented by calling the above Laplacian operator twice. |
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55 | We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. |
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56 | |
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57 | % ================================================================ |
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58 | % Direction of lateral Mixing |
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59 | % ================================================================ |
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60 | \section[Direction of lateral mixing (\textit{ldfslp.F90})]{Direction of lateral mixing (\protect\mdl{ldfslp})} |
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61 | \label{sec:LDF_slp} |
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62 | |
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63 | %%% |
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64 | \gmcomment{ |
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65 | we should emphasize here that the implementation is a rather old one. |
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66 | Better work can be achieved by using \citet{griffies.gnanadesikan.ea_JPO98, griffies_bk04} iso-neutral scheme. |
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67 | } |
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68 | |
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69 | A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. |
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70 | This occurs when $(a)$ horizontal mixing is required on tracer or momentum |
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71 | (\np{ln_traldf_hor}{ln\_traldf\_hor} or \np{ln_dynldf_hor}{ln\_dynldf\_hor}) in $s$- or mixed $s$-$z$- coordinates, |
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72 | and $(b)$ isoneutral mixing is required whatever the vertical coordinate is. |
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73 | This direction of mixing is defined by its slopes in the \textbf{i}- and \textbf{j}-directions at the face of |
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74 | the cell of the quantity to be diffused. |
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75 | For a tracer, this leads to the following four slopes: |
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76 | $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), |
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77 | while for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and |
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78 | $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. |
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79 | |
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80 | %gm% add here afigure of the slope in i-direction |
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81 | |
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82 | \subsection{Slopes for tracer geopotential mixing in the $s$-coordinate} |
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83 | |
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84 | In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between |
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85 | the geopotential and computational surfaces. |
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86 | Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when |
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87 | the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, |
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88 | \ie\ a linear function of $z_T$, the depth of a $T$-point. |
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89 | %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} |
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90 | |
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91 | \begin{equation} |
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92 | \label{eq:LDF_slp_geo} |
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93 | \begin{aligned} |
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94 | r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} |
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95 | \;\delta_{i+1/2}[z_t] |
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96 | &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \ \\ |
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97 | r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} |
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98 | \;\delta_{j+1/2} [z_t] |
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99 | &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \ \\ |
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100 | r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} |
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101 | &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] \\ |
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102 | r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} |
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103 | &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] |
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104 | \end{aligned} |
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105 | \end{equation} |
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106 | |
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107 | %gm% caution I'm not sure the simplification was a good idea! |
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108 | |
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109 | These slopes are computed once in \rou{ldf\_slp\_init} when \np{ln_sco}{ln\_sco}\forcode{=.true.}, |
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110 | and either \np{ln_traldf_hor}{ln\_traldf\_hor}\forcode{=.true.} or \np{ln_dynldf_hor}{ln\_dynldf\_hor}\forcode{=.true.}. |
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111 | |
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112 | \subsection{Slopes for tracer iso-neutral mixing} |
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113 | \label{subsec:LDF_slp_iso} |
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114 | |
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115 | In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces. |
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116 | Their formulation does not depend on the vertical coordinate used. |
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117 | Their discrete formulation is found using the fact that the diffusive fluxes of |
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118 | locally referenced potential density (\ie\ $in situ$ density) vanish. |
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119 | So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in |
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120 | the three directions to zero leads to the following definition for the neutral slopes: |
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121 | |
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122 | \begin{equation} |
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123 | \label{eq:LDF_slp_iso} |
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124 | \begin{split} |
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125 | r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} |
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126 | {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} \\ |
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127 | r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]} |
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128 | {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} \\ |
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129 | r_{1w} &= \frac{e_{3w}}{e_{1w}}\; |
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130 | \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}} |
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131 | {\delta_{k+1/2}[\rho]} \\ |
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132 | r_{2w} &= \frac{e_{3w}}{e_{2w}}\; |
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133 | \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}} |
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134 | {\delta_{k+1/2}[\rho]} |
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135 | \end{split} |
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136 | \end{equation} |
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137 | |
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138 | %gm% rewrite this as the explanation is not very clear !!! |
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139 | %In practice, \autoref{eq:LDF_slp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:LDF_slp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth. |
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140 | |
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141 | %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:LDF_slp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). |
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142 | |
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143 | %In the $z$-coordinate, the derivative of the \autoref{eq:LDF_slp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so the $in situ$ density can be used for its evaluation. |
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144 | |
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145 | As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to |
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146 | be evaluated at the same local pressure |
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147 | (which, in decibars, is approximated by the depth in meters in the model). |
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148 | Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, |
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149 | but further transformation is needed depending on the vertical coordinate used: |
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150 | |
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151 | \begin{description} |
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152 | |
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153 | \item[$z$-coordinate with full step: ] |
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154 | in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives are taken at the same depth, |
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155 | thus the $in situ$ density can be used. |
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156 | This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, |
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157 | where $N^2$ is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following \citet{mcdougall_JPO87} |
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158 | (see \autoref{subsec:TRA_bn2}). |
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159 | |
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160 | \item[$z$-coordinate with partial step: ] |
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161 | this case is identical to the full step case except that at partial step level, |
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162 | the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}. |
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163 | |
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164 | \item[$s$- or hybrid $s$-$z$- coordinate: ] |
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165 | in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if |
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166 | the Griffies scheme is used (\np{ln_traldf_triad}{ln\_traldf\_triad}\forcode{=.true.}; |
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167 | see \autoref{apdx:TRIADS}). |
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168 | In other words, iso-neutral mixing will only be accurately represented with a linear equation of state |
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169 | (\np{ln_seos}{ln\_seos}\forcode{=.true.}). |
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170 | In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} |
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171 | will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. |
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172 | |
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173 | %gm% |
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174 | Note: The solution for $s$-coordinate passes trough the use of different (and better) expression for |
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175 | the constraint on iso-neutral fluxes. |
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176 | Following \citet{griffies_bk04}, instead of specifying directly that there is a zero neutral diffusive flux of |
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177 | locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between |
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178 | the neutral direction diffusive fluxes of potential temperature and salinity: |
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179 | \[ |
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180 | \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) |
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181 | \] |
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182 | % gm{ where vector F is ....} |
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183 | |
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184 | This constraint leads to the following definition for the slopes: |
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185 | |
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186 | \[ |
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187 | % \label{eq:LDF_slp_iso2} |
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188 | \begin{split} |
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189 | r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac |
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190 | {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]} |
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191 | {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k} |
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192 | -\beta_u \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } \\ |
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193 | r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac |
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194 | {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]} |
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195 | {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k} |
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196 | -\beta_v \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} } \\ |
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197 | r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac |
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198 | {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2} |
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199 | -\beta_w \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} } |
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200 | {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\ |
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201 | r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac |
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202 | {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2} |
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203 | -\beta_w \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} } |
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204 | {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\ |
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205 | \end{split} |
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206 | \] |
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207 | where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in |
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208 | \autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points. |
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209 | In order to save computation time, they should be approximated by the mean of their values at $T$-points |
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210 | (for example in the case of $\alpha$: |
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211 | $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ and |
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212 | $\alpha_w=\overline{\alpha_T}^{k+1/2}$). |
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213 | |
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214 | Note that such a formulation could be also used in the $z$-coordinate and $z$-coordinate with partial steps cases. |
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215 | |
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216 | \end{description} |
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217 | |
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218 | This implementation is a rather old one. |
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219 | It is similar to the one proposed by \citet{cox_OM87}, except for the background horizontal diffusion. |
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220 | Indeed, the \citet{cox_OM87} implementation of isopycnal diffusion in GFDL-type models requires |
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221 | a minimum background horizontal diffusion for numerical stability reasons. |
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222 | To overcome this problem, several techniques have been proposed in which the numerical schemes of |
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223 | the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. |
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224 | Griffies's scheme is now available in \NEMO\ if \np{ln_traldf_triad}{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:TRIADS}. |
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225 | Here, another strategy is presented \citep{lazar_phd97}: |
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226 | a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of |
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227 | grid point noise generated by the iso-neutral diffusion operator (\autoref{fig:LDF_ZDF1}). |
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228 | This allows an iso-neutral diffusion scheme without additional background horizontal mixing. |
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229 | This technique can be viewed as a diffusion operator that acts along large-scale |
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230 | (2~$\Delta$x) \gmcomment{2deltax doesnt seem very large scale} iso-neutral surfaces. |
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231 | The diapycnal diffusion required for numerical stability is thus minimized and its net effect on the flow is quite small when compared to the effect of an horizontal background mixing. |
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232 | |
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233 | Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, |
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234 | contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. |
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235 | |
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236 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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237 | \begin{figure}[!ht] |
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238 | \centering |
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239 | \includegraphics[width=0.66\textwidth]{Fig_LDF_ZDF1} |
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240 | \caption{Averaging procedure for isopycnal slope computation} |
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241 | \label{fig:LDF_ZDF1} |
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242 | \end{figure} |
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243 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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244 | |
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245 | %There are three additional questions about the slope calculation. |
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246 | %First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes. |
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247 | %Second, numerical stability issues also require a bound on slopes. |
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248 | %Third, the question of boundary condition specified on slopes... |
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249 | |
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250 | %from griffies: chapter 13.1.... |
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251 | |
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252 | |
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253 | |
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254 | % In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04}, |
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255 | % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly |
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256 | % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the |
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257 | % surface motivates this flattening of isopycnals near the surface). |
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258 | |
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259 | For numerical stability reasons \citep{cox_OM87, griffies_bk04}, the slopes must also be bounded by |
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260 | the namelist scalar \np{rn_slpmax}{rn\_slpmax} (usually $1/100$) everywhere. |
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261 | This constraint is applied in a piecewise linear fashion, increasing from zero at the surface to |
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262 | $1/100$ at $70$ metres and thereafter decreasing to zero at the bottom of the ocean |
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263 | (the fact that the eddies "feel" the surface motivates this flattening of isopycnals near the surface). |
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264 | \colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.} |
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265 | |
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266 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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267 | \begin{figure}[!ht] |
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268 | \centering |
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269 | \includegraphics[width=0.66\textwidth]{Fig_eiv_slp} |
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270 | \caption[Vertical profile of the slope used for lateral mixing in the mixed layer]{ |
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271 | Vertical profile of the slope used for lateral mixing in the mixed layer: |
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272 | \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, |
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273 | which has to be adjusted at the surface boundary |
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274 | \ie\ it must tend to zero at the surface since there is no mixing across the air-sea interface: |
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275 | wall boundary condition). |
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276 | Nevertheless, |
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277 | the profile between the surface zero value and the interior iso-neutral one is unknown, |
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278 | and especially the value at the base of the mixed layer; |
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279 | \textit{(b)} profile of slope using a linear tapering of the slope near the surface and |
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280 | imposing a maximum slope of 1/100; |
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281 | \textit{(c)} profile of slope actually used in \NEMO: |
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282 | a linear decrease of the slope from zero at the surface to |
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283 | its ocean interior value computed just below the mixed layer. |
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284 | Note the huge change in the slope at the base of the mixed layer between |
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285 | \textit{(b)} and \textit{(c)}.} |
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286 | \label{fig:LDF_eiv_slp} |
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287 | \end{figure} |
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288 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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289 | |
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290 | \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} |
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291 | |
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292 | \subsection{Slopes for momentum iso-neutral mixing} |
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293 | |
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294 | The iso-neutral diffusion operator on momentum is the same as the one used on tracers but |
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295 | applied to each component of the velocity separately |
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296 | (see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). |
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297 | The slopes between the surface along which the diffusion operator acts and the surface of computation |
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298 | ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and |
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299 | \textit{vw}- points for the $v$-component. |
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300 | They are computed from the slopes used for tracer diffusion, |
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301 | \ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: |
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302 | |
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303 | \[ |
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304 | % \label{eq:LDF_slp_dyn} |
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305 | \begin{aligned} |
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306 | &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ |
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307 | &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ |
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308 | &r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ |
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309 | &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\ |
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310 | \end{aligned} |
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311 | \] |
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312 | |
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313 | The major issue remaining is in the specification of the boundary conditions. |
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314 | The same boundary conditions are chosen as those used for lateral diffusion along model level surfaces, |
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315 | \ie\ using the shear computed along the model levels and with no additional friction at the ocean bottom |
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316 | (see \autoref{sec:LBC_coast}). |
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317 | |
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318 | |
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319 | % ================================================================ |
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320 | % Lateral Mixing Coefficients |
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321 | % ================================================================ |
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322 | \section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t} \& \forcode{nn_ahm_ijk_t})]{Lateral mixing coefficient (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} |
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323 | \label{sec:LDF_coef} |
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324 | |
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325 | The specification of the space variation of the coefficient is made in modules \mdl{ldftra} and \mdl{ldfdyn}. |
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326 | The way the mixing coefficients are set in the reference version can be described as follows: |
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327 | |
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328 | \subsection[Mixing coefficients read from file (\forcode{=-20, -30})]{ Mixing coefficients read from file (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=-20, -30} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=-20, -30})} |
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329 | |
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330 | Mixing coefficients can be read from file if a particular geographical variation is needed. For example, in the ORCA2 global ocean model, |
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331 | the laplacian viscosity operator uses $A^l$~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ north and south and |
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332 | decreases linearly to $A^l$~= 2.10$^3$ m$^2$/s at the equator \citep{madec.delecluse.ea_JPO96, delecluse.madec_icol99}. |
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333 | Similar modified horizontal variations can be found with the Antarctic or Arctic sub-domain options of ORCA2 and ORCA05. |
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334 | The provided fields can either be 2d (\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=-20}, \np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=-20}) or 3d (\np{nn\_aht\_ijk\_t}\forcode{=-30}, \np{nn\_ahm\_ijk\_t}\forcode{=-30}). They must be given at U, V points for tracers and T, F points for momentum (see \autoref{tab:LDF_files}). |
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335 | |
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336 | %-------------------------------------------------TABLE--------------------------------------------------- |
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337 | \begin{table}[htb] |
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338 | \centering |
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339 | \begin{tabular}{|l|l|l|l|} |
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340 | \hline |
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341 | Namelist parameter & Input filename & dimensions & variable names \\ \hline |
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342 | \np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=-20} & \forcode{eddy_viscosity_2D.nc } & $(i,j)$ & \forcode{ahmt_2d, ahmf_2d} \\ \hline |
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343 | \np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=-20} & \forcode{eddy_diffusivity_2D.nc } & $(i,j)$ & \forcode{ahtu_2d, ahtv_2d} \\ \hline |
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344 | \np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=-30} & \forcode{eddy_viscosity_3D.nc } & $(i,j,k)$ & \forcode{ahmt_3d, ahmf_3d} \\ \hline |
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345 | \np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=-30} & \forcode{eddy_diffusivity_3D.nc } & $(i,j,k)$ & \forcode{ahtu_3d, ahtv_3d} \\ \hline |
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346 | \end{tabular} |
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347 | \caption{Description of expected input files if mixing coefficients are read from NetCDF files} |
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348 | \label{tab:LDF_files} |
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349 | \end{table} |
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350 | %-------------------------------------------------------------------------------------------------------------- |
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351 | |
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352 | \subsection[Constant mixing coefficients (\forcode{=0})]{ Constant mixing coefficients (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=0} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=0})} |
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353 | |
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354 | If constant, mixing coefficients are set thanks to a velocity and a length scales ($U_{scl}$, $L_{scl}$) such that: |
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355 | |
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356 | \begin{equation} |
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357 | \label{eq:LDF_constantah} |
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358 | A_o^l = \left\{ |
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359 | \begin{aligned} |
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360 | & \frac{1}{2} U_{scl} L_{scl} & \text{for laplacian operator } \\ |
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361 | & \frac{1}{12} U_{scl} L_{scl}^3 & \text{for bilaplacian operator } |
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362 | \end{aligned} |
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363 | \right. |
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364 | \end{equation} |
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365 | |
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366 | $U_{scl}$ and $L_{scl}$ are given by the namelist parameters \np{rn_Ud}{rn\_Ud}, \np{rn_Uv}{rn\_Uv}, \np{rn_Ld}{rn\_Ld} and \np{rn_Lv}{rn\_Lv}. |
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367 | |
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368 | \subsection[Vertically varying mixing coefficients (\forcode{=10})]{Vertically varying mixing coefficients (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=10} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=10})} |
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369 | |
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370 | In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which |
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371 | the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, |
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372 | and the transition takes place around z=500~m with a width of 200~m. |
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373 | This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. |
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374 | |
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375 | \subsection[Mesh size dependent mixing coefficients (\forcode{=20})]{Mesh size dependent mixing coefficients (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=20} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=20})} |
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376 | |
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377 | In that case, the horizontal variation of the eddy coefficient depends on the local mesh size and |
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378 | the type of operator used: |
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379 | \begin{equation} |
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380 | \label{eq:LDF_title} |
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381 | A_l = \left\{ |
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382 | \begin{aligned} |
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383 | & \frac{1}{2} U_{scl} \max(e_1,e_2) & \text{for laplacian operator } \\ |
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384 | & \frac{1}{12} U_{scl} \max(e_1,e_2)^{3} & \text{for bilaplacian operator } |
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385 | \end{aligned} |
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386 | \right. |
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387 | \end{equation} |
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388 | where $U_{scl}$ is a user defined velocity scale (\np{rn_Ud}{rn\_Ud}, \np{rn_Uv}{rn\_Uv}). |
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389 | This variation is intended to reflect the lesser need for subgrid scale eddy mixing where |
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390 | the grid size is smaller in the domain. |
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391 | It was introduced in the context of the DYNAMO modelling project \citep{willebrand.barnier.ea_PO01}. |
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392 | Note that such a grid scale dependance of mixing coefficients significantly increases the range of stability of |
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393 | model configurations presenting large changes in grid spacing such as global ocean models. |
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394 | Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to |
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395 | large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}), |
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396 | especially when using a bilaplacian operator. |
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397 | |
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398 | \colorbox{yellow}{CASE \np{nn_aht_ijk_t}{nn\_aht\_ijk\_t} = 21 to be added} |
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399 | |
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400 | \subsection[Mesh size and depth dependent mixing coefficients (\forcode{=30})]{Mesh size and depth dependent mixing coefficients (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=30} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=30})} |
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401 | |
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402 | The 3D space variation of the mixing coefficient is simply the combination of the 1D and 2D cases above, |
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403 | \ie\ a hyperbolic tangent variation with depth associated with a grid size dependence of |
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404 | the magnitude of the coefficient. |
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405 | |
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406 | \subsection[Velocity dependent mixing coefficients (\forcode{=31})]{Flow dependent mixing coefficients (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t}\forcode{=31} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=31})} |
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407 | In that case, the eddy coefficient is proportional to the local velocity magnitude so that the Reynolds number $Re = \lvert U \rvert e / A_l$ is constant (and here hardcoded to $12$): |
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408 | \colorbox{yellow}{JC comment: The Reynolds is effectively set to 12 in the code for both operators but shouldn't it be 2 for Laplacian ?} |
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409 | |
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410 | \begin{equation} |
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411 | \label{eq:LDF_flowah} |
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412 | A_l = \left\{ |
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413 | \begin{aligned} |
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414 | & \frac{1}{12} \lvert U \rvert e & \text{for laplacian operator } \\ |
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415 | & \frac{1}{12} \lvert U \rvert e^3 & \text{for bilaplacian operator } |
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416 | \end{aligned} |
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417 | \right. |
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418 | \end{equation} |
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419 | |
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420 | \subsection[Deformation rate dependent viscosities (\forcode{nn_ahm_ijk_t=32})]{Deformation rate dependent viscosities (\protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t}\forcode{=32})} |
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421 | |
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422 | This option refers to the \citep{smagorinsky_MW63} scheme which is here implemented for momentum only. Smagorinsky chose as a |
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423 | characteristic time scale $T_{smag}$ the deformation rate and for the lengthscale $L_{smag}$ the maximum wavenumber possible on the horizontal grid, e.g.: |
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424 | |
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425 | \begin{equation} |
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426 | \label{eq:LDF_smag1} |
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427 | \begin{split} |
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428 | T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2 } \\ |
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429 | L_{smag} & = \frac{1}{\pi}\frac{e_1 e_2}{e_1 + e_2} |
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430 | \end{split} |
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431 | \end{equation} |
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432 | |
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433 | Introducing a user defined constant $C$ (given in the namelist as \np{rn_csmc}{rn\_csmc}), one can deduce the mixing coefficients as follows: |
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434 | |
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435 | \begin{equation} |
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436 | \label{eq:LDF_smag2} |
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437 | A_{smag} = \left\{ |
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438 | \begin{aligned} |
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439 | & C^2 T_{smag}^{-1} L_{smag}^2 & \text{for laplacian operator } \\ |
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440 | & \frac{C^2}{8} T_{smag}^{-1} L_{smag}^4 & \text{for bilaplacian operator } |
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441 | \end{aligned} |
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442 | \right. |
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443 | \end{equation} |
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444 | |
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445 | For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that: |
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446 | \begin{equation} |
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447 | \label{eq:LDF_smag3} |
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448 | \begin{aligned} |
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449 | & C_{min} \frac{1}{2} \lvert U \rvert e < A_{smag} < C_{max} \frac{e^2}{ 8\rdt} & \text{for laplacian operator } \\ |
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450 | & C_{min} \frac{1}{12} \lvert U \rvert e^3 < A_{smag} < C_{max} \frac{e^4}{64 \rdt} & \text{for bilaplacian operator } |
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451 | \end{aligned} |
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452 | \end{equation} |
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453 | |
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454 | where $C_{min}$ and $C_{max}$ are adimensional namelist parameters given by \np{rn_minfac}{rn\_minfac} and \np{rn_maxfac}{rn\_maxfac} respectively. |
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455 | |
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456 | \subsection{About space and time varying mixing coefficients} |
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457 | |
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458 | The following points are relevant when the eddy coefficient varies spatially: |
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459 | |
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460 | (1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and |
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461 | divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). |
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462 | Although the eddy coefficient could be set to different values in these two terms, |
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463 | this option is not currently available. |
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464 | |
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465 | (2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of |
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466 | the horizontal divergence for operators acting along model-surfaces are no longer satisfied |
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467 | (\autoref{sec:INVARIANTS_dynldf_properties}). |
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468 | |
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469 | % ================================================================ |
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470 | % Eddy Induced Mixing |
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471 | % ================================================================ |
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472 | \section[Eddy induced velocity (\forcode{ln_ldfeiv})]{Eddy induced velocity (\protect\np{ln_ldfeiv}{ln\_ldfeiv})} |
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473 | |
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474 | \label{sec:LDF_eiv} |
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475 | |
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476 | %--------------------------------------------namtra_eiv--------------------------------------------------- |
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477 | |
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478 | \begin{listing} |
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479 | \nlst{namtra_eiv} |
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480 | \caption{\forcode{&namtra_eiv}} |
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481 | \label{lst:namtra_eiv} |
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482 | \end{listing} |
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483 | |
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484 | %-------------------------------------------------------------------------------------------------------------- |
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485 | |
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486 | |
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487 | %%gm from Triad appendix : to be incorporated.... |
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488 | \gmcomment{ |
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489 | Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. |
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490 | If none of the keys \key{traldf\_cNd}, N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and |
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491 | GM diffusivity $A_e$ are directly set by \np{rn_aeih_0}{rn\_aeih\_0} and \np{rn_aeiv_0}{rn\_aeiv\_0}. |
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492 | If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal |
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493 | scale factor according to \autoref{eq:title} |
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494 | \footnote{ |
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495 | Except in global ORCA $0.5^{\circ}$ runs with \key{traldf\_eiv}, |
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496 | where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$ |
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497 | }. |
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498 | In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in |
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499 | the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from |
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500 | the Held-Larichev parameterisation |
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501 | \footnote{ |
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502 | In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$, |
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503 | where $f_{20}$ is the value of $f$ at $20^{\circ}$~N |
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504 | } (\mdl{ldfeiv}) and \np{rn_aeiv_0}{rn\_aeiv\_0} is ignored unless it is zero. |
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505 | } |
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506 | |
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507 | When \citet{gent.mcwilliams_JPO90} diffusion is used (\np{ln_ldfeiv}{ln\_ldfeiv}\forcode{=.true.}), |
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508 | an eddy induced tracer advection term is added, |
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509 | the formulation of which depends on the slopes of iso-neutral surfaces. |
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510 | Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, |
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511 | \ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates, |
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512 | and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. |
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513 | |
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514 | If isopycnal mixing is used in the standard way, \ie\ \np{ln_traldf_triad}{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by: |
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515 | \begin{equation} |
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516 | \label{eq:LDF_eiv} |
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517 | \begin{split} |
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518 | u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ |
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519 | v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\ |
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520 | w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + \delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\ |
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521 | \end{split} |
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522 | \end{equation} |
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523 | where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{nn_aei_ijk_t}{nn\_aei\_ijk\_t} \nam{tra_eiv}{tra\_eiv} namelist parameter. |
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524 | The three components of the eddy induced velocity are computed in \rou{ldf\_eiv\_trp} and |
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525 | added to the eulerian velocity in \rou{tra\_adv} where tracer advection is performed. |
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526 | This has been preferred to a separate computation of the advective trends associated with the eiv velocity, |
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527 | since it allows us to take advantage of all the advection schemes offered for the tracers |
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528 | (see \autoref{sec:TRA_adv}) and not just the $2^{nd}$ order advection scheme as in |
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529 | previous releases of OPA \citep{madec.delecluse.ea_NPM98}. |
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530 | This is particularly useful for passive tracers where \emph{positivity} of the advection scheme is of |
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531 | paramount importance. |
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532 | |
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533 | At the surface, lateral and bottom boundaries, the eddy induced velocity, |
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534 | and thus the advective eddy fluxes of heat and salt, are set to zero. |
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535 | The value of the eddy induced mixing coefficient and its space variation is controlled in a similar way as for lateral mixing coefficient described in the preceding subsection (\np{nn_aei_ijk_t}{nn\_aei\_ijk\_t}, \np{rn_Ue}{rn\_Ue}, \np{rn_Le}{rn\_Le} namelist parameters). |
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536 | \colorbox{yellow}{CASE \np{nn_aei_ijk_t}{nn\_aei\_ijk\_t} = 21 to be added} |
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537 | |
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538 | In case of setting \np{ln_traldf_triad}{ln\_traldf\_triad}\forcode{ = .true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}. |
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539 | |
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540 | % ================================================================ |
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541 | % Mixed layer eddies |
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542 | % ================================================================ |
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543 | \section[Mixed layer eddies (\forcode{ln_mle})]{Mixed layer eddies (\protect\np{ln_mle}{ln\_mle})} |
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544 | \label{sec:LDF_mle} |
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545 | |
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546 | %--------------------------------------------namtra_eiv--------------------------------------------------- |
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547 | |
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548 | \begin{listing} |
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549 | \nlst{namtra_mle} |
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550 | \caption{\forcode{&namtra_mle}} |
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551 | \label{lst:namtra_mle} |
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552 | \end{listing} |
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553 | |
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554 | %-------------------------------------------------------------------------------------------------------------- |
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555 | |
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556 | If \np{ln_mle}{ln\_mle}\forcode{=.true.} in \nam{tra_mle}{tra\_mle} namelist, a parameterization of the mixing due to unresolved mixed layer instabilities is activated (\citet{foxkemper.ferrari_JPO08}). Additional transport is computed in \rou{ldf\_mle\_trp} and added to the eulerian transport in \rou{tra\_adv} as done for eddy induced advection. |
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557 | |
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558 | \colorbox{yellow}{TBC} |
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559 | |
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560 | \biblio |
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561 | |
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562 | \pindex |
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563 | |
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564 | \end{document} |
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