1 | |
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2 | % ================================================================ |
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3 | % Chapter Ñ Lateral Ocean Physics (LDF) |
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4 | % ================================================================ |
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5 | \chapter{Lateral Ocean Physics (LDF)} |
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6 | \label{LDF} |
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7 | \minitoc |
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8 | |
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9 | |
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10 | \newpage |
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11 | $\ $\newline % force a new ligne |
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12 | |
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13 | |
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14 | The lateral physics terms in the momentum and tracer equations have been |
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15 | described in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf} |
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16 | and \S\ref{DYN_ldf}). In this section we further discuss each lateral physics option. |
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17 | Choosing one lateral physics scheme means for the user defining, (1) the space |
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18 | and time variations of the eddy coefficients ; (2) the direction along which the |
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19 | lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal |
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20 | surfaces); and (3) the type of operator used (harmonic, or biharmonic operators, |
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21 | and for tracers only, eddy induced advection on tracers). These three aspects |
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22 | of the lateral diffusion are set through namelist parameters and CPP keys |
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23 | (see the \textit{nam\_traldf} and \textit{nam\_dynldf} below). Note |
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24 | that this chapter describes the default implementation of iso-neutral |
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25 | tracer mixing, and Griffies's implementation, which is used if |
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26 | \np{traldf\_grif}=true, is described in Appdx\ref{sec:triad} |
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27 | |
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28 | %-----------------------------------nam_traldf - nam_dynldf-------------------------------------------- |
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29 | \namdisplay{namtra_ldf} |
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30 | \namdisplay{namdyn_ldf} |
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31 | %-------------------------------------------------------------------------------------------------------------- |
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32 | |
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33 | |
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34 | % ================================================================ |
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35 | % Lateral Mixing Coefficients |
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36 | % ================================================================ |
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37 | \section [Lateral Mixing Coefficient (\textit{ldftra}, \textit{ldfdyn})] |
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38 | {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn}) } |
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39 | \label{LDF_coef} |
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40 | |
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41 | |
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42 | Introducing a space variation in the lateral eddy mixing coefficients changes |
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43 | the model core memory requirement, adding up to four extra three-dimensional |
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44 | arrays for the geopotential or isopycnal second order operator applied to |
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45 | momentum. Six CPP keys control the space variation of eddy coefficients: |
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46 | three for momentum and three for tracer. The three choices allow: |
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47 | a space variation in the three space directions (\key{traldf\_c3d}, \key{dynldf\_c3d}), |
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48 | in the horizontal plane (\key{traldf\_c2d}, \key{dynldf\_c2d}), |
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49 | or in the vertical only (\key{traldf\_c1d}, \key{dynldf\_c1d}). |
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50 | The default option is a constant value over the whole ocean on both momentum and tracers. |
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51 | |
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52 | The number of additional arrays that have to be defined and the gridpoint |
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53 | position at which they are defined depend on both the space variation chosen |
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54 | and the type of operator used. The resulting eddy viscosity and diffusivity |
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55 | coefficients can be a function of more than one variable. Changes in the |
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56 | computer code when switching from one option to another have been |
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57 | minimized by introducing the eddy coefficients as statement functions |
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58 | (include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions |
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59 | are replaced by their actual meaning during the preprocessing step (CPP). |
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60 | The specification of the space variation of the coefficient is made in |
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61 | \mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files |
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62 | \textit{traldf\_cNd.h90} and \textit{dynldf\_cNd.h90}, with N=1, 2 or 3. |
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63 | The user can modify these include files as he/she wishes. The way the |
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64 | mixing coefficient are set in the reference version can be briefly described |
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65 | as follows: |
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66 | |
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67 | \subsubsection{Constant Mixing Coefficients (default option)} |
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68 | When none of the \textbf{key\_dynldf\_...} and \textbf{key\_traldf\_...} keys are |
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69 | defined, a constant value is used over the whole ocean for momentum and |
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70 | tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist |
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71 | parameters. |
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72 | |
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73 | \subsubsection{Vertically varying Mixing Coefficients (\key{traldf\_c1d} and \key{dynldf\_c1d})} |
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74 | The 1D option is only available when using the $z$-coordinate with full step. |
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75 | Indeed in all the other types of vertical coordinate, the depth is a 3D function |
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76 | of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent |
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77 | mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation |
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78 | of the lateral mixing coefficient is introduced in which the surface value is |
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79 | \np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value, |
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80 | and the transition takes place around z=300~m with a width of 300~m |
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81 | ($i.e.$ both the depth and the width of the inflection point are set to 300~m). |
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82 | This profile is hard coded in file \hf{traldf\_c1d}, but can be easily modified by users. |
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83 | |
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84 | \subsubsection{Horizontally Varying Mixing Coefficients (\key{traldf\_c2d} and \key{dynldf\_c2d})} |
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85 | By default the horizontal variation of the eddy coefficient depends on the local mesh |
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86 | size and the type of operator used: |
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87 | \begin{equation} \label{Eq_title} |
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88 | A_l = \left\{ |
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89 | \begin{aligned} |
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90 | & \frac{\max(e_1,e_2)}{e_{max}} A_o^l & \text{for laplacian operator } \\ |
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91 | & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l & \text{for bilaplacian operator } |
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92 | \end{aligned} \right. |
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93 | \end{equation} |
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94 | where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked |
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95 | ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) |
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96 | namelist parameter. This variation is intended to reflect the lesser need for subgrid |
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97 | scale eddy mixing where the grid size is smaller in the domain. It was introduced in |
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98 | the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}. |
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99 | Note that such a grid scale dependance of mixing coefficients significantly increase |
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100 | the range of stability of model configurations presenting large changes in grid pacing |
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101 | such as global ocean models. Indeed, in such a case, a constant mixing coefficient |
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102 | can lead to a blow up of the model due to large coefficient compare to the smallest |
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103 | grid size (see \S\ref{STP_forward_imp}), especially when using a bilaplacian operator. |
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104 | |
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105 | Other formulations can be introduced by the user for a given configuration. |
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106 | For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian |
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107 | viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ |
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108 | north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s |
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109 | at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification |
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110 | can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. |
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111 | Similar modified horizontal variations can be found with the Antarctic or Arctic |
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112 | sub-domain options of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} |
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113 | defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}). |
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114 | |
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115 | \subsubsection{Space Varying Mixing Coefficients (\key{traldf\_c3d} and \key{dynldf\_c3d})} |
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116 | |
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117 | The 3D space variation of the mixing coefficient is simply the combination of the |
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118 | 1D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with |
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119 | a grid size dependence of the magnitude of the coefficient. |
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120 | |
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121 | \subsubsection{Space and Time Varying Mixing Coefficients} |
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122 | |
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123 | There is no default specification of space and time varying mixing coefficient. |
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124 | The only case available is specific to the ORCA2 and ORCA05 global ocean |
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125 | configurations (\key{orca\_r2} or \key{orca\_r05}). It provides only a tracer |
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126 | mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and |
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127 | eddy induced velocity (ORCA05) that depends on the local growth rate of |
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128 | baroclinic instability. This specification is actually used when an ORCA key |
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129 | and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. |
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130 | |
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131 | $\ $\newline % force a new ligne |
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132 | |
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133 | The following points are relevant when the eddy coefficient varies spatially: |
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134 | |
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135 | (1) the momentum diffusion operator acting along model level surfaces is |
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136 | written in terms of curl and divergent components of the horizontal current |
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137 | (see \S\ref{PE_ldf}). Although the eddy coefficient could be set to different values |
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138 | in these two terms, this option is not currently available. |
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139 | |
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140 | (2) with an horizontally varying viscosity, the quadratic integral constraints |
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141 | on enstrophy and on the square of the horizontal divergence for operators |
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142 | acting along model-surfaces are no longer satisfied |
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143 | (Appendix~\ref{Apdx_dynldf_properties}). |
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144 | |
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145 | (3) for isopycnal diffusion on momentum or tracers, an additional purely |
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146 | horizontal background diffusion with uniform coefficient can be added by |
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147 | setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal |
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148 | eddy viscosity or diffusivity coefficient (namelist parameters whose default |
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149 | values are $0$). However, the technique used to compute the isopycnal |
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150 | slopes is intended to get rid of such a background diffusion, since it introduces |
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151 | spurious diapycnal diffusion (see {\S\ref{LDF_slp}). |
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152 | |
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153 | (4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$, |
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154 | the eddy induced coefficient has to be defined. Its space variations are controlled |
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155 | by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$ |
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156 | \textbf{key\_traldf\_cNd}). |
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157 | |
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158 | (5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value. |
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159 | |
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160 | (6) it is possible to use both the laplacian and biharmonic operators concurrently. |
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161 | |
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162 | (7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap} = |
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163 | \np{ln\_dynldf\_bilap} = false). This is recommended when using the UBS advection |
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164 | scheme on momentum (\np{ln\_dynadv\_ubs} = true, see \ref{DYN_adv_ubs}) |
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165 | and can be useful for testing purposes. |
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166 | |
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167 | % ================================================================ |
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168 | % Direction of lateral Mixing |
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169 | % ================================================================ |
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170 | \section [Direction of Lateral Mixing (\textit{ldfslp})] |
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171 | {Direction of Lateral Mixing (\mdl{ldfslp})} |
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172 | \label{LDF_slp} |
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173 | |
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174 | %%% |
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175 | \gmcomment{ we should emphasize here that the implementation is a rather old one. |
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176 | Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. } |
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177 | |
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178 | A direction for lateral mixing has to be defined when the desired operator does |
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179 | not act along the model levels. This occurs when $(a)$ horizontal mixing is |
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180 | required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor}) |
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181 | in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required |
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182 | whatever the vertical coordinate is. This direction of mixing is defined by its |
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183 | slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the |
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184 | quantity to be diffused. For a tracer, this leads to the following four slopes : |
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185 | $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while |
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186 | for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for |
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187 | $u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. |
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188 | |
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189 | %gm% add here afigure of the slope in i-direction |
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190 | |
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191 | \subsection{slopes for tracer geopotential mixing in the $s$-coordinate} |
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192 | |
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193 | In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and |
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194 | $r_2$ are the slopes between the geopotential and computational surfaces. |
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195 | Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso} |
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196 | when the diffusive fluxes in the three directions are set to zero and $T$ is |
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197 | assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the |
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198 | depth of a $T$-point. |
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199 | %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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200 | |
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201 | \begin{equation} \label{Eq_ldfslp_geo} |
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202 | \begin{aligned} |
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203 | r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} |
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204 | \;\delta_{i+1/2}[z_t] |
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205 | &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] |
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206 | \\ |
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207 | r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} |
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208 | \;\delta_{j+1/2} [z_t] |
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209 | &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] |
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210 | \\ |
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211 | r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} |
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212 | &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] |
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213 | \\ |
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214 | r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} |
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215 | &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] |
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216 | \\ |
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217 | \end{aligned} |
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218 | \end{equation} |
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219 | |
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220 | %gm% caution I'm not sure the simplification was a good idea! |
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221 | |
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222 | These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True, |
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223 | and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True. |
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224 | |
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225 | \subsection{Slopes for tracer iso-neutral mixing}\label{LDF_slp_iso} |
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226 | In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral |
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227 | and computational surfaces. Their formulation does not depend on the vertical |
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228 | coordinate used. Their discrete formulation is found using the fact that the |
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229 | diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density) |
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230 | vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the |
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231 | diffusive fluxes in the three directions to zero leads to the following definition for |
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232 | the neutral slopes: |
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233 | |
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234 | \begin{equation} \label{Eq_ldfslp_iso} |
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235 | \begin{split} |
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236 | r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} |
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237 | {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} |
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238 | \\ |
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239 | r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]} |
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240 | {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} |
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241 | \\ |
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242 | r_{1w} &= \frac{e_{3w}}{e_{1w}}\; |
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243 | \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}} |
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244 | {\delta_{k+1/2}[\rho]} |
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245 | \\ |
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246 | r_{2w} &= \frac{e_{3w}}{e_{2w}}\; |
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247 | \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}} |
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248 | {\delta_{k+1/2}[\rho]} |
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249 | \\ |
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250 | \end{split} |
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251 | \end{equation} |
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252 | |
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253 | %gm% rewrite this as the explanation is not very clear !!! |
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254 | %In practice, \eqref{Eq_ldfslp_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 \eqref{Eq_ldfslp_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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255 | |
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256 | %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{McDougall1987}, therefore in \eqref{Eq_ldfslp_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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257 | |
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258 | %In the $z$-coordinate, the derivative of the \eqref{Eq_ldfslp_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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259 | |
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260 | As the mixing is performed along neutral surfaces, the gradient of $\rho$ in |
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261 | \eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which, |
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262 | in decibars, is approximated by the depth in meters in the model). Therefore |
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263 | \eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is |
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264 | needed depending on the vertical coordinate used: |
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265 | |
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266 | \begin{description} |
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267 | |
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268 | \item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities |
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269 | appearing in the $i$ and $j$ derivatives are taken at the same depth, thus |
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270 | the $in situ$ density can be used. This is not the case for the vertical |
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271 | derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$ |
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272 | is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following |
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273 | \citet{McDougall1987} (see \S\ref{TRA_bn2}). |
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274 | |
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275 | \item[$z$-coordinate with partial step : ] this case is identical to the full step |
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276 | case except that at partial step level, the \emph{horizontal} density gradient |
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277 | is evaluated as described in \S\ref{TRA_zpshde}. |
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278 | |
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279 | \item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO, |
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280 | iso-neutral mixing is only employed for $s$-coordinates if the |
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281 | Griffies scheme is used (\np{traldf\_grif}=true; see Appdx \ref{sec:triad}). |
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282 | In other words, iso-neutral mixing will only be accurately represented with a |
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283 | linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation |
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284 | of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} |
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285 | will include a pressure dependent part, leading to the wrong evaluation of |
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286 | the neutral slopes. |
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287 | |
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288 | %gm% |
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289 | Note: The solution for $s$-coordinate passes trough the use of different |
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290 | (and better) expression for the constraint on iso-neutral fluxes. Following |
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291 | \citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral |
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292 | diffusive flux of locally referenced potential density, we stay in the $T$-$S$ |
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293 | plane and consider the balance between the neutral direction diffusive fluxes |
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294 | of potential temperature and salinity: |
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295 | \begin{equation} |
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296 | \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) |
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297 | \end{equation} |
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298 | %gm{ where vector F is ....} |
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299 | |
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300 | This constraint leads to the following definition for the slopes: |
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301 | |
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302 | \begin{equation} \label{Eq_ldfslp_iso2} |
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303 | \begin{split} |
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304 | r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac |
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305 | {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]} |
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306 | {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k} |
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307 | -\beta_u \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } |
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308 | \\ |
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309 | r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac |
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310 | {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]} |
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311 | {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k} |
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312 | -\beta_v \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} } |
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313 | \\ |
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314 | r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac |
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315 | {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2} |
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316 | -\beta_w \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} } |
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317 | {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} |
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318 | \\ |
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319 | r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac |
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320 | {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2} |
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321 | -\beta_w \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} } |
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322 | {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} |
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323 | \\ |
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324 | \end{split} |
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325 | \end{equation} |
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326 | where $\alpha$ and $\beta$, the thermal expansion and saline contraction |
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327 | coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three |
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328 | velocity points. In order to save computation time, they should be approximated |
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329 | by the mean of their values at $T$-points (for example in the case of $\alpha$: |
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330 | $\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$ |
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331 | and $\alpha_w=\overline{\alpha_T}^{k+1/2}$). |
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332 | |
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333 | Note that such a formulation could be also used in the $z$-coordinate and |
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334 | $z$-coordinate with partial steps cases. |
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335 | |
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336 | \end{description} |
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337 | |
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338 | This implementation is a rather old one. It is similar to the one |
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339 | proposed by Cox [1987], except for the background horizontal |
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340 | diffusion. Indeed, the Cox implementation of isopycnal diffusion in |
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341 | GFDL-type models requires a minimum background horizontal diffusion |
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342 | for numerical stability reasons. To overcome this problem, several |
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343 | techniques have been proposed in which the numerical schemes of the |
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344 | ocean model are modified \citep{Weaver_Eby_JPO97, |
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345 | Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if |
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346 | \np{traldf\_grif\_iso} is set true; see Appdx \ref{sec:triad}. Here, |
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347 | another strategy is presented \citep{Lazar_PhD97}: a local |
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348 | filtering of the iso-neutral slopes (made on 9 grid-points) prevents |
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349 | the development of grid point noise generated by the iso-neutral |
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350 | diffusion operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an |
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351 | iso-neutral diffusion scheme without additional background horizontal |
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352 | mixing. This technique can be viewed as a diffusion operator that acts |
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353 | along large-scale (2~$\Delta$x) \gmcomment{2deltax doesnt seem very |
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354 | large scale} iso-neutral surfaces. The diapycnal diffusion required |
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355 | for numerical stability is thus minimized and its net effect on the |
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356 | flow is quite small when compared to the effect of an horizontal |
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357 | background mixing. |
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358 | |
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359 | Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, |
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360 | contrary to the \citet{Griffies_al_JPO98} operator which has that property. |
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361 | |
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362 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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363 | \begin{figure}[!ht] \begin{center} |
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364 | \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_LDF_ZDF1.pdf} |
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365 | \caption { \label{Fig_LDF_ZDF1} |
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366 | averaging procedure for isopycnal slope computation.} |
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367 | \end{center} \end{figure} |
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368 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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369 | |
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370 | %There are three additional questions about the slope calculation. |
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371 | %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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372 | %Second, numerical stability issues also require a bound on slopes. |
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373 | %Third, the question of boundary condition specified on slopes... |
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374 | |
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375 | %from griffies: chapter 13.1.... |
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376 | |
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377 | |
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378 | |
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379 | % In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04}, |
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380 | % the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly |
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381 | % to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the |
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382 | % surface motivates this flattening of isopycnals near the surface). |
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383 | |
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384 | For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also |
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385 | be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear |
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386 | fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter |
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387 | decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the |
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388 | surface motivates this flattening of isopycnals near the surface). |
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389 | |
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390 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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391 | \begin{figure}[!ht] \begin{center} |
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392 | \includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_eiv_slp.pdf} |
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393 | \caption { \label{Fig_eiv_slp} |
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394 | Vertical profile of the slope used for lateral mixing in the mixed layer : |
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395 | \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, |
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396 | which has to be adjusted at the surface boundary (i.e. it must tend to zero at the |
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397 | surface since there is no mixing across the air-sea interface: wall boundary |
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398 | condition). Nevertheless, the profile between the surface zero value and the interior |
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399 | iso-neutral one is unknown, and especially the value at the base of the mixed layer ; |
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400 | \textit{(b)} profile of slope using a linear tapering of the slope near the surface and |
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401 | imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in |
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402 | \NEMO: a linear decrease of the slope from zero at the surface to its ocean interior |
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403 | value computed just below the mixed layer. Note the huge change in the slope at the |
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404 | base of the mixed layer between \textit{(b)} and \textit{(c)}.} |
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405 | \end{center} \end{figure} |
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406 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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407 | |
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408 | \colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} |
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409 | |
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410 | \subsection{slopes for momentum iso-neutral mixing} |
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411 | |
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412 | The iso-neutral diffusion operator on momentum is the same as the one used on |
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413 | tracers but applied to each component of the velocity separately (see |
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414 | \eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the |
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415 | surface along which the diffusion operator acts and the surface of computation |
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416 | ($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the |
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417 | $u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component. |
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418 | They are computed from the slopes used for tracer diffusion, $i.e.$ |
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419 | \eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} : |
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420 | |
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421 | \begin{equation} \label{Eq_ldfslp_dyn} |
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422 | \begin{aligned} |
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423 | &r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ |
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424 | &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ |
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425 | &r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ |
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426 | &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\ |
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427 | \end{aligned} |
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428 | \end{equation} |
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429 | |
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430 | The major issue remaining is in the specification of the boundary conditions. |
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431 | The same boundary conditions are chosen as those used for lateral |
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432 | diffusion along model level surfaces, i.e. using the shear computed along |
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433 | the model levels and with no additional friction at the ocean bottom (see |
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434 | {\S\ref{LBC_coast}). |
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435 | |
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436 | |
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437 | % ================================================================ |
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438 | % Eddy Induced Mixing |
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439 | % ================================================================ |
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440 | \section [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})] |
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441 | {Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})} |
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442 | \label{LDF_eiv} |
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443 | |
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444 | When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), |
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445 | an eddy induced tracer advection term is added, the formulation of which |
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446 | depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral |
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447 | mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$ |
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448 | \eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo} |
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449 | + \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by: |
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450 | \begin{equation} \label{Eq_ldfeiv} |
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451 | \begin{split} |
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452 | 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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453 | 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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454 | 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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455 | \end{split} |
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456 | \end{equation} |
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457 | where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set |
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458 | through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter. |
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459 | The three components of the eddy induced velocity are computed and add |
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460 | to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a |
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461 | separate computation of the advective trends associated with the eiv velocity, |
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462 | since it allows us to take advantage of all the advection schemes offered for |
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463 | the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection |
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464 | scheme as in previous releases of OPA \citep{Madec1998}. This is particularly |
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465 | useful for passive tracers where \emph{positivity} of the advection scheme is |
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466 | of paramount importance. |
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467 | |
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468 | At the surface, lateral and bottom boundaries, the eddy induced velocity, |
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469 | and thus the advective eddy fluxes of heat and salt, are set to zero. |
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470 | |
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471 | |
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472 | |
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473 | |
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474 | |
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