1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | %% Local cmds |
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4 | \newcommand{\rML}[1][i]{\ensuremath{_{\mathrm{ML}\,#1}}} |
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5 | \newcommand{\rMLt}[1][i]{\tilde{r}_{\mathrm{ML}\,#1}} |
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6 | %% Move to ../../global/new_cmds.tex to avoid error with \listoffigures |
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7 | %\newcommand{\triad}[6][]{\ensuremath{{}_{#2}^{#3}{\mathbb{#4}_{#1}}_{#5}^{\,#6}} |
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8 | \newcommand{\triadd}[5]{\ensuremath{{}_{#1}^{#2}{\mathbb{#3}}_{#4}^{\,#5}}} |
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9 | \newcommand{\triadt}[5]{\ensuremath{{}_{#1}^{#2}{\tilde{\mathbb{#3}}}_{#4}^{\,#5}}} |
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10 | \newcommand{\rtriad}[2][]{\ensuremath{\triad[#1]{i}{k}{#2}{i_p}{k_p}}} |
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11 | \newcommand{\rtriadt}[1]{\ensuremath{\triadt{i}{k}{#1}{i_p}{k_p}}} |
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12 | |
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13 | \begin{document} |
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14 | % ================================================================ |
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15 | % Iso-neutral diffusion : |
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16 | % ================================================================ |
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17 | \chapter{Iso-Neutral Diffusion and Eddy Advection using Triads} |
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18 | \label{apdx:triad} |
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19 | |
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20 | \minitoc |
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21 | |
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22 | \newpage |
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23 | |
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24 | \section{Choice of \protect\ngn{namtra\_ldf} namelist parameters} |
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25 | %-----------------------------------------nam_traldf------------------------------------------------------ |
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26 | |
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27 | \nlst{namtra_ldf} |
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28 | %--------------------------------------------------------------------------------------------------------- |
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29 | |
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30 | Two scheme are available to perform the iso-neutral diffusion. |
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31 | If the namelist logical \np{ln\_traldf\_triad} is set true, |
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32 | \NEMO updates both active and passive tracers using the Griffies triad representation of iso-neutral diffusion and |
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33 | the eddy-induced advective skew (GM) fluxes. |
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34 | If the namelist logical \np{ln\_traldf\_iso} is set true, |
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35 | the filtered version of Cox's original scheme (the Standard scheme) is employed (\autoref{sec:LDF_slp}). |
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36 | In the present implementation of the Griffies scheme, |
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37 | the advective skew fluxes are implemented even if \np{ln\_traldf\_eiv} is false. |
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38 | |
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39 | Values of iso-neutral diffusivity and GM coefficient are set as described in \autoref{sec:LDF_coef}. |
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40 | Note that when GM fluxes are used, the eddy-advective (GM) velocities are output for diagnostic purposes using xIOS, |
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41 | even though the eddy advection is accomplished by means of the skew fluxes. |
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42 | |
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43 | The options specific to the Griffies scheme include: |
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44 | \begin{description} |
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45 | \item[\np{ln\_triad\_iso}] |
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46 | See \autoref{sec:taper}. |
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47 | If this is set false (the default), |
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48 | then `iso-neutral' mixing is accomplished within the surface mixed-layer along slopes linearly decreasing with |
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49 | depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec:lintaper}). |
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50 | This is the same treatment as used in the default implementation |
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51 | \autoref{subsec:LDF_slp_iso}; \autoref{fig:eiv_slp}. |
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52 | Where \np{ln\_triad\_iso} is set true, |
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53 | the vertical skew flux is further reduced to ensure no vertical buoyancy flux, |
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54 | giving an almost pure horizontal diffusive tracer flux within the mixed layer. |
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55 | This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec:Gerdes-taper} |
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56 | \item[\np{ln\_botmix\_triad}] |
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57 | See \autoref{sec:iso_bdry}. |
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58 | If this is set false (the default) then the lateral diffusive fluxes |
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59 | associated with triads partly masked by topography are neglected. |
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60 | If it is set true, however, then these lateral diffusive fluxes are applied, |
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61 | giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. |
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62 | \item[\np{rn\_sw\_triad}] |
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63 | blah blah to be added.... |
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64 | \end{description} |
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65 | The options shared with the Standard scheme include: |
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66 | \begin{description} |
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67 | \item[\np{ln\_traldf\_msc}] blah blah to be added |
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68 | \item[\np{rn\_slpmax}] blah blah to be added |
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69 | \end{description} |
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70 | |
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71 | \section{Triad formulation of iso-neutral diffusion} |
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72 | \label{sec:iso} |
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73 | |
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74 | We have implemented into \NEMO a scheme inspired by \citet{griffies.gnanadesikan.ea_JPO98}, |
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75 | but formulated within the \NEMO framework, using scale factors rather than grid-sizes. |
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76 | |
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77 | \subsection{Iso-neutral diffusion operator} |
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78 | |
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79 | The iso-neutral second order tracer diffusive operator for small angles between |
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80 | iso-neutral surfaces and geopotentials is given by \autoref{eq:iso_tensor_1}: |
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81 | \begin{subequations} |
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82 | \label{eq:iso_tensor_1} |
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83 | \begin{equation} |
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84 | D^{lT}=-\nabla \cdot\vect{f}^{lT}\equiv |
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85 | -\frac{1}{e_1e_2e_3}\left[\pd{i}\left (f_1^{lT}e_2e_3\right) + |
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86 | \pd{j}\left (f_2^{lT}e_2e_3\right) + \pd{k}\left (f_3^{lT}e_1e_2\right)\right], |
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87 | \end{equation} |
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88 | where the diffusive flux per unit area of physical space |
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89 | \begin{equation} |
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90 | \vect{f}^{lT}=-{A^{lT}}\Re\cdot\nabla T, |
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91 | \end{equation} |
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92 | \begin{equation} |
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93 | \label{eq:iso_tensor_2} |
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94 | \mbox{with}\quad \;\;\Re = |
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95 | \begin{pmatrix} |
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96 | 1 & 0 & -r_1 \rule[-.9 em]{0pt}{1.79 em} \\ |
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97 | 0 & 1 & -r_2 \rule[-.9 em]{0pt}{1.79 em} \\ |
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98 | -r_1 & -r_2 & r_1 ^2+r_2 ^2 \rule[-.9 em]{0pt}{1.79 em} |
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99 | \end{pmatrix} |
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100 | \quad \text{and} \quad\nabla T= |
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101 | \begin{pmatrix} |
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102 | \frac{1}{e_1} \pd[T]{i} \rule[-.9 em]{0pt}{1.79 em} \\ |
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103 | \frac{1}{e_2} \pd[T]{j} \rule[-.9 em]{0pt}{1.79 em} \\ |
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104 | \frac{1}{e_3} \pd[T]{k} \rule[-.9 em]{0pt}{1.79 em} |
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105 | \end{pmatrix} |
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106 | . |
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107 | \end{equation} |
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108 | \end{subequations} |
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109 | % \left( {{\begin{array}{*{20}c} |
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110 | % 1 \hfill & 0 \hfill & {-r_1 } \hfill \\ |
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111 | % 0 \hfill & 1 \hfill & {-r_2 } \hfill \\ |
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112 | % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ |
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113 | % \end{array} }} \right) |
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114 | Here \autoref{eq:PE_iso_slopes} |
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115 | \begin{align*} |
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116 | r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} |
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117 | \right) |
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118 | \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\ |
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119 | &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} + |
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120 | \beta\frac{\partial S }{\partial i} \right) \left( |
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121 | -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S |
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122 | }{\partial k} \right)^{-1} |
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123 | \end{align*} |
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124 | is the $i$-component of the slope of the iso-neutral surface relative to the computational surface, |
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125 | and $r_2$ is the $j$-component. |
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126 | |
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127 | We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write |
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128 | \[ |
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129 | % \label{eq:Fijk} |
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130 | \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right). |
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131 | \] |
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132 | Additionally, we will sometimes write the contributions towards the fluxes $\vect{f}$ and |
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133 | $\vect{F}_{\mathrm{iso}}$ from the component $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, |
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134 | with $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc. |
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135 | |
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136 | The off-diagonal terms of the small angle diffusion tensor |
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137 | \autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along |
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138 | the $i$- and $j$-directions resulting from the vertical tracer gradient: |
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139 | \begin{align} |
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140 | \label{eq:i13c} |
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141 | f_{13}=&+{A^{lT}} r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+{A^{lT}} r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\ |
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142 | \intertext{and in the k-direction resulting from the lateral tracer gradients} |
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143 | \label{eq:i31c} |
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144 | f_{31}+f_{32}=& {A^{lT}} r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+{A^{lT}} r_2\frac{1}{e_1}\frac{\partial T}{\partial i} |
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145 | \end{align} |
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146 | |
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147 | The vertical diffusive flux associated with the $_{33}$ component of the small angle diffusion tensor is |
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148 | \begin{equation} |
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149 | \label{eq:i33c} |
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150 | f_{33}=-{A^{lT}}(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}. |
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151 | \end{equation} |
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152 | |
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153 | Since there are no cross terms involving $r_1$ and $r_2$ in the above, |
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154 | we can consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$ planes, |
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155 | just adding together the vertical components from each plane. |
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156 | The following description will describe the fluxes on the $i$-$k$ plane. |
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157 | |
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158 | There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq:i13c}, |
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159 | as although it must be evaluated at $u$-points, |
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160 | it involves vertical gradients (both for the tracer and the slope $r_1$), defined at $w$-points. |
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161 | Similarly, the vertical skew flux, \autoref{eq:i31c}, |
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162 | is evaluated at $w$-points but involves horizontal gradients defined at $u$-points. |
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163 | |
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164 | \subsection{Standard discretization} |
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165 | |
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166 | The straightforward approach to discretize the lateral skew flux |
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167 | \autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA, |
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168 | \autoref{eq:tra_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from |
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169 | the average of the four surrounding vertical tracer gradients, and multiply this by a mean slope at the $u$-point, |
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170 | calculated from the averaged surrounding vertical density gradients. |
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171 | The total area-integrated skew-flux (flux per unit area in $ijk$ space) from tracer cell $i,k$ to $i+1,k$, |
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172 | noting that the $e_{{3}_{i+1/2}^k}$ in the area $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with |
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173 | the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:tra_ldf_iso} |
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174 | \[ |
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175 | \left(F_u^{13} \right)_{i+\frac{1}{2}}^k = {A}_{i+\frac{1}{2}}^k |
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176 | {e_{2}}_{i+1/2}^k \overline{\overline |
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177 | r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k}, |
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178 | \] |
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179 | where |
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180 | \[ |
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181 | \overline{\overline |
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182 | r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k} |
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183 | \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}}, |
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184 | \] |
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185 | and here and in the following we drop the $^{lT}$ superscript from ${A^{lT}}$ for simplicity. |
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186 | Unfortunately the resulting combination $\overline{\overline{\delta_k\bullet}}^{\,i,k}$ of a $k$ average and |
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187 | a $k$ difference of the tracer reduces to $\bullet_{k+1}-\bullet_{k-1}$, |
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188 | so two-grid-point oscillations are invisible to this discretization of the iso-neutral operator. |
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189 | These \emph{computational modes} will not be damped by this operator, and may even possibly be amplified by it. |
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190 | Consequently, applying this operator to a tracer does not guarantee the decrease of its global-average variance. |
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191 | To correct this, we introduced a smoothing of the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}). |
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192 | This technique works for $T$ and $S$ in so far as they are active tracers |
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193 | (\ie they enter the computation of density), but it does not work for a passive tracer. |
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194 | |
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195 | \subsection{Expression of the skew-flux in terms of triad slopes} |
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196 | |
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197 | \citep{griffies.gnanadesikan.ea_JPO98} introduce a different discretization of the off-diagonal terms that |
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198 | nicely solves the problem. |
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199 | % Instead of multiplying the mean slope calculated at the $u$-point by |
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200 | % the mean vertical gradient at the $u$-point, |
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201 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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202 | \begin{figure}[tb] |
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203 | \begin{center} |
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204 | \includegraphics[width=\textwidth]{Fig_GRIFF_triad_fluxes} |
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205 | \caption{ |
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206 | \protect\label{fig:ISO_triad} |
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207 | (a) Arrangement of triads $S_i$ and tracer gradients to |
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208 | give lateral tracer flux from box $i,k$ to $i+1,k$ |
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209 | (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from |
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210 | box $i,k$ to $i,k+1$. |
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211 | } |
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212 | \end{center} |
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213 | \end{figure} |
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214 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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215 | They get the skew flux from the products of the vertical gradients at each $w$-point surrounding the $u$-point with |
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216 | the corresponding `triad' slope calculated from the lateral density gradient across the $u$-point divided by |
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217 | the vertical density gradient at the same $w$-point as the tracer gradient. |
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218 | See \autoref{fig:ISO_triad}a, where the thick lines denote the tracer gradients, |
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219 | and the thin lines the corresponding triads, with slopes $s_1, \dotsc s_4$. |
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220 | The total area-integrated skew-flux from tracer cell $i,k$ to $i+1,k$ |
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221 | \begin{multline} |
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222 | \label{eq:i13} |
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223 | \left( F_u^{13} \right)_{i+\frac{1}{2}}^k = {A}_{i+1}^k a_1 s_1 |
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224 | \delta_{k+\frac{1}{2}} \left[ T^{i+1} |
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225 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} + {A} _i^k a_2 s_2 \delta |
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226 | _{k+\frac{1}{2}} \left[ T^i |
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227 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\ |
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228 | +{A} _{i+1}^k a_3 s_3 \delta_{k-\frac{1}{2}} \left[ T^{i+1} |
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229 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} +{A} _i^k a_4 s_4 \delta |
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230 | _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}, |
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231 | \end{multline} |
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232 | where the contributions of the triad fluxes are weighted by areas $a_1, \dotsc a_4$, |
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233 | and ${A}$ is now defined at the tracer points rather than the $u$-points. |
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234 | This discretization gives a much closer stencil, and disallows the two-point computational modes. |
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235 | |
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236 | The vertical skew flux \autoref{eq:i31c} from tracer cell $i,k$ to $i,k+1$ at |
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237 | the $w$-point $i,k+\frac{1}{2}$ is constructed similarly (\autoref{fig:ISO_triad}b) by |
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238 | multiplying lateral tracer gradients from each of the four surrounding $u$-points by the appropriate triad slope: |
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239 | \begin{multline} |
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240 | \label{eq:i31} |
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241 | \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} = {A}_i^{k+1} a_{1}' |
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242 | s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} |
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243 | +{A}_i^{k+1} a_{2}' s_{2}' \delta_{i+\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i+\frac{1}{2}}^{k+1} \\ |
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244 | + {A}_i^k a_{3}' s_{3}' \delta_{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k |
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245 | +{A}_i^k a_{4}' s_{4}' \delta_{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k. |
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246 | \end{multline} |
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247 | |
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248 | We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$ |
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249 | (appearing in both the vertical and lateral gradient), |
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250 | and the $u$- and $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the triad as follows |
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251 | (see also \autoref{fig:ISO_triad}): |
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252 | \begin{equation} |
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253 | \label{eq:R} |
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254 | _i^k \mathbb{R}_{i_p}^{k_p} |
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255 | =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} |
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256 | \ |
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257 | \frac |
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258 | { \alpha_i^k \ \delta_{i+i_p}[T^k] - \beta_i^k \ \delta_{i+i_p}[S^k] } |
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259 | { \alpha_i^k \ \delta_{k+k_p}[T^i] - \beta_i^k \ \delta_{k+k_p}[S^i] }. |
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260 | \end{equation} |
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261 | In calculating the slopes of the local neutral surfaces, |
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262 | the expansion coefficients $\alpha$ and $\beta$ are evaluated at the anchor points of the triad, |
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263 | while the metrics are calculated at the $u$- and $w$-points on the arms. |
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264 | |
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265 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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266 | \begin{figure}[tb] |
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267 | \begin{center} |
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268 | \includegraphics[width=\textwidth]{Fig_GRIFF_qcells} |
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269 | \caption{ |
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270 | \protect\label{fig:qcells} |
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271 | Triad notation for quarter cells. $T$-cells are inside boxes, |
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272 | while the $i+\fractext{1}{2},k$ $u$-cell is shaded in green and |
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273 | the $i,k+\fractext{1}{2}$ $w$-cell is shaded in pink. |
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274 | } |
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275 | \end{center} |
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276 | \end{figure} |
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277 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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278 | |
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279 | Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:qcells}) with the quarter cell that is |
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280 | the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell. |
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281 | Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:i13} and \autoref{eq:i31} in this notation, |
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282 | we have \eg \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. |
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283 | Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to |
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284 | calculate the lateral flux along its $u$-arm, at $(i+i_p,k)$, |
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285 | and then again as an $s'$ to calculate the vertical flux along its $w$-arm at $(i,k+k_p)$. |
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286 | Each vertical area $a_i$ used to calculate the lateral flux and horizontal area $a'_i$ used to |
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287 | calculate the vertical flux can also be identified as the area across the $u$- and $w$-arms of a unique triad, |
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288 | and we notate these areas, similarly to the triad slopes, |
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289 | as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, |
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290 | where \eg in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, |
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291 | and in \autoref{eq:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. |
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292 | |
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293 | \subsection{Full triad fluxes} |
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294 | |
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295 | A key property of iso-neutral diffusion is that it should not affect the (locally referenced) density. |
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296 | In particular there should be no lateral or vertical density flux. |
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297 | The lateral density flux disappears so long as the area-integrated lateral diffusive flux from |
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298 | tracer cell $i,k$ to $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the form |
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299 | \begin{equation} |
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300 | \label{eq:i11} |
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301 | \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} = |
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302 | - \left( {A}_i^{k+1} a_{1} + {A}_i^{k+1} a_{2} + {A}_i^k |
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303 | a_{3} + {A}_i^k a_{4} \right) |
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304 | \frac{\delta_{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, |
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305 | \end{equation} |
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306 | where the areas $a_i$ are as in \autoref{eq:i13}. |
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307 | In this case, separating the total lateral flux, the sum of \autoref{eq:i13} and \autoref{eq:i11}, |
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308 | into triad components, a lateral tracer flux |
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309 | \begin{equation} |
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310 | \label{eq:latflux-triad} |
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311 | _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - {A}_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p} |
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312 | \left( |
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313 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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314 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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315 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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316 | \right) |
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317 | \end{equation} |
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318 | can be identified with each triad. |
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319 | Then, because the same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and ${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both |
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320 | the density gradients in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients, |
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321 | the lateral density flux associated with each triad separately disappears. |
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322 | \begin{equation} |
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323 | \label{eq:latflux-rho} |
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324 | {\mathbb{F}_u}_{i_p}^{k_p} (\rho)=-\alpha _i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (S)=0 |
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325 | \end{equation} |
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326 | Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} + \left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from |
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327 | tracer cell $i,k$ to $i+1,k$ must also vanish since it is a sum of four such triad fluxes. |
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328 | |
---|
329 | The squared slope $r_1^2$ in the expression \autoref{eq:i33c} for the $_{33}$ component is also expressed in |
---|
330 | terms of area-weighted squared triad slopes, |
---|
331 | so the area-integrated vertical flux from tracer cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is |
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332 | \begin{equation} |
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333 | \label{eq:i33} |
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334 | \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = |
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335 | - \left( {A}_i^{k+1} a_{1}' s_{1}'^2 |
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336 | + {A}_i^{k+1} a_{2}' s_{2}'^2 |
---|
337 | + {A}_i^k a_{3}' s_{3}'^2 |
---|
338 | + {A}_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right], |
---|
339 | \end{equation} |
---|
340 | where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq:i31}. |
---|
341 | Then, separating the total vertical flux, the sum of \autoref{eq:i31} and \autoref{eq:i33}, |
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342 | into triad components, a vertical flux |
---|
343 | \begin{align} |
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344 | \label{eq:vertflux-triad} |
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345 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
---|
346 | &= {A}_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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347 | \left( |
---|
348 | {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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349 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
---|
350 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
351 | \right) \\ |
---|
352 | &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right) |
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353 | {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2} |
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354 | \end{align} |
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355 | may be associated with each triad. |
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356 | Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ associated with a triad then |
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357 | separately disappears (because the lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$ disappears). |
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358 | Consequently the total vertical density flux |
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359 | $\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} + \left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from |
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360 | tracer cell $i,k$ to $i,k+1$ must also vanish since it is a sum of four such triad fluxes. |
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361 | |
---|
362 | We can explicitly identify (\autoref{fig:qcells}) the triads associated with the $s_i$, $a_i$, |
---|
363 | and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq:i31}, |
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364 | \autoref{eq:i13}, \autoref{eq:i11} \autoref{eq:i33} and \autoref{fig:ISO_triad} to write out |
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365 | the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: |
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366 | %(\autoref{fig:ISO_triad}): |
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367 | \begin{flalign} |
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368 | \label{eq:iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv |
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369 | \sum_{\substack{i_p,\,k_p}} |
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370 | \begin{pmatrix} |
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371 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ |
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372 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\ |
---|
373 | \end{pmatrix}. |
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374 | \end{flalign} |
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375 | |
---|
376 | \subsection{Ensuring the scheme does not increase tracer variance} |
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377 | \label{subsec:variance} |
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378 | |
---|
379 | We now require that this operator should not increase the globally-integrated tracer variance. |
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380 | %This changes according to |
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381 | % \begin{align*} |
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382 | % &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
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383 | % &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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384 | % \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
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385 | % + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
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386 | % &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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387 | % {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
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388 | % + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
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389 | % \end{align*} |
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390 | Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across |
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391 | the $u$-point $i+i_p,k$ and a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the $w$-point $i,k+k_p$. |
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392 | The lateral flux drives a net rate of change of variance, |
---|
393 | summed over the two $T$-points $i+i_p-\fractext{1}{2},k$ and $i+i_p+\fractext{1}{2},k$, of |
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394 | \begin{multline} |
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395 | {b_T}_{i+i_p-1/2}^k\left(\frac{\partial T}{\partial t}T\right)_{i+i_p-1/2}^k+ |
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396 | \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial |
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397 | t}T\right)_{i+i_p+1/2}^k \\ |
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398 | \begin{aligned} |
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399 | &= -T_{i+i_p-1/2}^k{\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \quad + \quad T_{i+i_p+1/2}^k |
---|
400 | {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ |
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401 | &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:dvar_iso_i} |
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402 | \end{aligned} |
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403 | \end{multline} |
---|
404 | while the vertical flux similarly drives a net rate of change of variance summed over |
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405 | the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of |
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406 | \begin{equation} |
---|
407 | \label{eq:dvar_iso_k} |
---|
408 | _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
---|
409 | \end{equation} |
---|
410 | The total variance tendency driven by the triad is the sum of these two. |
---|
411 | Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with |
---|
412 | \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, it is |
---|
413 | \begin{multline*} |
---|
414 | -{A}_i^k\left \{ |
---|
415 | { } _i^k{\mathbb{A}_u}_{i_p}^{k_p} |
---|
416 | \left( |
---|
417 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
418 | - {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
---|
419 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\ |
---|
420 | - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p} |
---|
421 | \left( |
---|
422 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
423 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
424 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
425 | \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i] |
---|
426 | \right \}. |
---|
427 | \end{multline*} |
---|
428 | The key point is then that if we require $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$ to |
---|
429 | be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by |
---|
430 | \begin{equation} |
---|
431 | \label{eq:V-A} |
---|
432 | _i^k\mathbb{V}_{i_p}^{k_p} |
---|
433 | ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k} |
---|
434 | ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p}, |
---|
435 | \end{equation} |
---|
436 | the variance tendency reduces to the perfect square |
---|
437 | \begin{equation} |
---|
438 | \label{eq:perfect-square} |
---|
439 | -{A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
---|
440 | \left( |
---|
441 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
442 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
443 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
444 | \right)^2\leq 0. |
---|
445 | \end{equation} |
---|
446 | Thus, the constraint \autoref{eq:V-A} ensures that the fluxes |
---|
447 | (\autoref{eq:latflux-triad}, \autoref{eq:vertflux-triad}) associated with |
---|
448 | a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase the net variance. |
---|
449 | Since the total fluxes are sums of such fluxes from the various triads, this constraint, applied to all triads, |
---|
450 | is sufficient to ensure that the globally integrated variance does not increase. |
---|
451 | |
---|
452 | The expression \autoref{eq:V-A} can be interpreted as a discretization of the global integral |
---|
453 | \begin{equation} |
---|
454 | \label{eq:cts-var} |
---|
455 | \frac{\partial}{\partial t}\int\!\fractext{1}{2} T^2\, dV = |
---|
456 | \int\!\mathbf{F}\cdot\nabla T\, dV, |
---|
457 | \end{equation} |
---|
458 | where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the lateral and vertical fluxes/unit area |
---|
459 | \[ |
---|
460 | \mathbf{F}=\left( |
---|
461 | \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p}, |
---|
462 | \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
---|
463 | \right) |
---|
464 | \] |
---|
465 | and the gradient |
---|
466 | \[ |
---|
467 | \nabla T = \left( |
---|
468 | \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k}, |
---|
469 | \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p} |
---|
470 | \right) |
---|
471 | \] |
---|
472 | |
---|
473 | \subsection{Triad volumes in Griffes's scheme and in \NEMO} |
---|
474 | |
---|
475 | To complete the discretization we now need only specify the triad volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. |
---|
476 | \citet{griffies.gnanadesikan.ea_JPO98} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells, |
---|
477 | defined in terms of the distances between $T$, $u$,$f$ and $w$-points. |
---|
478 | This is the natural discretization of \autoref{eq:cts-var}. |
---|
479 | The \NEMO model, however, operates with scale factors instead of grid sizes, |
---|
480 | and scale factors for the quarter cells are not defined. |
---|
481 | Instead, therefore we simply choose |
---|
482 | \begin{equation} |
---|
483 | \label{eq:V-NEMO} |
---|
484 | _i^k\mathbb{V}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k, |
---|
485 | \end{equation} |
---|
486 | as a quarter of the volume of the $u$-cell inside which the triad quarter-cell lies. |
---|
487 | This has the nice property that when the slopes $\mathbb{R}$ vanish, |
---|
488 | the lateral flux from tracer cell $i,k$ to $i+1,k$ reduces to the classical form |
---|
489 | \begin{equation} |
---|
490 | \label{eq:lat-normal} |
---|
491 | -\overline{A}_{\,i+1/2}^k\; |
---|
492 | \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
493 | \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
494 | = -\overline{A}_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}\;\delta_{i+ 1/2}[T^k]}{{e_{1u}}_{\,i + 1/2}^{\,k}}. |
---|
495 | \end{equation} |
---|
496 | In fact if the diffusive coefficient is defined at $u$-points, |
---|
497 | so that we employ ${A}_{i+i_p}^k$ instead of ${A}_i^k$ in the definitions of the triad fluxes |
---|
498 | \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, |
---|
499 | we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above. |
---|
500 | |
---|
501 | \subsection{Summary of the scheme} |
---|
502 | |
---|
503 | The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that |
---|
504 | cross the $u$- and $w$-faces \autoref{eq:iso_flux}: |
---|
505 | \begin{subequations} |
---|
506 | % \label{eq:alltriadflux} |
---|
507 | \begin{flalign*} |
---|
508 | % \label{eq:vect_isoflux} |
---|
509 | \vect{F}_{\mathrm{iso}}(T) &\equiv |
---|
510 | \sum_{\substack{i_p,\,k_p}} |
---|
511 | \begin{pmatrix} |
---|
512 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ |
---|
513 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
---|
514 | \end{pmatrix}, |
---|
515 | \end{flalign*} |
---|
516 | where \autoref{eq:latflux-triad}: |
---|
517 | \begin{align} |
---|
518 | \label{eq:triadfluxu} |
---|
519 | _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - {A}_i^k{ |
---|
520 | \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
521 | \left( |
---|
522 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
523 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
---|
524 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
525 | \right),\\ |
---|
526 | \intertext{and} |
---|
527 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
---|
528 | &= {A}_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}} |
---|
529 | \left( |
---|
530 | {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
531 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
---|
532 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
533 | \right),\label{eq:triadfluxw} |
---|
534 | \end{align} |
---|
535 | with \autoref{eq:V-NEMO} |
---|
536 | \[ |
---|
537 | % \label{eq:V-NEMO2} |
---|
538 | _i^k{\mathbb{V}}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k. |
---|
539 | \] |
---|
540 | \end{subequations} |
---|
541 | |
---|
542 | The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at |
---|
543 | each tracer point: |
---|
544 | \[ |
---|
545 | % \label{eq:iso_operator} |
---|
546 | D_l^T = \frac{1}{b_T} |
---|
547 | \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k |
---|
548 | {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[ |
---|
549 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
---|
550 | \] |
---|
551 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
---|
552 | The diffusion scheme satisfies the following six properties: |
---|
553 | \begin{description} |
---|
554 | \item[$\bullet$ horizontal diffusion] |
---|
555 | The discretization of the diffusion operator recovers the traditional five-point Laplacian |
---|
556 | \autoref{eq:lat-normal} in the limit of flat iso-neutral direction: |
---|
557 | \[ |
---|
558 | % \label{eq:iso_property0} |
---|
559 | D_l^T = \frac{1}{b_T} \ |
---|
560 | \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; |
---|
561 | \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] \qquad |
---|
562 | \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
---|
563 | \] |
---|
564 | |
---|
565 | \item[$\bullet$ implicit treatment in the vertical] |
---|
566 | Only tracer values associated with a single water column appear in the expression \autoref{eq:i33} for |
---|
567 | the $_{33}$ fluxes, vertical fluxes driven by vertical gradients. |
---|
568 | This is of paramount importance since it means that a time-implicit algorithm can be used to |
---|
569 | solve the vertical diffusion equation. |
---|
570 | This is necessary since the vertical eddy diffusivity associated with this term, |
---|
571 | \[ |
---|
572 | \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
573 | {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: {A}_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
574 | \right\} = |
---|
575 | \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
576 | {b_u}_{i+i_p}^k\: {A}_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
577 | \right\}, |
---|
578 | \] |
---|
579 | (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large. |
---|
580 | |
---|
581 | \item[$\bullet$ pure iso-neutral operator] |
---|
582 | The iso-neutral flux of locally referenced potential density is zero. |
---|
583 | See \autoref{eq:latflux-rho} and \autoref{eq:vertflux-triad2}. |
---|
584 | |
---|
585 | \item[$\bullet$ conservation of tracer] |
---|
586 | The iso-neutral diffusion conserves tracer content, \ie |
---|
587 | \[ |
---|
588 | % \label{eq:iso_property1} |
---|
589 | \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 |
---|
590 | \] |
---|
591 | This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. |
---|
592 | |
---|
593 | \item[$\bullet$ no increase of tracer variance] |
---|
594 | The iso-neutral diffusion does not increase the tracer variance, \ie |
---|
595 | \[ |
---|
596 | % \label{eq:iso_property2} |
---|
597 | \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 |
---|
598 | \] |
---|
599 | The property is demonstrated in \autoref{subsec:variance} above. |
---|
600 | It is a key property for a diffusion term. |
---|
601 | It means that it is also a dissipation term, |
---|
602 | \ie it dissipates the square of the quantity on which it is applied. |
---|
603 | It therefore ensures that, when the diffusivity coefficient is large enough, |
---|
604 | the field on which it is applied becomes free of grid-point noise. |
---|
605 | |
---|
606 | \item[$\bullet$ self-adjoint operator] |
---|
607 | The iso-neutral diffusion operator is self-adjoint, \ie |
---|
608 | \begin{equation} |
---|
609 | \label{eq:iso_property3} |
---|
610 | \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
611 | \end{equation} |
---|
612 | In other word, there is no need to develop a specific routine from the adjoint of this operator. |
---|
613 | We just have to apply the same routine. |
---|
614 | This property can be demonstrated similarly to the proof of the `no increase of tracer variance' property. |
---|
615 | The contribution by a single triad towards the left hand side of \autoref{eq:iso_property3}, |
---|
616 | can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i} and \autoref{eq:dvar_iso_k}. |
---|
617 | This results in a term similar to \autoref{eq:perfect-square}, |
---|
618 | \[ |
---|
619 | % \label{eq:TScovar} |
---|
620 | - {A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
---|
621 | \left( |
---|
622 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
623 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
624 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
625 | \right) |
---|
626 | \left( |
---|
627 | \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
628 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
629 | \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
630 | \right). |
---|
631 | \] |
---|
632 | This is symmetrical in $T $ and $S$, so exactly the same term arises from |
---|
633 | the discretization of this triad's contribution towards the RHS of \autoref{eq:iso_property3}. |
---|
634 | \end{description} |
---|
635 | |
---|
636 | \subsection{Treatment of the triads at the boundaries} |
---|
637 | \label{sec:iso_bdry} |
---|
638 | |
---|
639 | The triad slope can only be defined where both the grid boxes centred at the end of the arms exist. |
---|
640 | Triads that would poke up through the upper ocean surface into the atmosphere, |
---|
641 | or down into the ocean floor, must be masked out. |
---|
642 | See \autoref{fig:bdry_triads}. |
---|
643 | Surface layer triads \triad{i}{1}{R}{1/2}{-1/2} (magenta) and \triad{i+1}{1}{R}{-1/2}{-1/2} (blue) that |
---|
644 | require density to be specified above the ocean surface are masked (\autoref{fig:bdry_triads}a): |
---|
645 | this ensures that lateral tracer gradients produce no flux through the ocean surface. |
---|
646 | However, to prevent surface noise, it is customary to retain the $_{11}$ contributions towards |
---|
647 | the lateral triad fluxes \triad[u]{i}{1}{F}{1/2}{-1/2} and \triad[u]{i+1}{1}{F}{-1/2}{-1/2}; |
---|
648 | this drives diapycnal tracer fluxes. |
---|
649 | Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:bdry_triads}b). |
---|
650 | Note that both near bottom triad slopes \triad{i}{k}{R}{1/2}{1/2} and \triad{i+1}{k}{R}{-1/2}{1/2} are masked when |
---|
651 | either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie the $i,k+1$ $u$-point is masked. |
---|
652 | The associated lateral fluxes (grey-black dashed line) are masked if \np{ln\_botmix\_triad}\forcode{ = .false.}, |
---|
653 | but left unmasked, giving bottom mixing, if \np{ln\_botmix\_triad}\forcode{ = .true.}. |
---|
654 | |
---|
655 | The default option \np{ln\_botmix\_triad}\forcode{ = .false.} is suitable when the bbl mixing option is enabled |
---|
656 | (\key{trabbl}, with \np{nn\_bbl\_ldf}\forcode{ = 1}), or for simple idealized problems. |
---|
657 | For setups with topography without bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary. |
---|
658 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
659 | \begin{figure}[h] |
---|
660 | \begin{center} |
---|
661 | \includegraphics[width=\textwidth]{Fig_GRIFF_bdry_triads} |
---|
662 | \caption{ |
---|
663 | \protect\label{fig:bdry_triads} |
---|
664 | (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots), |
---|
665 | and $i+1/2,1$ $u$-point (blue square). |
---|
666 | Triad slopes \triad{i}{1}{R}{1/2}{-1/2} (magenta) and \triad{i+1}{1}{R}{-1/2}{-1/2} (blue) poking through |
---|
667 | the ocean surface are masked (faded in figure). |
---|
668 | However, the lateral $_{11}$ contributions towards \triad[u]{i}{1}{F}{1/2}{-1/2} and |
---|
669 | \triad[u]{i+1}{1}{F}{-1/2}{-1/2} (yellow line) are still applied, |
---|
670 | giving diapycnal diffusive fluxes. |
---|
671 | \newline |
---|
672 | (b) Both near bottom triad slopes \triad{i}{k}{R}{1/2}{1/2} and |
---|
673 | \triad{i+1}{k}{R}{-1/2}{1/2} are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, |
---|
674 | \ie the $i,k+1$ $u$-point is masked. |
---|
675 | The associated lateral fluxes (grey-black dashed line) are masked if |
---|
676 | \protect\np{botmix\_triad}\forcode{ = .false.}, but left unmasked, |
---|
677 | giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.} |
---|
678 | } |
---|
679 | \end{center} |
---|
680 | \end{figure} |
---|
681 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
682 | |
---|
683 | \subsection{ Limiting of the slopes within the interior} |
---|
684 | \label{sec:limit} |
---|
685 | |
---|
686 | As discussed in \autoref{subsec:LDF_slp_iso}, |
---|
687 | iso-neutral slopes relative to geopotentials must be bounded everywhere, |
---|
688 | both for consistency with the small-slope approximation and for numerical stability \citep{cox_OM87, griffies_bk04}. |
---|
689 | The bound chosen in \NEMO is applied to each component of the slope separately and |
---|
690 | has a value of $1/100$ in the ocean interior. |
---|
691 | %, ramping linearly down above 70~m depth to zero at the surface |
---|
692 | It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to geopotentials |
---|
693 | (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials) |
---|
694 | \autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require |
---|
695 | \[ |
---|
696 | |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01. |
---|
697 | \] |
---|
698 | and then recalculate the slopes $r_i$ relative to coordinates. |
---|
699 | Each individual triad slope |
---|
700 | \begin{equation} |
---|
701 | \label{eq:Rtilde} |
---|
702 | _i^k\tilde{\mathbb{R}}_{i_p}^{k_p} = {}_i^k\mathbb{R}_{i_p}^{k_p} + \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
703 | \end{equation} |
---|
704 | is limited like this and then the corresponding $_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and |
---|
705 | combined to form the fluxes. |
---|
706 | Note that where the slopes have been limited, there is now a non-zero iso-neutral density flux that |
---|
707 | drives dianeutral mixing. |
---|
708 | In particular this iso-neutral density flux is always downwards, |
---|
709 | and so acts to reduce gravitational potential energy. |
---|
710 | |
---|
711 | \subsection{Tapering within the surface mixed layer} |
---|
712 | \label{sec:taper} |
---|
713 | |
---|
714 | Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer. |
---|
715 | When the Griffies triads are used, we offer two options for this. |
---|
716 | |
---|
717 | \subsubsection{Linear slope tapering within the surface mixed layer} |
---|
718 | \label{sec:lintaper} |
---|
719 | |
---|
720 | This is the option activated by the default choice \np{ln\_triad\_iso}\forcode{ = .false.}. |
---|
721 | Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below |
---|
722 | the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:eiv_slp}, to values |
---|
723 | \begin{equation} |
---|
724 | \label{eq:rmtilde} |
---|
725 | \rMLt = -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for } z>-h, |
---|
726 | \end{equation} |
---|
727 | and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to |
---|
728 | \[ |
---|
729 | % \label{eq:rm} |
---|
730 | \rML =\rMLt -\sigma_i \quad \text{ for } z>-h. |
---|
731 | \] |
---|
732 | Thus the diffusion operator within the mixed layer is given by: |
---|
733 | \[ |
---|
734 | % \label{eq:iso_tensor_ML} |
---|
735 | D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
736 | \mbox{with}\quad \;\;\Re =\left( {{ |
---|
737 | \begin{array}{*{20}c} |
---|
738 | 1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\ |
---|
739 | 0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\ |
---|
740 | {-\rML[1]}\hfill & {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill |
---|
741 | \end{array} |
---|
742 | }} \right) |
---|
743 | \] |
---|
744 | |
---|
745 | This slope tapering gives a natural connection between tracer in the mixed-layer and |
---|
746 | in isopycnal layers immediately below, in the thermocline. |
---|
747 | It is consistent with the way the $\tilde{r}_i$ are tapered within the mixed layer |
---|
748 | (see \autoref{sec:taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer. |
---|
749 | However, it gives a downwards density flux and so acts so as to reduce potential energy in the same way as |
---|
750 | does the slope limiting discussed above in \autoref{sec:limit}. |
---|
751 | |
---|
752 | As in \autoref{sec:limit} above, the tapering \autoref{eq:rmtilde} is applied separately to |
---|
753 | each triad $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. |
---|
754 | For clarity, we assume $z$-coordinates in the following; |
---|
755 | the conversion from $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as |
---|
756 | described above by \autoref{eq:Rtilde}. |
---|
757 | \begin{enumerate} |
---|
758 | \item |
---|
759 | Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in |
---|
760 | the slope definition. |
---|
761 | At each $i,j$ (simplified to $i$ in \autoref{fig:MLB_triad}), |
---|
762 | we define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, |
---|
763 | $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that |
---|
764 | the potential density ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, |
---|
765 | where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m. |
---|
766 | See the left side of \autoref{fig:MLB_triad}. |
---|
767 | We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems \eg with thin daytime mixed-layers. |
---|
768 | Currently we use the same $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to |
---|
769 | output the diagnosed mixed-layer depth $h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$, |
---|
770 | the depth of the $w$-point above the $i,k_{\mathrm{ML}}$ tracer point. |
---|
771 | \item |
---|
772 | We define `basal' triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as |
---|
773 | the slopes of those triads whose vertical `arms' go down from the $i,k_{\mathrm{ML}}$ tracer point to |
---|
774 | the $i,k_{\mathrm{ML}}-1$ tracer point below. |
---|
775 | This is to ensure that the vertical density gradients associated with |
---|
776 | these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline. |
---|
777 | The four basal triads defined in the bottom part of \autoref{fig:MLB_triad} are then |
---|
778 | \begin{align*} |
---|
779 | {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= |
---|
780 | {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, |
---|
781 | % \label{eq:Rbase} |
---|
782 | \\ |
---|
783 | \intertext{with \eg the green triad} |
---|
784 | {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&= |
---|
785 | {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. |
---|
786 | \end{align*} |
---|
787 | The vertical flux associated with each of these triads passes through |
---|
788 | the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth |
---|
789 | \[ |
---|
790 | % \label{eq:zbase} |
---|
791 | {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} |
---|
792 | \] |
---|
793 | one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in |
---|
794 | \autoref{eq:rmtilde}. |
---|
795 | \item |
---|
796 | Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within |
---|
797 | the mixed layer, by multiplying the appropriate ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by |
---|
798 | the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$. |
---|
799 | For instance the green triad centred on $i,k$ |
---|
800 | \begin{align*} |
---|
801 | {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &= |
---|
802 | \frac{{z_w}_{k-1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2} \\ |
---|
803 | \intertext{and more generally} |
---|
804 | {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= |
---|
805 | \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}. |
---|
806 | % \label{eq:RML} |
---|
807 | \end{align*} |
---|
808 | \end{enumerate} |
---|
809 | |
---|
810 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
811 | \begin{figure}[h] |
---|
812 | % \fcapside { |
---|
813 | \caption{ |
---|
814 | \protect\label{fig:MLB_triad} |
---|
815 | Definition of mixed-layer depth and calculation of linearly tapered triads. |
---|
816 | The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top. |
---|
817 | Tracer points are denoted by bullets, and black lines the edges of the tracer cells; |
---|
818 | $k$ increases upwards. |
---|
819 | \newline |
---|
820 | \hspace{5 em} |
---|
821 | We define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, |
---|
822 | $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that |
---|
823 | ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, |
---|
824 | where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m. |
---|
825 | We calculate the triad slopes within the mixed layer by linearly tapering them from zero |
---|
826 | (at the surface) to the `basal' slopes, |
---|
827 | the slopes of the four triads passing through the $w$-point $i,k_{\mathrm{ML}}-1/2$ (blue square), |
---|
828 | ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$. |
---|
829 | Triads with different $i_p,k_p$, denoted by different colours, |
---|
830 | (\eg the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.} |
---|
831 | % } |
---|
832 | \includegraphics[width=\textwidth]{Fig_GRIFF_MLB_triads} |
---|
833 | \end{figure} |
---|
834 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
835 | |
---|
836 | \subsubsection{Additional truncation of skew iso-neutral flux components} |
---|
837 | \label{subsec:Gerdes-taper} |
---|
838 | |
---|
839 | The alternative option is activated by setting \np{ln\_triad\_iso} = true. |
---|
840 | This retains the same tapered slope $\rML$ described above for the calculation of the $_{33}$ term of |
---|
841 | the iso-neutral diffusion tensor (the vertical tracer flux driven by vertical tracer gradients), |
---|
842 | but replaces the $\rML$ in the skew term by |
---|
843 | \begin{equation} |
---|
844 | \label{eq:rm*} |
---|
845 | \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i, |
---|
846 | \end{equation} |
---|
847 | giving a ML diffusive operator |
---|
848 | \[ |
---|
849 | % \label{eq:iso_tensor_ML2} |
---|
850 | D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
851 | \mbox{with}\quad \;\;\Re =\left( {{ |
---|
852 | \begin{array}{*{20}c} |
---|
853 | 1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\ |
---|
854 | 0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\ |
---|
855 | {-\rML[1]^*}\hfill & {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\ |
---|
856 | \end{array} |
---|
857 | }} \right). |
---|
858 | \] |
---|
859 | This operator |
---|
860 | \footnote{ |
---|
861 | To ensure good behaviour where horizontal density gradients are weak, |
---|
862 | we in fact follow \citet{gerdes.koberle.ea_CD91} and |
---|
863 | set $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$. |
---|
864 | } |
---|
865 | then has the property it gives no vertical density flux, and so does not change the potential energy. |
---|
866 | This approach is similar to multiplying the iso-neutral diffusion coefficient by |
---|
867 | $\tilde{r}_{\mathrm{max}}^{-2}\tilde{r}_i^{-2}$ for steep slopes, |
---|
868 | as suggested by \citet{gerdes.koberle.ea_CD91} (see also \citet{griffies_bk04}). |
---|
869 | Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$ |
---|
870 | |
---|
871 | In practice, this approach gives weak vertical tracer fluxes through the mixed-layer, |
---|
872 | as well as vanishing density fluxes. |
---|
873 | While it is theoretically advantageous that it does not change the potential energy, |
---|
874 | it may give a discontinuity between the fluxes within the mixed-layer (purely horizontal) and |
---|
875 | just below (along iso-neutral surfaces). |
---|
876 | % This may give strange looking results, |
---|
877 | % particularly where the mixed-layer depth varies strongly laterally. |
---|
878 | % ================================================================ |
---|
879 | % Skew flux formulation for Eddy Induced Velocity : |
---|
880 | % ================================================================ |
---|
881 | \section{Eddy induced advection formulated as a skew flux} |
---|
882 | \label{sec:skew-flux} |
---|
883 | |
---|
884 | \subsection{Continuous skew flux formulation} |
---|
885 | \label{sec:continuous-skew-flux} |
---|
886 | |
---|
887 | When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added. |
---|
888 | The associated velocity is the so called eddy induced velocity, |
---|
889 | the formulation of which depends on the slopes of iso-neutral surfaces. |
---|
890 | Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, |
---|
891 | \ie \autoref{eq:ldfslp_geo} is used in $z$-coordinate, |
---|
892 | and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. |
---|
893 | |
---|
894 | The eddy induced velocity is given by: |
---|
895 | \begin{subequations} |
---|
896 | % \label{eq:eiv} |
---|
897 | \begin{equation} |
---|
898 | \label{eq:eiv_v} |
---|
899 | \begin{split} |
---|
900 | u^* & = - \frac{1}{e_{3}}\; \partial_i\psi_1, \\ |
---|
901 | v^* & = - \frac{1}{e_{3}}\; \partial_j\psi_2, \\ |
---|
902 | w^* & = \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i \left( e_{2} \, \psi_1\right) |
---|
903 | + \partial_j \left( e_{1} \, \psi_2\right) \right\}, |
---|
904 | \end{split} |
---|
905 | \end{equation} |
---|
906 | where the streamfunctions $\psi_i$ are given by |
---|
907 | \begin{equation} |
---|
908 | \label{eq:eiv_psi} |
---|
909 | \begin{split} |
---|
910 | \psi_1 & = A_{e} \; \tilde{r}_1, \\ |
---|
911 | \psi_2 & = A_{e} \; \tilde{r}_2, |
---|
912 | \end{split} |
---|
913 | \end{equation} |
---|
914 | \end{subequations} |
---|
915 | with $A_{e}$ the eddy induced velocity coefficient, |
---|
916 | and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces. |
---|
917 | |
---|
918 | The traditional way to implement this additional advection is to add it to the Eulerian velocity prior to |
---|
919 | computing the tracer advection. |
---|
920 | This is implemented if \key{traldf\_eiv} is set in the default implementation, |
---|
921 | where \np{ln\_traldf\_triad} is set false. |
---|
922 | This allows us to take advantage of all the advection schemes offered for the tracers |
---|
923 | (see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme. |
---|
924 | This is particularly useful for passive tracers where |
---|
925 | \emph{positivity} of the advection scheme is of paramount importance. |
---|
926 | |
---|
927 | However, when \np{ln\_traldf\_triad} is set true, |
---|
928 | \NEMO instead implements eddy induced advection according to the so-called skew form \citep{griffies_JPO98}. |
---|
929 | It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity. |
---|
930 | For example in the (\textbf{i},\textbf{k}) plane, |
---|
931 | the tracer advective fluxes per unit area in $ijk$ space can be transformed as follows: |
---|
932 | \begin{flalign*} |
---|
933 | \begin{split} |
---|
934 | \textbf{F}_{\mathrm{eiv}}^T = |
---|
935 | \begin{pmatrix} |
---|
936 | {e_{2}\,e_{3}\; u^*} \\ |
---|
937 | {e_{1}\,e_{2}\; w^*} |
---|
938 | \end{pmatrix} \; T |
---|
939 | &= |
---|
940 | \begin{pmatrix} |
---|
941 | { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\ |
---|
942 | {+ \partial_i \left( e_{2} \, \psi_1 \right) \; T \;} |
---|
943 | \end{pmatrix} \\ |
---|
944 | &= |
---|
945 | \begin{pmatrix} |
---|
946 | { - \partial_k \left( e_{2} \, \psi_1 \; T \right) \;} \\ |
---|
947 | {+ \partial_i \left( e_{2} \,\psi_1 \; T \right) \;} |
---|
948 | \end{pmatrix} |
---|
949 | + |
---|
950 | \begin{pmatrix} |
---|
951 | {+ e_{2} \, \psi_1 \; \partial_k T} \\ |
---|
952 | { - e_{2} \, \psi_1 \; \partial_i T} |
---|
953 | \end{pmatrix} |
---|
954 | \end{split} |
---|
955 | \end{flalign*} |
---|
956 | and since the eddy induced velocity field is non-divergent, |
---|
957 | we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space: |
---|
958 | \begin{equation} |
---|
959 | \label{eq:eiv_skew_ijk} |
---|
960 | \textbf{F}_\mathrm{eiv}^T = |
---|
961 | \begin{pmatrix} |
---|
962 | {+ e_{2} \, \psi_1 \; \partial_k T} \\ |
---|
963 | { - e_{2} \, \psi_1 \; \partial_i T} |
---|
964 | \end{pmatrix} |
---|
965 | \end{equation} |
---|
966 | The total fluxes per unit physical area are then |
---|
967 | \begin{equation} |
---|
968 | \label{eq:eiv_skew_physical} |
---|
969 | \begin{split} |
---|
970 | f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T \\ |
---|
971 | f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T \\ |
---|
972 | f^*_3 & = -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T + e_{1} \psi_2 \partial_j T \right\}. |
---|
973 | \end{split} |
---|
974 | \end{equation} |
---|
975 | Note that \autoref{eq:eiv_skew_physical} takes the same form whatever the vertical coordinate, |
---|
976 | though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:eiv_psi} are relative to |
---|
977 | geopotentials. |
---|
978 | The tendency associated with eddy induced velocity is then simply the convergence of the fluxes |
---|
979 | (\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so |
---|
980 | \[ |
---|
981 | % \label{eq:skew_eiv_conv} |
---|
982 | \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 } \left[ |
---|
983 | \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) |
---|
984 | + \frac{\partial}{\partial j} \left( e_1 \; |
---|
985 | \psi_2 \partial_k T\right) |
---|
986 | - \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T |
---|
987 | + e_{1} \psi_2 \partial_j T \right) \right] |
---|
988 | \] |
---|
989 | It naturally conserves the tracer content, as it is expressed in flux form. |
---|
990 | Since it has the same divergence as the advective form it also preserves the tracer variance. |
---|
991 | |
---|
992 | \subsection{Discrete skew flux formulation} |
---|
993 | |
---|
994 | The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}), |
---|
995 | like the off-diagonal terms (\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor, |
---|
996 | are best expressed in terms of the triad slopes, as in \autoref{fig:ISO_triad} and |
---|
997 | (\autoref{eq:i13}, \autoref{eq:i31}); |
---|
998 | but now in terms of the triad slopes $\tilde{\mathbb{R}}$ relative to geopotentials instead of |
---|
999 | the $\mathbb{R}$ relative to coordinate surfaces. |
---|
1000 | The discrete form of \autoref{eq:eiv_skew_ijk} using the slopes \autoref{eq:R} and |
---|
1001 | defining $A_e$ at $T$-points is then given by: |
---|
1002 | |
---|
1003 | \begin{subequations} |
---|
1004 | % \label{eq:allskewflux} |
---|
1005 | \begin{flalign*} |
---|
1006 | % \label{eq:vect_skew_flux} |
---|
1007 | \vect{F}_{\mathrm{eiv}}(T) &\equiv \sum_{\substack{i_p,\,k_p}} |
---|
1008 | \begin{pmatrix} |
---|
1009 | {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T) \\ \\ |
---|
1010 | {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T) \\ |
---|
1011 | \end{pmatrix}, |
---|
1012 | \end{flalign*} |
---|
1013 | where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:latflux-triad}, |
---|
1014 | \autoref{eq:triadfluxu}): |
---|
1015 | \begin{align} |
---|
1016 | \label{eq:skewfluxu} |
---|
1017 | _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \fractext{1}{4} {A_e}_i^k{ |
---|
1018 | \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
1019 | \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \ |
---|
1020 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, \\ |
---|
1021 | \intertext{ |
---|
1022 | and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign |
---|
1023 | to be consistent with \autoref{eq:eiv_skew_ijk}: |
---|
1024 | } |
---|
1025 | _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) |
---|
1026 | &= -\fractext{1}{4} {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} |
---|
1027 | {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw} |
---|
1028 | \end{align} |
---|
1029 | \end{subequations} |
---|
1030 | |
---|
1031 | Such a discretisation is consistent with the iso-neutral operator as it uses the same definition for the slopes. |
---|
1032 | It also ensures the following two key properties. |
---|
1033 | |
---|
1034 | \subsubsection{No change in tracer variance} |
---|
1035 | |
---|
1036 | The discretization conserves tracer variance, \ie it does not include a diffusive component but is a `pure' advection term. |
---|
1037 | This can be seen %either from Appendix \autoref{apdx:eiv_skew} or |
---|
1038 | by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. |
---|
1039 | For, following \autoref{subsec:variance} and \autoref{eq:dvar_iso_i}, |
---|
1040 | the associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ drives a net rate of change of variance, |
---|
1041 | summed over the two $T$-points $i+i_p-\fractext{1}{2},k$ and $i+i_p+\fractext{1}{2},k$, of |
---|
1042 | \begin{equation} |
---|
1043 | \label{eq:dvar_eiv_i} |
---|
1044 | _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], |
---|
1045 | \end{equation} |
---|
1046 | while the associated vertical skew-flux gives a variance change summed over |
---|
1047 | the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of |
---|
1048 | \begin{equation} |
---|
1049 | \label{eq:dvar_eiv_k} |
---|
1050 | _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
---|
1051 | \end{equation} |
---|
1052 | Inspection of the definitions (\autoref{eq:skewfluxu}, \autoref{eq:skewfluxw}) shows that |
---|
1053 | these two variance changes (\autoref{eq:dvar_eiv_i}, \autoref{eq:dvar_eiv_k}) sum to zero. |
---|
1054 | Hence the two fluxes associated with each triad make no net contribution to the variance budget. |
---|
1055 | |
---|
1056 | \subsubsection{Reduction in gravitational PE} |
---|
1057 | |
---|
1058 | The vertical density flux associated with the vertical skew-flux always has the same sign as |
---|
1059 | the vertical density gradient; |
---|
1060 | thus, so long as the fluid is stable (the vertical density gradient is negative) |
---|
1061 | the vertical density flux is negative (downward) and hence reduces the gravitational PE. |
---|
1062 | |
---|
1063 | For the change in gravitational PE driven by the $k$-flux is |
---|
1064 | \begin{align} |
---|
1065 | \label{eq:vert_densityPE} |
---|
1066 | g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) |
---|
1067 | &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k |
---|
1068 | {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k |
---|
1069 | {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ |
---|
1070 | \intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:skewfluxw}, gives} |
---|
1071 | % and separating out |
---|
1072 | % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, |
---|
1073 | % gives two terms. The |
---|
1074 | % first $\rtriad{R}$ term (the only term for $z$-coordinates) is: |
---|
1075 | &=-\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} |
---|
1076 | \frac{ -\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{\,i + i_p}^{\,k} } \notag \\ |
---|
1077 | &=+\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
1078 | \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) {_i^k\mathbb{R}_{i_p}^{k_p}} |
---|
1079 | \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, |
---|
1080 | \end{align} |
---|
1081 | using the definition of the triad slope $\rtriad{R}$, \autoref{eq:R} to |
---|
1082 | express $-\alpha _i^k\delta_{i+ i_p}[T^k]+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of |
---|
1083 | $-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$. |
---|
1084 | |
---|
1085 | Where the coordinates slope, the $i$-flux gives a PE change |
---|
1086 | \begin{multline} |
---|
1087 | \label{eq:lat_densityPE} |
---|
1088 | g \delta_{i+i_p}[z_T^k] |
---|
1089 | \left[ |
---|
1090 | -\alpha _i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (S) |
---|
1091 | \right] \\ |
---|
1092 | = +\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
1093 | \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
1094 | \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) |
---|
1095 | \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, |
---|
1096 | \end{multline} |
---|
1097 | (using \autoref{eq:skewfluxu}) and so the total PE change \autoref{eq:vert_densityPE} + |
---|
1098 | \autoref{eq:lat_densityPE} associated with the triad fluxes is |
---|
1099 | \begin{multline*} |
---|
1100 | % \label{eq:tot_densityPE} |
---|
1101 | g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + |
---|
1102 | g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ |
---|
1103 | = +\fractext{1}{4} g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
1104 | \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)^2 |
---|
1105 | \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}. |
---|
1106 | \end{multline*} |
---|
1107 | Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+ |
---|
1108 | \beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative. |
---|
1109 | |
---|
1110 | \subsection{Treatment of the triads at the boundaries} |
---|
1111 | \label{sec:skew_bdry} |
---|
1112 | |
---|
1113 | Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries |
---|
1114 | in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes, |
---|
1115 | as described in \autoref{sec:iso_bdry} and \autoref{fig:bdry_triads}. |
---|
1116 | Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked, |
---|
1117 | and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when |
---|
1118 | either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie the $i,k+1$ $u$-point is masked. |
---|
1119 | The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes. |
---|
1120 | |
---|
1121 | \subsection{Limiting of the slopes within the interior} |
---|
1122 | \label{sec:limitskew} |
---|
1123 | |
---|
1124 | Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$, |
---|
1125 | exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:limit}. |
---|
1126 | Each individual triad \rtriadt{R} is so limited. |
---|
1127 | |
---|
1128 | \subsection{Tapering within the surface mixed layer} |
---|
1129 | \label{sec:taperskew} |
---|
1130 | |
---|
1131 | The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R}) |
---|
1132 | are always tapered linearly from their value immediately below the mixed layer to zero at the surface |
---|
1133 | \autoref{eq:rmtilde}, as described in \autoref{sec:lintaper}. |
---|
1134 | This is option (c) of \autoref{fig:eiv_slp}. |
---|
1135 | This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by |
---|
1136 | the value of \np{ln\_triad\_iso}. |
---|
1137 | |
---|
1138 | The justification for this linear slope tapering is that, for $A_e$ that is constant or varies only in |
---|
1139 | the horizontal (the most commonly used options in \NEMO: see \autoref{sec:LDF_coef}), |
---|
1140 | it is equivalent to a horizontal eiv (eddy-induced velocity) that is uniform within the mixed layer |
---|
1141 | \autoref{eq:eiv_v}. |
---|
1142 | This ensures that the eiv velocities do not restratify the mixed layer \citep{treguier.held.ea_JPO97,danabasoglu.ferrari.ea_JC08}. |
---|
1143 | Equivantly, in terms of the skew-flux formulation we use here, |
---|
1144 | the linear slope tapering within the mixed-layer gives a linearly varying vertical flux, |
---|
1145 | and so a tracer convergence uniform in depth |
---|
1146 | (the horizontal flux convergence is relatively insignificant within the mixed-layer). |
---|
1147 | |
---|
1148 | \subsection{Streamfunction diagnostics} |
---|
1149 | \label{sec:sfdiag} |
---|
1150 | |
---|
1151 | Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.}, |
---|
1152 | diagnosed mean eddy-induced velocities are output. |
---|
1153 | Each time step, streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at |
---|
1154 | $uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ (integer $i$, integer +1/2 $j$, integer +1/2 $k$) |
---|
1155 | points (see Table \autoref{tab:cell}) respectively. |
---|
1156 | We follow \citep{griffies_bk04} and calculate the streamfunction at a given $uw$-point from |
---|
1157 | the surrounding four triads according to: |
---|
1158 | \[ |
---|
1159 | % \label{eq:sfdiagi} |
---|
1160 | {\psi_1}_{i+1/2}^{k+1/2}={\fractext{1}{4}}\sum_{\substack{i_p,\,k_p}} |
---|
1161 | {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}. |
---|
1162 | \] |
---|
1163 | The streamfunction $\psi_1$ is calculated similarly at $vw$ points. |
---|
1164 | The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:eiv_v}: |
---|
1165 | \[ |
---|
1166 | % \label{eq:eiv_v_discrete} |
---|
1167 | \begin{split} |
---|
1168 | {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right), \\ |
---|
1169 | {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right), \\ |
---|
1170 | {w^*}_{i,j}^{k+1/2} & = \frac{1}{e_{1t}e_{2t}}\; \left\{ |
---|
1171 | {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} - |
---|
1172 | {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\ |
---|
1173 | \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\}, |
---|
1174 | \end{split} |
---|
1175 | \] |
---|
1176 | |
---|
1177 | \biblio |
---|
1178 | |
---|
1179 | \pindex |
---|
1180 | |
---|
1181 | \end{document} |
---|