1 | % ================================================================ |
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2 | % Iso-neutral diffusion : |
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3 | % ================================================================ |
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4 | \chapter{Griffies's iso-neutral diffusion} |
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5 | \label{sec:triad} |
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6 | \minitoc |
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7 | |
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8 | \section{Griffies's formulation of iso-neutral diffusion} |
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9 | \label{sec:triad:iso} |
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10 | |
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11 | We define a scheme inspired by \citet{Griffies_al_JPO98}, but formulated within the \NEMO |
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12 | framework, using scale factors rather than grid-sizes. |
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13 | |
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14 | \subsection{The iso-neutral diffusion operator} |
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15 | The iso-neutral second order tracer diffusive operator for small |
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16 | angles between iso-neutral surfaces and geopotentials is given by |
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17 | \eqref{Eq_PE_iso_tensor}: |
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18 | \begin{subequations} \label{eq:triad:PE_iso_tensor} |
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19 | \begin{equation} |
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20 | D^{lT}=-\Div\vect{f}^{lT}\equiv |
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21 | -\frac{1}{e_1e_2e_3}\left[\pd{i}\left (f_1^{lT}e_2e_3\right) + |
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22 | \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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23 | \end{equation} |
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24 | where the diffusive flux per unit area of physical space |
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25 | \begin{equation} |
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26 | \vect{f}^{lT}=-\Alt\Re\cdot\grad T, |
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27 | \end{equation} |
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28 | \begin{equation} |
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29 | \label{eq:triad:PE_iso_tensor:c} |
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30 | \mbox{with}\quad \;\;\Re = |
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31 | \begin{pmatrix} |
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32 | 1&0&-r_1\mystrut \\ |
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33 | 0&1&-r_2\mystrut \\ |
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34 | -r_1&-r_2&r_1 ^2+r_2 ^2\mystrut |
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35 | \end{pmatrix} |
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36 | \quad \text{and} \quad\grad T= |
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37 | \begin{pmatrix} |
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38 | \frac{1}{e_1}\pd[T]{i}\mystrut \\ |
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39 | \frac{1}{e_2}\pd[T]{j}\mystrut \\ |
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40 | \frac{1}{e_3}\pd[T]{k}\mystrut |
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41 | \end{pmatrix}. |
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42 | \end{equation} |
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43 | \end{subequations} |
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44 | % \left( {{\begin{array}{*{20}c} |
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45 | % 1 \hfill & 0 \hfill & {-r_1 } \hfill \\ |
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46 | % 0 \hfill & 1 \hfill & {-r_2 } \hfill \\ |
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47 | % {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ |
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48 | % \end{array} }} \right) |
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49 | Here \eqref{Eq_PE_iso_slopes} |
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50 | \begin{align*} |
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51 | r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} |
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52 | \right) |
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53 | \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\ |
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54 | &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} + |
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55 | \beta\frac{\partial S }{\partial i} \right) \left( |
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56 | -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S |
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57 | }{\partial k} \right)^{-1} |
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58 | \end{align*} |
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59 | is the $i$-component of the slope of the iso-neutral surface relative to the computational |
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60 | surface, and $r_2$ is the $j$-component. |
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61 | |
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62 | We will find it useful to consider the fluxes per unit area in $i,j,k$ |
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63 | space; we write |
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64 | \begin{equation} |
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65 | \label{eq:triad:Fijk} |
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66 | \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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67 | \end{equation} |
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68 | Additionally, we will sometimes write the contributions towards the |
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69 | fluxes $\vect{f}$ and $\vect{F}_\mathrm{iso}$ from the component |
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70 | $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, with |
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71 | $f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc. |
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72 | |
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73 | The off-diagonal terms of the small angle diffusion tensor |
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74 | \eqref{Eq_PE_iso_tensor}, \eqref{eq:triad:PE_iso_tensor:c} produce skew-fluxes along the |
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75 | $i$- and $j$-directions resulting from the vertical tracer gradient: |
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76 | \begin{align} |
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77 | \label{eq:triad:i13c} |
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78 | f_{13}=&+\Alt r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+\Alt r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\ |
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79 | \intertext{and in the k-direction resulting from the lateral tracer gradients} |
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80 | \label{eq:triad:i31c} |
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81 | f_{31}+f_{32}=& \Alt r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+\Alt r_2\frac{1}{e_1}\frac{\partial T}{\partial i} |
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82 | \end{align} |
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83 | |
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84 | The vertical diffusive flux associated with the $_{33}$ |
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85 | component of the small angle diffusion tensor is |
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86 | \begin{equation} |
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87 | \label{eq:triad:i33c} |
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88 | f_{33}=-\Alt(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}. |
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89 | \end{equation} |
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90 | |
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91 | Since there are no cross terms involving $r_1$ and $r_2$ in the above, we can |
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92 | consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$ |
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93 | planes, just adding together the vertical components from each |
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94 | plane. The following description will describe the fluxes on the $i$-$k$ |
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95 | plane. |
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96 | |
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97 | There is no natural discretization for the $i$-component of the |
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98 | skew-flux, \eqref{eq:triad:i13c}, as |
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99 | although it must be evaluated at $u$-points, it involves vertical |
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100 | gradients (both for the tracer and the slope $r_1$), defined at |
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101 | $w$-points. Similarly, the vertical skew flux, \eqref{eq:triad:i31c}, is evaluated at |
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102 | $w$-points but involves horizontal gradients defined at $u$-points. |
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103 | |
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104 | \subsection{The standard discretization} |
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105 | The straightforward approach to discretize the lateral skew flux |
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106 | \eqref{eq:triad:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 |
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107 | into OPA, \eqref{Eq_tra_ldf_iso}, is to calculate a mean vertical |
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108 | gradient at the $u$-point from the average of the four surrounding |
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109 | vertical tracer gradients, and multiply this by a mean slope at the |
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110 | $u$-point, calculated from the averaged surrounding vertical density |
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111 | gradients. The total area-integrated skew-flux (flux per unit area in |
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112 | $ijk$ space) from tracer cell $i,k$ |
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113 | to $i+1,k$, noting that the $e_{{3}_{i+1/2}^k}$ in the area |
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114 | $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with |
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115 | the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer |
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116 | gradient, is then \eqref{Eq_tra_ldf_iso} |
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117 | \begin{equation*} |
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118 | \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k |
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119 | {e_{2}}_{i+1/2}^k \overline{\overline |
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120 | r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k}, |
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121 | \end{equation*} |
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122 | where |
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123 | \begin{equation*} |
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124 | \overline{\overline |
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125 | r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k} |
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126 | \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}}, |
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127 | \end{equation*} |
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128 | and here and in the following we drop the $^{lT}$ superscript from |
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129 | $\Alt$ for simplicity. |
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130 | Unfortunately the resulting combination $\overline{\overline{\delta_k |
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131 | \bullet}}^{\,i,k}$ of a $k$ average and a $k$ difference %of the tracer |
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132 | reduces to $\bullet_{k+1}-\bullet_{k-1}$, so two-grid-point oscillations are |
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133 | invisible to this discretization of the iso-neutral operator. These |
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134 | \emph{computational modes} will not be damped by this operator, and |
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135 | may even possibly be amplified by it. Consequently, applying this |
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136 | operator to a tracer does not guarantee the decrease of its |
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137 | global-average variance. To correct this, we introduced a smoothing of |
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138 | the slopes of the iso-neutral surfaces (see \S\ref{LDF}). This |
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139 | technique works for $T$ and $S$ in so far as they are active tracers |
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140 | ($i.e.$ they enter the computation of density), but it does not work |
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141 | for a passive tracer. |
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142 | \subsection{Expression of the skew-flux in terms of triad slopes} |
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143 | \citep{Griffies_al_JPO98} introduce a different discretization of the |
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144 | off-diagonal terms that nicely solves the problem. |
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145 | % Instead of multiplying the mean slope calculated at the $u$-point by |
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146 | % the mean vertical gradient at the $u$-point, |
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147 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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148 | \begin{figure}[h] \begin{center} |
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149 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_triad_fluxes} |
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150 | \caption{ \label{fig:triad:ISO_triad} |
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151 | (a) Arrangement of triads $S_i$ and tracer gradients to |
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152 | give lateral tracer flux from box $i,k$ to $i+1,k$ |
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153 | (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from |
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154 | box $i,k$ to $i,k+1$.} |
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155 | \end{center} \end{figure} |
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156 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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157 | They get the skew flux from the products of the vertical gradients at |
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158 | each $w$-point surrounding the $u$-point with the corresponding `triad' |
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159 | slope calculated from the lateral density gradient across the $u$-point |
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160 | divided by the vertical density gradient at the same $w$-point as the |
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161 | tracer gradient. See Fig.~\ref{fig:triad:ISO_triad}a, where the thick lines |
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162 | denote the tracer gradients, and the thin lines the corresponding |
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163 | triads, with slopes $s_1, \dotsc s_4$. The total area-integrated |
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164 | skew-flux from tracer cell $i,k$ to $i+1,k$ |
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165 | \begin{multline} |
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166 | \label{eq:triad:i13} |
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167 | \left( F_u^{13} \right)_{i+\frac{1}{2}}^k = \Alts_{i+1}^k a_1 s_1 |
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168 | \delta _{k+\frac{1}{2}} \left[ T^{i+1} |
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169 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} + \Alts _i^k a_2 s_2 \delta |
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170 | _{k+\frac{1}{2}} \left[ T^i |
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171 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\ |
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172 | +\Alts _{i+1}^k a_3 s_3 \delta _{k-\frac{1}{2}} \left[ T^{i+1} |
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173 | \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} +\Alts _i^k a_4 s_4 \delta |
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174 | _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}, |
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175 | \end{multline} |
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176 | where the contributions of the triad fluxes are weighted by areas |
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177 | $a_1, \dotsc a_4$, and $\Alts$ is now defined at the tracer points |
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178 | rather than the $u$-points. This discretization gives a much closer |
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179 | stencil, and disallows the two-point computational modes. |
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180 | |
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181 | The vertical skew flux \eqref{eq:triad:i31c} from tracer cell $i,k$ to $i,k+1$ at the |
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182 | $w$-point $i,k+\hhalf$ is constructed similarly (Fig.~\ref{fig:triad:ISO_triad}b) |
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183 | by multiplying lateral tracer gradients from each of the four |
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184 | surrounding $u$-points by the appropriate triad slope: |
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185 | \begin{multline} |
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186 | \label{eq:triad:i31} |
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187 | \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} = \Alts_i^{k+1} a_{1}' |
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188 | s_{1}' \delta _{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} |
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189 | +\Alts_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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190 | + \Alts_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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191 | +\Alts_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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192 | \end{multline} |
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193 | |
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194 | We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$ |
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195 | (appearing in both the vertical and lateral gradient), and the $u$- and |
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196 | $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the |
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197 | triad as follows (see also Fig.~\ref{fig:triad:ISO_triad}): |
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198 | \begin{equation} |
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199 | \label{eq:triad:R} |
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200 | _i^k \mathbb{R}_{i_p}^{k_p} |
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201 | =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} |
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202 | \ |
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203 | \frac |
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204 | {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } |
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205 | {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] }. |
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206 | \end{equation} |
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207 | In calculating the slopes of the local neutral |
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208 | surfaces, the expansion coefficients $\alpha$ and $\beta$ are |
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209 | evaluated at the anchor points of the triad \footnote{Note that in \eqref{eq:triad:R} we use the ratio $\alpha / \beta$ |
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210 | instead of multiplying the temperature derivative by $\alpha$ and the |
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211 | salinity derivative by $\beta$. This is more efficient as the ratio |
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212 | $\alpha / \beta$ can to be evaluated directly}, while the metrics are |
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213 | calculated at the $u$- and $w$-points on the arms. |
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214 | |
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215 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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216 | \begin{figure}[h] \begin{center} |
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217 | \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_qcells} |
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218 | \caption{ \label{fig:triad:qcells} |
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219 | Triad notation for quarter cells.$T$-cells are inside |
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220 | boxes, while the $i+\half,k$ $u$-cell is shaded in green and the |
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221 | $i,k+\half$ $w$-cell is shaded in pink.} |
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222 | \end{center} \end{figure} |
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223 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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224 | |
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225 | Each triad $\{_i^k\:_{i_p}^{k_p}\}$ is associated (Fig.~\ref{fig:triad:qcells}) with the quarter |
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226 | cell that is the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ |
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227 | $u$-cell and the $i,k+k_p$ $w$-cell. Expressing the slopes $s_i$ and |
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228 | $s'_i$ in \eqref{eq:triad:i13} and \eqref{eq:triad:i31} in this notation, we have |
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229 | e.g.\ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. Each triad slope $_i^k |
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230 | \mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to calculate the |
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231 | lateral flux along its $u$-arm, at $(i+i_p,k)$, and then again as an |
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232 | $s'$ to calculate the vertical flux along its $w$-arm at |
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233 | $(i,k+k_p)$. Each vertical area $a_i$ used to calculate the lateral |
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234 | flux and horizontal area $a'_i$ used to calculate the vertical flux |
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235 | can also be identified as the area across the $u$- and $w$-arms of a |
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236 | unique triad, and we notate these areas, similarly to the triad |
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237 | slopes, as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, |
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238 | $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, where e.g. in \eqref{eq:triad:i13} |
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239 | $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, and in \eqref{eq:triad:i31} |
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240 | $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. |
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241 | |
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242 | \subsection{The full triad fluxes} |
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243 | A key property of iso-neutral diffusion is that it should not affect |
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244 | the (locally referenced) density. In particular there should be no |
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245 | lateral or vertical density flux. The lateral density flux disappears so long as the |
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246 | area-integrated lateral diffusive flux from tracer cell $i,k$ to |
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247 | $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the |
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248 | form |
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249 | \begin{equation} |
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250 | \label{eq:triad:i11} |
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251 | \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} = |
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252 | - \left( \Alts_i^{k+1} a_{1} + \Alts_i^{k+1} a_{2} + \Alts_i^k |
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253 | a_{3} + \Alts_i^k a_{4} \right) |
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254 | \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, |
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255 | \end{equation} |
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256 | where the areas $a_i$ are as in \eqref{eq:triad:i13}. In this case, |
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257 | separating the total lateral flux, the sum of \eqref{eq:triad:i13} and |
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258 | \eqref{eq:triad:i11}, into triad components, a lateral tracer |
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259 | flux |
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260 | \begin{equation} |
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261 | \label{eq:triad:latflux-triad} |
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262 | _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - \Alts_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p} |
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263 | \left( |
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264 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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265 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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266 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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267 | \right) |
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268 | \end{equation} |
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269 | can be identified with each triad. Then, because the |
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270 | same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and |
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271 | ${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both the density gradients |
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272 | in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients, the lateral |
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273 | density flux associated with each triad separately disappears. |
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274 | \begin{equation} |
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275 | \label{eq:triad:latflux-rho} |
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276 | {\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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277 | \end{equation} |
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278 | Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} + |
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279 | \left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from tracer cell $i,k$ |
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280 | to $i+1,k$ must also vanish since it is a sum of four such triad fluxes. |
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281 | |
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282 | The squared slope $r_1^2$ in the expression \eqref{eq:triad:i33c} for the |
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283 | $_{33}$ component is also expressed in terms of area-weighted |
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284 | squared triad slopes, so the area-integrated vertical flux from tracer |
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285 | cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is |
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286 | \begin{equation} |
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287 | \label{eq:triad:i33} |
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288 | \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = |
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289 | - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2 |
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290 | + \Alts_i^{k+1} a_{2}' s_{2}'^2 |
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291 | + \Alts_i^k a_{3}' s_{3}'^2 |
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292 | + \Alts_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right], |
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293 | \end{equation} |
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294 | where the areas $a'$ and slopes $s'$ are the same as in |
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295 | \eqref{eq:triad:i31}. |
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296 | Then, separating the total vertical flux, the sum of \eqref{eq:triad:i31} and |
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297 | \eqref{eq:triad:i33}, into triad components, a vertical flux |
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298 | \begin{align} |
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299 | \label{eq:triad:vertflux-triad} |
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300 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
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301 | &= \Alts_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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302 | \left( |
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303 | {_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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304 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
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305 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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306 | \right) \\ |
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307 | &= - \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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308 | {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:triad:vertflux-triad2} |
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309 | \end{align} |
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310 | may be associated with each triad. Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$ |
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311 | associated with a triad then separately disappears (because the |
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312 | lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$ |
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313 | disappears). Consequently the total vertical density flux $\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} + |
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314 | \left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from tracer cell $i,k$ |
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315 | to $i,k+1$ must also vanish since it is a sum of four such triad |
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316 | fluxes. |
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317 | |
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318 | We can explicitly identify (Fig.~\ref{fig:triad:qcells}) the triads associated with the $s_i$, $a_i$, and $s'_i$, $a'_i$ used in the definition of |
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319 | the $u$-fluxes and $w$-fluxes in |
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320 | \eqref{eq:triad:i31}, \eqref{eq:triad:i13}, \eqref{eq:triad:i11} \eqref{eq:triad:i33} and |
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321 | Fig.~\ref{fig:triad:ISO_triad} to write out the iso-neutral fluxes at $u$- and |
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322 | $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: |
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323 | %(Fig.~\ref{Fig_ISO_triad}): |
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324 | \begin{flalign} \label{Eq_iso_flux} \vect{F}_\mathrm{iso}(T) &\equiv |
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325 | \sum_{\substack{i_p,\,k_p}} |
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326 | \begin{pmatrix} |
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327 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ |
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328 | \\ |
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329 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\ |
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330 | \end{pmatrix}. |
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331 | \end{flalign} |
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332 | \subsection{Ensuring the scheme does not increase tracer variance} |
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333 | \label{sec:triad:variance} |
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334 | |
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335 | We now require that this operator should not increase the |
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336 | globally-integrated tracer variance. |
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337 | %This changes according to |
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338 | % \begin{align*} |
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339 | % &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\ |
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340 | % &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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341 | % \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] |
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342 | % + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\ |
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343 | % &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ |
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344 | % {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] |
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345 | % + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\ |
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346 | % \end{align*} |
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347 | Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux |
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348 | $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across the $u$-point $i+i_p,k$ and |
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349 | a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the |
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350 | $w$-point $i,k+k_p$. The lateral flux drives a net rate of change of |
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351 | variance, summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of |
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352 | \begin{multline} |
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353 | {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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354 | \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial |
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355 | t}T\right)_{i+i_p+1/2}^k \\ |
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356 | \begin{split} |
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357 | &= -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 |
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358 | {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ |
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359 | &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:triad:dvar_iso_i} |
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360 | \end{split} |
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361 | \end{multline} |
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362 | while the vertical flux similarly drives a net rate of change of |
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363 | variance summed over the $T$-points $i,k+k_p-\half$ (above) and |
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364 | $i,k+k_p+\half$ (below) of |
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365 | \begin{equation} |
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366 | \label{eq:triad:dvar_iso_k} |
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367 | _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
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368 | \end{equation} |
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369 | The total variance tendency driven by the triad is the sum of these |
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370 | two. Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and |
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371 | $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with \eqref{eq:triad:latflux-triad} and |
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372 | \eqref{eq:triad:vertflux-triad}, it is |
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373 | \begin{multline*} |
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374 | -\Alts_i^k\left \{ |
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375 | { } _i^k{\mathbb{A}_u}_{i_p}^{k_p} |
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376 | \left( |
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377 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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378 | - {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
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379 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\ |
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380 | - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p} |
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381 | \left( |
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382 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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383 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
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384 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
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385 | \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i] |
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386 | \right \}. |
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387 | \end{multline*} |
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388 | The key point is then that if we require |
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389 | $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$ |
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390 | to be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by |
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391 | \begin{equation} |
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392 | \label{eq:triad:V-A} |
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393 | _i^k\mathbb{V}_{i_p}^{k_p} |
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394 | ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k} |
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395 | ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p}, |
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396 | \end{equation} |
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397 | the variance tendency reduces to the perfect square |
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398 | \begin{equation} |
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399 | \label{eq:triad:perfect-square} |
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400 | -\Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
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401 | \left( |
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402 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
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403 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
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404 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
405 | \right)^2\leq 0. |
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406 | \end{equation} |
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407 | Thus, the constraint \eqref{eq:triad:V-A} ensures that the fluxes (\ref{eq:triad:latflux-triad}, \ref{eq:triad:vertflux-triad}) associated |
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408 | with a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase |
---|
409 | the net variance. Since the total fluxes are sums of such fluxes from |
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410 | the various triads, this constraint, applied to all triads, is |
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411 | sufficient to ensure that the globally integrated variance does not |
---|
412 | increase. |
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413 | |
---|
414 | The expression \eqref{eq:triad:V-A} can be interpreted as a discretization |
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415 | of the global integral |
---|
416 | \begin{equation} |
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417 | \label{eq:triad:cts-var} |
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418 | \frac{\partial}{\partial t}\int\!\half T^2\, dV = |
---|
419 | \int\!\mathbf{F}\cdot\nabla T\, dV, |
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420 | \end{equation} |
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421 | where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the |
---|
422 | lateral and vertical fluxes/unit area |
---|
423 | \[ |
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424 | \mathbf{F}=\left( |
---|
425 | \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p}, |
---|
426 | \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p} |
---|
427 | \right) |
---|
428 | \] |
---|
429 | and the gradient |
---|
430 | \[\nabla T = \left( |
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431 | \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k}, |
---|
432 | \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p} |
---|
433 | \right) |
---|
434 | \] |
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435 | \subsection{Triad volumes in Griffes's scheme and in \NEMO} |
---|
436 | To complete the discretization we now need only specify the triad |
---|
437 | volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. \citet{Griffies_al_JPO98} identify |
---|
438 | these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter |
---|
439 | cells, defined in terms of the distances between $T$, $u$,$f$ and |
---|
440 | $w$-points. This is the natural discretization of |
---|
441 | \eqref{eq:triad:cts-var}. The \NEMO model, however, operates with scale |
---|
442 | factors instead of grid sizes, and scale factors for the quarter |
---|
443 | cells are not defined. Instead, therefore we simply choose |
---|
444 | \begin{equation} |
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445 | \label{eq:triad:V-NEMO} |
---|
446 | _i^k\mathbb{V}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k, |
---|
447 | \end{equation} |
---|
448 | as a quarter of the volume of the $u$-cell inside which the triad |
---|
449 | quarter-cell lies. This has the nice property that when the slopes |
---|
450 | $\mathbb{R}$ vanish, the lateral flux from tracer cell $i,k$ to |
---|
451 | $i+1,k$ reduces to the classical form |
---|
452 | \begin{equation} |
---|
453 | \label{eq:triad:lat-normal} |
---|
454 | -\overline\Alts_{\,i+1/2}^k\; |
---|
455 | \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
456 | \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
457 | = -\overline\Alts_{\,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}}. |
---|
458 | \end{equation} |
---|
459 | In fact if the diffusive coefficient is defined at $u$-points, so that |
---|
460 | we employ $\Alts_{i+i_p}^k$ instead of $\Alts_i^k$ in the definitions of the |
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461 | triad fluxes \eqref{eq:triad:latflux-triad} and \eqref{eq:triad:vertflux-triad}, |
---|
462 | we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above. |
---|
463 | |
---|
464 | \subsection{Summary of the scheme} |
---|
465 | The iso-neutral fluxes at $u$- and |
---|
466 | $w$-points are the sums of the triad fluxes that cross the $u$- and |
---|
467 | $w$-faces \eqref{Eq_iso_flux}: |
---|
468 | \begin{subequations}\label{eq:triad:alltriadflux} |
---|
469 | \begin{flalign}\label{eq:triad:vect_isoflux} |
---|
470 | \vect{F}_\mathrm{iso}(T) &\equiv |
---|
471 | \sum_{\substack{i_p,\,k_p}} |
---|
472 | \begin{pmatrix} |
---|
473 | {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ |
---|
474 | \\ |
---|
475 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) |
---|
476 | \end{pmatrix}, |
---|
477 | \end{flalign} |
---|
478 | where \eqref{eq:triad:latflux-triad}: |
---|
479 | \begin{align} |
---|
480 | \label{eq:triad:triadfluxu} |
---|
481 | _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - \Alts_i^k{ |
---|
482 | \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
483 | \left( |
---|
484 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
485 | -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ |
---|
486 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
487 | \right),\\ |
---|
488 | \intertext{and} |
---|
489 | _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) |
---|
490 | &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}} |
---|
491 | \left( |
---|
492 | {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
493 | -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ |
---|
494 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
495 | \right),\label{eq:triad:triadfluxw} |
---|
496 | \end{align} |
---|
497 | with \eqref{eq:triad:V-NEMO} |
---|
498 | \begin{equation} |
---|
499 | \label{eq:triad:V-NEMO2} |
---|
500 | _i^k{\mathbb{V}}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k. |
---|
501 | \end{equation} |
---|
502 | \end{subequations} |
---|
503 | |
---|
504 | The divergence of the expression \eqref{Eq_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at |
---|
505 | each tracer point: |
---|
506 | \begin{equation} \label{eq:triad:iso_operator} D_l^T = \frac{1}{b_T} |
---|
507 | \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k |
---|
508 | {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[ |
---|
509 | {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} |
---|
510 | \end{equation} |
---|
511 | where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. |
---|
512 | The diffusion scheme satisfies the following six properties: |
---|
513 | \begin{description} |
---|
514 | \item[$\bullet$ horizontal diffusion] The discretization of the |
---|
515 | diffusion operator recovers \eqref{eq:triad:lat-normal} the traditional five-point Laplacian in |
---|
516 | the limit of flat iso-neutral direction : |
---|
517 | \begin{equation} \label{eq:triad:iso_property0} D_l^T = \frac{1}{b_T} \ |
---|
518 | \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; |
---|
519 | \overline\Alts^{\,i} \; \delta_{i+1/2}[T] \right] \qquad |
---|
520 | \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0 |
---|
521 | \end{equation} |
---|
522 | |
---|
523 | \item[$\bullet$ implicit treatment in the vertical] Only tracer values |
---|
524 | associated with a single water column appear in the expression |
---|
525 | \eqref{eq:triad:i33} for the $_{33}$ fluxes, vertical fluxes driven by |
---|
526 | vertical gradients. This is of paramount importance since it means |
---|
527 | that a time-implicit algorithm can be used to solve the vertical |
---|
528 | diffusion equation. This is necessary |
---|
529 | since the vertical eddy |
---|
530 | diffusivity associated with this term, |
---|
531 | \begin{equation} |
---|
532 | \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
533 | {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
534 | \right\} = |
---|
535 | \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{ |
---|
536 | {b_u}_{i+i_p}^k\: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 |
---|
537 | \right\}, |
---|
538 | \end{equation} |
---|
539 | (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large. |
---|
540 | |
---|
541 | \item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of |
---|
542 | locally referenced potential density is zero. See |
---|
543 | \eqref{eq:triad:latflux-rho} and \eqref{eq:triad:vertflux-triad2}. |
---|
544 | |
---|
545 | \item[$\bullet$ conservation of tracer] The iso-neutral diffusion |
---|
546 | conserves tracer content, $i.e.$ |
---|
547 | \begin{equation} \label{eq:triad:iso_property1} \sum_{i,j,k} \left\{ D_l^T \ |
---|
548 | b_T \right\} = 0 |
---|
549 | \end{equation} |
---|
550 | This property is trivially satisfied since the iso-neutral diffusive |
---|
551 | operator is written in flux form. |
---|
552 | |
---|
553 | \item[$\bullet$ no increase of tracer variance] The iso-neutral diffusion |
---|
554 | does not increase the tracer variance, $i.e.$ |
---|
555 | \begin{equation} \label{eq:triad:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T |
---|
556 | \ b_T \right\} \leq 0 |
---|
557 | \end{equation} |
---|
558 | The property is demonstrated in |
---|
559 | \S\ref{sec:triad:variance} above. It is a key property for a diffusion |
---|
560 | term. It means that it is also a dissipation term, $i.e.$ it |
---|
561 | dissipates the square of the quantity on which it is applied. It |
---|
562 | therefore ensures that, when the diffusivity coefficient is large |
---|
563 | enough, the field on which it is applied becomes free of grid-point |
---|
564 | noise. |
---|
565 | |
---|
566 | \item[$\bullet$ self-adjoint operator] The iso-neutral diffusion |
---|
567 | operator is self-adjoint, $i.e.$ |
---|
568 | \begin{equation} \label{eq:triad:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T |
---|
569 | \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} |
---|
570 | \end{equation} |
---|
571 | In other word, there is no need to develop a specific routine from |
---|
572 | the adjoint of this operator. We just have to apply the same |
---|
573 | routine. This property can be demonstrated similarly to the proof of |
---|
574 | the `no increase of tracer variance' property. The contribution by a |
---|
575 | single triad towards the left hand side of \eqref{eq:triad:iso_property3}, can |
---|
576 | be found by replacing $\delta[T]$ by $\delta[S]$ in \eqref{eq:triad:dvar_iso_i} |
---|
577 | and \eqref{eq:triad:dvar_iso_k}. This results in a term similar to |
---|
578 | \eqref{eq:triad:perfect-square}, |
---|
579 | \begin{equation} |
---|
580 | \label{eq:triad:TScovar} |
---|
581 | - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} |
---|
582 | \left( |
---|
583 | \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
584 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
585 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
586 | \right) |
---|
587 | \left( |
---|
588 | \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } |
---|
589 | -{\:}_i^k\mathbb{R}_{i_p}^{k_p} |
---|
590 | \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } |
---|
591 | \right). |
---|
592 | \end{equation} |
---|
593 | This is symmetrical in $T $ and $S$, so exactly the same term arises |
---|
594 | from the discretization of this triad's contribution towards the |
---|
595 | RHS of \eqref{eq:triad:iso_property3}. |
---|
596 | \end{description} |
---|
597 | \subsection{Treatment of the triads at the boundaries}\label{sec:triad:iso_bdry} |
---|
598 | The triad slope can only be defined where both the grid boxes centred at |
---|
599 | the end of the arms exist. Triads that would poke up |
---|
600 | through the upper ocean surface into the atmosphere, or down into the |
---|
601 | ocean floor, must be masked out. See Fig.~\ref{fig:triad:bdry_triads}. Surface layer triads |
---|
602 | $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and |
---|
603 | $\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) that require density to be |
---|
604 | specified above the ocean surface are masked (Fig.~\ref{fig:triad:bdry_triads}a): this ensures that lateral |
---|
605 | tracer gradients produce no flux through the ocean surface. However, |
---|
606 | to prevent surface noise, it is customary to retain the $_{11}$ contributions towards |
---|
607 | the lateral triad fluxes $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and |
---|
608 | $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$; this drives diapycnal tracer |
---|
609 | fluxes. Similar comments apply to triads that would intersect the |
---|
610 | ocean floor (Fig.~\ref{fig:triad:bdry_triads}b). Note that both near bottom |
---|
611 | triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and |
---|
612 | $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ |
---|
613 | or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is |
---|
614 | masked. The associated lateral fluxes (grey-black dashed line) are |
---|
615 | masked if \nlv{ln\_botmix\_grif=.false.}, but left unmasked, |
---|
616 | giving bottom mixing, if \nlv{ln\_botmix\_grif=.true.}. |
---|
617 | |
---|
618 | The default option \nlv{ln\_botmix\_grif=.false.} is suitable when the |
---|
619 | bbl mixing option is enabled (\key{trabbl}, with \nlv{nn\_bbl\_ldf=1}), |
---|
620 | or for simple idealized problems. For setups with topography without |
---|
621 | bbl mixing, \nlv{ln\_botmix\_grif=.true.} may be necessary. |
---|
622 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
623 | \begin{figure}[h] \begin{center} |
---|
624 | \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_bdry_triads} |
---|
625 | \caption{ \label{fig:triad:bdry_triads} |
---|
626 | (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer |
---|
627 | points (black dots), and $i+1/2,1$ $u$-point (blue square). Triad |
---|
628 | slopes $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$ |
---|
629 | (blue) poking through the ocean surface are masked (faded in |
---|
630 | figure). However, the lateral $_{11}$ contributions towards |
---|
631 | $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ |
---|
632 | (yellow line) are still applied, giving diapycnal diffusive |
---|
633 | fluxes.\\ |
---|
634 | (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and |
---|
635 | $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ |
---|
636 | or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point |
---|
637 | is masked. The associated lateral fluxes (grey-black dashed |
---|
638 | line) are masked if \smnlv{ln\_botmix\_grif=.false.}, but left |
---|
639 | unmasked, giving bottom mixing, if \smnlv{ln\_botmix\_grif=.true.}} |
---|
640 | \end{center} \end{figure} |
---|
641 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
642 | \subsection{ Limiting of the slopes within the interior}\label{sec:triad:limit} |
---|
643 | As discussed in \S\ref{LDF_slp_iso}, iso-neutral slopes relative to |
---|
644 | geopotentials must be bounded everywhere, both for consistency with the small-slope |
---|
645 | approximation and for numerical stability \citep{Cox1987, |
---|
646 | Griffies_Bk04}. The bound chosen in \NEMO is applied to each |
---|
647 | component of the slope separately and has a value of $1/100$ in the ocean interior. |
---|
648 | %, ramping linearly down above 70~m depth to zero at the surface |
---|
649 | It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to |
---|
650 | geopotentials (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to |
---|
651 | geopotentials) \eqref{Eq_PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate |
---|
652 | surfaces, so we require |
---|
653 | \begin{equation*} |
---|
654 | |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01. |
---|
655 | \end{equation*} |
---|
656 | and then recalculate the slopes $r_i$ relative to coordinates. |
---|
657 | Each individual triad slope |
---|
658 | \begin{equation} |
---|
659 | \label{eq:triad:Rtilde} |
---|
660 | _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}} |
---|
661 | \end{equation} |
---|
662 | is limited like this and then the corresponding |
---|
663 | $_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and combined to form the fluxes. |
---|
664 | Note that where the slopes have been limited, there is now a non-zero |
---|
665 | iso-neutral density flux that drives dianeutral mixing. In particular this iso-neutral density flux |
---|
666 | is always downwards, and so acts to reduce gravitational potential energy. |
---|
667 | \subsection{Tapering within the surface mixed layer} |
---|
668 | Additional tapering of the iso-neutral fluxes is necessary within the |
---|
669 | surface mixed layer. When the Griffies triads are used, we offer two |
---|
670 | options for this. |
---|
671 | \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:triad:lintaper} |
---|
672 | This is the option activated by the default choice |
---|
673 | \nlv{ln\_triad\_iso=.false.}. Slopes $\tilde{r}_i$ relative to |
---|
674 | geopotentials are tapered linearly from their value immediately below the mixed layer to zero at the |
---|
675 | surface, as described in option (c) of Fig.~\ref{Fig_eiv_slp}, to values |
---|
676 | \begin{subequations} |
---|
677 | \begin{equation} |
---|
678 | \label{eq:triad:rmtilde} |
---|
679 | \rMLt = |
---|
680 | -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for } z>-h, |
---|
681 | \end{equation} |
---|
682 | and then the $r_i$ relative to vertical coordinate surfaces are appropriately |
---|
683 | adjusted to |
---|
684 | \begin{equation} |
---|
685 | \label{eq:triad:rm} |
---|
686 | \rML =\rMLt -\sigma_i \quad \text{ for } z>-h. |
---|
687 | \end{equation} |
---|
688 | \end{subequations} |
---|
689 | Thus the diffusion operator within the mixed layer is given by: |
---|
690 | \begin{equation} \label{eq:triad:iso_tensor_ML} |
---|
691 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
692 | \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} |
---|
693 | 1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\ |
---|
694 | 0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\ |
---|
695 | {-\rML[1]}\hfill & {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill |
---|
696 | \end{array} }} \right) |
---|
697 | \end{equation} |
---|
698 | |
---|
699 | This slope tapering gives a natural connection between tracer in the |
---|
700 | mixed-layer and in isopycnal layers immediately below, in the |
---|
701 | thermocline. It is consistent with the way the $\tilde{r}_i$ are |
---|
702 | tapered within the mixed layer (see \S\ref{sec:triad:taperskew} below) |
---|
703 | so as to ensure a uniform GM eddy-induced velocity throughout the |
---|
704 | mixed layer. However, it gives a downwards density flux and so acts so |
---|
705 | as to reduce potential energy in the same way as does the slope |
---|
706 | limiting discussed above in \S\ref{sec:triad:limit}. |
---|
707 | |
---|
708 | As in \S\ref{sec:triad:limit} above, the tapering |
---|
709 | \eqref{eq:triad:rmtilde} is applied separately to each triad |
---|
710 | $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the |
---|
711 | $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. For clarity, we assume |
---|
712 | $z$-coordinates in the following; the conversion from |
---|
713 | $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as described |
---|
714 | above by \eqref{eq:triad:Rtilde}. |
---|
715 | \begin{enumerate} |
---|
716 | \item Mixed-layer depth is defined so as to avoid including regions of weak |
---|
717 | vertical stratification in the slope definition. |
---|
718 | At each $i,j$ (simplified to $i$ in |
---|
719 | Fig.~\ref{fig:triad:MLB_triad}), we define the mixed-layer by setting |
---|
720 | the vertical index of the tracer point immediately below the mixed |
---|
721 | layer, $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) |
---|
722 | such that the potential density |
---|
723 | ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is |
---|
724 | the tracer gridbox within which the depth reaches 10~m. See the left |
---|
725 | side of Fig.~\ref{fig:triad:MLB_triad}. We use the $k_{10}$-gridbox |
---|
726 | instead of the surface gridbox to avoid problems e.g.\ with thin |
---|
727 | daytime mixed-layers. Currently we use the same |
---|
728 | $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is |
---|
729 | used to output the diagnosed mixed-layer depth |
---|
730 | $h_\mathrm{ML}=|z_{W}|_{k_\mathrm{ML}+1/2}$, the depth of the $w$-point |
---|
731 | above the $i,k_\mathrm{ML}$ tracer point. |
---|
732 | |
---|
733 | \item We define `basal' triad slopes |
---|
734 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ as the slopes |
---|
735 | of those triads whose vertical `arms' go down from the |
---|
736 | $i,k_\mathrm{ML}$ tracer point to the $i,k_\mathrm{ML}-1$ tracer point |
---|
737 | below. This is to ensure that the vertical density gradients |
---|
738 | associated with these basal triad slopes |
---|
739 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ are |
---|
740 | representative of the thermocline. The four basal triads defined in the bottom part |
---|
741 | of Fig.~\ref{fig:triad:MLB_triad} are then |
---|
742 | \begin{align} |
---|
743 | {\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p} &= |
---|
744 | {\:}^{k_\mathrm{ML}-k_p-1/2}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase} |
---|
745 | \\ |
---|
746 | \intertext{with e.g.\ the green triad} |
---|
747 | {\:}_i{\mathbb{R}_\mathrm{base}}_{1/2}^{-1/2}&= |
---|
748 | {\:}^{k_\mathrm{ML}}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2}. \notag |
---|
749 | \end{align} |
---|
750 | The vertical flux associated with each of these triads passes through the $w$-point |
---|
751 | $i,k_\mathrm{ML}-1/2$ lying \emph{below} the $i,k_\mathrm{ML}$ tracer point, |
---|
752 | so it is this depth |
---|
753 | \begin{equation} |
---|
754 | \label{eq:triad:zbase} |
---|
755 | {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} |
---|
756 | \end{equation} |
---|
757 | (one gridbox deeper than the |
---|
758 | diagnosed ML depth $z_\mathrm{ML})$ that sets the $h$ used to taper |
---|
759 | the slopes in \eqref{eq:triad:rmtilde}. |
---|
760 | \item Finally, we calculate the adjusted triads |
---|
761 | ${\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p}$ within the mixed |
---|
762 | layer, by multiplying the appropriate |
---|
763 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ by the ratio of |
---|
764 | the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_\mathrm{base}}_{\,i}$. For |
---|
765 | instance the green triad centred on $i,k$ |
---|
766 | \begin{align} |
---|
767 | {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,1/2}^{-1/2} &= |
---|
768 | \frac{{z_w}_{k-1/2}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2} |
---|
769 | \notag \\ |
---|
770 | \intertext{and more generally} |
---|
771 | {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p} &= |
---|
772 | \frac{{z_w}_{k+k_p}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}.\label{eq:triad:RML} |
---|
773 | \end{align} |
---|
774 | \end{enumerate} |
---|
775 | |
---|
776 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
777 | \begin{figure}[h] |
---|
778 | \fcapside {\caption{\label{fig:triad:MLB_triad} Definition of |
---|
779 | mixed-layer depth and calculation of linearly tapered |
---|
780 | triads. The figure shows a water column at a given $i,j$ |
---|
781 | (simplified to $i$), with the ocean surface at the top. Tracer points are denoted by |
---|
782 | bullets, and black lines the edges of the tracer cells; $k$ |
---|
783 | increases upwards. \\ |
---|
784 | \hspace{5 em}We define the mixed-layer by setting the vertical index |
---|
785 | of the tracer point immediately below the mixed layer, |
---|
786 | $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point) |
---|
787 | such that ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, |
---|
788 | where $i,k_{10}$ is the tracer gridbox within which the depth |
---|
789 | reaches 10~m. We calculate the triad slopes within the mixed |
---|
790 | layer by linearly tapering them from zero (at the surface) to |
---|
791 | the `basal' slopes, the slopes of the four triads passing through the |
---|
792 | $w$-point $i,k_\mathrm{ML}-1/2$ (blue square), |
---|
793 | ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$. Triads with |
---|
794 | different $i_p,k_p$, denoted by different colours, (e.g. the green |
---|
795 | triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}} |
---|
796 | {\includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_triad_MLB}} |
---|
797 | \end{figure} |
---|
798 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
799 | |
---|
800 | \subsubsection{Additional truncation of skew iso-neutral flux components} |
---|
801 | The alternative option is activated by setting \nlv{ln\_triad\_iso = |
---|
802 | .true.}. This retains the same tapered slope $\rML$ described above for the |
---|
803 | calculation of the $_{33}$ term of the iso-neutral diffusion tensor (the |
---|
804 | vertical tracer flux driven by vertical tracer gradients), but |
---|
805 | replaces the $\rML$ in the skew term by |
---|
806 | \begin{equation} |
---|
807 | \label{eq:triad:rm*} |
---|
808 | \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i, |
---|
809 | \end{equation} |
---|
810 | giving a ML diffusive operator |
---|
811 | \begin{equation} \label{eq:triad:iso_tensor_ML2} |
---|
812 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
813 | \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} |
---|
814 | 1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\ |
---|
815 | 0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\ |
---|
816 | {-\rML[1]^*}\hfill & {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\ |
---|
817 | \end{array} }} \right). |
---|
818 | \end{equation} |
---|
819 | This operator |
---|
820 | \footnote{To ensure good behaviour where horizontal density |
---|
821 | gradients are weak, we in fact follow \citet{Gerdes1991} and set |
---|
822 | $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.} |
---|
823 | then has the property it gives no vertical density flux, and so does |
---|
824 | not change the potential energy. |
---|
825 | This approach is similar to multiplying the iso-neutral diffusion |
---|
826 | coefficient by $\tilde{r}_\mathrm{max}^{-2}\tilde{r}_i^{-2}$ for steep |
---|
827 | slopes, as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}). |
---|
828 | Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$ |
---|
829 | |
---|
830 | In practice, this approach gives weak vertical tracer fluxes through |
---|
831 | the mixed-layer, as well as vanishing density fluxes. While it is |
---|
832 | theoretically advantageous that it does not change the potential |
---|
833 | energy, it may give a discontinuity between the |
---|
834 | fluxes within the mixed-layer (purely horizontal) and just below (along |
---|
835 | iso-neutral surfaces). |
---|
836 | % This may give strange looking results, |
---|
837 | % particularly where the mixed-layer depth varies strongly laterally. |
---|
838 | % ================================================================ |
---|
839 | % Skew flux formulation for Eddy Induced Velocity : |
---|
840 | % ================================================================ |
---|
841 | \section{Eddy induced advection and its formulation as a skew flux} |
---|
842 | |
---|
843 | \subsection{The continuous skew flux formulation}\label{sec:triad:continuous-skew-flux} |
---|
844 | |
---|
845 | When Gent and McWilliams's [1990] diffusion is used (\key{traldf\_eiv} defined), |
---|
846 | an additional advection term is added. The associated velocity is the so called |
---|
847 | eddy induced velocity, the formulation of which depends on the slopes of iso- |
---|
848 | neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used |
---|
849 | here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo} |
---|
850 | is used in $z$-coordinate, and the sum \eqref{Eq_ldfslp_geo} |
---|
851 | + \eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates. |
---|
852 | |
---|
853 | The eddy induced velocity is given by: |
---|
854 | \begin{equation} \label{eq:triad:eiv_v} |
---|
855 | \begin{split} |
---|
856 | u^* & = - \frac{1}{e_{3}}\; \partial_k \left( A_{e} \; \tilde{r}_1 \right) \\ |
---|
857 | v^* & = - \frac{1}{e_{3}}\; \partial_k \left( A_{e} \; \tilde{r}_2 \right) \\ |
---|
858 | w^* & = \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i \left( e_{2} \, A_{e} \; \tilde{r}_1 \right) |
---|
859 | + \partial_j \left( e_{1} \, A_{e} \;\tilde{r}_2 \right) \right\} |
---|
860 | \end{split} |
---|
861 | \end{equation} |
---|
862 | where $A_{e}$ is the eddy induced velocity coefficient, and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces. |
---|
863 | |
---|
864 | The traditional way to implement this additional advection is to add it to the Eulerian |
---|
865 | velocity prior to computing the tracer advection. This allows us to take advantage of |
---|
866 | all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not just |
---|
867 | a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers |
---|
868 | where \emph{positivity} of the advection scheme is of paramount importance. |
---|
869 | |
---|
870 | \citet{Griffies_JPO98} introduces another way to implement the eddy induced advection, |
---|
871 | the so-called skew form. It is based on a transformation of the advective fluxes |
---|
872 | using the non-divergent nature of the eddy induced velocity. |
---|
873 | For example in the (\textbf{i},\textbf{k}) plane, the tracer advective |
---|
874 | fluxes per unit area in $ijk$ space can be |
---|
875 | transformed as follows: |
---|
876 | \begin{flalign*} |
---|
877 | \begin{split} |
---|
878 | \textbf{F}_\mathrm{eiv}^T = |
---|
879 | \begin{pmatrix} |
---|
880 | {e_{2}\,e_{3}\; u^*} \\ |
---|
881 | {e_{1}\,e_{2}\; w^*} \\ |
---|
882 | \end{pmatrix} \; T |
---|
883 | &= |
---|
884 | \begin{pmatrix} |
---|
885 | { - \partial_k \left( e_{2} \, A_{e} \; \tilde{r}_1 \right) \; T \;} \\ |
---|
886 | {+ \partial_i \left( e_{2} \, A_{e} \; \tilde{r}_1 \right) \; T \;} \\ |
---|
887 | \end{pmatrix} \\ |
---|
888 | &= |
---|
889 | \begin{pmatrix} |
---|
890 | { - \partial_k \left( e_{2} \, A_{e} \; \tilde{r}_1 \; T \right) \;} \\ |
---|
891 | {+ \partial_i \left( e_{2} \, A_{e} \; \tilde{r}_1 \; T \right) \;} \\ |
---|
892 | \end{pmatrix} |
---|
893 | + |
---|
894 | \begin{pmatrix} |
---|
895 | {+ e_{2} \, A_{e} \; \tilde{r}_1 \; \partial_k T} \\ |
---|
896 | { - e_{2} \, A_{e} \; \tilde{r}_1 \; \partial_i T} \\ |
---|
897 | \end{pmatrix} |
---|
898 | \end{split} |
---|
899 | \end{flalign*} |
---|
900 | and since the eddy induced velocity field is non-divergent, we end up with the skew |
---|
901 | form of the eddy induced advective fluxes per unit area in $ijk$ space: |
---|
902 | \begin{equation} \label{eq:triad:eiv_skew_ijk} |
---|
903 | \textbf{F}_\mathrm{eiv}^T = \begin{pmatrix} |
---|
904 | {+ e_{2} \, A_{e} \; \tilde{r}_1 \; \partial_k T} \\ |
---|
905 | { - e_{2} \, A_{e} \; \tilde{r}_1 \; \partial_i T} \\ |
---|
906 | \end{pmatrix} |
---|
907 | \end{equation} |
---|
908 | The total fluxes per unit physical area are then |
---|
909 | \begin{equation}\label{eq:triad:eiv_skew_physical} |
---|
910 | \begin{split} |
---|
911 | f^*_1 & = \frac{1}{e_{3}}\; A_{e} \; \tilde{r}_1 \partial_k T \\ |
---|
912 | f^*_2 & = \frac{1}{e_{3}}\; A_{e} \; \tilde{r}_2 \partial_k T \\ |
---|
913 | f^*_3 & = -\frac{1}{e_{1}e_{2}}\; A_{e} \left\{ e_{2} \tilde{r}_1 \partial_i T |
---|
914 | + e_{1} \tilde{r}_2 \partial_j T \right\}. \\ |
---|
915 | \end{split} |
---|
916 | \end{equation} |
---|
917 | Note that Eq.~ \eqref{eq:triad:eiv_skew_physical} takes the same form whatever the |
---|
918 | vertical coordinate, though of course the slopes |
---|
919 | $\tilde{r}_i$ are relative to geopotentials. |
---|
920 | The tendency associated with eddy induced velocity is then simply the convergence |
---|
921 | of the fluxes (\ref{eq:triad:eiv_skew_ijk}, \ref{eq:triad:eiv_skew_physical}), so |
---|
922 | \begin{equation} \label{eq:triad:skew_eiv_conv} |
---|
923 | \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 } \left[ |
---|
924 | \frac{\partial}{\partial i} \left( e_2 A_{e} \; \tilde{r}_1 \partial_k T\right) |
---|
925 | + \frac{\partial}{\partial j} \left( e_1 A_{e} \; |
---|
926 | \tilde{r}_2 \partial_k T\right) |
---|
927 | - \frac{\partial}{\partial k} A_{e} \left( e_{2} \tilde{r}_1 \partial_i T |
---|
928 | + e_{1} \tilde{r}_2 \partial_j T \right) \right] |
---|
929 | \end{equation} |
---|
930 | It naturally conserves the tracer content, as it is expressed in flux |
---|
931 | form. Since it has the same divergence as the advective form it also |
---|
932 | preserves the tracer variance. |
---|
933 | |
---|
934 | \subsection{The discrete skew flux formulation} |
---|
935 | The skew fluxes in (\ref{eq:triad:eiv_skew_physical}, \ref{eq:triad:eiv_skew_ijk}), like the off-diagonal terms |
---|
936 | (\ref{eq:triad:i13c}, \ref{eq:triad:i31c}) of the small angle diffusion tensor, are best |
---|
937 | expressed in terms of the triad slopes, as in Fig.~\ref{fig:triad:ISO_triad} |
---|
938 | and Eqs~(\ref{eq:triad:i13}, \ref{eq:triad:i31}); but now in terms of the triad slopes |
---|
939 | $\tilde{\mathbb{R}}$ relative to geopotentials instead of the |
---|
940 | $\mathbb{R}$ relative to coordinate surfaces. The discrete form of |
---|
941 | \eqref{eq:triad:eiv_skew_ijk} using the slopes \eqref{eq:triad:R} and |
---|
942 | defining $A_e$ at $T$-points is then given by: |
---|
943 | |
---|
944 | |
---|
945 | \begin{subequations}\label{eq:triad:allskewflux} |
---|
946 | \begin{flalign}\label{eq:triad:vect_skew_flux} |
---|
947 | \vect{F}_\mathrm{eiv}(T) &\equiv |
---|
948 | \sum_{\substack{i_p,\,k_p}} |
---|
949 | \begin{pmatrix} |
---|
950 | {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T) \\ |
---|
951 | \\ |
---|
952 | {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T) \\ |
---|
953 | \end{pmatrix}, |
---|
954 | \end{flalign} |
---|
955 | where the skew flux in the $i$-direction associated with a given |
---|
956 | triad is (\ref{eq:triad:latflux-triad}, \ref{eq:triad:triadfluxu}): |
---|
957 | \begin{align} |
---|
958 | \label{eq:triad:skewfluxu} |
---|
959 | _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \quarter {A_e}_i^k{ |
---|
960 | \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
961 | \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \ |
---|
962 | \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, |
---|
963 | \\ |
---|
964 | \intertext{and \eqref{eq:triad:triadfluxw} in the $k$-direction, changing the sign |
---|
965 | to be consistent with \eqref{eq:triad:eiv_skew_ijk}:} |
---|
966 | _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) |
---|
967 | &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} |
---|
968 | {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:triad:skewfluxw} |
---|
969 | \end{align} |
---|
970 | \end{subequations} |
---|
971 | |
---|
972 | Such a discretisation is consistent with the iso-neutral |
---|
973 | operator as it uses the same definition for the slopes. It also |
---|
974 | ensures the following two key properties. |
---|
975 | \subsubsection{No change in tracer variance} |
---|
976 | The discretization conserves tracer variance, $i.e.$ it does not |
---|
977 | include a diffusive component but is a `pure' advection term. This can |
---|
978 | be seen either from Appendix \ref{Apdx_eiv_skew} or by considering the |
---|
979 | fluxes associated with a given triad slope |
---|
980 | $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. For, following |
---|
981 | \S\ref{sec:triad:variance} and \eqref{eq:triad:dvar_iso_i}, the |
---|
982 | associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ |
---|
983 | drives a net rate of change of variance, summed over the two |
---|
984 | $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of |
---|
985 | \begin{equation} |
---|
986 | \label{eq:triad:dvar_eiv_i} |
---|
987 | _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], |
---|
988 | \end{equation} |
---|
989 | while the associated vertical skew-flux gives a variance change summed over the |
---|
990 | $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of |
---|
991 | \begin{equation} |
---|
992 | \label{eq:triad:dvar_eiv_k} |
---|
993 | _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. |
---|
994 | \end{equation} |
---|
995 | Inspection of the definitions (\ref{eq:triad:skewfluxu}, \ref{eq:triad:skewfluxw}) |
---|
996 | shows that these two variance changes (\ref{eq:triad:dvar_eiv_i}, \ref{eq:triad:dvar_eiv_k}) |
---|
997 | sum to zero. Hence the two fluxes associated with each triad make no |
---|
998 | net contribution to the variance budget. |
---|
999 | |
---|
1000 | \subsubsection{Reduction in gravitational PE} |
---|
1001 | The vertical density flux associated with the vertical skew-flux |
---|
1002 | always has the same sign as the vertical density gradient; thus, so |
---|
1003 | long as the fluid is stable (the vertical density gradient is |
---|
1004 | negative) the vertical density flux is negative (downward) and hence |
---|
1005 | reduces the gravitational PE. |
---|
1006 | |
---|
1007 | For the change in gravitational PE driven by the $k$-flux is |
---|
1008 | \begin{align} |
---|
1009 | \label{eq:triad:vert_densityPE} |
---|
1010 | g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) |
---|
1011 | &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k |
---|
1012 | {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k |
---|
1013 | {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ |
---|
1014 | \intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from |
---|
1015 | \eqref{eq:triad:skewfluxw}, gives} |
---|
1016 | % and separating out |
---|
1017 | % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, |
---|
1018 | % gives two terms. The |
---|
1019 | % first $\rtriad{R}$ term (the only term for $z$-coordinates) is: |
---|
1020 | &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} |
---|
1021 | \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 \\ |
---|
1022 | &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
1023 | \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}} |
---|
1024 | \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}}, |
---|
1025 | \end{align} |
---|
1026 | using the definition of the triad slope $\rtriad{R}$, |
---|
1027 | \eqref{eq:triad:R} to express $-\alpha _i^k\delta_{i+ i_p}[T^k]+ |
---|
1028 | \beta_i^k\delta_{i+ i_p}[S^k]$ in terms of $-\alpha_i^k \delta_{k+ |
---|
1029 | k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$. |
---|
1030 | |
---|
1031 | Where the coordinates slope, the $i$-flux gives a PE change |
---|
1032 | \begin{multline} |
---|
1033 | \label{eq:triad:lat_densityPE} |
---|
1034 | g \delta_{i+i_p}[z_T^k] |
---|
1035 | \left[ |
---|
1036 | -\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) |
---|
1037 | \right] \\ |
---|
1038 | = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
1039 | \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} |
---|
1040 | \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) |
---|
1041 | \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}}, |
---|
1042 | \end{multline} |
---|
1043 | (using \eqref{eq:triad:skewfluxu}) and so the total PE change |
---|
1044 | \eqref{eq:triad:vert_densityPE} + \eqref{eq:triad:lat_densityPE} associated with the triad fluxes is |
---|
1045 | \begin{multline} |
---|
1046 | \label{eq:triad:tot_densityPE} |
---|
1047 | g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + |
---|
1048 | g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ |
---|
1049 | = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k |
---|
1050 | \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 |
---|
1051 | \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}}. |
---|
1052 | \end{multline} |
---|
1053 | Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+ |
---|
1054 | \beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative. |
---|
1055 | |
---|
1056 | \subsection{Treatment of the triads at the boundaries}\label{sec:triad:skew_bdry} |
---|
1057 | Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes |
---|
1058 | are masked at the boundaries in exactly the same way as are the triad |
---|
1059 | slopes \rtriad{R} used for the iso-neutral diffusive fluxes, as |
---|
1060 | described in \S\ref{sec:triad:iso_bdry} and |
---|
1061 | Fig.~\ref{fig:triad:bdry_triads}. Thus surface layer triads |
---|
1062 | $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are |
---|
1063 | masked, and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ |
---|
1064 | and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the |
---|
1065 | $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ |
---|
1066 | $u$-point is masked. The namelist parameter \nlv{ln\_botmix\_grif} has |
---|
1067 | no effect on the eddy-induced skew-fluxes. |
---|
1068 | |
---|
1069 | \subsection{ Limiting of the slopes within the interior}\label{sec:triad:limitskew} |
---|
1070 | Presently, the iso-neutral slopes $\tilde{r}_i$ relative |
---|
1071 | to geopotentials are limited to be less than $1/100$, exactly as in |
---|
1072 | calculating the iso-neutral diffusion, \S \ref{sec:triad:limit}. Each |
---|
1073 | individual triad \rtriadt{R} is so limited. |
---|
1074 | |
---|
1075 | \subsection{Tapering within the surface mixed layer}\label{sec:triad:taperskew} |
---|
1076 | The slopes $\tilde{r}_i$ relative to |
---|
1077 | geopotentials (and thus the individual triads \rtriadt{R}) are always tapered linearly from their value immediately below the mixed layer to zero at the |
---|
1078 | surface \eqref{eq:triad:rmtilde}, as described in \S\ref{sec:triad:lintaper}. This is |
---|
1079 | option (c) of Fig.~\ref{Fig_eiv_slp}. This linear tapering for the |
---|
1080 | slopes used to calculate the eddy-induced fluxes is |
---|
1081 | unaffected by the value of \nlv{ln\_triad\_iso}. |
---|
1082 | |
---|
1083 | The justification for this linear slope tapering is that, for $A_e$ |
---|
1084 | that is constant or varies only in the horizontal (the most commonly |
---|
1085 | used options in \NEMO: see \S\ref{LDF_coef}), it is |
---|
1086 | equivalent to a horizontal eiv (eddy-induced velocity) that is uniform |
---|
1087 | within the mixed layer \eqref{eq:triad:eiv_v}. This ensures that the |
---|
1088 | eiv velocities do not restratify the mixed layer \citep{Treguier1997, |
---|
1089 | Danabasoglu_al_2008}. Equivantly, in terms |
---|
1090 | of the skew-flux formulation we use here, the |
---|
1091 | linear slope tapering within the mixed-layer gives a linearly varying |
---|
1092 | vertical flux, and so a tracer convergence uniform in depth (the |
---|
1093 | horizontal flux convergence is relatively insignificant within the mixed-layer). |
---|
1094 | |
---|
1095 | \subsection{Streamfunction diagnostics}\label{sec:triad:sfdiag} |
---|
1096 | Where the namelist parameter \nlv{ln\_botmix\_grif=.true.}, diagnosed |
---|
1097 | mean eddy-induced velocities are output. Each time step, |
---|
1098 | streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at |
---|
1099 | $uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ |
---|
1100 | (integer $i$, integer +1/2 $j$, integer +1/2 $k$) points (see Table |
---|
1101 | \ref{Tab_cell}) respectively. We follow \citep{Griffies_Bk04} and |
---|
1102 | calculate the streamfunction at a given $uw$-point from the |
---|
1103 | surrounding four triads according to: |
---|
1104 | \begin{equation} |
---|
1105 | \label{eq:triad:sfdiagi} |
---|
1106 | {\psi_{[i]}}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}} |
---|
1107 | {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} |
---|
1108 | \end{equation} |
---|
1109 | |
---|
1110 | \newpage %force an empty line |
---|
1111 | % ================================================================ |
---|
1112 | % Discrete Invariants of the skew flux formulation |
---|
1113 | % ================================================================ |
---|
1114 | \subsection{Discrete Invariants of the skew flux formulation} |
---|
1115 | \label{Apdx_eiv_skew} |
---|
1116 | |
---|
1117 | |
---|
1118 | Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. |
---|
1119 | |
---|
1120 | This have to be moved in an Appendix. |
---|
1121 | |
---|
1122 | The continuous property to be demonstrated is : |
---|
1123 | \begin{align*} |
---|
1124 | \int_D \nabla \cdot \textbf{F}_\mathrm{eiv}(T) \; T \;dv \equiv 0 |
---|
1125 | \end{align*} |
---|
1126 | The discrete form of its left hand side is obtained using \eqref{eq:triad:allskewflux} |
---|
1127 | \begin{align*} |
---|
1128 | \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; |
---|
1129 | \delta_i &\left[ |
---|
1130 | {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} |
---|
1131 | \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] |
---|
1132 | \right] \; T_i^k \\ |
---|
1133 | - \delta_k &\left[ |
---|
1134 | {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} |
---|
1135 | \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] |
---|
1136 | \right] \; T_i^k \ \Biggr\} |
---|
1137 | \end{align*} |
---|
1138 | apply the adjoint of delta operator, it becomes |
---|
1139 | \begin{align*} |
---|
1140 | \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\; |
---|
1141 | &\left( |
---|
1142 | {e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k} |
---|
1143 | \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}] |
---|
1144 | \right) \; \delta_{i+1/2}[T^{k}] \\ |
---|
1145 | - &\left( |
---|
1146 | {e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2} |
---|
1147 | \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] |
---|
1148 | \right) \; \delta_{k+1/2}[T_{i}] \ \Biggr\} |
---|
1149 | \end{align*} |
---|
1150 | Expending the summation on $i_p$ and $k_p$, it becomes: |
---|
1151 | \begin{align*} |
---|
1152 | \begin{matrix} |
---|
1153 | &\sum\limits_{i,k} \Bigl\{ |
---|
1154 | &+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} |
---|
1155 | &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
1156 | &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} |
---|
1157 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}} &\delta_{k-1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
1158 | &&+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k} |
---|
1159 | &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
1160 | &&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:} |
---|
1161 | &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ |
---|
1162 | % |
---|
1163 | &&-{e_{2u}}_i^{k+1} &{A_{e}}_i^{k+1} |
---|
1164 | &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}} &\delta_{i-1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
1165 | &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} |
---|
1166 | &{\ \ \;_i^k \mathbb{R}_{-1/2}^{+1/2}} &\delta_{i-1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
1167 | &&-{e_{2u}}_i^{k+1 } &{A_{e}}_i^{k+1} |
---|
1168 | &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}} &\delta_{i+1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\ |
---|
1169 | &&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:} |
---|
1170 | &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] |
---|
1171 | &\Bigr\} \\ |
---|
1172 | \end{matrix} |
---|
1173 | \end{align*} |
---|
1174 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the |
---|
1175 | same but of opposite signs, they cancel out. |
---|
1176 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. |
---|
1177 | The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the |
---|
1178 | same but both of opposite signs and shifted by 1 in $k$ direction. When summing over $k$ |
---|
1179 | they cancel out with the neighbouring grid points. |
---|
1180 | Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the |
---|
1181 | $i$ direction. Therefore the sum over the domain is zero, $i.e.$ the variance of the |
---|
1182 | tracer is preserved by the discretisation of the skew fluxes. |
---|
1183 | |
---|
1184 | %%% Local Variables: |
---|
1185 | %%% TeX-master: "../../NEMO_book-luatex.tex" |
---|
1186 | %%% End: |
---|